Technical communiqueRobust Minkowski–Lyapunov functions☆
Section snippets
Background
Linear inclusions represent a common and convenient mathematical model for handling parametrically uncertain linear dynamical systems, for which the parametric uncertainty arises due to, for instance, approximations of more complex and even locally linearized nonlinear dynamics, imperfect and finite precision arithmetics, or simply incomplete knowledge of the related parameters. Naturally, and without any doubt, stability analysis of linear inclusions is a theme that has received considerable
Robust Minkowski–Lyapunov inequality
In this section, we first introduce the robust Minkowski–Lyapunov inequality, and then derive finite systems of the set inclusions, which are in a one to one correspondence with the robust Minkowski–Lyapunov inequality.
Fundamental property
In this Section, we first formalize the fundamental property of robust Minkowski–Lyapunov functions, and subsequently derive its alternative interpretation that links in an essential way robust Minkowski–Lyapunov functions and robust positively invariant proper –sets.
In light of the preceding analysis, Minkowski function of a proper –set in verifies the robust Minkowski–Lyapunov inequality (2.2) if and only if the finite system of the set inclusions (2.10) or, equivalently, (2.11)
Existence
In this Section, we identify a necessary and sufficient condition for existence of robust Minkowski–Lyapunov functions. The necessary and sufficient condition is revealed by showing that, on one hand, existence of robust Minkowski–Lyapunov functions implies strict stability of the considered polytopic linear inclusion, while, on the other hand, strict stability of the considered polytopic linear inclusion implies existence of robust Minkowski–Lyapunov functions.
On one hand, when a Minkowski
Inequality aftermath and equation problem
It is a common knowledge that stability analysis of polytopic linear inclusions with strictly convex quadratic Lyapunov functions is conservative in the sense that it might fail to certify strict stability of strictly stable polytopic linear inclusions. In a stark contrast to this troublesome deficiency of strictly convex quadratic Lyapunov functions, topological properties of Minkowski functions of proper –sets in render them natural candidate Lyapunov functions for stability
Acknowledgements
The author is grateful to the Editor, Associate Editor and Referee/s for a timely and constructive review as well as very helpful comments.
References (20)
Stability of discrete linear inclusion
Linear Algebra and its Applications
(1995)- et al.
Criteria of asymptotic stability of differential and difference inclusions encountered in control theory
Systems & Control Letters
(1989) The Minkowski–Lyapunov equation
Automatica
(2017)Polarity of stability and robust positive invariance
Automatica
(2020)- et al.
The Minkowski–Lyapunov equation for linear dynamics: Theoretical foundations
Automatica
(2014) - et al.
Corrigendum to “The Minkowski—Lyapunov equation for linear dynamics: Theoretical foundation”
Automatica
(2019)Automatica
(2014) Lyapunov indicators of discrete inclusions I, II, and III
Automation and Remote Control
(1988)- et al.
Set–theoretic methods in control
(2008) - et al.
Sets of matrices all infinite products of which converge
Linear Algebra and its Applications
(1992) - et al.
Dual matrix inequalities in stability and performance analysis of linear differential/difference inclusions
Cited by (4)
Robust control Minkowski–Lyapunov functions
2021, AutomaticaCitation Excerpt :An illustrative example in Section 7.3 highlights that the proposed framework can be also applied within the setting of constrained parametrically uncertain linear discrete time systems. Analysis in what follows is made as close as possible to the one performed in the predecessor article (Raković, 2020a) concerned with deterministic linear discrete time systems; This is done deliberately so as to utilize results derived in Raković (2020a) as a natural base for the generalizations reported herein. Section 2 provides a concise formulation of the robust control Minkowski–Lyapunov inequality, and it details the setting utilized for the subsequent technical analysis.
Minkowski–Bellman inequality and equation
2021, AutomaticaCitation Excerpt :The theoretically flexible and computationally potent notions of Minkowski–Lyapunov functions (Raković, 2017, 2020a) and control Minkowski–Lyapunov functions (Raković, 2020b) have been recently developed through a methodical symbiosis of the fundamental notions of Lyapunov and control Lyapunov functions (Artstein, 1983; LaSalle, 1986; Lyapunov, 1992) on the one hand and Minkowski functions (Rockafellar, 1970; Rockafellar & Wets, 2009; Schneider, 1993) on the other hand.
The Robust Minkowski-Lyapunov Equation
2022, IEEE Transactions on Automatic Control
- ☆
The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Rifat Sipahi under the direction of Editor André L. Tits.