Technical communiqueTime-invariant uncertain systems: A necessary and sufficient condition for stability and instability via homogeneous parameter-dependent quadratic Lyapunov functions☆
Introduction
Various methods have been proposed for the stability of linear systems with time-invariant uncertainty constrained in a polytope. Generally, these methods exploit parameter-dependent Lyapunov functions and LMIs; see e.g. Leite and Peres (2003), which considers Lyapunov functions with linear dependence, Bliman (2004), which proposes Lyapunov functions with polynomial dependence, Chesi, Garulli, Tesi, and Vicino (2005), which introduces the class of HPD-QLFs, Scherer (2006), which proposes a general framework for LMI relaxations, Oliveira and Peres (2007), where homogeneous solutions are characterized, Lavaei and Aghdam (2008), which addresses the case of semi-algebraic sets, and Oishi (2009) and Peaucelle and Sato (2009), where matrix-dilation approaches are considered.
Some of these methods provide necessary and sufficient conditions for robust stability. However, the necessity is achieved for an unknown degree of the polynomials used. This implies that, if the system is unstable, no conclusion can be reached. This paper addresses this problem via homogeneous parameter-dependent quadratic Lyapunov functions (HPD-QLFs) for the case of polynomial dependence on the uncertainty. It is shown that a sufficient condition for establishing either stability or instability can be obtained by solving a generalized eigenvalue problem, and that this condition is also necessary by using a sufficiently large degree of the HPD-QLF. The idea behind this condition is to exploit the LMI relaxation introduced in Chesi et al. (2005) via the square matrical representation (SMR)1 in order to characterize the instability via the presence of suitable vectors in certain eigenspaces.
Before proceeding, it is worth explaining that the proposed approach differs from Chesi (2005), which proposes a non-Lyapunov method for establishing stability and instability of uncertain systems, from Chesi (2007), which investigates robust performance via eigenvalue problems, from Ebihara, Onishi, and Hagiwara (2009), which exploits -scaling in the case of rational dependence on the uncertainty, from Masubuchi and Scherer (2009), which derives a recursive algorithm based on the linear fractional representation, and from Goncalves et al., 2006, Goncalves et al., 2007, which propose a branch-and-bound method in the case of linear dependence on the uncertainty.
Section snippets
Preliminaries
Notation. : real and complex numbers; : ; : identity matrix; : symmetric positive definite matrix; : Kronecker’s product; , , : transpose, trace and determinant of ; : vector with the columns of stacked below each other; ; ; with ; CT, DT: continuous-time and discrete-time; s.t.: subject to.
Let us consider the uncertain system
Stability and instability condition
Let us define where is the matrix satisfying (see also (18) for a key property of ) and define Let us define where are optimal values of in (12), and let be the eigenvectors of the non-positive eigenvalues of , i.e.
Theorem 2 The setis stable if and only if there existssuch that. Moreover,is unstable if and only if there exist
Examples
Here we present some illustrative examples. The computational time on a standard computer with Matlab of the proposed condition for all these examples (and of Goncalves et al. (2007) for the examples in Section 4.1) is lesser than 5 s.
Conclusion
We have proposed a condition via HPD-QLFs for establishing the stability and instability of linear systems with polynomial dependence on uncertainties constrained in the simplex. This condition is necessary and sufficient.
Acknowledgements
The author would like to thank the Associate Editor and the Reviewers for their comments, and the authors of Goncalves et al., 2006, Goncalves et al., 2007 for providing the code of their method.
References (20)
Establishing stability and instability of matrix hypercubes
Systems and Control Letters
(2005)Establishing tightness in robust H-infinity analysis via homogeneous parameter-dependent Lyapunov functions
Automatica
(2007)On the non-conservatism of a novel LMI relaxation for robust analysis of polytopic systems
Automatica
(2008)- et al.
New strategy for robust stability analysis of discrete-time uncertain systems
Systems and Control Letters
(2007) - et al.
A recursive algorithm of exactness verification of relaxations for robust sdps
Systems and Control Letters
(2009) LMI relaxations in robust control
European Journal of Control
(2006)A convex approach to robust stability for linear systems with uncertain scalar parameters
SIAM Journal on Control and Optimization
(2004)On the gap between positive polynomials and SOS of polynomials
IEEE Transactions on Automatic Control
(2007)- et al.
Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: An LMI approach
IEEE Transactions on Automatic Control
(2005) - et al.
Solving quadratic distance problems: An LMI-based approach
IEEE Transactions on Automatic Control
(2003)
Cited by (14)
Exact robust stability analysis of uncertain systems with a scalar parameter via LMIs
2013, AutomaticaCitation Excerpt :Most of these methods provide in general sufficient conditions for robust asymptotic stability based on LMIs; see, e.g. some of the pioneering works (Boyd, El Ghaoui, Feron, & Balakrishnan, 1994; Neto, 1999). In some cases such as Bliman (2004), Chesi (2010a), Chesi, Garulli, Tesi, and Vicino (2005), Oliveira and Peres (2007) and Scherer (2006) the conservatism can be reduced by increasing the degree of some polynomials used to build the LMIs, for instance representing a Lyapunov function candidate or a multiplier. Interestingly, this sometimes allows complete elimination of the conservatism, which means that these conditions are not only sufficient but also necessary.
Stability analysis of linear time-varying systems: Improving conditions by adding more information about parameter variation
2011, Systems and Control LettersCitation Excerpt :Therefore, robust stability analysis of time-varying uncertain systems is a major concern that has been handled with Lyapunov functions and LMIs over the past decades. When no bounds are available for the parameter variation strategies for time-invariant systems can be applied at the cost of some conservatism; see [1–5] and references therein. In the context of time-varying linear systems a usual approach to assess robust stability is to consider a Lyapunov function that depends on the time-varying parameters.
LMI conditions for time-varying uncertain systems can be non-conservative
2011, AutomaticaCitation Excerpt :It is worth observing that homogeneous polynomial Lyapunov functions have been exploited also in the case of uncertain systems with rational dependence on the uncertainty, e.g. in Chesi (2010b). While non-conservative conditions have been obtained in terms of LMIs for the case of time-invariant uncertainty (see e.g. Chesi, 2008, 2010c), till now it has been unclear whether and how sufficient conditions that are also necessary can be obtained for the case of time-varying uncertainty through LMIs. This paper provides an answer to this question, in particular showing that the LMI condition proposed in Chesi et al. (2003, 2009) through the use of homogeneous polynomial Lyapunov functions and the square matrix representation (SMR), is not only sufficient, but also necessary for a sufficiently large degree of the function.
Uniform exponential stability criteria with verification for nonautonomous switched nonlinear systems with uncertainty
2022, International Journal of Robust and Nonlinear ControlComments on 'robust stability and stabilization of fractional-order interval systems with the fractional order α: the 0<α<1 case'
2015, IEEE Transactions on Automatic ControlSign stability analysis of polytopic uncertain systems
2015, Kongzhi Lilun Yu Yingyong/Control Theory and Applications
- ☆
This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Mikael Johansson, under the direction of Editor André L. Tits.