A new approach for the H&lt;inf&gt;∞&lt;/inf&gt; control of Markov jump linear systems with partial information

Todorov, Marcos G.; Fragoso, Marcelo D.; Costa, Oswaldo Luiz do Valle

doi:10.1109/cdc.2015.7402775

Cited by 13 publications

(23 citation statements)

References 24 publications

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“…Remark The set of conditions and is different from the one obtained in the work of Todorov et al As we are going to see in Section 4.4, this formulation will be convenient to tackle the joint mixed

H_{2} false/ H_{\infty}

problem, since it will allow us to obtain the same set of controller gains for both the

H_{2}

and

H_{\infty}

costs.…”

Section: Resultsmentioning

confidence: 99%

“…Proof See the work of Todorov et al Consider the following inequality set for all

i \in double-struckN

l \in M_{i}

and some given δ>0:

R_{i} > \sum_{l \in M_{i}} α_{i l} M_{i l},

([], \begin{matrix} array M_{i l} & array • & array • & array • \\ array 0 & array {normalδ}^{2} I & array • & array • \\ array A_{i l} & array J_{i} & array {scriptE}_{i} {(R)}^{- 1} & array • \\ array C_{i l}^{(\infty)} & array E_{i}^{(\infty)} & array 0 & array I \end{matrix}) > 0 .

The following proposition shows that Equations and are sufficient conditions for Equations and . Notice that for the case of complete observation (α i l =α i i ), Equations and become the inequalities of the bounded‐real lemma for the MJLS given in the work of Seiler and Sengupta …”

Section: Preliminariesmentioning

confidence: 99%

“…We consider the control law in the following form:

u false(k false) = K_{\overset{θ}{true} false(k false)} x false(k false),

that is, state‐feedback controllers that depend only on the estimated parameter

\overset{θ}{true} false(k false)

, for instance, as in the works of Costa et al and Todorov et al Briefly, the main goal is to find

K_{l}, l \in double-struckM

, while minimizing a multiple performance criteria related to the

H_{2}

and

H_{\infty}

control. From Equations and , we have the closed‐loop system given as follows:

G_{K} : ({, \begin{matrix} array x (k + 1) & array = A_{θ (k) \hat{normalθ} (k)} x (k) + J_{θ (k)} w (k), \\ array z_{2} (k) & array = C_{θ (k) \hat{normalθ} (k)}^{(2)} x (k) + E_{θ (k)}^{(2)} w (k), \\ array z_{\infty} (k) & array = C_{θ (k) \hat{normalθ} (k)}^{(\infty)} x (k) + E_{θ (k)} \end{matrix})

…”

Section: Preliminariesmentioning

confidence: 99%

“…We also consider the case in which Υ possesses the polytopic uncertainty given in Equation . The second example is based on the unmanned aerial vehicle subject to failures as considered in the works of Costa et al and Todorov et al whose dynamics were originally taken from the work of Ducard . The goal is to study the behavior of δ as we vary γ and some parameters of Υ in Problem .…”

Section: Numerical Examplesmentioning

confidence: 99%

“…In this context, it is shown in the work of Costa et al that the detector approach generalizes the complete observation, the cluster, and the mode‐independent cases that were previously discussed. Thereby, under this detector approach, the

H_{2}

and

H_{\infty}

state‐feedback control and filtering problems were studied respectively in the works of Costa et al and Todorov et al and in our previous works() for discrete‐time MJLS and in the work of Stadtmann and Costa for the

H_{2}

continuous‐time MJLS state‐feedback case.…”

Section: Introductionmentioning

confidence: 99%

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Mixed control of hidden Markov jump systems

Oliveira

Costa

2017

Intl J Robust & Nonlinear

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Summary In this work, we study the mixed H2false/H∞ control for Markov jump linear systems with hidden Markov parameters. The hidden Markov process is denoted by false(normalθfalse(kfalse),trueθ^false(kfalse)false), where the nonobservable component θ(k) represents the mode of operation of the system, whereas trueθ^false(kfalse) represents the observable component provided by a detector. The goal is to obtain design techniques for mixed H2false/H∞ control problems, with the controllers depending only on the estimate trueθ^false(kfalse), for problems formulated in 3 different forms: (i) minimizing an upper bound on the H2 norm subject to a given restriction on the H∞ norm; (ii) minimizing an upper bound on the H∞ norm, while limiting the H2 norm; and (iii) minimizing a weighted combination of upper bounds of both the H2 and H∞ norms. We propose also new conditions for synthesizing robust controllers under parametric uncertainty in the detector probabilities and in the transition probabilities. The so‐called cluster case for the mixed H2false/H∞ control problem is also analyzed under the detector approach. The results are illustrated by means of 2 numerical examples.

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H_{2} false/ H_{\infty}

problem, since it will allow us to obtain the same set of controller gains for both the

H_{2}

and

H_{\infty}

costs.…”

Section: Resultsmentioning

confidence: 99%

“…Proof See the work of Todorov et al Consider the following inequality set for all

i \in double-struckN

l \in M_{i}

and some given δ>0:

R_{i} > \sum_{l \in M_{i}} α_{i l} M_{i l},

([], \begin{matrix} array M_{i l} & array • & array • & array • \\ array 0 & array {normalδ}^{2} I & array • & array • \\ array A_{i l} & array J_{i} & array {scriptE}_{i} {(R)}^{- 1} & array • \\ array C_{i l}^{(\infty)} & array E_{i}^{(\infty)} & array 0 & array I \end{matrix}) > 0 .

Section: Preliminariesmentioning

confidence: 99%

“…We consider the control law in the following form:

u false(k false) = K_{\overset{θ}{true} false(k false)} x false(k false),

that is, state‐feedback controllers that depend only on the estimated parameter

\overset{θ}{true} false(k false)

, for instance, as in the works of Costa et al and Todorov et al Briefly, the main goal is to find

K_{l}, l \in double-struckM

, while minimizing a multiple performance criteria related to the

H_{2}

and

H_{\infty}

control. From Equations and , we have the closed‐loop system given as follows:

G_{K} : ({, \begin{matrix} array x (k + 1) & array = A_{θ (k) \hat{normalθ} (k)} x (k) + J_{θ (k)} w (k), \\ array z_{2} (k) & array = C_{θ (k) \hat{normalθ} (k)}^{(2)} x (k) + E_{θ (k)}^{(2)} w (k), \\ array z_{\infty} (k) & array = C_{θ (k) \hat{normalθ} (k)}^{(\infty)} x (k) + E_{θ (k)} \end{matrix})

…”

Section: Preliminariesmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

H_{2}

and

H_{\infty}

H_{2}

continuous‐time MJLS state‐feedback case.…”

Section: Introductionmentioning

confidence: 99%

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Mixed control of hidden Markov jump systems

Oliveira

Costa

2017

Intl J Robust & Nonlinear

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Robust dissipative control for semilinear Markovian jump distributed parameter systems with time‐varying delay and incomplete transition probabilities

Song

Wang

Song

et al. 2019

Asian Journal of Control

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This paper investigates the robust stabilization problem of semilinear Markovian jump distributed parameter systems (which are modeled by parabolic partial differential equations) with time-varying delay and incomplete transition probabilities. Based on Takagi-Sugeno (T-S) fuzzy theory, a T-S fuzzy model is obtained to describe the nonlinear systems. Furthermore, by constructing a novel Lyapunov functional candidate, several sufficient delay-dependent conditions, which ensure the considered systems stochastically stable and strictly dissipative, are established in terms of linear matrix inequalities. Finally, a fuzzy controller design approach is proposed, and two examples are presented to demonstrate the effectiveness of the designed controller.

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