High-level canonical piecewise linear representation using a simplicial partition

Julián, P.; Desages, A.; Agamennoni, Osvaldo

doi:10.1109/81.754847

Cited by 202 publications

(189 citation statements)

References 25 publications

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“…Julian, Guivant, and Desages [20] and Julian [19] present a linear programming problem to construct piecewise affine Lyapunov functions for autonomous piecewise affine systems. This method can be used for autonomous, nonlinear systems if some a posteriori analysis of the generated Lyapunov function is done.…”

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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“…For instance, rectangular regions with sides parallel to the coordinate axes are used in [9], while simplices (i.e. polytopes with d + 1 corners, where d is the dimension of the domain) are considered in [23] and [40]. This approach drastically simplifies the estimation of the linear/affine submodels, since standard linear identification techniques can be used to estimate the submodels, given enough data points in each region.…”

Section: Remark 31mentioning

confidence: 99%

Identification of Hybrid Systems A Tutorial

Paoletti

Juloski

Ferrari-Trecate

et al. 2007

European Journal of Control

355

271

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“…As advocated in [16], [20], [22], [24]- [27], it can be very beneficial for the eventual (circuit) implementation to use canonical PWA controllers that are based on regular partitions using e.g. regular simplices [20], [22], [24]- [27] or (multiscale) hypercubes [16].…”

Section: B Motivationmentioning

confidence: 99%

“…regular simplices [20], [22], [24]- [27] or (multiscale) hypercubes [16]. Essentially, any desired polytopic shape can be chosen in our approximation method that follows.…”

Section: B Motivationmentioning

confidence: 99%

“…The approximate control law is defined on a regular (possibly multiscale) partition, which can be chosen in a desirable way, e.g. based on multiscale rectangles as in [16] or regular simplices as in [20], [22], [24]- [27]. Due to the choice of a regular partition, the resulting regular PWA functions inherently result in efficient, low-complexity on-line implementations in terms of both computational effort and memory requirements.…”

Section: Introductionmentioning

confidence: 99%

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Approximation of explicit model predictive control using regular piecewise affine functions: an input-to-state stability approach

Genuit

Heemels

2012

IET Control Theory &Amp; Applications

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Piecewise affine (PWA) feedback control laws defined on general polytopic partitions, as for instance obtained by explicit MPC, will often be prohibitively complex for fast systems. In this work we study the problem of approximating these high-complexity controllers by low-complexity PWA control laws defined on more regular partitions, facilitating faster on-line evaluation. The approach is based on the concept of input-to-state stability (ISS). In particular, the existence of an ISS Lyapunov function (LF) is exploited to obtain a priori conditions that guarantee asymptotic stability and constraint satisfaction of the approximate low-complexity controller. These conditions can be expressed as local semidefinite programs (SDPs) or linear programs (LPs), in case of 2-norm or 1, ∞-norm based ISS, respectively, and apply to PWA plants. In addition, as ISS is a prerequisite for our approximation method, we provide two tractable computational methods for deriving the necessary ISS inequalities from nominal stability. A numerical example is provided that illustrates the main results.The authors are with the Hybrid and

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High-level canonical piecewise linear representation using a simplicial partition

Cited by 202 publications

References 25 publications

Revised CPA method to compute Lyapunov functions for nonlinear systems

Revised CPA method to compute Lyapunov functions for nonlinear systems

Identification of Hybrid Systems A Tutorial

Approximation of explicit model predictive control using regular piecewise affine functions: an input-to-state stability approach

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