Convex invariant sets for discrete-time Lur’e systems

Álamo, T.; Cepeda, A.; Fiacchini, Mirko; Camacho, Eduardo F.

doi:10.1016/j.automatica.2008.11.013

Cited by 38 publications

(36 citation statements)

References 13 publications

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“…The bounding functionsφ(·) andφ(·) are used to determine an upper bounding function of η T f (x) for any η ∈ R n , with f (·) characterizing the saturated system (1).…”

Section: Invariance For Symmetric Convex Setsmentioning

confidence: 99%

“…Many results regarding invariance and related topics have been provided in literature: see, for instance, the notable pioneering contribution [4], the works [10], [15], concerning the maximal invariant set, and [18] regarding the minimal one. The problem of obtaining invariant sets for discrete-time nonlinear systems is dealt with using ellipsoids in [16], parallelotopes in [7], and polytopes in [1], [8]. Invariance of polytopes for continuoustime nonlinear systems has been considered in [9].…”

Section: Introductionmentioning

confidence: 99%

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Invariance of symmetric convex sets for discrete-time saturated systems

Fiacchini

Tarbouriech

Prieur

2011

IEEE Conference on Decision and Control and European Control Conference

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“…The bounding functionsφ(·) andφ(·) are used to determine an upper bounding function of η T f (x) for any η ∈ R n , with f (·) characterizing the saturated system (1).…”

Section: Invariance For Symmetric Convex Setsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Invariance of symmetric convex sets for discrete-time saturated systems

Fiacchini

Tarbouriech

Prieur

2011

IEEE Conference on Decision and Control and European Control Conference

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“…The results derived in this paper apply to any Lure-type nonlinear system with a nonlinearity that can be expressed as a time-varying parameterised linear function '.H x/ D H x with the parameter in a bounded interval (cf. (15)). More generally, extensions of the results are currently being 1096 M. M. SERON AND J.…”

Section: Remark 32mentioning

confidence: 99%

“…The following result compares the matrices ƒ NL and ƒ LP V in (28) and (29) and gives conditions for them to be equal. (10), (15) and c given in (22), consider the matrices ƒ NL and ƒ LP V given in (28)-(29), with V and D K defined in (25). Then…”

Section: Achieving Identical Ultimate-bound Invariant Sets Via Both Mmentioning

confidence: 99%

On invariant sets and closed‐loop boundedness of Lure‐type nonlinear systems by LPV‐embedding

Serón

Doná

2015

Intl J Robust & Nonlinear

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Summary We address the problem of achieving trajectory boundedness and computing ultimate bounds and invariant sets for Lure‐type nonlinear systems with a sector‐bounded nonlinearity. Our first contribution is to compare two systematic methods to compute invariant sets for Lure systems. In the first method, a linear‐like bound is considered for the nonlinearity, and this bound is used to compute an invariant set by regarding the nonlinear system as a linear system with a nonlinear perturbation. In the second method, the sector‐bounded nonlinearity is treated as a time‐varying parameterised linear function with bounded parameter variations, and then invariant sets are computed by embedding the nonlinear system into a convex polytopic linear parameter varying (LPV) system. We show that under some conditions on the system matrices, these approaches give identical invariant sets, the LPV‐embedding method being less conservative in the general case. The second contribution of the paper is to characterise a class of Lure systems, for which an appropriately designed linear state feedback achieves bounded trajectories of the closed‐loop nonlinear system and allows for the computation of an invariant set via a simple, closed‐form expression. The third contribution is to show that, for disturbances that are ‘aligned’ with the control input, arbitrarily small ultimate bounds on the system states can be achieved by assigning the eigenvalues of the linear part of the system with ‘large enough’ negative real part. We illustrate the results via examples of a pendulum system, a Josephson junction circuit and the well‐known Chua circuit. Copyright © 2015 John Wiley & Sons, Ltd.

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“…The methods to obtain an estimation of an inner approximation of the domain of attraction have been extended to nonlinear systems too. An estimation of the domain of attraction of a class of discrete time Lur'e system is presented in [7], in [8] an algorithm to obtain control invariant sets for constrained nonlinear systems is proposed. DC functions are used in [9] to provide a method to compute robust control invariant sets for non linear discrete-time systems.…”

Section: Introductionmentioning

confidence: 99%

Oracle based approach to admissible invariant sets

Redondo

Bravo

Álamo

2012

2012 American Control Conference (ACC)

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Given a nonlinear system and a control law, a new method to obtain cutting planes excluding points not belonging to the maximal admissible invariant set is proposed. The method uses linearization along the trajectory of the close-loop system to obtain a set of cutting planes. The main advantage of the approach is that if an inner approximation of the maximal admissible invariant set is available, the provided cutting planes do not intersect this initial approximation. A scaled version of an initial invariant set is used to improve the cutting planes obtained by linearization. The proposed approach can be used in the context of nonlinear model predictive control. The method is illustrated by an example. * M. J. Redondo is with the

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Convex invariant sets for discrete-time Lur’e systems

Cited by 38 publications

References 13 publications

Invariance of symmetric convex sets for discrete-time saturated systems

Invariance of symmetric convex sets for discrete-time saturated systems

On invariant sets and closed‐loop boundedness of Lure‐type nonlinear systems by LPV‐embedding

Oracle based approach to admissible invariant sets

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