Multi-Mode Multi-Dimensional Systems with Poissonian Sequencing

Verriest, Erik I.

doi:10.4310/cis.2009.v9.n1.a4

Cited by 11 publications

(17 citation statements)

References 20 publications

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“…This avoids the explicit computation of the perturbations and certain gradients as for instance in Egerstedt et al (2003). Moreover, we present for the first time the multi-dimensional aspect in its full generality, thus extending preliminary results first presented in Verriest (2006Verriest ( , 2010. We avoid expressing reset dynamics as additive impulsive terms as e.g.…”

Section: Optimal Control Of M 3 D Systemsmentioning

confidence: 92%

“…Transitions are quite detectable! Recently, we introduced the class of M 3 D systems (Verriest 2006(Verriest , 2009a to capture such behavior. These are switched mode systems belonging to a general class of hybrid systems, discussed in Branicky et al (1998), Sussmann (1999, and van der Schaft and Schumacher (2000), but where the state space and signal space associated with each mode may have a different dimension.…”

Section: D Systemsmentioning

confidence: 99%

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Pseudo-continuous multi-dimensional multi-mode systems

Verriest

2011

Discrete Event Dyn Syst

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The dynamics of multi-mode multi-dimensional (M 3 D) hybrid systems is described. Such systems have modes of different dimensions, for which the state space is defined as a fibre bundle. The implications of the behavior at the mode transitions is investigated in detail, for which pseudo-continuity is introduced. An M 3 D system is pseudo-continuous if instantaneous switching via higher dimensional modes does not have any effect. Canonical forms and parameterizations are derived for pseudo-continuous M 3 D systems. The system may be actively controlled (exo-M 3 D), or passively switched via a fixed switching surface (auto-M 3 D). M 3 D systems are of interest in the approximate and reduced order modeling for nonlinear systems and the remote control over one-way communication channels. The optimal timing (switching) control for such M 3 D systems is solved in the general case. Necessary conditions for a stationary solution are derived and shown to extend those of the equal dimension case (Egerstedt et al. 2003). We also give a specific solution for the linear quadratic problem, involving a generalization of the Riccati equation. This problem is of interest in deriving neighboring extremal solutions for the control under small perturbations of s nominal solution. Some suggestions towards determining the optimal mode sequence are given. We illustrate the problem with the optimal control for a spring assisted high jump (aka man on a trampoline).

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Section: Optimal Control Of M 3 D Systemsmentioning

confidence: 92%

Section: D Systemsmentioning

confidence: 99%

Pseudo-continuous multi-dimensional multi-mode systems

Verriest

2011

Discrete Event Dyn Syst

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“…Proof: Throughout this proof, we use V σ(t) (t) to denote V σ(t) (t, x σ(t) (t)) for simplicity. Assuming a certain switching sequence on (t 0 , t f ] to be t 1 , t 2 , ..., t k , ..., k ≤ N (t 0 , t f ), then by integrating (15) on both sides from t 0 to t k+1 , and applying (16), one obtains…”

Section: A Stability Conditions For Nonlinear Mdssmentioning

confidence: 99%

“…To date, related works on the MDSS are in a minority. A series of early discussions can be found in [16]- [19], in which the terminology of "multi-dimensional multi-mode (M 3 D) system" was coined to describe switched systems with dimension inconsistence. They also introduced the concept of pseudo-continuity ((see Definition 2.1 of [18])) to describe the transition behaviors of the systems at the switching instant, such that a continuous-like state space can be obtained.…”

Section: Introductionmentioning

confidence: 99%

Stability of Multi-Dimensional Switched Systems with an Application to Open Multi-Agent Systems

Xue¹,

Tang²,

Ren³

et al. 2020

Preprint

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A multi-dimensional switched system (MDSS) contains subsystems of different state dimensions. By modeling the state transition at certain switching instant as the combination of dimension variations and state impulses, the stability of the MDSS is investigated. Sufficient conditions featuring the new transition-dependent average dwell time (TDADT) and the piecewise TDADT methods are provided to guarantee the corresponding stability performance under two different impulse types and different stability conditions for subsystems. By virtue of the proposed MDSS framework, the consensus problems for the open multi-agent system (MAS) featuring agent circulation are studied. It shows that the agent circulation that results in a switching and size-varying communication topology can be well captured by the state transition mechanism for the MDSS. Therefore, the consensus problems of an open MAS can be readily converted to the corresponding stability problems of the MDSS. The simulation example illustrates the effectiveness of the proposed methods as well as the feasibility of the application.

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“…Based on variational arguments, necessary conditions of optimality are derived and used in a first order scheme, see Verriest et al (2005) for an epidemic control application. Another contribution by the same authors is the class of hybrid systems introduced in Verriest (2006). The class includes systems with variable state space dimensions where the transitions are time triggered and can thus be viewed as a general class of switched systems.…”

Section: Introductionmentioning

confidence: 99%

On an Optimization Problem for a Class of Impulsive Hybrid Systems

Attia

Azhmyakov

Raisch

2009

Discrete Event Dyn Syst

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This contribution addresses the problem of optimal control for a class of hybrid systems, where discrete transitions are accompanied by instantaneous changes in the continuous state variables, and where these changes can be considered as control variables. Based on a variational approach, necessary conditions of optimality are first established. The problem is then cast as a parametric optimization problem for which gradient information is derived. Finally, we discuss assumptions that guarantee convergence of a conceptual algorithm to a stationary solution. A brief discussion on the main implementation issues is also included.

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Multi-Mode Multi-Dimensional Systems with Poissonian Sequencing

Cited by 11 publications

References 20 publications

Pseudo-continuous multi-dimensional multi-mode systems

Pseudo-continuous multi-dimensional multi-mode systems

Stability of Multi-Dimensional Switched Systems with an Application to Open Multi-Agent Systems

On an Optimization Problem for a Class of Impulsive Hybrid Systems

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