Robust control for incremental quadratic constrained nonlinear time-delay systems subject to actuator saturation☆
Introduction
In the real world, most systems are nonlinear. Thus, the investigation of nonlinear systems is always a hot but difficult subject, whether in stabilization [1], [2] or observation [3], [4]. After the Lipschitz constraint was presented, many scholars have made outstanding achievements in the study of Lipschitz nonlinear systems [5], [6], [7], [8]. Rajamani [5] and [6] both focused on the observer design for nonlinear systems, while [7] and [8] studied the filtering problem for fuzzy nonlinear systems. Chang et al. [7] designed an filter for nonlinear discrete-time systems with measurement quantization. In the case of multi-path data packet dropouts, a design method of fuzzy peak-to-peak filter for nonlinear systems was given in [8]. However, if the Lipschitz constant is relatively large, feasible solutions for sufficient conditions may not exist. Then, the one-sided Lipschitz (OSL) constraint was proposed in [9], and it was proved that the OSL constraint is more general than the Lipschitz constraint. In order to overcome the weighted bilinear form in dealing with OSL functions, Abbaszadeh and Marquez [10] introduced the condition of quadratic inner-boundedness (QIB) and designed an observer for OSL nonlinear systems satisfying QIB. Following this, Zhang et al. [11] presented a less conservative observer design method for OSL nonlinear systems. In [12], the definition of incremental quadratic constraint (IQC) was introduced to nonlinear systems, which is a more general description of nonlinearities. By introducing an incremental multiplier matrix (IMM), many nonlinear functions can be described by IQC, such as Lipschitz nonlinear functions [13], OSL nonlinear functions [14] and conic-type nonlinear functions [15], [16] (more details can be found in [17]). Li et al. [13] addressed the energy-to-peak filtering problem for Lipschitz nonlinear systems. Li and Yuan [14] discussed strong convergence problem for OSL nonlinear systems. He et al. [15] and [16] considered sliding mode control and dissipative filtering design for conic nonlinear systems, respectively. In recent years, investigation of IQC nonlinear systems has attracted significant research interest, such as [18], [19], [20], [21] herein.
As we know, time delays always appear in practical engineering, such as power systems [22], [23], neural network systems [24], multi-agent systems [25], etc. Time delay can not be ignored in the design of control systems, and it may weaken the performance of systems and even cause instability. Due to the physical restrictions of the actuator, actuator saturation phenomenon is another common problem existing in control systems [26], [27], [28], [29]. Actuator saturation has typical nonlinear characteristics, so it is necessary to deal with this nonlinearity in the design of the controller. Hu et al. [30] presented the linear convex hull method to handle saturation phenomenon by introducing an auxiliary matrix, and gave the concept of estimating the domain of attraction (DA) by quadratic Lyapunov function (QLF). In order to further enlarge the DA for saturation systems, Cao and Lin [31] extended the QLF to the saturation-dependent Lyapunov function (SDLF). However, the method in [31] is also somewhat conservative, because the estimated DA is only an intersection of a set of ellipsoids. Therefore, the convex hull Lyapunov function (CHLF) was introduced in [32] and it was proved that CHLF is less conservative than SDLF in estimating the DA. For saturation systems with time delays, many valuable achievements have also been made. By introducing a method via auxiliary time delay feedback, Chen et al. [33] investigated the saturation control problem for neutral time-delay systems. In [34], saturation controllers were designed for time-delay systems with stochastic disturbance. Recent interesting works on saturation systems with time delay can be found in [35], [36], [37], [38], [39], [40], [41]. It is noteworthy that the aforementioned results focused on the saturation control problem for linear time-delay systems. To the authors’ knowledge, the saturation controller design for IQC nonlinear time-delay systems has not yet been reported.
Based on the above discussion, this article investigates the saturation control problem for nonlinear systems satisfying IQC with time delay. The main challenge of this article includes the following two aspects: One is using a proper differential inclusion to describe the saturated control system, and employing the CHLF to derive less conservative conditions under which the closed-loop system is exponentially stable. The other is dealing with the nonlinear function satisfying IQC properly, and obtaining sufficient conditions formulated by the form of linear matrix inequalities. The contributions are mainly reflected in three components:
(i) The IQC nonlinear function with time delay is proposed for the first time and the nonlinear term is dealt with by the S-procedure.
(ii) The saturated feedback law is constructed and the considered systems are proved to be exponentially stable under the given conditions.
(iii) An optimization problem of maximizing the estimation of DA is derived by CHLF approach, which is less conservative than that derived by QLF.
The remainder of this paper is organized as follows: Section 2 gives the problem statement and preliminaries. In Section 3, sufficient conditions are proposed to ensure that the closed-loop system satisfies exponential stability and an optimization problem is given for maximizing the estimation of DA. Section 4 presents an algorithm for computing IMMs of some commonly encountered nonlinearities. In Section 5, the validity of the method is demonstrated by two examples, and the comparisons with existing works illustrate that the proposed method can reduce the conservatism in estimating the DA.
The following standard notations will be adopted in this article. represents the symbol function and stands for the set of integers . Let and is the -norm of which means that . is the -level set of and is an ellipsoid.
Section snippets
Problem statement and preliminaries
In this article, the following system is taken into consideration:where are the state vectors, and is the time delay. and represent nonlinear terms. is the saturated input. and are given matrices. For the sake of simplicity, the variable is omitted and the variable is defined as . Therefore, (1) is equivalent toDenote the
Saturated linear feedback for robust stabilization
In this section, the CHLF is employed to obtain sufficient conditions to ensure that system (3) is exponentially stable. Furthermore, the maximal estimation of DA is determined by solving an optimization problem. Theorem 3.1 Assume that nonlinear functions and satisfy the condition of IQC. Let for . If there exist for and such thatwhere is the ith row of and
An algorithm of computing IMM
In this section, a method for calculating IMMs of some common nonlinear functions satisfying IQC is introduced. Based on the results in [12], the nonlinear functions are assumed to be continuously differentiable. Denote as a convex set of vertex matrices and the nonlinear function satisfies . According to the expression of (4), one can getBy substituting the decomposition of into (51), we have
Example 1
Consider system (2) with the following parametersand the nonlinear functions are given byUsing the results in Section 4, the vertex matrices of and areBy substituting the expressions of and into LMIs (53), the IMMs of the nonlinear functions and are determined by
Conclusion
In this article, we have studied the saturation control problem for nonlinear systems satisfying IQC with time delay. By using the theory of convex set, the saturated nonlinear system is transformed into the form of convex set, and CHLF has been constructed to analyze the stability of the closed-loop system. Meanwhile, we have presented an optimization problem to maximize the estimation of DA and demonstrated that the maximal ellipsoid of DA derived by the CHLF approach is larger than those
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