Multi-rate sampled-data composite control of linear singularly perturbed systems

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Abstract

This paper addresses the multi-rate stabilization problem for linear singularly perturbed systems. The proposed multi-rate sampled-data control law is based on the discretization in multi-rate fashion on the continuous-time composite control law obtained from the singular perturbation theory. The sampling times of the slow and fast state variables are allowed to be asynchronous and nonuniformly spaced. A new time-dependent Lyapunov functional is introduced to analyze the closed-loop stability of the considered system with the multi-rate feedback. With the use of the Lyapunov functional, a sufficient condition for exponential stability of the closed-loop system is derived in terms of linear matrix inequalities. Further, a robust stabilizability condition of the proposed multi-rate control law with respect to uncertain singular perturbation parameter is also obtained. Three numerical examples are presented to show the effectiveness of the developed methodology.

Introduction

Many engineering systems possess slow and fast modes, such as electrical power system [1], battery systems [2], mechanical systems with electrical components [3], and unmanned aerial vehicles [4], etc. The mathematical models of such systems can be represented in the form of singularly perturbed systems, in which a small positive parameter is used to characterize the separation degree between slow and fast dynamics. The presence of small singular perturbation parameters usually leads to ill-conditioning numerical problems when applying the standard design methods for regular systems. To alleviate such difficulties caused by small singular perturbation parameters, a widely used approach for solving the control issues of singularly perturbed systems is the reduction technique based on time-scale separation. The main idea of the technique is to decompose the overall system into two reduced-order systems in different time scales. Then the controller for the overall system can be designed in a composite form of two controllers designed for the reduced-order systems. Systematical study on singular perturbation approach in control started at the 1950s by Tikhonov [5]. From then on, singular perturbations and time-scale technique became an important means for analysis and design of control systems, a lot of results have been reported in the literature [6], [7], [8], [9], [10], [11], [12].

With the applications of network and embedded control systems in control engineering field, sampled-data control has received increasing attention during the past decades. A sampled-data system consists of a continuous-time plant and a piecewise constant input signal. In the literature, three main approaches have been proposed for analysis and design of aperiodic sampled-data systems: the discrete-time approach [13], [14], [15], the input delay approach [16], [17], [18], [19], and the impulsive system approach [20], [21]. In contrast to regular systems, only a few results are available in the literature about sampled-data control of singularly perturbed systems. Due to the time-scale difference between the slow and fast dynamics, multi-rate sampled-data designs were suggested in [22], [23] for digital control of singularly perturbed systems. Moreover, it was reported in [27] that the use of multi-rate controllers can provide better closed-loop control performance than the single-rate controllers. In [24], a multi-rate sampled-data control strategy based on discrete-time approach was proposed for linear singularly perturbed systems. The optimal regulator problem of a class of nonlinear singularly perturbed systems using multi-rate measurements was studied in [25]. In [26], three distinct multi-rate composite control strategies for nonlinear singularly perturbed systems were investigated. In [27], a new discretization scheme based on Euler’s methodology was proposed for a class of nonlinear singularly perturbed systems. A multi-rate digital control scheme via discrete-time observers was developed therein. In [28], a multi-rate control strategy by discretizing the continuous-time composite feedback for nonlinear singularly perturbed systems was studied. Sufficient conditions that preserve the closed-loop exponential stability were derived in terms of Lyapunov functions of the reduced-order systems. However, the multi-rate control results in [24], [25], [26], [27], [28] assume constant sampling periods. In [29], the multi-rate state feedback H control problem for linear singularly perturbed systems with nonumiform sampling was solved in the framework of the input-delay approach and the input-output approach. The same problem was revisited in [30] by applying a time-dependent Lyapunov functional based method. It should be pointed out that the multi-rate control designs presented in [29], [30] are not based on the reduced-order technique as adopted in [24], [25], [26], [27], [28]. Moreover, although the results in [29], [30] allow both the slow and fast sampling intervals to be uncertain and varying with time, the ratio of the slow and the corresponding fast sampling intervals is required to be constant. Such restriction cannot be strictly satisfied in practical applications.

In this paper, the multi-rate sampled-data control of linear singularly perturbed systems is investigated. The multi-rate sampled-data control law is generated by discretizing the continuous-time composite control law based on the reduced-order technique, where the slow and fast state variables are measured at the slow and the fast sampling rates, respectively. Unlike [29], [30], the ratio between the slow and the corresponding fast sampling intervals is no longer required to be constant. To guarantee the stability of the closed-loop system with multi-rate composite feedback, a time-dependent Lyapunov functional is constructed by considering the asynchronous sampling mechanism. By employing the Lyapunov functional combined with the use of some matrix inequalities techniques, sufficient conditions for exponential stability of the closed-loop system is derived in terms of linear matrix inequalities (LMIs). The LMI-based conditions reveal the relationship between the slow and fast sampling rates. Given the upper bound of the slow sampling intervals, the maximum allowable sampling period for fast variables can be obtained by solving a set of LMIs. The main contributions of this paper can be summarized in two aspects: (1) a systematic procedure for designing multi-rate composite control laws is proposed, by which the sampling rates of the sensors are allowed to be uncertain and nonuniform; (2) Exploiting the two-time-scale characteristics of the closed-loop system, a time-scale-dependent time-varying Lyapunov functional is introduced to prove exponential stability. The developed approach presents a new efficient tool for analysis and design problems of multi-rate sampled-data control systems.

The rest of this paper is organised as follows. In Section 2, preliminaries are introduced and the problem to be studied is described. Section 3 deals with stability analysis of the closed-loop system with multi-rate composite feedback and the sufficient conditions for exponential stability are also given in this section. In Section 4, three numerical examples are presented to illustrate the usefulness of the proposed method. Finally, we conclude the paper in Section 5.Notation. We use N to denote the set of positive integers, and set N0={0}N. I and In, respectively, denote a identity matrix of suitable dimension and a n × n identity matrix. For any real square matrix A, He(A) denotes the symmetric matrix with the form of A+AT. ‖ · ‖ represents the Euclidean norm of a vector and the spectral norm of a matrix. For a real symmetric matrix A, λmax(A) and λmin(A) denote the maximum and minimum eigenvalue of A, respectively. We further denote a positive-definite (positive-semidefinite, negative, negative-semidefinite) matrix P by P > ( ≥ ,  < ,  ≤ ) 0.

Section snippets

Preliminaries and problem formulation

Consider a linear singular perturbed system described by the following differential equation:{x˙1(t)=A11x1(t)+A12x2(t)+B1u(t)εx˙2(t)=A21x1(t)+A22x2(t)+B2u(t),where x1(t)Rn1, x2(t)Rn2, and u(t)Rm are the slow state vector, the fast state vector, and the control input vector, respectively, ε is a small positive scalar indicating degree of separation between fast and slow dynamics, AijRni×nj, i, j ∈ 1, 2, are constant matrices of appropriate dimensions. Initial conditions on state vectors are

Main result

In this section, we establish an exponential stability condition for system (13) via a time-dependent Lyapunov functional. Before proceeding, we introduce the following notation:n=n1+n2,A1=[A¯11A¯12B1K1B1K2HB1K2],A20=[0n2×n1A¯22B2K1B2K2HB2K2],A21=HA1,I1=[In10n1×2n],I2=[0n2×n1In20n2×(n+n1)],I30=[0n1×nIn10n1×n],I31=[0n1×(n+n1)In10n1×n2],I32=[0n2×(n+2n1)In2],I˜30=I1I30,I˜3i=IiI3i,i=1,2.Define an augmented state vector η(t) asη(t)=col(z1(t),z2(t),e1s(t),e1f(t),e2f(t)).Then, system (13) can be

Numerical examples

In this section, we use three examples to illustrate the effectiveness of the double-rate sampled-data control.

Example 1

Consider the singularly perturbed system defined in Eq. (1) with the following matrix parameters ([5]):A11=[00.400],A12=[000.3450],B1=[00],A21=[00.52400],A22=[0.4650.26201],B2=[01].

Direct computation givesA0=[00.400.3964],B0=[00.1982].ChooseK0=[0.870.57],K2=[0.180.05].Then, the composite control law (4) withK1=[1.00350.8643]stabilizes the system for small enough ε > 0. In the

Conclusion

A design method for multi-rate stabilization of linear singularly perturbed systems has been proposed. It is based on discretizing an available continuous-time composite control law obtained using the singular perturbation theory. In the proposed multi-rate composite control law, the slow and fast state variables are measured at different sampling rates, and their sampling intervals and have aperiodic sampling intervals. The stability analysis of the closed-loop system has been carried out

Declaration of Competing Interest

The authors declared that they have no conflicts of interest to this work.

Acknowledgement

This work was supported in part by the National Natural Science Foundation of China under Grants 61973095 and 61633011, the Guangxi Natural Science Foundation under Grant 2018GXNSFDA281055, the Innovation Project of Guangxi Graduate Education (YCSW2019044).

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