On Converses to the Stability Theorems for Difference Equations

Gordon, Sheldon P.

doi:10.1137/0310007

Cited by 19 publications

(7 citation statements)

References 6 publications

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“…Converse Lyapunov theorems such as [1, Theorem 5.12.5], [9], [13], [16], and [21, Theorem 1.7.6] state that if system (1) is asymptotically stable, then there exists a Lyapunov function. These results are, in general, not constructive as the value of the Lyapunov function at each point is given by a function of the system solution forward in time from that point.…”

Section: Converse Theorem Construction Of Lyapunov Functionsmentioning

confidence: 99%

Computation of continuous and piecewise affine Lyapunov functions for discrete-time systems

Hafstein

Kellett

2015

Journal of Difference Equations and Applications

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In this paper, we present a new approach for computing Lyapunov functions for nonlinear discrete-time systems with an asymptotically stable equilibrium at the origin. Given a suitable triangulation of a compact neighbourhood of the origin, a continuous and piecewise ane function can be parameterised by the values at the vertices of the triangulation. If these vertex values satisfy system-dependent linear inequalities, the parameterised function is a Lyapunov function for the system. We propose calculating these vertex values using constructions from two classical converse Lyapunov theorems originally due to Yoshizawa and Massera. Numerical examples are presented to illustrate the proposed approach.

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Section: Converse Theorem Construction Of Lyapunov Functionsmentioning

confidence: 99%

Computation of continuous and piecewise affine Lyapunov functions for discrete-time systems

Hafstein

Kellett

2015

Journal of Difference Equations and Applications

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“…Based on converse Lyapunov function theorems such as [1, Theorem 5.12.5], [8], [13], and [17, Theorem 1.7.6], if system (1) is asymptotically stable, then there exists a Lyapunov function. Among these results, we are interested in a particular construction of the Lyapunov function which was originally proposed in [18] in continuous-time and extended to the discrete-time case in [13].…”

Section: Yoshizawa Construction Of Lyapunov Functionsmentioning

confidence: 99%

Computation of Lyapunov functions for discrete-time systems using the Yoshizawa construction

Hafstein

Kellett

2014

53rd IEEE Conference on Decision and Control

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Abstract-In this paper, we present a new approach for computing Lyapunov functions for nonlinear discrete-time systems with an asymptotically stable equilibrium at the origin. The proposed method constructs a continuous piecewise affine (CPA) function on a compact subset of the state space containing the origin, given a suitable triangulation or partition of the compact set and values at the vertices of the triangulation. Here, the vertex values are fixed using a function from a classical converse Lyapunov theorem originally due to Yoshizawa. Several numerical examples are presented to illustrate the proposed approach.

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“…A lot of researchers take their interest in this area and draw many good results. Many conclusions about stability, asmptopically stability or exponential stability have been obtained, see [1][2][3][4].The exponential stability of the nonlinear discrete system has been studied widely .For example, [5][6][7][8][9][10]. The exponential stability of the nonlinear system can be discussed by the Lyapunov function method.…”

Section: Introductionmentioning

confidence: 99%