A control problem for affine dynamical systems on a full-dimensional polytope

Habets, L.C.G.J.M.; Schuppen, Jan H. van

doi:10.1016/j.automatica.2003.08.001

Cited by 180 publications

(208 citation statements)

References 18 publications

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“…By choosing M to be C T q C q and Q = S, we obtain a solution to (9). If we choose M q = P B q B T q P , then using the fact that S = 1 γ P and Lemma 16, we get that ∀q ∈ Q : A q S −1 + S −1 A T q + B q B T q γ 2 < 0.…”

Section: Proof Of Lemma 1 (Ii) Implies (I) It Is Clear That If ∀Q ∈mentioning

confidence: 99%

“…(10)) has a positive definite solution, see Example 1 of Section 6. This is due to the fact that in (9) and (10), we require inequalities instead of equalities.…”

Section: System-theoretic Interpretation Of Grammians and Their Singumentioning

confidence: 99%

“…A rich literature covers the subject of model reduction for switched systems, [6,7,8,9,10,11,12,13,14,15,16]. In particular, balanced truncation was explored in [1,2,14,13,15,12,16].…”

Section: Related Workmentioning

confidence: 99%

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Balanced truncation for linear switched systems

Petreczky

Wiśniewski

Leth

2013

Nonlinear Analysis: Hybrid Systems

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In this paper, we present a theoretical analysis of the model reduction algorithm for linear switched systems from [1,2]. This algorithm is a reminiscence of the balanced truncation method for linear parameter varying systems [3]. Specifically in this paper, we provide a bound on the approximation error in L 2 norm for continuous-time and l 2 norm for discrete-time linear switched systems. We provide a system theoretic interpretation of grammians and their singular values. Furthermore, we show that the performance of balanced truncation depends only on the input-output map and not on the choice of the state-space representation. For a class of stable discrete-time linear switched systems (so called strongly stable systems), we define nice controllability and nice observability grammians, which are genuinely related to reachability and controllability of switched systems. In addition, we show that quadratic stability and LMI estimates of the L 2 and l 2 gains depend only on the input-output map.

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Section: Proof Of Lemma 1 (Ii) Implies (I) It Is Clear That If ∀Q ∈mentioning

confidence: 99%

“…(10)) has a positive definite solution, see Example 1 of Section 6. This is due to the fact that in (9) and (10), we require inequalities instead of equalities.…”

Section: System-theoretic Interpretation Of Grammians and Their Singumentioning

confidence: 99%

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Balanced truncation for linear switched systems

Petreczky

Wiśniewski

Leth

2013

Nonlinear Analysis: Hybrid Systems

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“…gene deletion) are now available and allow to modify the production or degradation terms of some genes of the networks. This leads to problems of mathematical control of piecewise affine genetic networks, similar to more general problems for hybrid affine systems [14]. The global problem is to control the trajectories through some prescribed sequence of rectangular regions.…”

Section: Challenges In Pl Models Analysismentioning

confidence: 99%

Piecewise-Linear Models of Genetic Regulatory Networks: Theory and Example

Grognard

Jong

Gouzé

2007

Biology and Control Theory: Current Challenges

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Summary. The experimental study of genetic regulatory networks has made tremendous progress in recent years resulting in a huge amount of data on the molecular interactions in model organisms. It is therefore not possible anymore to intuitively understand how the genes and interactions together influence the behavior of the system. In order to answer such questions, a rigorous modeling and analysis approach is necessary. In this chapter, we present a family of such models and analysis methods enabling us to better understand the dynamics of genetic regulatory networks. We apply such methods to the network that underlies the nutritional stress response of the bacterium E. coli.The functioning and development of living organisms is controlled by large and complex networks of genes, proteins, small molecules, and their interactions, socalled genetic regulatory networks. The study of these networks has recently taken a qualitative leap through the use of modern genomic techniques that allow for the simultaneous measurement of the expression levels of all genes of an organism. This has resulted in an ever growing description of the interactions in the studied genetic regulatory networks. However, it is necessary to go beyond the simple description of the interactions in order to understand the behavior of these networks and their relation with the actual functioning of the organism. Since the networks under study are usually very large, an intuitive approach for their understanding is out of question. In order to support this work, mathematical and computer tools are necessary: the unambiguous description of the phenomena that mathematical models provide allows for a detailed analysis of the behaviors at play, though they might not exactly represent the exact behavior of the networks.In this chapter, we will be mostly interested in the modeling of the genetic regulatory networks by means of differential equations. This classical approach allows precise numerical predictions of deterministic dynamic properties of genetic regulatory networks to be made. However, for most networks of biological interest the application of differential equations is far from straightforward. First, the biochemical reaction mechanisms underlying the interactions are usually not or incompletely

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“…In this paper, we utilize very recent results [5], [10] that will guarantee that the discrete strategy of the pursuer will be feasibly executed by the dynamic model of the pursuer. Under suitable conditions, a feedback controller is constructed for each triangle, steering the pursuer between adjacent triangles.…”

Section: Introductionmentioning

confidence: 99%

Hybrid control for visibility-based pursuit-evasion games

Isler¹,

Belta

Daniilidis

et al.

2004 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (IEEE Cat. No.04CH37566)

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Abstract-Pursuit-evasion games in complex environments have a rich but disconnected history. Continuous or differential pursuit-evasion games focus on optimal control methods, and rely on very intense computations in order to provide locally optimal controls. Discrete pursuit-evasion games on graphs are algorithmically much more appealing, but completely ignore the physical dynamics of the players, resulting in possibly infeasible motions. In this paper, we present a provable and algorithmically feasible solution for visibility-based pursuit-evasion games in simplyconnected environments, for players with dynamic constraints. This is achieved by combining two recent but distant results.

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A control problem for affine dynamical systems on a full-dimensional polytope

Cited by 180 publications

References 18 publications

Balanced truncation for linear switched systems

Balanced truncation for linear switched systems

Piecewise-Linear Models of Genetic Regulatory Networks: Theory and Example

Hybrid control for visibility-based pursuit-evasion games

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