A Parameter Varying Lyapunov Function Approach for Tracking Control for Takagi-Sugeno Class of Nonlinear Systems

Transcription

1 2 8th IEEE International Conference on Control and Automation Xiamen China June 9-2 WeD6.6 A Parameter Varying yapunov Function Approach for Tracking Control for Takagi-Sugeno Class of Nonlinear Systems M. Ezzeldin A. Jokic and P.P.J. van den Bosch Abstract A reference model tracking control technique for nonlinear systems based on the Takagi-Sugeno model is proposed. The control design synthesis is aimed to reduce the tracking error for all bounded reference inputs and disturbances and to guarantee 2 gain performance. A nonlinear static output feedback controller is proposed to tackle this problem. Unlike the approaches using a single quadratic yapunov function a parameter varying quadratic yapunov function is employed in our approach. The controller synthesis is formulated in terms of a feasibility problem of a set of linear matrix inequalities which can be efficiently solved. A simulation example of a twolink robot system demonstrates the tracking performance and the validity of the proposed approach. I. INTRODUCTION The Takagi-Sugeno (T-S) model-based approach has nowadays become popular since it showed its efficiency to control complex nonlinear systems and has been used for many applications. The Takagi-Sugeno (T-S) models were introduced in [] with the purpose of approximating complex nonlinear systems and can be described as the weighted sum of local linear models. The weights depend on the working point of the nonlinear system. The T-S models can be seen as a generalization of piece-wise linear systems. Based on the so called Universal Approximation theorem [2] appropriate weighting of the local models can reproduce the original nonlinear system arbitrary well. T-S models have led to the design of a feedback controller for each local model which are further combined into a single overall controller. Stability and stabilization analyses of the T-S model have been investigated through yapunov direct method [3]-[5]. The key point of the proposed approaches is to achieve conditions in terms of MIs (linear matrix inequalities). For different approaches on this subject the interested reader is referred to [6]-[]. Tracking control design is an important issue for practical applications for example in robotic tracking control missile tracking control and attitude tracking control of aircraft. However there are very few studies concerning tracking control design based on the T-S model []-[4]. Most of the available tracking control techniques using T-S models This work has been carried out as part of the OCTOPUS project with Océ Technologies B.V. under the responsibility of the Embedded Systems Institute. This project is partially supported by the Netherlands Ministry of Economic Affairs under the Bsik program. M. Ezzeldin A. Jokic and P.P.J. van den Bosch are with the Department of Electrical Engineering Eindhoven University of Technology P.O. Box MB Eindhoven The Netherlands. m.ezz@tue.nl a.jokic@tue.nl and p.p.j.v.d.bosch@tue.nl are either based on state or observer based output feedback and based on the conventional single quadratic yapunov function. These techniques depend on finding a constant positive definite matrix of a quadratic yapunov function to satisfy the stability condition of the system. In [4] a fuzzy tracking controller design for discrete time systems is proposed using the feedback linearization technique. Similarly [5] has established a simple necessary and sufficient condition to determine local stability for the fuzzy systems and has derived a fuzzy tracking controller. In [] an H performance related to the tracking error for bounded reference inputs is formulated and an observer-based fuzzy controller is developed. In this paper we consider the tracking problem for nonlinear systems described by a T-S model with external disturbances. A reference model tracking based static output feedback controller is proposed to achieve minimal tracking error in terms of the desired 2 gain tracking performance. Instead of using the conventional quadratic yapunov function in our approach a parameter varying quadratic yapunov function is employed. The parameter varying yapunov function approach for stabilization problem was already studied in [7] where the considered yapunov function is a weighted sum of several quadratic yapunov functions. The weights are the same ones as used in the T-S model. Note that this yapunov function which is used in our approach as well presents a significantly wider class of functions than the conventional quadratic one. Initial formulation of the considered 2 gain tracking problem is necessarily given in terms of a nonlinear matrix inequality. A transformation is proposed to transform this nonlinear inequality into a set of MIs for which efficient optimization techniques are available. Furthermore the stability of the proposed controller is guaranteed for a general class of T-S systems. The main contributions of this paper are summarized as follows. The 2 gain static output feedback control design for the model tracking control is extended from linear system toward nonlinear system using nonlinear control. Moreover based on the proposed approach a simple algorithm in terms of MI optimization technique is developed. The remainder of the paper is organized as follows. Section II presents basic definitions of the T-S model and the problem formulation while Section III introduces 2 gain based tracking control. Section IV presents the MI formulation of the proposed approach. A simulation example is given in //$26. 2 IEEE 428

2 section V to illustrate the design effectiveness. Concluding remarks are collected in section VI. II. PROBEM FORMUATION In this section we recall the definition and some properties of the T-S model which is used to approximate a class of nonlinear system in the form: ẋ(t)= f(x(t))+g(x(t))u(t)+b w w(t) y(t)=cx(t) where x(t) R n denotes the state vector u(t) R m denotes the control input w(t) R m w denotes the unknown but bounded disturbance the vector fields f(x(t)) and g(x(t)) are smooth functions with f()= and y(t) R p is the output of the system which depends linearly on the states with C R p n. Next we will formally present the T-S model as a weighed average of several linear models through weighting functions. First consider a set of linear models of the form: ẋ(t)=a i x(t)+b i u(t)+b wi w(t) y(t)=c i x(t) for i = where A i R n n B i R n m C i R p n and is the number of linear models. The T-S model is defined as: ẋ(t)= h i (z(t))(a i x(t)+b i u(t)+b iw w(t)) Σ : i= y(t)= (3) h i (z(t)) C i x(t) i= with h i (z(t)) for i = i= h i(z(t)) = and z(t) are triggering variables which depend on the current value of system states x(t). In this paper we will assume that C i = C for i = what is in conformance with the output of the original nonlinear system (). Remark : An example for the membership functions h(z(t)) is shown in Fig.. Since in this paper we are concerned with the output feedback control the triggering variables z(t) are considered to be dependent only on the measured output y(t). () (2) The T-S model (3) is a general nonlinear time-varying dynamic system. Without going into more details the T-S model (3) can capture the piecewise linear system (PW) accurately if h i (z(t)) belongs to the discrete set {}. To define the output tracking problem we first introduce a linear time invariant reference model given by: x r = A r x r (t)+b r r(t) y r = C r x r (t) where x r (t) is the reference state A r is an asymptotically stable matrix r(t) is a bounded reference input and y r (t) is the reference output. It is assumed that for all t y r (t) represents a desired trajectory for y(t) to follow. The choice of the reference model parameters mainly depends on the desired closed loop behavior e.g. system overshoot settling time etc. For more details about how to choose the reference model see [6]. III. OPTIMIZED 2 GAIN BASED TRACKING CONTRO In this paper we are concerned with the controller synthesis which incorporates the reference model tracking performance criteria. Next we recall the definition of dissipativity and the dissipativity characterization of the upper bounds for the 2 gain. Definition : The system Σ with state x R n input w R m w and output y R p is dissipative with respect to the supply function s : R m w R p R if there exists a positive definite continuous function V : R n R + called a storage function such that V() = and V(x(t )) V(x(t )) t t s(w(t)y(t))dt for all t t and for all corresponding solutions (wxy) to the system Σ on the interval [t t ]. Proposition : The system Σ has a finite 2 -gain from input w to the output y smaller than or equal to ρ if it is dissipative with respect to the supply function s(wy)=ρ 2 w w y y. (4) h(z(t)) Next we present the structure of the output feedback controller considered in this paper. Consider a set of static feedback controllers of the following form: u(t)=k j [y(t) y r (t)] for j =. Then the overall controller is defined by: z(t) Fig.. Normalized membership functions. u(t)= j= h j (z(t))k j [y(t) y r (t)] (5) where K j are the controller gains for j = and the functions h j (z(t)) for j = are the same ones as in (3). The closed loop system can be expressed as the following 429

3 augmented system: ẋ(t) = x r (t) h i (z(t))h j (z(t)) i= j= Ai + B ( i K j C B i K j C r x(t) Bwi w(t) + ) A r x r (t) B r r(t) (6) In the sequel the following abbreviations are adopted: x :=[x(t) [ x r (t) ] w :=[w(t) ] r(t) [] C :=[C C r ] Ai + B à ij := i K j C B i K j C r Biw E A i := r B r ] and the output tracking error ỹ(t) := y(t) y r (t). Therefore the closed loop system (6) can be expressed in the following compact form: x = i= j= ỹ = C x. h i h j (à ij x+e i w) For a given ρ R ρ > the objective is to design a static feedback controller of the form (5) such that the corresponding closed loop system (7) has 2 gain from w to ỹ less than or equal to ρ. The 2 gain tracking performance criteria for the closed loop system is defined in Proposition with a supply function selected as follows: s( wỹ)=ρ 2 w w ỹ ỹ = ρ 2 w w x Q x (8) [ C where Q = C C ] C r Cr C Cr. C r Theorem : Suppose that h k λ k λ k >. Given the closed loop system (7) if there exist symmetric positive definite matrices P k as a solution of the following matrix inequalities : à T ij P k + P k à ij + ˆP λ + ρ 2 P ke i E T i P k + Q < i jk = with ˆP λ = λ k(p k + Y) and where λ k are scalars Y is a symmetric matrix then the 2 gain tracking control performance in (8) is guaranteed for a prescribed value of ρ. Proof: Consider the candidate yapunov function: V( x(t)) = (7) (9) h k (z(t)) x (t)p k x(t) () where the function h k (z(t)) for k = are the same ones as in (3). The time derivative of V(t) along the systems trajectories is given by: + h k x (t)p k x(t)+ h k x (t)p k x(t). h k x (t)p k x(t) Furthermore using (7) we have i= j= h i h j h k ( x (t)ã ij P k x(t)+ x (t)p k à ij x(t) + w (t)e i P k x(t)+ x (t)p k E i w(t))+ h k x (t)p k x(t). () Since h k = hence h k = it follows that h k Y = for arbitrary matrix Y. Adding h k Y to () with Y R 2n 2n and using the inequality h k λk it follows that: V( x(t)) i= j= + ρ 2 P ke i Ei P k ) x(t)+ ρ 2 w (t) w(t) h k h i h j x (t)(ã ij P k + P k à ij inequality (2) further implies V( x(t)) λ k x (t)(p k +Y) x(t)+ ( ) ( ρ E i P k x(t) ρ w ρ P kei i= j= ) x(t) ρ w h k h i h j x (t)( ˆP λ + à ijp k + P k à ij + ρ 2 P ke i E i P k ) x(t)+ρ 2 w (t) w(t). (2) with the abbreviation ˆP λ := λ k(p k +Y). Using (9) this further implies the following inequality V( x(t)) i=k i= j= h k h i h j x (t)( Q) x(t)+ρ 2 w (t) w(t). (3) From the properties of h i (z(t)) the inequality (3) implies the following inequality: V( x(t)) x (t)( Q) x(t)+ρ 2 w (t) w(t). Integrating both sides from to t f we obtain t f x (t) Q x(t)dt x ()P x()+ρ 2 tf w (t) w(t)dt Therefore according to Proposition the 2 gain performance is achieved with a prescribed ρ. To obtain a better tracking performance the tracking control problem can be formulated as the following minimization problem: min P k K j ρ ρ2 subject to P k > k = and (9) (4) Theorem 2: Given the closed loop system (7). If there exist symmetric positive definite matrices P k k = for the minimization problem (4) then the closed loop system (7) is quadratically stable. Proof: Set w = and consider the candidate yapunov function V( x(t)) = h k x (t)p k x(t). 43

4 Then we can write the following equality h k ( x (t)p k x(t)+ x (t)p k x(t)+ h k x T (t)p k x(t)) i= j= h k h i h j x (t)(ã ij P k + P k à ij + ˆP λ ) x(t). Using (9) from the above inequality we deduce that V( x(t)) < In the next section we will present a constructive algorithm to compute the controller gains based on Theorem in terms of an MI feasibility problem. IV. MI FORMUATION A class of numerical optimization problems called linear matrix inequality (MI) problems has received significant attention over the last decade [7]-[9]. These optimization problems can be solved in polynomial time via the interior-point methods [8] and hence are tractable at least in a theoretical sense. Note that (9) is not an MI in P k and K j as variables. Therefore a transformation method is proposed to reformulate (9) as a feasibility problem of a set of MIs. This transformation method is summarized in the following emma. emma : The following MI implies the inequality (9): Pk Q Q P k + ˆP λ + Q ( ) E 2ρ 2 i Ei P k + A ij + Q+ P ρ 2 k 2 < (5) I ρ 2 for i jk =. Proof: Pre- and Post-multiplying (5) by [ I I Pk respectively yields à ij P k + P k à ij + ˆP λ + ρ 2 P ke i E i P k + Q < P k ] and Note that (5) is an MI in P k and K j and can therefore be efficiently solved by a dedicated software. According to the above analysis the tracking control via nonlinear static output feedback is now summarized as follows: Design Procedure: ) Select weighing functions h i (z(t))i = 2 and construct a T-S model as in (3). 2) Select an initial value of ρ 2 and λ k. 3) Solve the MI problem in (5). 4) Decrease ρ 2 and repeat steps 3-4 until the MI problem can not be solved. 5) Construct the controller (5) and verify whether h k λ k. Remark 2: The time derivative of () contains information on the time derivative of the weighting functions h k. It s assumed that the time derivative of the weighting functions h k are upper bounded with a given value which may lead to some conservatism in the MI problem. However the use of structural information on T-S model to construct the yapunov function is an advantage in comparison with disregarding the time derivative of the weighting functions. Moreover the stability proof is based on this assumption hence it should be checked that this assumption is satisfied to have a complete stability proof. Furthermore note that the number of MIs (5) are 3 which will have a high computational time however the number of the MIs can be further reduced to taking into account that not all the T-S subsystems are triggered at the same time [5]. V. SIMUATION EXAMPE To illustrate the proposed design approach and show its efficiency the results are applied to the two-link robot system. The dynamic equations of the two-link robot system are given as follows []: M(q) q+c(q q) q+g(q)=t (6) where [ (m M(q)= + m 2 )l ] m 2 l l 2 (s s 2 + c c 2 ) m 2 l l 2 (s s 2 + c c 2 ) m 2 l2 2 q2 C(q q)=m 2 l l 2 (c s 2 s c 2 ) q (m + m G(q)= 2 )l gs 4 m 2 l 2 gs 2 and the system output y = q =[q q 2 ] q q 2 are generalized coordinates M(q) is the moment of inertia C(q q) includes Coriolis centrifugal forces and G(q) is the gravitational force. Other quantities are: link mass m m 2 [kg] link length l l 2 [m] angular position q q 2 [rad] applied torques T =[T T 2 ] [Nm] the gravity g = 9.8[m/s 2 ] and s = sin(q )s 2 = sin(q 2 )c = cos(q )c 2 = cos(q 2 ) et x = q x 2 = q x 3 = q 2 and x 4 = q 2 then the Takagi- Sugeno model which approximates the nonlinear system (6) can be represented using the following set of linearized models []: R If π 2 x and π 2 x 3 then ẋ = A x+b u+w z = C x R2 If π 2 x and π 2 x 3 π 2 then ẋ = A 2 x+b 2 u+w z = C 2 x R3 If π 2 x and x 3 π 2 then ẋ = A 3 x+b 3 u+wz = C 3 x R4 If π 2 x π 2 and π 2 x 3 then ẋ = A 4 x+b 4 u+w z = C 4 x R5 If π 2 x π 2 and π 2 x 3 π 2 then ẋ = A 5 x+b 5 u+w z = C 5 x R6 If π 2 x π 2 and x 3 π 2 then ẋ = A 6 x+b 6 u+w z = C 6 x 43

5 R7 If x π 2 and π 2 x 3 then ẋ = A 7 x+b 7 u+w z = C 7 x R8 If x π 2 and π 2 x 3 π 2 then ẋ = A 8 x+b 8 u+w z = C 8 x R9 If x π 2 and x 3 π 2 then ẋ = A 9 x+b 9 u+w z = C 9 x where x = x x 2 x 3 x 4 u = T T 2 and w =. w w 2 w 3 w 4 A = A 2 = A 3 = A 4 = A 5 = A 6 = A 7 = A 8 = A 9 = B = B 2 =.5 B 3 = 2 2 B 4 =.5 B 5 = B 6 =.5 2 B 7 = B 8 =.5 B 9 = 2 2 C i = for i = 9 The triangle type of weighting functions for the triggering variables x and x 3 in each rule are adopted for the T-S model as illustrated in Fig. and λ k =.8 for k =. The reference model (4) parameters are given by [] A r = 6 5 B r = and 6 5 C r = C i. Using Theorem and the MI optimization toolbox in Matlab the following controller gains K j for j = are obtained K = K 2 = K 3 = K 4 = K 5 = K 6 = K 7 = K 8 = K 9 = Different reference and disturbance signals are applied in simulation of the closed loop system. Fig. 2-3 present the simulation results for the proposed tracking control for T-S model. The simulation results indicate that the closed loop system can efficiently track the reference model in the presence of external disturbances and the desired performance can be achieved. It is found that the tracking error between the reference output y r (t) and actual closed loop output y(t) is always less than.3 and h k.4 for k =. 432

6 y(rad) Time(sec) Fig. 2. The trajectories of the output variable y (the proposed nonlinear control: dashed line) reference output y r (solid line) and external disturbance w (dotted line) y2(rad) Time(sec) Fig. 3. The trajectories of the output variable y 2 (the proposed nonlinear control: dashed line) reference output y 2r (solid line) and external disturbance w 2 (dotted line) VI. CONCUSION In this paper the problem of reference model tracking of nonlinear systems has been considered. Based on the Takagi- Sugeno model a nonlinear static output feedback controller is developed to reduce the tracking error by minimizing the attenuation level in terms of 2 gain. The novelty of the presented approach is that it is based on a parameter varying yapunov function. The advantage of the proposed tracking control design is that a simple static output feedback controller is used without feedback linearization technique and a complicated adaptive scheme. The complete synthesis problem has been formulated in terms of MI feasibility which can be efficiently solved using a dedicated software. Simulation results based on a two-link robot arm have demonstrated the efficiency of the proposed algorithm. REFERENCES [] T. Takagi and M. Sugeno Fuzzy identification of systems and its applications to modeling and control IEEE trans. on Systems Man and Cybernetics vol. 5 no. pp [2]. Wang A course in fuzzy systems and control Prentice Hall ondon UK 997. [3] K. Tanaka and M. Sugeno Stability analysis and design of fuzzy control systems Fuzzy Sets Syst. vol. 45 no. 2 pp Jan [4] S. Cao N. Rees and G. Feng Analysis and design of fuzzy control systems using dynamic fuzzy global models Fuzzy Sets and Syst. vol. 75 no. pp Oct [5]. Tanaka and H. Wang Fuzzy control systems design and analysis: A linear matrix inequality approach John Wiley & Sons Inc.2. [6] K. Tanaka T.Ikeada and H. Wang Fuzzy regulators and fuzzy observers: Relaxed stability conditions and MI-based designs IEEE trans. on Fuzzy Systems vol. 6 no.2 pp. -6 May 998. [7] K. Tanaka T.Hori and H.O. Wang A multiple yapunov function approach to stabilization of fuzzy systems IEEE Trans. on Fuzzy Systems vol. no. 4 pp [8] B. Ding H. Sun and P. Yang Further studies on MI-based relaxed stabilization conditions for nonlinear systems in Takagi-Sugeno s form Automatica vol. 42 no. 3 pp Mar. 26 [9] M. Ezzeldin H.M. Emara A. Elshafei and A. Bahgat Observer based fuzzy-control design using relaxed MI conditions IEEE Conference on Decision and Control pp Dec. 27. [] H. Fang Y. iu W. Kau. Hong and H. ee A new MIbased approach to relaxed quadratic stabilization of T-S fuzzy control systems IEEE Trans. Fuzzy Syst. vol. 4 no. 3 pp Jun. 26 [] C. S. Tseng B. S. Chen and H. J. Uang Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model IEEE Trans. Fuzzy Syst. vol. 9 no. 3 pp Jun. 2. [2] C. S. Tseng and B. S. Chen H Decentralized fuzzy modeling reference tracking control design for nonlinear interconnected systems IEEE Trans. Fuzzy Syst. vol. 9 no. 6 pp Dec. 2. [3] C. S. Tseng and B. S. Chen Output Tracking Control for Fuzzy Via Output Feedback Design IEEE Trans. Fuzzy Syst. vol. 9 no. 6 pp Dec. 2. [4] C.C. Kung and H.H. i Tracking control of non linear systems by fuzzy model-based controller Proc. IEEE Int. Conf. pp Jul [5] H. Ying Analytical analysis and feedback linearization tracking control of the general Takagi-Sugeno fuzzy dynamic systems IEEE Trans. Syst. Man Cybern. pp May 999 [6] andua and D. Youan. Adaptive Control: The Model Reference Approach New York: Dekker 979. [7] S. Boyd. El Ghaoui E. Feron and V. Balakrishnan inear Matrix Inequalities in System and Control Theory Philadelphia (PA:SIAM)994. [8] Stephen Boyd and ieven Vandenberghe Convex Optimization 24 (Cambridge University Press) Internet site http: [9] H.D.Tuan P. Apkarian T.NariKiyo and Y.Yamamoto Parameterized linear matrix inequality techniques in fuzzy system design IEEE Trans. Fuzzy Syst. vol. 9 no.2 pp Apr