SlideShare a Scribd company logo
1 of 16
ISA
                                                                                                          TRANSACTIONS®
                                                ISA Transactions 43 ͑2004͒ 361–376




          Flexible-structure control: A strategy for releasing input
                                 constraints
                                                Leonardo L. Giovanini*
 Industrial Control Centre, University of Strathclyde, Graham Hills Building, 50 George Street, Glasgow G1 1QE, United Kingdom
                                        ͑Received 26 July 2002; accepted 5 September 2003͒



Abstract
   In this paper output unreachability under input saturation phenomenon is studied: under a large disturbance or
setpoint change, the process output may never reach the set point even when the manipulated variable has driven to
saturation. The process output can be brought back to the set point only by activating an auxiliary manipulated variable.
A new control structure for designing and implementing a control system capable of solving this problem is proposed
by transferring the control from one variable to another and taking into account the different dynamics involved in the
system. The control structure, called flexible-structure control due to its ability to adapt the control structure to the
operating conditons, is a generalization of the split-range control. It can be summarized as two controllers connected
through a piecewise linear function. This function decides, based on the value of one manipulated variable, when and
how the control structure changes. Its parameters control the interaction between both manipulated variables and leave
the capability for handling the balance between control quality and other goals to the operator. © 2004 ISA—The
Instrumentation, Systems, and Automation Society.

Keywords: Multi-input systems; PID control; cooperative control; Manipulated variable constraint



1. Introduction                                                           Antiwindup methods may work well under the
                                                                       nominal conditions under which the controller was
   Quite frequently process control engineers face                     designed. However, it is possible that under a large
problems in which hard constraints restrict the ma-                    set-point change, load disturbance, or component
nipulated variables to a finite operating range. The                    failure, the manipulated variable will reach its
constraints may also come from process con-                            limit while the system output still cannot reach its
straints intended to avoid damage to the system or                     set point at the steady state. This phenomenon is
the material being proceeded ͓1͔. There are many                       known as output unreachability under input con-
control techniques to deal with this problem. They                     straint ͓9͔. It is directly associated with the size of
can be divided into two categories: antiwindup                         the operability space of the system. Sometimes
compensation reduces the adverse effects of the                        this problem is solved by modifying the control
constraints on the closed-loop performance ͓1– 4͔,                     structure of the system, but many times the pro-
and cooperative control schemes that use a com-                        cess itself requires substantial changes. On the
bination of inputs to achieve the reference ͓5–9͔.                     other hand, cooperative control schemes solve the
                                                                       output-unreachability problem by activating auxil-
                                                                       iary manipulated variables, and many times they
  *Tel: ϩ44 141 548 2666; fax: 44 141 552 2487. E-mail                 introduce a degree of optimality in the solution
address: leonardo.giovanini@eee.strath.ac.uk                           ͓8͔. Nevertheless, manipulated constraints are

0019-0578/2004/$ - see front matter © 2004 ISA—The Instrumentation, Systems, and Automation Society.
362                          Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376


never completely eliminated because these tech-
niques only solve the output-unreachability prob-
lem for the steady state. Therefore during the tran-
sient the manipulated variables could temporarily
reach their limits and the system becomes uncon-
trollable during this period. Hence the need for
finding a way for broadening controlled operation
spaces to provide all the available process flexibil-
ity while preserving good performances has stimu-
                                                                          Fig. 1. Basic process structure.
lated the search for new control structures.
   This paper proposed a new control structure
called flexible-structure control, due to its ability
to adapt the control structure to the operating con-         namic elements Gp 1 ( s ) and Gp 2 ( s ) . The second
ditons. It is a generalization of the split-range con-       feature is that assumed to be u 1 is the primary
trol ͓6͔ and it can be summarized as two control-            manipulated variable because Gp 1 ( s ) is faster
lers connected through a piecewise linear function.          and with smaller time delay than Gp 2 ( s ) . A hard
This function decides, based on the value of one             constraint might become active at some given ex-
manipulated variable, when and how the control               treme values of this variable. If eventually u 1 satu-
structure changes. Its parameters control the inter-         rates, u 2 may be used as an auxiliary manipulated
action between both manipulated variables. They              variable to keep the system under regulation. The
can be selected attending not only to the process            process may be also subject to many disturbances.
structure but also following some optimization cri-          For linear systems, they can be collectively repre-
teria.                                                       sented by one disturbance d entering the process
   The paper is organized as follow: in Section 2,           at the output.
the requirements that must satisfy the process to               In normal operation, the system is designed so
apply the proposed control strategy and some ex-             that for any moderate set-point or load disturbance
amples are presented. In Section 3 the flexible-              change, the primary manipulated variable u 1 can
structure control is proposed. The structure can be          regulate the process output to achieve a zero out-
summarized as two controllers connected with a               put steady-state error working within its working
piecewise-linear function. The parameters of the             range ͓ u 1 min ,u1 max͔, while u 2 is kept unchanged.
nonlinear function control the interaction between           However, it is possible that under a large set point,
both inputs and the steady-state values of each ma-          load disturbance change, or component failure,
nipulated variable. In Section 4 two tuning proce-           there is no u 1 ෈ ͓ u 1 min ,u1 max͔ so that y ( ϱ )
dures for the controller are proposed. The first              ϭr ( ϱ ) if u 2 is unchanged, leading to output un-
method is based on the PID controller and it is              reachability.
obtained from an IMC parametrization. The sec-                  This problem is frequently found in practice, so
ond procedure employes more sophisticated con-               it deserves special research. For example, consider
trollers. Section 5 addresses the closed-loop stabil-        the problem of controlling a continuous stirred
ity analysis. In Section 6 some guidelines for the           tank reactor where an irreversible exothermic re-
selection of the decision function are presented.            action is carried out at a constant volume em-
Finally, Section 7 presents results obtained from            ployed by Moningred et al. ͓10͔ to test nonlinear
the application of the proposed algorithm to a lin-          control algorithms. The reactor is cooled by water
ear system. Conclusions are presented in Section             through a surrounding jacket. The concentration
8.                                                           can be controlled either by manipulating the flow
                                                             rates of the coolant or the process stream. From
                                                             the process perspective, the use of cooling water is
2. Process conditions                                        preferred over the process stream. However, the
                                                             dynamic response of the reactor concentration to a
  Figure 1 shows a sketch of the process structure           change in coolant flow rate shows a nonlinear be-
considered in this work. The first special feature to         havior. In practice, the nominal flowrates of the
be noted is that the output variable y may be con-           process stream is determined in advance, and wa-
trolled by either u 1 or u 2 through different dy-           ter is employed as the primary manipulated vari-
Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376                          363


able, while the process stream acts as the auxiliary         3. Flexible-structure control
manipulated variable and is normally kept con-
stant.                                                          The examples of output unreachability provided
   Another example is the control of the tempera-            in the previous section are only few of many in-
ture in the reactor ͓11͔. The reactor is cooled by           dustrial cases. Such cases require changes in the
both cooling water flowing through a surrounding              auxiliary manipulated variable. To perform these
jacket and condensing vapor that boils off the re-           changes several control strategies have been de-
actor in a heat exchanger cooled by a refrigerant.           veloped in the past decades. Shinkey ͓6͔ proposed
From an energy-saving point of view, the use of              the valve-position control. This technique uses a
cooling water is preferred over the refrigerant due          maximum signal selector to change u 2 to keep all
to cost. However, the dynamic response of the                the actuators from exceeding a preset limit. The
plant’s temperature to a change in refrigerant flow           algorithm has to wait for the process variable to
is faster than a change in cooling water. In prac-           reach its steady state, then regulate u 2 iteratively
tice, the nominal flow rates of both are determined           until the steady-state error becomes zero. The
in advance. Water is utilized as the primary ma-             main drawbacks of the method are that it is time
nipulated variable, while the refrigerant acts as an         consuming and the results largely depend on the
auxiliary manipulated variable and it is normally            engineer experience.
kept constant. If there is a disturbance or setpoint            Another alternative control structure is the coop-
change such that the temperature cannot return to            erative control, proposed by Wang et al. ͓9͔. Simi-
the set point even when the cooling water valve              larly to valve-position control, cooperative control
hits its limitation, the flow of refrigerant has to be        activates u 2 using constant levels which are com-
adjusted to make the temperature reach the desired           puted base on a disturbance estimation and output
steady-state value.                                          steady-state prediction. The use of these features
   Another example can be found in the tempera-              improves the closed-loop performance by reduc-
ture control of thermical integrated chemical pro-           ing the settling time and simplifying the imple-
cess ͓12͔. A simple configuration of this type of             mentation of this control scheme.
                                                                Both techniques solve the output unreachability
system is given by two heat exchangers in series:
                                                             problem only for the steady-state of the multiple-
one heat exchanger and a service equipment. This
                                                             input single-output ͑MISO͒ system with a control
arrangement is very common in practice when be-
                                                             structure that remains unchanged. Therefore the
sides the task of reaching a final temperature target
                                                             transient behavior of the closed-loop system will
on a process stream there is an extra goal like
                                                             be poor because the control laws generated by
maximum energy recovery. The heat exchanger is
                                                             both control strategies do not take account of the
specifically designed for recovering the exceeding            dynamic part of Gp 2 . Many times the closed-loop
energy in the process stream, and the service                response is driven in an open-loop manner during
equipment completes the thermic conditioning                 the transient. This situation is significantly impor-
through a utility stream ͓13͔. The heat exchanger            tant when a fault, like a frozen valve, occurs or
operates at a constant flowrate that maximizes the            during the saturation of the manipulated variables.
energy recovering on nominal conditions and the              Hence the above control problem requires an ap-
service is designed to cope with the long term               propriate design that would be able to transfer the
variations on the inlet stream conditions or having          control from one variable to another, takes account
the capability of changing stream temperature tar-           of the different dynamics involved in the system
gets. In the case of a disturbance or set-point              and, if it is possible, leaves the capability for han-
change, the steady state may deviate. There could            dling the balance between control quality and
exist a situation in which the effect is so large that       other goals to the operator. Hence, under this
the temperature cannot return to the set point even          framework, solving a manipulated constraint prob-
when the service hits its limits. In such a case, the        lem requires a process engineering approach ca-
auxiliary manipulated variable, i.e., the flow rate           pable of combining control strategy and process
of the heat exchanger, has to be adjusted to make            efficiency.
the temperature’s steady state reach the desired                Note that if there is a controller C 1 ( s ) handling
value.                                                       u 1 ( s ) to control y ( s ) at a given set-point value
364                                Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376




                        Fig. 2. Block diagram of ͑a͒ preventive protection, and ͑b͒ reactive protection.




r 0 , the stationary value expected for the controlled             able, a preventive protection for disturbance can
variable is Gp 1 ( 0 ) u 1 ( 0 ) , and if an integral mode         also be included by computing x as follows:
is present, the manipulated variable goes to
u 1 ( 0 ) ϭGp Ϫ1 ( 0 ) r 0 . Let us assume now that just
                 1                                                  x ͑ s ͒ ϭ ␩ Gp Ϫ1 ͑ 0 ͒ r ͑ s ͒ ϩ ␩ Gp Ϫ1 ͑ 0 ͒ G d ͑ 0 ͒ w ͑ s ͒ .
                                                                                   1                       1
a fraction of this output y can be handled through
the manipulated u 1 ͑this implies that a control                   For the second case, Fig. 2͑b͒ shows that the re-
constraint is active at a given level͒, and that an                active protection introduces a new feedback loop
additional capacity can be provided through                        that results from combining both controllers,
u 2 ( s ) . Then, there are two ways of defining the                C 1 ( s ) and C 2 ( s ) , in a single controller, C ( s )
protection for regulation and operability:                         ϭC 1 ( s ) C 2 ( s ) . As long as the primary manipu-
   1. Feedforward or preventive protection: given a                lated variable is not saturated, the output is con-
total         output          steady-state       requirement       trolled by C 1 and the secondary control loop is no
Gp 1 ( 0 ) u 1 ( 0 ) , a fraction ␩ Gp 1 ( 0 ) u 1 ( 0 ) , ␩ у0,   operative because the auxiliar signal x ( t ) is zero.
is permanently provided by the process part Gp 2 ,                 When u 1 ( t ) Ͼu 1 max , the original control loop re-
in order to keep controlled operability.                           mains open and consequently not operative as
   2. Feedback or reactive protection: given the in-               long as the primary manipulated is saturated. Thus
stant manipulated variable u 1 ( t ) at an operating               the process part represented by Gp 2 ( s ) is now in
point such that ͉ u 1 ( t ) ͉ Ͼu 1 max , the process part          charge of the regulation. As soon as the structure
Gp 2 ( s ) provides the complementary output                       of the system changes, due to the saturation of u 1 ,
Gp 1 ( 0 ) ͓ u 1 ( t ) Ϫu 1 max͔ to reach the target r 0 .         the structure of the controller is modified from
   These two types of protections are schematized                  C 1 ( s ) to C ( s ) following the change of the system
in Figs. 2͑a͒ and 2͑b͒ respectively, where a second                and adapting the structure and parameters of the
controller C 2 ( s ) is included for better dynamic ad-            controller to the dynamic of the system. The con-
justment of the secondary manipulated variable                     troller C ( s ) must include an antiwindup scheme to
u 2 . It is clear from the block diagrams that in the              mitigate the effect of the constraint on u 2 , the
case of preventive action, C 2 ( s ) is a feedforward              effect of constraint on u 1 is compensated by the
controller, which does not affect the closed-loop                  control structure.
stability, but it cannot protect the system from the                 The switching element is implemented through
effect of nonmeasurable disturbances, uncertain-                   a simple nonlinear decision function such that the
ties, and/or faults. If the disturbance w ( s ) , or an            signal x ( t ) entering the controller C 2 ( s ) is given
estimation w ( s ) , and a model of G d ( s ) are avail-
                 ˆ                                                 by
Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376                                       365




                               Fig. 3. Plot of the nonlinear switching function for some parameters.




     f ͑ u1͒ϭ   ͭ   u 1 ͑ t ͒ Ϫu 1 max
                    0
                                         u 1 ͑ t ͒ Ͼu 1 max ,
                         u 1 ͑ t ͒ рu 1 max .
                                                                ͑1͒
                                                                      modifying the switching function ͑1͒ through the
                                                                      inclusion of the linear term

Observe that a similar protection can be developed                                           ␩ u 1͑ t ͒ ,   ␩ у0,
for a lower constraint but it must actuate on a dif-                  for the active range of u 1 . This term defines a
ferent process part, let us say Gp 3 ( s ) . This means               permanent and increasing level of protection,
that a third manipulated variable u 3 must be avail-                  when the control variable u 1 ( t ) approaches the
able and become active. In this case the switching                    constraints. For the upper limit case, function ͑1͒,
function may be written




                                                                                 ͭ
                                                                      the switching function is given by

     f ͑ u1͒ϭ   ͭ   0    u 1 ͑ t ͒ уu 1 min ,
                    u 1 ͑ t ͒ Ϫu 1 min   u 1 ͑ t ͒ Ͻu 1 min .
                                                                ͑2͒
                                                                                     u 1 ͑ t ͒ Ϫ ͑ 1Ϫ ␩ ͒ u 1 max,
                                                                                                                   u 1 ͑ t ͒ Ͼu 1 max ,
                                                                      f ͑ u1͒ϭ
The switching functions ͑1͒ and ͑2͒ can be easily                                    ␩ u 1͑ t ͒ ,   u 1 minрu 1 ͑ t ͒ рu 1 max ,
implemented using a dead zone of width u 1 max                                       0,     u 1 ͑ t ͒ рu 1 min .
Ϫu1 min with a selector connected in series, such
that it generates the proper signal for each auxil-                                                                     ͑3͒
iary loop.                                                            If ␩ ϭ0 the split-range control ͓6͔ is recovered,
   In spite that a nonlinearity is introduced in the                  the decision function ͑3͒ becomes Eq. ͑1͒, and the
closed loop to transfer the control from Gp 1 to                      auxiliary variable u 2 is used just to cover output
Gp 2 , the stability properties of the system remain                  demands when u 1 saturates only. When ␩ Ͼ0 the
unaffected because, as was explained previously,                      auxiliary variable u 2 is used to prevent the satura-
there is no interaction between both control loops.                   tion of u 1 , by providing the fraction ␩ r of y ( t )
When one manipulated variable is active, said that                    while u 1 Ͻu 1 max . Both manipulated variables are
the output is controlled with u 1 , the other is inac-                simultaneously acting on the system, but with dif-
tive and vice versa.                                                  ferent gains: Kp 1 for u 1 and ␩ Kp 2 for u 2 , where
   Functions ͑1͒ and ͑2͒ represent the feedback                       Kp i iϭ1,2 is the gain of Gp i ( s ) , respectively.
protection exclusively, however, a single structure                   This fact leads to interactions between both con-
can be developed to also include the feedforward                      trol loops that can upset each other, resulting in a
protection into the control loop. It is achieved by                   deterioration of the closed-loop performance com-
366                                 Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376


pared with the previous situation.1 However, it                      This fact means when a major fault in the actuator
might be a desirable feature since it tends to keep                  of u 1 happens, like a actuator frozen at a given a
the fastest loop working in a wider range. When                      value, the flexible structure is able to transfer the
u 1 ( t ) уu 1 max the control system changes the                    control of the system output to u 2 without any
structure and only Gp 2 controls the system output.                  extra information than a measurement of the sys-
Therefore the closed-loop performance will dete-                     tem output y. In the case of a minor fault, like a
riorate a bit more. The gain of the system jumps                     loss of a actuator sensibility, it can lead to an out-
from ␩ Kp 2 to Kp 2 , producing a peak in the con-                   put unreachability problem that is overcome as has
trol signal u 2 if ideal PID’s are employed. In the                  been explained in the previous paragraph.
case of ␩ ϭu 2 max /u1 max both controllers work si-
multaneously, the fastest control loop is active in                  3.1. Smith predictor for flexible-structure control
the whole operational range maintaining a good
control quality in the entire operational space.                       Time delay is a common feature in most of the
   A second parameter can be included in the                         process models. Control of systems with dominant
above formulation to start the preventive protec-                    time delay are notoriously difficult. It is also a
tion from a given value u 1 ( t ) ϭ ␬ in ahead, instead              topic on which there are many different opinions
of doing it from u 1 min as indicated by Eq. ͑3͒, i.e.,              concerning PID control. For open-loop stable pro-




           ͭ
                                                                     cesses, the response to command signals can be
               u 1 ͑ t ͒ Ϫu 1 maxϩ ␩ ͑ u 1 maxϪ ␬ ͒ ,
                          ¯                                          improved substantially by introducing dead time
                                            u 1 ͑ t ͒ Ͼu 1 max ,     compensation ͓15͔. The dead time compensator,
f ͑ u1͒ϭ                                                             or Smith predictor, is built by implementing a lo-
               ␩ ͑ u 1 ͑ t ͒ Ϫ ␬ ͒ , ␬ рu 1 ͑ t ͒ рu 1 max ,         cal loop around the controller with the difference
               0, u 1 ͑ t ͒ Ͻ ␬ .                                    between the model of the process without and with
                                                               ͑4͒   time delay ͓see Fig. 4͑a͔͒.
                                                                       Now, the structure of the Smith predictor is
Figure 3 shows a sketch of this function for differ-                 modified to consider the proposed control struc-
ent parameter values. Note that a similar function                   ture. In this case a second loop is introduced to
can be used for saturations at the lower constraint.                 represent the effect of u 2 over the system output.
Both parameters of the decision function, ␩ and ␬,                   This fact leads to a new structure that include the
can be used as tuning parameters to satisfy addi-                                                ˜         ˜
                                                                     models of both process ( G p 1 and G p 2 ) and the
tional process goals than control ones like process                  constraints in the manipulated variables ͓see Fig.
efficiency.                                                           4͑b͔͒. The nonlinearities are required to follow the
   Remark 1. Regarding the work of the actuators,                    changes in the structure of the system. This struc-
it is interesting to observe that for ␩ ϭ0 the con-                  ture works for time delays with different values as
trol action is executed over a divided range ͓6,14͔,                 we can see in the example.
that is, the second actuator starts moving once the                    One can notice that a Smith predictor can be
first one saturates. For positive values of ␩, the                    coupled with a robust controller ( H ϱ , QFT, robust
control variable is executed over a common range,                    pole placement, etc.͒ to cope with parametric
that is, both actuators work together until one of                   variations.
them saturates. The amplitude of the common
range is handled through the parameter ␬ while
the intensity is given by ␩.                                         4. Controller design and tuning
   The capability of transferring the control from
one input to another provides an implicit fault-                       For designing and tuning the controllers in-
tolerant capabilities to flexible-structure control.                  volved in the flexible structure it is necessary to
                                                                     analyze each control condition separately: ͑i͒ the
  1                                                                  first control condition—or control structure—is
   If Gp 1 is much faster than Gp 2 , C 1 will be able to
compensate the effect of the interactions and the closed-
                                                                     when C 1 ( s ) is in charge of regulation of y ( t ) , i.e.,
loop performance will not be affected. However, if Gp 1 is           ␩ ϭ0 and u 1 ( t ) is not saturated. ͑ii͒ The second
not fast enough, the closed-loop response will show an               control structure is defined by the secondary loop
overshoot and an increment of the closed-loop settling time          only, that is, ␩ ϭ0 and u 1 ( t ) is saturated; the con-
due to the interaction between both control loops.                   troller in this case is the combination C ( s )
Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376                             367




            Fig. 4. Block diagram of Smith predictor ͑a͒ for normal system, and ͑b͒ for cooperative control.



ϭC1(s)C2(s). ͑iii͒ The third control condition ap-             the IMC strategy provides a rapid parametrization
pears when including preventive protection, i.e.,              of traditional PI or PID controllers ͓16,17͔. If two
␩ Ͼ0 while u 1 ( t ) is not yet saturated.                     PID algorithms with time delay compensation are
   The above decomposition of the problem indi-                proposed for controllers C 1 ( s ) and C 2 ( s ) , recall
cates that C 1 ( s ) must be adjusted for high quality         that they work in series, i.e., the outlet of C 1 ( s ) is
control when the process part Gp 1 ( s ) handles the           the input to C 2 ( s ) through the switching function
regulation, i.e., ᭙t:u 1 minрu1(t)рu1 max , and this is        f ( u 1 ) . The following few hypotheses and practi-
essentially the traditional tuning problem for a               cal reasons allow the selection of some important
single feedback loop. When u 1 ( t ) Ͼu 1 max , the            terms and the elimination of others:
process part Gp 2 ( s ) must provide the comple-                  1. The double integration term is not necessary
mentary effect on the controlled variable, which               since offset elimination is required for set-point
means that C 2 ( s ) must be combined with C 1 ( s )           changes only.
such to obtain the best possible performance.                     2. A single integral mode is necessary just in
   In the following subsections two approaches for             C 1 ( s ) , because offset elimination is desired under
designing and tuning controllers C 1 ( s ) and C 2 ( s )       any working condition and this controller is the
are presented. One is based on IMC parametriza-                one which is always active.
tion of the controllers, the other is based on can-               3. The control system structure assumes that
celation design criteria. In the IMC design the or-            Gp 1 ( s ) is faster and with smaller time delay than
der of the process will be constrained to one and              Gp 2 ( s ) ; this could be a main argument for select-
two, leading to PI and PID controllers. In the can-            ing u 1 ( s ) as primary manipulated variable, but it
celation design, the constraint in the order of the            also suggests that if a derivative term is desired,
systems is removed and the controllers can be de-              this should be in C 2 ( s ) , i.e., the slower plant dy-
signed by any controller design techniques.                    namics.
                                                                  These arguments support the selection, for in-
4.1. IMC design                                                stance, of C 1 ( s ) as a PI controller,

  For simplicity, let us assume stable plants, such
that first- or second-order plus time delay models
are adequate for describing the dynamics. Then,
                                                                                            ͩ
                                                                              C 1 ͑ s ͒ ϭK C 1 1ϩ
                                                                                                     1
                                                                                                    T I1s   ͪ   ,    ͑5͒
368                                  Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376


and C 2 ( s ) as a PD controller,                                    Kp 2 , ␶ p2 , and T d 2 , to adjust the dynamic model
                                                                     Gp 2 ( s ) to the correspondent physical data.
                                 ͑ 1ϩT D 2 s ͒
                                                                       4. Follow the IMC parametrization procedure
              C 2 ͑ s ͒ ϭK C 2                         .    ͑6͒
                                 ͑ 1ϩT F s ͒                         for Gp 2 ( s ) , which gives a PID controller based
                                                                     on a Smith predictor structure—the combined
Hence the combination C ( s ) ϭC 1 ( s ) C 2 ( s ) results           controller—whose parameters K C , T I , and T D are
to be a PID controller,                                              calculated by the following relationships:

                  ͩ
      C ͑ s ͒ ϭK C 1ϩ
                            1
                           T Is
                                ϩT D s
                                           1
                                            ͪ
                                       ͑ 1ϩT F s ͒
                                                   ,        ͑7͒                             K Cϭ
                                                                                                    ␶ p1 ϩ ␶ p2
                                                                                                     Kp 2 ␭ 2
                                                                                                                ,                ͑11a͒

whose parameters are                                                                        T I ϭ ␶ p1 ϩ ␶ p2 ,                  ͑11b͒

              K C ϭK C 1 K C 2 1ϩͩ        T D2
                                            T I1   ͪ   ,   ͑8a͒                             T Dϭ
                                                                                                     ␶ p1 ␶ p2
                                                                                                    ␶ p1 ϩ ␶ p2
                                                                                                                .                ͑11c͒

                      T I ϭT I 1 ϩT D 2 ,                  ͑8b͒        Comparing relationships ͑8a͒–͑8c͒ with ͑11a͒–
                                                                     ͑11c͒ and taking in account Eqs. ͑9a͒ and ͑9b͒, the
                              T I1T D2                               parameters of C 2 ( s ) are given by
                      T Dϭ                  .              ͑8c͒
                             T I 1 ϩT D 2                                                       ␶ p1       Kp 1 ␭ 1
                                                                                  K C2ϭ                  ϭ          ,            ͑12a͒
                                                                                            K C1 Kp 2 ␭ 2 Kp 2 ␭ 2
The forms of Eqs. ͑8a͒–͑8c͒ lead to the following
adjustment procedure:                                                                           T D 2 ϭ ␶ p2 .                   ͑12b͒
   1. Approach the dynamics relating y ( t ) and
u 1 ( t ) with a first-order plus time-delay transfer                 Observe the procedure leaves three parameters,
function                                                             ␭ 1 ,␭ 2 , and T F , for adjusting both controllers to
                                                                     achieve robust performance of the closed-loop
                                      e ϪT d1 s                      system. Since both transfers are connected through
               Gp 1 ͑ s ͒ ХKp 1                 .
                                     ␶ p1 sϩ1                        an interactive control scheme, the tuning must be
                                                                     coupled, because the uncertainties of each model,
  2. Use the IMC ͓16͔ parametrization to define                       lm 1 ( s ) and lm 2 ( s ) , affect both controllers simul-
the parameters of controller C 1 ( s ) , which is based              taneously. If there are uncertainties, the system
on the Smith predictor structure, i.e.,                              output y ( s ) controlled by the flexible-structure
                                                                     control is given by
                                2 ␶ p1
                      K C1ϭ             ,                  ͑9a͒
                               Kp 1 ␭ 1                                 y ͑ s ͒ ϭ ͕ G p 1 ͑ s ͓͒ 1ϩlm 1 ͑ s ͔͒ ϩC 2 ͑ s ͒ G p 2 ͑ s ͒
                                                                                    ˜                                     ˜

                         T I 1 ϭ ␶ p1 .                    ͑9b͒                  ϫ ͓ 1ϩlm 2 ͑ s ͔͒ ͖ u 1 ͑ s ͒ .

   3. Approach the dynamics relating variables                       Using the approximation ͑10͒ for G p 2 ( s ) and the
                                                                                                      ˜
y ( t ) and u 2 ( t ) with a second-order plus time-                 tuned the procedure proposed in this section for
delay model,                                                         C 2 , based on IMC design procedure,

                                   e ϪT d2 s                                       C 2 ͑ s ͒ ϭ ͕ G p * ͑ s ͒ ͖ Ϫ1 f 2 ͑ s ͒ ,
                                                                                                 ˜             ϩ
       Gp 2 ͑ s ͒ ХKp 2                            ,       ͑10͒
                          ͑ ␶ p1 sϩ1 ͒͑ ␶ p2 sϩ1 ͒
                                                                     the system output is given by
where one of the time constants is arbitrarily made
                                                                             y ͑ s ͒ ϭGp 1 ͑ s ͒ ͕ ͓ 1ϩlm 1 ͑ s ͔͒ ϩ f 2 ͑ s ͒
equal to ␶ p1 , the time constant determined for the
model Gp 1 ( s ) . This condition is just a convenient                                 ϫ ͓ 1ϩlm 2 ͑ s ͔͒ ͖ u 1 ͑ s ͒ .
way to get consistent individual and combined ad-
justments for C 1 ( s ) and C ( s ) , respectively. No-              From this expression we can identify the overall
tice that this still leaves three parameters,                        uncertainty that will suffer C 1 ( s ) ,
Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376                            369




       Fig. 5. Nonlinearities and its approximations for stability analysis: ͑a͒ decision function, and ͑b͒ saturation.



          lm ͑ s ͒ ϭlm 1 ͑ s ͒ ϩ f 2 ͑ s ͒ lm 2 ͑ s ͒ .             nonlinearities involved in the system ͑saturations
                                                                    and decision function͒ are approximated through a
Then, the last expressions lead to the following                    conic sector and then the resulting linear system is
tuning procedure for the controllers parameters:                    analyzed using robust control tools. In the second
   1. Tune the controller C 2 ( s ) to obtain the robust            approach, the stability of each linear system is
stability of Eq. ͑16͒, modifying ␭ 2 , for lm 2 ( s ) as            guaranteed and then the stability of the switching
if C ( s ) is an ideal PID.                                         between them is analyzed using describing func-
   2. Tune the controller C 1 ( s ) to obtain the robust            tion techniques ͓21͔.
stability of Eq. ͑15͒, modifying ␭ 1 , for lm ( s ) .                  Since the nonlinearities involved in the flexible-
   3. Finally, the parameter T F is determined such                 structure control, the decision function, and the
that the sensitivity function is attenuated in the                  saturations, satisfied the sector nonlinearity condi-
high-frequency range.                                               tion ͓20͔

4.2. Cancellation design
                                                                           ͯ         u  ͯ
                                                                               h ͑ u ͒ Ϫcu 2 2
                                                                                            Ͻr Ϫ␧,      ᭙u 0,␧Ͼ0, ͑14͒
  As a general rule and from some of the above
arguments it can be concluded that selecting the                    the nonlinearities can be approximated by ͑see
combined controller C ( s ) as being of equal or                    Fig. 5͒
higher order than C 1 ( s ) will always give a realiz-
able C 2 ( s ) . Hence any available design and ad-                                    h ͑ u ͒ Ϸ ͑ cϮr ͒ u,
justment procedure can be followed to completely                    where c is the gain of the linear model and r is the
define C 1 ( s ) and C ( s ) for controlling Gp 1 ( s ) and          uncertainty of the linear model. Then, the control-
Gp 2 ( s ) , respectively, as if they were not related to           lers are tuned to obtain the robust stability of the
each other. Then, the second controller C 2 ( s ) is                closed-loop system for the overall uncertainties
determined by                                                       ͓17,18͔: the model and the approximation errors r.
                C 2 ͑ s ͒ ϭC ͑ s ͒ C Ϫ1 ͑ s ͒ .
                                     1                    ͑13͒      This approach leads to an over conservative tune
                                                                    due to the consevatiness of the process gain, intro-
There are no stability problems in this design tech-                duced by the approximation ͑14͒. Therefore the
nique because the zeros and poles of C 1 ( s ) and                  resulting closed-loop performance will be poor
C ( s ) are stable and known.                                       and the response will be sluggish.
                                                                       The stability analysis of the flexible-structure
5. Stability analysis                                               control from the switching system point of view
                                                                    implies the stability analysis of every single or
  A general stability analysis for the flexible con-                 combined loop and the stability analysis for a gen-
trol system can be performed using two different                    eral switching sequence between these loops.
frameworks: ͑i͒ a stability analysis of the resulting                  In a first step, the stability of each closed-loop
nonlinear system ͓18͔, or ͑ii͒ a stability analysis of              system is studied. While u 1 ( t ) Ͻ ␬ the characteris-
switched linear system ͓19͔. In the first case the                   tic equation for the primary control loop includes
370                                Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376


Gp 1 ( s ) and C 1 ( s ) only, i.e., the stability condi-                     ͑ ␶ 1 sϩ1 ͒ K 1        ͑ ␶ 1 sϩ1 ͒ K 1 ␭ 1
tion may be written as follows:                                          1ϩ                       ϩ␩
                                                                                K 1 ␭ 1 s ␶ 1 sϩ1      K 1␭ 1s K 2␭ 2
         1ϩC 1 ͑ s ͒ Gp 1 ͑ s ͒ 0,        ᭙s෈C ϩ ,         ͑15͒                   ͑ ␶ 2 sϩ1 ͒          K2
          ϩ                                                                   ϫ                                      ϭ0. ͑20͒
where C stands for the right-side complex plane.                                  ͑ T F sϩ1 ͒ ͑ ␶ 1 sϩ1 ͒͑ ␶ 2 sϩ1 ͒
The second important stability condition is for the
secondary loop which works through the manipu-                      Operating with this equation, the characteristic
lated u 2 when u 1 ( t ) Ͼu 1 max ,                                 equation is

      1ϩC 2 ͑ s ͒ C 1 ͑ s ͒ Gp 2 ͑ s ͒ 0,     ᭙s෈C ϩ .                 ␭ 1 ␭ 2 T F s 2 ϩ␭ 2 ͑ ␭ 1 ϩT F ͒ sϩ ͑ ␭ 2 ϩ␭ 1 ␩ ͒ ϭ0,
                                                           ͑16͒                                                              ͑21͒
Finally, for the intermediate case when ␩ Ͼ0 and                    and the poles of the closed-loop system p are
␬ Ͻu 1 ( t ) Ͻu 1 max , two control paths coexist si-               given by
multaneously: one through u 1 and the other
through u 2 . Then, the condition takes the form
  1ϩC 1 ͑ s ͒ Gp 1 ͑ s ͒ ϩ ␩ C 1 ͑ s ͒ C 2 ͑ s ͒ Gp 2 ͑ s ͒ 0,
                                                                        p 1,2ϭϪ
                                                                                ␭ 1 ϩT F
                                                                                2␭ 1 T F
                                                                                         Ϯ        ͱ   ͑ ␭ 1 ϪT F ͒ 2
                                                                                                             2 2
                                                                                                         4␭ 1 T F
                                                                                                                     Ϫ
                                                                                                                         ␩
                                                                                                                       ␭ 2T F
                                                                                                                              .
                                                                                                                               ͑22͒
                         ᭙s෈C ϩ .                          ͑17͒
                                                                    The stability only depends on the real part of these
The first two conditions can be satisfied sequen-                     poles, therefore it is clear that the stability of the
tially, Eq. ͑15͒ while adjusting controller C 1 ( s ) ,             closed-loop system only depends on the values of
Eq. ͑16͒ while adjusting controller C 2 ( s ) . Hence               the controllers parameters ( ␭ 1 ,␭ 2 and T F ) and the
the stability problem of Eq. ͑17͒ is automatically                  parameter ␩ of the decision function.
satisfied for the nominal system.                                      Remark 2. If ␭ 2 T F ӷ ␩ the closed-loop poles
   Theorem 5.1. Given a two-inputs and one-                         will be approximately located at p 1 ϭ␭ Ϫ1 and p 2
                                                                                                               1
                                                                        Ϫ1
output system with stable transfer function be-                     ϭT F . However, if
tween the inputs, u 1 ( s ) and u 2 ( s ) , and output,
Gp 1 ( s ) and Gp 2 ( s ) , and ␩ у0, the closed-loop                                            ͑ ␭ 1 ϪT F ͒ 2
                                                                                        ␩ Ͼ␭ 2                  ,
system resulting from the application of the flex-                                                   4␭ 2 T F
                                                                                                        1
ible structure control is stable if the following con-
ditions are satisfied:                                               ␭ 1 and T F can be used to fix the damping ratio of
                                                   ϩ                the closed-loop response and ␭ 2 and ␩ can be
         1ϩGp 1 ͑ s ͒ C 1 ͑ s ͒ 0,        ᭙s෈C ,
                                                                    employed to fix the natural frequency of the
      1ϩGp 2 ͑ s ͒ C 2 ͑ s ͒ C 1 ͑ s ͒ 0,     ᭙s෈C ϩ .              closed-loop response.
                                                                       Finally, the stability analysis of the switching
  Proof: Given the closed-loop equation                             sequence implies the analysis of the following sys-
                                                                    tems:
  1ϩC 1 ͑ s ͒ Gp 1 ͑ s ͒ ϩ ␩ C 1 ͑ s ͒ C 2 ͑ s ͒ Gp 2 ͑ s ͒ ϭ0,
                                                             ͑18͒       1ϩC 1 ͑ s ͒ Gp 1 ͑ s ͒ ϩS ͑ s ͒ ␩ C ͑ s ͒ Gp 2 ͑ s ͒ ϭ0,
                                                                                                                              ͑23a͒
both transfer functions
                                   e ϪT d 1 s                        1ϩS ͑ s ͒ C 1 ͑ s ͒ Gp 1 ͑ s ͒ ϩC ͑ s ͒ Gp 2 ͑ s ͒ ϪS ͑ s ͒͑ 1
                 Gp 1 ͑ s ͒ ϭK 1              ,           ͑19a͒
                                   ␶ 1 sϩ1                                  Ϫ ␩ ͒ C ͑ s ͒ Gp 2 ͑ s ͒ ϭ0,                       ͑23b͒
                                    e ϪT d 2 s                       1ϩS ͑ s ͒ C 1 ͑ s ͒ Gp 1 ͑ s ͒ ϩ ͓ 1ϪS ͑ s ͔͒ C ͑ s ͒ Gp 2 ͑ s ͒
          Gp 2 ͑ s ͒ ϭK 2                          ,     ͑19b͒
                            ͑ ␶ 1 sϩ1 ͒͑ ␶ 2 sϩ1 ͒
                                                                        ϭ0,                                                    ͑23c͒
and the controllers C 1 ( s ) and C 2 ( s ) tuned accord-
ing with the procedure described in the previous                    that represent all possible changes in the system.
section characteristic equation of the closed-loop                  In these expressions S ( s ) is the Fourier transform
system is given by                                                  of the switching function S ( t ) given by
Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376                                  371


                             1                                          everywhere the magnitude of the Bode plot of the
                    S ͑ t ͒ ϭ ͓ 1Ϫg ͑ t ͔͒ ,                   ͑24͒     resulting linear system exceeds unity.
                             2
where g ( t ) is a scalar signal that only assumes the
                                                                        6. Choice of ␩ and ␬
value Ϫ1 or ϩ1. Eq. ͑23a͒ represents the changes
in the system when u 1 ( t ) Ͼ ␬ , therefore the sec-
                                                                           In this control scheme ␩ represents the amount
ond loop becomes active. The second equation,
                                                                        of u 2 requires to prevent the saturation of u 1 while
Eq. ͑23b͒, represents the change in the system
                                                                        ␬ is the value of u 1 at which the protection begins.
when u 1 ( t ) Ͼu 1 max , then only the second loop
                                                                        However, both parameters can be employed to sat-
controls the system output. Finally, Eq. ͑23c͒ rep-
                                                                        isfy some additional objective than control objec-
resents the direct transition from u 1 ( t ) Ͻ ␬ to
                                                                        tives.
u 1 ( t ) Ͼu 1 max , this fact means that the main loop
                                                                           If the objective is to obtain a good transient
becomes inactive at the same time that the second-
                                                                        behavior, Gp 1 must be kept active in the whole
ary loop is activated.
                                                                        operating range. Hence the parameters of the de-
   For switching sequences slower than the system
                                                                        cision function ͑4͒ must be fixed to
dynamic, the stability of the overall closed-loop
system is guaranteed ͓19͔. The stability of an ar-                                                  u 2 max
bitrary switching sequence between these systems                                              ␩ϭ            ,                   ͑26a͒
                                                                                                    u 1 max
can be analyzed using describing function tech-
niques and harmonic balance explained by Leith et                                                 ␬ ϭ0.                         ͑26b͒
al. ͓19͔ First, the nonlinearities are approximate
through a Fourier series,                                               This selection implies the use of u 2 to prevent the
                                                                        saturation of u 1 for any value. This might be a
          S ͑ t ͒ ϭ f 0 ϩ f 1 cos͑ ␻ tϩ ␾ ͒ ϩh.o.t,                     desirable feature for the control system since it
                                                                        keeps the fastest loop working in order to maintain
then, due to the low-pass characteristic of the sys-                    a better control quality. This protection is clearly
tem S ( t ) may be approximate,                                         done at the expense of u 2 . In the opposite case, if
               S ͑ t ͒ Ϸ f 0 ϩ f 1 cos͑ ␻ tϩ ␾ ͒ .                      the objective is to minimize the amount of u 2 em-
                                                                        ployed to control the system, Gp 2 must be kept
For the controllers resulting from the IMC design,                      inactive as much as possible. Therefore the param-
the transfer function of the equivalent linear sys-                     eters of the decision function must be fixed to ␩
tem of Eqs. ͑23a͒–͑23c͒ are given by                                    ϭ0. This selection implies the use of the auxiliary
                                                                        variable to cover output demands when u 1 satu-
    H 1 ͑ s ͒ ϭϪ ͑ f 1 ͒ / ͕ ␭ 1 ␭ 2 T F s 2                            rates only. Finally, if the objectives are a combi-
                ϩ␭ 2 ͑ ␭ 1 ϩT F ͒ sϩ ͑ ␭ 2 ϩ ␩ ␭ 1 f 0 ͒ ͖ ,            nation of previous ones, Gp 1 must be kept active
                                                                        in the range such that it can handle the most fre-
                                                              ͑25a͒     quent changes. Therefore the parameters of the de-
                                                                        cision function ͑4͒ are given by
 H 2 ͑ s ͒ ϭϪ ͕ f 1 ͓ ␭ 2 T F sϩ ͑ ␭ 2 Ϫ␭ 1 ͔͒ ͖ / ͕ ␭ 1 ␭ 2 T F s 2
                                                                                     1
            ϩ␭ 2 ͑ ␭ 1 ϩT F f 0 ͒ s                                            ␩у        „max͕ var͓ r ͑ t ͔͒ ,var͓ d ͑ t ͔͒ ͖
                                                                                    Kp 2
            ϩ ͓͑ 1Ϫ ␩ Ϫ f 0 ͒ ␭ 1 ϩ f 0 ␭ 2 ͔ ͖ ,             ͑25b͒                 ϪKp 1 u 1 max…,                             ͑27a͒
H 3 ͑ s ͒ ϭϪ ͕ f 1 ͓ ␭ 2 T F sϩ ͑ ␭ 2 Ϫ␭ 1 ͔͒ ͖ / ͕ ␭ 1 ␭ 2 T F s   2
                                                                                                ␬ уu 10 ,                       ͑27b͒
            ϩ␭ 2 ͑ ␭ 1 ϩT F f 0 ͒ sϩ ͓͑ 1Ϫ f 0 ͒ ␭ 1 ϩ␭ 2 ͔ ͖ .         where u 10 is the steady-state value of u 1 for the
                                                              ͑25c͒     nominal set-point value. This selection implies the
                                                                        use of u 2 to prevent the saturation of u 1 for the
Finally, the resulting transfer functions are ana-                      most frequent changes only when u 1 Ͼ ␬ .
lyzed using the method of harmonic balance to                             To explain these concepts, let us consider the
determine the stability of the switching sequence.                      case of two heat exchangers in series: one heat
The harmonic balance method predicts instability                        exchanger and a service equipment with a direct
372                          Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376




                             Fig. 6. Bode plots associated with approximated analysis.


bypass on the controlled stream. In this configura-                                           2e Ϫ10s
tion the service equipment works intermittently                          Gp 2 ͑ s ͒ ϭ                     ,       ͑29͒
                                                                                        ͑ 6sϩ1 ͒͑ 17sϩ1 ͒
such that recovered energy is maximized. In this
problem, the parameters ␩ and ␬ correspond to the            and the disturbance d ( s ) are modeled by
service energy level required to avoid the control-
lability problems, which implies a loss of process                                            1
                                                                              G d͑ s ͒ ϭ           .              ͑30͒
efficiency, and the level at which the protection                                           5s ϩsϩ1
                                                                                             2

begins, respectively. If the objective is to obtain a
good dynamic behavior, the parameters must be                The primary and the auxiliary manipulated vari-
set to ␩ ϭ1 and ␬ ϭ0, respectively. In the opposite          ables u 1 and u 2 are constrained to u 1,2෈ ͓ Ϫ1,
case, if the objective is to maximize the amount of          ϩ1 ͔ .
energy recovered, the parameters must be fixed to                The controllers will be developed using a Smith
␩ ϭ0. Finally, if the objective is to maximize the           predictor structure to compensate the time delays
amount of energy recovered while keeping good                of the process. First, the controller C 1 ( s ) is de-
closed-loop performance, the parameters must be              signed using the formulas ͑9a͒ and ͑9b͒ and obtain
fixed to 0Ͻ ␩ Ͻ1 ͑according to the disturbances               a response without offset. This means that C 1 ( s )
and set-point variance͒ and ␬ уu 10 , where u 10 is          is a PI controller, whose parameters are
the steady-state value u 1 for the nominal set-point
                                                                                             2.75
value.                                                                               K C1ϭ        ,             ͑31a͒
                                                                                              ␭1
7. Simulation and results                                                        T I 1 ϭ ␶ p1 ϭ2.75.            ͑31b͒

  In this section a simulation example is consid-            To tune the controller C 2 ( s ) , we approximate
ered to show the effectiveness of the control                Gp 2 using Eq. ͑10͒. The result is
scheme proposed in this work. We consider a lin-
ear system previously used by Wang et al. ͓9͔ to                                            2e Ϫ10s
evaluate the cooperative control ͓9͔. The system is
                                                                     Gp 2 ͑ s ͒ Ӎ                           .     ͑32͒
                                                                                    ͑ 2.75sϩ1 ͒͑ 18.75sϩ1 ͒
given by
                                                             First, we design the combined controller C ( s )
                        e Ϫ4s     e Ϫ5s                      ϭC 1 ( s ) C 2 ( s ) that controls this model. In this ap-
        Gp 1 ͑ s ͒ ϭ           Х        ,         ͑28͒
                     ͑ 2sϩ1 ͒ 2 2.75sϩ1                      plication, a PID controller is adopted for C ( s ) .
Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376                         373




                    Fig. 7. Set-point responses obtained using flexible structure and cooperative control.


This means that C 1 ( s ) is a PD controller, whose              the constraint in the primary manipulated variable
parameters are given by Eqs. ͑12a͒ and ͑12b͒,                    ( u 1 ) . The parameters of the cooperative control
                                                                 are observation time T o and a margin ␤. The value
                        Kp 1 ␭ 1   ␭1                            of these parameters are the same to that one em-
               K C2ϭ             ϭ    ,              ͑33a͒
                        Kp 2 ␭ 2 2␭ 2                            ployed by Wang et al. to obtain the best perfor-
                                                                 mance,
                 T D 2 ϭ ␶ p2 ϭ18.75.               ͑33b͒
                                                                                   T o ϭ0.275,     ␤ ϭ2.
Since there is no uncertainty, the parameters
␭ 1 ,␭ 2 , and T F are determined to obtain the best             In order to evaluate the performances of both con-
possible performance. The values of these param-
                                                                 trol schemes, a sequence of reference and distur-
eters are
                                                                 bance changes are introduce at several times. The
        ␭ 1 ϭ1.5,      ␭ 2 ϭ0.56,    T F ϭ30.2.       ͑34͒       set point r was changed in intervals of 200 sec
                                                                 from 0.5 to 0.75 and then steps to 1.5, and finally
An IMC antiwindup scheme is included in the                      the disturbance w changes from 0 to 0.5 at 450
C ( s ) controller to compensate for the effect of the           sec.
constraint in u 2 . Finally, the parameters of the de-             In Fig. 7 we see the responses obtained by both
cision function ͑1͒ were chosen to obtain a good                 control schemes. This figure shows the superior
performance, therefore ␩ and ␬ were fixed to                      performance of the proposed scheme when the
                ␬ ϭ0.25,       ␩ ϭ1.0.                ͑35͒       system needs to modify the value of the auxiliary
                                                                 manipulated variable. The better performance is
Finally, the stability of the system for the switch is           due to the fact that the proposed scheme takes ac-
analyzed. Fig. 6 shows the Bode plots for the                    count of the dynamics of the auxiliary manipu-
transfer functions ͑25a͒–͑25c͒ for parameters ͑34͒               lated variable and the main loop is still working.
and ͑35͒. It is easy to see the resulting linear sys-            This fact leads to the temporary saturation of the
tem does not exceed unity, therefore all possible                primary manipulated variable of cooperative con-
switching between these systems are stable.                      trol during the transition period ͑Fig. 8͒, that leads
   In this example the proposed control structure is             to open-loop behavior of the closed-loop system
compared with the cooperative algorithm pro-                     that deteriorates the closed-loop performance.
posed by Wang et al. ͓9͔. The controller employed                However, the cooperative control scheme shows a
by the cooperative schemes is C 1 ( s ) , with an IMC            better performance when a change in the auxiliary
antiwindup scheme, to compensate the effect of                   variable is not needed. It is clear that the interac-
374                           Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376


                                                                  To address the effects of the protection level ␩
                                                               and the start of the protection ␬, the proposed al-
                                                               gorithm is simulated with different values of them.
                                                               The results with different ␩ are shown in Fig. 9. It
                                                               can be seen that the higher the level of ␩ the better
                                                               the performance, since both transfer functions
                                                               control the system output. When ␩ ϭ0 the system
                                                               output is first controlled by Gp 1 and then by Gp 2 .
                                                               In this case the system output shows a large over-
                                                               shoot and large settling time due to the change of
                                                               the process structure. The system output is now
                                                               controlled by the portion of the system that has the
                                                               slower dynamics and the bigger time delay, there-
                                                               fore closed-loop performance is the poorest.
                                                                  Fig. 10 shows the control actions associated to
                                                               the responses of Fig. 9. In this figure it is easy to
                                                               see that u 2 works to avoid the saturation u 1 or
                                                               takes full control of the system output, depending
                                                               on the value of ␩. If ␩ Ͼ0, u 2 prevents the satu-
                                                               ration of u 1 by providing a level ␩ r of the output.
                                                               This fact keeps Gp 1 active for a wider output
                                                               range, therefore good closed-loop responses are
Fig. 8. Primary and auxiliary variables corresponding to       obtained. When ␩ ϭ0, u 2 takes full control of the
set-point and disturbance changes of Fig. 7.                   system output when u 1 is saturated, therefore the
                                                               slowest dynamic is in charge of the output regula-
                                                               tion and poor closed-loop responses are obtained.
tion between both loops improves the performance               In these figures we can also see that the auxiliary
of the closed-loop performance when the control                variable exhibits peaks. This fact is due to the in-
is transferred from u 1 to u 2 , but it deteriorates the       crement in the gain of C 2 ( s ) , which is a PD con-
performance when this change is not required.                  troller.




                             Fig. 9. Set-point responses obtained for different values of ␩.
Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376                            375




Fig. 10. Primary and auxiliary variables corresponding to       Fig. 12. Primary and auxiliary variables corresponding to
set-point and disturbance changes of Fig. 9.                    set-point and disturbance changes of Fig. 11.


  The results with different ␬ are shown in Figs.               When ␬ ϭ0 the protective action is applied all the
11 and 12. It can be seen that higher the level of ␬            time. This fact leads to a good closed-loop perfor-
the smaller the value of u 2 , however, similar                 mance. If ␬ Ͼ0 the protective action is only ap-
closed-loop responses are obtained in both cases.               plied when u 2 у ␬ , the closed-loop performance is




                             Fig. 11. Set-point responses obtained for different values of ␩.
376                             Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376


similar to the case of ␬ ϭ0. This fact leads to a                      trolling the operation of heat exchanger networks.
conservative use of u 2 .                                              AIChE J. 44, 1090–1104 ͑1998͒.
                                                                 ͓9͔   Wang, Q., Zhang, Y., Cai, W., Bi, Q., and Hang, C.,
                                                                       Co-operative control of multi-input single-output pro-
8. Conclusions                                                         cesses: On-line strategy for releasing input saturation.
                                                                       Control Eng. Pract. 9, 491–500 ͑2001͒.
   A new control structure proposed for dealing                 ͓10͔   Morningred, D., Paden, B., and Mellichamp, D., An
with constraints on manipulated variables process                      adaptive nonlinear predictive controller. Chem. Eng.
shows at least two input variables associated to the                   Sci. 47, 755–765 ͑1992͒.
                                                                ͓11͔   Luyben, W., Process Modelling, Simulation and Con-
control target. Flexible-structure control refers to                   trol for Chemical Engineers. McGraw-Hill Interna-
the resulting control system, which switches from                      tional, New York, 1990.
one closed loop to another in order to keep con-                ͓12͔   Marselle, D., Morari, M., and Rudd, D., Design of
trollability. This flexible characteristic of the con-                  resilient processing plants: II. Design and control of
trol system is particularly useful when the optimal                    energy management system. Chem. Eng. Sci. 37, 259–
operating point locates near a limit constraint of                     270 ͑1982͒.
                                                                ͓13͔   Mathisen, K. Integrated Design and Control of Heat
the main manipulated variable. Issues related to                       Exchanger Networks. Ph.D. thesis, University of
design, analysis, and tuning a flexible-structure                       Trondheim, Trondheim, Norway, 1994.
control system are discussed. The application to a              ͓14͔        ¨                  ¨
                                                                       Åstrom, K. and Hagglund, T., PID Controllers:
linear system shows the tradeoff between process                       Theory, Design and Tuning, 2nd ed. Instrument Soci-
efficiency and controllability.                                         ety of America, 1995.
                                                                ͓15͔   Smith, O., Close control of loops with dead time.
                                                                       Chem. Eng. Prog. 53, 217–219 ͑1957͒.
Acknowledgments                                                 ͓16͔   Chien, I. and Fruehauf, P., Consider IMC tuning to
                                                                       improve controller performance. Chem. Eng. Prog. 86,
  We are grateful for the financial support for this                    33– 41 ͑1990͒.
work provided by the Engineering and Physical                   ͓17͔   Morari, M. and Zafiriou, E., Robust Process Control,
Science Research Council ͑EPSRC͒ Industrial                            2nd ed. Prentice-Hall, Englewood Cliffs, NJ, 1989.
                                                                ͓18͔   Sofanov, M. and Athans, M., A multiloop generalisa-
Non-linear Control and Applications Grant No.
                                                                       tion of the circle criterion for stability analysis. IEEE
GR/R04683/01. Thanks are due to Dr. Hao Xia for                        Trans. Autom. Control 26, 415– 422 ͑1981͒.
valuable discussions and the anonymous referees                 ͓19͔   Leith, D., Shorten, R., Leithead, W., Mason, O., and
for the suggestions to improve the paper.                              Curran, P., Issues in the design of switched linear con-
                                                                       trol systems: A benchmark study. Int. J. Adapt. Control
References                                                             Signal Process. 17, 103–118 ͑2003͒.
                                                                ͓20͔   Desoer, C. and Vidyasagar, M., Feedback Systems:
 ͓1͔ Chen, C. and Peng, S., Learning control of process                Input-Output Systems. Academic, New York, 1975.
     systems with hard constraints. J. Process Control 9,       ͓21͔   Khalil, K., Nonlinear System Analysis. Macmillan,
     151–160 ͑1999͒.                                                   New York, 1992.
 ͓2͔ Doyle, F., III, An auto-windup input-output lineariza-
     tion scheme for SISO systems. J. Process Control 9,
     213–220 ͑1999͒.
                                                                                                         Leonardo L. Giovanini was
 ͓3͔ Kothare, M., Campo, P., Morari, M., and Nett, C., A                                                 born in Argentina in 1969. He
     unified framework for the study of anti-windup design.                                               received the BSc ͑Honors͒ de-
     Automatica 30, 1869–1883 ͑1994͒.                                                                    gree in Electronic Engineering
 ͓4͔ Wang, Q. and Zhang, Y., A method to incorporate con-                                                from the Universidad Tecno-
     troller output constraints into controller design. ISA                                              logica Nacional, Regional
     Trans. 37, 403– 416 ͑1998͒.                                                                         School at Villa Maria in 1995,
                                                                                                         and the Ph.D. degree in Engi-
 ͓5͔ Stephanopoulos, G., Chemical Process Control: An In-                                                neering Science from Univer-
     troduction to Theory and Practice. Prentice-Hall,                                                   sidad Nacional del Litoral,
     Englewood Cliffs, NJ, 1984.                                                                         School of Water Resource in
 ͓6͔ Shinskey, F., Process Control Systems: Application,                                                 2000. Since then he has been
     Design and Tuning, 3rd ed. McGraw-Hill, New York,                                                   Lecturer in the Department of
     1988.                                                                                               Electronic Engineering at Uni-
                                                                versidad Tecnologica Nacional, Regional School at Villa Maria. Cur-
 ͓7͔ Gao, Z., Stability of the pseudo-inverse method for        rently, he is working as a research fellow in the Industrial Control
     reconfigurable control system. Int. J. Control 8, 469–      Center, University of Strathclyde. His research interests include pre-
     474 ͑1991͒.                                                dictive control for non-linear systems, fault detection and isolation,
 ͓8͔ Aguilera, N. and Marchetti, J., Optimizing and con-        fault tolerant control and distributed control systems.

More Related Content

What's hot

Ch2 mathematical modeling of control system
Ch2 mathematical modeling of control system Ch2 mathematical modeling of control system
Ch2 mathematical modeling of control system Elaf A.Saeed
 
C0325010015
C0325010015C0325010015
C0325010015theijes
 
Simulation analysis of Series Cascade control Structure and anti-reset windup...
Simulation analysis of Series Cascade control Structure and anti-reset windup...Simulation analysis of Series Cascade control Structure and anti-reset windup...
Simulation analysis of Series Cascade control Structure and anti-reset windup...IOSR Journals
 
Control Algorithms for Evaluating Seismic Performance of Structures
Control Algorithms for Evaluating Seismic Performance of StructuresControl Algorithms for Evaluating Seismic Performance of Structures
Control Algorithms for Evaluating Seismic Performance of StructuresIRJET Journal
 
SLIDING MODE CONTROL AND ITS APPLICATION
SLIDING MODE CONTROL AND ITS APPLICATIONSLIDING MODE CONTROL AND ITS APPLICATION
SLIDING MODE CONTROL AND ITS APPLICATIONBindutesh Saner
 
Auto tuning of cascade control systems
Auto tuning of cascade control systemsAuto tuning of cascade control systems
Auto tuning of cascade control systemsISA Interchange
 
Dynamic Matrix Control (DMC) on jacket tank heater - Rishikesh Bagwe
Dynamic Matrix Control (DMC) on jacket tank heater - Rishikesh BagweDynamic Matrix Control (DMC) on jacket tank heater - Rishikesh Bagwe
Dynamic Matrix Control (DMC) on jacket tank heater - Rishikesh BagweRishikesh Bagwe
 
Addressing control applications using wireless hart devices
Addressing control applications using wireless hart devicesAddressing control applications using wireless hart devices
Addressing control applications using wireless hart devicesEmerson Exchange
 
Advanced sliding mode control for mechanical systems
Advanced sliding mode control for mechanical systemsAdvanced sliding mode control for mechanical systems
Advanced sliding mode control for mechanical systemsAlejandro Todoroff
 
Sliding Mode Methodology Vs. Computed Torque Methodology Using MATLAB/SIMULIN...
Sliding Mode Methodology Vs. Computed Torque Methodology Using MATLAB/SIMULIN...Sliding Mode Methodology Vs. Computed Torque Methodology Using MATLAB/SIMULIN...
Sliding Mode Methodology Vs. Computed Torque Methodology Using MATLAB/SIMULIN...CSCJournals
 
Synthesis of the Decentralized Control System for Robot-Manipulator with Inpu...
Synthesis of the Decentralized Control System for Robot-Manipulator with Inpu...Synthesis of the Decentralized Control System for Robot-Manipulator with Inpu...
Synthesis of the Decentralized Control System for Robot-Manipulator with Inpu...ITIIIndustries
 
Sliding Mode Controller for Robotic Flexible Arm Joint
Sliding Mode Controller  for Robotic Flexible Arm JointSliding Mode Controller  for Robotic Flexible Arm Joint
Sliding Mode Controller for Robotic Flexible Arm Jointomkarharshe
 
Formulation of Generalized Block Backstepping Control Law of Underactuated Me...
Formulation of Generalized Block Backstepping Control Law of Underactuated Me...Formulation of Generalized Block Backstepping Control Law of Underactuated Me...
Formulation of Generalized Block Backstepping Control Law of Underactuated Me...Shubhobrata Rudra
 

What's hot (18)

Ch2 mathematical modeling of control system
Ch2 mathematical modeling of control system Ch2 mathematical modeling of control system
Ch2 mathematical modeling of control system
 
C0325010015
C0325010015C0325010015
C0325010015
 
Simulation analysis of Series Cascade control Structure and anti-reset windup...
Simulation analysis of Series Cascade control Structure and anti-reset windup...Simulation analysis of Series Cascade control Structure and anti-reset windup...
Simulation analysis of Series Cascade control Structure and anti-reset windup...
 
WM12_6.PDF
WM12_6.PDFWM12_6.PDF
WM12_6.PDF
 
Control Algorithms for Evaluating Seismic Performance of Structures
Control Algorithms for Evaluating Seismic Performance of StructuresControl Algorithms for Evaluating Seismic Performance of Structures
Control Algorithms for Evaluating Seismic Performance of Structures
 
B010511015
B010511015B010511015
B010511015
 
SLIDING MODE CONTROL AND ITS APPLICATION
SLIDING MODE CONTROL AND ITS APPLICATIONSLIDING MODE CONTROL AND ITS APPLICATION
SLIDING MODE CONTROL AND ITS APPLICATION
 
Auto tuning of cascade control systems
Auto tuning of cascade control systemsAuto tuning of cascade control systems
Auto tuning of cascade control systems
 
Dynamic Matrix Control (DMC) on jacket tank heater - Rishikesh Bagwe
Dynamic Matrix Control (DMC) on jacket tank heater - Rishikesh BagweDynamic Matrix Control (DMC) on jacket tank heater - Rishikesh Bagwe
Dynamic Matrix Control (DMC) on jacket tank heater - Rishikesh Bagwe
 
B04450517
B04450517B04450517
B04450517
 
Addressing control applications using wireless hart devices
Addressing control applications using wireless hart devicesAddressing control applications using wireless hart devices
Addressing control applications using wireless hart devices
 
C010411722
C010411722C010411722
C010411722
 
Advanced sliding mode control for mechanical systems
Advanced sliding mode control for mechanical systemsAdvanced sliding mode control for mechanical systems
Advanced sliding mode control for mechanical systems
 
Sliding Mode Methodology Vs. Computed Torque Methodology Using MATLAB/SIMULIN...
Sliding Mode Methodology Vs. Computed Torque Methodology Using MATLAB/SIMULIN...Sliding Mode Methodology Vs. Computed Torque Methodology Using MATLAB/SIMULIN...
Sliding Mode Methodology Vs. Computed Torque Methodology Using MATLAB/SIMULIN...
 
Synthesis of the Decentralized Control System for Robot-Manipulator with Inpu...
Synthesis of the Decentralized Control System for Robot-Manipulator with Inpu...Synthesis of the Decentralized Control System for Robot-Manipulator with Inpu...
Synthesis of the Decentralized Control System for Robot-Manipulator with Inpu...
 
B044050811
B044050811B044050811
B044050811
 
Sliding Mode Controller for Robotic Flexible Arm Joint
Sliding Mode Controller  for Robotic Flexible Arm JointSliding Mode Controller  for Robotic Flexible Arm Joint
Sliding Mode Controller for Robotic Flexible Arm Joint
 
Formulation of Generalized Block Backstepping Control Law of Underactuated Me...
Formulation of Generalized Block Backstepping Control Law of Underactuated Me...Formulation of Generalized Block Backstepping Control Law of Underactuated Me...
Formulation of Generalized Block Backstepping Control Law of Underactuated Me...
 

Viewers also liked

PortalEIR: Tools for tutoring residents
PortalEIR: Tools for tutoring residentsPortalEIR: Tools for tutoring residents
PortalEIR: Tools for tutoring residentsJosé Luis de la Rosa
 
Ibon Arechalde "Electromagnetic Disturbances and Blocking Problems in PLC Tec...
Ibon Arechalde "Electromagnetic Disturbances and Blocking Problems in PLC Tec...Ibon Arechalde "Electromagnetic Disturbances and Blocking Problems in PLC Tec...
Ibon Arechalde "Electromagnetic Disturbances and Blocking Problems in PLC Tec...TECNALIA Research & Innovation
 
Capa 2 - Enllaç de dades i adreça MAC
Capa 2 - Enllaç de dades i adreça MACCapa 2 - Enllaç de dades i adreça MAC
Capa 2 - Enllaç de dades i adreça MACAitor Mendoza
 
Connected Collaboration
Connected CollaborationConnected Collaboration
Connected CollaborationAlan Lepofsky
 
#Elviajedemitarjeta prensa (ES)
#Elviajedemitarjeta prensa (ES)#Elviajedemitarjeta prensa (ES)
#Elviajedemitarjeta prensa (ES)Michiel Das
 
Cinema per-a-estudiants-2015-16
Cinema per-a-estudiants-2015-16Cinema per-a-estudiants-2015-16
Cinema per-a-estudiants-2015-16Antonio Bravo
 
Les muralles de Tarragona
Les muralles de TarragonaLes muralles de Tarragona
Les muralles de Tarragonaclorent3
 
2989340 informatica-dicionario-internet
2989340 informatica-dicionario-internet2989340 informatica-dicionario-internet
2989340 informatica-dicionario-internetNilton Reder
 
10 Tendencias de Consumo para 2010
10 Tendencias de Consumo para 201010 Tendencias de Consumo para 2010
10 Tendencias de Consumo para 2010María José López
 
Trabajo Técnicas de venta:Manual de ventas trias completo
Trabajo Técnicas de venta:Manual de ventas trias completoTrabajo Técnicas de venta:Manual de ventas trias completo
Trabajo Técnicas de venta:Manual de ventas trias completoJaime Rodriguez
 
El sol fuente de energía quimica
El sol fuente  de energía quimicaEl sol fuente  de energía quimica
El sol fuente de energía quimicaMSMSANDOVAL
 
From Big to Smart Data - Smart Data Innovation Lab Overview
From Big to Smart Data - Smart Data Innovation Lab OverviewFrom Big to Smart Data - Smart Data Innovation Lab Overview
From Big to Smart Data - Smart Data Innovation Lab OverviewPlamen Kiradjiev
 

Viewers also liked (20)

Distress Technology in Boating - Nichole Kalil, ACR Electronics
Distress Technology in Boating - Nichole Kalil, ACR ElectronicsDistress Technology in Boating - Nichole Kalil, ACR Electronics
Distress Technology in Boating - Nichole Kalil, ACR Electronics
 
Vinos y playas Wines & Beaches
Vinos y playas Wines & BeachesVinos y playas Wines & Beaches
Vinos y playas Wines & Beaches
 
PortalEIR: Tools for tutoring residents
PortalEIR: Tools for tutoring residentsPortalEIR: Tools for tutoring residents
PortalEIR: Tools for tutoring residents
 
Ibon Arechalde "Electromagnetic Disturbances and Blocking Problems in PLC Tec...
Ibon Arechalde "Electromagnetic Disturbances and Blocking Problems in PLC Tec...Ibon Arechalde "Electromagnetic Disturbances and Blocking Problems in PLC Tec...
Ibon Arechalde "Electromagnetic Disturbances and Blocking Problems in PLC Tec...
 
Preporuka OCD SOT
Preporuka OCD SOTPreporuka OCD SOT
Preporuka OCD SOT
 
Wu Tang
Wu TangWu Tang
Wu Tang
 
Capa 2 - Enllaç de dades i adreça MAC
Capa 2 - Enllaç de dades i adreça MACCapa 2 - Enllaç de dades i adreça MAC
Capa 2 - Enllaç de dades i adreça MAC
 
Connected Collaboration
Connected CollaborationConnected Collaboration
Connected Collaboration
 
El alzheimer y las embolias
El alzheimer y las emboliasEl alzheimer y las embolias
El alzheimer y las embolias
 
#Elviajedemitarjeta prensa (ES)
#Elviajedemitarjeta prensa (ES)#Elviajedemitarjeta prensa (ES)
#Elviajedemitarjeta prensa (ES)
 
Cinema per-a-estudiants-2015-16
Cinema per-a-estudiants-2015-16Cinema per-a-estudiants-2015-16
Cinema per-a-estudiants-2015-16
 
Elpais
ElpaisElpais
Elpais
 
Cmap
CmapCmap
Cmap
 
Les muralles de Tarragona
Les muralles de TarragonaLes muralles de Tarragona
Les muralles de Tarragona
 
El tren de la vida 1
El tren de la vida 1El tren de la vida 1
El tren de la vida 1
 
2989340 informatica-dicionario-internet
2989340 informatica-dicionario-internet2989340 informatica-dicionario-internet
2989340 informatica-dicionario-internet
 
10 Tendencias de Consumo para 2010
10 Tendencias de Consumo para 201010 Tendencias de Consumo para 2010
10 Tendencias de Consumo para 2010
 
Trabajo Técnicas de venta:Manual de ventas trias completo
Trabajo Técnicas de venta:Manual de ventas trias completoTrabajo Técnicas de venta:Manual de ventas trias completo
Trabajo Técnicas de venta:Manual de ventas trias completo
 
El sol fuente de energía quimica
El sol fuente  de energía quimicaEl sol fuente  de energía quimica
El sol fuente de energía quimica
 
From Big to Smart Data - Smart Data Innovation Lab Overview
From Big to Smart Data - Smart Data Innovation Lab OverviewFrom Big to Smart Data - Smart Data Innovation Lab Overview
From Big to Smart Data - Smart Data Innovation Lab Overview
 

Similar to Flexible structure control a strategy for releasing input constraints

Modified smith predictor based cascade control of unstable time delay processes
Modified smith predictor based cascade control of unstable time delay processesModified smith predictor based cascade control of unstable time delay processes
Modified smith predictor based cascade control of unstable time delay processesISA Interchange
 
Improving performance using cascade control and a Smith predictor
Improving performance using cascade control and a Smith predictorImproving performance using cascade control and a Smith predictor
Improving performance using cascade control and a Smith predictorISA Interchange
 
Guidelines for the tuning and the evaluation of decentralized and decoupling ...
Guidelines for the tuning and the evaluation of decentralized and decoupling ...Guidelines for the tuning and the evaluation of decentralized and decoupling ...
Guidelines for the tuning and the evaluation of decentralized and decoupling ...ISA Interchange
 
Implementation of closed loop control technique for improving the performance...
Implementation of closed loop control technique for improving the performance...Implementation of closed loop control technique for improving the performance...
Implementation of closed loop control technique for improving the performance...IJERA Editor
 
Output feedback trajectory stabilization of the uncertainty DC servomechanism...
Output feedback trajectory stabilization of the uncertainty DC servomechanism...Output feedback trajectory stabilization of the uncertainty DC servomechanism...
Output feedback trajectory stabilization of the uncertainty DC servomechanism...ISA Interchange
 
Self tuning adaptive control for an industrial weigh belt feeder
Self tuning adaptive control for an industrial weigh belt feederSelf tuning adaptive control for an industrial weigh belt feeder
Self tuning adaptive control for an industrial weigh belt feederISA Interchange
 
A hybrid bacterial foraging and modified particle swarm optimization for mode...
A hybrid bacterial foraging and modified particle swarm optimization for mode...A hybrid bacterial foraging and modified particle swarm optimization for mode...
A hybrid bacterial foraging and modified particle swarm optimization for mode...IJECEIAES
 
Maximizing the return on your control investment meet the experts sessions part2
Maximizing the return on your control investment meet the experts sessions part2Maximizing the return on your control investment meet the experts sessions part2
Maximizing the return on your control investment meet the experts sessions part2Emerson Exchange
 
Process control examples and applications
Process control examples and applications Process control examples and applications
Process control examples and applications Amr Seif
 
Types of Controllers PID PD I PD
Types of Controllers PID PD I PDTypes of Controllers PID PD I PD
Types of Controllers PID PD I PDAnaseem Hanini
 
Commissioning highly interactive process an approach for tuning control loops
Commissioning highly interactive process an approach for tuning control loopsCommissioning highly interactive process an approach for tuning control loops
Commissioning highly interactive process an approach for tuning control loopsEmerson Exchange
 
A New Control Method for the Multi-Area LFC System Based on Port-Hamiltonian ...
A New Control Method for the Multi-Area LFC System Based on Port-Hamiltonian ...A New Control Method for the Multi-Area LFC System Based on Port-Hamiltonian ...
A New Control Method for the Multi-Area LFC System Based on Port-Hamiltonian ...IRJET Journal
 
Simple robust autotuning rules for 2-DoF PI controllers
Simple robust autotuning rules for 2-DoF PI controllersSimple robust autotuning rules for 2-DoF PI controllers
Simple robust autotuning rules for 2-DoF PI controllersISA Interchange
 
4 combined gain scheduling and multimodel control of a reactive distillation ...
4 combined gain scheduling and multimodel control of a reactive distillation ...4 combined gain scheduling and multimodel control of a reactive distillation ...
4 combined gain scheduling and multimodel control of a reactive distillation ...nazir1988
 
Fuzzy controlled mine drainage system based on embedded system
Fuzzy controlled mine drainage system based on embedded systemFuzzy controlled mine drainage system based on embedded system
Fuzzy controlled mine drainage system based on embedded systemIRJET Journal
 
Recurrent fuzzy neural network backstepping control for the prescribed output...
Recurrent fuzzy neural network backstepping control for the prescribed output...Recurrent fuzzy neural network backstepping control for the prescribed output...
Recurrent fuzzy neural network backstepping control for the prescribed output...ISA Interchange
 
Process control 4 chapter
Process control 4 chapterProcess control 4 chapter
Process control 4 chapterSrinivasa Rao
 

Similar to Flexible structure control a strategy for releasing input constraints (20)

Modified smith predictor based cascade control of unstable time delay processes
Modified smith predictor based cascade control of unstable time delay processesModified smith predictor based cascade control of unstable time delay processes
Modified smith predictor based cascade control of unstable time delay processes
 
Improving performance using cascade control and a Smith predictor
Improving performance using cascade control and a Smith predictorImproving performance using cascade control and a Smith predictor
Improving performance using cascade control and a Smith predictor
 
Guidelines for the tuning and the evaluation of decentralized and decoupling ...
Guidelines for the tuning and the evaluation of decentralized and decoupling ...Guidelines for the tuning and the evaluation of decentralized and decoupling ...
Guidelines for the tuning and the evaluation of decentralized and decoupling ...
 
Implementation of closed loop control technique for improving the performance...
Implementation of closed loop control technique for improving the performance...Implementation of closed loop control technique for improving the performance...
Implementation of closed loop control technique for improving the performance...
 
Output feedback trajectory stabilization of the uncertainty DC servomechanism...
Output feedback trajectory stabilization of the uncertainty DC servomechanism...Output feedback trajectory stabilization of the uncertainty DC servomechanism...
Output feedback trajectory stabilization of the uncertainty DC servomechanism...
 
Self tuning adaptive control for an industrial weigh belt feeder
Self tuning adaptive control for an industrial weigh belt feederSelf tuning adaptive control for an industrial weigh belt feeder
Self tuning adaptive control for an industrial weigh belt feeder
 
A hybrid bacterial foraging and modified particle swarm optimization for mode...
A hybrid bacterial foraging and modified particle swarm optimization for mode...A hybrid bacterial foraging and modified particle swarm optimization for mode...
A hybrid bacterial foraging and modified particle swarm optimization for mode...
 
Maximizing the return on your control investment meet the experts sessions part2
Maximizing the return on your control investment meet the experts sessions part2Maximizing the return on your control investment meet the experts sessions part2
Maximizing the return on your control investment meet the experts sessions part2
 
Process control examples and applications
Process control examples and applications Process control examples and applications
Process control examples and applications
 
Types of Controllers PID PD I PD
Types of Controllers PID PD I PDTypes of Controllers PID PD I PD
Types of Controllers PID PD I PD
 
Commissioning highly interactive process an approach for tuning control loops
Commissioning highly interactive process an approach for tuning control loopsCommissioning highly interactive process an approach for tuning control loops
Commissioning highly interactive process an approach for tuning control loops
 
Power system operation and control
Power system operation and controlPower system operation and control
Power system operation and control
 
A New Control Method for the Multi-Area LFC System Based on Port-Hamiltonian ...
A New Control Method for the Multi-Area LFC System Based on Port-Hamiltonian ...A New Control Method for the Multi-Area LFC System Based on Port-Hamiltonian ...
A New Control Method for the Multi-Area LFC System Based on Port-Hamiltonian ...
 
Simple robust autotuning rules for 2-DoF PI controllers
Simple robust autotuning rules for 2-DoF PI controllersSimple robust autotuning rules for 2-DoF PI controllers
Simple robust autotuning rules for 2-DoF PI controllers
 
4 combined gain scheduling and multimodel control of a reactive distillation ...
4 combined gain scheduling and multimodel control of a reactive distillation ...4 combined gain scheduling and multimodel control of a reactive distillation ...
4 combined gain scheduling and multimodel control of a reactive distillation ...
 
Fuzzy controlled mine drainage system based on embedded system
Fuzzy controlled mine drainage system based on embedded systemFuzzy controlled mine drainage system based on embedded system
Fuzzy controlled mine drainage system based on embedded system
 
Recurrent fuzzy neural network backstepping control for the prescribed output...
Recurrent fuzzy neural network backstepping control for the prescribed output...Recurrent fuzzy neural network backstepping control for the prescribed output...
Recurrent fuzzy neural network backstepping control for the prescribed output...
 
chapter_1.ppt
chapter_1.pptchapter_1.ppt
chapter_1.ppt
 
Design and Analysis of Adaptive Sliding Mode with Exponential Reaching Law Co...
Design and Analysis of Adaptive Sliding Mode with Exponential Reaching Law Co...Design and Analysis of Adaptive Sliding Mode with Exponential Reaching Law Co...
Design and Analysis of Adaptive Sliding Mode with Exponential Reaching Law Co...
 
Process control 4 chapter
Process control 4 chapterProcess control 4 chapter
Process control 4 chapter
 

More from ISA Interchange

An optimal general type-2 fuzzy controller for Urban Traffic Network
An optimal general type-2 fuzzy controller for Urban Traffic NetworkAn optimal general type-2 fuzzy controller for Urban Traffic Network
An optimal general type-2 fuzzy controller for Urban Traffic NetworkISA Interchange
 
Embedded intelligent adaptive PI controller for an electromechanical system
Embedded intelligent adaptive PI controller for an electromechanical  systemEmbedded intelligent adaptive PI controller for an electromechanical  system
Embedded intelligent adaptive PI controller for an electromechanical systemISA Interchange
 
State of charge estimation of lithium-ion batteries using fractional order sl...
State of charge estimation of lithium-ion batteries using fractional order sl...State of charge estimation of lithium-ion batteries using fractional order sl...
State of charge estimation of lithium-ion batteries using fractional order sl...ISA Interchange
 
Fractional order PID for tracking control of a parallel robotic manipulator t...
Fractional order PID for tracking control of a parallel robotic manipulator t...Fractional order PID for tracking control of a parallel robotic manipulator t...
Fractional order PID for tracking control of a parallel robotic manipulator t...ISA Interchange
 
Fuzzy logic for plant-wide control of biological wastewater treatment process...
Fuzzy logic for plant-wide control of biological wastewater treatment process...Fuzzy logic for plant-wide control of biological wastewater treatment process...
Fuzzy logic for plant-wide control of biological wastewater treatment process...ISA Interchange
 
Design and implementation of a control structure for quality products in a cr...
Design and implementation of a control structure for quality products in a cr...Design and implementation of a control structure for quality products in a cr...
Design and implementation of a control structure for quality products in a cr...ISA Interchange
 
Model based PI power system stabilizer design for damping low frequency oscil...
Model based PI power system stabilizer design for damping low frequency oscil...Model based PI power system stabilizer design for damping low frequency oscil...
Model based PI power system stabilizer design for damping low frequency oscil...ISA Interchange
 
A comparison of a novel robust decentralized control strategy and MPC for ind...
A comparison of a novel robust decentralized control strategy and MPC for ind...A comparison of a novel robust decentralized control strategy and MPC for ind...
A comparison of a novel robust decentralized control strategy and MPC for ind...ISA Interchange
 
Fault detection of feed water treatment process using PCA-WD with parameter o...
Fault detection of feed water treatment process using PCA-WD with parameter o...Fault detection of feed water treatment process using PCA-WD with parameter o...
Fault detection of feed water treatment process using PCA-WD with parameter o...ISA Interchange
 
Model-based adaptive sliding mode control of the subcritical boiler-turbine s...
Model-based adaptive sliding mode control of the subcritical boiler-turbine s...Model-based adaptive sliding mode control of the subcritical boiler-turbine s...
Model-based adaptive sliding mode control of the subcritical boiler-turbine s...ISA Interchange
 
A Proportional Integral Estimator-Based Clock Synchronization Protocol for Wi...
A Proportional Integral Estimator-Based Clock Synchronization Protocol for Wi...A Proportional Integral Estimator-Based Clock Synchronization Protocol for Wi...
A Proportional Integral Estimator-Based Clock Synchronization Protocol for Wi...ISA Interchange
 
An artificial intelligence based improved classification of two-phase flow patte...
An artificial intelligence based improved classification of two-phase flow patte...An artificial intelligence based improved classification of two-phase flow patte...
An artificial intelligence based improved classification of two-phase flow patte...ISA Interchange
 
New Method for Tuning PID Controllers Using a Symmetric Send-On-Delta Samplin...
New Method for Tuning PID Controllers Using a Symmetric Send-On-Delta Samplin...New Method for Tuning PID Controllers Using a Symmetric Send-On-Delta Samplin...
New Method for Tuning PID Controllers Using a Symmetric Send-On-Delta Samplin...ISA Interchange
 
Load estimator-based hybrid controller design for two-interleaved boost conve...
Load estimator-based hybrid controller design for two-interleaved boost conve...Load estimator-based hybrid controller design for two-interleaved boost conve...
Load estimator-based hybrid controller design for two-interleaved boost conve...ISA Interchange
 
Effects of Wireless Packet Loss in Industrial Process Control Systems
Effects of Wireless Packet Loss in Industrial Process Control SystemsEffects of Wireless Packet Loss in Industrial Process Control Systems
Effects of Wireless Packet Loss in Industrial Process Control SystemsISA Interchange
 
Fault Detection in the Distillation Column Process
Fault Detection in the Distillation Column ProcessFault Detection in the Distillation Column Process
Fault Detection in the Distillation Column ProcessISA Interchange
 
Neural Network-Based Actuator Fault Diagnosis for a Non-Linear Multi-Tank System
Neural Network-Based Actuator Fault Diagnosis for a Non-Linear Multi-Tank SystemNeural Network-Based Actuator Fault Diagnosis for a Non-Linear Multi-Tank System
Neural Network-Based Actuator Fault Diagnosis for a Non-Linear Multi-Tank SystemISA Interchange
 
A KPI-based process monitoring and fault detection framework for large-scale ...
A KPI-based process monitoring and fault detection framework for large-scale ...A KPI-based process monitoring and fault detection framework for large-scale ...
A KPI-based process monitoring and fault detection framework for large-scale ...ISA Interchange
 
An adaptive PID like controller using mix locally recurrent neural network fo...
An adaptive PID like controller using mix locally recurrent neural network fo...An adaptive PID like controller using mix locally recurrent neural network fo...
An adaptive PID like controller using mix locally recurrent neural network fo...ISA Interchange
 
A method to remove chattering alarms using median filters
A method to remove chattering alarms using median filtersA method to remove chattering alarms using median filters
A method to remove chattering alarms using median filtersISA Interchange
 

More from ISA Interchange (20)

An optimal general type-2 fuzzy controller for Urban Traffic Network
An optimal general type-2 fuzzy controller for Urban Traffic NetworkAn optimal general type-2 fuzzy controller for Urban Traffic Network
An optimal general type-2 fuzzy controller for Urban Traffic Network
 
Embedded intelligent adaptive PI controller for an electromechanical system
Embedded intelligent adaptive PI controller for an electromechanical  systemEmbedded intelligent adaptive PI controller for an electromechanical  system
Embedded intelligent adaptive PI controller for an electromechanical system
 
State of charge estimation of lithium-ion batteries using fractional order sl...
State of charge estimation of lithium-ion batteries using fractional order sl...State of charge estimation of lithium-ion batteries using fractional order sl...
State of charge estimation of lithium-ion batteries using fractional order sl...
 
Fractional order PID for tracking control of a parallel robotic manipulator t...
Fractional order PID for tracking control of a parallel robotic manipulator t...Fractional order PID for tracking control of a parallel robotic manipulator t...
Fractional order PID for tracking control of a parallel robotic manipulator t...
 
Fuzzy logic for plant-wide control of biological wastewater treatment process...
Fuzzy logic for plant-wide control of biological wastewater treatment process...Fuzzy logic for plant-wide control of biological wastewater treatment process...
Fuzzy logic for plant-wide control of biological wastewater treatment process...
 
Design and implementation of a control structure for quality products in a cr...
Design and implementation of a control structure for quality products in a cr...Design and implementation of a control structure for quality products in a cr...
Design and implementation of a control structure for quality products in a cr...
 
Model based PI power system stabilizer design for damping low frequency oscil...
Model based PI power system stabilizer design for damping low frequency oscil...Model based PI power system stabilizer design for damping low frequency oscil...
Model based PI power system stabilizer design for damping low frequency oscil...
 
A comparison of a novel robust decentralized control strategy and MPC for ind...
A comparison of a novel robust decentralized control strategy and MPC for ind...A comparison of a novel robust decentralized control strategy and MPC for ind...
A comparison of a novel robust decentralized control strategy and MPC for ind...
 
Fault detection of feed water treatment process using PCA-WD with parameter o...
Fault detection of feed water treatment process using PCA-WD with parameter o...Fault detection of feed water treatment process using PCA-WD with parameter o...
Fault detection of feed water treatment process using PCA-WD with parameter o...
 
Model-based adaptive sliding mode control of the subcritical boiler-turbine s...
Model-based adaptive sliding mode control of the subcritical boiler-turbine s...Model-based adaptive sliding mode control of the subcritical boiler-turbine s...
Model-based adaptive sliding mode control of the subcritical boiler-turbine s...
 
A Proportional Integral Estimator-Based Clock Synchronization Protocol for Wi...
A Proportional Integral Estimator-Based Clock Synchronization Protocol for Wi...A Proportional Integral Estimator-Based Clock Synchronization Protocol for Wi...
A Proportional Integral Estimator-Based Clock Synchronization Protocol for Wi...
 
An artificial intelligence based improved classification of two-phase flow patte...
An artificial intelligence based improved classification of two-phase flow patte...An artificial intelligence based improved classification of two-phase flow patte...
An artificial intelligence based improved classification of two-phase flow patte...
 
New Method for Tuning PID Controllers Using a Symmetric Send-On-Delta Samplin...
New Method for Tuning PID Controllers Using a Symmetric Send-On-Delta Samplin...New Method for Tuning PID Controllers Using a Symmetric Send-On-Delta Samplin...
New Method for Tuning PID Controllers Using a Symmetric Send-On-Delta Samplin...
 
Load estimator-based hybrid controller design for two-interleaved boost conve...
Load estimator-based hybrid controller design for two-interleaved boost conve...Load estimator-based hybrid controller design for two-interleaved boost conve...
Load estimator-based hybrid controller design for two-interleaved boost conve...
 
Effects of Wireless Packet Loss in Industrial Process Control Systems
Effects of Wireless Packet Loss in Industrial Process Control SystemsEffects of Wireless Packet Loss in Industrial Process Control Systems
Effects of Wireless Packet Loss in Industrial Process Control Systems
 
Fault Detection in the Distillation Column Process
Fault Detection in the Distillation Column ProcessFault Detection in the Distillation Column Process
Fault Detection in the Distillation Column Process
 
Neural Network-Based Actuator Fault Diagnosis for a Non-Linear Multi-Tank System
Neural Network-Based Actuator Fault Diagnosis for a Non-Linear Multi-Tank SystemNeural Network-Based Actuator Fault Diagnosis for a Non-Linear Multi-Tank System
Neural Network-Based Actuator Fault Diagnosis for a Non-Linear Multi-Tank System
 
A KPI-based process monitoring and fault detection framework for large-scale ...
A KPI-based process monitoring and fault detection framework for large-scale ...A KPI-based process monitoring and fault detection framework for large-scale ...
A KPI-based process monitoring and fault detection framework for large-scale ...
 
An adaptive PID like controller using mix locally recurrent neural network fo...
An adaptive PID like controller using mix locally recurrent neural network fo...An adaptive PID like controller using mix locally recurrent neural network fo...
An adaptive PID like controller using mix locally recurrent neural network fo...
 
A method to remove chattering alarms using median filters
A method to remove chattering alarms using median filtersA method to remove chattering alarms using median filters
A method to remove chattering alarms using median filters
 

Recently uploaded

EPC Contractors aspects Presentation.pdf
EPC Contractors  aspects Presentation.pdfEPC Contractors  aspects Presentation.pdf
EPC Contractors aspects Presentation.pdfGiuseppe Tommasone
 
The Smart Bridge Interview now Veranda Learning
The Smart Bridge Interview now Veranda LearningThe Smart Bridge Interview now Veranda Learning
The Smart Bridge Interview now Veranda LearningNaval Singh
 
14 march 2024-capital-markets-update eni.pdf
14 march 2024-capital-markets-update eni.pdf14 march 2024-capital-markets-update eni.pdf
14 march 2024-capital-markets-update eni.pdfEni
 
L-1 VISA Business (Plan Sample) - Plan Writers
L-1 VISA Business (Plan Sample) - Plan WritersL-1 VISA Business (Plan Sample) - Plan Writers
L-1 VISA Business (Plan Sample) - Plan WritersPlan Writers
 
Shravan Kumaran and sanjay kumaran.pdf..
Shravan Kumaran and sanjay kumaran.pdf..Shravan Kumaran and sanjay kumaran.pdf..
Shravan Kumaran and sanjay kumaran.pdf..ranjithapriya2
 
The 10 Most Influential Women Making Difference In 2024.pdf
The 10 Most Influential Women Making Difference In 2024.pdfThe 10 Most Influential Women Making Difference In 2024.pdf
The 10 Most Influential Women Making Difference In 2024.pdfInsightsSuccess4
 
Pitch Deck Teardown: SuperScale's $5.4M Series A deck
Pitch Deck Teardown: SuperScale's $5.4M Series A deckPitch Deck Teardown: SuperScale's $5.4M Series A deck
Pitch Deck Teardown: SuperScale's $5.4M Series A deckHajeJanKamps
 
Meet Raj Shamani: A Trailblazing Entrepreneur
Meet Raj Shamani: A Trailblazing EntrepreneurMeet Raj Shamani: A Trailblazing Entrepreneur
Meet Raj Shamani: A Trailblazing Entrepreneurramya202104
 
Mist Cooling & Fogging System Company in Saudi Arabia
Mist Cooling & Fogging System Company in Saudi ArabiaMist Cooling & Fogging System Company in Saudi Arabia
Mist Cooling & Fogging System Company in Saudi Arabiaopstechsanjanasingh
 
Presented by Sabri international .......
Presented by Sabri international .......Presented by Sabri international .......
Presented by Sabri international .......SABRI INTERNATIONAL
 
Bus Eth ch3 ppt.ppt business ethics and corporate social responsibilities ppt
Bus Eth ch3 ppt.ppt business ethics and corporate social responsibilities pptBus Eth ch3 ppt.ppt business ethics and corporate social responsibilities ppt
Bus Eth ch3 ppt.ppt business ethics and corporate social responsibilities pptendeworku
 
Shopclues: Failure & Solutions in Business Model
Shopclues: Failure & Solutions in Business ModelShopclues: Failure & Solutions in Business Model
Shopclues: Failure & Solutions in Business ModelBhaviniSharma12
 
Business Models and Business Model Innovation
Business Models and Business Model InnovationBusiness Models and Business Model Innovation
Business Models and Business Model InnovationMichal Hron
 
Record of Module Forensic photography in
Record of Module Forensic photography inRecord of Module Forensic photography in
Record of Module Forensic photography inalexademileighpacal
 
How The Hustle Milestone Referral Program Got 300K Subscribers
How The Hustle Milestone Referral Program Got 300K SubscribersHow The Hustle Milestone Referral Program Got 300K Subscribers
How The Hustle Milestone Referral Program Got 300K SubscribersFlyyx Tech
 
Dashboards y paneles - CP Home - Area de Operaciones
Dashboards y paneles - CP Home - Area de OperacionesDashboards y paneles - CP Home - Area de Operaciones
Dashboards y paneles - CP Home - Area de OperacionesLPI ONG
 
Unleashing the Power of Fandom: A Short Guide to Fan Business
Unleashing the Power of Fandom: A Short Guide to Fan BusinessUnleashing the Power of Fandom: A Short Guide to Fan Business
Unleashing the Power of Fandom: A Short Guide to Fan Businesstompeter3736
 
NewBase 14 March 2024 Energy News issue - 1707 by Khaled Al Awadi_compress...
NewBase  14 March  2024  Energy News issue - 1707 by Khaled Al Awadi_compress...NewBase  14 March  2024  Energy News issue - 1707 by Khaled Al Awadi_compress...
NewBase 14 March 2024 Energy News issue - 1707 by Khaled Al Awadi_compress...Khaled Al Awadi
 
CXO 2.0 Conference (Event Information Deck | Dec'24-Mar'25)
CXO 2.0 Conference (Event Information Deck | Dec'24-Mar'25)CXO 2.0 Conference (Event Information Deck | Dec'24-Mar'25)
CXO 2.0 Conference (Event Information Deck | Dec'24-Mar'25)CXO 2.0 Conference
 
Reframing Requirements: A Strategic Approach to Requirement Definition, with ...
Reframing Requirements: A Strategic Approach to Requirement Definition, with ...Reframing Requirements: A Strategic Approach to Requirement Definition, with ...
Reframing Requirements: A Strategic Approach to Requirement Definition, with ...Jake Truemper
 

Recently uploaded (20)

EPC Contractors aspects Presentation.pdf
EPC Contractors  aspects Presentation.pdfEPC Contractors  aspects Presentation.pdf
EPC Contractors aspects Presentation.pdf
 
The Smart Bridge Interview now Veranda Learning
The Smart Bridge Interview now Veranda LearningThe Smart Bridge Interview now Veranda Learning
The Smart Bridge Interview now Veranda Learning
 
14 march 2024-capital-markets-update eni.pdf
14 march 2024-capital-markets-update eni.pdf14 march 2024-capital-markets-update eni.pdf
14 march 2024-capital-markets-update eni.pdf
 
L-1 VISA Business (Plan Sample) - Plan Writers
L-1 VISA Business (Plan Sample) - Plan WritersL-1 VISA Business (Plan Sample) - Plan Writers
L-1 VISA Business (Plan Sample) - Plan Writers
 
Shravan Kumaran and sanjay kumaran.pdf..
Shravan Kumaran and sanjay kumaran.pdf..Shravan Kumaran and sanjay kumaran.pdf..
Shravan Kumaran and sanjay kumaran.pdf..
 
The 10 Most Influential Women Making Difference In 2024.pdf
The 10 Most Influential Women Making Difference In 2024.pdfThe 10 Most Influential Women Making Difference In 2024.pdf
The 10 Most Influential Women Making Difference In 2024.pdf
 
Pitch Deck Teardown: SuperScale's $5.4M Series A deck
Pitch Deck Teardown: SuperScale's $5.4M Series A deckPitch Deck Teardown: SuperScale's $5.4M Series A deck
Pitch Deck Teardown: SuperScale's $5.4M Series A deck
 
Meet Raj Shamani: A Trailblazing Entrepreneur
Meet Raj Shamani: A Trailblazing EntrepreneurMeet Raj Shamani: A Trailblazing Entrepreneur
Meet Raj Shamani: A Trailblazing Entrepreneur
 
Mist Cooling & Fogging System Company in Saudi Arabia
Mist Cooling & Fogging System Company in Saudi ArabiaMist Cooling & Fogging System Company in Saudi Arabia
Mist Cooling & Fogging System Company in Saudi Arabia
 
Presented by Sabri international .......
Presented by Sabri international .......Presented by Sabri international .......
Presented by Sabri international .......
 
Bus Eth ch3 ppt.ppt business ethics and corporate social responsibilities ppt
Bus Eth ch3 ppt.ppt business ethics and corporate social responsibilities pptBus Eth ch3 ppt.ppt business ethics and corporate social responsibilities ppt
Bus Eth ch3 ppt.ppt business ethics and corporate social responsibilities ppt
 
Shopclues: Failure & Solutions in Business Model
Shopclues: Failure & Solutions in Business ModelShopclues: Failure & Solutions in Business Model
Shopclues: Failure & Solutions in Business Model
 
Business Models and Business Model Innovation
Business Models and Business Model InnovationBusiness Models and Business Model Innovation
Business Models and Business Model Innovation
 
Record of Module Forensic photography in
Record of Module Forensic photography inRecord of Module Forensic photography in
Record of Module Forensic photography in
 
How The Hustle Milestone Referral Program Got 300K Subscribers
How The Hustle Milestone Referral Program Got 300K SubscribersHow The Hustle Milestone Referral Program Got 300K Subscribers
How The Hustle Milestone Referral Program Got 300K Subscribers
 
Dashboards y paneles - CP Home - Area de Operaciones
Dashboards y paneles - CP Home - Area de OperacionesDashboards y paneles - CP Home - Area de Operaciones
Dashboards y paneles - CP Home - Area de Operaciones
 
Unleashing the Power of Fandom: A Short Guide to Fan Business
Unleashing the Power of Fandom: A Short Guide to Fan BusinessUnleashing the Power of Fandom: A Short Guide to Fan Business
Unleashing the Power of Fandom: A Short Guide to Fan Business
 
NewBase 14 March 2024 Energy News issue - 1707 by Khaled Al Awadi_compress...
NewBase  14 March  2024  Energy News issue - 1707 by Khaled Al Awadi_compress...NewBase  14 March  2024  Energy News issue - 1707 by Khaled Al Awadi_compress...
NewBase 14 March 2024 Energy News issue - 1707 by Khaled Al Awadi_compress...
 
CXO 2.0 Conference (Event Information Deck | Dec'24-Mar'25)
CXO 2.0 Conference (Event Information Deck | Dec'24-Mar'25)CXO 2.0 Conference (Event Information Deck | Dec'24-Mar'25)
CXO 2.0 Conference (Event Information Deck | Dec'24-Mar'25)
 
Reframing Requirements: A Strategic Approach to Requirement Definition, with ...
Reframing Requirements: A Strategic Approach to Requirement Definition, with ...Reframing Requirements: A Strategic Approach to Requirement Definition, with ...
Reframing Requirements: A Strategic Approach to Requirement Definition, with ...
 

Flexible structure control a strategy for releasing input constraints

  • 1. ISA TRANSACTIONS® ISA Transactions 43 ͑2004͒ 361–376 Flexible-structure control: A strategy for releasing input constraints Leonardo L. Giovanini* Industrial Control Centre, University of Strathclyde, Graham Hills Building, 50 George Street, Glasgow G1 1QE, United Kingdom ͑Received 26 July 2002; accepted 5 September 2003͒ Abstract In this paper output unreachability under input saturation phenomenon is studied: under a large disturbance or setpoint change, the process output may never reach the set point even when the manipulated variable has driven to saturation. The process output can be brought back to the set point only by activating an auxiliary manipulated variable. A new control structure for designing and implementing a control system capable of solving this problem is proposed by transferring the control from one variable to another and taking into account the different dynamics involved in the system. The control structure, called flexible-structure control due to its ability to adapt the control structure to the operating conditons, is a generalization of the split-range control. It can be summarized as two controllers connected through a piecewise linear function. This function decides, based on the value of one manipulated variable, when and how the control structure changes. Its parameters control the interaction between both manipulated variables and leave the capability for handling the balance between control quality and other goals to the operator. © 2004 ISA—The Instrumentation, Systems, and Automation Society. Keywords: Multi-input systems; PID control; cooperative control; Manipulated variable constraint 1. Introduction Antiwindup methods may work well under the nominal conditions under which the controller was Quite frequently process control engineers face designed. However, it is possible that under a large problems in which hard constraints restrict the ma- set-point change, load disturbance, or component nipulated variables to a finite operating range. The failure, the manipulated variable will reach its constraints may also come from process con- limit while the system output still cannot reach its straints intended to avoid damage to the system or set point at the steady state. This phenomenon is the material being proceeded ͓1͔. There are many known as output unreachability under input con- control techniques to deal with this problem. They straint ͓9͔. It is directly associated with the size of can be divided into two categories: antiwindup the operability space of the system. Sometimes compensation reduces the adverse effects of the this problem is solved by modifying the control constraints on the closed-loop performance ͓1– 4͔, structure of the system, but many times the pro- and cooperative control schemes that use a com- cess itself requires substantial changes. On the bination of inputs to achieve the reference ͓5–9͔. other hand, cooperative control schemes solve the output-unreachability problem by activating auxil- iary manipulated variables, and many times they *Tel: ϩ44 141 548 2666; fax: 44 141 552 2487. E-mail introduce a degree of optimality in the solution address: leonardo.giovanini@eee.strath.ac.uk ͓8͔. Nevertheless, manipulated constraints are 0019-0578/2004/$ - see front matter © 2004 ISA—The Instrumentation, Systems, and Automation Society.
  • 2. 362 Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376 never completely eliminated because these tech- niques only solve the output-unreachability prob- lem for the steady state. Therefore during the tran- sient the manipulated variables could temporarily reach their limits and the system becomes uncon- trollable during this period. Hence the need for finding a way for broadening controlled operation spaces to provide all the available process flexibil- ity while preserving good performances has stimu- Fig. 1. Basic process structure. lated the search for new control structures. This paper proposed a new control structure called flexible-structure control, due to its ability to adapt the control structure to the operating con- namic elements Gp 1 ( s ) and Gp 2 ( s ) . The second ditons. It is a generalization of the split-range con- feature is that assumed to be u 1 is the primary trol ͓6͔ and it can be summarized as two control- manipulated variable because Gp 1 ( s ) is faster lers connected through a piecewise linear function. and with smaller time delay than Gp 2 ( s ) . A hard This function decides, based on the value of one constraint might become active at some given ex- manipulated variable, when and how the control treme values of this variable. If eventually u 1 satu- structure changes. Its parameters control the inter- rates, u 2 may be used as an auxiliary manipulated action between both manipulated variables. They variable to keep the system under regulation. The can be selected attending not only to the process process may be also subject to many disturbances. structure but also following some optimization cri- For linear systems, they can be collectively repre- teria. sented by one disturbance d entering the process The paper is organized as follow: in Section 2, at the output. the requirements that must satisfy the process to In normal operation, the system is designed so apply the proposed control strategy and some ex- that for any moderate set-point or load disturbance amples are presented. In Section 3 the flexible- change, the primary manipulated variable u 1 can structure control is proposed. The structure can be regulate the process output to achieve a zero out- summarized as two controllers connected with a put steady-state error working within its working piecewise-linear function. The parameters of the range ͓ u 1 min ,u1 max͔, while u 2 is kept unchanged. nonlinear function control the interaction between However, it is possible that under a large set point, both inputs and the steady-state values of each ma- load disturbance change, or component failure, nipulated variable. In Section 4 two tuning proce- there is no u 1 ෈ ͓ u 1 min ,u1 max͔ so that y ( ϱ ) dures for the controller are proposed. The first ϭr ( ϱ ) if u 2 is unchanged, leading to output un- method is based on the PID controller and it is reachability. obtained from an IMC parametrization. The sec- This problem is frequently found in practice, so ond procedure employes more sophisticated con- it deserves special research. For example, consider trollers. Section 5 addresses the closed-loop stabil- the problem of controlling a continuous stirred ity analysis. In Section 6 some guidelines for the tank reactor where an irreversible exothermic re- selection of the decision function are presented. action is carried out at a constant volume em- Finally, Section 7 presents results obtained from ployed by Moningred et al. ͓10͔ to test nonlinear the application of the proposed algorithm to a lin- control algorithms. The reactor is cooled by water ear system. Conclusions are presented in Section through a surrounding jacket. The concentration 8. can be controlled either by manipulating the flow rates of the coolant or the process stream. From the process perspective, the use of cooling water is 2. Process conditions preferred over the process stream. However, the dynamic response of the reactor concentration to a Figure 1 shows a sketch of the process structure change in coolant flow rate shows a nonlinear be- considered in this work. The first special feature to havior. In practice, the nominal flowrates of the be noted is that the output variable y may be con- process stream is determined in advance, and wa- trolled by either u 1 or u 2 through different dy- ter is employed as the primary manipulated vari-
  • 3. Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376 363 able, while the process stream acts as the auxiliary 3. Flexible-structure control manipulated variable and is normally kept con- stant. The examples of output unreachability provided Another example is the control of the tempera- in the previous section are only few of many in- ture in the reactor ͓11͔. The reactor is cooled by dustrial cases. Such cases require changes in the both cooling water flowing through a surrounding auxiliary manipulated variable. To perform these jacket and condensing vapor that boils off the re- changes several control strategies have been de- actor in a heat exchanger cooled by a refrigerant. veloped in the past decades. Shinkey ͓6͔ proposed From an energy-saving point of view, the use of the valve-position control. This technique uses a cooling water is preferred over the refrigerant due maximum signal selector to change u 2 to keep all to cost. However, the dynamic response of the the actuators from exceeding a preset limit. The plant’s temperature to a change in refrigerant flow algorithm has to wait for the process variable to is faster than a change in cooling water. In prac- reach its steady state, then regulate u 2 iteratively tice, the nominal flow rates of both are determined until the steady-state error becomes zero. The in advance. Water is utilized as the primary ma- main drawbacks of the method are that it is time nipulated variable, while the refrigerant acts as an consuming and the results largely depend on the auxiliary manipulated variable and it is normally engineer experience. kept constant. If there is a disturbance or setpoint Another alternative control structure is the coop- change such that the temperature cannot return to erative control, proposed by Wang et al. ͓9͔. Simi- the set point even when the cooling water valve larly to valve-position control, cooperative control hits its limitation, the flow of refrigerant has to be activates u 2 using constant levels which are com- adjusted to make the temperature reach the desired puted base on a disturbance estimation and output steady-state value. steady-state prediction. The use of these features Another example can be found in the tempera- improves the closed-loop performance by reduc- ture control of thermical integrated chemical pro- ing the settling time and simplifying the imple- cess ͓12͔. A simple configuration of this type of mentation of this control scheme. Both techniques solve the output unreachability system is given by two heat exchangers in series: problem only for the steady-state of the multiple- one heat exchanger and a service equipment. This input single-output ͑MISO͒ system with a control arrangement is very common in practice when be- structure that remains unchanged. Therefore the sides the task of reaching a final temperature target transient behavior of the closed-loop system will on a process stream there is an extra goal like be poor because the control laws generated by maximum energy recovery. The heat exchanger is both control strategies do not take account of the specifically designed for recovering the exceeding dynamic part of Gp 2 . Many times the closed-loop energy in the process stream, and the service response is driven in an open-loop manner during equipment completes the thermic conditioning the transient. This situation is significantly impor- through a utility stream ͓13͔. The heat exchanger tant when a fault, like a frozen valve, occurs or operates at a constant flowrate that maximizes the during the saturation of the manipulated variables. energy recovering on nominal conditions and the Hence the above control problem requires an ap- service is designed to cope with the long term propriate design that would be able to transfer the variations on the inlet stream conditions or having control from one variable to another, takes account the capability of changing stream temperature tar- of the different dynamics involved in the system gets. In the case of a disturbance or set-point and, if it is possible, leaves the capability for han- change, the steady state may deviate. There could dling the balance between control quality and exist a situation in which the effect is so large that other goals to the operator. Hence, under this the temperature cannot return to the set point even framework, solving a manipulated constraint prob- when the service hits its limits. In such a case, the lem requires a process engineering approach ca- auxiliary manipulated variable, i.e., the flow rate pable of combining control strategy and process of the heat exchanger, has to be adjusted to make efficiency. the temperature’s steady state reach the desired Note that if there is a controller C 1 ( s ) handling value. u 1 ( s ) to control y ( s ) at a given set-point value
  • 4. 364 Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376 Fig. 2. Block diagram of ͑a͒ preventive protection, and ͑b͒ reactive protection. r 0 , the stationary value expected for the controlled able, a preventive protection for disturbance can variable is Gp 1 ( 0 ) u 1 ( 0 ) , and if an integral mode also be included by computing x as follows: is present, the manipulated variable goes to u 1 ( 0 ) ϭGp Ϫ1 ( 0 ) r 0 . Let us assume now that just 1 x ͑ s ͒ ϭ ␩ Gp Ϫ1 ͑ 0 ͒ r ͑ s ͒ ϩ ␩ Gp Ϫ1 ͑ 0 ͒ G d ͑ 0 ͒ w ͑ s ͒ . 1 1 a fraction of this output y can be handled through the manipulated u 1 ͑this implies that a control For the second case, Fig. 2͑b͒ shows that the re- constraint is active at a given level͒, and that an active protection introduces a new feedback loop additional capacity can be provided through that results from combining both controllers, u 2 ( s ) . Then, there are two ways of defining the C 1 ( s ) and C 2 ( s ) , in a single controller, C ( s ) protection for regulation and operability: ϭC 1 ( s ) C 2 ( s ) . As long as the primary manipu- 1. Feedforward or preventive protection: given a lated variable is not saturated, the output is con- total output steady-state requirement trolled by C 1 and the secondary control loop is no Gp 1 ( 0 ) u 1 ( 0 ) , a fraction ␩ Gp 1 ( 0 ) u 1 ( 0 ) , ␩ у0, operative because the auxiliar signal x ( t ) is zero. is permanently provided by the process part Gp 2 , When u 1 ( t ) Ͼu 1 max , the original control loop re- in order to keep controlled operability. mains open and consequently not operative as 2. Feedback or reactive protection: given the in- long as the primary manipulated is saturated. Thus stant manipulated variable u 1 ( t ) at an operating the process part represented by Gp 2 ( s ) is now in point such that ͉ u 1 ( t ) ͉ Ͼu 1 max , the process part charge of the regulation. As soon as the structure Gp 2 ( s ) provides the complementary output of the system changes, due to the saturation of u 1 , Gp 1 ( 0 ) ͓ u 1 ( t ) Ϫu 1 max͔ to reach the target r 0 . the structure of the controller is modified from These two types of protections are schematized C 1 ( s ) to C ( s ) following the change of the system in Figs. 2͑a͒ and 2͑b͒ respectively, where a second and adapting the structure and parameters of the controller C 2 ( s ) is included for better dynamic ad- controller to the dynamic of the system. The con- justment of the secondary manipulated variable troller C ( s ) must include an antiwindup scheme to u 2 . It is clear from the block diagrams that in the mitigate the effect of the constraint on u 2 , the case of preventive action, C 2 ( s ) is a feedforward effect of constraint on u 1 is compensated by the controller, which does not affect the closed-loop control structure. stability, but it cannot protect the system from the The switching element is implemented through effect of nonmeasurable disturbances, uncertain- a simple nonlinear decision function such that the ties, and/or faults. If the disturbance w ( s ) , or an signal x ( t ) entering the controller C 2 ( s ) is given estimation w ( s ) , and a model of G d ( s ) are avail- ˆ by
  • 5. Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376 365 Fig. 3. Plot of the nonlinear switching function for some parameters. f ͑ u1͒ϭ ͭ u 1 ͑ t ͒ Ϫu 1 max 0 u 1 ͑ t ͒ Ͼu 1 max , u 1 ͑ t ͒ рu 1 max . ͑1͒ modifying the switching function ͑1͒ through the inclusion of the linear term Observe that a similar protection can be developed ␩ u 1͑ t ͒ , ␩ у0, for a lower constraint but it must actuate on a dif- for the active range of u 1 . This term defines a ferent process part, let us say Gp 3 ( s ) . This means permanent and increasing level of protection, that a third manipulated variable u 3 must be avail- when the control variable u 1 ( t ) approaches the able and become active. In this case the switching constraints. For the upper limit case, function ͑1͒, function may be written ͭ the switching function is given by f ͑ u1͒ϭ ͭ 0 u 1 ͑ t ͒ уu 1 min , u 1 ͑ t ͒ Ϫu 1 min u 1 ͑ t ͒ Ͻu 1 min . ͑2͒ u 1 ͑ t ͒ Ϫ ͑ 1Ϫ ␩ ͒ u 1 max, u 1 ͑ t ͒ Ͼu 1 max , f ͑ u1͒ϭ The switching functions ͑1͒ and ͑2͒ can be easily ␩ u 1͑ t ͒ , u 1 minрu 1 ͑ t ͒ рu 1 max , implemented using a dead zone of width u 1 max 0, u 1 ͑ t ͒ рu 1 min . Ϫu1 min with a selector connected in series, such that it generates the proper signal for each auxil- ͑3͒ iary loop. If ␩ ϭ0 the split-range control ͓6͔ is recovered, In spite that a nonlinearity is introduced in the the decision function ͑3͒ becomes Eq. ͑1͒, and the closed loop to transfer the control from Gp 1 to auxiliary variable u 2 is used just to cover output Gp 2 , the stability properties of the system remain demands when u 1 saturates only. When ␩ Ͼ0 the unaffected because, as was explained previously, auxiliary variable u 2 is used to prevent the satura- there is no interaction between both control loops. tion of u 1 , by providing the fraction ␩ r of y ( t ) When one manipulated variable is active, said that while u 1 Ͻu 1 max . Both manipulated variables are the output is controlled with u 1 , the other is inac- simultaneously acting on the system, but with dif- tive and vice versa. ferent gains: Kp 1 for u 1 and ␩ Kp 2 for u 2 , where Functions ͑1͒ and ͑2͒ represent the feedback Kp i iϭ1,2 is the gain of Gp i ( s ) , respectively. protection exclusively, however, a single structure This fact leads to interactions between both con- can be developed to also include the feedforward trol loops that can upset each other, resulting in a protection into the control loop. It is achieved by deterioration of the closed-loop performance com-
  • 6. 366 Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376 pared with the previous situation.1 However, it This fact means when a major fault in the actuator might be a desirable feature since it tends to keep of u 1 happens, like a actuator frozen at a given a the fastest loop working in a wider range. When value, the flexible structure is able to transfer the u 1 ( t ) уu 1 max the control system changes the control of the system output to u 2 without any structure and only Gp 2 controls the system output. extra information than a measurement of the sys- Therefore the closed-loop performance will dete- tem output y. In the case of a minor fault, like a riorate a bit more. The gain of the system jumps loss of a actuator sensibility, it can lead to an out- from ␩ Kp 2 to Kp 2 , producing a peak in the con- put unreachability problem that is overcome as has trol signal u 2 if ideal PID’s are employed. In the been explained in the previous paragraph. case of ␩ ϭu 2 max /u1 max both controllers work si- multaneously, the fastest control loop is active in 3.1. Smith predictor for flexible-structure control the whole operational range maintaining a good control quality in the entire operational space. Time delay is a common feature in most of the A second parameter can be included in the process models. Control of systems with dominant above formulation to start the preventive protec- time delay are notoriously difficult. It is also a tion from a given value u 1 ( t ) ϭ ␬ in ahead, instead topic on which there are many different opinions of doing it from u 1 min as indicated by Eq. ͑3͒, i.e., concerning PID control. For open-loop stable pro- ͭ cesses, the response to command signals can be u 1 ͑ t ͒ Ϫu 1 maxϩ ␩ ͑ u 1 maxϪ ␬ ͒ , ¯ improved substantially by introducing dead time u 1 ͑ t ͒ Ͼu 1 max , compensation ͓15͔. The dead time compensator, f ͑ u1͒ϭ or Smith predictor, is built by implementing a lo- ␩ ͑ u 1 ͑ t ͒ Ϫ ␬ ͒ , ␬ рu 1 ͑ t ͒ рu 1 max , cal loop around the controller with the difference 0, u 1 ͑ t ͒ Ͻ ␬ . between the model of the process without and with ͑4͒ time delay ͓see Fig. 4͑a͔͒. Now, the structure of the Smith predictor is Figure 3 shows a sketch of this function for differ- modified to consider the proposed control struc- ent parameter values. Note that a similar function ture. In this case a second loop is introduced to can be used for saturations at the lower constraint. represent the effect of u 2 over the system output. Both parameters of the decision function, ␩ and ␬, This fact leads to a new structure that include the can be used as tuning parameters to satisfy addi- ˜ ˜ models of both process ( G p 1 and G p 2 ) and the tional process goals than control ones like process constraints in the manipulated variables ͓see Fig. efficiency. 4͑b͔͒. The nonlinearities are required to follow the Remark 1. Regarding the work of the actuators, changes in the structure of the system. This struc- it is interesting to observe that for ␩ ϭ0 the con- ture works for time delays with different values as trol action is executed over a divided range ͓6,14͔, we can see in the example. that is, the second actuator starts moving once the One can notice that a Smith predictor can be first one saturates. For positive values of ␩, the coupled with a robust controller ( H ϱ , QFT, robust control variable is executed over a common range, pole placement, etc.͒ to cope with parametric that is, both actuators work together until one of variations. them saturates. The amplitude of the common range is handled through the parameter ␬ while the intensity is given by ␩. 4. Controller design and tuning The capability of transferring the control from one input to another provides an implicit fault- For designing and tuning the controllers in- tolerant capabilities to flexible-structure control. volved in the flexible structure it is necessary to analyze each control condition separately: ͑i͒ the 1 first control condition—or control structure—is If Gp 1 is much faster than Gp 2 , C 1 will be able to compensate the effect of the interactions and the closed- when C 1 ( s ) is in charge of regulation of y ( t ) , i.e., loop performance will not be affected. However, if Gp 1 is ␩ ϭ0 and u 1 ( t ) is not saturated. ͑ii͒ The second not fast enough, the closed-loop response will show an control structure is defined by the secondary loop overshoot and an increment of the closed-loop settling time only, that is, ␩ ϭ0 and u 1 ( t ) is saturated; the con- due to the interaction between both control loops. troller in this case is the combination C ( s )
  • 7. Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376 367 Fig. 4. Block diagram of Smith predictor ͑a͒ for normal system, and ͑b͒ for cooperative control. ϭC1(s)C2(s). ͑iii͒ The third control condition ap- the IMC strategy provides a rapid parametrization pears when including preventive protection, i.e., of traditional PI or PID controllers ͓16,17͔. If two ␩ Ͼ0 while u 1 ( t ) is not yet saturated. PID algorithms with time delay compensation are The above decomposition of the problem indi- proposed for controllers C 1 ( s ) and C 2 ( s ) , recall cates that C 1 ( s ) must be adjusted for high quality that they work in series, i.e., the outlet of C 1 ( s ) is control when the process part Gp 1 ( s ) handles the the input to C 2 ( s ) through the switching function regulation, i.e., ᭙t:u 1 minрu1(t)рu1 max , and this is f ( u 1 ) . The following few hypotheses and practi- essentially the traditional tuning problem for a cal reasons allow the selection of some important single feedback loop. When u 1 ( t ) Ͼu 1 max , the terms and the elimination of others: process part Gp 2 ( s ) must provide the comple- 1. The double integration term is not necessary mentary effect on the controlled variable, which since offset elimination is required for set-point means that C 2 ( s ) must be combined with C 1 ( s ) changes only. such to obtain the best possible performance. 2. A single integral mode is necessary just in In the following subsections two approaches for C 1 ( s ) , because offset elimination is desired under designing and tuning controllers C 1 ( s ) and C 2 ( s ) any working condition and this controller is the are presented. One is based on IMC parametriza- one which is always active. tion of the controllers, the other is based on can- 3. The control system structure assumes that celation design criteria. In the IMC design the or- Gp 1 ( s ) is faster and with smaller time delay than der of the process will be constrained to one and Gp 2 ( s ) ; this could be a main argument for select- two, leading to PI and PID controllers. In the can- ing u 1 ( s ) as primary manipulated variable, but it celation design, the constraint in the order of the also suggests that if a derivative term is desired, systems is removed and the controllers can be de- this should be in C 2 ( s ) , i.e., the slower plant dy- signed by any controller design techniques. namics. These arguments support the selection, for in- 4.1. IMC design stance, of C 1 ( s ) as a PI controller, For simplicity, let us assume stable plants, such that first- or second-order plus time delay models are adequate for describing the dynamics. Then, ͩ C 1 ͑ s ͒ ϭK C 1 1ϩ 1 T I1s ͪ , ͑5͒
  • 8. 368 Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376 and C 2 ( s ) as a PD controller, Kp 2 , ␶ p2 , and T d 2 , to adjust the dynamic model Gp 2 ( s ) to the correspondent physical data. ͑ 1ϩT D 2 s ͒ 4. Follow the IMC parametrization procedure C 2 ͑ s ͒ ϭK C 2 . ͑6͒ ͑ 1ϩT F s ͒ for Gp 2 ( s ) , which gives a PID controller based on a Smith predictor structure—the combined Hence the combination C ( s ) ϭC 1 ( s ) C 2 ( s ) results controller—whose parameters K C , T I , and T D are to be a PID controller, calculated by the following relationships: ͩ C ͑ s ͒ ϭK C 1ϩ 1 T Is ϩT D s 1 ͪ ͑ 1ϩT F s ͒ , ͑7͒ K Cϭ ␶ p1 ϩ ␶ p2 Kp 2 ␭ 2 , ͑11a͒ whose parameters are T I ϭ ␶ p1 ϩ ␶ p2 , ͑11b͒ K C ϭK C 1 K C 2 1ϩͩ T D2 T I1 ͪ , ͑8a͒ T Dϭ ␶ p1 ␶ p2 ␶ p1 ϩ ␶ p2 . ͑11c͒ T I ϭT I 1 ϩT D 2 , ͑8b͒ Comparing relationships ͑8a͒–͑8c͒ with ͑11a͒– ͑11c͒ and taking in account Eqs. ͑9a͒ and ͑9b͒, the T I1T D2 parameters of C 2 ( s ) are given by T Dϭ . ͑8c͒ T I 1 ϩT D 2 ␶ p1 Kp 1 ␭ 1 K C2ϭ ϭ , ͑12a͒ K C1 Kp 2 ␭ 2 Kp 2 ␭ 2 The forms of Eqs. ͑8a͒–͑8c͒ lead to the following adjustment procedure: T D 2 ϭ ␶ p2 . ͑12b͒ 1. Approach the dynamics relating y ( t ) and u 1 ( t ) with a first-order plus time-delay transfer Observe the procedure leaves three parameters, function ␭ 1 ,␭ 2 , and T F , for adjusting both controllers to achieve robust performance of the closed-loop e ϪT d1 s system. Since both transfers are connected through Gp 1 ͑ s ͒ ХKp 1 . ␶ p1 sϩ1 an interactive control scheme, the tuning must be coupled, because the uncertainties of each model, 2. Use the IMC ͓16͔ parametrization to define lm 1 ( s ) and lm 2 ( s ) , affect both controllers simul- the parameters of controller C 1 ( s ) , which is based taneously. If there are uncertainties, the system on the Smith predictor structure, i.e., output y ( s ) controlled by the flexible-structure control is given by 2 ␶ p1 K C1ϭ , ͑9a͒ Kp 1 ␭ 1 y ͑ s ͒ ϭ ͕ G p 1 ͑ s ͓͒ 1ϩlm 1 ͑ s ͔͒ ϩC 2 ͑ s ͒ G p 2 ͑ s ͒ ˜ ˜ T I 1 ϭ ␶ p1 . ͑9b͒ ϫ ͓ 1ϩlm 2 ͑ s ͔͒ ͖ u 1 ͑ s ͒ . 3. Approach the dynamics relating variables Using the approximation ͑10͒ for G p 2 ( s ) and the ˜ y ( t ) and u 2 ( t ) with a second-order plus time- tuned the procedure proposed in this section for delay model, C 2 , based on IMC design procedure, e ϪT d2 s C 2 ͑ s ͒ ϭ ͕ G p * ͑ s ͒ ͖ Ϫ1 f 2 ͑ s ͒ , ˜ ϩ Gp 2 ͑ s ͒ ХKp 2 , ͑10͒ ͑ ␶ p1 sϩ1 ͒͑ ␶ p2 sϩ1 ͒ the system output is given by where one of the time constants is arbitrarily made y ͑ s ͒ ϭGp 1 ͑ s ͒ ͕ ͓ 1ϩlm 1 ͑ s ͔͒ ϩ f 2 ͑ s ͒ equal to ␶ p1 , the time constant determined for the model Gp 1 ( s ) . This condition is just a convenient ϫ ͓ 1ϩlm 2 ͑ s ͔͒ ͖ u 1 ͑ s ͒ . way to get consistent individual and combined ad- justments for C 1 ( s ) and C ( s ) , respectively. No- From this expression we can identify the overall tice that this still leaves three parameters, uncertainty that will suffer C 1 ( s ) ,
  • 9. Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376 369 Fig. 5. Nonlinearities and its approximations for stability analysis: ͑a͒ decision function, and ͑b͒ saturation. lm ͑ s ͒ ϭlm 1 ͑ s ͒ ϩ f 2 ͑ s ͒ lm 2 ͑ s ͒ . nonlinearities involved in the system ͑saturations and decision function͒ are approximated through a Then, the last expressions lead to the following conic sector and then the resulting linear system is tuning procedure for the controllers parameters: analyzed using robust control tools. In the second 1. Tune the controller C 2 ( s ) to obtain the robust approach, the stability of each linear system is stability of Eq. ͑16͒, modifying ␭ 2 , for lm 2 ( s ) as guaranteed and then the stability of the switching if C ( s ) is an ideal PID. between them is analyzed using describing func- 2. Tune the controller C 1 ( s ) to obtain the robust tion techniques ͓21͔. stability of Eq. ͑15͒, modifying ␭ 1 , for lm ( s ) . Since the nonlinearities involved in the flexible- 3. Finally, the parameter T F is determined such structure control, the decision function, and the that the sensitivity function is attenuated in the saturations, satisfied the sector nonlinearity condi- high-frequency range. tion ͓20͔ 4.2. Cancellation design ͯ u ͯ h ͑ u ͒ Ϫcu 2 2 Ͻr Ϫ␧, ᭙u 0,␧Ͼ0, ͑14͒ As a general rule and from some of the above arguments it can be concluded that selecting the the nonlinearities can be approximated by ͑see combined controller C ( s ) as being of equal or Fig. 5͒ higher order than C 1 ( s ) will always give a realiz- able C 2 ( s ) . Hence any available design and ad- h ͑ u ͒ Ϸ ͑ cϮr ͒ u, justment procedure can be followed to completely where c is the gain of the linear model and r is the define C 1 ( s ) and C ( s ) for controlling Gp 1 ( s ) and uncertainty of the linear model. Then, the control- Gp 2 ( s ) , respectively, as if they were not related to lers are tuned to obtain the robust stability of the each other. Then, the second controller C 2 ( s ) is closed-loop system for the overall uncertainties determined by ͓17,18͔: the model and the approximation errors r. C 2 ͑ s ͒ ϭC ͑ s ͒ C Ϫ1 ͑ s ͒ . 1 ͑13͒ This approach leads to an over conservative tune due to the consevatiness of the process gain, intro- There are no stability problems in this design tech- duced by the approximation ͑14͒. Therefore the nique because the zeros and poles of C 1 ( s ) and resulting closed-loop performance will be poor C ( s ) are stable and known. and the response will be sluggish. The stability analysis of the flexible-structure 5. Stability analysis control from the switching system point of view implies the stability analysis of every single or A general stability analysis for the flexible con- combined loop and the stability analysis for a gen- trol system can be performed using two different eral switching sequence between these loops. frameworks: ͑i͒ a stability analysis of the resulting In a first step, the stability of each closed-loop nonlinear system ͓18͔, or ͑ii͒ a stability analysis of system is studied. While u 1 ( t ) Ͻ ␬ the characteris- switched linear system ͓19͔. In the first case the tic equation for the primary control loop includes
  • 10. 370 Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376 Gp 1 ( s ) and C 1 ( s ) only, i.e., the stability condi- ͑ ␶ 1 sϩ1 ͒ K 1 ͑ ␶ 1 sϩ1 ͒ K 1 ␭ 1 tion may be written as follows: 1ϩ ϩ␩ K 1 ␭ 1 s ␶ 1 sϩ1 K 1␭ 1s K 2␭ 2 1ϩC 1 ͑ s ͒ Gp 1 ͑ s ͒ 0, ᭙s෈C ϩ , ͑15͒ ͑ ␶ 2 sϩ1 ͒ K2 ϩ ϫ ϭ0. ͑20͒ where C stands for the right-side complex plane. ͑ T F sϩ1 ͒ ͑ ␶ 1 sϩ1 ͒͑ ␶ 2 sϩ1 ͒ The second important stability condition is for the secondary loop which works through the manipu- Operating with this equation, the characteristic lated u 2 when u 1 ( t ) Ͼu 1 max , equation is 1ϩC 2 ͑ s ͒ C 1 ͑ s ͒ Gp 2 ͑ s ͒ 0, ᭙s෈C ϩ . ␭ 1 ␭ 2 T F s 2 ϩ␭ 2 ͑ ␭ 1 ϩT F ͒ sϩ ͑ ␭ 2 ϩ␭ 1 ␩ ͒ ϭ0, ͑16͒ ͑21͒ Finally, for the intermediate case when ␩ Ͼ0 and and the poles of the closed-loop system p are ␬ Ͻu 1 ( t ) Ͻu 1 max , two control paths coexist si- given by multaneously: one through u 1 and the other through u 2 . Then, the condition takes the form 1ϩC 1 ͑ s ͒ Gp 1 ͑ s ͒ ϩ ␩ C 1 ͑ s ͒ C 2 ͑ s ͒ Gp 2 ͑ s ͒ 0, p 1,2ϭϪ ␭ 1 ϩT F 2␭ 1 T F Ϯ ͱ ͑ ␭ 1 ϪT F ͒ 2 2 2 4␭ 1 T F Ϫ ␩ ␭ 2T F . ͑22͒ ᭙s෈C ϩ . ͑17͒ The stability only depends on the real part of these The first two conditions can be satisfied sequen- poles, therefore it is clear that the stability of the tially, Eq. ͑15͒ while adjusting controller C 1 ( s ) , closed-loop system only depends on the values of Eq. ͑16͒ while adjusting controller C 2 ( s ) . Hence the controllers parameters ( ␭ 1 ,␭ 2 and T F ) and the the stability problem of Eq. ͑17͒ is automatically parameter ␩ of the decision function. satisfied for the nominal system. Remark 2. If ␭ 2 T F ӷ ␩ the closed-loop poles Theorem 5.1. Given a two-inputs and one- will be approximately located at p 1 ϭ␭ Ϫ1 and p 2 1 Ϫ1 output system with stable transfer function be- ϭT F . However, if tween the inputs, u 1 ( s ) and u 2 ( s ) , and output, Gp 1 ( s ) and Gp 2 ( s ) , and ␩ у0, the closed-loop ͑ ␭ 1 ϪT F ͒ 2 ␩ Ͼ␭ 2 , system resulting from the application of the flex- 4␭ 2 T F 1 ible structure control is stable if the following con- ditions are satisfied: ␭ 1 and T F can be used to fix the damping ratio of ϩ the closed-loop response and ␭ 2 and ␩ can be 1ϩGp 1 ͑ s ͒ C 1 ͑ s ͒ 0, ᭙s෈C , employed to fix the natural frequency of the 1ϩGp 2 ͑ s ͒ C 2 ͑ s ͒ C 1 ͑ s ͒ 0, ᭙s෈C ϩ . closed-loop response. Finally, the stability analysis of the switching Proof: Given the closed-loop equation sequence implies the analysis of the following sys- tems: 1ϩC 1 ͑ s ͒ Gp 1 ͑ s ͒ ϩ ␩ C 1 ͑ s ͒ C 2 ͑ s ͒ Gp 2 ͑ s ͒ ϭ0, ͑18͒ 1ϩC 1 ͑ s ͒ Gp 1 ͑ s ͒ ϩS ͑ s ͒ ␩ C ͑ s ͒ Gp 2 ͑ s ͒ ϭ0, ͑23a͒ both transfer functions e ϪT d 1 s 1ϩS ͑ s ͒ C 1 ͑ s ͒ Gp 1 ͑ s ͒ ϩC ͑ s ͒ Gp 2 ͑ s ͒ ϪS ͑ s ͒͑ 1 Gp 1 ͑ s ͒ ϭK 1 , ͑19a͒ ␶ 1 sϩ1 Ϫ ␩ ͒ C ͑ s ͒ Gp 2 ͑ s ͒ ϭ0, ͑23b͒ e ϪT d 2 s 1ϩS ͑ s ͒ C 1 ͑ s ͒ Gp 1 ͑ s ͒ ϩ ͓ 1ϪS ͑ s ͔͒ C ͑ s ͒ Gp 2 ͑ s ͒ Gp 2 ͑ s ͒ ϭK 2 , ͑19b͒ ͑ ␶ 1 sϩ1 ͒͑ ␶ 2 sϩ1 ͒ ϭ0, ͑23c͒ and the controllers C 1 ( s ) and C 2 ( s ) tuned accord- ing with the procedure described in the previous that represent all possible changes in the system. section characteristic equation of the closed-loop In these expressions S ( s ) is the Fourier transform system is given by of the switching function S ( t ) given by
  • 11. Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376 371 1 everywhere the magnitude of the Bode plot of the S ͑ t ͒ ϭ ͓ 1Ϫg ͑ t ͔͒ , ͑24͒ resulting linear system exceeds unity. 2 where g ( t ) is a scalar signal that only assumes the 6. Choice of ␩ and ␬ value Ϫ1 or ϩ1. Eq. ͑23a͒ represents the changes in the system when u 1 ( t ) Ͼ ␬ , therefore the sec- In this control scheme ␩ represents the amount ond loop becomes active. The second equation, of u 2 requires to prevent the saturation of u 1 while Eq. ͑23b͒, represents the change in the system ␬ is the value of u 1 at which the protection begins. when u 1 ( t ) Ͼu 1 max , then only the second loop However, both parameters can be employed to sat- controls the system output. Finally, Eq. ͑23c͒ rep- isfy some additional objective than control objec- resents the direct transition from u 1 ( t ) Ͻ ␬ to tives. u 1 ( t ) Ͼu 1 max , this fact means that the main loop If the objective is to obtain a good transient becomes inactive at the same time that the second- behavior, Gp 1 must be kept active in the whole ary loop is activated. operating range. Hence the parameters of the de- For switching sequences slower than the system cision function ͑4͒ must be fixed to dynamic, the stability of the overall closed-loop system is guaranteed ͓19͔. The stability of an ar- u 2 max bitrary switching sequence between these systems ␩ϭ , ͑26a͒ u 1 max can be analyzed using describing function tech- niques and harmonic balance explained by Leith et ␬ ϭ0. ͑26b͒ al. ͓19͔ First, the nonlinearities are approximate through a Fourier series, This selection implies the use of u 2 to prevent the saturation of u 1 for any value. This might be a S ͑ t ͒ ϭ f 0 ϩ f 1 cos͑ ␻ tϩ ␾ ͒ ϩh.o.t, desirable feature for the control system since it keeps the fastest loop working in order to maintain then, due to the low-pass characteristic of the sys- a better control quality. This protection is clearly tem S ( t ) may be approximate, done at the expense of u 2 . In the opposite case, if S ͑ t ͒ Ϸ f 0 ϩ f 1 cos͑ ␻ tϩ ␾ ͒ . the objective is to minimize the amount of u 2 em- ployed to control the system, Gp 2 must be kept For the controllers resulting from the IMC design, inactive as much as possible. Therefore the param- the transfer function of the equivalent linear sys- eters of the decision function must be fixed to ␩ tem of Eqs. ͑23a͒–͑23c͒ are given by ϭ0. This selection implies the use of the auxiliary variable to cover output demands when u 1 satu- H 1 ͑ s ͒ ϭϪ ͑ f 1 ͒ / ͕ ␭ 1 ␭ 2 T F s 2 rates only. Finally, if the objectives are a combi- ϩ␭ 2 ͑ ␭ 1 ϩT F ͒ sϩ ͑ ␭ 2 ϩ ␩ ␭ 1 f 0 ͒ ͖ , nation of previous ones, Gp 1 must be kept active in the range such that it can handle the most fre- ͑25a͒ quent changes. Therefore the parameters of the de- cision function ͑4͒ are given by H 2 ͑ s ͒ ϭϪ ͕ f 1 ͓ ␭ 2 T F sϩ ͑ ␭ 2 Ϫ␭ 1 ͔͒ ͖ / ͕ ␭ 1 ␭ 2 T F s 2 1 ϩ␭ 2 ͑ ␭ 1 ϩT F f 0 ͒ s ␩у „max͕ var͓ r ͑ t ͔͒ ,var͓ d ͑ t ͔͒ ͖ Kp 2 ϩ ͓͑ 1Ϫ ␩ Ϫ f 0 ͒ ␭ 1 ϩ f 0 ␭ 2 ͔ ͖ , ͑25b͒ ϪKp 1 u 1 max…, ͑27a͒ H 3 ͑ s ͒ ϭϪ ͕ f 1 ͓ ␭ 2 T F sϩ ͑ ␭ 2 Ϫ␭ 1 ͔͒ ͖ / ͕ ␭ 1 ␭ 2 T F s 2 ␬ уu 10 , ͑27b͒ ϩ␭ 2 ͑ ␭ 1 ϩT F f 0 ͒ sϩ ͓͑ 1Ϫ f 0 ͒ ␭ 1 ϩ␭ 2 ͔ ͖ . where u 10 is the steady-state value of u 1 for the ͑25c͒ nominal set-point value. This selection implies the use of u 2 to prevent the saturation of u 1 for the Finally, the resulting transfer functions are ana- most frequent changes only when u 1 Ͼ ␬ . lyzed using the method of harmonic balance to To explain these concepts, let us consider the determine the stability of the switching sequence. case of two heat exchangers in series: one heat The harmonic balance method predicts instability exchanger and a service equipment with a direct
  • 12. 372 Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376 Fig. 6. Bode plots associated with approximated analysis. bypass on the controlled stream. In this configura- 2e Ϫ10s tion the service equipment works intermittently Gp 2 ͑ s ͒ ϭ , ͑29͒ ͑ 6sϩ1 ͒͑ 17sϩ1 ͒ such that recovered energy is maximized. In this problem, the parameters ␩ and ␬ correspond to the and the disturbance d ( s ) are modeled by service energy level required to avoid the control- lability problems, which implies a loss of process 1 G d͑ s ͒ ϭ . ͑30͒ efficiency, and the level at which the protection 5s ϩsϩ1 2 begins, respectively. If the objective is to obtain a good dynamic behavior, the parameters must be The primary and the auxiliary manipulated vari- set to ␩ ϭ1 and ␬ ϭ0, respectively. In the opposite ables u 1 and u 2 are constrained to u 1,2෈ ͓ Ϫ1, case, if the objective is to maximize the amount of ϩ1 ͔ . energy recovered, the parameters must be fixed to The controllers will be developed using a Smith ␩ ϭ0. Finally, if the objective is to maximize the predictor structure to compensate the time delays amount of energy recovered while keeping good of the process. First, the controller C 1 ( s ) is de- closed-loop performance, the parameters must be signed using the formulas ͑9a͒ and ͑9b͒ and obtain fixed to 0Ͻ ␩ Ͻ1 ͑according to the disturbances a response without offset. This means that C 1 ( s ) and set-point variance͒ and ␬ уu 10 , where u 10 is is a PI controller, whose parameters are the steady-state value u 1 for the nominal set-point 2.75 value. K C1ϭ , ͑31a͒ ␭1 7. Simulation and results T I 1 ϭ ␶ p1 ϭ2.75. ͑31b͒ In this section a simulation example is consid- To tune the controller C 2 ( s ) , we approximate ered to show the effectiveness of the control Gp 2 using Eq. ͑10͒. The result is scheme proposed in this work. We consider a lin- ear system previously used by Wang et al. ͓9͔ to 2e Ϫ10s evaluate the cooperative control ͓9͔. The system is Gp 2 ͑ s ͒ Ӎ . ͑32͒ ͑ 2.75sϩ1 ͒͑ 18.75sϩ1 ͒ given by First, we design the combined controller C ( s ) e Ϫ4s e Ϫ5s ϭC 1 ( s ) C 2 ( s ) that controls this model. In this ap- Gp 1 ͑ s ͒ ϭ Х , ͑28͒ ͑ 2sϩ1 ͒ 2 2.75sϩ1 plication, a PID controller is adopted for C ( s ) .
  • 13. Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376 373 Fig. 7. Set-point responses obtained using flexible structure and cooperative control. This means that C 1 ( s ) is a PD controller, whose the constraint in the primary manipulated variable parameters are given by Eqs. ͑12a͒ and ͑12b͒, ( u 1 ) . The parameters of the cooperative control are observation time T o and a margin ␤. The value Kp 1 ␭ 1 ␭1 of these parameters are the same to that one em- K C2ϭ ϭ , ͑33a͒ Kp 2 ␭ 2 2␭ 2 ployed by Wang et al. to obtain the best perfor- mance, T D 2 ϭ ␶ p2 ϭ18.75. ͑33b͒ T o ϭ0.275, ␤ ϭ2. Since there is no uncertainty, the parameters ␭ 1 ,␭ 2 , and T F are determined to obtain the best In order to evaluate the performances of both con- possible performance. The values of these param- trol schemes, a sequence of reference and distur- eters are bance changes are introduce at several times. The ␭ 1 ϭ1.5, ␭ 2 ϭ0.56, T F ϭ30.2. ͑34͒ set point r was changed in intervals of 200 sec from 0.5 to 0.75 and then steps to 1.5, and finally An IMC antiwindup scheme is included in the the disturbance w changes from 0 to 0.5 at 450 C ( s ) controller to compensate for the effect of the sec. constraint in u 2 . Finally, the parameters of the de- In Fig. 7 we see the responses obtained by both cision function ͑1͒ were chosen to obtain a good control schemes. This figure shows the superior performance, therefore ␩ and ␬ were fixed to performance of the proposed scheme when the ␬ ϭ0.25, ␩ ϭ1.0. ͑35͒ system needs to modify the value of the auxiliary manipulated variable. The better performance is Finally, the stability of the system for the switch is due to the fact that the proposed scheme takes ac- analyzed. Fig. 6 shows the Bode plots for the count of the dynamics of the auxiliary manipu- transfer functions ͑25a͒–͑25c͒ for parameters ͑34͒ lated variable and the main loop is still working. and ͑35͒. It is easy to see the resulting linear sys- This fact leads to the temporary saturation of the tem does not exceed unity, therefore all possible primary manipulated variable of cooperative con- switching between these systems are stable. trol during the transition period ͑Fig. 8͒, that leads In this example the proposed control structure is to open-loop behavior of the closed-loop system compared with the cooperative algorithm pro- that deteriorates the closed-loop performance. posed by Wang et al. ͓9͔. The controller employed However, the cooperative control scheme shows a by the cooperative schemes is C 1 ( s ) , with an IMC better performance when a change in the auxiliary antiwindup scheme, to compensate the effect of variable is not needed. It is clear that the interac-
  • 14. 374 Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376 To address the effects of the protection level ␩ and the start of the protection ␬, the proposed al- gorithm is simulated with different values of them. The results with different ␩ are shown in Fig. 9. It can be seen that the higher the level of ␩ the better the performance, since both transfer functions control the system output. When ␩ ϭ0 the system output is first controlled by Gp 1 and then by Gp 2 . In this case the system output shows a large over- shoot and large settling time due to the change of the process structure. The system output is now controlled by the portion of the system that has the slower dynamics and the bigger time delay, there- fore closed-loop performance is the poorest. Fig. 10 shows the control actions associated to the responses of Fig. 9. In this figure it is easy to see that u 2 works to avoid the saturation u 1 or takes full control of the system output, depending on the value of ␩. If ␩ Ͼ0, u 2 prevents the satu- ration of u 1 by providing a level ␩ r of the output. This fact keeps Gp 1 active for a wider output range, therefore good closed-loop responses are Fig. 8. Primary and auxiliary variables corresponding to obtained. When ␩ ϭ0, u 2 takes full control of the set-point and disturbance changes of Fig. 7. system output when u 1 is saturated, therefore the slowest dynamic is in charge of the output regula- tion and poor closed-loop responses are obtained. tion between both loops improves the performance In these figures we can also see that the auxiliary of the closed-loop performance when the control variable exhibits peaks. This fact is due to the in- is transferred from u 1 to u 2 , but it deteriorates the crement in the gain of C 2 ( s ) , which is a PD con- performance when this change is not required. troller. Fig. 9. Set-point responses obtained for different values of ␩.
  • 15. Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376 375 Fig. 10. Primary and auxiliary variables corresponding to Fig. 12. Primary and auxiliary variables corresponding to set-point and disturbance changes of Fig. 9. set-point and disturbance changes of Fig. 11. The results with different ␬ are shown in Figs. When ␬ ϭ0 the protective action is applied all the 11 and 12. It can be seen that higher the level of ␬ time. This fact leads to a good closed-loop perfor- the smaller the value of u 2 , however, similar mance. If ␬ Ͼ0 the protective action is only ap- closed-loop responses are obtained in both cases. plied when u 2 у ␬ , the closed-loop performance is Fig. 11. Set-point responses obtained for different values of ␩.
  • 16. 376 Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376 similar to the case of ␬ ϭ0. This fact leads to a trolling the operation of heat exchanger networks. conservative use of u 2 . AIChE J. 44, 1090–1104 ͑1998͒. ͓9͔ Wang, Q., Zhang, Y., Cai, W., Bi, Q., and Hang, C., Co-operative control of multi-input single-output pro- 8. Conclusions cesses: On-line strategy for releasing input saturation. Control Eng. Pract. 9, 491–500 ͑2001͒. A new control structure proposed for dealing ͓10͔ Morningred, D., Paden, B., and Mellichamp, D., An with constraints on manipulated variables process adaptive nonlinear predictive controller. Chem. Eng. shows at least two input variables associated to the Sci. 47, 755–765 ͑1992͒. ͓11͔ Luyben, W., Process Modelling, Simulation and Con- control target. Flexible-structure control refers to trol for Chemical Engineers. McGraw-Hill Interna- the resulting control system, which switches from tional, New York, 1990. one closed loop to another in order to keep con- ͓12͔ Marselle, D., Morari, M., and Rudd, D., Design of trollability. This flexible characteristic of the con- resilient processing plants: II. Design and control of trol system is particularly useful when the optimal energy management system. Chem. Eng. Sci. 37, 259– operating point locates near a limit constraint of 270 ͑1982͒. ͓13͔ Mathisen, K. Integrated Design and Control of Heat the main manipulated variable. Issues related to Exchanger Networks. Ph.D. thesis, University of design, analysis, and tuning a flexible-structure Trondheim, Trondheim, Norway, 1994. control system are discussed. The application to a ͓14͔ ¨ ¨ Åstrom, K. and Hagglund, T., PID Controllers: linear system shows the tradeoff between process Theory, Design and Tuning, 2nd ed. Instrument Soci- efficiency and controllability. ety of America, 1995. ͓15͔ Smith, O., Close control of loops with dead time. Chem. Eng. Prog. 53, 217–219 ͑1957͒. Acknowledgments ͓16͔ Chien, I. and Fruehauf, P., Consider IMC tuning to improve controller performance. Chem. Eng. Prog. 86, We are grateful for the financial support for this 33– 41 ͑1990͒. work provided by the Engineering and Physical ͓17͔ Morari, M. and Zafiriou, E., Robust Process Control, Science Research Council ͑EPSRC͒ Industrial 2nd ed. Prentice-Hall, Englewood Cliffs, NJ, 1989. ͓18͔ Sofanov, M. and Athans, M., A multiloop generalisa- Non-linear Control and Applications Grant No. tion of the circle criterion for stability analysis. IEEE GR/R04683/01. Thanks are due to Dr. Hao Xia for Trans. Autom. Control 26, 415– 422 ͑1981͒. valuable discussions and the anonymous referees ͓19͔ Leith, D., Shorten, R., Leithead, W., Mason, O., and for the suggestions to improve the paper. Curran, P., Issues in the design of switched linear con- trol systems: A benchmark study. Int. J. Adapt. Control References Signal Process. 17, 103–118 ͑2003͒. ͓20͔ Desoer, C. and Vidyasagar, M., Feedback Systems: ͓1͔ Chen, C. and Peng, S., Learning control of process Input-Output Systems. Academic, New York, 1975. systems with hard constraints. J. Process Control 9, ͓21͔ Khalil, K., Nonlinear System Analysis. Macmillan, 151–160 ͑1999͒. New York, 1992. ͓2͔ Doyle, F., III, An auto-windup input-output lineariza- tion scheme for SISO systems. J. Process Control 9, 213–220 ͑1999͒. Leonardo L. Giovanini was ͓3͔ Kothare, M., Campo, P., Morari, M., and Nett, C., A born in Argentina in 1969. He unified framework for the study of anti-windup design. received the BSc ͑Honors͒ de- Automatica 30, 1869–1883 ͑1994͒. gree in Electronic Engineering ͓4͔ Wang, Q. and Zhang, Y., A method to incorporate con- from the Universidad Tecno- troller output constraints into controller design. ISA logica Nacional, Regional Trans. 37, 403– 416 ͑1998͒. School at Villa Maria in 1995, and the Ph.D. degree in Engi- ͓5͔ Stephanopoulos, G., Chemical Process Control: An In- neering Science from Univer- troduction to Theory and Practice. Prentice-Hall, sidad Nacional del Litoral, Englewood Cliffs, NJ, 1984. School of Water Resource in ͓6͔ Shinskey, F., Process Control Systems: Application, 2000. Since then he has been Design and Tuning, 3rd ed. McGraw-Hill, New York, Lecturer in the Department of 1988. Electronic Engineering at Uni- versidad Tecnologica Nacional, Regional School at Villa Maria. Cur- ͓7͔ Gao, Z., Stability of the pseudo-inverse method for rently, he is working as a research fellow in the Industrial Control reconfigurable control system. Int. J. Control 8, 469– Center, University of Strathclyde. His research interests include pre- 474 ͑1991͒. dictive control for non-linear systems, fault detection and isolation, ͓8͔ Aguilera, N. and Marchetti, J., Optimizing and con- fault tolerant control and distributed control systems.