Intelligent Positioning Plate Predictive Control and Concept of Diagnosis System Design

Transcription

1 Intelligent Positioning Plate Predictive Control and Concept of Diagnosis System Design Matej Oravec - Anna Jadlovská Department of Cybernetics and Artificial Intelligence, Faclty of Electrical Engineering and Informatics, TU Košice, matej.oravec@tke.sk Department of Cybernetics and Artificial Intelligence, Faclty of Electrical Engineering and Informatics, TU Košice, anna.jadlovska@tke.sk Keywords predictive control design, ARX model of the system, mechatronic system, diagnosis system Abstract This paper presents design of the predictive control algorithms, which are verified sing the simlation and laboratory model of the Intelligent Positionig Plate. The reslts of the predictive control algorithm verification are presented also in this article. The created tool called the IPPtools is based on the designed predictive control algorithms and it is shortly presented. A part of the paper is dedicated to the concept of the diagnosis system, which is designed and implemented into the 5-level Distribted Control System of the Department of the Cybernetics and Artificial Intelligence. Possibility how to modify the predictive control algorithms sing diagnosis system is also stated in the last section of this paper. History Received 8 March 7 / Revised 9 March 7 / Accepted 5 May 7 Article Category Citation Original Scientific Paper Matej Oravec, Anna Jadlovská (7). Jornal of Manfactring and Indstrial Engineering, 5(-):-9, Introdction Predictive control methods are nowadays widely sed for the control of the varios dynamic systems. Methods of the predictive control provide highly efficient control of the dynamic systems, which is able to control dring long periods of the time with hardly any intervention. Disadvantage of these predictive control methods is more complex derivation than that of the classical PID controllers, bt the reslting control law is easy to implement. Different approaches to model predictive control design are presented in books [],[]. At present, several research grops deal with the predictive control of mechatronic systems. In the [3], [4] athors present the reslts of the design and verification of the predictive control algorithms for the Ball and Beam system. The papers [3], [4] present the predictive control of the Ball and Plate system sing Internet-based control. The varios methods of the Ball and Plate system control sch as fzzy or sliding mode control, which are sed for the reference trajectory tracking goal, are presented in [5], [6]. In general, the model application called Intelligent Positioning Plate (shortly IPP) is very similar to the Ball and Plate system presented in [5], [6]. The IPP model application belongs to fnd of the laboratory models of the Department of Cybernetics and Artificial Intelligence (DCAI), FEEI, TU and it is sitated in the Laboratory of Modern Control Techniqes of Physical Systems ( The IPP model application is specific by its constrction and consists of two servomotors, which commnicate with the control PC via serial link with the single microchip compter. The actal ball position is captred by the camera. This paper is focsed on the selected approaches to predictive control sing linear predictor for prediction of the dynamic system ftre behavior. According to the selected predictive control approaches are designed predictive control algorithms, which are implemented into the Matlab environment and verified sing the IPP model application. The paper follows the Center of Modern Control Techniqes and Indstrial Informatics ( members previos pblications. The paper [] presents reslts of the predictive control of the Hmsoft Helicopter CE5 laboratory model. Some obtained reslts of the LQ control of the IPP model application was listed in []. The one section of the paper is related to the tasks solved within the project Research and Development Operational Program for project: University Science Park TECHNICOM for innovative applications with knowledge technology spport with sbactivity Center for Nondestrctive Diagnostics of Technological Processes and is oriented to the concept design of the diagnosis system within the infrastrctre of Distribted Control System (DCS) within the DCAI, FEEI, TU. Model application of the Intelligent positioning plate The Intelligent positioning plate (IPP) represents an intelligent positioning mechatronic system based on the concept of position control of an object moving on an adjstable plate (Fig. ). This application provides a platform for mathematical modeling and sbseqent simlation of the mechatronic system in Matlab/Simlink, bt also for the implementation of designed control algorithms with the falt diagnosis algorithms realized via the technological PC or the PLC [9]. The IPP model application ( modely/gnk.php) consists of the two servomotors and the plate with a ball whose position coordinates on the plate are determined sing a fixed camera. The camera is connected directly to control PC where image processing algorithm is implemented (Fig. ) for calclating of the ball position coordinates in programming langage C# sing emgcv libraries []. Commnication between servomotors and the control PC is realized via a microcontroller throgh serial connection. The specific design soltion of the model enables the implementation and verification of proposed control algorithms

2 Oravec M., Jadlovská A. in varios programming langages (C/C++/C#) or simlation tools (Matlab/Simlin [9]. The mathematical model of the IPP is derived with the assmption, that originally MIMO system is decomposed to the two SISO systems for the axis x and axis y (Fig. 3). Next, the SISO system consists of the sbsystem Ball on beam and sbsystem Servomotor. The sbsystem Ball on beam is described by the nonlinear differential eqation: 5 x (t ) g sin (t ) () 7 Actal tilt of the plate represented by angle α(t) is otpt of the sbsystem Servomotor, which is described by transfer fnction: Fs ( s ) Kx (s), x ( s ) Tx s () where parameters Kx, Tx are identified from the step response of the sbsystem Servomotor []. Figre Workplace of the Intelligent positioning plate model application Image processing camera initialization t < simlation time - Figre 3 System decomposition of the IPP model application + Transfer fnction () can be expressed in form of the linear differential eqation: (t ) ( K x x (t ) (t )) (3) Tx Nonlinear differential eqation () of the Ball on beam sbsystem is adjsted into the canonical form sing sbstittion t) = t) and α (t) = x3(t): image captre transformation from RGB to binary image Gassian filter Hogh transformation comptation of ball coordinates x (t ) x (t ) x (t ) End Figre Flowchart of the image processing algorithm for comptation of the ball position coordinates Inpts for the servomotors and for whole system are reference angles αt), βy(t) of the plate tilt. Otpts of the model application are coordinates t), y(t) of the ball position on plate. Actal tilt of the plate is represented by angles α(t) and β(t). Physical variables and parameters of the IPP model application are listed in Tab., Tab.. Table Parameters of the IPP model application Description servomotor gain axis x time constant of servomotor axis x servomotor gain axis y time constant of servomotor axis y Label Kx Tx Ky Ty Table Physical variables of the IPP model application Description Label t) ball position axis x y(t) ball position axis y α(t) plate tilt axis x αt) reference angle axis x β(t) plate tilt axis y βy(t) reference angle axis y [s] [s] [m] [m] [rad] [rad] [rad] [rad] 5 g sin x3 (t ) 7 (4) and linear differential eqation (3) of the Servomotor sbsystem is also modified into form: x 3 (t ) ( K x x (t ) x3 (t )), Tx (5) where t) = t) represents coordinate of the ball position, ẋ (t) = t) is velocity of the ball and α (t) = x3(t) is angle of the plate tilt. In general, the mathematical model of the IPP for axis x incldes the Ball on beam sbsystem and Servomotor sbsystem represented by eqation (5) and (6) can be expressed: x (t ) f ( x (t ), (t ), t ) y (t ) h( x (t ), t ) (6) where x (t) = [t), t), x3(t)]t a (t) = [αt)]. For the reference trajectory tracking control target, the mathematical model (6) of the IPP is linearized arond the eqilibrim point xep=[x=, x=, x3=]. The reslt of the linearization is the state space model of the system for the axis x: x (t ) At ) B (t ) y (t ) Ct ) (7) The matrix of dynamic A, inpt matrix B and otpt matrix C are defined:

3 Oravec M., Jadlovská A. f x f A x f 3 x f x f x f x 3 f x3 f3 x 3 f 3 x3 xep 5 g 7 T x f f B, C K x f 3 Tx xep xep Transfer fnction is derived from the state space model of the system by relation: F ( s) C( si A) B, (9) where s is Laplace operator. For the predictive control algorithm design prpose, it is necessary to discretize the state space model of the system (7) with selected sample period T s. Discretization of the state space model (7) is done sing the Matlab fnction cd with sample period T s =,5 s. Discrete transfer fnction F(z - ) is obtained from the continos transfer fnction (9) by discretization with selected sample period T s in form: xep, (8) Bz ( z ) Fz ( z ), () A ( z ) z Following the same procedre as for the axis x, discrete linear state space model and transfer fnction of the IPP is obtained in the direction of the axis y. Simlation model of the IPP is created according to mathematical model of the IPP model application in Matlab/Simlink environment. The time responses to the same step inpt signal for the nonlinear simlation and laboratory model IPP are compared in Fig. 4, Fig. 5. Figre 4 Comparison of the IPP simlation and real model time responses to the same step inpt direction of the axis x Figre 5 Comparison of the IPP simlation and real model time responses to the same step inpt direction of the axis y In the next sections of the article is simlation model of the IPP sed for verification of the designed predictive control algorithms. Simlation model of the IPP is also connected to the 3D virtal model (Fig. 6) in Simlink environment. The 3D virtal model, which is created in Simlink 3D Animation toolbox, is sed for visalization of the IPP control simlation. Figre 6 Virtal model of the IPP created sing Simlink 3D Animation toolbox Design of predictive control algorithm based on linear discrete state space model of the dynamic system The varios predictive control methods propose different cost fnctions for derivation of the control law. The presented predictive control algorithm is based on the solving optimization tasks and in general minimizes the cost fnction: 3 J N p i N i Q[ yˆ( k i) w( k i)] R[ ( k i )] () where N p, N represents prediction horizon and control horizon, ŷ( is predicted vale of the system otpt, w( denotes reference trajectory, ( is control inpt and Q, R are weight matrices []. For the design of the predictive control algorithm is sed the linear predictor, which is derived by iteration from the linear discrete state space model of the system: k ) Ad Bd( () y( C and the linear predictor has form: yˆ ( V G(, (3) where: ŷ(=[y(k+),..., y(k+n p )] T, (=[(, (k+),..., (k+n p -)] T In (3), the term V represents the free response and the term G( is the forced response of the system. Matrices of the free response V and forced response G, which are obtained by recrsive comptation, can be expressed in form: CBd CAd CAd Bd CBd V, G (4) N p CA d N p CAd Bd CAd Bd CBd Selection of the control horizon N (N N p ) affects the size of forced response matrix G. According to [], the comptation of the optimal control inpt opt ( of the presented predictive control algorithm is based on the minimization of the cost fnction () with respect to the condition:

4 Oravec M., Jadlovská A. J MPC! (k ) (5) Figre 7 Flowchart of the designed predictive control algorithm based on the discrete state space model of the dynamic system Finally, the control law has form: opt (k ) H g, (6) where the Hessian H and the gradient g are defined as follows: H (G T QG R) (7) g T (Vk ) w (k ) )T QG The optimal control seqence opt( can be compted in Matlab environment sing qadprog fnction (from Optimization toolbox) with respect to constraints of physical variables of the controlled system. Fnction qadprog comptes opt( by formla: min ( H g T ) (8) with consideration of constraints of the controlled system, which are composed to ineqality Uconopt vcon. For the constraints of the control inpt opt ϵ <min,max> or system otpt y(ϵ <ymin,ymax> has matrix Ucon and vector vcon form: I U con D, v con max, (9) I min D U con y y G,, v con max G y min y Figre 8 The predictive control algorithm based on the discrete state space model of the dynamic system implemented to the control strctre For verification of the designed predictive control algorithm, two experiments were done sing the IPP model application. The experiments were realized for the circle reference trajectory tracking goal. Simlation and sampling time in experiments were set to T = s, Ts =,5s. The first experiment of the IPP control was realized sing the simlation model with parameters, which are listed in Tab. 4. Time responses of the ball position coordinates t), y(t) and control inpts αt), βy(t) are shown in Fig. 9. Table 3 Parameters sed for the predictive control of the IPP model application () where is a nit vector and ID is an nit matrix [8]. Design of the state predictive control algorithm is shown in flowchart, Fig. 7. This algorithm is implemented to the Matlab environment with respect to selected control strctre illstrated in Fig. 8 as m-file called ssmpccon. For prpose of the predictive control algorithm is also created parammpcc fnction: [H,G,V,Ucon,vcon]=paramMPCc(Ad,Bd,C,D,Q,R,Np,N) Fnction parammpcc comptes Hessian H, matrix of forced response G, matrix of free response V, constraints matrix and vector Ucon, vcon for the predictive control algorithm. axis x axis y Parameter Np N Q R Np N Q R Vale I, I 75 I, I ssmpccon simlation time T, sample period Ts system matrices Ad, Bd, C weight matrices Q, R, Np, N horizons <min,max>, <ymin,ymax> constrains parammpcc k t( t( T - + actal vale of the dynamic system state variables gradient g comptation (7) optimal control seqence opt( comptation (8) ( first vale opt( control inpt ( application to dynamic system inpt t(k+) t(+ts k k+ Figre 9 State predictive control of the IPP simlation model time responses of the ball position coordinates t), y(t) and control inpts αt), βy(t) End 4

5 Oravec M., Jadlovská A. The second experiment was realized with the real laboratory model of the IPP sing the parameters presented in Tab. 3. Obtained reslts of the IPP laboratory model predictive control are represented by time responses of the ball position coordinates t), y(t) and control inpts αt), βy(t) (Fig. ). According to the reslts, the designed predictive control algorithm is characterized by the small divergence of the ball position from desired reference trajectory sing the optimal control inpt. In the case of the real laboratory model, the control inpt is represented by higher oscillations than the case of the simlation model. The lower control qality of the IPP laboratory model is inflenced by: - ncertainty in modeling of the IPP mechanical parts - nmeasrable distrbances effect Despite the inflence of nmeasrable distrbances or modeling ncertainties, the designed predictive control algorithm flfills the control reqirements with sfficient qality. X ( k ) y ( k n ) y (k ) y (k ) A (k ) X y (k n ) y (k ) a a a y (k ) n n () (k m ) (k ) b b b (k ) n n U (k ) B y (k n ) y (k ) ( ) C y (k ) X (k ) In general, matrix form () can be written into psedo-state space model: X (k ) A X (k ) BU (k ) y (k ) C X (k ) (3) The system otpt prediction ŷ( = [y(k+),..., y(k+np)] sing psedo-state space model () can be expressed: y (k ) C A y (k n ) (k m ), (4) G y (k N ) C A N p (k N ) y ( k ) p p where the matrix Ḡ has form: C B, B G C Bi b b C B n n N p B i A B i B, Figre State predictive control of the IPP laboratory model time responses of the ball position coordinates t), y(t) and control inpts αt), βy(t) Design of predictive control algorithm based on regressive ARX model of the dynamic system This section is dedicated to the predictive control algorithm design, which is based on the predictive control approach presented in []. In paper [3], predictive control based on the ARX model of the system is sed for control of the Ball and Beam, which is similar to the IPP model application. Presented algorithm of the predictive control assmes that the cost fnction () it is minimized, bt the control law is derived sing the regressive ARX model of the dynamic system: m y (k ) n b (k i ) a y(k i ) i i i () i Comptation of the system otpt y(k+) in () can be arranged, according to [], into the matrix form: b (5) i,, N p It is possible to express the predictor of the modified predictive control algorithm from the psedo-state space model () according to [] in form: y (k ) (k ) yˆ (k ), ( k ), y (k N ) (k N ) p p C A y (k n ) y (k ) C A N p y ( k ) (6) ( k m ) G (:,:m ), (k ) G G (:, m:m N p ), yˆ (k ) y (k ) G(k ), where : convention is sed for the selection of part of the matrix Ḡ in predictor (6) and meaning of : convention is 5

6 Oravec M., Jadlovská A. taken from Matlab environment. With respect to constraints composed in form (9) or (), the optimal control inpt seqence opt ( is compted sing fnction qadprog of Matlab environment based on formla (8) [8]. In case of the predictive control based on the regressive ARX model of the system, the Hessian H and the gradient g have form: T T T H ( G Q QG R R) (7) T T T g ( y w() Q QG The designed modified predictive control algorithm based on the regressive ARX model of the system is shown in the flowchart, Fig.. According to the flowchart (Fig. ), designed predictive control algorithm is implemented as m-file called iogpccon to the Matlab environment with respect to control strctre, which is shown in Fig.. For comptation of the Hessian H, matrices of the free response y and matrix of the forced response G is also created fnction: [H,y_AC,y_G,G,Ucon,vcon]= paramgpcc(bz,az,q,r,np,n) The designed predictive control algorithm based on the ARX model of the dynamic system was also verified by experiments sing the IPP model application. The goal of the predictive control experiments was the circle reference trajectory tracking. In the first experiment, the simlation model of the IPP was sed and the predictive control parameters are also listed in Tab. 3. Time responses of the ball position coordinates t), y(t) and control inpts α x (t), β y (t) are shown in Fig. 3. iogpccon simlation time T, sample period Ts Az(z - ), Bz(z - ) polynomials weight matrices Q, R, Np, N horizons <min,max>, <ymin,ymax> constrains paramgpcc k t( + t( T y( actal vale of the dynamic system otpt free response y comptation (6) gradient g comptation (7) optimal control seqence opt( comptation (8) - Figre 3 Predictive control of the IPP simlation model time responses of the ball position coordinates t), y(t) and control inpts α t), β y(t) The second experiment was realized sing the real laboratory model with the same parameters of the predictive control algorithm like the first experiment. The reslts of the IPP laboratory model predictive control are characterized by the oscillations of the control inpt, bt with the acceptable divergence of the ball position from the desired position. The time responses of the ball position coordinates t), y(t) and control inpts α x (t), β y (t) are shown in Fig. 4. ( first vale opt( control inpt ( application to dynamic system inpt t(k+) t(+ts k k+ End Figre Flowchart of the designed predictive control algorithm based on the regressive ARX model of the dynamic system Figre The predictive control algorithm based on the regressive ARX model of the dynamic system implemented to the control strctre 6

7 Oravec M., Jadlovská A. Figre 4 Predictive control of the IPP laboratory model time responses of the ball position coordinates t), y(t) and control inpts α t), β y(t) Both designed predictive control algorithms are a part of the tool for the control of the IPP model application, which is called IPPtools. The IPPtools also incldes the generator of different reference trajectories like a circle, sqare or spiral. Diagnosis system concept design of the dynamic system Dynamic systems are often inflenced by falts of their components. The task of the falt diagnosis research area is to design algorithms for detection of the falt, localization of the place of the falt occrrence (falt isolation) and determining its magnitde [5], [6]. Selected falt diagnosis methods and approaches can be implemented in the diagnosis system, which is sed for the monitoring of the controlled system. In general, the control design of the dynamic system is very important step to the correct fnctionality of the diagnosis system. Controlled dynamic system withot falt occrrence is considered as nominal system, which is sed for the falt detection algorithm design. The actators and sensors are the most sensitive parts of the dynamic system to the falt occrrence. Condition of these parts of the dynamic system can be monitored by the diagnosis system which consists of the algorithms for falt detection, isolation and estimation. Information from the diagnosis system can be sed for the falt accommodation, what means to adapt the controller parameters to the properties of the falty dynamic system (Fig. 5) []. The actators falts matrix F a = B d and vector of the actators falts f a can be expressed: f ( ( Γ I) ( (, (9) a where the vector f ( = [ f,..., fp ] T corresponds to the effect of an additive actators falts and Γ( represents the effect of a mltiplicative actators falts. The matrix of the mltiplicative actators falts Γ has form: Γ j (3) p If the j-th actator is falty then γ j < or fj [5]. Varios types of the actator falts are described in Tab. 5. Table 3 Different types of the actator falts [5] fj = fj γ j = falt-free bias < γ j < loss of effectivness loss of effectivness γ j = loss of inpt actator blocked Also the presence of the sensors falts can be reflected in the linear model of the dynamic system: k ) Ad Bd ( (3) y( C F f ( s s where F s = C and the vector f s represents the sensors falts. The falty models of the dynamic system are the basic assmption for the falt diagnosis algorithms design and they can be implemented as a part of the complex diagnosis system (Fig. 5) into the 5-level Distribted Control System (DCS) of the DCAI. Concept scheme of the diagnosis system implemented into the lowest three levels of the DCS of the DCAI is shown in Fig. 6. The design of the diagnosis system is very important step to improve the presented predictive control algorithms to the form that expands capability of these algorithms to adapt to the changed properties of the controlled dynamic system. f Figre 5 Concept scheme of the diagnosis system The sensors falt diagnosis method presented in [6] ses the grop of the falt estimation filters, which are based on the Kalman filtering principles. This method can be sed for the actators falt diagnosis after the modification of the falty model of the dynamic system. Other method for actators or sensors falt diagnosis is presented in [5] and its derivation is based on the SVD principle, bt they are some similarity with the falt diagnosis method in [6]. Both methods can be sefl for implementation to the diagnosis system strctre shown in Fig. 5. The algorithms based on these methods can be implemented in the Matlab/Simlink enivornment, bt also in the single-chip microcompter or in the PLC. The selected falt diagnosis methods se linear model of the system for the falt diagnosis. The presence of the actators falts can be reflected in the linear model of the dynamic system: k ) Ad Bd( Fa fa ( (8) y( C Figre 6 Concept of the diagnosis system implemented in the 5 - level DCS of the DCAI In article [7] is presented possibility of changing the predictive control algorithm in case of the actator falt what can be reflected in the algorithm by changing vales min, max of the optimal control inpt opt constraints (9) according to the estimated magnitde of the actator falt. If the j-th actator loss effectiveness, it affects constraints: ( jmax fj ) v con (3) ( jmin fj ) In case of the j-th actator blocking, constraints have form: 7

8 Oravec M., Jadlovská A. ( fj ) v con, ( ) (33) fj where fj = const. For loss of the j-th actator inpt are constraints modified to form: v con (34) According to the presented diagnosis system concept design, it is planned the fnctionality of the IPPtools extend to the falt diagnosis algorithms, which shold be sed for the monitoring of the IPP actators falts occrrence. Conclsion This article presents the design of the two predictive control algorithms, which are implemented in Matlab environment. For verification of the designed predictive control algorithms, series of the experiments were realized sing the Intelligent Positioning Plate (IPP) model application. For realizing the experiments, it was sed simlation and real laboratory model of the IPP model application. The reslts of the experiments illstrate that designed predictive control algorithms are sitable for the reference trajectory tracking goal of the control. The best reslts of the IPP model application predictive control were obtained by the state predictive control algorithm. The tilt of the plate was more flently sing the state predictive control algorithm and the control inpts for servomotors were withot strong oscillations. The best reslts of the predictive control of the laboratory model were achieved with the prediction horizon N p = and control horizon N =. The selection of the sample period is mainly limited by the camera, which is capable to captre maximal 3 images per second. The IPP is similar to the Hmsoft CE5 Ball & Plate laboratory model ( gnp.php), which is also one of the laboratory models of the DCAI. There are some differences in the constrction of the both laboratory models and more details of the constrction differences are stated in []. The constrction of the IPP model application provides better portfolio for or soltion like the Hmsoft CE5 Ball & Plate laboratory model. The main advantage of the IPP model application is possibility to commnicate with the Matlab environment withot sing any toolbox. On the other side, it was necessary to create own image processing application in C# langage. The designed predictive control algorithms are the part of the IPPtools, which also incldes generator for the circle, sqare or spiral reference trajectories. Created m-files (ssmpcc, iogpcc) and fnctions (parammpcc, paramgpcc) for the predictive control inclded in the IPPtools are cstomized to the IPP laboratory model, becase m-files and fnctions of the IPPtools incldes necessary transformations of the control inpts to form sitable for the single-chip microcompter, which is sed for control of the servomotors. For these reasons is comptation of the control inpts of the predictive control algorithms for the servomotors more effective like sing fnctions of the Model Predictive Control toolbox of Matlab environment. Also this paper presents conceptal design of the diagnosis system, which is implemented in the lowest three levels of the Distribted Control System of the DCAI. According to this concept, the diagnosis system for the selected laboratory models of the DCAI shold be created where the IPP model application is one of them. Information from the diagnosis system can be se for the elimination of the falts inflence to the control of the dynamic system. In respect to the goals of the project USP TECHNICOM for Innovation Applications Spported by Knowledge Technology with sbactivity Center for Nondestrctive Diagnostics of Technological Processes, it is planned to design algorithms for the diagnosis system and implement them in the Matlab/Simlink environment. Also, one of the ftre research goals is modified the predictive control algorithms to the falt tolerant form, which can extend the IPPtools tool. Acknowledgement This pblication is the reslt of the Project implementation: University Science Park TECHNICOM for Innovation Applications Spported by Knowledge Technology - II. phase, ITMS: 33D3 spported by the Operational Programme Research & Development fnded by the ERDF (5%), grant TUKE FEI-5-33: Research Laboratory of Nonlinear Underactated Systems (3%) and grant KEGA TUKE- 4/5 (%). References [] E. F. Camacho, C. Bordons (3) Model predictive control, Springer Science & Bsiness Media, ISBN [] A. W. Ordys, D. W. Clarke (993) A state-space description for GPC controllers, International Jornal of Systems Science, Vol. 4, Isse 9, pp ISSN [3] K. Belda, J. Böhm (6) Adaptive Generalized Predictive Control for Mechatronic Systems, WSEAS Transactions on Systems, Vol. 5, Isse 8, Agst 6, Athens, Greece, pp [4] N. Hara, M. Takahashi, K. Konishi () Experimental evalation of model predictive control of ball and beam systems, Proceedings of the American Control Conference, pp ISSN [5] H. 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