Recent works have shown that the $L_1$ and $L_\infty$-gains are natural performance criteria for linear positive systems as they can be characterized using linear programs. Those performance measures have also been extended to linear positive impulsive and switched systems through the concept of hybrid $L_1\times\ell_1$-gain. For LTI positive systems, the $L_\infty$-gain is known to coincide with the $L_1$-gain of the transposed system and, as a consequence, one can use linear copositive Lyapunov functions for characterizing the $L_\infty$-gain of LTI positive systems. Unfortunately, this does not hold in the time-varying setting and one cannot characterize the hybrid $L_\infty\times\ell_\infty$-gain of a linear positive impulsive system in terms of the hybrid $L_1\times\ell_1$-gain of the transposed system. An alternative approach based on the use of linear copositive max-separable Lyapunov functions is proposed. We first prove very general necessary and sufficient conditions characterizing the exponential stability and the $L_\infty\times\ell_\infty$- and $L_1\times\ell_1$-gains using linear max-separable copositive and linear sum-separable copositive Lyapunov functions. Results characterizing the stability and the hybrid $L_\infty\times\ell_\infty$-gain of linear positive impulsive systems under arbitrary, constant, minimum, and range dwell-time constraints are then derived from the previously obtained general results. These conditions are then exploited to yield constructive convex stabilization conditions via state-feedback. By reformulating linear positive switched systems as impulsive systems with multiple jump maps, stability and stabilization conditions are also obtained for linear positive switched systems. It is notably proven that the obtained conditions generalize existing ones of the literature.
$L_1/\ell_1$-to-$L_1/\ell_1$ analysis of linear positive impulsive systems with application to the $L_1/\ell_1$-to-$L_1/\ell_1$ interval observation of linear impulsive and switched systems
Stability and $L_1/\ell_1$-to-$L_1/\ell_1$ performance analysis of uncertain impulsive linear positive systems with applications to the interval observation of impulsive and switched systems with constant delays
For general time-varying or switched (nonlinear) systems, converse Lyapunov theorems for stability are not available. In these cases, the integral input-to-state stability (iISS) property is not equivalent to the existence of an iISS-Lyapunov function but can still be characterized as the combination of global uniform asymptotic stability under zero input (0-GUAS) and uniformly bounded energy input-bounded state (UBEBS). For impulsive systems, asymptotic stability can be weak (when the asymptotic decay depends only on elapsed time) or strong (when such a decay depends also on the number of imp…
In this paper, we consider the efficient and robust reconstruction of signals and images via $\ell_{1}-α\ell_{2}~(0<α\leq 1)$ minimization in impulsive noise case. To achieve this goal, we introduce two new models: the $\ell_1-α\ell_2$ minimization with $\ell_1$ constraint, which is called $\ell_1-α\ell_2$-LAD, the $\ell_1-α\ell_2$ minimization with Dantzig selector constraint, which is called $\ell_1-α\ell_2$-DS. We first show that sparse signals or nearly sparse signals can be exactly or stably recovered via $\ell_{1}-α\ell_{2}$ minimization under some conditions based on the restricted $1$-…
Uniform Exponential Stability Analysis of Impulsive Linear Time-Invariant Systems on Banach and Hilbert Spaces: Non-Coercive and Coercive Stability Conditions
School of Life Sciences, University of Applied Sciences, Muttenz, Switzerland（瑞士穆滕兹应用科学大学生命科学学院）；Department of Engineering, University of Perugia, Perugia, Italy（意大利佩鲁贾大学工程学院）；Université Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, Grenoble, France（法国格勒诺布尔阿尔卑斯大学、法国国家科学研究中心、格勒诺布尔理工学院、GIPSA实验室）
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
We design an algorithm for control of a linear switched system by means of Geometric Algebra. More precisely, we develop a switching path searching algorithm for a two-dimensional linear dynamical switched system with non-singular matrix whose integral curves are formed by two sets of centralised ellipses. Then it is natural to represent them as elements of Geometric Algebra for Conics (GAC) and construct the switching path by calculating the switching points, i.e. intersections and contact points. For this, we use symbolic algebra operations, more precisely the wedge and inner products, that …
In this paper, we present a geometric approach for computing controlled invariant sets for hybrid control systems. While the problem is well studied in the ellipsoidal case, this family is quite conservative for constrained or switched linear systems. We reformulate the invariance of a set as an inequality for its support function that is valid for any convex set. This produces novel algebraic conditions for the invariance of sets with polynomial or piecewise quadratic support function.
Efficient computation of trajectories of switched affine systems becomes possible, if for any such hybrid system, we can manage to efficiently compute the sequence of switching times. Once the switching times have been computed, we can easily compute the trajectories between two successive switches as the solution of an affine ODE. Each switching time can be seen as a positive real root of an analytic function, thereby allowing for efficient computation by using root finding algorithms. These algorithms require a finite interval, within which to search for the switching time. In this paper, we…
