Dwell-time stability and stabilization conditions for linear positive impulsive and switched systems

Abstract

Several results regarding the stability and the stabilization of linear impulsive positive systems under arbitrary, constant, minimum, maximum and range dwell-time are obtained. The proposed stability conditions characterize the pointwise decrease of a linear copositive Lyapunov function and are formulated in terms of finite-dimensional or semi-infinite linear programs. To be applicable to uncertain systems and to control design, a lifting approach introducing a clock-variable is then considered in order to make the conditions affine in the matrices of the system. The resulting stability and stabilization conditions are stated as infinitedimensional linear programs for which three asymptotically exact computational methods are proposed and compared with each other on numerical examples. Similar results are then obtained for linear positive switched systems by exploiting the possibility of reformulating a switched system as an impulsive system. Some existing stability conditions are retrieved and extended to stabilization using the proposed lifting approach. Several examples are finally given for illustration.

Equivalence of several stability conditions for switched linear systems with dwell time

“…The SOS approach has been utilized in some papers to solve infinite‐dimensional problem (see the work of Briat). Through letting the infinite‐dimensional matrix‐valued functions S i ( τ ) to be matrix‐valued polynomial functions, the infinite‐dimensional conditions in statement (d) of Theorem are transformed into the following SOS program, which can be solved with the help of SOSTOOLs (see the work of Papachristodoulou) and SDP solver SeDuMi (see the work of Sturm).…”

“…e. There exist N differentiable matrix-valued functions S i ( ): [0, * ]  → d(n,m) , S i ( ) > 0, N vector-valued functions i ( ): [0, * ]  → ℝ d par (n,m) , and N(N+1) 2 vectors i ∈ ℝ d par (n,m) such that, ∀(i × ) ∈ ( × ), i ≠ j, (10), (11), and (12) hold. f. There exist N differentiable matrix-valued functions S i ( ): [0, * ]  → d(n,m) , S i ( ) > 0, N vector-valued functions i ( ): [0, * ]  → ℝ d par (n,m) , and N(N+1) 2 vectors i ∈ ℝ d par (n,m) such that, ∀(i × ) ∈ ( × ), i ≠ j, (13), (14), and (15) hold.…”

Exponential stability of impulsive positive switched systems with discrete and distributed time‐varying delays

“…Recently, stability problems of impulsive switched systems have been studied in other works . Moreover, there are also some results in the framework of positive systems . Particularly, Wang et al addressed the exponential stability problem of continuous‐time impulsive positive (nonswitched) systems with mixed time‐varying delays.…”

“…[33][34][35] Moreover, there are also some results in the framework of positive systems. [36][37][38][39] Particularly, Wang et al 36 addressed the exponential stability problem of continuous-time impulsive positive (nonswitched) systems with mixed time-varying delays. In addition, Li and Xiang 40 studied the problems of exponential stability and L 1 -gain control for continuous-time positive impulsive switched systems with mixed time-varying delays.…”

Stability of delayed neural networks with impulsive strength-dependent average impulsive intervals

“…Although, the average impulsive interval approach has been applied to many existing literatures, there are no results concerning the time-varying impulses, where both stabilizing and destabilizing impulses are considered simultaneously. In addition, in most existing results on impulsive systems with average impulsive approach [4,15,16,35], it is assumed that the average impulsive interval is independent of the impulsive strength (we call it as strength-independent average impulsive interval), and the average impulsive intervals for all impulsive sequences are assumed to be equal to each other, which may not be anticipated. Actually, when considering time-varying impulsive effects, there are many impulses with different impulsive strengths.…”