Robust Observer for Uncertain T S model of a Synchronous Machine

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1 Recent Advances in Circuits Communications Signal Processing Robust Observer for Uncertain T S model of a Synchronous Machine OUAALINE Najat ELALAMI Noureddine Laboratory of Automation Computer Engineering LAI University Mohammed V Agdal Mohammadia School of Engineers Box: 765 Rabat Morocco n_ouaaline@yahoo.com; prelalami@gmail.com; elalami@emi.ac.ma. Abstract: - This paper deals with the problem of fuzzy observer of nonlinear uncertain systems to study the continuous model of a synchronous machine in order to get an estimation of the immeasurable states vector. The uncertain non-linear model is first represented by a Takagi-Sugeno fuzzy modeling. Next we develop a technique for designing a multimodel observer corresponding to this system show its asymptotic convergence. The gain s estimation is obtained by solving a set of linear matrix inequalities LMIs. This observer scheme gives the estimated states converging to the real states exponentially. Finally the simulation results demonstrate the effectiveness of the proposed method Key-Words: - Continuous T-S systems fuzzy observer synchronous machine uncertain LMI. 1 Introduction Design of robust tracking control for uncertain nonlinear systems has attracted great attention in the past few years. Among nonlinear control theory the Takagi Sugeno T S fuzzy model-based approach has nowadays become popular since it showed its efficiency to control complex nonlinear systems has been used for many applications; see e.g Indeed Takagi Sugeno have proposed a fuzzy model to describe nonlinear models 6 as a collection of linear time invariant models blended together with nonlinear functions. A control law called parallel distributed compensation PDC can be synthesized as a collection of feedback gains that are connected using the same nonlinear functions 36. In this paper for Synchronous Machine we consider robust stability stabilization of uncertain affine T S fuzzy systems where uncertainties come into the state input matrices. The majority of T-S fuzzy controller design was developed by using the fuzzy observer design based on PDC controller of a Synchronous Machine. The nonlinear model is represented by T-S fuzzy model 9. The stability of the whole closed-loop model is investigated using the well-known quadratic Lyapunov function. The main contribution of the paper is the proposition of a LMI formulation to derive the proposed robust output feedback control law. The Lyapunov stability criterion. Moreover the PDC control scheme 9 is used in this paper to attack the problem of robust stability design problems for the perturbed system. Then a simple fuzzy controller is designed via the parallel distributed compensation PDC technique to realize the stabilization of the perturbed affine T S fuzzy models. The robust stability of the closed-loop synchronous machine system is proved based on Lyapunov criterion. Based on the Linear Matrix Inequalities LMI technique one can find a suitable common positive definite matrix for the stability conditions then to obtain a stable fuzzy controller for the closed-loop T S fuzzy models. Finally by using simulation we show that the proposed method can be successfully used to control a synchronous machine it is very robust to the TS model with uncertainties parameter. The paper is organized as follows. Section is dedicated to the description of the uncertain continuous-time T-S model. In section 3 we illustrated a Fuzzy observer based on PDC type T-S fuzzy control robust stability conditions reformulated into solving an LMI problem. Section deals with the description of the mathematical model of synchronous machine which is transformed to a TS fuzzy model in section. To illustrate the ISBN:

2 Recent Advances in Circuits Communications Signal Processing effectiveness of the proposed methods simulation results are given. In the rest of the paper the following useful notation is used: XT denotes the transpose of the matrix X X denotes a symmetric positive definite semidefinite matrix I denotes an identity matrix +. 6 The structured uncertainties considered here are norm-bounded in the form: The uncertainties matrices are represented for i=1 as follows : Where Where Dia Dib Eia Eib are known real constant matrices of appropriate dimension are unknown matrix functions satisfying : Uncertain T-S fuzzy model Consider an uncertain T-S fuzzy continuous model with time delay for a nonlinear system as follows: Model Rule i : {. / The following lemma commonly used in several papers is recalled. 1 Lemma 17: Given constant matrices D E unknown constant matrix F of appropriate dimension satisfying constraint. The following two propositions are equivalent: Where i=1 r. r is the number of IF-THEN rules. The normalized activation function corresponding to the ith submodel is such that 3 1: i ii Where Mij i = 1 r; j = 1 p is the fuzzy set r is the number of model rules; is the state vector is the input vector are constant real matrices z1t zpt are known premise variables. are time-varying matrices representing parametric uncertainties in the plant model. These uncertainties are admissibly norm-bounded structured. The final outputs of the fuzzy systems are: 3 Robust fuzzy observer design In this paper we assume that all of the state variables are not measurable. Fuzzy state observer for T-S fuzzy model with parametric uncertainties 1 is formulated as follows: { Where Since We have +. / From we obtain: We have the following property: ISBN: The fuzzy observer design is to determine the local gains Li in the consequent part. Where is the estimated state vector the activation function is the same than that used in the TS model verifying. One considers the following state estimation error defined as: 8 When the estimated state is available we can consider the global control law with PDC technique as follows:

3 Recent Advances in Circuits Communications Signal Processing The final fuzzy model is described by only two rules 1. The premise vector is defined by with For a premise terms define Next calculate the minimum maximum values of under. They are obtained as follows: } 1 The membership functions are respectively 11 The Takagi-Sugeno fuzzy model of the synchronous machine are described by the four matrices Ai Bi Ci i=1... as follows: 1 The following result based on the works 81 proposes stabilization conditions designing an observer-based robust fuzzy control of uncertain fuzzy models 1 by introducing supplementary semidefinite matrices tacking account the numbers of sub-models simultaneously activated r. Theorem 1 7: the state estimation error converges asymptotically to zero if there exists positive symmetric matrices P gains Ki positive scalars ε1 ε satisfying the following conditions : P ε1 ε 13 Where: The gain of observer is given by - - ISBN: With; And attenuation level is derived from 77 -

4 Recent Advances in Circuits Communications Signal Processing The T.S fuzzy model exactly represents the nonlinear systems. Notice that this fuzzy model has the commons B C matrix. The uncertainties matrices represented for i=1 as follows : Where state vector x1 x x3 x x5 5 Simulation Results - To visualize the fuzzy controller obtained from the proposed method we consider the parameter values given in Table 1 which are retained from the literature 1. Here we have employed for the simulation a initial conditions Many tests have been performed to prove the goodness of the proposed fuzzy control system. Some results obtained by means of the SIMULINK program of MATLAB are reported in what follows. For these simulations the model rules are chosen for -. - We presents only the results for the value of a =.8. Every set of LMIs was solved via the MATLAB LMI toolbox. The synthesis of the controller consists of finding the feedback gains of the conclusion parts Ki which guaranteed the robust asymptotic stability of the closed loop TS uncertain of the synchronous machine. ISBN: The trajectories of the state vector were illustrated by the simulation results in the case of robust control stability. Figs.1 displays overall simulation results of the state vector the estimated vector. are Time Estimed State vector Time 3 Fig.1 The simulations results displays the trajectory of the state vector the estimated state vector 5 Conclusion This paper presents a fuzzy observer for a synchronous machine with damper infinite bus described by uncertain T-S models with measurable decision variables. The method proposed here are developed in order to estimate the state of the system despite the presence of modeling uncertainties. The convergence conditions of the observer are established by using the quadratic function of Lyapunov. Simulation results have verified confirmed the effectiveness of our approach in controlling nonlinear systems with parametric uncertainties. 78

5 Recent Advances in Circuits Communications Signal Processing Appendix 3 N. OUAALINE N. ELALAMI. Observer Based Stabilization of T-S Systems for a synchronous machine without damper. ICGST International Conference on Computer Science Engineering CSE December 11 Istanbul Turkey ; ; ; + ; N. OUAALINE N. ELALAMI. Exponential stability stabilization of T-S System for synchronous machine without Damper. International Journal Of Mathematica Models Method In Applied Sciences Issue 8 Volume 5 11 pp : ; 5 N. OUAALINE N. ELALAMI " Statis Output Feedback For T-S Fuzzy Model Of a Synchronous Machine" WSEAS International Conferences Cambridge UK February H.-N.Wu Robust H fuzzy output feedback control for discrete-time nonlinear systems with parametric uncertainties Internat.J.Approx. Reason K. Zhou J. C. Doyle. Essentials of robust control. Prentice-Hall S. K. Hong R. Langari An LMI-based H fuzzy control system design with T-S framework Information science no. 13 pp M. Akar U. Ozguner Decentralized techniques for the analysis control of T-S fuzzy systems IEEE Trans. on Fuzzy Systems vol. 8 no. 6 pp K. S. Narendra J. Balakrishnan A common Lyapunov function for stable LTI systems with commuting A-matrices IEEE Trans. on Automatic Control vol. 39 no. 1 pp W.L. Chiang T.W. Chen M.Y. Liu C.J. Hsu Application robust H control of PDC fuzzy controller for nonlinear systems with external disturbance Journal of Marine Science Technology Vol. 9 No. 1 pp T. Takagi M. Sugeno Fuzzy identification of systems its application to modelling control IEEE Transactions on Systems Man Cybernetics Vol.15 No pp T. A. Johansson Computation of Lyapunov functions for smooth nonlinear systems using convex optimisation Automatica 3611 pp H.O. Wang K. Tanaka M. F. Griffin An approach to fuzzy control of Nonlinear systems: ;. TABLE I Machine Synchronous data capacity power VA Symbol Quantity d - axis magnetic reactance d -axis transient reactance d -axis subtransient reactance q - axis magnetic reactance q -axis subtransient reactance field leakage reactance magnetic dipole moment d -axis transient open circuit time constant d -axis subtransient open circuit time constant q -axis subtransient open circuit time constant synchronous rotor angular Value p.u References: 1 N.OUAALINE N. ELALAMI. Takagi-Sugeno fuzzy control of a synchronous machine. In Control Fault-Tolerant Systems SysTol 1 Conference on pp N. OUAALINE N. ELALAMI. Observer Design via Takagi-Sugeno Fuzzy of a Synchronous Machine. SIL 1 ENSA Marrakech & 1 October 1. ISBN:

6 Recent Advances in Circuits Communications Signal Processing Stability Design Issues IEEE Transactions on Fuzzy Systems Vol. No pp C.-W. Park LMI-based robust stability analysis for fuzzy feedback linearization regulators with its applications Information Sciences Vol. 15 No 1 3 pp S. Boyd L. El Ghaoui E. Feron V. Balakrishnan Linear Matrix Inequalities in System Control Theory SIAM Philadelphia PA K.Tanaka M.Sugeno Stability Analysis Design of Fuzzy Control Systems Fuzzy Sets Systems Vol. 5 No. 199 pp K. Tanaka T. Ikeda H.O. Wang Fuzzy regulators fuzzy observers: relaxed stability conditions LMI-based designs IEEE Trans. on Fuzzy Systems Vol.6 No pp W.L. Chiang T.W. Chen M.Y. Liu C.J. Hsu Application robust H control of PDC fuzzy controller for nonlinear systems with external disturbance Journal of Marine Science Technology 9 1 pp A.BOUZIT Stabilité et Stabilisation des systèmes Dynamiques Application au cas d un réseau électrique Thèse 199 Université de Perpignan. 1 Yann MORERE Mise en oeuvre de lois de comme pour les modèles flous de type Takagi-Sugeno LAMIH-SF Thèse 1 L'Université de Valenciennes et du Hainaut- Cambrésis. M. Chadli On the Stability Analysis of Uncertain Fuzzy Models International Journal of Fuzzy Systems Vol. 8 No. December 6. ISBN: