Neuro-predictive control based self-tuning of PID controllers

Transcription

1 Nero-predictive control based self-tning of PID controllers Corneli Lazar, Sorin Carari, Dragna Vrabie, Maris Kloetzer Gh. Asachi Technical Universit of Iasi, Department of Atomatic Control Blvd. D. Mangeron 53A, 75 Iasi, Romania Abstract: In this paper we present a new self-tning procedre for PID controllers based on nero-predictive control. A finite horizon optimal control problem is solved on-line, permitting to calclate the tning parameters of the PID controller. The proposed method is implemented on a level-flow pilot plant and a comparison with conventional ato-tning methods is also given. 1. Introdction Althogh most of the indstrial processes are complex nonlinear sstems, the are still controlled with classical PID control strctres, which are tned to give good reslts onl arond a fixed operating point. Under these circmstances, in order to obtain the optimal response over the entire operating range, on-line adaptation or selftning of the controller is reqired, and several methods have been proposed in the last decade, e.g. [1], [2], [3]. In [1], the existent tpes of adaptive techniqes are classified based on the fact that if the process dnamics are varing, then the controller shold compensate these variations b adapting its parameters. There are two tpes of process dnamics variations: predictable and npredictable. The predictable ones are tpicall cased b nonlinearities and can be handled sing a gain schedle, which means that the controller parameters are fond for different operating conditions with an ato-tning procedre that is emploed thereafter to bild a schedle. In this paper, a new self-tning method for PID controllers designed to control processes with predictable dnamics variations is presented. The gain schedling principle is replaced b sing a neral network based model that is capable to captre the predictable dnamics variations of the process. The neral network model is also sed to develop a neral strctre that predicts the ftre control error cased b process dnamics variations. The controller tning parameters are calclated solving a finite horizon optimal control problem that minimizes the predicted control error. Real-life experimental reslts are given for a level-flow pilot plant, which demonstrate the practical benefits of this self-tning method. 2. Description of the self-tning procedre The proposed self-tning approach is based on two parallel control strctres (see Fig. 1) that are snchronized with the reference clock of the predictable dnamics

2 process in closed-loop with a PID controller. r(k+i) e pred (k+i) + - Adaptive controller (k+i) Neral predictors (k+i) Nero-predictive control loop r + - q opt Controller x Process Real-time control loop Figre 1: Nero-predictive strctre The pper strctre ses a predictive control loop consisting in a neral predictor and a PID controller with adaptive tning parameters. The predictive strctre, with the sampling rate T p, works faster than the real-time control loop sppling the predicted control error over a finite ftre time horizon. The tning parameters are calclated at each sample time instant throgh the minimization of the predicted control error and the obtained vales are sed to pdate the tning parameters of the real-time control loop. Ths, the controller parameters are adapted based on the predictive optimization of the control sstem behavior and the desired performances can be achieved over the entire operating range. 2.1 Process neral model Nero-predictive control loop contains a neral network model, which models the real process with predictable dnamic variations. The se of neral networks for nonlinear process modeling and identification is stified b their capabilit to approximate nknown non-linear sstems. A nonlinear model that incldes a large class of non-linear processes is the following NARMAX model: k ( ) = f[ k ( 1),..., k ( n), k ( d 1),..., k ( d m)] (1) where f(.) is some nonlinear fnction, d is the dead time, n and m are the orders of the nonlinear sstem model, and being the inpt and the otpt of the process. A neral network based model, NNARMAX, corresponding to the NARMAX model, ma be obtained b adsting the weights of a mlti-laer perceptron architectre with adeqatel delaed inpts [4]. The neral network otpt will be given b: N ( k) = f [ ( k d 1), ( ], (2) where f N denotes the inpt-otpt transfer fnction of the neral network, which replaces the non-linear fnction f in (1) and, (k-d-1) and (k-1) have the following strctre: T ( k d 1) = [ ( k d 1) ( k d 2)... ( k d m)] T ( = [ ( ( k 2)... ( k n)]. (3) For a two laer network, the following expression is obtained from eqation (2):

3 n k ( ) = wσ ( w ( k d 1) + w ( k 1) + b) + b, (4) = 1 where n is the nmber of nerons in the hidden laer, σ is the activation fnction for the -th neron from the hidden laer, w the weight vector for the -th neron with respect to the inpts stored in (k-d-1), w the weight vector for the -th neron with respect to the inpts stored in (k-1), b the bias for the -th neron from the hidden laer, w the weight for the otpt laer corresponding to the -th neron from the hidden laer and b the bias for the otpt laer. Sch strctres with a single hidden laer are considered satisfactor for most of the cases. Since all the indstrial processes are working in closed-loop, a closed-loop identification method has been sed to obtain the neral model of the process. In order to captre all the nonlinear dnamics of the process, the training data had to be attained arond several different operating points sch that the entire variation range of the process otpt to be covered. For this reason, a stepwise reference was chosen and then smmed with a psedo random binar signal generated with a shifting register [5]. 2.2 Nero-predictive control loop In order to obtain the predictable dnamics variations at the time instants k, a neral predictor based on the neral-based model of the process was sed. A seqential algorithm based on the knowledge of crrent vales of and together with the neral network sstem model gives the i-step ahead neral predictor: n ( k + i) = w σ ( w ( k d + i 1) + w ( k + i 1) + b ) + b. (5) = 1 The ftre control (k-d+i-1) from (5) is obtained rnning the nero-predictive control loop. Ths, at time instant k, the predicted otpt (k+i) is determined, for i = N 1, N 2 where N 1 and N 2 are the prediction horizons. If T p is the sampling time with which the predictive control loop operates, this mst satisf: ( N2 N1) Tp << T. Placing the neral model of the process to operate in the nero-predictive control loop allows for transferring the crrent state x of the process to the neral predictor (Fig. 1) at each time instant k. Ths, at each time instant k, the predicted behavior of the process is obtained in the vector form: T pred = [ ( k + N1 ) ( k + N1 + 1)... ( k + N 2 )]. (6) The process otpt pred, predicted b the neral predictor, is sed to calclate the predicted control error based on the controller set-point. Considering the discrete form of a PID controller, k) = ( + q e( k) + q e( + q e( k 2), (7) ( 1 2

4 and the model (5), ields the following eqation for the predicted control error: n e pred ( k + i) = w ( ( k d i 1) ( k i 1) b ) b r( k + i) σ w + + w + + +, (8) = 1 where the vector (k-d+i-1) is a fnction of the tning parameters vector q=[q q 1 q 2 ]. Minimizing the cost fnction N J = e pred ( k + i) (9) 2 i= N1 with respect to the variable q, ields the optimal tning parameters q opt, which compensate the process predictable dnamics variations. At the time instant k+1, the tning parameters q opt are transferred to the real-time control loop. The self-tning method has the advantage that it does not impl a pre-tne phase becase the vector q opt is available after the first sampling period. 3. Pilot plant stdies The nero-predictive self-tning approach for PID controllers was tested on a level-flow pilot plant. The schematic diagram of the pilot plant is presented in Fig. 2. r 1 1=r 2 LC FC LT FT h q i A A 1 water LC - Level Controller FC - Flow Controller LT - Level Transdcer FT - Flow Transdcer A - cross section area A 1- circlar orifice area q i - flow rate Figre 2: Schematic diagram of the level-flow pilot plant The level is controlled sing a cascade control strctre that has as internal variable the feed water flow. The inner loop controls the feed water flow and reects the distrbances cased b the pressre variations in the water pipe. The oter loop controller, a PI controller, ields the feed water flow reference signal r 2 for the inner loop based on the received water level signal. The tank and the inner loop represent the plant for the oter controller, which was modelled with a neral network in order to implement the developed self-tning method for the level controller. The following non-linear model is available for the tank: dh A = qi Cd A1 2gh. (1) dt The model parameters are: A = 23.4 cm 2, A 1 = 2.26 cm 2, h max = 13.5 cm, C d =.6.

5 3.1 Neral model of the plant In order to estimate the parameters of the neral model, a training seqence was bilt sch that the process otpt explores its whole operating range. Ths, a stepwise reference smmed with a psedo random binar signal was applied to the real time control loop and, b monitoring the control signal 1 and the process otpt 1, a training seqence was obtained. Using the training seqence, a two laer neral network was trained off-line. Model parameters m, n and d were estimated based on the collected inpt-otpt data and on the phsical strctre of the process (the inner flow loop and the tank). With a sampling rate of 2sec, it was fond that the process has a dela d=2 and m=2, n=2. For the training and the validation of the neral network that models the process, the software instrments presented in [6] were sed. In Figre 3, the reslts of a closed-loop experimental validation of the neral model are plotted. 8 6 [%] 4 2 neral model otpt process otpt reference Time [sec] Figre 3: Process neral model validation 3.2 Adaptive procedre reslts The parameters q and q 1 of the PI controller were obtained b minimizing an obective fnction sing an optimization approach. The cost fnction was taken as the mean sqared prediction error over a finite prediction horizon, as in eqation (9). The neral network model identified in the previos section was sed to predict the ftre otpts. In order to avoid the satration of the actator, the minimization took into accont the constraints imposed on the control inpt and controller parameters. In order to do a fair comparison with the existing ato-tning methods, the PI level controller was first tned sing the rela method of Astrom and Hagglnd [3]. Figre 4 shows the reslting real-time closed-loop stepwise responses obtained with fixed tning parameters of the PI controller and with continos adaptation of tning parameters based on the self-tning method presented in this paper. In the same figre, the control inpts are depicted together with the tning parameters of the PI controller. As seen in the figre, the fixed parameters controller gave a slggish control response. With the nero-predictive tning approach, the controller had continos

6 adaptation of the tning parameters reslting in a faster control. Process otpt [%] reference adaptive constant Control inpt [%] PI parameters adaptive constant Time [sec] Figre 4: Comparative reslts and parameters for the adaptive discrete PI controller q q1 4. Conclsions A nero-predictive control based self-tning procedre for PID controllers has been developed. The main advantage of the method consists in the on-line adaptation of the controller parameters and in the possibilit to track different process operating regimes. The proposed method has been implemented on a benchmark real-life sstem with good reslts and a comparison with a classical ato-tning method for PID controllers has also been given. References 1. Hagglnd T., Astrom K.J., Indstrial Adaptive Controllers Based on Freqenc Response Techniqes, Atomatica, vol.27, no.4, (1991). 2. Cheng-Ching Y, Atotning of PID Controllers, Springer-Verlag, London (1999). 3. Astrom K.J. Hagglnd T., PID Controllers: Theor, Design and Tning, Instrment Societ of America: Research Triangle Park, (1995). 4. Chen S, Billings S.A. and P.M. Grant, Nonlinear sstem identification sing neral networks, Int. J. Control, vol 49, no.3, (199). 5. Landa I.D., Sstem Identification and Control Design, Prentice Hall, (199). 6. Norgaard, M, Neral Network Based Sstem Identification Toolbox, Technical Report -E- 891, Department of Atomation, Technical Universit of Denmark, (2).