Efficient Polynomial-Time Outer Bounds on State Trajectories for Uncertain Polynomial Systems Using Skewed Structured Singular Values

Kishida, Masako; Rumschinski, Philipp; Findeisen, Rolf; Braatz, Richard D.

doi:10.1109/tac.2014.2321230

Cited by 17 publications

(10 citation statements)

References 38 publications

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“…15,22,23 Moreover, we will iteratively compute optimal solutions 56 rather than feasible solutions. Especially, we will investigate outer approximations of domains of attraction of polynomial systems, 22,57 uncertain polynomial systems, 58,59 uncertain polynomial switched systems 60,61 and even polynomial hybrid systems, 38 where there may not exist common equilibria (eg, switched affine systems 45 ) or may exist multiple common equilibria. 62 ORCID Zhikun She http://orcid.org/0000-0003-2762-8730…”

Section: Discussionmentioning

confidence: 99%

Computing multiple Lyapunov‐like functions for inner estimates of domains of attraction of switched hybrid systems

Zheng

She

et al. 2018

Intl J Robust & Nonlinear

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Summary Domain of attraction plays an important role in stability analysis and safety verification of nonlinear control systems. In this paper, based on the concept of multiple Lyapunov‐like functions, we propose iteration algorithms for computing inner estimates of domains of attraction for a class of switched hybrid systems, where the state space is composed of several regions and each region is described by polyhedral sets. Starting with an initial inner estimate of domain of attraction, we firstly present a theoretical framework for obtaining a larger inner estimate by iteratively computing multiple Lyapunov‐like functions. Successively, the theoretical framework is underapproximatively realized by using S‐procedure and sums of squares programming, associated with the coordinatewise iteration method. Afterwards, for obtaining a required initial inner estimate of domain of attraction, we propose an alternative higher‐order truncation and linear semidefinite programming based method for computing a common Lyapunov function. Especially, a bisection method based improvement is proposed for obtaining better estimates in each iteration step. Finally, we implement proposed algorithms and test them on numerical examples with comparisons. These computation and comparison results show that the advantages of our multiple Lyapunov‐like functions based algorithm. Especially, we provide alternative underapproximations for avoiding the possible numerical problem.

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Section: Discussionmentioning

confidence: 99%

Computing multiple Lyapunov‐like functions for inner estimates of domains of attraction of switched hybrid systems

Zheng

She

et al. 2018

Intl J Robust & Nonlinear

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show abstract

“…It is also interesting in our further work to improve our proposed iteration algorithms for convergence and investigate inner approximations of domains of attraction of uncertain polynomial systems and even uncertain polynomial switched systems …”

Section: Discussionmentioning

confidence: 99%

Inner approximations of domains of attraction for a class of switched systems by computing Lyapunov‐like functions

Zheng

She

Liang

et al. 2017

Intl J Robust & Nonlinear

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Summary Domain of attraction plays an important role in control systems analysis, which is usually estimated by sublevel sets of Lyapunov functions. In this paper, based on the concept of common Lyapunov‐like functions, we propose an iteration method for estimating domains of attraction for a class of switched systems, where the state space is divided into several regions, each region is described by polynomial inequalities, and any region has no intersection among with each other. Starting with an initial inner estimate of domain of attraction, we first present a theoretical framework for obtaining a larger inner estimate by iteratively computing common Lyapunov‐like functions. Then, for obtaining a required initial inner estimate of domain of attraction, we propose a higher‐order truncation and linear semidefinite programming–based method for computing a common Lyapunov function. Successively, the theoretical framework is under‐approximatively realized by using S‐procedure and sum‐of‐squares programming, associated with a coordinatewise iteration idea. Finally, we implement our method and test it on some examples with comparisons. These computation and comparison results show the advantages of our method.

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“…A number of estimation techniques based on semi-definite programming (SDP) have been proposed for the development of computational method for the evaluation of state and parameter estimate uncertainty sets. [13][14][15] The SDP-based methods can be used to provide very precise uncertainty descriptions. For example, the work proposed in Kishida et al [15] provides various computational techniques that can be used to estimate state uncertainty sets.…”

Section: Introductionmentioning

confidence: 99%

“…[13][14][15] The SDP-based methods can be used to provide very precise uncertainty descriptions. For example, the work proposed in Kishida et al [15] provides various computational techniques that can be used to estimate state uncertainty sets. Interval algorithms [16] have also been proposed for the evaluation of parameter uncertainty sets for a general class of nonlinear systems.…”

Section: Introductionmentioning

confidence: 99%

Set‐based adaptive estimation for a class of uncertain nonlinear systems with output dependent nonlinearities

Dhaliwal

Guay

2014

Can J Chem Eng

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In this article, we consider the problem of parameter identification and state estimation of an uncertain continuous-time linearly parameterized nonlinear system with output dependent nonlinearities subject to exogenous disturbances. A set-based adaptive estimation is proposed in which the parameters and the states of the system are estimated along with an uncertainty set guaranteed to contain the true unknown values. The setupdate approach is such that the sets are updated only when an improvement in the precision of the parameter estimates and the state estimates can be guaranteed. The formulation provides robustness to parameter estimation error and bounded disturbances. The adaptive estimation technique can be viewed as an adaptive interval observer. Simulation examples are used to illustrate the effectiveness of the developed procedure and ascertain the theoretical results.

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Efficient Polynomial-Time Outer Bounds on State Trajectories for Uncertain Polynomial Systems Using Skewed Structured Singular Values

Cited by 17 publications

References 38 publications

Computing multiple Lyapunov‐like functions for inner estimates of domains of attraction of switched hybrid systems

Computing multiple Lyapunov‐like functions for inner estimates of domains of attraction of switched hybrid systems

Inner approximations of domains of attraction for a class of switched systems by computing Lyapunov‐like functions

Set‐based adaptive estimation for a class of uncertain nonlinear systems with output dependent nonlinearities

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