Technical communiqueStability analysis of linear systems with time-varying delay via intermediate polynomial-based functions☆
Introduction
Time delay is a natural phenomenon in practical systems (Fridman and Dambrine, 2009, Li et al., 2019b, Yan et al., 2019), such as power system (Schiffer, Dörfler, & Fridman, 2017), robotic airship (Wang et al., 2019), and vehicle suspension system (Li, Jing, & Karimi, 2014). Time delay brings about sensitive dependence of system performance and stability on delay intervals (Li et al., 2019a, Zhang and Han, 2015). In addition, the input delay idea is extensively used to model variable sampling intervals (Fridman, 2010) and network-induced delays (Freirich and Fridman, 2016, Liu et al., 2015, Zhang and Han, 2013). Accordingly, stability analysis of time delay systems has received considerable attentions.
As is well known, general Lyapunov–Krasovskii functionals (LKFs) with more information on time delay are helpful for reducing conservatism (Li et al., 2014, Li et al., 2019a, Yan et al., 2019). A relaxed stability condition is suggested that some Lyapunov matrices need not to be positive definite (Lee, Park, & Xu, 2017). Dividing the delay interval by delay-central-point (DCP) method, novel LKF is introduced with delay-dependent matrices (Fridman, Shaked, & Liu, 2009). Moreover, a generalized delay partitioning approach is developed in Yue, Tian, and Zhang (2009) with equally spaced subintervals, in which the number of decision variables grows dramatically as the delay partitioning segments increase. In Liu and Li (2015), an optimal delay division approach is proposed by using a variable parameter. However, this adjustable parameter can only vary within the delay range. In Lee and Park (2017), the matrix-refined-functions (MRFs) are proposed in single integral augmentation form to provide impressive flexibility. Furthermore, by introducing a couple of orthogonal polynomials, new auxiliary polynomial-based functions (APFs) are presented to produce single integral with the 1st order scalar function (Li et al., 2019b). Based on a novel double integral inequality, an improved inequality-based functions (IBFs) approach is given to offer both of single and double integrals (Li et al., 2019a). It is noted that only the single integrals are considered in Lee and Park (2017), and the matrices in Li et al. (2019a) and Li et al. (2019b) are incomplete with some zero components for numerical tractability, which results in under-utilization of full relationships on system information.
For making use of delay information, the double integral of quadratics concerned with system state is developed to present delay-dependent stability criterion (Fridman & Shaked, 2003). Thus, the conservatism significantly depends on estimation of integral terms (Zhang, Han et al., 2017, Zhang et al., 2016). Recently, the Bessel–Legendre inequality (BLI) (Seuret & Gouaisbaut, 2015) and new double integral inequalities (Li, Bai, Huang, Yan, & Mu, 2018) are successively established to capture analytical superiorities over the Jensen inequality (Fridman, 2010, Li et al., 2014, Zhang and Han, 2013) and Wirtinger-based inequality (WBI) (Seuret & Gouaisbaut, 2013) in single and double integral forms.
In this note, the stability analysis for linear system with time-varying delay is investigated inspired by the works (Li et al., 2019a, Li et al., 2019b). Firstly, novel fractional order intermediate polynomial-based functions (IPFs) with variable parameters are proposed in delay-product types, which allow adoption of slack matrices and double integrals of state. Considering the combined effect of all involved matrices, the positive definite requirement of IPFs is guaranteed by an equation instead of inequality in Li et al. (2019a), which eliminates the gap between the IBFs and its lower bound, and makes for reducing conservatism. Then, an improved stability condition is derived for time delay system. Different from Li et al. (2019a) and Li et al. (2019b) with some zero matrices, based on the fractional order polynomials, the relationships among various states are fully coupled and essentially consolidated, and the characteristics of improved bounding inequality are reflected. Most importantly, unlike (Li et al., 2019a, Li et al., 2019b) with constant coefficients for slack variables, in virtue of tunable parameters, the fixed restriction on matrices is converted into a dynamic one, and the feasible space is significantly relaxed. Finally, a numerical example is presented to demonstrate the advantages of the proposed approach.
Notations: is -dimensional Euclidean space. is the transpose of matrix . represents a term induced by symmetry. diag denotes a block diagonal matrix. means being a positive definite matrix. col is a column vector. {M} M M. is Kronecker product of matrices and . and stand for the minimum and maximum eigenvalues of , respectively.
Section snippets
System description and preliminaries
Let us consider a class of linear systems with time delay: where is system state with initial condition ; and are system matrices; represents time-varying delay satisfying and with and being constants. Nextly, some necessary lemmas are recalled here.
Lemma 1 The following inequality is true for the matrix , and any differentiable function : where Seuret & Gouaisbaut, 2015 Second Order Bessel–Legendre Inequality (SOBLI)
Intermediate polynomial-based functions
For simplicity, , and stand for and .
Proposition 1 Let scalars and , and matrices and of compatible dimensions. The functions defined as follows are positive definite: where
Main results
For brevity, denote
Theorem 1 For given scalars , and , the time delay system (1) is asymptotically stable, if there exist
Numerical examples
Example 1 Consider the time delay system (1) with For various , the comparative upper bounds of and number of decision variables (NoDVs) by Theorem 1 and some existing approaches are given in Table 1. Considering a tradeoff of feasible delay range and time consumed, and assuming and , the optimal MADBs at by Theorem 1 are presented. In Kim (2016), the time-varying delay case of BLI are proposed. For treating single integrals with time
Conclusion
In this note, by fractional order intermediate polynomials, the IPFs are formulated, which substantially enhance the relationships among extra-states. Then, an improved stability criterion is derived. By involving slack matrices, some additional degree of freedom is exploited. By feat of variable parameters, the feasible space is further relaxed. Accordingly, multiple synergy for reducing conservatism is achieved in the proposed approach without excessive computational complexity, which is
Acknowledgments
This work is supported by National Natural Science Foundation of China (61803159, 61673178, 61773289, 61922063), Shanghai Chenguang Project (18CG31), Shanghai Shuguang Project (18SG18), Shanghai Sailing Program (18YF1406400), Program of Shanghai Academic Research Leader (19XD1421000), Shanghai Natural Science Foundation (18ZR1409600, 17ZR1444700, 17ZR1445800), China Postdoctoral Science Foundation (2019T120311, 2018M032042), Fundamental Research Funds for the Central Universities (222201814040
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The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Emilia Fridman under the direction of Editor André L. Tits