Design of fuzzy based virtual actuator for a class of nonlinear systems

CONTROL ENGINEERING VOLUME: 10 | NUMBER: 2 | 2012 | JUNE DESIGN OF FUZZY BASED VIRTUAL ACTUATOR FOR A CLASS OF NONLINEAR SYSTEMS Anna FILASOVA1, Vladimir SERBAK1 1Department of Cybernetics and Artificial Intelligence, Faculty electrical Engineering and Informatics, Technical University of Kosice, Letna 9, 042 00 Kosice, Slovakia [email protected], [email protected] Abstract. The paper presents new conditions suitable in design of a virtual actuator for a class of continuous-time nonlinear systems represented by Takagi-Sugeno models, and measurable premise variables. Simulation results illustrate the design procedure and demonstrate the basic performances of the proposed control design method. Keywords Actuator faults, fault tolerant control, fuzzy control, fuzzy virtual actuator, Takagi - Sugeno models. 1. Introduction All technological systems are subject to faults, due to both component malfunctions and unforeseen external influences. The complexity of control systems requires fault tolerance schemes to provide control of the faulty system. Fault tolerant systems (FTS) are that the fruitful applications with potential significance for domains in which control of systems must proceed while the system is operative and testing opportunities are limited by operational conditions. The real problem is usually to fix the system with faults so that it can continue its mission for some time with some limitations of functionality. The main task to be tackled in achieving fault tolerance is design of controllers with such suitable reconfigurable structure, which guarantees the stability, the satisfactory performance, and the plant operation economy in nominal operational conditions. To achieve the fault tolerance, used methods rely on employing online fault diagnosis schemes, which activate an alternative control - reconfigurable control structure - that is supposed to handle a fault. Among these structures can be quoted control systems with adaptation to faults, the virtual-based control structures, as well as the output control reconfiguration algorithms [1]. In order to solve the complexity problems, the control reconfiguration has to satisfy the requirement that the control reconfiguration has to be performed on line after the fault has been detected [2], [3]. This requires simple and fast algorithms that work reliably without manual interventions and without the controller parameters tuning in a fault case. It is sufficient to store a parametric model of the system (including all faults) and a reconfiguration algorithm. The new control structure is generated on demand after the fault has been detected. Bibliographical reviews can be found in [4], [5], new developments in fault-tolerant control methods are presented e.g. in [2], [6], [7]. For nonlinear system design, various control schemes were introduced including exact feedback linearization and adaptive control. The technique of exact feedback linearization needs perfect knowledge of the nonlinear system and uses that knowledge to cancel the nonlinearities of the system. Since perfect knowledge of the system is almost impossible, the technique of exact feedback linearization cannot be generally used for nonlinear system control design. Also, adaptive control schemes, which were introduced to deal with nonlinear systems, exploit complicated parameter update laws and so the adaptive control algorithms posse hard limitations. Since a generic controller design method for all types of nonlinear systems has not been developed yet, an alternative to design a controller for nonlinear systems is e.g. fuzzy approach, which benefits from the advantages of the approximation techniques approximating nonlinear system model equations. Using the Takagi-Sugeno (TS) fuzzy model [8] the nonlinear system is represented as a collection of fuzzy rules, where each rule utilizes the local dynamics by a linear system model. Since TS fuzzy models can well approximate a large class of nonlinear systems, and the TS model based approach can apprehend the nonlinear behavior of a system while keeping the simplicity of the linear models, by employing the TS fuzzy model, a control design methodology exploits fully advantage of the modern control theory, especially in the state space optimal and robust control. The main idea of the TS model-based controller © 2012 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 75 CONTROL ENGINEERING VOLUME: 10 | NUMBER: 2 | 2012 | JUNE design is to derive control rules so as to compensate each rule of a fuzzy system, determining the local feedback gains [9]. It is known that the separate stabilization of these local models does not ensure the stability of the overall fuzzy model, and global design conditions have to be used to guarantee the global stability and control performance. Therefore, a range of stability conditions have been developed for TS fuzzy systems [10], most of them relying on the feasibility of an associated system of linear matrix inequalities (LMI) [11]. Therefore, the state control based on fuzzy TS systems model gives control structures, which can be designed using technique also based on equivalent LMIs. The idea behind this type of design is that the TS model based fuzzy control provides a user-friendly formalism for representing, implementing and achieving high-performance control structures [12]. This paper is concerned with the problem of the fuzzy virtual actuator design. Focusing on procedures for nonlinear dynamic system control, numerical example and system simulation are presented in advance. 2. On Takagi-Sugeno Fuzzy Models The systems under consideration are one class of multiinput and multi-output nonlinear (MIMO) dynamic systems, represented in the state-space form as , (1) )())((=)( tButqatq   , (2) )(=)( tCqty where , , , are vectors of the state, input, and output variables, respectively, and , and are real finite values matrices. n Rtq )( rnC r Rtu )( nm R  m Rty )( RB It is assumed that is a vector function, is bounded in associated sectors, i.e. in the fuzzy regions where it is assumed the system will operate within, and takes the value . It is considered that the number of the nonlinear terms in the nonlinear part of the model is , and that there exists a set of nonlinear sector functions of these properties ))(( tqa 0=(0)a ))(( tqa p , (3)         ))((=))(( ))((1=))(( ,1,2,=,,1,2,=)),(( 2= 1 twtw twtw plkjtw jljlj lj k j l lj     where k is the number of sector functions, and (4) 12 ()= () () () q ttt     t   is the vector of premise variables. It is supposed in the next that premise variables are measurable, and a premise variable can represent the state variable. Using a TS model, the conclusion part of a single rule consists no longer of a fuzzy set, but determines a function with state variables as arguments, and the corresponding function is a local function for the fuzzy region that is described by the premise part of the rule. Thus, using linear functions, a system state is described locally (in fuzzy regions) by linear models, and at the boundaries between regions an interpolation is used between the corresponding local models. Thus, the normalized aggregated function set can be constructed from all combinations of the sector functions which implies {(()), =1,2, ,=2} k i hti ss   =1 =1 (()) ()= ( ()) (), ( ())= (()) si iii s ii i wt qt h t t h t wt      , (5) , (6) ()= () () ii tAqtBut where is the i-th aggregated (normalized) membership function satisfying conditions ))(( thi  . (7) =1 0(())1, (())=1 {1,, s ii i ht ht i s    } i Therefore, the approximation of (1) leads to (5), (6), where is the Jacobian matrix of with respect to , and is the center of the i-th fuzzy region described by the set of sector functions (3). nn iRA   tq =)( ))(( tqa i qi q  Assumption 1: The matrices B, C are the same for all local models.  Assumption 2: The pair is locally controllable and (B, C) is of full column (row) rank, where ( ( ( )), )aqt B . (8) =1 (())= (()) s i i aqt h t A   Now the TS fuzzy model for (1), (2) takes form , (9) =1 ()= ( ())( () ()) s ii i qt h t Aqt But     , (10) )(=)( tCqty and can be interpreted as a combination of linear submodels through the set of normalized membership functions to approximate the nonlinear system. s },1,2,=)),(({ sithi  3. Stabilizing Fuzzy Controller 3.1. Standard Fuzzy Control Design Considering (9), i.e. © 2012 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 76 CONTROL ENGINEERING VOLUME: 10 | NUMBER: 2 | 2012 | JUNE , (11) =1 ()= ( ())( () ()) s ii i qt h t Aqt But     and using the same set of membership function, the nonlinear fuzzy state controller is defined as . (12) )())((=)( 1= tqKthtu jj s j  Substituting (12) into (11) results in . (13) =1 =1 ()= ( ())( () ( ()) ()) ss ii j j ij qt h t Aqt h t BK qt     Since it yields },{1,1=))(( 1= sithi s i   . (14) =1 =1 ()= ( ()) ( ())( ) () ss ij ij ij qt h t h t A BK qt     The equilibrium of the fuzzy system (9), (10), controlled by the fuzzy controller (12) is globally asymptotically stable [2] if there exists positive definite matrix and matrices such that nn RX  nr jRY   , (15) 0>= T XX , (16) <0 TTT ii j j XA AX Y B BY  for , , and . 0))(())((  thth ji 0tsji ,1,2,=,  The set of the control law gain matrices is given as . (17) sjXYK jj ,1,2,=,= 1  4. Design of Fuzzy Based Nonlinear Virtual Actuator Problem of the interest is to design an asymptotically stable fuzzy virtual actuator using Takagi – Sugeno fuzzy model of the nonlinear system (11). An actuator fault is modeled by changing the input matrix B towards Bf. Columns of Bf that correspond to faulty actuators are scaled in case of actuator degradation, or set to zero in the case of actuator failure. The faulty dynamic system is now given by the set of equations , (18) =1 ()= ( ())( () ()) s fiifff i qt h tAqt But     , (19) )(=)( tCqty ff where , , , are vectors of the state, input, and output variables, respectively, and are matrix parameters describing the faulty system. The controller is a nonlinear fuzzy state feedback controller in the form (12). n fRtq )( , nn Rf BR r fRtu )( , nrm RC  m fRty )( n i A The stabilization requires the reconfigured control loop to be stable while the signals of the controller are not affected by the fault. Since the idea of the reconfiguration is to make the faulty plant behaviour like the nominal plant ones, the state of the model of the nominal plant can be used as a reference. Thus, the fuzzy virtual actuator is given as =1 =1 ()= ( ()) ( ())( )() () ss ij ifj ij et h t h t A BM et But    , (20) . (21) =1 ()= ( ()) ( () ()) s fjjf j ut h tMqt qt    The structure of the reconfigured loop with a virtual actuator is in Fig. 1, where for simplicity =1 =1 = ( ()) , = ( ()) ss ii j ij j A htAM htM    . (22) Fig. 1: Block diagram of the virtual actuator. Theorem: The equilibrium of the system (11) with fuzzy virtual actuator is globally asymptotically stable if the pair (a(q(t)), Bf) be robust stabilizable on the prescribed area of the premise variables and there exist positive definite symmetric matrix and matrices j=1, 2,…s such that ,0X, nn RX   , mn jRZ   , (23) 0>= T XX , (24) <0 TTT ii jf fj XA AX Z B B Z  for (()) (()) 0 ij htht    , 0  t, and . sji ,1,2,=,  The set of the control law gain matrices is given as © 2012 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 77 CONTROL ENGINEERING VOLUME: 10 | NUMBER: 2 | 2012 | JUNE . (25) sjXZM jj ,1,2,=,= 1  Proof: Introducing the error between the state vector of the nominal and faulty system in the form )()(=)( tqtqte f  , (26) then using TS fuzzy model (11) and (18), and taking the time derivative of e(t) it can be directly obtained , (27) =1 =1 ()= ( ()) ( ())( ) () () ss ijifj ij et h t h t A BM et Bu t    c The equations (11), (12), and (27) can be expressed in the matrix form =1 =1 0 () () =(())(()) () () ss ij ij jifj ij ABK x tx ht ht et BK A BM et                 t   n ) T 1 = . (28) It is evident from (28) that the separation principle can be used and to design separately the fuzzy controller and the fuzzy virtual actuator. Since autonomous error dynamics of the closed loop system with fuzzy virtual actuator is given by , (29) =1 =1 ()= ( ()) ( ())( )() ss ijifj ij et h t h t A BM et     defining the Lyapunov function of the form , (30) (()) () () 0 T vet e tPet where , P > 0 is asymmetric positive definite matrix, then the Lyapunov function derivative takes the form Tn PP R   ,(31) =1 =1 ( ( )) ( ( )) ( ( )) ( ) ( ) 0 ssT ij v ij vet h t h t e tPet     where . (32) ()(T vifjifj PPABM ABMP  Thus, (31) implies that (29) is asymptotically stable if , (33) 0)()(  PMBAMBAP T jfijfi for , , and . (()) (()) 0 ij htht  0tsji ,1,2,=,  Since P is a regular matrix, then pre-multiplying left-hand side and right-hand side of (33) by P-1 gives , (34) 0)()( 11   T jfijfi MBAPPMBA , (35) 11 11 <0 TT iifj jf AP P A B M P P M B     respectively. Then, with the notations 1 , j j P XMPZ   , (36) (35) implies (24). This concludes the proof. 5. Illustrative Example The nonlinear dynamics of the system (1), (2) is represented by the six order model with the parameters 3 01 00,0500,05 2 ( ) 0,08 0,01 0,08 0,01 0 0,05 0 ( ) 0 0,05 ()=0,08 0,01 2 0,5 0,08 0,01 00,05 00,050 1 0,08 0,01 0,08 0,01 ( ) 2 xt pt A pt                          66 100000 =001000, 000010 T BCI        where the matrices B, C are the same for all local models. The interpretation of the nonlinear system in a TS fuzzy system gives 01 00,0500,05 2 1 0,08 0,01 0,08 0,01 00,050 1 0 0,05 =0,08 0,01 2 0,5 0,08 0,01 00,0500,050 1 0,08 0,01 0,08 0,01 1 2 l k i k b c A c                         with the associations 2)=2,=(4=2)=1,=(3= 1)=2,=(2=1)=1,=(1= klikli klikli    Evidently, vector of the premise variables can be chosen as follows     12 3 ()= () () = () ()tttqtp  t, and the premise variables and  0,3)(tp     1,4)( 3tq are bounded on the prescribed sectors. Thus, the set of nonlinear sector functions 4=1,=, )( =))(( 21 21 31 311 bb bb tqb tqw   , ))((1= )( =))(( 311 21 23 312 tqw bb btq tqw   , 3=0,=, )( =))(( 21 21 1 21 cc cc tpc tpw  , ))((1= )( =))(( 21 21 2 22 tpw cc ctp tpw   , implies the next set of normalized membership functions , ))(())((=))(),(( 2131131 tpwtqwtptqh , ))(())((=))(),(( 2131232 tpwtqwtptqh , ))(())((=))(),(( 2231133 tpwtqwtptqh © 2012 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 78 CONTROL ENGINEERING VOLUME: 10 | NUMBER: 2 | 2012 | JUNE , ))(())((=))(),(( 2231234 tpwtqwtptqh 010 20 30 40 0 1 2 3 time [s] p(t) Fig. 2: Time response of p(t). 0 5 10 15 20 25 30 -1 0 1 2 3 time [s] q 1 q 3 q 5 q 1 q 3 q 5 p(t) Fig. 3: Response of the fault-free system with fuzzy state controller. 5.1. Fuzzy Control Design Thus, solving (15), (16) with respect to the LMI matrix variables X and ,using Self-Dual-Minimization (SeDuMi) package for Matlab, then according to (17) the matrices were obtained as j Y 1 K 1,2,3,4=j 432 ,,, KKK 1 7,093 15,609 0,421 0,068 0,101 0,061 = 0,216 0,131 5,342 9,275 0,735 0,040 0,077 0,192 0,319 0,897 27,764 13,995 K            2 7,017 15,340 0,413 0,058 0,166 0,094 = 0,196 0,158 5,314 9,202 0,772 0,059 0,073 0,184 0,316 0,867 27,084 13,579 K           3 7,108 15,642 0,435 0,121 0,203 0,121 = 0,183 0,221 5,473 9,622 0,919 0,134 0,113 0,057 0,300 0,933 27,474 13,804 K         4 6,947 15,107 0,413 0,071 0,137 0,065 = 0,211 0,153 5,415 9,471 0,859 0,114 0,109 0,083 0,281 0,992 27,807 14,003 K         Figure 3 gives the simulation result of the fuzzy control for the fault-free systems, and to show the effectiveness of the fuzzy state control application for the nonlinear systems, where external signal was from prescribed sector (see Fig. 2). )(tp 5.2. The Failure of the Actuator As can see in Fig. 4, if the actuator failure occurs, the nominal fuzzy controller is not be able to stabilize the system. To stabilize the system, the fuzzy virtual actuator has to be incorporated into the control. 0 5 10 15 20 25 30 35 -5 0 5 10 time [s] q 1 q 3 q 5 q 1 q 3 q 5 Fig. 4: Response of the system with the first actuator fault (tf =26 s). 010 20 30 40 50 -3 -2 -1 0 1 2 3 time [s] q 1 q 3 q 5 q 1 q 3 q 5 Fig. 5: Response of the system with the first actuator fault (tf =26 s) and fuzzy virtual actuator action starting at tva = 30 s. 5.3. Fuzzy Virtual Actuator Design If the first actuator fault occurs, the matrix B is changed into matrix Bf as follow            100000 001000 000000 T f B Thus, solving (23), (24) with respect to the LMI matrix variables X and , using Self-DualMinimization (SeDuMi) package for Matlab the next matrix parameter were obtained j Z 2,M 1,2,3,4=j 43,MM 1,M 4 1 0000 0 0 = 0,088 0,056 0,03 0,007 0,001 0,001 10 1,284 0,801 0,047 0,033 0,002 0,01 M        4 2 0000 0 0 = 0,088 0,056 0,03 0,007 0,001 0,001 10 1,284 0,801 0,047 0,033 0,002 0,01 M        4 3 0000 0 0 = 0,088 0,056 0,03 0,007 0,001 0,001 10 1,284 0,801 0,047 0,033 0,002 0,01 M        4 4 0000 0 0 = 0,088 0,056 0,03 0,007 0,001 0,001 10 1,284 0,801 0,047 0,033 0,002 0,01 M        and was applied in the fuzzy virtual actuator design. © 2012 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 79 CONTROL ENGINEERING NUMBER: 2 | 2012 | JUNE VOLUME: 10 | © 2012 IN ELECTRICAL AND ELECTRONIC ENGINEERING 80 ADVANCES Simulation results (Fig. 5.) show on the effectiveness of the fuzzy virtual actuator application for faulty nonlinear systems. The presented results show that the nonlinear fuzzy controlled system with the failure of an actuator can be stabilized using the fuzzy virtual actuator. 6. Conclusion The paper presents new conditions suitable in design of a fuzzy virtual actuator for a class of continuous-time nonlinear systems represented by Takagi-Sugeno models, and measurable premise variables. Simulation results show on the effectiveness the fuzzy state controller application for the nonlinear systems. If after the actuator failure the fuzzy controller is not able to stabilize the system, the faulty system can be stabilized by incorporating the fuzzy virtual actuator into the control structure. The presented results show that such instable nonlinear system with an actuator failure can be stabilized using the fuzzy controller and fuzzy virtual actuator. Acknowledgements The work presented in this paper was supported by VEGA, Grant Agency of Ministry of Education and Academy of Science of Slovak Republic under Grant No. 1/0256/11 by Research & Development Operational Programme Grant No. 26220120030 realized in Development of Centre of Information and Communication Technologies for Knowledge Systems. These supports are very gratefully acknowledged. References [1] DING, S.X. Model-Based Fault Diagnosis Techniques. Design Schemes, Algorithms, and Tools. Berlin: Springer-Verlag, 2008. ISBN 978-3-540-76303-1. [2] KROKAVEC, D., A. FILASOVA, and V. HLADKY. Stabilizing fuzzy control for a class of nonlinear systems. In: Proceedings of the 10th IEEE International Symposium on Applied Machine Intelligence and Informatics SAMI 2012. Herlany, Slovakia, 2012, pp. 53-58. ISBN 978-1-4577-0197-9. [3] CHEN, J. and R. J. PATON. Robust Model-Based Fault Diagnosis for Dynamic Systems. Norwell: Kluwer Academic Publishers, 1999. ISBN 978-0792384113. 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Generic two-degree-of-freedom linear and fuzzy controllers for integral processes. Journal of The Franklin Institute. Engineering and Applied Mathematics. 2009, vol. 346, iss. 10, pp. 988-1003. ISSN 0016-0032. [10] ABDELMALEK, I., N. GOLEA, and M. L. HADJILI. A new fuzzy Lyapunov approach to non-quadratic stabilization of Takagi-Sugeno fuzzy models. International Journal of Applied Mathematics and Computer Science. 2007, vol. 17, no. 1, pp. 3951. ISSN 1641-876X. [11] BOYD, D., EL GHAOUI, L., PERON E. and V. BALAKRISHNAN. Linear Matrix Inequalities in System and Control Theory. Philadelphia: Society for Industrial and Applied Mathematics SIAM, 1994. ISBN 0-89871-334-X. [12] YANG, F. and Y. LI. Set-membership fuzzy filtering for nonlinear discrete-time systems. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics. 2010, vol. 40, iss. 1, pp. 116-124. ISSN 1083-4419. About Authors Anna FILASOVA was born in Kosice, Slovakia. She graduated in technical cybernetics and received M.Sc. degree in 1975, and Ph.D. degree in 1993 both from the Faculty of Electrical Engineering and Informatics, TU of Kosice, Slovakia. In 1999 she was appointed Associated Professor from the Technical University in Kosice in Technical Cybernetics. Her main research interests are in robust control, decentralized control, large-scale system optimization, and control reconfiguration. Vladimir SERBAK was born in Sobrance, Slovakia. He graduated in Cybernetics and received M.Sc. degree in 2011 from the Faculty of Electrical Engineering and Informatics, Technical University of Kosice, Slovakia. Since 2011 he is a PhD. student in the same study branch with the Department of Cybernetics and Artificial Intelligence, FEI TU Kosice, focusing on fault tolerant control systems design.