Multi-model adaptive control of a simulated pH neutralization processIntroductionMulti-model adaptive PID controlProcess model and unmeasured disturbancesForgetting of past dataController designSimulationsThe nonlinear pH model simulationConclusionsAcknowledgmentsReferencesUNCORRECTED PROOFControl Engineering Practice ] (]]]]) ]]]–]]]Multi-model adaptive control of a simulated pH neutralization processJari M. Bo¨linga,, Dale E. Seborgb, Joa˜o P. HespanhacaProcess Control Laboratory, A˚bo Akademi University, FinlandbDepartment of Chemical Engineering, University of California, Santa Barbara, USAcDepartment of Electrical and Computer Engineering, University of California, Santa Barbara, USAReceived 27 February 2006; accepted 24 November 2006AbstractA multi-model adaptive PID controller is developed and evaluated in a simulation study for a nonlinear pH neutralization process.The performance and robustness characteristics of the multi-model controller are compared to those for conventional PID controllersand an alternative ‘‘multi-model interpolation’’ controller.r 2006 Published by Elsevier Ltd.Keywords: Adaptive control; Switching control; Multiple models; PID control1. IntroductionMany indust rial processes inevitably change over timefor a variety of reasons that include: equipment changes,different operating conditions, or changing economicconditions. Consequently, a fundamental control problemis how to provide effective control of complex processeswhere significant process changes can occur, but cannot bemeasured or anticipated. The conventional solution isconservative controller tuning for worst case conditions.However, this approach can result in poor control systemperformance for more typical conditions. Alternatively,adaptive control strategies are available where the con-troller parameters and/or control structure are modifiedonline as conditions change (A˚stro¨m & Wittenmark, 1995).This paper is concerned with a special class of adaptivecontrol strategies referred to as switching control or multi-model adaptive control ( Angeli & Mosca, 2002; Freidovi ch& Khalil, 2003; Hespanha, 2001; Johansen & Murray-Smith, 1997; Morse, 1996; Narendra & Balakrishnan,1997). The motivation for multi-model control is that formany complex technical processes, the local behavior canbe captured at least approximately by a set of relativelysimple models. Also, a corresponding feedback controllercan be designed for each individual model. For thesesituations, an adaptive control approach based on selectingthe best model (and controller) for the current conditionsprovides a promising approach. Selection of the perfor-mance criterion and switching strategy are key designissues.In multi-model adaptive control, a bank of candidatemodels (and/or controllers) are specified a priori. Then asupervisory controller selects the most appropriate model(or controller) for the current conditions. For each model,a suitable controller can be designed off-line. The onlinecontroller switching is based on the performance evalua-tion of the bank of models (and/or controllers). Controlproblems involving transitions between know n operatingregimes are readily handled by a multi-model approach(Johansen & Murray-Smith, 1997). Multi-model control isalso applicable to more general control problems whereoperating regimes cannot be determined a priori (Hespan-ha, 2001; Narendra & Balakrishnan, 1997). For example,the capabilities of multi-model control have been success-fully demonstrated for drug infusion control wherevariability and unpredictability are key issues (He, Kauf-man, & Roy, 1986; Schott & Beque tte, 1997). Otherreported applications include control of pH (Dougherty &Cooper, 2003b; Gala´n, Romagnoli, & Palazoglu, 2004),distillation columns (Dougherty & Cooper, 2003a; Rodri-guez, Romagnoli, & Goodwin, 2003), power systemsARTICLE IN PRESSwww.elsevier.com/locate/conengprac135791113151719212325272931333537394143454749515355575961636567697173757779818385878:07f=WðJul162004Þþ modelCONPRA : 2141Prod:Type:FTPpp:1210ðcol:fig::NILÞED:SwarnaR:PAGN:sree SCAN:0967-0661/$ - see front matter r 2006 Published by Elsevier Ltd.doi:10.1016/j.conengprac.2006.11.008Corresponding author. Tel.: +358 2 2153238; fax: +358 2 2154479.E-mail address: jboling@abo.fi (J.M. Bo¨ling).Please cite this article as: Bo¨ling, J. M., et al. Multi-model adaptive control of a simulated pH neutralization process. Control Engineering Practice,(2006), doi:10.1016/j.conengprac.2006.11.008UNCORRECTED PROOF(Chadouri, Majumder, & Pal, 2004), and chemical reactors(Tian & Hoo, 2005). In (Rodrigues, Theilliol, Adam-Medina, & Sauter, 2006) multiple models are used for faultdetection and isolation, with the models representingnormal or faulty situati ons.2. Multi-model adaptive PID controlIn this paper, a multi-model adaptive strategy for PIDcontrollers that is based on a set of simple linear dynamicmodels is considered. Each model has the same structurebut different values of the model parameters. Grids ofparameter values are assigned based on an assumed rangefor each model parameter. The ranges can be determinedfrom a priori knowledge of expected operating conditions.For example, ranges for process gains and time constantscan be specified based on physical knowledge such as themaximum and minimum values of temperatures andproduct flow rate. The grid spacing does not have to beconstant.In Section 3 the multi-model strategy is compared to anovel adapti ve control strategy where the controller isautomatically re-tuned after poor performance is detected(Wojsznis & Blevins, 2002; Wojsznis, Blevins, & Wojsznis,2003). The re-tuning is based on re-estimating modelparameters from recent input/output data.A block diagra m for the multi-model control strategyconsidered in this paper is shown in Fig. 1, where u is theinput, y is the output, d is the unmeasured dist urbance, andyspis the setpoint. The model parameters^y for the currentconditions are determined by calculating a performanceindex pifor each model i . For example, the following low-pass filtered squared prediction error can be used as suchan index:piðkÞ¼lpiðk  1Þþð1  lÞe2iðkÞ; i ¼ 1; 2; ...; M. (1)Here M is the number of models in the model bank, andeiðkÞ¼yðkÞyiðkÞ denotes the one-step prediction errorfor model i at time k. The filter constant 0plp1 can beinterpreted as a forgetting factor, as will be discussed later.At each time k, the