A self-tuning control method for Wiener nonlinear systems and its application to process control problems☆

Ping Yuan ,Bi Zhang ,*,Zhizhong Mao ,2

1 College of Information Science&Engineering,Northeastern University,Shenyang 110819,China

2 State Key Laboratory of Synthetical Automation for Process Industries,Northeastern University,Shenyang 110819,China

1.Introduction

Since the dynamics of most chemical processes are nonlinear in nature,accurate and fast modeling of these complex systems remains to be a challenging problem.The traditional linear model is generally not favored in many process control problems.In the past years,more advanced techniques such as NARMAX models[1],multi-model representations[2],Volterra series[3],fuzzy logic systems[4],neural networks[5],and support vector machines[6]have been found applicable to chemical engineering.

As an alternative,interests in modeling and control of block-oriented nonlinear models have grown since last century.Among them,one typical model structure is Wiener systems,which consists of a linear dynamic block followed by a static nonlinear block.Such systems have been proven to be useful as simple nonlinear models for capturing the nonlinear dynamic properties of typical chemical processes,for instance,chemical reactor,pH neutralization,distillation column,polymerization reactor,and so on[7].Compared with the aforementioned representations,the Wiener cascade models are more attractive due to the following advantages[8]:(1)ease of model development;(2)good approximation accuracy;(3)possibility to incorporate a prior process knowledge;and(4)possibility of being used for control.

There exists a vast literature on the control problem of Wiener-type processes,a conventional approach is to design a linear-like controller based on the separation characteristics of nonlinear gain and dynamic sector inside a Wiener model[9,10].Later,researches have been further carried out for the purpose of better control performance,such as the linearizing feed for ward-feedback control[11],the PID control[12],the model predictive control[13–15].Meanwhile some robust control methods have also been developed for Wiener-type processes with perturbations[16–18].In sum,the above control schemes are mostly derived from fixed Wiener model parameters and may become less attractive when processes show a strong nonlinear behavior and time varying characteristics during a wide operation range.

To copy with more highly nonlinear properties,a more favored approach may be adaptive controlsince the closed-loop stability is guaranteed despite of large uncertainties.In such controllers,a parameterized model of the process is continuously identified,utilizing some recursive algorithms.This model is then used to update the controller parameters[19].In the literature,various adaptive control schemes have been proposed for uncertain Wiener systems:Pajunen[20]employed an inverse function to approximate the nonlinear block and established a discrete time self-tuning control scheme.Later,these ideas are extended to Wiener systems described by piecewise linear basis functions[21]and neural networks[22].A recent paper[23]derived a novel adaptive control scheme by the use of internal variables estimations but the method is only applicable in a local stable domain.Despite the salient features of these methods,it still faces some important challenges:(1)They are mostly derived from the squared tracking error criterion which ignores control efforts.It is a fact that control performance can be improved if control efforts are considered adequately.However,the control scheme derived from such criterion has always been a challenge,especially for Wiener systems;(2)The unstable zero dynamics problem remains unsolved.The tracking control of non-minimum phase process is a challenge due to the existence of inverse response behavior.To copy with this problem,a Clarke criterion index[24]is available,which can establish a stable extended system.

To this end,this paper proposes a novel self-tuning control scheme for uncertain Wiener nonlinear systems,which contains many further research fruits compared with[20].The parameterization model is put forward based on the inverse of the nonlinear function block.Then the control method is derived from a modified Clarke criterion function[24],which penalizes the excessive control action and accounts for unstable zero dynamics problems.The uncertain parameters are updated by a recursive least squares algorithm and the control law exhibits an explicit form.Finally two groups of simulation examples including the composition control problem in a chemical reactor are studied to show the wide applicability of the proposed method.

The main merits of the proposed adaptive control are as follows:

?It is motivated by self-tuning control and possesses reliable stability properties.

?It is applicable to processes with unstable zero-dynamics.

?It can adaptively deal with system uncertainties.

?It is straightforward in form and has some potential in applications.

Brie fly,this paper is organized as follows.The problem formulation is described in Section 2.The main results are presented in Section 3.Some simulation studies are conducted in Section 4.A brief summary is given in Section 5.

2.Problem Formulation

Consider the discrete-time mathematical description of the Wiener system shown in Fig.1.It consists of two cascade blocks,a linear time invariant(LTI)subsystemG(z?1)followed by a static(memory-less)nonlinearityf[·].The system input signalu(t)and the output signaly(t)are measurable,whereas the intermediate signalv(t)are unmeasurable.

Fig.1.The discrete-time SISO Wiener model.

The linear block is described by

whereA(z?1)andB(z?1)are polynomials in the unit time delay operatorz?1[z?1u(t)=u(t?1)]which are defined as follows

The nonlinear block is described by

To parameterize the controlled plant,we assume that the nonlinear functionf[·]is continuous and invertible[20].Therefore the inverse of the nonlinear function block can be described by a polynomial of a known basis

where the parameters(a1,a2,...ana,b1,b2,...bnb,c1,c2,...cp)are uncertain but the ordersna,nb,pare pre-specified.

Based on Eq.(5),the Wiener system can be further rewritten as

3.Adaptive Control Method

3.1.A modi fi ed performance index

Introduce the following Clarke performance index[24]:

wherey*(t)is a bounded reference,the weighting polynomialsP(z?1)=p0+p1z?1+...+pnpz?npandQ(z?1)=q0+q1z?1+...+qnqz?nqare with ordersnpandnq;Ris a weighting matrix.The above weighting terms should be properly chosen to meet the following conditions[25]:

Remark1.Eq.(8)can guarantee the same growth rate amongu(t),y(t)andv(t).Eq.(9)can eliminate steady-state tracking error betweeny(t)andy*(t).

Remark 2.To satisfyQ(1)=0,the weighting polynomialQ(z?1)is usually of the convenient formQ(z?1)=K(z?1)(1?z?1)for a polynomialK(z?1).Meanwhile,the ordersnpandnqare often chosen asnp≤naandnq≤nbin order to reduce the computational complexity.

3.2.Recursive estimator

De fine the output Υ(t)of the extended system

Suitable manipulation of Eqs.(6)and(10)yields that

Note that for the ease of identification,P(z?1)andQ(z?1)are chosen such that the orders satisfyng=naandnh=nb.

Then the above extended system can be rewritten as the following compact form

where the parameter vector θ and the regress or vector φ(t)are

Then we have the prediction out put Υ（t+1）?φT（t+1）^θ（t）,with^θ（t）de fined as the estimation of the parameter vector θ

The vector^θ（t）is updated by the following weighted recursive least squares algorithm[26]:

wheree(t)is the extended model error;anda(t)is a nonnegative weighted factor.

3.3.Control law design

According to the certainty equivalence principle[26],we obtain the following one-step-ahead adaptive control law:

The control signalu(t)is implicit in Eq.(19)and can be obtained by solving the following equation:

3.4.Overall adaptive control scheme

In sum,the proposed Wiener model structure based adaptive control(W-AC)approach can be used according to the following steps:

Step 1.Collect{u(t),y(t)}and construct φ(t)by Eq.(14).

Step 2.Calculateby Eqs.(16),(18).

Step 3.Calculateu(t)by solving the Eq.(20).

Step 4.Lett=t+1,and applyu(t)to the system.

3.5.Closed-loop system stability properties

Lemma 1.[26,Lemma 3.3.6]

Lemma 2.For the Wiener system Eq.(6)with the proposed adaptive controller,there exist positive constantsK1,K2such that

Proof.Please see Appendix 0 for details.

3.6.Stochastic case

In the above content,only the deterministic case is considered.In practice,there may existprocess noise(disturbing the intermediate signalv(t))and measurement noise(disturbing the output signaly(t))in the Wiener system.In such a stochastic case,the extended system can no longer be written as Eq.(12).

Now suppose that the system can be described as[27]

where the variables Υ(t), θ and φ(t)have the same definitions as Eqs.(10),(13)and(14).Note that the term η(t)denotes the sum of the noise,namely,(1)the process noise and(2)the transformation of the measurement noise to the internal place.

Since η(t)is no longer subject to Gaussian distribution,the recursive algorithm Eqs.(16),(18)may not guarantee the robustness.Now a much more reasonable assumption on η(t)is that|η(t)|≤Δ for known operation regions.Then the robust recursive least squares algorithm with a deadzone weighted factor is used[26]:

wheree(t)is the extended model error;a(t)is a nonnegative weighted factor;? is a user-chosen coefficient;Δ is the deadzone width.

This slight modification can enhance the system robustness effectively despite of stochastic noise.Meanwhile,this technique can also improve control performance when perturbations exist in Wienertype processes.Other controller design procedures are similar to Subsection 3.4.

4.Simulation Studies

4.1.Application to synthetic Wiener systems

To con firm the obtained results,simulations for synthetic Wiener systems will be conducted in this subsection.Consider the following Wiener system.

The linear subsystem has unstable zero dynamics

Two types of output nonlinearities are considered.They are as follows:(1)the type one nonlinearity Eq.(28).It simulates a saturation characteristic.

(2)The type two nonlinearity Eq.(29).It consists of two nonlinear functions and exhibits distinct features in different sides of the origin.

The above Wiener structures are borrowed from the valve for control of fluid flow[8,28,29]with contrived parameters.The nonlinear characteristics of the nonlinear functions Eqs.(28),(29)are plotted in Fig.2.

Fig.2.Two types of nonlinearities.

A reference trajectory is to be followed by means of the proposed W-AC method.A three-order polynomial function is used to approximate the inverse output nonlinearity.The weighting polynomials are chosen as:P(z?1)=1?0.4z?1,Q(z?1)=0.4(1?z?1),R=0.6.The regress or vector φ(t)is chosen as

and the initial condition isy(0)=v(0)=u(0)=0.

Two groups of simulations results are shown in Figs.3,4 with the system outputy(t),inputu(t)and parameter estimates^θ（t）.It is seen that the W-AC method is applicable for non-minimum phase processes since the closed-loop system is stable,the system output tracks the reference rapidly,and the parameter estimates approach a bounded interval.Interestingly,the control scheme is still applicable even though the controlled Wiener system has complex output nonlinearities Eq.(29).In sum,these results verify the theoretical analysis in Section 4.

Fig.3.Control effects for Simulation 5.1.

Fig.4.Control effects for Simulation 5.2.

4.2.Application to a CSTR system

To show the wide applicability of the proposed method,the continuous stirred tank reactor(CSTR)system given in[10,30,31]will be considered in this subsection.

4.2.1.Process description

This system consists of a constant volume reactor cooled by a single coolant stream flowing in a concurrent fashion,as shown in Fig.5.An irreversible,exothermal reactionA→Boccurs in the tank.The mass and energy balances of the reaction are described as follows:

Fig.5.CSTR system structure.

whereCAis the concentration of the product compoundA;TAis the temperature of the tank;the coolant flow rateqcis the control input;and the other parameters of the system are given in Table 1.

Table 1The explanation of the symbols for CSTR

The control objective is to control the concentrationCA(system output)by manipulating the coolant flow rateqc(control input).Open-loop output responses of this process are given in Fig.6 with+10%and?10%steps.It is obvious that the gain and damping of the system vary widely over the operation[10].This non-symmetrical curve indicates that the system exhibits highly nonlinear behavior.Moreover,static nonlinear mapping of this process is also given in Fig.7.From this figure,it is also seen that this process exhibits distinct nonlinear behaviors during a wide operation range.Interestingly,it has been proven in[7,32,33]that Wiener model structures can properly describe such kind of nonlinear properties when the system dynamics and gains both vary with the operation points.Thus it is reasonable to consider this CSTR process as a typical Wiener system.

4.2.2.Control schemes

The proposed W-AC method is used to control this process which is sampled every 0.1 min.The input and output of the Wiener model are normalized asu=(qc?100)/10 andy=(CA?0.1)/0.1 respectively.The weighting polynomials are chosen asP(z?1)=1?0.65z?1,Q(z?1)=0.035(1?z?1),R=0.35.The constanta(t)=0.5.The regress or vector φ(t)is chosen as

Fig.6.Open-loop output responses of the CSTR system.

Fig.7.Static nonlinear characteristics of the CSTR system.

For comparisons,two other adaptive control methods are applied to the CSTR system:(1)the multiple model based adaptive control(MM-AC)in[25].This method is composed of a linear controller,a nonlinear controller and a switching rule.As shown in[25],the MM-AC method is applicable for the CSTR system Eqs.(30)–(32).However,based on a simple linearized model,the control effects may deteriorate for wide operation regions.(2)The Wiener model structure and inverse neural networks based adaptive control(W-IVE-NN-AC)in[22].This method describes the Wiener system based on online training neural networks.The approximation accuracy is shown to be satisfactory.However,additional modeling errors may be brought in since the control signal is derived by training another inverse network.Meanwhile,the computational burden and the robustness performance are also major concerns.

4.2.3.Set-point tracking

The tracking results for large set-point changes are shown in Fig.8 withCA,TA,qcand(CA*?CA).It can be seen that all these methods can guarantee the system stability in the nominal condition.However,the regulation time of the MM-AC scheme is unattractive due to that a simple linear model incorporated with neural networks can hardly capture the nonlinear dynamics of this process within a short time.What are worse,severe fluctuations happen during 8th and 14th minutes.Relatively speaking,the W-IVE-NN-AC method improves the tracking properties greatly,but it still requires longer settling time than the proposed W-AC method.In sum,the W-AC method leads to the mildest oscillation and achieves the most satisfactory output tracking performance in the nominal condition.

4.2.4.Parametric uncertainties

To test the adaptive ability,assume there exist uncertainties in the system parametersk0,k1which are both associated with the nonlinear termCAe?E/(RTA).Under such a situation,the process is now described as

Fig.8.Set-point tracking results under the nominal condition.

Before the 2nd minute,the system is under the nominal condition and the term Δ1is 1;At the 2nd minute,the term Δ1becomes 1.01.Simulation results are shown in Fig.9 with the regulation results and the parameter estimates of the W-AC method.

Fig.9.Regulation results under the parametric uncertainties.

As expected,the W-IVE-NN-AC approach is sensitive to parametric uncertainties,leads to unsatisfactory oscillation and can hardly ensure zero tracking error.This is caused by the online training of neural networks and the inverse function.Thus its robustness performance is a major concern.Meanwhile,the W-AC and MM-AC methods both lead to good regulation results.The settling time is short and the tracking error is reduced.Furthermore,the parameter estimation results of the W-AC method also verify the reliable convergence properties.

4.2.5.Actuator arrearage faults

Another group of comparison study considers the actuator arrearage faults[34,35].Assume the CSTR system is originally in the nominal condition and operating at the previous state Eq.(36).At the 2nd minute,the valve deadzone phenomenonDdz(·)happens around the pointqc=103.41 L·min?1:

Fig.10.Regulation results under the fault condition.

It can be seen that the W-IVE-NN-AC method leads to severe oscillation and requires long settling time under the fault condition.Similar to the previous results,the W-AC and MM-AC schemes overcome the deadzone effectively.Despite a better transitional performance of the MM-AC scheme,the proposed W-AC method also has reliable robustness performance.

Fig.11.Control effects in a noisy environment.

From Figs.8–10,we conclude that these three methods have distinct scopes of applications:(1)when the operation region varies largely but uncertainties are few,the W-IVE-NN-AC method is applicable.(2)When the process operates at a narrowly range but uncertainties can't be neglected,the MM-AC method is more attractive.(3)The W-AC method has wider applicability than the other two methods and guarantees satisfactory performance in many complex situations.

4.2.6.Stochastic case

To test the applicability in a stochastic case,both measurement noise(which is subject to N(0,0.0012)and added onCA)and process noise(which is subject to N(0,12)and added onTA)are considered.

The modified W-AC method with the robust estimator Eqs.(23),(26)is applied to this process.The coefficients are?=0.5,andΔ=0.05.The control effects are shown in Fig.11 withCA,TA,qcand^θ（t）.It is seen that the modified method is applicable in a noisy case:the closed-loop system is stable,the system output tracks the set-point,the noise rejection ability is acceptable,the system robustness is guaranteed,and the recursive parameter estimates are convergent.

5.Conclusions

This paper proposes a self-tuning control method for Wiener nonlinear systems with uncertainties.The proposed method is derived from a new criterion function,which penalizes the excessive controlaction and accounts for unstable zero dynamics problems.Comparison results show the merits of the proposed control method over some existing ones.In the future,it is expected that the proposed control method could be applied to other industrial processes[36–39].

Appendix A

Before the proof of Lemma 2,the following lemma is given.

Lemma 3.For the Wiener system,the terms{g1(y(t)),g2(y(t)),...gp(y(t)),v(t)}grow at the same rate.

Proof.Please see[12,Lemmas 3–5]for details.

Appendix A.(Proof of Lemma 2).

Proof.From Eqs.(6),(18),(19),we have the input–output dynamics

From the condition(8)and Lemma 3,we know that there exist positive constantsK3,K4,K5,K6,K7,K8,K9,K10such that

Together with(14),Lemma 2 is obtained.

[1]M.Pottmann,R.K.Pearson,Block-oriented NARMAX models with output multiplicities,AIChE J.44(1)(1998)131–140.

[2]J.M.B?ling,D.E.Seborg,J.P.Hespanha,Multi-model adaptive control of a simulated pH neutralization process,Control.Eng.Pract.15(6)(2007)663–672.

[3]J.K.Gruber,D.R.Ramirez,D.Limon,T.Alamo,Computationally efficient nonlinear Min–Max Model Predictive Control based on Volterra series models— Application to a pilot plant,J.Process Control23(4)(2013)543–560.

[4]C.X.Yang,X.F.Yan,A fuzzy-based adaptive genetic algorithm and its case study in chemical engineering,Chin.J.Chem.Eng.19(2)(2011)299–307.

[5]A.Khazaie Poul,M.Soleimani,S.Salahi,Solubility prediction of disperse dyes in supercritical carbon dioxide and ethanol as co-solvent using neural network,Chin.J.Chem.Eng.24(4)(2016)491–498.

[6]P.Wang,C.Yang,X.Tian,D.Huang,Adaptive nonlinear model predictive control using an on-line support vector regression updating strategy,Chin.J.Chem.Eng.22(7)(2014)774–781.

[7]F.Giri,E.W.Bai,Block-oriented nonlinear system identification,Springer,London,2010.

[8]M.?awryńczuk,Nonlinear predictive control for Hammerstein–Wiener systems,ISA Trans.55(2015)49–62.

[9]H.Duwaish,M.Karim,V.Chandrasekar,Use of multilayer feedforward neural networks in identification and control of Wiener model,IEE Proc.Control Theory Appl.143(3)(1996)255–258.

[10]S.Norquay,A.Palazogu,J.Romagnoli,Model predictive control based on Wiener models,Chem.Eng.Sci.53(1)(1998)75–84.

[11]A.Kalafatis,L.Wang,W.Cluett,Linearizing feedforward-feedback control of pH processes based on the Wiener model,J.Process Control15(2005)103–112.

[12]Z.Zhusubaliyev,A.Medvedev,M.Silva,Bifurcation analysis of PID-controlled neuromuscular blockade in closed-loop anesthesia,J.Process Control25(2015)152–163.

[13]M.?awryńczuk,Computationally efficient nonlinear predictive control based on neural Wiener models,Neurocomputing74(2010)401–417.

[14]H.Bloemen,C.Chou,T.van den Boom,V.Verdult,M.Verhaegen,T.Backx,Wiener model identification and predictive control for dual composition control of a distillation column,J.Process Control11(6)(2001)601–620.

[15]H.Zhang,G.Chen,Z.Chen,A novel dual-mode predictive control strategy for constrained Wiener systems,Int.J.Robust Nonlinear Control20(9)(2010)975–986.

[16]S.Biagiola,J.Figueroa,Robust model predictive control of Wiener systems,Int.J.Control.84(3)(2011)432–444.

[17]F.Khani,M.Haeri,Robust model predictive control of nonlinear processes represented by Wiener or Hammerstein models,Chem.Eng.Sci.129(2015)223–231.

[18]K.Kim,E.Ríos-Patrónc,R.Braatza,Robust nonlinear internal model control of stable Wiener systems,J.Process Control22(2012)1468–1477.

[19]G.Gregor?i?,G.Lightbody,Nonlinear model-based control of highly nonlinear processes,Comput.Chem.Eng.34(2010)1268–1281.

[20]G.Pajunen,Adaptive control of Wiener type-nonlinear systems,Automatica28(4)(1992)781–785.

[21]J.Figueroa,J.Cousseau,S.Werner,T.Laakso,Adaptive control of a Wiener type system:application of a pH neutralization reactor,Int.J.Control.80(2)(2007)231–240.

[22]J.Peng,R.Dubay,J.Hernandez,M.Abu-Ayyad,A Wiener neural network-based identification and adaptive generalized predictive control for nonlinear SISO systems,Ind.Eng.Chem.Res.50(12)(2011)7388–7397.

[23]B.Zhang,Z.Z.Mao,Adaptive control of stochastic Hammerstein–Wiener non-linear systems with measurement noise,Int.J.Syst.Sci.47(1)(2016)162–178.

[24]D.W.Clarke,M.A.Phil,P.J.Gawthrop,Self-tuning controller,Proc.Inst.Electr.Eng.122(1975)929–934.

[25]Y.Fu,T.Y.Chai,Indirect self-tuning control using multiple models for non-af fine nonlinear systems,Int.J.Control.84(6)(2011)1031–1040.

[26]G.C.Goodwin,K.S.Sin,Adaptive filtering,prediction and control,Prentice Hall,Englewood Cliffs,New Jersey,1984.

[27]A.Hagenblad,L.Ljung,A.Wills,Maximum likelihood identification of Wiener models,Automatica44(2008)2697–2705.

[28]T.Wigren,Recursive prediction error identification using the nonlinear Wiener model,Automatica29(4)(1993)1011–1025.

[29]Y.Fang,T.Chow,Orthogonal wavelet neural networks applying to identification of Wiener model,IEEE Trans.Circuits Syst.I Fundam.Theory Appl.47(4)(2000)591–593.

[30]G.Lightbody,G.Irwin,Direct neural model reference adaptive control,IEE Proc.Control Theory Appl.142(1)(1995)31–43.

[31]S.Ge,C.Hang,T.Zhang,Nonlinear adaptive control using neural networks and its application to CSTR systems,J.Process Control9(1998)313–323.

[32]J.Hahn,T.F.Edgar,A gramian based approach to nonlinearity quantification and model classification,Ind.Eng.Chem.Res.40(2001)5724–5731.

[33]L.A.Aguirre,M.C.S.Coelho,M.V.Corrêa,On the interpretation and practice of dynamical differences between Hammerstein and Wiener models,IEE Proc.Control Theory Appl.152(4)(2005)349–356.

[34]C.Kravaris,S.Palanki,Robust nonlinear state feedback under structured uncertainty,AIChE J.34(7)(1988)1119–1127.

[35]Y.Wang,D.Zhou,F.Gao,Generalized predictive control of linear systems with actuator arrearage faults,J.Process Control19(2009)803–815.

[36]Z.Zou,D.Zhao,X.Liu,Y.Guo,C.Guan,W.Feng,N.Guo,Pole-placement self-tuning control of nonlinear Hammerstein system and its application to pH process control,Chin.J.Chem.Eng.23(8)(2015)1364–1368.

[37]P.Cao,X.Luo,Soft sensor model derived from Wiener model structure:modeling and identification,Chin.J.Chem.Eng.22(5)(2014)538–548.

[38]Y.Wang,X.Pang,Z.Piao,J.Fang,J.Fu,T.Chai,Intelligent decoupling PID control for the forced-circulation evaporation system,Chin.J.Chem.Eng.23(12)(2015)2075–2086.

[39]Q.Li,D.Li,L.Cao,Closed-loop identification of systems using hybrid box—Jenkins structure and its application to PID tuning,Chin.J.Chem.Eng.23(12)(2015)1997–2004.