362 Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376never 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 temporarilyreach their limits and the system becomes uncon-trollable during this period. Hence the need forﬁnding a way for broadening controlled operationspaces 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 structurecalled ﬂexible-structure control, due to its abilityto adapt the control structure to the operating con- namic elements Gp 1 ( s ) and Gp 2 ( s ) . The secondditons. It is a generalization of the split-range con- feature is that assumed to be u 1 is the primarytrol ͓6͔ and it can be summarized as two control- manipulated variable because Gp 1 ( s ) is fasterlers connected through a piecewise linear function. and with smaller time delay than Gp 2 ( s ) . A hardThis 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 manipulatedaction between both manipulated variables. They variable to keep the system under regulation. Thecan 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 soapply the proposed control strategy and some ex- that for any moderate set-point or load disturbanceamples are presented. In Section 3 the ﬂexible- change, the primary manipulated variable u 1 canstructure 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 workingpiecewise-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, soond procedure employes more sophisticated con- it deserves special research. For example, considertrollers. Section 5 addresses the closed-loop stabil- the problem of controlling a continuous stirredity 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 nonlinearthe application of the proposed algorithm to a lin- control algorithms. The reactor is cooled by waterear system. Conclusions are presented in Section through a surrounding jacket. The concentration8. 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 is2. 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 thebe 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 363able, while the process stream acts as the auxiliary 3. Flexible-structure controlmanipulated 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 theboth cooling water ﬂowing through a surrounding auxiliary manipulated variable. To perform thesejacket 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 proposedFrom an energy-saving point of view, the use of the valve-position control. This technique uses acooling water is preferred over the refrigerant due maximum signal selector to change u 2 to keep allto cost. However, the dynamic response of the the actuators from exceeding a preset limit. Theplant’s temperature to a change in refrigerant ﬂow algorithm has to wait for the process variable tois faster than a change in cooling water. In prac- reach its steady state, then regulate u 2 iterativelytice, the nominal ﬂow rates of both are determined until the steady-state error becomes zero. Thein advance. Water is utilized as the primary ma- main drawbacks of the method are that it is timenipulated variable, while the refrigerant acts as an consuming and the results largely depend on theauxiliary 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 controlhits 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 outputsteady-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 unreachabilitysystem 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 controlarrangement is very common in practice when be- structure that remains unchanged. Therefore thesides the task of reaching a ﬁnal temperature target transient behavior of the closed-loop system willon a process stream there is an extra goal like be poor because the control laws generated bymaximum energy recovery. The heat exchanger is both control strategies do not take account of thespeciﬁcally designed for recovering the exceeding dynamic part of Gp 2 . Many times the closed-loopenergy in the process stream, and the service response is driven in an open-loop manner duringequipment 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 oroperates 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 thevariations on the inlet stream conditions or having control from one variable to another, takes accountthe capability of changing stream temperature tar- of the different dynamics involved in the systemgets. 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 andexist a situation in which the effect is so large that other goals to the operator. Hence, under thisthe 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 processof 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 ) handlingvalue. 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 canvariable 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 tou 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 1a fraction of this output y can be handled throughthe 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 loopadditional 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 noGp 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. Thusstant manipulated variable u 1 ( t ) at an operating the process part represented by Gp 2 ( s ) is now inpoint such that ͉ u 1 ( t ) ͉ Ͼu 1 max , the process part charge of the regulation. As soon as the structureGp 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 systemin Figs. 2͑a͒ and 2͑b͒ respectively, where a second and adapting the structure and parameters of thecontroller 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 tou 2 . It is clear from the block diagrams that in the mitigate the effect of the constraint on u 2 , thecase of preventive action, C 2 ( s ) is a feedforward effect of constraint on u 1 is compensated by thecontroller, which does not affect the closed-loop control structure.stability, but it cannot protect the system from the The switching element is implemented througheffect of nonmeasurable disturbances, uncertain- a simple nonlinear decision function such that theties, and/or faults. If the disturbance w ( s ) , or an signal x ( t ) entering the controller C 2 ( s ) is givenestimation 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 termObserve 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 aferent 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 theable 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 ͑ u1͒ϭThe 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, suchthat it generates the proper signal for each auxil- ͑3͒iary 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 theclosed loop to transfer the control from Gp 1 to auxiliary variable u 2 is used just to cover outputGp 2 , the stability properties of the system remain demands when u 1 saturates only. When Ͼ0 theunaffected 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 arethe 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 aprotection into the control loop. It is achieved by deterioration of the closed-loop performance com-
366 Leonardo L. Giovanini / ISA Transactions 43 (2004) 361–376pared with the previous situation.1 However, it This fact means when a major fault in the actuatormight be a desirable feature since it tends to keep of u 1 happens, like a actuator frozen at a given athe fastest loop working in a wider range. When value, the ﬂexible structure is able to transfer theu 1 ( t ) уu 1 max the control system changes the control of the system output to u 2 without anystructure 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 ariorate 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 hastrol 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 controlthe whole operational range maintaining a goodcontrol 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 dominantabove formulation to start the preventive protec- time delay are notoriously difﬁcult. It is also ation from a given value u 1 ( t ) ϭ in ahead, instead topic on which there are many different opinionsof 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 isFigure 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 tocan 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 thecan be used as tuning parameters to satisfy addi- ˜ ˜ models of both process ( G p 1 and G p 2 ) and thetional 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 astrol 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 beﬁrst one saturates. For positive values of , the coupled with a robust controller ( H ϱ , QFT, robustcontrol variable is executed over a common range, pole placement, etc.͒ to cope with parametricthat is, both actuators work together until one of variations.them saturates. The amplitude of the commonrange is handled through the parameter whilethe intensity is given by . 4. Controller design and tuning The capability of transferring the control fromone 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 tocompensate 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 secondnot fast enough, the closed-loop response will show an control structure is deﬁned by the secondary loopovershoot 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 parametrizationpears 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 ) , recallcates that C 1 ( s ) must be adjusted for high quality that they work in series, i.e., the outlet of C 1 ( s ) iscontrol when the process part Gp 1 ( s ) handles the the input to C 2 ( s ) through the switching functionregulation, 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 importantsingle 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 necessarymentary effect on the controlled variable, which since offset elimination is required for set-pointmeans 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 underdesigning and tuning controllers C 1 ( s ) and C 2 ( s ) any working condition and this controller is theare 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 thatcelation design criteria. In the IMC design the or- Gp 1 ( s ) is faster and with smaller time delay thander 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 itcelation 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, suchthat ﬁrst- or second-order plus time delay modelsare 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) 361–376and 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 combinedHence the combination C ( s ) ϭC 1 ( s ) C 2 ( s ) results controller—whose parameters K C , T I , and T D areto 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 , ͑11a͒whose 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 2The forms of Eqs. ͑8a͒–͑8c͒ lead to the followingadjustment procedure: T D 2 ϭ p2 . ͑12b͒ 1. Approach the dynamics relating y ( t ) andu 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 systemon 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 fordelay 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 bywhere 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 themodel 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 overalltice 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 aThen, the last expressions lead to the following conic sector and then the resulting linear system istuning 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 isstability of Eq. ͑16͒, modifying 2 , for lm 2 ( s ) as guaranteed and then the stability of the switchingif 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 thethat the sensitivity function is attenuated in the saturations, satisﬁed the sector nonlinearity condi-high-frequency range. tion ͓20͔4.2. Cancellation design ͯ u ͯ h ͑ u ͒ Ϫcu 2 2 Ͻr Ϫ, ᭙u 0,Ͼ0, ͑14͒ As a general rule and from some of the abovearguments it can be concluded that selecting the the nonlinearities can be approximated by ͑seecombined controller C ( s ) as being of equal or Fig. 5͒higher 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 thedeﬁne 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 theeach other. Then, the second controller C 2 ( s ) is closed-loop system for the overall uncertaintiesdetermined 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 thenique because the zeros and poles of C 1 ( s ) and resulting closed-loop performance will be poorC ( s ) are stable and known. and the response will be sluggish. The stability analysis of the ﬂexible-structure5. 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-loopnonlinear 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) 361–376Gp 1 ( s ) and C 1 ( s ) only, i.e., the stability condi- ͑ 1 sϩ1 ͒ K 1 ͑ 1 sϩ1 ͒ K 1 1tion 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. ͑20͒where C stands for the right-side complex plane. ͑ T F sϩ1 ͒ ͑ 1 sϩ1 ͒͑ 2 sϩ1 ͒The second important stability condition is for thesecondary loop which works through the manipu- Operating with this equation, the characteristiclated 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 ͑21͒Finally, 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 bymultaneously: one through u 1 and the otherthrough 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 theseThe ﬁrst two conditions can be satisﬁed sequen- poles, therefore it is clear that the stability of thetially, Eq. ͑15͒ while adjusting controller C 1 ( s ) , closed-loop system only depends on the values ofEq. ͑16͒ while adjusting controller C 2 ( s ) . Hence the controllers parameters ( 1 , 2 and T F ) and thethe 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 Ϫ1output system with stable transfer function be- ϭT F . However, iftween 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 1ible 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, ͑23a͒both 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, ͑23c͒and 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 transformsystem 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. 2where g ( t ) is a scalar signal that only assumes the 6. Choice of and value Ϫ1 or ϩ1. Eq. ͑23a͒ represents the changesin the system when u 1 ( t ) Ͼ , therefore the sec- In this control scheme represents the amountond loop becomes active. The second equation, of u 2 requires to prevent the saturation of u 1 whileEq. ͑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 transientbecomes inactive at the same time that the second- behavior, Gp 1 must be kept active in the wholeary loop is activated. operating range. Hence the parameters of the de- For switching sequences slower than the system cision function ͑4͒ must be ﬁxed todynamic, the stability of the overall closed-loopsystem is guaranteed ͓19͔. The stability of an ar- u 2 maxbitrary switching sequence between these systems ϭ , ͑26a͒ u 1 maxcan be analyzed using describing function tech-niques and harmonic balance explained by Leith et ϭ0. ͑26b͒al. ͓19͔ First, the nonlinearities are approximatethrough 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 maintainthen, due to the low-pass characteristic of the sys- a better control quality. This protection is clearlytem 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 keptFor 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…, ͑27a͒H 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 theFinally, 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 thedetermine the stability of the switching sequence. case of two heat exchangers in series: one heatThe 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 Ϫ10stion the service equipment works intermittently Gp 2 ͑ s ͒ ϭ , ͑29͒ ͑ 6sϩ1 ͒͑ 17sϩ1 ͒such that recovered energy is maximized. In thisproblem, the parameters and correspond to the and the disturbance d ( s ) are modeled byservice energy level required to avoid the control-lability problems, which implies a loss of process 1 G d͑ s ͒ ϭ . ͑30͒efﬁciency, and the level at which the protection 5s ϩsϩ1 2begins, respectively. If the objective is to obtain agood 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 delaysamount 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 obtainﬁxed 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 arethe steady-state value u 1 for the nominal set-point 2.75value. K C1ϭ , ͑31a͒ 17. 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 approximateered to show the effectiveness of the control Gp 2 using Eq. ͑10͒. The result isscheme proposed in this work. We consider a lin-ear system previously used by Wang et al. ͓9͔ to 2e Ϫ10sevaluate 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 variableparameters 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 ﬁnallyAn IMC antiwindup scheme is included in the the disturbance w changes from 0 to 0.5 at 450C ( 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 bothcision function ͑1͒ were chosen to obtain a good control schemes. This ﬁgure shows the superiorperformance, 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 isFinally, 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 thetem 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 systemcompared 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 aby the cooperative schemes is C 1 ( s ) , with an IMC better performance when a change in the auxiliaryantiwindup 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 areFig. 8. Primary and auxiliary variables corresponding to obtained. When ϭ0, u 2 takes full control of theset-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 auxiliaryof 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 375Fig. 10. Primary and auxiliary variables corresponding to Fig. 12. Primary and auxiliary variables corresponding toset-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 the11 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) 361–376similar 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., Anwith 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 oftrollability. This ﬂexible characteristic of the con- resilient processing plants: II. Design and control oftrol 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 Heatthe main manipulated variable. Issues related to Exchanger Networks. Ph.D. thesis, University ofdesign, 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. IEEEGR/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., andfor the suggestions to improve the paper. Curran, P., Issues in the design of switched linear con- trol systems: A benchmark study. Int. J. Adapt. ControlReferences 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.