NEURAL CONTROLLERS FOR NONLINEAR SYSTEMS IN MATLAB

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1 NEURAL CONTROLLERS FOR NONLINEAR SYSTEMS IN MATLAB S.Kajan Institte of Control and Indstrial Informatics, Faclt of Electrical Engineering and Information Technolog, Slovak Universit of Technolog in Bratislava, Slovak Repblic Abstract This paper deals ith design of neral controllers for nonlinear sstems control. For the prpose of neral control strctres a direct and inverse neral model of a nonlinear dnamic sstem sing three-laer perceptron netork as created. These neral models ere sed in folloing control strctres: direct inverse control, internal model control and predictive control. The performance tests for particlar controllers ere realized in the simlation environment Matlab/Simlink sing selected tpes of nonlinear dnamic processes. Introdction For prposes of nonlinear sstem control, it is important to have accrate models. Thanks to a ver good approximating abilit of mlti-laer perceptron netorks (MLP) e are able to create accrate neral models of nonlinear processes. For prpose of control of nonlinear dnamic sstems, several control strctres sing neral models and inverse neral models have been developed, and this article deals ith them. Neral models Neral model of process is represented b three-laer artificial neral netork of MLP tpe. The objective of MLP netork is to approximate the relation of sstem otpt in k-th step on basis of past vales of sstem otpt and inpt and ths get feed-forard neral model. Than e can describe nonlinear dnamic sstem b folloing model: ˆ ( k + ) = f [ ( k), ( k ), K, ( k n + ), ( k), ( k ), K, ( k m + )] () here - process inpt, - process otpt, n- order of process otpt, m- order of process inpt, f- nonlinear fnction, k- discrete time ( t = k * Tvz, T vz is sampling period). For prpose of sstem control, e sed inverse neral model, hich e get b exact inversion of model from eqation () b expressing controller otpt (k): ( k) = f [ ˆ( k + ), ( k), ( k ), K, ( k n + ), ( k ), K, ( k m + )] () pertrbance P r o c e s s p Model m + - L e a r n i n g alg orithm Fig. The process modelling block scheme sing artificial neral netork A block scheme of the artificial neral netork process model is in Fig.. []. The neral model is located parallel to process, and prediction error is sed as netork training signal for the learning algorithm. The Levemberg-Marqart method has been sed for training the MLP netork [].

2 3 Neral controllers for nonlinear sstems Neral controllers for control of nonlinear processes are sing inverse neral models. We ve chosen folloing neral controllers to compare the control performance of nonlinear processes: - Robst direct inverse neral control [4], - Internal model control ith neral models [], - Predictive neral control [3, 5, 6] 3. Robst direct inverse neral control In direct inverse control, inverse model described b eqation () is sed. We sed predicted vale ˆ ( k +) knon from the set of inpt-otpt data hile training the inverse model. In closed-loop circit the predicted vale ˆ ( k +) is replaced b reference vale ( k +), and thereb e get closed-loop neral controller ot of the inverse model. In Fig. is shon a block scheme of direct inverse control. (k+) (k-i) i=0,,,n- (k-j) Inverse Neral Controller (k) Sstem (k+) j=,,m- B (bias) Fig. The scheme of direct inverse neral control Direct inverse control as shon in Fig. can not remove permanent reference error hen the sstem parameters changed, or disrption occrs. Therefore an adapting block hich adapts neron threshold vale in otpt laer of neral netork is added to control. [4]. (k-j) j=,,m- (k-i) i=0,,,n- Inverse Neral Controller (k) Sstem (k+) (k+) B (bias) B = + β ( ) Fig. 3 The scheme of robst direct inverse neral control

3 In Fig.3 is displaed a block scheme of robst inverse control, here the adapter is as a simple integrator in form: ( ) B = + β (3) here β is adaptive parameter from range 0 to. 3. Internal model control ith neral models The IMC control strctre ses inverse neral model of sstem as a controller. It ses negative feedback of difference beteen the sstem otpt and otpt of neral model to sppress the reference error. (see Fig.4). A filter to attenate step changes of differences ma be connected in the negative feedback. (k+) - d f (k+) (k-i) i=0,,,n- (k-j) Inverse Neral Controller (k) Sstem (k+) j=,,m- Neral Model - ŷ(k+) d(k+) Filter Fig. 4 The scheme of internal model control ith neral models 3.3 Predictive neral control Connection of this control strctre is shon in Fig.5. It ses direct neral model to predict ftre otpts of process assming the control variable ill be. This control variable is optimized in ˆ k + i step reaches reference vale. each step of control process, so that predicted vale of otpt ( ) Optimization Sstem ư Neral Model ŷ Fig. 5 The scheme of predictive neral control

4 4 Simlation reslts Testing of control qalit of selected nonlinear sstems ith several tpes of neral controllers as realized in simlation environment of Matlab Simlink. For the prpose of testing, e sed simlation models of nonlinear dnamic sstems described b folloing differential eqations, here is otpt and is inpt of the sstem: Sstem A). ''+0.7' =0 (3) Sstem B). ''+'( +) ^-0.=0 (4) Sstem C). ''+5' (+)=0 (5) Listed nonlinear sstems have nonlinear transfer characteristic and dnamics of the sstem changes according to operating point, here the range of sstem inpt is 0 to 0. In Matlab environment, e sed simlation models of sstems to generate training and testing data to create neral and inverse neral model of sstem, described b eqations () and (), here e pt m and n parameters eqal. Neral model as created sing Neral Toolbox, here e sed MLP netork ith one hidden laer ith 9 nerons and tansig activation fnction for modelling. We sed Levemberg-Marqart method for training of the MLP netork []. We created a simlation scheme for each tpe of neral controller (see Fig. 6 to 9). Simlation scheme for predictive control has been created b modifing an existing scheme in Neral toolbox of Matlab [6]. For each sstem, e performed a simlation of time responses of control signal and sstem otpt for step changes of reference vale. Time responses of sstem otpt and reference vale of some sstems for individal tpes of control are depicted in Figres 0a), a) and a). Time responses of sstem otpt and reference vale of some sstems ith pertrbation occrring in time t=30s and t=50s and vale of 0. are depicted in Figres 0b), b) and b). Nmerical comparisons of control qalit criteria are shon in tables Tab., Tab. and Tab 3. We evalated qalit criteria like, control time and integral of absolte vales of control error. variable Control variable Random p{} {} Inverse Neral Model Sstem Otpt sstem (controled) variable (k-) (k-) (k) Pertrbance Delaed (k-) (k-) (k) Delaed Clock Fig. 6 The simlation scheme of direct inverse neral control (INC) t Time

5 Random B Adaptive item B=+beta(sm(-)) variable p{} {} B Inverse Neral Model Sstem Otpt sstem (controled) variable (k-) (k) (k-) Delaed (k-) (k-) Pertrbance (k) Control variable Delaed Clock t Time Fig. 7 The simlation scheme of robst direct inverse neral control (RINC) variable Random p{} {} Inverse Neral Model Sstem Sstem otpt (controled) variable (k-) Pertrbance (k-) (k) (k-3) Delaed (k-) Control variable (k-) (k) (k-3) Delaed Clock t Time p{} {} Neral Model e Model error m s+ Filter Model otpt Fig. 8 The simlation scheme of internal model control ith neral models (IMNC) Random Plant Otpt NN Predictive Controller Optim. NN Model Control Signal Control variable Sstem Clock Sstem otpt t Time X(Y) Graph variable Fig. 9 The simlation scheme of predictive neral control (PNC)

6 Comparison of neral controllers - INC - RINC 3 - IMNC 4 - PNC - sstem otpt a) INC - RINC 3 - IMNC 4 - PNC - sstem otpt Fig. 0 Time responses comparison of some neral controllers for A sstem a) ith pertrbation b) b)

7 - sstem otpt Comparison of neral controllers INC RINC IMNC PNC a) sstem otpt INC RINC IMNC PNC Fig. Time responses comparison of some neral controllers for B sstem a) ith pertrbation b) b)

8 Comparison of neral controllers - INC - RINC 3 - IMNC 4 - PNC - sstem otpt a) sstem otpt.5 - INC - RINC 3 - IMNC 4 - PNC Fig. Time responses comparison of some neral controllers for C sstem a) ith pertrbation b) b)

9 Tab. Control performance vales for A sstem Withot pertrbation (Fig.0a) With pertrbation (Fig.0b) control time [s] pertrbation. time [s] INC, ,,6 0 0,5,5-3 RINC 3,55 5, 36,9 5 9,6, 0 0,5 4, 7,5 IMNC, ,4,8,85 0, - 5, 8,4 PNC 3,89 5,6 5,3 3 3,6,84 0, - 3,6 4, Tab. Control performance vales for B sstem Withot pertrbation (Fig.a) With pertrbation (Fig.b) control time [s] pertrbation. time [s] INC 7,3 4, 33,4,4 7,,35 0,5 -,,4 RINC 7,3 5,3 33,6,6 7,47 0 0,5,3,8 IMNC 6,78 5,6 6, 3,6 7 3,6 0 0,5 5, 6,8 PNC 7,77,5 3, 4,6 6,95 0, - 5, 5,8 Tab. 3 Control performance vales for C sstem Withot pertrbation (Fig.a) With pertrbation (Fig.b) control time [s] pertrbation. time [s] INC ,9 0. 0,8 0,9 0, RINC,4,5 4, 0.4 4,3 0,53 0 0,5,,8 IMNC,05 0, 38,45 0,, 0,8 0, -,8 4, PNC,56,3 7,, ,5 -,8,3 5 Conclsion The main objective of this article as to compare particlar strctres of neral controllers on selected tpes of nonlinear processes. We sed Matlab Simlink to evalate qalit control criteria and behavior of particlar vales to generalize for particlar tpes of neral controllers. Acknoledgment The ork has been spported b the grant agenc VEGA No. /300/06. s [] A. Jadlovská. Modelovanie a riadenie dnamických procesov s vžitím nerónových sietí. Edícia vedeckých spisov FEI TU Košice, ISBN , 003 [] M. T. Hagan, M. B. Menhaj. Training Feedforard Netorks ith the Marqardt Algorithm. Sbmitted to the IEEE Proceedings on Neral Ne, 994

10 [3] R. Keser, O. Pastravan, D. On. MATLAB Neral Netork Toolbox based softare environment for nonlinear identification and control, IFAC Workshop Ne Trends in Design of Control Sstems, Smolenice, Slovakia, 994 [4] L. Pastoreková, A. Mészáros, P. Brian. Inteligentné riadenie sstémov na báze inverzných nerónových modelov. ATP jornal Pls7, 005, pp [5] S. Kajan, Š. Kozák. Adaptive predictive control of nonlinear dnamical sstems sing neral netorks, nd International Prediction Conference, Nostradams 99, October Zlin 999, pp.3-37 [6] The Mathorks. Neral Netork Toolbox,User s Gide, 00 Ing. Slavomír Kajan, PhD.: Institte of Control and Indstrial Informatics, FEI STU in Bratislava, slavomir.kajan@stba.sk