Optimal Control for Continuous-Time Linear Quadratic Problems with Infinite Markov Jump Parameters

Fragoso, Marcelo D.; Baczynski, Jack

doi:10.1137/s0363012900367485

Cited by 73 publications

(23 citation statements)

References 29 publications

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“…It is worth noticing that, due to the presence of a product such as ρ 2α in (9), the problem of maximizing ρ > 0 subjects to (9), which corresponds to determineρ in (11), is non-convex. However, for each given ρ > 0, (9) is affine in P andα.…”

Section: Remarkmentioning

confidence: 99%

“…However, for each given ρ > 0, (9) is affine in P andα. Bearing this in mind, we propose here the following method to solve the optimization problem, by means of a bisectional procedure, which avoids the scaling optimization problem.…”

Section: Remarkmentioning

confidence: 99%

“…As far as we are aware, until now, no attention has been given to this scenario. Also, different from all previous work on this subject, (except from [18,20]) we assume that the Markov process takes values on a countably infinite set (the so-called infinite case, which was previously considered, for example, in [9,19,21]). It is very important to emphasize, however, that every contribution presented here remains original even in the finite case.…”

Section: Introductionmentioning

confidence: 99%

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On the stability radii of continuous-time infinite Markov jump linear systems

Todorov

Fragoso

2010

Math. Control Signals Syst.

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In this paper we introduce the subject of stability radii for continuous-time infinite Markov jump linear systems (MJLS) with respect to unstructured perturbations. By means of the small-gain approach, a lower bound for the complex radius is derived along with a linear matrix inequality (LMI) optimization method which is new in this context. In this regard, we propose an algorithm to solve the optimization problem, based on a bisectional procedure, which is tailored in such a way that avoids the issue of scaling optimization. In addition, an easily computable upper bound for the real and complex stability radii is devised, with the aid of a spectral characterization of the problem. This seems to be a novel approach to the problem of robust stability, even when restricted to the finite case, which in turn allows us to obtain explicit formulas for the stability radii of two-mode scalar MJLS. We also introduce a connection between stability radii and a certain margin of stability with respect to perturbations on the transition rates of the Markov process. The applicability of the main results is illustrated with some numerical examples.

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Section: Remarkmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the stability radii of continuous-time infinite Markov jump linear systems

Todorov

Fragoso

2010

Math. Control Signals Syst.

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“…When there is an underlying, at least partial, knowledge of the dynamics of the parameters, LPV systems specialize to jump linear (JL) systems [16], [17], [10], [13], [12] and linear dynamically varying (LDV) systems [5]- [8], [18]. In this paper we explore the hitherto hidden links between the last two systems.…”

Section: Introductionmentioning

confidence: 99%

Relationships between Linear Dynamically Varying Systems and Jump Linear Systems

Bohacek

Jonckheere

2003

Mathematics of Control, Signals, and Systems (MCSS)

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The connection between linear dynamically varying (LDV) systems and jump linear (JL) systems is explored. LDV systems have been used to model the error in nonlinear tracking problems. Some nonlinear systems, for example Axiom A systems, admit Markov partitions and their dynamics can be quantized as Markov chains. In this case the tracking error can be approximated by a JL system. It is shown that (i) JL controllers for arbitrarily fine partitions exist if and only if the LDV controller exists; (ii) the JL controller stabilizes the nonlinear dynamical system; (iii) JL controllers provide approximations of the LDV controller. Finally, it is shown that this process is robust against such errors in the probability structure as the inaccurate assumption that an easily constructed partition is Markov.

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“…Nessa classe de sistemas, as matrizes de parâmetros estão associadas a uma variável de salto admitindo valores a cada instante em um conjunto finito, situação na qual se configura o objeto de estudo desta tese. O caso em que cadeia de Markov admite valores em um conjunto infinito enumerável também é investigado na literatura, veja, por exemplo, [23,31,32,52,53,55], porém não será alvo da pesquisa deste trabalho. Um amplo conteúdo sobre SLSM pode ser encontrado em [33,86].…”

Section: Introductionunclassified

Controle e filtragem para sistemas lineares discretos incertos sujeitos a saltos Markovianos

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Optimal Control for Continuous-Time Linear Quadratic Problems with Infinite Markov Jump Parameters

Cited by 73 publications

References 29 publications

On the stability radii of continuous-time infinite Markov jump linear systems

On the stability radii of continuous-time infinite Markov jump linear systems

Relationships between Linear Dynamically Varying Systems and Jump Linear Systems

Controle e filtragem para sistemas lineares discretos incertos sujeitos a saltos Markovianos

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