PREDICTIVE CONTROL OF NONLINEAR SYSTEMS. Received February 2008; accepted May 2008

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1 ICIC Express Letters ICIC International c 2008 ISSN X Volume 2, Number 3, September 2008 pp PREDICTIVE CONTROL OF NONLINEAR SYSTEMS Martin Janík, Eva Miklovičová and Marián Mrosko Faculty of Electrical Engineering and Information Technology Slovak University of Technology Ilkovičova 3, Bratislava, Slovak Republic {eva.miklovicova; marian.mrosko}@stuba.sk Received February 2008; accepted May 2008 Abstract. The paper deals with the predictive control design for nonlinear systems. When linear model based controller is employed on nonlinear processes, its application is limited to relatively small operating regions. Adding an adaptive mechanism to linear predictive controller is one of the possible solutions to this problem. Two adaptive strategies are presented in the paper: multiple model adaptive control and self-tuning control strategy. Both controllers are based on dynamic matrix control design which allows for the incorporation of physical constraints directly in the objective function. The capabilities of the proposed control strategies are demonstrated through simulations. Keywords: Nonlinear system, Model predictive control, Multiple models, Adaptive control 1. Introduction. Model predictive control (MPC) has gained wide popularity in industrial process control due to the possibility of reformulating the control problem as an optimisation problem in which physical constraints can be allowed for [1]. Dynamic matrix control is the most popular MPC algorithm used in the process industry today [2]. The reason is its ability to obtain good performances starting from rather intuitive design principles and simple linear models [3,4]. The quadratic performance objective function possibly subject to a set of linear inequalities representing plant constraints is minimised over a prediction horizon to compute the optimal controller output moves. Many industrial processes have nonlinear dynamics such as operating regions with distinctly different input-output behaviour. The most common way to control nonlinear plants is simply by ignoring nonlinearities and applying the linear controller, which is often sufficient for applications with a single operating point. Such control strategy may result in considerable deterioration of the control performances in situations where the operating points change frequently and span a wide range of nonlinear process dynamics. Much recent research focus on nonlinear MPC where a nonlinear plant model is used for prediction [5]. However, predictive control problem with linear model, linear constraints and quadratic objective function results in a convex optimisation problem easily solved using quadratic programming. With nonlinear models the optimisation problem is no longer convex and convergence of the global optimum is not guaranteed. Also for large scale systems the computation time may be a significant fraction of the sample interval. To maintain the controllerperformances over a wide range of operating levels, a multiple model adaptive control (MMAC) strategy for DMC has been proposed [6]. The method employs multiple linear DMC controllers, eachwiththeirownstepresponsemodeldescribing process dynamics at a specific operation level, while the final controller output is an interpolation of the individual controller outputs weighted based on the current value of the measured process variable. However, the input/output constraints have not been handled in the control design. In our paper a modification of the MMAC strategy based 239

2 240 M. JANÍK, E. MIKLOVIČOVÁ AND M. MROSKO on the DMC design is proposed in which the plant constraints are taken into account explicitly. Self-tuning control (STC) represents an alternative way to deal with the plant nonlinearities. This control approach is based on the certainty equivalence principle. It consists in solving the online control design problem by combining a parameter estimator with the control design procedure. In this paper we propose a self-tuning controller in which the DMC control design is used and the plant parameters are updated each sampling period using the least squares method. Both proposed control algorithms are evaluated in simulations using a simple example. Resulting control performances are compared to those of the non-adaptive DMC controller. 2. Dynamic Matrix Control. The DMC has been discussed extensively by past researchers and is briefly summarized here for convenience of reader. The DMC objective consists in minimizing the following linear quadratic cost function: J (t, ph, ch, ρ) = ³R (t + ph) Ŷ (t + ph/t) T ³R (t + ph) Ŷ (t + ph/t) (1) +ρ U (t + ch 1) T U (t + ch 1) with Ŷ (t + ph/t) =[ŷ (t +1/t),...,ŷ (t + ph/t)] T R (t + ph) =[r (t +1),...,r(t + ph)] T U (t + ch 1) = [ u (t/t),..., u (t + ch 1/t)] T where ŷ (t + j/t) isaj-step ahead prediction of y(t) based on data available in time t; r(t) represents a reference signal; sh, ph, ch ph denote starting horizon, prediction horizon and control horizon, respectively; ρ is the nonnegative control weighting scalar. The output j-step ahead prediction is based on the non-parametric step response (SR) model of the process that is composed of the output signal samples responding to a unit input step. Three types of process constraints can be enforced in DMC design by formulating them as linear inequalities, namely manipulated variable constraints, manipulated variable rate constraints and output variable constraints: u min u (t + j/t) u max j =0, 1,...,ch 1 (2a) u (t + j/t) u max j =0, 1,...,ch 1 (2b) y min ŷ (t + j/t) y max j =0, 1,...,ph (2c) These constraints can be combined into one convenient expression: C U U (t + ch 1) c (t) (3) and the DMC problem with the quadratic objective (1) subject to a set of linear inequalities (2) can be converted to a standard quadratic program (QP) form: min x T Hx g T x C x where x : U (t + ch 1) (4) x c The implementation of DMC is done in a receding horizon fashion. This implies that the QP is solved in each sampling period resulting the vector of ch future control moves U (t + ch 1), but only the first component of this vector will be used for control at the time t and the whole process will be repeated in the next sampling period with updated data.

3 ICIC EXPRESS LETTERS, VOL.2, NO.3, Multiple Model Adaptive Strategy. The key idea of this approach is to construct a small set of SR models that span the range of expected system operation. By combining these models to form the nonlinear approximation of the plant, the true plant behaviour can be approached. The accuracy of the nonlinear approximation can be increased by combining more models. However, this can be expensive because each model requires the collection of plant data at a different operation level. Thus the number of process models used in a particular implementation is a decision to be made by designer. Then the linear DMC controller described above is designed for each level of operation. All controllers use the same value of sampling period T andcontroldesignparameters: T =min(t j ) for j =1,...,k (5a) ph =max(ph j ) for j =1,..., k (5b) ch =max(ch j ) for j =1,...,k (5c) where k is the number of SR models. This ensures that when the process is operating in the level with fastest dynamic, the sample time is fast enough to capture the process behaviour and the horizons will always be long enough to capture the slowest dynamics in the range of operation. The final controller output is a weighted average of each controller output move: kx u MMAC (t) = x j (t) u j (t) (6) j=1 if y (t) y 3 then x 1 =0;x 2 =0;x 3 =1 if y 2 <y(t) <y 3 then x 1 =0;x 2 =1 x 3 ; x 3 = y (t) y 2 y 3 y 2 if y 1 <y(t) <y 2 then x 1 =1 x 2 ; x 3 = y (t) y (7) 1 ; x 3 =0 y 2 y 1 if y (t) y 1 then x 1 =1;x 2 =0;x 3 =0. where y j, j =1, 2, 3 are the values of output variable at selected operating levels. The weighting factors x j (t) are from the range of [0, 1]. For clarity, expressions (7) involve designing and combining three non-adaptive DMC controllers (k = 3). However, they can easily be expanded to include as many local linear controllers as the designer would like. The value of the MMAC controller output finally implemented is calculated as follows: u MMAC (t) =u MMAC (t 1) + u MMAC (t) (8) 4. Self-tuning Strategy. In self-tuning controllers recursive identification is used to update the parameters of the process model as a new plant measurements become available at each sampling period. Based on the updated model parameters, the controller parameters are subsequently recalculated and new value of controller output at time t is obtained. In the proposed approach the system dynamics is first approximated by parametric model (transfer function) using standard least squares technique and the parametric model is then used to generate the dynamic matrix needed for DMC design. 5. Example. Consider a simple nonlinear system a gravity drained cylindrical tank described by its material balance: S dh dt = Q 1 μs 0 p 2gh (9) where h is the liquid level, Q 1 is the inlet flow rate, S is the tank cross section area, S 0 is the outlet cross section area and μ is a coefficient in the range of [0, 1]. The parameter values are in Table 1.

4 242 M. JANÍK, E. MIKLOVIČOVÁ AND M. MROSKO Table 1. Tank parameter values S=1m 2 S 0 = 100cm 2 μ =0.62 ρ = 1000kgm 3 g=9.81ms 2 h max =2m The controller output manipulates the inlet flow rate into the tank and the output variable is the liquid level of the tank. The maximum possible value of the inlet flow rate is m 3 /s. The second inlet flow rate Q 3 =0.004m 3 /s acts as a disturbance from the time 1500 s. The process displays a nonlinear behaviour in that the process gain and time vary significantly over the range of operation. Three DMC controllers have been designed to maintain the liquid level set-point tracking and input flow disturbance rejection. The non-adaptive DMC has been based on the SR model associated with the operating level corresponding to the input flow rate Q 1 = 0.02 m 3 /s. In the MMAC DMC control six SR models have been used to span the range of operation of the tank. In the self-tuning DMC control the parameters of the first order transfer function have been approximated and used for the dynamic matrix calculation. The tuning parameters for the three controllers are summarized in Table 2. Table 2. Tuning parameters Method Parameters T ph ch ρ 1 ρ 2 ρ 3 ρ 4 ρ 5 ρ 6 DMC MMAC STC The performances of the proposed controllers are compared in Figure 1 to Figure 3. It can be seen that both adaptive controllers outperform the non-adaptive one. The STC exhibit not only the best tracking and disturbance rejection capabilities but also requires less computation time than the MMAC. The non-adaptive version has least computation time and also acceptable control performances can be obtained but it requires suitable choice of the operation point for the SR model identification as well as careful controller parameters tuning. Figure 1. Simulation results

5 ICIC EXPRESS LETTERS, VOL.2, NO.3, Figure 2. Simulation results - detail Figure 3. Simulation results - detail 6. Conclusions. Two adaptive control strategies for predictive control of nonlinear systems which significantly change their dynamics with respect to the operating level have been proposed. Both adaptive controllers are based on linear DMC design that allows incorporating input/output constraints explicitly into the design procedure. The benefits of these strategies have been demonstrated through simple simulation example. Both adaptive controllers are able to maintain better set point tracking and disturbance rejection comparing to the non-adaptive DMC controller. The best results have been obtained using the STC version. The MMAC design requires suitable choice of the number of process models and data collection at the corresponding operation levels. In STC version correct estimation of plant model parameters is an important issue. For all controllers also the proper choice of sampling period and control design parameters is essential. Acknowledgment. This work has been supported by the Slovak Scientific Grant Agency, Grant No.1/3841/06.

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