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, ᭙sC 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, ᭙sC 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
᭙sC 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, ᭙sC ,
employed to fix the natural frequency of the
1ϩGp 2 ͑ s ͒ C 2 ͑ s ͒ C 1 ͑ s ͒ 0, ᭙sC 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.
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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
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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-
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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.