Robust PID Controller Design for Nonlinear Systems

Robust PID Controller Design for Nonlinear Systems Part II Amin Salar 8700884 Final Project Nonlinear Control Course Dr H.D. Taghirad 1

About the Project In part one we discussed about auto tuning techniques for PID controllers, and a number of frequency response methods have been introduced, such as: Prototype Frequency Response Approach (Nonparametric) Parametric Approach Using Finite Number of Frequency Response Data In the first two topics which have been mentioned above we explained two ways for finding PID parameters to control nonlinear systems, after searching through the internet we found that like authors in [1] said, most of the tuning methods mentioned in papers and articles are only suitable to linear systems. They introduced a fuzzy modeling approach for nonlinear ones. So in part 2 of the project we will explain this technique that is one of the few main ways for designing PID controllers for nonlinear systems, and we will show the results of our simulations with simulink software. Abstract The design problem of proportional and proportional plus integral (PI) controllers design for nonlinear systems via fuzzy modeling approach is studied in this chapter. First, the Takagi Sugeno (T S) fuzzy model with parameter uncertainties is used to approximate the nonlinear systems. Then a numerically tractable algorithm based on the technique of iterative linear matrix inequalities is developed to design a proportional (static output feedback) controller for the robust stabilization of the system in T S fuzzy model. Third, we transform the problem of PI controller design to that of proportional controller design for an augmented system and thus bring the solution of the former problem into the configuration of the developed algorithm. Finally, the proposed method is applied to the design of robust stabilizing controllers for the excitation control of power systems. Simulation results show that the transient stability can be improved by using a fuzzy PI controller, compared to the conventional PI controller designed by using linearization method around the steady state. I. INTRODUCTION Despite the developments of various kinds of modern or postmodern control theories, such as LQG or LQR optimal control, control, and analysis and synthesis, classical proportional plus integral (PI) or proportional plus integral derivative (PID) controllers are widely used in industry due to their relatively simple structure, ready in hand implementation, and perhaps, being easily understood. Therefore, it is often the case that in practical applications one first considers PID controllers unless evidence shows that they are insufficient to meet specifications. Because of the popularity of PID controllers in the real world, many approaches have been developed to determine the parameters of PID controllers. The first systematic tuning method for PID parameters was proposed by Ziegler and Nichols in the early 1940s. Then came the well known formulae such as the Cohen Coon method, integral absolute error (IAE) optimum method, integral time weighted absolute error (ITAE) optimum method, internal model control (IMC) method, and relay auto tuning method. Recently, many modified tuning methods have also been proposed associated with different performance specifications or different methods used accordingly. Feng Zheng, Qing Guo Wang, Tong Heng Lee, and Xiaogang Huang in [1], declared that all of the aforementioned tuning methods are only suitable to linear systems and introduced a fuzzy modeling approach for nonlinear ones (which we are going to explain) but in some articles 2

including [2] authors introduced a way for Robust self tuning PID controller for nonlinear systems which I have explained completely in the first part of my project. But it is true that compared with the voluminous references on tuning PID controllers for linear systems, few results have been reported in the literature on PID controller tuning for nonlinear systems. Indeed, many industrial processes can be approximated sufficiently well in concerned operating region of state space by linear systems. However, many other plants exist whose dynamics must be described by nonlinear systems. Therefore, it is highly desirable to develop effective methods to determine the parameters of PID controllers for nonlinear systems. As I said so apart from the above works, the only reference discussing this problem is the monograph by Freeman and Kokotovic [3], where backstepping and control Lyapunov function methods are used to design PI control, but it is supposed that all state variables are available. In this chapter as told before, we will try to take a different approach to design PI controllers for nonlinear systems. As shown later, our controllers will use output variables only or partial state variables. As in the design methodologies of many other types of controllers, a crucial step for the design of PID controllers is the modeling of the nonlinear plants. Since the 1980s, fuzzy technique has been widely adopted to model complex nonlinear plants. The so called T S or T S K fuzzy model, first proposed by Takagi and Sugeno [4] and further developed by Sugeno and Kang [5], is one of most successful models in this direction. The basic idea in this approach is first to decompose the model of a nonlinear system or other kinds of complex systems into linear systems in accordance with the cases for which linear models are suitable to describe and then to aggregate (fuzzy blend) each individual model (linear model) into a single nonlinear model in terms of their membership function. Thus, the relatively complex consequence part allows the number of fuzzy rules (local models) to be quite small in many applications. Consequently, the T S fuzzy model is less prone to the curse of dimensionality than other fuzzy models. As is well known, the key problem in this approach is to what degree the nonlinear system can be approximated by a convex (fuzzy) blending of several linear systems. Theoretical justification of T S fuzzy model as a universal approximator has been given by Wang and Mendel [6]. Now T S model has found wide applications in the control of complex systems, e.g., in the control of robot manipulators and time delay systems. In this chapter, we will apply T S fuzzy modeling approach to the design of robust PI controllers for nonlinear systems. This chapter is organized as follows. The problem formulation is introduced in Section II. In Section III, proportional (static output feedback) controller design for nonlinear systems using T S fuzzy models is presented. PI controller design for nonlinear systems is given in Section IV. A numerical example is provided in Section V to show the design procedure and the effectiveness of the proposed method. Finally, concluding remarks are drawn in Section VI. II. PROBLEM FORMULATION Consider the following uncertain nonlinear system:, Δ,,, (1) Where vector of state variables vector of control inputs vector of outputs vector of uncertain parameters which is restricted to a prescribed bounding set nonlinear functions of (x,t), representing the nominal model of the system under consideration, are continuously differentiable with respect to (x,t) 3

Δ represents the model uncertainties constant matrix Ignoring the uncertainty term, we obtain the nominal model of the system (1) as follows:,, (2),(3) To deal with the problem for the system (2) to run at different operating points or in different conditions, Takagi and Sugeno [4] proposed an effective method to represent the system (2). The main feature of a T S fuzzy model is to express the join dynamics of each fuzzy implication (rule) by a linear system. The ith rule of the T S fuzzy model is of the following form. Plant Rule i (i=1,,p): IF θ 1 is μ i1 and and θ p is μ ip THEN Where θ j (x) (j=1,,p) are the premise variables, which are functions of state variables x; μ ij (i=1, r,j=1,,p ) are fuzzy sets; r is the number of the IF THEN rules; p is the number of the premise variables; and A i and B i are constant matrices of compatible dimensions. It is assumed that the premise variables are independent of the input variables u(t). The overall fuzzy model is achieved by fuzzy blending (aggregation) of each individual rule (model) as follows: (4) Where θ = [θ 1,, θ p ], and ω i : R p > [0,1], i=1,,r, are the membership function of the system belonging to plant rule i. Define The system (4) admits the following form: (5) with the constraints 0, 1 (6) Stripping off the fuzzy cover of system (5), one can see that the model (5) is nothing but the convex combination of several linear systems. It has been shown [6], [7] that a nonlinear system can be approximated by means of the above fuzzy basis functions to desired accuracy inside an arbitrarily large compact subspace of R n. Correspondingly, we can model uncertain nonlinear system (1) as the following uncertain fuzzy system: Plant Rule i (i=1,,r): IF θ 1 is μ i1 and and θ p is μ ip THEN Δ,, (7) where ΔA i represents the uncertainties in system matrix. We assume that ΔA i admits the following form: Δ,, Τ,, 1,2,, (8) where D i and E i are known real constant matrices of dimensions nxn Di and n Ei xn, respectively, and the uncertainty T i (x,t,ξ), an unknown matrix valued function of (x,t,ξ), belongs to the following bounded set: Γ,, :,, 1 1,2,, (9) Fuzzy blending of each individual model yields the overall fuzzy model as follows: 4

Δ,, (10) Here, the output equation is also added into the overall model for the convenience of later citation. We assume that all the triples (A i, B i, C), =1,2,,r, are controllable and observable. It should be noted that the parameter uncertainty structure in (8) and (9) describes how the uncertainties enter the system model and has been widely used in the study of the problem of robust stability and stabilization of uncertain linear systems and it can represent parameter uncertainty in many physical cases. Actually, any norm bounded parameter uncertainty can be expressed in the form of (8) and (9). Our objective in this chapter is to design a PI controller of the following form: (11) such that the closed loop system (10) and (11) is asymptotically stable, where F P and F I are constant matrices. Controller (11) is an ideal PI controller. Readers are referred to [8] for how to change the ideal PI controllers into practical ones and the relationships between the parameters of the two kinds of controllers. III. PROPORTIONAL CONTROL In this section, we first study the stabilization of system (10) by a proportional controller, i.e., by the following controller: (12) where F is a constant matrix. To do this, we need the following proposition. Proposition 1: System (10) is asymptotically stabilized via controller (12) if there exists a matrix such that the following matrix inequalities hold: Δ Δ 0,, Γ, 1,2,, (13) Proof: First notice that system (10) is asymptotically stabilized via controller (12) if there exists a matrix P>0 such that Δ Δ 0,, Γ, 1,2,, (14) To show this claim, we choose V(x):=x T P 1 x(t) as a Lyapunov function candidate for system (10). Since Г i is a closed set and r is finite, there exists a positive definite matrix Q 0 such that Δ Δ,, Γ, 1,2,, if matrix inequality (14) holds. Then, noticing the fact that α i (x) 0, i=1,2,..,r, and 1, we have Δ Δ Thus follows the claim. Now, following the same procedure as the proof of [9, Theorem 1], we can prove that matrix inequality (14) holds if and only if matrix inequality (13) holds. This completes the proof. 5

Theorem 1: Consider the uncertain nonlinear system (10). Suppose there exist positive definite matrices P, a matrix F, and positive numbers ε i, i=1,2,,r, such that the following matrix inequalities hold: 0 1,2,, (15) Then, system (10) can be robustly stabilized by controller u(t)=fy(t). To prove this theorem, we need the following lemma. Lemma 1: Let D, E, and T be real matrices of appropriate dimensions with T satisfying T 1. Then for any real number ε>0, we have (16) Proof: The assumption T 1 leads to I-TT T 0, which again yields the following inequality: 0 by applying Schur complements [10]. Then, for any two real numbers ε 1 and ε 2 we will have 0 (17) If ε 1 ε 2 =1, the inequality (17) will be of the following form: 0 which is equivalent to the inequality (16). From (8) and (9) and applying Lemma 1, we have Δ Δ,,,, where ε i, i=1,2,,r, are real positive numbers. Thus, it follows that matrix inequalities (13) hold if so do matrix inequalities (15). Applying Proposition 1, we arrive at this theorem. One cannot directly use matrix inequalities (15) to calculate the required feedback matrix due to the existence of the term PBB T P in (15), which makes the solution of (15) very complicated. Neither is it convex, nor can it be transformed into a convex problem in the space of unknown parameters (P,F, ε i, i=1,2,,r). In [9], an iterative linear matrix inequality (ILMI) algorithm was developed to solve the similar problem. Following the same idea as in [9], we can also develop an ILMI algorithm to solve matrix inequalities (15), which is summarized in the sequel. Algorithm 1 (ILMI Algorithm for Proportional Nonlinear Controllers): Initial data: systems state space parameters (Ai, Bi, Ci, Di, Ei, i=1,2,,r) Step 1. Choose P 0 >0. Set j=1 and X 1 =P 0. Step 2. Solve the following optimization problem for P j, F, and α j. OP1: Minimize subject to the following LMI constraints Σ 0 0 (18) 0 0 1,2,, (19) Where Σ Denote by α * j the minimized value of α j. Step 3.If α * j 0, the obtained matrices (Pj, F, ε i, i=1,2,,r) solve the problem. Stop. Otherwise go to Step 4. Step 4.Solve the following optimization problem for Pj, F and ε i, i=1,2,,r. 6

OP2: Minimize tr(p j ) subject to LMI constraints (18) and (19) with α j =α * j, where tr stands for the trace of a square matrix. Denote by P * j the optimal P j. Step 5. If Xj P * j <δ, where δ is a prescribed tolerance, go to Step 6; otherwise set j:=j+1, Xj=P * j and go to Step 2. Step 6. It cannot be decided by this algorithm whether the problem is solvable for the system. Stop. The form of inequality (18) is different from the form of inequalities (15) in that there is an additional term α j P j in (18). The introduction of this term is to guarantee LMI (18) has solutions for a fixed X j, which is evident for a sufficiently large positive number α j. The optimization problem OP1 in Step 2 is a generalized eigenvalue problem (see [7, pp. 10 11]) in LMI language, which can be solved by the command gevp in LMI tool box of Matlab [11]; while the optimization problem OP2 in Step 4 is an eigenvalue problem in LMI language, which can be solved by the command mincx in LMI tool box of Matlab [11]. In Theorem 1, a common feedback matrix F is to be found for the T matrix inequalities. This constraint is not necessary and sometimes it is difficult to find such a common feedback matrix F. For some systems such as the one we will study later, all of the T matrices B i, i=1,2,,r are identical, i.e., B 1 =B 2 = =B r :=B. In this case, we can design a controller of the following form: (20) to stabilize system (10). To this end, we first establish the following. Theorem 2: Consider the uncertain nonlinear system (10) with B 1 =B 2 = =B r :=B. If there exist a positive definite matrix P, matrices F i and positive numbers ε i, i=1,2,,r, such that the following matrix inequalities hold: 0 1,2,, then system (10) can be robustly stabilized by controller (20). The proof of this theorem is similar to that of Theorem 1, hence it is omitted. A corresponding algorithm can also be developed accordingly to solve feedback matrices F i, i=1,2,,r. This is only to change LMI (18) to Σ 0 0 0 1,2,, (21) Then Algorithm 1 applies verbatim. IV. PI CONTROL We again consider system (10), but now we use PI controller (11) instead of proportional controller (12). Our objective here is to design the feedback matrices F p and F I such that system (10) is robustly stabilized by the controller. Let Denote z=[z T 1, z T 2 ] T. The variable z can be viewed as the state vector of a new system, whose dynamics is governed by 7

i.e. Where Δ,, Δ,, 0 0, 0 Δ,,,, Define 0 0 0 0 1 And denote : 0, : 0, :, : Then the problem of PI controller design for system (10) is reduced to that of proportional controller design for the following system: Δ,, Thus, the feedback matrices F P and F I can be calculated by applying Algorithm 1 to system (22). In the case where all B i, i=1,2,,r, are equal, the following kind of controller can be also designed by applying Algorithm 1 and Theorem 2: V. NUMERICAL EXAMPLE: EXCITATION CONTROL OF POWER SYSTEMS Most existing excitation controllers of power systems are designed by application of linear control theory to the approximate linearized models of power systems. Thus, they work well only when the disturbance caused by faults in the systems is relatively small. Under large disturbance, the systems might be out of synchronization. In this section, we will apply the results obtained in the previous sections to the design of excitation controllers for a thermal turbine synchronous generator. The simplified dynamical model of the single machine infinite bus power system with a siliconcontrolled rectifier direct excitor is as follows: 2 8

Where 1 cos. sin where δ is the angular position of the rotor of generator (G) with respect to a synchronously rotating reference, which is selected here to be the infinite bus; ω is the angular velocity of the rotor; P e and P m are the active power and mechanical power of G, respectively, E q is the electromotive force (EMF) in the q axis of G; E q is transient EMF in the q axis of G; E f is the equivalent EMF in the excitation winding of G; x d is the d axis reactance of G; x d is the d axis transient reactance of G; x T and x L are the reactances of the transformer and the transmission line, respectively x d = x d + x T + x L, x d = x d + x T + x L, x ad is the mutual reactance between the excitation winding and the stator winding; R f is the resistance of the excitation winding; k A is the gain of the amplifier, V s is the bus voltage; u is the control input; and h(t) denotes the external disturbance. Suppose δ 0 is the value of δ under steady operating condition. For this δ 0, let 1 sin cos Define new state variables and control input as Δ The dynamical model of the power system (23) can be rewritten as 2 : sin sin cos : cos Δ : (24) The parameters of the system are as follows: f 0 = 50 Hz ω 0 = 1 p.u. δ 0 = 60 o H = 8 s D = 0.8 P m = 0.79 p.u. V s = 1 p.u. x d =1.5 p.u. x d = 0.3 p.u. x ad = 1.3 p.u. x L = (0.8+ΔxL) p.u. Δx L = 0.0008 p.u. x T = 0.01 p.u. k A = 10 T d0 = 3 s R f = 0.0045 p.u. These parameters yield E q0 = 1.2723 p.u. u 0 = 7.2942*10^ 4 p.u. 9

Notice that all parameters except x L are supposed to be known here. The parameter x L is supposed to have some perturbation. The reason why we choose the perturbation in x L to illustrate the robustness of our controller is that x L is one of the system parameters which are most uncertain and difficult to measure. Generally, the state variable δ is difficult to measure. Hence, the system output equation is as follows: 0 1 0 0 0 1 The physical limit of the excitation voltage leads to the constraint on the control input Δu as follows: Δ 3.0.. Or equivalently 0.0018.. Δ 0.00031.. The local fuzzy models of the system are obtained through the linearization of the system (24) around the points x 1 = [ 30, 0, 0] T, x 2 = [0, 0, 0] T and x 3 = [+30, 0, 0] T, respectively. In terms of IF THEN rules, the fuzzy models admit the following form: Rule 1. IF x 1 (or δ ) is small (i.e., x 1 is about 30 o, or equivalently, δ is about 30 o ) THEN Δ Δ Rule 2. IF x 1 (or δ ) is middle (i.e., x 1 is about 0 o, or equivalently, δ is about 60 o ) THEN Δ Δ Rule 3. IF x 1 (or δ ) is large (i.e., x 1 is about +30 o, or equivalently, δ is about 90 o ) THEN Δ Δ where ΔA i (i=1,2,3 ) account for the parameter perturbation in x L. By using Teixeira Zak s formula [12] 0 0 1,2,3, 1,2,3 matrices A i and B i are obtained as follows (noticing that we set Δx L =0 while solving A i and B i ): 0 314.16 0 0.1002 0.1 0.0563 0.2519 0 0.6937 0 314.16 0 0.1009 0.1 0.0975 0.3121 0 0.6937 0 314.16 0 0.085 0.1 0.1126 0.3441 0 0.6937 10

0 0 962.963 There are many options to assign membership functions. For the sake of convenience in computation, we select triangular function as our membership functions, which are 1 6 3 3 6 3 6 0 3 0 3 3 2 3 2 3 1 2 1 Figure 1 Membership functions of the fuzzy models for power system (24). Since the right hand side of the first equation of the power system (24) includes no uncertainties and it is also linear function of the state variables, we let the first row of ΔA i be zero. To account for the parameter perturbation and approximation error caused by linearization, we let the second and third rows be 0.1% times the corresponding elements of ΔA i i.e. 0 0 0 Δ 0.1002 0.1 0.056310 0.2519 0 0.6937 0 0 0 Δ 0.1009 0.1 0.097510 0.3121 0 0.6937 0 0 0 Δ 0.085 0.1 0.112610 0.3441 0 0.6937 Since B 1 =B 2 =B 3, we can use Theorem 2 combined with Algorithm 1 to calculate the feedback matrices of proportional and PI controllers, which are as follows. For proportional controller 0.0398 0.0071 11

0.0353 0.0212 0.0065 0.0027 For PI controller 0.0163 0.0021 0.0037 0.0125 0.0171 0.0028 0.0028 0.0125 0.0241 0.0033 0.005 0.0125 12

Simulations with Simulink 1) With Proportional control and switching A 1, A 2 and A 3, according to x 1 we have: A1x1 A1x2 A1x3 A2x1 scope A2x1 Switch 1 A2x2 Switch3 A2x3 Switch 2 A3x1 x scope u scope A3x2 A3x3 u MATLAB Function x2 & x3 scope 13

X1: Yellow X2: Pink X3: Blue Figure 2) X1: Yellow X2: Pink X3: Blue for P controller with switching 14

X2: Pink X3: Blue Figure 3) X2: Pink X3: Blue for P controller with switching 15

The input u: Figure 4) input u for P controller with switching 16

2) With Proportional plus Integral control and switching A 1, A 2 and A 3, according to x 1 we have: A1x1 A1x2 A1x3 u scope A2x1 Switch1 Matrix Multiply alfaifij *(y/s) alfa 1Fi1 Integrator 1 1 s Integrator 2 1 s Fi1 Pu MATLAB Function y MATLAB Function A2x2 A2x3 A3x1 Switch 2 Switch3 x scope alfa 2Fi2 alfa 3Fi3 [-.0037 -.0125 ] Fi2 [-.0028 -.0125 ] Fi3 alfa MATLAB Function A3x2 A3x3 x2 & x3 scope [.005 -.0125 ] 17

X1: Yellow X2: Pink X3: Blue Figure 5) X1: Yellow X2: Pink X3: Blue for PI controller with switching 18

X2: Pink X3: Blue Figure 6) X2: Pink X3: Blue for PI controller with switching 19

The input u: Figure 7) input u for P controller with switching 20

3) With Proportional control and without switching A 1, A 2 and A 3 we have: A1x1 A1x2 A1x3 x u Pu MATLAB Function 21

X1: Yellow X2: Pink X3: Blue Figure 8) X1: Yellow X2: Pink X3: Blue for P controller without switching 22

The input u: Figure 9) input u for P controller without switching 23

4) With Proportional plus Integral control and without switching A 1, A 2 and A 3 we have: A1x1 A1x2 A1x3 u scope x scope Pu Matrix Multiply alfaifij*(y/s) Integrator1 1 s Integrator2 MATLAB Function y MATLAB Function 1 s alfa1fi1 alfa2fi2 alfa3fi3 Fi1 [-.0037 -.0125] Fi2 [-.0028 -.0125] Fi3 [.005 -.0125] alfa MATLAB Function 24

X1: Yellow X2: Pink X3: Blue Figure 10) X1: Yellow X2: Pink X3: Blue for PI controller without switching 25

The input u: Figure 11) input u for PI controller without switching 26

VI. CONCLUDING REMARKS As you saw the responses in switching mode have some jumps and nonsmoothnesses because of switching phenomenon but they finally became stable, the results without switching obviously shows more smoothness in responses. The design problem of proportional and PI controllers for nonlinear systems has been studied in this chapter. First, Takagi Sugeno (T S) fuzzy model with parameter uncertainties is used to model the nonlinear systems. The introduction of parameter uncertainties in the T S fuzzy model can account for both approximation error of fuzzy model and the actual parameter uncertainties in the practical process. Second, an algorithm based on ILMI is developed to design a proportional controller to robustly stabilize the system which can be expressed by a T S fuzzy model. Third, we transform the problem of PI controller design to that of proportional controller design for an augmented system and thus bring the solution of the former problem into the configuration of the developed algorithm. The characteristics of the developed method is that it is numerically tractable and can be applied to general multi variable nonlinear systems. Finally, the proposed method is applied to the robust stabilizing controller design for the excitation control of power systems. Simulation results show that the transient stability can be improved by using a fuzzy PI controller when large faults appear in the system, compared to the conventional PI controller designed by using linearization method around the steady state. Notice that the controllers proposed in this paper use only output variables (in the case of common feedback matrix corresponding to all the fuzzy rules) or output variables plus some other state variables needed in the fuzzy premise (in the case of different feedback matrix corresponding to each fuzzy rule accordingly). In the latter case, the state variables needed in the fuzzy premise should be available normally. VII. References [1] Feng Zheng, Qing-Guo Wang, Tong Heng Lee and Xiaogang Huang Robust PI Controller Design for Nonlinear Systems via Fuzzy Modeling Approach IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART A: SYSTEMS AND HUMANS, VOL. 31, NO. 6, NOVEMBER 2001 [2] K.K. Tan*, S. Huang, R. Ferdous Robust self-tuning PID controller for nonlinear systems Journal of Process Control 12 (2002) 753 761 [3] R. A. Freeman and P. K. Kokotovic, Robust Nonlinear Control Design. Cambridge, MA: Birkhaüser, 1996 [4] T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst., Man, Cybern., vol. SMC-15, pp. 116 132, 1985. [5] M. Sugeno and G. T. Kang, Structure identification of fuzzy model, Fuzzy Sets Syst., vol. 28, pp. 15 33, 1988. [6] L. X.Wang and J. M. Mendel, Fuzzy basis functions, universal approximatiors and orthogonal least-square learning, IEEE Trans. Neural Networks, vol. 3, pp. 807 814, Sept. 1992. [7] T. A. Johansen and B. A. Foss, Constructing NARMAX models using ARMAX models, Int. J. Control, vol. 58, pp. 1125 1153, 1993. [8] PID Controllers: Theory, Design, and Tuning. Research Triangle Park, NC: Instrument Society of America, 1995. [9] Y.-Y. Cao, J. Lam, and Y.-X. Sun, Static output feedback stabilization: An ILMI approach, Automatica, vol. 34, pp. 1641 1645, 1998. [10] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [11] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox: The Math Works Inc., 1995. 27

[12] M. C. M. Teixeira and S. H. Zak, Stabilizing controller design for uncertain nonlinear systems using fuzzy models, IEEE Trans. Fuzzy Syst., vol. 7, pp. 133 142, Apr. 1999. 28