Technical communiqueA novel -dependent Lyapunov function and its application to singularly perturbed systems☆
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) . 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 -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 -dependent Lyapunov method, a key point is how to construct an -dependent Lyapunov function (-DLF). In the early days, the -DLF is constructed as follows (Fridman, 2002, Saksena and Kokotovic, 1981): where with being the slow state and being the fast state in SPSs. Some control problems for SPSs have been solved by using the -DLF . For example, the state feedback and static output feedback 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 -DLF was improved in Yang and Zhang (2009) as where the -DLF 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 -DLF , the -DLF is more general, due to the fact that the -DLF can reduce to the -DLF by letting . Additionally, since more information of the slow and fast modes is exploited in the -DLF by introducing matrices and , less conservative results can be achieved by utilizing the -DLF than utilizing the -DLF (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 -DLF, a new -DLF was presented in Wang et al. (2019) as Compared with the -DLF , the -DLF is more general and it can reduce to the -DLF by letting . This means that less conservative results are obtainable by using the -DLF than using the -DLF (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 -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 -DLF be established with the following two features?
- (i)
The existing three classes of common -DLFs, i.e., , and , can be seen as special cases of the novel -DLF;
- (ii)
The established novel -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 -DLF with the above two features for SPSs. To show its superiority, the established novel -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 -dissipative for all , where is the upper bound of the singular perturbation parameter . 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 -DLF.
Notation : the real -dimensional Euclidean space. : the set of real numbers. : the identity matrix of order . : the block diagonal matrix. “”: the symmetric block matrices. : the transpose of the matrix . .
Section snippets
Main results
In this section, a novel -DLF will be given. Before proceeding further, some useful lemmas are first introduced.
Lemma 1 For a scalar and symmetric matrices with appropriate dimensions, if , and , then holds for all .Yang & Zhang, 2009
Remark 1 Lemma 1 plays a key role to guarantee the positive definiteness of the -DLFs and and to obtain the corresponding main results for SPSs when using the -DLFs and . To establish a novel -DLF, a more
Application to analysis of singularly perturbed Lur’e systems
To show the superiority of the novel -DLF, the following singularly perturbed Lur’e system adopted from Yang et al. (2011) is considered in this section: where with being the slow state and being the fast state; is the controlled output vector; is the exogenous disturbance signal; ; is the SPP; , , , and are known real matrices. Lur’e nonlinearity is a continuous
Conclusion
A novel generalized -DLF for SPSs has been proposed in this paper, which encompasses three classes of common -DLFs in the existing literature. To demonstrate its advantages, the novel generalized -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.