Linear programming based Lyapunov function computation for differential inclusions

Baier, Robert; Grüne, Lars; Hafstein, Sigurður

doi:10.3934/dcdsb.2012.17.33

Cited by 51 publications

(44 citation statements)

References 15 publications

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“…In [13] it was proved that for exponentially stable equilibria this method is always capable of generating a Lyapunov function V : C\N − →R,w h e r eN⊂Cis an arbitrary small, a priori determined neighborhood of the origin. In [14], these results were generalized to asymptotically stable systems, in [15] to asymptotically stable, arbitrary switched, nonautonomous systems, and in [1] to asymptotically stable differential inclusions.…”

Section: Introductionmentioning

confidence: 99%

Revised CPA method to compute Lyapunov functions for nonlinear systems

Giesl

Hafstein

2014

Journal of Mathematical Analysis and Applications

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The CPA method uses linear programming to compute Continuous and Piecewise Affine Lyapunov function for nonlinear systems with asymptotically stable equilibria. In [14] it was shown that the method always succeeds in computing a CPA Lyapunov function for such a system. The size of the domain of the computed CPA Lyapunov function is only limited by the equilibrium's basin of attraction. However, for some systems, an arbitrary small neighborhood of the equilibrium had to be excluded from the domain a priori. This is necessary, if the equilibrium is not exponentially stable, because the existence of a CPA Lyapunov function in a neighborhood of the equilibrium is equivalent to its exponential stability as shown in [11]. However, if the equilibrium is exponentially stable, then this was an artifact of the method. In this paper we overcome this artifact by developing a revised CPA method. We show that this revised method is always able to compute a CPA Lyapunov function for a system with an exponentially stable equilibrium. The only conditions on the system are that it is C 2 and autonomous. The domain of the CPA Lyapunov function can be any a priori given compact neighborhood of the equilibrium which is contained in its basin of attraction. Whereas in a previous paper [10] we have shown these results for planar systems, in this paper we cover general n-dimensional systems.

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Section: Introductionmentioning

confidence: 99%

Revised CPA method to compute Lyapunov functions for nonlinear systems

Giesl

Hafstein

2014

Journal of Mathematical Analysis and Applications

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“…C ∞ , ISS-Lyapunov function for (1). In this following, we prove that this theorem makes sure our algorithm terminates successfully.…”

Section: Notations and Preliminariesmentioning

confidence: 84%

“…We are ready to formulate the linear programming algorithm for computing an ISS Lyapunov function V for system (1). In this algorithm, the values V (x i ), σ are introduced as optimization variables.…”

Section: The Algorithmmentioning

confidence: 99%

“…3. Load the information about the constraints and variables into a matrix which can be read by the GNU Linear Programming Kit (GLPK) 1 . Set the objective of the linear optimization problem: min r. …”

Section: Examplesmentioning

confidence: 99%

“…Furthermore, this result was extended to asymptotically stable systems [8], to asymptotically stable, arbitrarily switched, non-autonomous systems [9], and to asymptotically stable differential inclusions [1]. The approaches proposed in these papers deliver true Lyapunov functions on compact subsets of the state space except possibly on arbitrarily small neighbourhood of the asymptotically stable equilibrium.…”

Section: Introductionmentioning

confidence: 99%

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Computation of local ISS Lyapunov functions for discrete-time systems via linear programming

Grüne

2016

Journal of Mathematical Analysis and Applications

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This paper presents a numerical algorithm for computing ISS Lyapunov functions for discrete-time systems which are input-to-state stable (ISS) on compact subsets of the state space. The algorithm relies on solving a linear optimization problem and delivers a continuous and piecewise affine ISS Lyapunov function on a suitable triangulation covering the given compact set excluding a small neighbourhood of the origin. The objective of the linear optimization problem is to minimize the ISS gain. It is shown that for every ISS system there exist a suitable triangulation such that the proposed algorithm terminates successfully.

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Design of Lyapunov functions for a class of homogeneous systems: Generalized forms approach

Sánchez

Moreno

2018

Intl J Robust & Nonlinear

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Summary In this paper, we provide a method to design Lyapunov functions (LFs) for a class of homogeneous systems described by functions that we call generalized forms (GFs). Homogeneous polynomial systems and several high‐order sliding modes are included in the class. The LF candidate is chosen from the same class of functions and it is parameterized in its coefficients. Since the derivative of the LF candidate along the system's trajectories is also a GF, the problem is reduced to verify positive definiteness of two GFs. We establish a procedure to represent a GF with a finite set of polynomials. Thus, the problem is changed to determine positive definiteness of a set of polynomials. Such a problem can be solved by means of Pólya's theorem or the sum of squares representation.

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Linear programming based Lyapunov function computation for differential inclusions

Cited by 51 publications

References 15 publications

Revised CPA method to compute Lyapunov functions for nonlinear systems

Revised CPA method to compute Lyapunov functions for nonlinear systems

Computation of local ISS Lyapunov functions for discrete-time systems via linear programming

Design of Lyapunov functions for a class of homogeneous systems: Generalized forms approach

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