Stability andL₁× ℓ₁-to-L₁× ℓ₁performance analysis of uncertain impulsive linear positive systems with applications to the interval observation of impulsive and switched systems with constant delays

Abstract

Solutions to the interval observation problem for delayed impulsive and switched systems with L1performance are provided. The approach is based on first obtaining stability and L1/ 1-to-L1/ 1 performance analysis conditions for uncertain linear positive impulsive systems in linear fractional form with norm-bounded uncertainties using a scaled small-gain argument involving time-varying D-scalings. Both range and minimum dwell-time conditions are formulated -the case of constant and maximum dwell-times can be directly obtained as corollaries. The conditions are stated as timer/clock-dependent conditions taking the form of infinite-dimensional linear programs that can be relaxed into finite-dimensional ones using polynomial optimization techniques. It is notably shown that under certain conditions, the scalings can be eliminated from the stability conditions to yield equivalent stability conditions on the so-called worst-case system, which is obtained by replacing the uncertainties by the identity matrix. These conditions are then applied to the special case of linear positive systems with delays, where the delays are considered as uncertainties, similarly to as in [1]. As before, under certain conditions, the scalings can be eliminated from the conditions to obtain conditions on the worst-case system, coinciding here with the zero-delay system -a result that is consistent with all the existing ones in the literature on linear positive systems with delays. Finally, the case of switched systems with delays is considered. The approach also encompasses standard continuous-time and discrete-time systems, possibly with delays and the results are flexible enough to be extended to cope with multiple delays, time-varying delays, distributed/neutral delays and any other types of uncertain systems that can be represented as a feedback interconnection of a known system with an uncertainty. the design of interval observers. In particular, the design of structured state-feedback controllers is convex in this setup [23,24], the L p -gains for p = 1, 2, ∞ of linear positive systems can be exactly computed by solving linear or semidefinite programs [24][25][26], optimal state-feedback controllers and observers gains enjoy an interesting invariance property with respect to the input and output matrices of a linear positive system, respectively [4,27]. Linear positive systems with discrete-delays are also stable provided that their zero-delay counterparts are also stable; see e.g. [24,[28][29][30][31][32][33][34]. In particular, all those exact results have been shown to be consequences of small-gain arguments using various L p -gains in [1] by exploiting the robust analysis results reported in [25,26,35]. Some extensions have also been provided, notably pertaining on the analysis of linear positive systems with time-varying delays.Before delving into the objective and the contributions of this paper, it seems important to motivate the consideration of uncertain linear impulsive systems. Impulsive systems are, in fact, a powerful cla...

Interval estimator design for switched systems with its application

“…Unlike [24,25] in which the multiple square Lyapunov functions approach was used, here, the multiple Lyapunov functions are constituted by co-positive Lyapunov functions, which offers a new tool and new testing criterion to force the interval estimation and simplifies the condition solving. Different from [12] in which the dwell time and range dwell time was conducted on the switching signal, here, the more general average dwell time constrain is performed on the switching signal.…”

“…It must be noted that those usual square Lyapunov function approaches do not exploit the positiveness of the interval estimation error dynamics. In fact, for the study of positive systems, the welcome Lyapunov function method is the co-positive Lyapunov function which better reflects the positiveness features of positive systems and simplifies the calculation than the usual square Lyapunov functions [12][13][14]. Whereas, the co-positive Lyapunov function method has not been adopt to investigate the interval estimation issue.…”

Hybrid gain performance‐based random event‐triggered filter of positive semi‐Markovian jump systems with intermittent sensor faults

“…▪ Remark 3. Most of existing literature 16,17,[47][48][49][50] have verified the effectiveness of linear approach for positive systems. In the linear approach, linear Lyapunov functions and linear programming are commonly used for positive systems rather than traditional quadratic Lyapunov functions and linear matrix inequalities.…”

Stability Analysis and Stabilization of Linear Symmetric Matrix-Valued Continuous, Discrete, and Impulsive Dynamical Systems -- A Unified Approach for the Stability Analysis of Linear Systems

“…The stability properties do not depend on the value of the delays and the system is exponentially stable for all possible values of the delays if and only if the system with zero-delays is exponentially stable. It also admits an extension to the periodic systems case using a periodic Lyapunov-Krasovskii functional under the same assumptions as in [10]. This extension is omitted for brevity.…”