- Open Access
Output feedback stabilization of nonlinear discrete-time systems with time-delay
© Dong and Wei; licensee Springer. 2012
- Received: 24 December 2011
- Accepted: 31 May 2012
- Published: 31 May 2012
This article considers both the static output feedback stabilization issue and output-feedback guaranteed cost controller design of a class of discrete-time nonlinear systems with time-delay. First, by static output feedback controller, the new sufficient conditions for static output feedback stabilization of a class of discrete-time nonlinear systems with time-delay are presented. Then, we establish the new delay-independent sufficient conditions for existence of the guaranteed cost control by static output-feedback controller in terms of matrix inequalities. Finally, two examples are given to show the effectiveness of our proposed approaches.
2000 MSC 93D15; 93C55; 34K20; 34D23.
- Linear Matrix Inequality
- Positive Definite Matrice
- Output Feedback Controller
- Static Output Feedback
- Guarantee Cost Control
Time delay exists commonly in dynamic systems due to measurement, transmission and transport lags, computational delays, or unmodeled inertia of system components, which has been generally regarded as a main source of instability and poor performance. Therefore, considerable attention has been devoted to the problem of analysis and synthesis for time-delayed systems, and many research results have been reported in the literature. To mention a few, the stability analysis result is reported in [1, 2], the stabilization problem for switched nonlinear time-delay systems is solved in , and the model filtering problems are solved in , and the design problem of a hybrid output feedback controller is also considered in .
The static output feedback problem for linear and nonlinear systems is an important problem not yet completely solved and continuously investigated by many people. In practice, it is not always possible to have full access to the state vector and only the partial information through a measured output is available. Introducing all of the results is not easy because there exist various unconnected approaches. However, among the proposed results, we can distinguish stability conditions expressed in terms of linear matrix inequalities for discrete-time switched linear systems with average dwell time , constructive approaches based on the resolution of Riccati equations , linear matrix inequality approach to static output-feedback stabilization of discrete-time networked control systems  or optimization techniques [9, 10], pole or eigenstructure assignment techniques [11, 12].
Recently, much effort has been directed towards finding a feedback controller in order to guarantee robust stability, see [13, 14]. On the other hand, when controlling a real plant, it is also desirable to design a control systems which is not only asymptotically stable, but also guarantees an adequate level of performance index. One way to address the robust performance problem is to consider a linear quadratic cost function. This approach is the so-called guaranteed cost control [15, 16]. In recent years, with the development of robust control theory and H ∞ control theory, the robust guaranteed cost control approach to the design of state feedback control laws for uncertain systems has been a subject of intensive research . Quadratic guaranteed cost control for linear systems with norm-bounded uncertainty was dealt with in . However, all this research has been done on uncertain system without time delay or continuous-time delay systems. Little attention has been paid towards discrete-time systems with delay.
This article is concerned with both the static output feedback stabilization and output-feedback guaranteed cost controller design for a class of discrete-time nonlinear systems with time-delay. The new sufficient conditions for static output feedback stabilization of a class of discrete-time nonlinear systems with time-delay are presented. Then, sufficient LMI conditions for guaranteed cost control by static output feedback are given. Finally, examples are given to show the effectiveness of our proposed approaches.
The rest of the article is organized as follows. In Section 2, we present the main results concerning the static output feedback stabilization problem for a class of nonlinear discrete-time systems with time-delay. In Section 3, we deal with the problem of guaranteed cost control via static output feedback for a class of nonlinear discrete-time systems with time-delay. Two numerical examples are given in Section 4 to illustrate the proposed results. Finally, we draw some conclusions in Section 5.
The following notations will be used throughout this article. R is the set of all real numbers. Z + is the set of all non-negative integers. Z+ is the set of all positive integers. R n denotes the n-dimensional Euclidean space. R n×m is the set of all (n×m)-dimensional real matrices. I denotes an identity matrix with appropriate dimension. The superscript 'T' represents the matrix transposition. If a matrix is invertible, the superscript '-1' represents the matrix inverse. X > 0(X ≥ 0) means that X is a real symmetric and positive-definite (semi-definite) matrix. The notation ||·|| refers to the Euclidean norm. For an arbitrary matrix B and two symmetric matrices A and C, the symmetric term in a symmetric matrix is denoted by an asterisk, i.e., .
Our first objective is to give sufficient linear matrix inequality conditions for stabilization of system (2.1) by a static output controller u k =Ky k We introduce the following key lemmas which will be used in setting the proofs of the next statements.
where v1 and v2 are the numbers of the convex hull matrices of the Jacobian of f(x k ) and g(x k ), respectively.
Then, from the result of Lemma 2.1, we can get (2.12) are satisfied if conditions (2.5) are verified. This ends the proof.
Similar to the proof of Theorem 2.3, we can easily get the following corollary for system (2.1).
Remark 2.5. In , the stabilization problem for a class of discrete-time linear systems with time-delay is investigated by state feedback. But, the state vector is not often available for feedback. In this article, the static output feedback is used and the system is nonlinear. Compare to , the results obtained in this article have a greater range of applications.
where U =UT ≥ 0 and R = RT > 0 are some prescribed real matrices and P = PT,Q =QT are positive definite matrices to be determined.
for all initial conditions ϕ k ∈R n -d ≤ k ≤ 0.
is verified then the static output feedback u k =Ky k minimizes the criterion (3.1). From result of Lemma 2.1, we can get that (3.3) are satisfied if conditions (3.2) are verified. So, we get
This ends the proof.
for all initial conditions ϕ k ∈R n ,-d ≤ k ≤ 0.
Proof. By Theorem 3.1 and the Schur complement lemma, the condition of the corollary follows readily.
Remark 3.3. The study  considered the static output feedback and guaranteed cost control for a class of discrete-time nonlinear systems. But, the time-delay system did not deal with in . In this article, we investigated the static output feedback stabilization and output-feedback guaranteed cost controller design for a class of discrete-time nonlinear systems with time-delay. So, the results obtained in this article have a greater range of applications.
for all initial conditions ϕ k ∈R n ,-d ≤ k ≤ 0, where U, R, P, and Q are given by (4.3) and (4.4).
Both the problems of the static output feedback stabilization and output-feedback guaranteed cost controller design for a class of discrete-time nonlinear systems with time-delay are investigated in this article. The new static output feedback stabilization conditions are proposed, which are independent of the time delay. We develop a quadratic guaranteed cost control method for stabilization via static output feedback. Two numerical examples are provided to show the applicability of the developed results.
This study was supported by the Natural Science Foundation of Tianjin under Grant 11JCYBJC06800.
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