Max-type copositive Lyapunov functions for switching positive linear systems

Diagonal stability of a class of discrete‐time positive switched systems with delay

“…In particular, the result of Theorem 4.1 relaxes the condition required in [8] for the existence of a common diagonal Lyapunov functional, thereby giving a less conservative stability criterion. It also provides an extension of Theorem 3 of [11] to nonlinear timedelayed systems. The result of Theorem 5.1 describes conditions for a switched diagonal Lyapunov functional to exist for the same class of nonlinear switched systems, and gives an alternative type of condition to the LMIs described in [14].…”

“…Finally, on a theoretical level, it is of course interesting to understand the degree to which properties of basic LTI systems can be extended to more complex system classes and results characterising the existence of diagonal L-K functionals allow us to understand the, somewhat intricate, relationship between the various types of Lyapunov function available. In this context, the interesting work of [11] on the links between linear, max-type and diagonal Lyapunov functions for switched positive linear systems is noteworthy; in fact, the work of this latter paper has inspired some of the results to be presented here.…”

Further Results on Diagonally Invariant Exponential Stability of Switching Linear Systems

“…For any ∈ J , introduce (similarly to (15-H)) ( ( )) * = max ∈C max (A ( )), C * = { * ∈ C | max (A * ( )) = ( ( )) * } and notice that there exists * ∈ C * such that matrix A * ( ) is irreducible (since all A ( ), ∈ C are essentially positive and, inherently, irreducible). Hence, for any ∈ J , we apply case 1 However the proof of Theorem 4 together with Lemma 1 in [22] also allows addressing quantitative aspects, in the sense that there exist pairs (k, ), with k ≫ 0, satisfying inequalities (10-S), with ≥ ( ) * as close to ( ) * as we want, and, respectively, inequalities (10-H), with ≥ ( ) * as close to ( ) * as we want. Equivalently, this means that ( ) * and ( ) * , respectively, represent the best (fastest) contraction rate for all the invariant sets defined by inequality (8-S) and (8-H), respectively, with = 1.…”

“…where , ∈ [0, 1] are parameters, inspired by the numerical example presented in [27]. Note that matrices A 1 , A 2 (22) are nonnegative, and therefore A = A , = 1,2; i.e., A = A. The DIES analysis is organized mutatis mutandis as in example 1, by taking a grid constructed for ( , ) ∈ [0, 1] × [0, 1], with a step of 0.02.…”

Stability analysis of switched positive nonlinear systems: an invariant ray approach

“…Switched positive systems have also attracted attention lately; they have important applications, such as investigating epidemic spread over time-varying networks [21] and the mitigation of HIV mutations [22,23]. Certain studies have made in-depth investigations of stability analysis and control design of switched positive linear systems [21][22][23][24][25][26][27][28][29][30][31][32], in which the co-positive Lyapunov function method is particularly emphasized [24,26,27,30]. Indeed the nonnegativity of the states facilitates the construction of such types of Lyapunov functions.…”