COMPENSATION OF VALVE DEADZONE USING MIXED INTEGER PREDICTIVE CONTROL

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1 COMPENSAION OF VALVE DEADZONE USING MIXED INEGER PREDICIVE CONROL Jakub Novak and Petr Chalupa omas Bata University Faulty of Applied Informatis Centre for Seurity Information and Advaned ehnologies nam. GM 5555 Zlín 76 Czeh Republi KEYWORDS Preditive ontrol Mixed-integer quadrati programming Valve deadzone Hybrid system. ABSRAC Stition is a nonlinear frition phenomenon that auses poor performane of ontrol loops in the proess industries. In this work we develop a mixed-integer MPC (Model Preditive Control) formulation inluding valve dynamis for a stiky valve in order to improve ontrol loop performane. he introdution of the valve nonlinearity into the model prevents the MPC from requesting physially unrealisti ontrol ations due to valve stition. Simulation studies using a two-tank systems show that if the deadband value is known a- priori the Mixed-Integer Quadrati Programming (MIQP) an effetively improve the losed-loop performane in the presene of valve stition. INRODUCION Model Preditive Control has beome one of the most widespread modern ontrol strategies whih have been suessfully applied in many industrial appliations. he idea of MPC is to determine an optimal ontrol at the urrent time instant by solving an optimal ontrol problem on a predition horizon. he main reason for the wide-sale adoption of MPC is its ability to handle onstraints on inputs and states that arise in most appliations. MPC also naturally handles the multidimensional systems. Control valves are neessary elements in many hemial proess ontrol systems and are equipped for manipulating mass flows energy flows or pressure. In general they ontain stati and dynami nonlinearities inluding saturation baklash stition deadband and hysteresis. If these nonlinearities are present the valve is not apable of following the ommand signals provided by the ontroller thus leading to ontrol performane degradation. A limit yle is typially produed around the steady-state operating points if these nonlinearities are not inluded expliitly in the ontroller design. o ompensate for valve malfuntion several methods based on MPC has been developed. he framework developed in (Rodrıguez and Heath ) used the inverse of the nonlinearity in series with the original nonlinearity to overome the problem. General ompensation formulation that inludes dynamis of stiky valve and additional onstraints on inputs rate of hange have been introdued in (Durand and Christofides 6). A signifiant ontribution in the area of valve nonlinearities ompensation has been done in the work by (Zabiri and Samyudia 6) whih used a hybrid formulation of the input onstraint within the MPC design to express the atuator baklash. he ontroller was used only in the only in the proximity of steady state operating points. Strategy that uses twomove stition ompensation have been revised in (Bai di Capai et al 6) and suessfully applied to the pilot plant. In this paper nonlinear model of the proess with defined valve dead-zone was developed and used for optimization of the ontrol signal via MPC strategy. he valve nonlinearity is expressed in terms of IF HEN onditions resulting in the hybrid model. Hybrid systems represent a unified framework for modelling suh proesses that ombine ontinuous and disrete dynami with logial rules and appear in various appliations like roboti systems and automotives. A speial lass of hybrid systems alled Mixed Logial Dynamial (MLD) systems has been introdued in (Bemporad and Morari 999). Hybrid systems an effetively model a variety of systems: hybrid automata nonlinear systems with the nonlinearity represented by the piee-wise affine funtions linear systems with onstraints et. MPC is a general approah for ontrol of suh systems. However the optimization problem is no longer quadrati programming (QP) problem but a Mixed-Integer Quadrati Programming problem. he inlusion of integer variables turns the easily solved QP problem into an NP-hard problem (Borrelli et al 6). VALVE NONLINEARIY DESCRIPION A ommon proess nonlinearity affeting the performane of ontrol iruits with ontrol valves (Fig. ) is known as stition whih exhibits a range of nonlinear behaviour inluding hysteresis bakslash and deadzones both dynami and stati. In this paper the model that inludes deadband in every valve move was hosen. In Figure the signal u is the proess input that is the valve output y is the proess output umpc is the MPC output w and v are white Gaussian noises. Proeedings 3st European Conferene on Modelling and Simulation ECMS Zita Zoltay Paprika Péter Horák Kata Váradi Péter amás Zwierzyk Ágnes Vidovis-Dans János Péter Rádis (Editors) ISBN: / ISBN: (CD)

2 Propositional logi is translated into an equivalent linear inequalities using the strategy desribed in (Bemporad and Morari 999). For example the first relation of () an be translated into two linear inequalities: Figure : he Closed-loop System with the (stiky) Control Valve followed by the Proess. he stiky valve has a nonlinear dynamis expressed by the following relations: uimpc ( k): uimpc d ui ( k) ui MPC ( k): ui MPC d () ui ( k ): otherwise he deadband model involves a set of logial rules that represent three regions of the deadband defined by parameter d (Figure ). Figure : Stiky Valve Nonlinearity. he input hange uimpc ( k) with respet to the deadband size d would determine what the output would be. It should be noted that the deadzone is ative if uk ( ) uk ( ) i.e. the input signal is travelling within the deadband. he nonlinearity is formed by a set of three relatively simple linear relations thus onstituting a sort of swithing multiple model sheme. MODELING HE VALVE NONLINEARIY he nonlinearity () of the valve an be modelled using a set of logial variables ij : i ( k) ui( k) di i ( k) ui( k) di i3( k) ui( k) uimin ui( k) uimax () u u ( k) u imin i imax 3 ij ( k) j ui( k) ( M d) i( k) M u ( k) ( md ) ( k) d i where Mm are upper bounds and lower bounds of uk ( ) and is a small positive salar. his equivalene permits the assignment of binary variables to dynamial onstraints whih may define the different operation modes of hybrid system. he linear proess dynamis is expressed by a state-spae model i x( k ) Apx( k) Bpu( k) y( k) C x( k) p n* n n* r m* n where Ap R Bp R Cp R and n is the proess order r and m are number of input and outputs respetively. he resulting MLD system whih inludes dynamis of the proess is desribed by the following relations: x(k) Ax(k) Bu(k) B (k) B3z(k) B y(k) Cx(k) Du(k) D (k) D3z(k) D E (k) E3z(k) Eu(k) E x(k) E 4 5 (3) (4) (5) where A B B B3 B C D D D3 D E E E3 E4 E5 are matrixes of appropriate dimension and zk ( ) are anillary ontinuous variables. PREDICIVE CONROL OF HE MLD SYSEM Bemporad and Morari introdued a model preditive ontrol of hybrid systems using mixed logial dynamial (MLD) system desription and a mixed integer linear program solver in (Bemporad and Morari 998). he quadrati objetive funtion in ase of ontrol to a setpoint may be written: min Ju ( z ) x(k N) x N j P x(k j) x u(k j) S Q R (6) Subjeted to equations (5) whih define the MLD system state preditions. In (6) the matries QRS indiate weighting matries and y is the erene signal and Np is the predition horizon.

3 he problem (6) an be rewritten to a general MIQP programing form min.5x Hx f x xi n ni x x R xi N x x a b xi j... m x d xi j... m j j e j j i (7) where n and n i define the numbers of ontinuous and integer variables H is a positive definite matrix f is the n-dimensional vetor. he n-dimensional vetors a j and j and vetors b and d are used to set up the linearity and nonlinearity onstraints. he numbers of equality and inequality onstraints are speified with m e and m i respetively. he equality and inequality onstraints define a feasible region in whih the solution to the problem must be loated in order for the onstraints to be satisfied. he problem an be solved either by brute fore or using the existing tools for MIQP programming based on branh-and-bound or branh-and-ut strategies. For MLD systems by definition of the vetor n i uk ( ) V ( k) (8) uk ( N) zk ( ) zk ( N) the problem (7) an be rewritten in the ompat form: min.5 V( k) HV(k) f V( k) AV ( k) b CV( k) d he MPC algorithm in the paper uses inremental form whih is insensitive to slowly varying system and measurement trends and theore has integral ation (Di Rusio 3). Given the proess model x( k ) Ax( k) Bu( k) v yk ( ) Cxk ( ) w (9) () n r where x( k) R is the state vetor uk ( ) Ris the m ontrol input vetor y( k) R is the output vetor AB and C are system matries of appropriate dimensions. he augmented model whih is independent of the unknown disturbanes is then given x( k ) A x( k) B ( k) ( k) ( k) u y C Iy SIMULAION EXAMPLE () In other to evaluate the presented approah the simulation example of the two-tank system is used. Figure 3 shows the sheme of the system. Figure 3: wo-tank System Sheme he hanges of water levels an be desribed by the set of differential balane equations of the two tanks: dh S q usv g h h dt () dh S usv g hh usv g h dt where S is ross-setion area of the tanks Sv is ross-setion area of the valves g is gravity onstant q is volume flow through pump u is inputs signal to valve u is inputs signal to valve h is water level in tank h is water level in tank. he parameters of the simulation system (able ) were taken from (Chalupa and Novak 3). able Parameters of the System Par. Value S.54m S 5e-5m v g 9.8m/s q e-4m 3 /s he levels in both tanks are ontrolled using two ontrol valves. he seond ontrol valve u has a deadband

4 nonlinearity desribed by the relations () of the size d.5. here are also limits on both ontrol signals and ontrol inrements: uk ( ). uk ( ). (3) he system was linearized with a sampling time of s in steady-state onditions h.3m h.5m. he performane of the MPC ontrol with (Np= Q=I R= S=I) that is unaware of the nonlinearity is presented in Figure 4 and 5. h [m] h [m] he nonlinearities have negative effets on set-point traking and result in osillations in ontrol loops. When a valve has deadband the valve output does not hange in response to hanges in the ontrol signal to the valve until the ontrol signal overomes the deadband. In order to improve the performane the nonlinearity () is translated into the set linear inequalities using () that onstrain the optimization problem and introdue binary variables. hus originally quadrati optimization problem with ontinuous variables is replaed by a mixed-integer quadrati optimization problem. In order to redue the number of binary variables the ontrol horizon Np is redued to 3 steps whih represents 9 binary variables. he resulting quadrati problem has Np*3 binary variables and Np*r+N*(m+n) ontinuous variables (inputs and predited states of the augmented model). he problem is onstrained by limits for input variables binary variables and relations defining the evolution of the predited states. he problem is solved using the branh-and-bound strategy and interior point method is used for solution of the relaxed problems (Novak and Chalupa 5). Using the suessive linearization strategy the parameters of augmented model are obtained at eah sampling period using the urrent state x( k) and linearization of (). he steadystate target for eah step of the erene signal is given: Figure 4: Output Response for Step-hange of the Level in the First ank (dotted line - erene) y h h (4) he resulting ontrol ourses for step-hanges of both outputs with the same weighting matries are presented in Figure 6 and 7. Figure 5: Controller Output Response for Step-hange of the Level in the First ank Figure 6: Output Response for MIQP-based MPC Formulation (dotted line - erene)

5 u [-] u [-] [s] Figure 7: Input Response for MIQP-based MPC Formulation It an be observed that stition embedding MPC guarantees very good traking performane and also an effetive stition ompensation. he number of relaxed quadrati problems that are solved during solution of MIQP problem are presented in Figure 8. Solution of the MIQP problem with 9 binary variables by brutefore would require solution of 9 relaxed QP problems. Beause of the effiieny of the branh-and-bound algorithm less than 4 QP problems are solved at eah sampling point. Figure 8: Number of Evaluations of the Relaxed QP Problems CONCLUSION his paper has presented a possible formulation of MPC to fae deadband nonlinearity in ontrol valves for industrial proesses. he improved performane of MIQP-based MPC ontroller is possible at the expense of using more omplex and omputationally more demanding mixed-integer optimization algorithm. However the MIQP-based MPC requires the knowledge of the deadband a-priori. ACKNOWLEDGMEN his work was supported by the Ministry of Eduation Youth and Sports of the Czeh Republi within the National Sustainability Programme projet No. LO33 (MSM-7778/4). REFERENCES Bai di Capai R.; Sali C.; and B. Huang. 6. A revised tehnique of stition ompensation for ontrol valves. In Proeedings of the th IFAC DYCOPS (rondheim Norway) Bemporad A. and M. Morari Control of systems integrating logi dynamis and onstraints. Automatia Vol Bemporad A. and M. Morari Preditive ontrol of onstrained hybrid systems. In Proeedings of Int. Symposium on Nonlinear Model Preditive Control (Asona Switzerland) 8-7. Borrelli F.; A. Bemporad; M. Fodor; D. Hrovat 6. An MPC/hybrid system approah to tration ontrol IEEE ransations on Control Systems ehnology Chalupa P. and J. Novak. 3. Modeling and Model Preditive Control of Nonlinear Hydrauli System. Computers & Mathematis with Appliations 66() Di Rusio D. 3. Model Preditive Control with Integral Ation: A simple MPC algorithm. Modeling Identifiation and Control Vol. 34 No Durand H. and P. Christofides 6. Atuator stition ompensation via model preditive ontrol for nonlinear proesses. AIChE Journal Vol Novak J. and P. Chalupa. 5. Implementation of Mixedinteger Programming on Embedded System. Proedia Engineering Vol Rodrıguez M. and W. Heath.. MPC for plants subjet to saturation and deadzone baklash or stition. In Proeedings of the 4th IFAC Nonlinear Model Preditive Control Conferene (Noordwijkerhout he Netherlands) Zabiri H. and Y. Samyudia. 6. A hybrid formulation and design of model preditive ontrol for systems under atuator saturation and baklash. Journal of Proes Control Vol AUHOR BIOGRAPHIES JAKUB NOVAK was born in Zlin Czeh Republi and reeived the Ph.D. degree from the omas Bata University Zlin Czeh Republi in 7. He is now a researher at the CEBIA-ECH researh enter at Faulty of Applied Informatis at the university. His researh interests are multiple model strategies and preditive ontrol. His address is : jnovak@fai.utb.z PER CHALUPA was born in Zlin Czeh Republi in 976. He graduated from Brno University of ehnology in 999. He obtained his Ph.D. in ehnial Cybernetis at omas Bata University in Zlin in 3. He works as a researher at CEBIA-ECH researh enter at omas Bata University in Zlin. His researh interests are adaptive and preditive ontrol of real-time systems. You an ontat him on address halupa@fai.utb.z