Convergent piecewise affine systems: analysis and design Part I: continuous case

Pavlov, Alexey; Pogromsky, Alexander Yu.; Wouw, Nathan van de; Nijmeijer, Henk; Rooda, K.

doi:10.1109/cdc.2005.1583019

Cited by 34 publications

(26 citation statements)

References 19 publications

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“…Combining (31) and (32) yields (19) for this particular trajectory. Global existence of any trajectory can also be proven via Theorem 2 page 78 [8] by using the bound (19) (that shows that there cannot be "finite escape times.")…”

Section: Which Implies |X(t)| χ(|U(t)|)mentioning

confidence: 99%

“…The approach of designing the observer and the state feedback controller separately (as in the linear case), and applying ISS-interconnection results to prove the stability of the closed loop system was used for Lipschitz continuous nonlinear systems in e.g. [3,2] and for continuous piecewise affine systems using smooth ISS Lyapunov functions in [19].…”

Section: Introductionmentioning

confidence: 99%

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Input-to-state stability and interconnections of discontinuous dynamical systems

Heemels

Weiland

2008

Automatica

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In this paper we will extend the input-to-state stability (ISS) framework introduced by Sontag to continuous-time discontinuous dynamical systems adopting Filippov's solution concept and using non-smooth ISS Lyapunov functions. The main motivation for investigating non-smooth ISS Lyapunov functions is the recent focus on "multiple Lyapunov functions" for which feasible computational schemes are available that have proven useful in the stability theory for hybrid systems. This paper proposes an extension of the well-known Filippov's solution concept, that is appropriate for 'open' systems so as to allow interconnections of hybrid systems. It is proven that the existence of a non-smooth ISS Lyapunov function for a discontinuous system implies ISS. In addition, a (small gain) ISS interconnection theorem is derived for two discontinuous dynamical systems that both admit a non-smooth ISS Lyapunov function. This result is constructive in the sense that an explicit ISS Lyapunov function for the interconnected system is given. It is shown how these results can be applied to piecewise linear (PWL) systems. In particular, it is shown how piecewise quadratic ISS Lyapunov functions can be constructed for PWL systems via linear matrix inequalities (LMIs). The theory will be illustrated by an example of an extended 'flower' system, which also demonstrates the computational machinery.

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Section: Which Implies |X(t)| χ(|U(t)|)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Input-to-state stability and interconnections of discontinuous dynamical systems

Heemels

Weiland

2008

Automatica

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“…In this paper we continue the study originated in [33,31] on convergency of piece-wise affine (PWA) systems. This class of systems attracted a lot of attention over the last years, see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…[3,17,18,38] and references therein. In this paper two extensions are given to the convergency theory as presented in [33,31]. Each of the extensions is discussed in the context of a suitable application area.…”

Section: Introductionmentioning

confidence: 99%

“…First, we present a convergence based approach to the anti-windup controller design for marginally stable systems with input saturation. For such systems one cannot directly use the results on quadratic convergency of PWA continuous systems presented in [33], since there exists no common quadratic Lyapunov function for this kind of system. To tackle the problem we developed a method that allows to establish uniform, but not necessary quadratic convergence.…”

Section: Introductionmentioning

confidence: 99%

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Design of Convergent Switched Systems

Wouw

Павлов²,

Nijmeijer

Group Coordination and Cooperative Control

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Summary. In this paper we deal with the problem of rendering hybrid/nonlinear systems into convergent closed-loop systems by means of a feedback law or switching rules. We illustrate our approach to this problem by means of two examples: the antiwindup design for a marginally stable system with input saturation, and the design of a switching rule for a piece-wise affine system operating in different modes.

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Observer‐based control of linear complementarity systems

Heemels

Camlibel

Schumacher

et al. 2010

Intl J Robust & Nonlinear

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Çamlıbel, Mehmet Kanat (Dogus Author)In this paper we present observer and output-based controller design methods for linear complementarity systems (LCS) employing a passivity approach. Due to various inherent properties of LCS, such as the presence of state jumps, mode dynamics described by differential and algebraic equations (DAEs), and regions for certain modes being lower dimensional, various observer and control design schemes that have been proposed for other classes of (hybrid) dynamical systems do not apply to LCS. In particular, we present an observer design method for LCS which is effective even in the presence of state jumps. We show the well-posedness of the observer, in the sense of existence and uniqueness of solution trajectories for the estimated state, and prove the global exponential stability of the observation error. These two properties guarantee that the estimated state exponentially recovers the state of the system. For the problem of stabilization based on output measurements only, we adopt an observer-based control approach in which we apply a state feedback law to the estimated state obtained from the observer. We prove that the resulting closed-loop system is well-posed and globally exponentially stable. In order to show the well-posedness of the closed loop, novel well-posedness results for LCS based on low-index properties are presented

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Convergent piecewise affine systems: analysis and design Part I: continuous case

Cited by 34 publications

References 19 publications

Input-to-state stability and interconnections of discontinuous dynamical systems

Input-to-state stability and interconnections of discontinuous dynamical systems

Design of Convergent Switched Systems

Observer‐based control of linear complementarity systems

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