Affine Parameter-dependent Lyapunov Function Research Articles

Geometric approach to robust stability analysis of Linear Parameter-Varying systems: Computational trade-offs between the exact and the simplex convex hulls

Robust stability is investigated for continuous-time, affine, Linear Parameter-Varying (LPV) models, with bounded variation rates of uncertainty. Towards this end, affine Parameter Dependent Lyapunov Functions (PDLF) are considered to certificate stability in the Lyapunov sense. In the literature, the search for PDLFs amounts to a non conservative Linear Matrix Inequality (LMI) problem, at expense of a factorial growth in its complexity. Remedies to overcome this complexity have been proposed recently, exploiting geometrical aspects of the problem, however, they can be conservative. This paper presents new stability tests that are a trade-off between the exact and the fastest solutions. We offer an analytical procedure to indicate when the proposed tests are prone to reduce conservativeness. Also, a simple procedure is introduced to possibly improve existing and new tests, without impacts on the computation effort. Numerical simulations illustrate the improvements of the proposed strategies.

Robust D-stability LMI conditions of matrix polytopes via affine parameter-dependent Lyapunov functions

The problem of the robust D-stability analysis for linear systems with parametric uncertainties is addressed. For matrix polytopes, new conditions via the affine parameter-dependent Lyapunov function of uncertain systems are developed with the benefit of the scalar multi-convex function. To be convenient for applications, such conditions are simplified into new linear matrix inequality (LMI) conditions, which can be solved by the powerful LMI toolbox. Numerical examples are provided to indicate that this new approach is less conservative than previous results for Hurwitz stability, Schur stability and D-stability of uncertain systems under certain circumstances.

Model predictive control for linear parameter varying systems using path-dependent Lyapunov functions

This paper is concerned with the design of Model Predictive Control (MPC) for Linear Parameter Varying (LPV) discrete-time systems. Sufficient linear matrix inequality (LMI) conditions are provided for the existence of a path-dependent Lyapunov function which generalizes previous results based on affine parameter-dependent Lyapunov functions. At each sampling time the control law is obtained from a convex optimization problem under LMI constraints. As illustrated by examples, the proposed approach yields less conservative results than other available methods for MPC.

Robust Stability of Polytopic Systems via Affine Parameter-Dependent Lyapunov Functions

This paper studies robust stability of linear systems with polytopic uncertainty. New necessary and sufficient conditions for the existence of an affine parameter-dependent Lyapunov function assuring the Hurwitz or the Schur stability of a polytopic system are presented. These conditions are composed of a family of linear matrix inequality conditions of increasing precision. At each step, a set of linear matrix inequalities provides sufficient conditions for the existence of the affine parameter-dependent Lyapunov function, and necessity is asymptotically attained. Compared with the existing results in the literature, it is shown that the new stability conditions provide less conservative tests at each step. Numerical examples are given to illustrate the effectiveness of the new results.

Comparisons among robust stability criteria for linear systems with affine parameter uncertainties

This paper is concerned with the problem of comparisons among robust stability criteria for a class of uncertain linear systems, where the system state matrices considered are affinely dependent on the uncertain parameters. At first, a robust stability criterion for the class of systems to be affinely quadratically stable (AQS) is derived based on the vertex separator approach, where affine parameter-dependent Lyapunov functions are exploited to prove stability. Then comparison results between the robust stability criterion and the existing tests for AQS are given in terms of degree of conservatism. A numerical example is given to illustrate the results.

Stability of polytopes of matrices via affine parameter-dependent Lyapunov functions: Asymptotically exact LMI conditions

This paper investigates necessary and sufficient conditions for the existence of an affine parameter-dependent Lyapunov function assuring the Hurwitz (or Schur) stability of a polytope of matrices. A systematic procedure for constructing a family of linear matrix inequalities conditions of increasing precision is given. At each step, a set of linear matrix inequalities provides sufficient conditions for the existence of the affine parameter-dependent Lyapunov function. Necessity is asymptotically attained through a relaxation based on a generalization of Pólya’s Theorem to the case of matrix valued functions. Numerical experiments illustrate the results.

ROBUST STABILITY ANALYSIS OF LINEAR SYSTEMS WITH AFFINE PARAMETER UNCERTAINTIES

This paper is concerned with the problem of robust stability of a class of uncertain linear systems, where the system state-matrices considered are affinely dependent on the uncertain parameters. Affine parameter-dependent Lyapunov functions are exploited to prove stability, and a robust stability criterion for the above class of systems to be affinely quadratically stable (AQS) is given in terms of linear matrix inequalities (LMIs). A comparison with the existing tests for AQS is given, and it is shown that the robust stability criterion provides a test that is not more conservative than the existing tests. Numerical examples are given to illustrate the results.

Analysis of linear parameter‐varying systems using a non‐smooth dissipative systems framework

AbstractThis paper presents new finite‐dimensional linear matrix inequality formulations for several important analysis problems of linear parameter‐varying systems. The approach is based on a non‐smooth dissipative systems framework using a continuous, piecewise‐affine parameter‐dependent Lyapunov function. The new analysis approach yields guaranteed and less conservative results than previously published techniques that are based on affine parameter‐dependent Lyapunov functions or Luré–Postnikov Lyapunov functions. Conservatism is reduced in this new approach because the analysis uses a more general class of parameter‐dependent Lyapunov functions (PDLFs). The new approach also provides an explicit trade‐off between conservatism and computational effort of the analysis technique. Note that, in contrast to the gridding approach typically used to develop a computationally feasible algorithm, this proposed approach guarantees the analysis results. This paper also shows that the numerical results of our approach, while computationally intensive, could be used to develop many new insights into the selection of PDLFs in the other analysis techniques. Copyright © 2002 John Wiley & Sons, Ltd.

Parameter-dependent state observer design for affine LPV systems

In this paper, the design of a parameter-dependent state observer that allows to estimate the state of an affine linear parameter-varying (LPV) system is investigated. The observer has the property of being parameter-dependent since the corresponding state space matrices are scheduled using an interpolation method. Moreover, the stability of the estimation error is based on the existence of an affine parameter-dependent Lyapunov function. The main contribution of this paper is that the problem of the observer design and the existence of such a Lyapunov function is interpreted as a flexible LMI feasibility condition. Some illustrative examples are presented at the end of this paper.