Non-fragile observer-based robust control for uncertain systems via aperiodically intermittent control
Introduction
Stability is the primary condition to ensure the normal operation of systems. In practice, however, the unavoidable and inherent system modeling errors/uncertainties may affect the performance and stability of systems. In this case, robust stability and stabilization problems are essential for uncertain systems. When all the system state variables are available, state feedback control can be achieved to realize the stabilization of control systems and to obtain excellent dynamic performance as well. In practical engineering applications, it is unable to measure all the system state variables directly. Then, the state observer is proposed to estimate the system state so that one can use the estimated state to realize state feedback control. For these reasons, the observer-based robust control has attracted considerable attention of researchers, and many results have been obtained [1], [2], [11], [12], [30]. From the point view of practical requirement, it is necessary to study the problem of observer-based control of systems.
In the problem of observer-based control, the continuous control strategy is usually considered, which demands that the control inputs should be activated incessantly. The continuous control method is favorable for control achievement but not for reducing control cost and energy saving. Intermittent control, as one kind of discontinuous control, is more economical than the traditional continuous control since the control inputs of intermittent control are continuous imposed on the systems over certain time intervals and is off during other time intervals [28], [48]. Different from impulsive control that is activated at isolated instants [33], the control width of intermittent control is non zero, which means an easier implementation of intermittent control. Especially for the systems with unobservable system state, intermittent control is more reasonable in practical applications. Therefore, intermittent control, a transition between continuous control and impulsive control, has the superiorities of these two control schemes, and compensates for their disadvantages as well. From the aspects of reducing energy consumption and wear of the controller in practical situation, intermittent control shows especial advantages than the traditional continuous control and impulsive control [27]. As a result, intermittent control has different practical applications in economy, multi-agent systems, air-quality control and so on [13], [35], [36], [38], [43], [45]. Recently, intermittent control has been applied in the observer-based control issues [7], [39], [41], [42]. For example, a distributed observer-based intermittent controller has been designed in [41] for multi-agent systems with switching topologies. In [39], an observer-based periodically intermittent control method has been proposed for a class of linear systems.
Note that the controller designed in [7], [39], [41], [42] is periodically intermittent controller with fixed control width and control period. The periodicity limits the flexibility of the controller design, and is unnecessary for the requirements of practical applications. To overcome these disadvantages, the authors extend the periodically intermittent control to the aperiodic one [47]. Compared with periodically intermittent control, the aperiodically intermittent control strategy can provide more flexibility in controller design and more generality in applications [25], [26], [25], [26], [37], [44]. Recently, the aperiodically intermittent control method has been also considered to solve the observer-based feedback control problem [8], [20], [31]. For example, in [20], an observer-based intermittent controller with variable control periods and control widths has been constructed for nonlinear stochastic systems. In [8], a disturbance-observer-based aperiodically intermittent controller has been designed for the robust stabilization problems of uncertain systems.
Although the above results realize the design of observer-based intermittent controller, it can be noticed that these results mainly concentrate on the deterministic systems. In fact, however, uncertainties that occur in control systems are unavoidable and usually lead to instability of systems [15], [18], [34]. In addition to the system uncertainty, due to the additive unknown noise, inherent inaccuracy in analog systems and the influence of the implementation environment, the parameters in the observer-based controller may also have some slight variations[21], [22], [23], [19]. This brings about an inaccurate realization of the controller. Under this circumstance, it is necessary to consider both robustness performance of systems and the robust and non-fragile property of controllers. In [8], [14], the authors have discussed the robust control problems for a class of uncertain systems, and constructed observer-based intermittent controllers. It can be seen that in these results, only the uncertainties in systems or in the feedback control gain are considered, but the uncertainties in the observer gain are ignored. Therefore, it is meaningful to design the non-fragile observer-based aperiodically intermittent controller containing uncertainties both in the control and observer gains to realize robust stabilization and to derive more general robust stabilization conditions for uncertain systems. This motivates the current work.
On the other hand, the Lyapunov function method is well known as an effective tool to analyze the stability of systems. The Lyapunov function method is also applied to investigate the problems of stability analysis and stabilization for intermittent control systems. From the structure of intermittent control, the control inputs are imposed intermittently in the systems. The dynamical system under the periodically/aperiodically intermittent control with open-loop and closed-loop modes can be regarded as a time-dependent switched systems consisting of a stable subsystem and an unstable subsystem [24]. In the existing results, the common Lyapunov function method that the two modes sharing the same function is usually employed to analyze the stability of intermittent control systems. It can be noticed that sharing the same function is unable to catch the unique and essential feature of intermittent control. The results acquired by the common Lyapunov function method have conservativeness and lack generality. Recently, the piecewise Lyapunov function method has been successfully used in analyzing the stability of intermittent control systems [9], [39]. In comparison with the traditional Lyapunov function, the piecewise Lyapunov function method can effectively capture the information of the two different modes and result in less conservative results to a certain extent. Hence, it is principal to adopt the piecewise Lyapunov function method to solve the robust stabilization problems when obtaining more general robust stabilization criteria, which gives rise to another motivation of this study.
Motivated by the aforementioned discussions, we investigate the non-fragile observer-based robust control problem for a class of uncertain systems based on the aperiodically intermittent control. The robust stability criteria are derived, and the observer-based intermittent controller design criteria are provided. The main contributions lie in the following aspects.
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On the basis of the structure characteristics of intermittent control systems, the description of the non-fragile observer-based aperiodically intermittent controller is presented. The designed controller takes the periodic one as a special case. Thus, the robust stabilization criteria obtained in this paper are more general and are also available for the case of periodically intermittent control scheme.
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Compared with the existing non-fragile observer that only the feedback control gain contains uncertainties, both of the feedback control and observer gains of the controller designed in this paper are considered to have uncertainties. Since the uncertainties may be associated with the change of parameters, the cases of additive and multiplicative uncertainties are also taken into consideration.
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On account of the special features that the intermittent control systems have two dynamic modes, the piecewise Lyapunov function is constructed to establish the robust stabilization criteria, which can fully consider the dynamic characteristics and derive the criteria with less conservatism. By decoupling the stability conditions into a set of tractable linear matrix inequalities (LMIs), the controller design becomes simple and straightforward. Some simple controller design steps are provided to better demonstrate how to design the appropriate observer-based intermittent controller.
In the remainder of this paper, some preliminaries and the problem formulation are introduced in Section 2. In Section 3, the robust asymptotic stability conditions are derived, and the existence and design method of the controller are presented. Section 4 displays a numerical example to illustrate the effectiveness of the obtained results. Conclusions are drawn in Section 5.
Notations: Throughout this paper, is the -dimensional Euclidean space, and refers to the set of all real matrices. is the set of non-negative integer. The superscripts and mean the transpose and the inverse of a matrix, respectively. Matrix means that X is a real symmetric and positive-definite (negative-definite, positive semi-definite, negative semi-definite) matrix. Matrix I stands for an appropriately dimensioned identity matrix. The vector norm is defined by , while a matrix norm is defined by is an eigenvalue of . and denote the maximum and minimum eigenvalue of square matrix X, respectively. Symmetric term in a symmetric matrix is represented by , and . stands for a time sequence. For any time-varying function , let .
Section snippets
Problem description preliminaries
Consider the following uncertain systemswhere is the system state with initial value, ; and are the control input and the measured output, respectively; , and C are the nominal parts of system matrices with appropriate dimensions, and W is a constant matrix, represents the admissible uncertainties satisfyingwith and being known constant matrices, and being unknown matrix and
Main-results
In this section, the design method of the aforementioned stabilization scheme is developed step by step, including the asymptotical stability criterion for system (12), the stabilization design criteria for the control scheme with different uncertainties in the gains, and the controller design procedure.
In order to give the conditions that the control scheme are required to stabilize system (1), the following asymptotical stability criterion of system (12) is derived at first. Theorem 1 Suppose that
A numerical example
This section provides an example to demonstrate the effectiveness of the proposed method. Example 1 Consider the uncertain system (1) with the following parameters
Take the nonlinear functions as . It is easy to see that satisfies (4) with and . Fig. 1 shows the state
Conclusions
This paper has investigated the non-fragile observer-based robust control problem of uncertain systems via the aperiodically intermittent control with variable control periods and control widths. Based on the feature of the intermittent control scheme, a general state space representation of non-fragile nonlinear observer coupled with the intermittent controller is given. The corresponding robust asymptotic stability condition is derived by constructing the piecewise Lyapunov function. By using
CRediT authorship contribution statement
Ying Yang: Conceptualization, Methodology, Software, Writing - original draft. Yong He: Methodology, Writing - review & editing, Funding acquisition.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
This work was supported by the National Natural Science Foundation of China under Grants 61973284 and 61573325, by the Hubei Provincial Natural Science Foundation of China under Grant 2015CFA010, and by the 111 project under Grant B17040.
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