Stability of Switched Polynomial Systems

Abstract

This paper investigates the stability of (switched) polynomial systems. Using semi-tensor product of matrices, the paper develops two tools for testing the stability of a (switched) polynomial system. One is to convert a product of multi-variable polynomials into a canonical form, and the other is an easily verifiable sufficient condition to justify whether a multi-variable polynomial is positive definite. Using these two tools, the authors construct a polynomial function as a candidate Lyapunov function and via testing its derivative the authors provide some sufficient conditions for the global stability of polynomial systems.

Exponential Feedback Passivity of Switched Polynomial Nonlinear Systems

“…, if there exist positive definite symmetric matrices , positive definite functions ( ) = (1/2) , symmetrical polynomial parameter matrices ( , ), and positive constants̃, system (1) can be transformed into the form of (12) and, for ∀ ∈ + ,…”

“…wherẽ, 1 , , and ( ) are the same as Theorem 9 and is a reassigned average dwell time. Then, the robust stabilization problem is solvable for system (1) under the parameterized controller (25) and switching law with average dwell time .…”

“…Then, the robust stabilization problem is solvable for system (1) under the parameterized controller (25) and switching law with average dwell time .…”

“…Switched polynomial nonlinear systems (SPNSs) [1][2][3] are a class of dynamical systems consisting of nonlinear subsystems with polynomial structure and a switching law, which defines a specific subsystem being activated during a certain interval of time. Due to the theoretical development as well as practical applications, analysis and synthesis of switched systems have recently gained considerable attention.…”

Controllability, Reachability, and Stabilizability of Finite Automata: A Controllability Matrix Method

“…Up to now, STP has been successfully applied to many research fields related to finite-valued systems like Boolean networks [10][11][12][13][14][15][16][17][18][19][20], multivalued logical networks [21][22][23], game theory [24,25], finite automata [5,26], and so on [27][28][29][30][31][32][33][34][35]. The main feature of STP is to convert a finite-valued system into an equivalent algebraic form [22].…”