High-order maximum principles for the stability analysis of positive bilinear control systems

Abstract

We consider a continuous-time positive bilinear control system (PBCS), i.e. a bilinear control system with Metzler matrices. The positive orthant is an invariant set of such a system, and the corresponding transition matrix C(t) is entrywise nonnegative for all time t ≥ 0. Motivated by the stability analysis of positive linear switched systems (PLSSs) under arbitrary switching laws, we fix a final time T > 0 and define a control as optimal if it maximizes the spectral radius of C(T ). A recent paper [2] developed a first-order necessary condition for optimality in the form of a maximum principle (MP). In this paper, we derive higher-order necessary conditions for optimality for both singular and bang-bang controls. Our approach is based on combining results on the secondorder derivative of the spectral radius of a nonnegative matrix with the generalized Legendre-Clebsch condition and the Agrachev-Gamkrelidze second-order optimality condition. Index TermsPositive switched systems, stability under arbitrary switching laws, variational approach, high-order maximum principles, Perron-Frobenius theory.

State‐bounding observer design for uncertain positive systems under ℓ1 performance

“…Theorem 5. For a stable positive system (1) with interval parameters (2), there exists an upper-bounding observer (8) such that the augmented system (14) is positive and robustly stable with the 1 -induced performance ||ẑ|| 1 < γ u ||w|| 1 if and only if one can find matricesX P ,X Q , andX R and vectors p 1 ≥≥ 0 and p 2 ≥≥ 0 satisfying…”

“…Recently, many theoretical results concerning positive systems such as behavioral analysis and synthesis have been reported . For instance, the state‐space characterization of positive systems has been derived in the work of Benvenuti and Farina .…”

“…Recently, many theoretical results concerning positive systems such as behavioral analysis and synthesis have been reported. [3][4][5][6][7][8][9][10][11][12][13][14] For instance, the state-space characterization of positive systems has been derived in the work of Benvenuti and Farina. 15 The problem of reachability and controllability for positive systems has been thoroughly addressed in the works of Fanti et al 16 and Valcher.…”