Proportional PDC Design-Based Robust Stabilization and Tracking Control Strategies for Uncertain and Disturbed T-S Model

*is paper presents a proportional parallel distributed compensation (PPDC) design to the robust stabilization and tracking control of the nonlinear dynamic system, which is described by the uncertain and perturbed Takagi–Sugeno (T-S) fuzzy model. *e proposed PPDC control design can greatly reduce the number of adjustable parameters involved in the original PDC and separate them from the feedback gain. Furthermore, the process of finding the common quadratic Lyapunov matrix is simplified. *en, the global asymptotic stability with decay rate and disturbance attenuation of the closed-loop T-S model affected by uncertainties and external disturbances are discussed using the H∞ synthesis and linear matrix inequality (LMI) tools. Finally, to illustrate the merit of our purpose, numerical simulation studies of stabilizing and tracking an inverted pendulum system are presented.


Introduction
During the last decade, the fuzzy logic control has attracted rapidly growing attention from both the academic and industrial communities [1][2][3]. Specially, the Takagi-Sugeno (T-S) fuzzy model has been extensively used to investigate nonlinear control systems [4,5]. is model is described by a set of fuzzy If-then rules with fuzzy sets in the antecedents and linear dynamics models in the consequent [6][7][8]. e overall model of the complex system is achieved by fuzzy interpolating these linear models through nonlinear fuzzy membership functions. Moreover, the stability study of this class of systems has been usually based on the use of the Lyapunov direct method. e obtained stability conditions are in general given in terms of linear matrix inequalities (LMI), which can efficiently be solved by convex programming techniques [9][10][11]. e overall controller of the T-S fuzzy model used the parallel distributed compensation (PDC) approach which used multiple linear state feedback controllers corresponding to the local models via fuzzy rules [12,13]. roughout this work, the PPDC control scheme is employed to design the fuzzy controllers from the T-S fuzzy model. On the one hand, most of the results obtained have focused on the stabilization and the tracking problems by using the normal PDC approach and very few of them are concerned with the stability problem of satisfying the decay rate and the disturbance attenuation problem [14][15][16][17]. On the other hand, for r rules, the total number of unknown parameters is reduced to r + mn, compared with the normal PDC rmn where r, m, and n are the numbers of the rules, the inputs, and the state variables, respectively [18].
In addition, uncertainties, unknown parameters, and external disturbances are frequently a source of instability and encountered in various complex systems [19][20][21]. In particular, the design of the robust fuzzy control of the T-S fuzzy model has received considerable interest in the literature [3,[22][23][24]. Based on the PDC approach, some researchers treated the stability analysis, stabilization, and tracking control problems of this class of systems [25][26][27].
In this study, H ∞ state feedback synthesis and reference model tracking control schemes are expanded to include nonlinear systems described as uncertain and disturbed the T-S fuzzy model by using the PPDC approach. However, we consider both the stability problem of satisfying the decay rate and the disturbance attenuation. Hence, we obtain sufficient conditions, expressed in LMI terms, for the existence of robust fuzzy controllers. e main contributions of this paper are summarized as follows: (i) Compared with the existing results obtained with the normal PDC approach [9,13,22,23,[28][29][30][31], our control design is carried out based on the T-S fuzzy model via the PPDC scheme, which can significantly reduce the number of parameters in PDC. (ii) In the proposed PPDC control design, the proportional coefficients are first assigned. en, the common quadratic Lyapunov matrix and the feedback gains are directly obtained from the LMI constraints that consider not only stability but also other control performances such as speed of response, attenuation of the disturbances effect, and structured uncertainties. As a result, the solution of the LMI is more flexible and the controller design is more feasible. is paper is organized as follows: Section 2 introduces the uncertain T-S fuzzy model and the proportional PDCbased robust fuzzy H ∞ stabilization design scheme. In Section 3, sufficient stability conditions are developed to ensure the stability of the augmented system with a reference model tracking. A simulation example is considered in Section 4 to illustrate the merit of the designed H ∞ controllers.

Proportional PDC-Based Robust Fuzzy H ' Stabilization Design Scheme
We consider a nonlinear affine system: where f and g are nonlinear functions of the state x ∈ R n and u ∈ R m is the control input. en, its fuzzy dynamic model can be described by fuzzy If-then rules which represented local linear input-output relations [7,8,32,33]. e ith rule is of the following form.

Plant Rule i
If x 1 is M i1 x 1 and . . . and x n is M in x n , for i � 1, 2, . . . r, where w ∈ R q is the disturbance input, z ∈ R p is the controlled output, M ij is the fuzzy set, and r is the number of If-then rules.A i ∈ R n×n , B 1i ∈ R n×m , B 2i ∈ R n×q , C i ∈ R p×n , D 1i ∈ R p×m , and D 2i ∈ R p×q are real matrices verifying in which ΔA i (t), ΔB 1i (t), and ΔB 2i (t) represent the timevarying parameter uncertainties defined as follows: where H i , E 1i , E 2i , and E 3i are known constant real matrices of appropriate dimensions. F(t) is a matrix function, which is bounded by: F T (t)F(t) ≤ I where I is the matrix identity of an appropriate dimension. e resulting state _ x and the final output z of the T-S fuzzy model are inferred by using the center of gravity method for defuzzification as follows: It should be noted that For the fuzzy controller design, we supposed that all the states are measurable and the studied system is locally controllable. en, we applied a proportional PDCbased compensator for each local fuzzy model (2) as follows.

Control Rule i
If x 1 is M i1 x 1 and . . . and x n is M in x n , then u � − k i Kx, where K ∈ R m×n is the local state feedback matrix and k i , for i � 1, 2, . . . , r, represent the proportional adjustable coefficients which differ with different control rules. en, the overall fuzzy controller u is defined by Substituting controller (8) into the T-S fuzzy system given by (5), the final closed-loop model is described by Consequently, we presented in eorem 1 sufficient stability conditions that guarantee the stability of the considered T-S model (9) and achieve a prescribed level of disturbance attenuation c for all admissible uncertainties such that where ‖z‖ 2 � ∞ 0 z T zdt.

Theorem 1.
e equilibrium of the closed-loop fuzzy model (9) described by using the proportional PDC controller (8) is globally asymptotically stable with decay rate α and satisfying the performance objective (10) if there exists a common positive definite matrix X, a matrix R, and positive constants τ 1 ∼ τ 3 verifying the LMI formulation: where Proof 1. In order to prove eorem 1, we used the quadratic Lyapunov function given by and verifying the control performance where P � P T > 0 and α > 0 is the largest Lyapunov exponent.

□
Additionally, the stability of the closed-loop T-S fuzzy model is satisfied under the H ∞ performance, given in (10), According to equations (9), (13), and (14), the development of the above matrix inequality leads to where As all h i (x) ∈ [0, 1], condition (16) gives ϖ ij < 0. en, by denoting X � P − 1 , pre-and postmultiplying to it by the positive definite matrix diag(X, I), respectively, and using the Schur complement, see Appendix A, we obtain It is clear that the matrix inequality (17) contains certain Ψ ij and uncertain ΔΨ ij parts. ereby, it can be transformed into the following form: Using the uncertainties defined in (3), ΔΨ ij becomes It is worth pointing out that ΔΨ ij contained antidiagonal terms. However, to transform them into diagonal terms, we used the appropriate lemma, as presented in Appendix B. By denoting R � K X, it follows that where Furthermore, according to Ψ ij and (20), the matrix inequality (18) can be transformed as After some manipulations using the Schur complement, we complete the proof and we get an optimization problem involving LMI, as illustrated in (11).

Design of Robust Fuzzy H ' Tracking Control
In this section, we treated the robust fuzzy H ∞ tracking problem for the considered global model (5) to the following reference model [11,13,30]: where A e ∈ R n×n is an asymptotically stable matrix, B e ∈ R n is an input matrix, x e ∈ R n is the state of the reference model, and e ∈ R is a bounded reference input. We applied then the following local control law.

Control Rule i
If x 1 is M i1 x 1 and . . . and x n is M in x n , then u � − l i Lε, where ε � x e − x is the tracking error. en, the final fuzzy controller u is defined as Substituting controller (24) into the state x of the T-S model (5), we obtained the following augmented model: where Consequently, we presented in eorem 2 sufficient stability conditions in terms of LMI that guarantee the stability of the augmented T-S model (25) and ensure a good H ∞ tracking performance as Theorem 2. e equilibrium of the augmented closed-loop fuzzy model (25) is globally asymptotically stable, and the H ∞ tracking control performance, shown in (26), is guaranteed if there exist symmetric positive definite matrices X 1 and P 2 , a matrix V, and positive constants c and μ 1 ∼ μ 4 verifying the LMI formulation: maximize X 1 ,P 2 ,V,c α subject to: θ 22 � P 2 A e + ( * ) + 2αP 2 + μ 1 E T 1i E 1i . Furthermore, the common feedback matrix L, shown in (24), is given by Proof 2. In order to prove eorem 2, we considered the following candidate Lyapunov function: and verifying the control performance where P � P T > 0 and α > 0.

□ 4 Complexity en, the corresponding closed-loop model (25) is globally asymptotically stable and the decay rate is at least
where Q � Q 0 0 0 . e control problem is to minimize Using (25), (29), and (30), the condition above becomes where As all h i (x) ∈ [0, 1], the matrix inequality (33) gives Ω ij < 0. en, by assuming that P � diag(P 1 P 2 ), we obtain ereafter, the inequality matrix (34) can be transformed into the following form: Using the uncertainties defined in (3), the matrix ΔΣ ij (t) can be rewritten as By using the lemma, as presented in Appendix B, ΔΣ ij is increased as follows: where erefore, according to Σ ij and (37), the matrix inequality (35) can be transformed as By denoting X 1 � P − 1 1 and V � LX 1 and pre-and postmultiplying to (38) by the positive definite matrix diag(X 1 , I, I, I), respectively, we obtain Using then the Schur complement, we complete the proof and we get an optimization problem involving LMI, as illustrated in (27).
Remark. It should be mentioned that the above results on the design problem for T-S fuzzy models are based on an implicit assumption that the controller will be implemented exactly. However, uncertainties or inaccuracies do occur in the implementation of a designed filter or controller. In recent years, there are considerable studies on the robust nonfragile H ∞ filtering problem of T-S fuzzy models [20,21,24].

Numerical Simulation
In this section, the proposed control strategies are verified for an inverted pendulum system. Its mathematical model is described by the following nonlinear equations [11,13,34]: where x 1 � θ(rad) is the angle from the vertical position and x 2 � _ θ(rad/s) is the angular velocity, g � 9.81 m/s 2 is the gravity constant, M � 2.4 kg is the mass of the cart, m � 0.23 kg is the mass of the pendulum, l � 0.36 m is the pole length, J � 0.099 kg m 2 is the moment of inertia of the pole, and d � 0.005 Nms/rad is the pendulum damping coefficient.
For simplicity, we suppose that m 2 l 2 cos 2 (x 1 ) ≪ (m + M)(J + ml 2 ) and we use two rules: Using eorem 1 with c � 0.5, τ 1 � 2.9, τ 2 � 1.7, τ 3 � 0.9, k 1 � 0.62, and k 2 � 0.85, we obtain α � e responses of the state variables x 1 (t) � θ(t)(rad) and x 2 (t) � _ θ(t)(rad/s) are shown in Figures 1 and 2, respectively. e evolution of the force F(t)(N) is depicted in Figure 3, while the disturbance input signal is given by It is shown from the simulation results that the designed controller u � − 2 i�1 h i (x)k i Kx is efficient and guarantees the rapid global stability of the closed-loop fuzzy model (9).

Complexity
It is observed that in contrast with the normal PDC approach, the PPDC one can rapidly achieve desired responses despite the presence of uncertainties and disturbances. Moreover, magnitudes of the quadratic error tracking using the PDC design are larger than those of PPDC.

Conclusion
is paper has presented the robust state feedback synthesis and model reference tracking control for the nonlinear dynamic system described as the T-S fuzzy model affected by uncertainties and external disturbances using the proportional PDC design. Its interest is to reduce the number of adjustable parameters in the normal PDC one. Based on the quadratic Lyapunov function with a decay rate, sufficient stability conditions have been obtained using the H ∞ criterion and LMI tools that guarantee the stability of the closed-loop T-S model and ensure a good robust performance. Moreover, the solution of finding the common positive definite matrix was simplified and proportional coefficient design was separated from the feedback matrix parameters. Finally, an inverted pendulum system was considered to show the effectiveness of the designed H ∞ fuzzy controllers. 8 Complexity