Computing multiple Lyapunov‐like functions for inner estimates of domains of attraction of switched hybrid systems

Abstract

Summary Domain of attraction plays an important role in stability analysis and safety verification of nonlinear control systems. In this paper, based on the concept of multiple Lyapunov‐like functions, we propose iteration algorithms for computing inner estimates of domains of attraction for a class of switched hybrid systems, where the state space is composed of several regions and each region is described by polyhedral sets. Starting with an initial inner estimate of domain of attraction, we firstly present a theoretical framework for obtaining a larger inner estimate by iteratively computing multiple Lyapunov‐like functions. Successively, the theoretical framework is underapproximatively realized by using S‐procedure and sums of squares programming, associated with the coordinatewise iteration method. Afterwards, for obtaining a required initial inner estimate of domain of attraction, we propose an alternative higher‐order truncation and linear semidefinite programming based method for computing a common Lyapunov function. Especially, a bisection method based improvement is proposed for obtaining better estimates in each iteration step. Finally, we implement proposed algorithms and test them on numerical examples with comparisons. These computation and comparison results show that the advantages of our multiple Lyapunov‐like functions based algorithm. Especially, we provide alternative underapproximations for avoiding the possible numerical problem.

Average dwell time based stability analysis for nonautonomous continuous‐time switched systems

“…It is worthy to investigate how to solve bilinear problems caused by the constraints and in our future work with YALMIP . Moreover, it is also worthy to investigate the computation issue for more general nonautonomous switched systems based on existing methods . Besides, it is also interesting to extend the results to the stochastic systems and the discrete‐time switched systems …”

“…49 Moreover, it is also worthy to investigate the computation issue for more general nonautonomous switched systems based on existing methods. [50][51][52][53][54][55][56] Besides, it is also interesting to extend the results to the stochastic systems 57 and the discrete-time switched systems. 28,[58][59][60][61]…”

Corrigendum to “Computing multiple Lyapunov‐like functions for inner estimates of domains of attraction of switched hybrid systems” [International Journal of Robust and Nonlinear Control, 2018;28(17):5191–5212. DOI: 10.1002/rnc.4280]

“…For constraint (12) in Section 4.1 and constraint (28) in Section 5.2 in the original paper, s 9, i , j ( x ) and s 10, i , j ( x ) ∈ ∑ should be both corrected as positive definite polynomials s 9, i , j ( x ) and s 10, i , j ( x ) ∈ ∑ or nonzero polynomials s 9, i , j ( x ) and s 10, i , j ( x ) ∈ ∑ with degree 0 (ie, positive constant polynomials s 9, i , j ( x ) and s 10, i , j ( x ) ∈ ∑ ).…”

“…Note that, in the original paper, except the case d = 2 in Example 1, s 9, i , j ( x ) and s 10, i , j ( x ) were both preassumed to be nonzero polynomials in ∑ with degree 0 for all other examples when applying SOSTOOLS for implementation.…”

Sum-of-Squares Flight Control Synthesis for Deep-Stall Recovery

“…The thus modified algorithm, to which we will refer as Algorithm 1b, does not need an ellipsoidal shape as input. Similar approaches have recently been employed for analysis of reachable sets of polynomial systems [26] and the region of attraction of hybrid systems [27]. While Algorithm 1b allows the region-of-attraction estimate to grow in all directions rather than the one given by E, we will see later that is unsuitable for the purpose of deep-stall recovery.…”