Elsevier

Automatica

Volume 133, November 2021, 109749
Automatica

Technical communique
A novel ε-dependent Lyapunov function and its application to singularly perturbed systems

https://doi.org/10.1016/j.automatica.2021.109749Get rights and content

Abstract

The ε-dependent Lyapunov method is an effective technique to analyze both standard and nonstandard singularly perturbed systems, and the constructed ε-dependent Lyapunov function has an impact on the conservatism of the obtained results. This paper proposes a novel ε-dependent Lyapunov function, which includes three classes of common ε-dependent Lyapunov functions in the existing literature and can lead to less conservative results. Firstly, the novel ε-dependent Lyapunov function is introduced, and then it is used to analyze the stability and dissipativity of singularly perturbed Lur’e systems. Some stability and dissipativity analysis criteria are obtained for the singularly perturbed Lur’e systems by using the novel ε-dependent Lyapunov function. The superiority of the obtained analysis criteria is shown by an illustrative example.

Introduction

Real systems, such as mechanical systems, power systems and chemical reaction processes, may involve two-time-scale behaviors because of some parasitic parameters, e.g., inertia, small-time constants, inductances, resistances, capacitances, in these real systems (Naidu, 2002, Rejeb et al., 2018, Zhang et al., 2014). An effective way to mathematically describe such kind of real systems is to model them as singularly perturbed systems (SPSs), in which the state variables are separated into slow and fast types by a small singular perturbation parameter (SPP) ε. Theories and applications of SPSs have enjoyed a rapid development and many results have been reported in the literature (Feng, 1988, Li et al., 2018, Li et al., 2020, Shen et al., 2019, Song et al., 2020, Wang et al., 2018).

Up to now, several techniques have been proposed for the stability analysis of SPSs. A typical one is the so-called reduction method (Saberi & Khalil, 1984). Based on this method, the original SPS is first decomposed into two reduced-order subsystems in different time scales, and then the stability of the original SPS is deduced from the stability of its two reduced-order subsystems. It should be pointed out, however, that this method is only available for standard SPSs. To analyze both standard SPSs and nonstandard SPSs, the state transformation-based Lyapunov method (Chen et al., 2012, Shao, 2004) and the so-called ε-dependent Lyapunov method can be employed (Fridman, 2002, Saksena and Kokotovic, 1981, Wang et al., 2019, Yang et al., 2013, Yang and Zhang, 2009, Yang et al., 2011). When using the ε-dependent Lyapunov method, a key point is how to construct an ε-dependent Lyapunov function (ε-DLF). In the early days, the ε-DLF is constructed as follows (Fridman, 2002, Saksena and Kokotovic, 1981): V1ε,xt=xTtZ1εZ3,1TεZ3,1εZ4,1xtwhere xt=[xsTtxfTt]T with xstRns being the slow state and xftRnf being the fast state in SPSs. Some control problems for SPSs have been solved by using the ε-DLF V1ε,xt. For example, the state feedback and static output feedback H control problems for fuzzy SPSs were studied in Chen, Sun, Min, Sun, and Zhang (2013) and Liu, Sun, and Hu (2005), respectively.

To achieve less conservative results, the ε-DLF V1ε,xt was improved in Yang and Zhang (2009) as V2ε,xt=xTtZ1+εZ2,1εZ3,1TεZ3,1εZ4,1+ε2Z4,2xtwhere the ε-DLF V2ε,xt is positive definite under certain conditions, and the reader can see Yang et al., 2013, Yang and Zhang, 2009 and Yang et al. (2011) for more details. Compared with the ε-DLF V1ε,xt, the ε-DLF V2ε,xt is more general, due to the fact that the ε-DLF V2ε,xt can reduce to the ε-DLF V1ε,xt by letting Z2,1=Z4,2=0. Additionally, since more information of the slow and fast modes is exploited in the ε-DLF V2ε,xt by introducing matrices Z2,1 and Z4,2, less conservative results can be achieved by utilizing the ε-DLF V2ε,xt than utilizing the ε-DLF V1ε,xt (Yang et al., 2013, Yang and Zhang, 2009, Yang et al., 2011).

Recently, to exploit more information of the slow and fast states as well as their interactions in the ε-DLF, a new ε-DLF was presented in Wang et al. (2019) as V3ε,xt=xTtZ1+εZ2,1+ε2Z2,2εZ3,1T+ε2Z3,2TεZ3,1+ε2Z3,2εZ4,1+ε2Z4,2xt. Compared with the ε-DLF V2ε,xt, the ε-DLF V3ε,xt is more general and it can reduce to the ε-DLF V2ε,xt by letting Z2,2=Z3,2=0. This means that less conservative results are obtainable by using the ε-DLF V3ε,xt than using the ε-DLF V2ε,xt (Wang et al., 2019).

Motivated by the works in Wang et al., 2019, Yang et al., 2013, Yang and Zhang, 2009 and Yang et al. (2011), one can see that when constructing the ε-DLF for SPSs, more information of the slow and fast modes and their interactions should be taken into account to obtain less conservative results. Therefore, a natural question to ask is: can a novel ε-DLF be established with the following two features?

  • (i)

    The existing three classes of common ε-DLFs, i.e., V1ε,xt, V2ε,xt and V3ε,xt, can be seen as special cases of the novel ε-DLF;

  • (ii)

    The established novel ε-DLF can reveal more information of the slow and fast modes and their interactions as much as possible such that the obtained results are less conservative.

This paper aims to establish the novel ε-DLF with the above two features for SPSs. To show its superiority, the established novel ε-DLF is used to analyze the stability and dissipativity of singularly perturbed Lur’e systems adopted from Wang et al. (2019) and Yang et al. (2011). The stability and dissipativity analysis criteria are established to determine if the singularly perturbed Lur’e systems are absolutely stable or strictly Q,S,R-dissipative for all ε0,ε̄, where ε̄ is the upper bound of the singular perturbation parameter ε. An illustrative example borrowed from Wang et al. (2019) and Yang et al. (2011) is presented to show the superiority of the stability and dissipativity analysis criteria derived by the novel ε-DLF.

Notation

Rn: the real n-dimensional Euclidean space. R: the set of real numbers. In: the identity matrix of order n. diag: the block diagonal matrix. “”: the symmetric block matrices. ΩT: the transpose of the matrix Ω. HeΩΩ+ΩT.

Section snippets

Main results

In this section, a novel ε-DLF will be given. Before proceeding further, some useful lemmas are first introduced.

Lemma 1

Yang & Zhang, 2009

For a scalar ε̄>0 and symmetric matrices Sll=1,2,3 with appropriate dimensions, if S10, S1+ε̄S2>0 and S1+ε̄S2+ε̄2S3>0, then S1+εS2+ε2S3>0 holds for all ε0,ε̄.

Remark 1

Lemma 1 plays a key role to guarantee the positive definiteness of the ε-DLFs V2ε,xt and V3ε,xt and to obtain the corresponding main results for SPSs when using the ε-DLFs V2ε,xt and V3ε,xt. To establish a novel ε-DLF, a more

Application to analysis of singularly perturbed Lur’e systems

To show the superiority of the novel ε-DLF, the following singularly perturbed Lur’e system adopted from Yang et al. (2011) is considered in this section: Eεẋt=Axt+Bϕσt+D1ϖtσt=C1xst+C2xftzt=Fxt+D2ϖtwhere xt=[xsTtxfTt]T with xstRns being the slow state and xftRnf being the fast state; ztRnz is the controlled output vector; ϖtRnϖ is the exogenous disturbance signal; EεdiagIns,εInf; ε is the SPP; A, B, C[C1C2], D1 and D2 are known real matrices. Lur’e nonlinearity ϕσtRr is a continuous

Conclusion

A novel generalized ε-DLF for SPSs has been proposed in this paper, which encompasses three classes of common ε-DLFs in the existing literature. To demonstrate its advantages, the novel generalized ε-DLF has been used to analyze the stability and dissipativity of singularly perturbed Lur’e systems and some effective analysis criteria have been deduced. Then, an illustrative example has been employed to show the superiority of the obtained criteria over the existing ones. Finally, the obtained

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    This work was supported in part by the National Natural Science Foundation of China under Grant 62073166 and Grant 62022042, and in part by the Australian Research Council under Grant DP120104986. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Tingshu Hu under the direction of Editor André L. Tits.

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