FUZZY EVENT-TRIGGERED DISTURBANCE REJECTION CONTROL OF NONLINEAR SYSTEMS

. The problem of fuzzy based event-triggered disturbance rejection control for nonlinear systems is addressed in this paper. A new fuzzy event based anti rejection controller is designed and a fuzzy reduced disturbance observer is constructed. Suﬃcient conditions for the closed loop system to be asymptotically stable under an H ∞ performance index are derived. Based on these conditions, the design of a fuzzy event-triggered state feedback controller is formulated and solved. Numerical results are presented to demonstrate the correctness and eﬀectiveness of our theoretical ﬁndings.


1.
Introduction. To design an effective controller for a dynamic system, some knowledge of system model or structure needs to be known. It is well known that classic control theory is providing a large variety of methods for solving model-based controller design problems, particularly for those with linear structured systems. However, most of processes of practical significance are highly nonlinear and contain uncertain parameters so that conventional control theory is unable to solve them satisfactorily. To remedy this, the concept of fuzzy model based system has been introduced and applied to nonlinear systems successfully. In this approach, a system is assume to be fuzzy and controllers satisfying fuzzy rules are sought for the system. Clearly, this approach provides a practical controller design method even when only some rough knowledge of a process is available.
Over the past several years, many successful applications of the aforementioned fuzzy approach have been obtained, especially for control problems using Takagi-Sugeno (T-S) [15] fuzzy model, and the fuzzy method has been successful for investigating nonlinear systems [13,17,5,16,14,3,24,25,1]. Furthermore, the method has also been applied to some complex biotechnological processes [23]. However, all these results are under the assumption that the signal of a controller is transmitted periodically. In practice, it turns out that even if the state has minor changes, the controller is still updated. Event triggered controller design is a better approach to this type of problem, in which comparison between the real state and the recent past one is carried out and the signal is transmitted only when the gap is sufficiently large, see [19,20,21].
On the other hand, noise or disturbance is still a big challenge for a system, especially when its distribution can not be described exactly. Much work has been done to find methods for solving complex matrix equations [10,11,12,22]. The disturbance observer based control (DOBC) is a good way to estimate the disturbance which occurs in the input channel. This disturbance will be rejected during controller design which will decrease its affect of the entire system. Some work has been done on disturbance rejection control problem. For instance, in [6], a nonlinear system is considered and disturbance rejection approach is addressed. In [4], some work for linear uncertain and time delay systems has been done, and extensively studies have been done in many areas, see [7,8,18].
In this paper, the fuzzy H ∞ control problem with nonlinear systems in discretetime domain is studied. The T-S fuzzy model is employed to describe the nonlinear system in terms of IF-THEN rules and the reduced order observer is constructed to estimate disturbance. The rest of the paper is organized as follows: Problem statement and preliminary results are presented in Section 2. In Section 3, stability analysis of the resulting closed loop fuzzy system is given. In Section 4, H ∞ performance for the error dynamic system is analyzed and the fuzzy H ∞ controller is designed such that the associated error dynamic system is asymptotically stable. A numerical example is given to illustrate the effectiveness of our approach in Section 5. Finally, some concluding remarks are made in Section 6.
Notation. Throughout the paper, R n is used to denote the n-dimensional Euclidean space, A T denotes the transpose of the matrix A. A positive (negative) definite matrix P is written as P > 0 (P < 0) and * indicates a symmetric element in a symmetric matrix.
2. Problem statement and preliminaries. We consider a discrete-time nonlinear system which can be described by the following fuzzy model: where i ∈ {S} = {1, 2, 3, . . . , v}, M ij is a given fuzzy set for any feasible i and j, v is the number of IF-THEN rules, θ 1k , · · · , θ gk are the premise variables, x(k) ∈ R l is the state vector of the system, u(k) ∈ R t is the input vector of the system, is the external disturbance vector of the system, and A i , B i and H i are constant matrices with appropriate dimensions at the working instant k.
The input disturbance d 1 (k) in system (1) is given as follows.
where W i , M i and V i are constant matrices with appropriate dimensions.
Assumption 1. For systems (1) and (2), it holds that 1) The discrete time fuzzy system is inferred as follows: , and Then, we have: Therefore, system (1) is rewritten as: Under the assumption that all of the system states are available, we need to estimate d 1 (k). If θ 1k is M i1 , · · · , and θ gk is M ig , then, a reduced-order observer is constructed below: whered 1 (k) andŵ (k) are estimations of d 1 (k) and w (k), respectively. And the fuzzy reduced-order observer is obtained as: (6) To reduce the effects of disturbance, controller is constructed as Let To reduce network based transmission load, by event-triggered theory, letx k be a new signal applied to the controller in the time interval (k, k + 1] witĥ if event condition is not satisfied, The following decision condition for signal transmission is given by the event generator: where σ > 0. Then, combing (8) and (9), we have an event-triggered based controller given below Let (1), (2) and (5), we obtain a fuzzy based error estimation system: The reference output of system (11) is set as: To proceed further, some definitions are needed in developing our main results in the paper.
Then, system (11) is said to be asymptotically stable and K i is the gain matrix of the controller.

Lemma 2.2. [2]
Let R > 0 be a given symmetric matrix, and let W t , t = 1, 2, · · · , h be matrices with appropriate dimensions, if 0 ≤ ε t ≤ 1 and h t=1 ε t = 1, then Definition 2.3. For a given constant γ > 0, system (11) is said to be asymptotically stable and satisfy an H ∞ performance index γ, if it is asymptotically stable and the following condition is satisfied: The aim of our work is to design an event trigger based fuzzy anti-disturbance controller to ensure that the error system (11) is asymptotically stable and an H ∞ performance index is satisfied.
3. Stability analysis. In this section, sufficient conditions are given under which system (11) is asymptotically stable.
Applying the lossless S-procedure [9], for a given κ > 0 and combining conditions (15) and (16), we finally obtain from condition (14) that Thus, system (11) is asymptotically stable and the proof is completed.
4. H ∞ performance analysis and controller design. Based on the conditions established in Theorem 3.1, we now consider performance analysis and controller design.
Theorem 4.1. Let σ be given. If there exist a positive definite symmetric matrix P , and a constant κ such that then system (11) is said to be asymptotically stable and an H ∞ performance index is satisfied.
Proof. To establish the H ∞ performance for the system (11), the following cost function is introduced: In the case of zero initial condition, J(T ) can be written as Thus, we have Then, for each fuzzy model, it follows that J(T ) ≤ 0 whenever Θ ij < 0. Recalling condition (16) and following an argument similar to that in the proof of Theorem 3.1, we can show that system (11) is asymptotically stable and the following condition is also satisfied.
Note that the inequality in Theorem 4.1 is unsolvable and we need to transform it to a solvable one.
then system (11) is asymptotically stable and satisfies an H ∞ performance index.
Proof. By the Schur complement, multiply from the left hand side and the right hand side of inequality (18) by diag G T , I, G T , I, I, I and diag {G, I, G, I, I, I}, respectively, where G ∈ R n×n is a positive definite diagonal matrix. Noting that (23) is obtained which is linear in the variables G, Q andK ij . Once a solution of (23) is obtained, the feedback gain can be calculated as K ij =K ij G −1 .
5. Numerical example. Consider a discrete-time fuzzy model based nonlinear system with the following system and other parameters: Our purpose is to design a fuzzy H ∞ reduced order observer for system (1) such that the resulting error system (11)  we obtain the following fuzzy based matrices of the observer and controller, under which the system is asymptotically stable with an H ∞ performance index satisfied. For the case of σ = 0.01, the corresponding trajectories are shown in Figure  1. The disturbance d 1 (t), the estimation disturbanced 1 (t), and the error disturbance d 1 (t) −d 1 (t) are illustrated in Figure 2. Obviously, the system concerned is asymptotically stable under such a controller. Conclusion. In this paper, the problem of fuzzy event-based disturbance rejection control of nonlinear systems is studied. Sufficient conditions are given under which the closed-loop system is stable. Based on these conditions, fuzzy controller and rejection observer are formulated and solved. A numerical example illustrates the effectiveness of the proposed design procedure and the performance of the resulting closed-loop system.