Improved H2 and H&lt;FONT FACE=Symbol&gt;¥&lt;/FONT&gt; conditions for robust analysis and control synthesis of linear systems

Trofino, Alexandre; Coutinho, Daniel; Barbosa, Karina A.

doi:10.1590/s0103-17592005000400004

Cited by 8 publications

(8 citation statements)

References 7 publications

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“…It should be noted that a direct application of the Elimination Lemma to the parameter‐dependent matrix inequality provides a synthesis condition based on LMIs parameterized in terms of a scalar parameter that belongs to an unbounded domain. () However, to provide parameterized LMI conditions with a search on a scalar parameter that belongs to a bounded domain, analogous to the discrete‐time case,() the change of variables (similar to those in the work of Trofino et al) given by

\begin{matrix} right {\tilde{A}}_{cl} (α) & left : = ε A_{cl} (α) - I / (2 ε), & right \\ right {\hat{A}}_{cl} (α) & left : = ε A_{cl} (α) + I / (2 ε), \\ right {\bar{B}}_{1} (α) & left : = ε B_{1} (α), \\ right {\bar{C}}_{cl} (α) & left : = ε C_{cl} (α), \end{matrix}

with ε≠0, is introduced so that inequality becomes

([], \begin{matrix} array {\hat{A}}_{cl} (α) W (α) {\hat{A}}_{cl} {(α)}^{normalT} - {\tilde{A}}_{cl} (α) W (α) {\tilde{A}}_{cl} ( \end{matrix})

…”

Section: Robust State Feedback Control Designmentioning

confidence: 99%

“…It should be noted that a direct application of the Elimination Lemma to the parameter-dependent matrix inequality (7) provides a synthesis condition based on LMIs parameterized in terms of a scalar parameter that belongs to an unbounded domain. 21,22 However, to provide parameterized LMI conditions with a search on a scalar parameter that belongs to a bounded domain, analogous to the discrete-time case, 13,23 the change of variables (similar to those in the work of Trofino et al 24 ) given byÃ…”

Section: Lemma 3 (Elimination Lemma)mentioning

confidence: 99%

“…To construct the polytopic representation described in Equation 5, for the mechanical system (23)- (24) with k in the interval [0.5, 2.0], the uncertain parameter α = (α 1 , α 2 ), which lies in the unit simplex Λ, is defined as…”

Section: Figure 1 Spring-mass Systemmentioning

confidence: 99%

“…Output z(t) of the closed-loop system (23)-(24) with the state feedback gains K[10] and K Th2 , for k = 1.0, null initial conditions,and exogenous disturbance defined in Equation 25 [Colour figure can be viewed at wileyonlinelibrary.com] using ζ = 10.0 and K Th2 = [ −9.9420 −0.9935 −29.5539 −16.0522 ]…”

mentioning

confidence: 99%

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Parameterized LMIs for robust and state feedback control of continuous‐time polytopic systems

Rodrigues

Oliveira

Camino

2017

Intl J Robust & Nonlinear

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Summary This paper presents new extended linear matrix inequality (LMI) characterizations for the synthesis of robust H∞ and H2 state feedback controllers for continuous‐time linear time‐invariant systems with polytopic uncertainty. Based on a suitable change of variables and the Elimination Lemma, the proposed robust control design techniques are stated as extended LMI conditions parameterized in terms of 2 scalar parameters. One parameter is shown to belong to a bounded domain, thus limiting the scalar search domain. For the other parameter, a bounded subset is provided from numerical experiments. The benefits of the methodology are illustrated through numerical simulations performed on an uncertain model borrowed from the literature. It is shown that the proposed LMI relaxations can provide less conservative results with fewer scalar searches than some existing methods in the literature.

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\begin{matrix} right {\tilde{A}}_{cl} (α) & left : = ε A_{cl} (α) - I / (2 ε), & right \\ right {\hat{A}}_{cl} (α) & left : = ε A_{cl} (α) + I / (2 ε), \\ right {\bar{B}}_{1} (α) & left : = ε B_{1} (α), \\ right {\bar{C}}_{cl} (α) & left : = ε C_{cl} (α), \end{matrix}

with ε≠0, is introduced so that inequality becomes

([], \begin{matrix} array {\hat{A}}_{cl} (α) W (α) {\hat{A}}_{cl} {(α)}^{normalT} - {\tilde{A}}_{cl} (α) W (α) {\tilde{A}}_{cl} ( \end{matrix})

…”

Section: Robust State Feedback Control Designmentioning

confidence: 99%

Section: Lemma 3 (Elimination Lemma)mentioning

confidence: 99%

Section: Figure 1 Spring-mass Systemmentioning

confidence: 99%

mentioning

confidence: 99%

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Parameterized LMIs for robust and state feedback control of continuous‐time polytopic systems

Rodrigues

Oliveira

Camino

2017

Intl J Robust & Nonlinear

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“…Statistical numerical comparisons among the main available methods can be found in [20] for continuous-time systems and in [14] for discrete-time systems. Regarding the extensions of the stabilization conditions to cope with performance criteria based on the H 2 and H ∞ norms, in general they present the same sources of conservativeness that affect the stabilization conditions [16,17,[21][22][23][24][25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Robust stabilization and ℋ<inf>∞</inf> control by means of state-feedback for polytopic linear systems using LMIs and scalar searches

Vieira

Oliveira

Peres

2015

2015 American Control Conference (ACC)

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The problems of robust stabilization and robust H∞ control by means of state feedback for uncertain linear systems in polytopic domains are investigated in this paper. New synthesis conditions based on LMI relaxations and search of scalar variables are proposed for both continuous and discretetime systems. The main novelty comes from the fact that the closed-loop stability is certified by means of polynomially parameter-dependent Lyapunov matrices. It is also shown that the proposed conditions contain as particular cases the less conservative conditions available in the literature for this class of systems. Numerical experiments illustrate the efficiency of the proposed relaxations.

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New iterative linear matrix inequality based procedure for H₂ and H_∞ state feedback control of continuous‐time polytopic systems

Jaimoukha

2020

Intl J Robust & Nonlinear

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This article provides new linear matrix inequality (LMI) sufficient conditions for a generalized robust state feedback control synthesis problem for linear continuous-time polytopic systems. This generalized problem includes the robust stability, H 2-norm, and H ∞-norm problems as special cases. Using a novel general separation result, which separates the state feedback gain from the Lyapunov matrix but with the state feedback gain synthesized from the slack variable, then allows the formulation of LMI sufficient conditions for the generalized problem. Compared to existing parameterized LMI based conditions, where auxiliary scalar parameters are introduced in order to include the quadratic stability conditions (ie, assuming a constant Lyapunov matrix) as a special case, the proposed new conditions are true LMIs and contain as a particular case the optimal quadratic stability solution. Utilizing any initial solution derived by the quadratic or some existing methods as a starting solution, we propose an algorithm based on an iterative procedure, which is recursively feasible in each update, to compute a sequence of nonincreasing upper bounds for the H 2-norm and H ∞-norm. In addition, if no feasible initial solution can be found for some uncertain systems using any existing methods, another algorithm is presented that offers the possibility of obtaining a robust stabilizing gain. Numerical examples from the literature demonstrate that our algorithms can provide less conservative results than existing methods, and they can also find feasible solutions where all other methods fail. K E Y W O R D S H 2 and H ∞ norms, LMI relaxations, polytopic uncertainties, robust state feedback control, slack variables, uncertain linear systems 1 INTRODUCTION The design of robust H 2-norm and H ∞-norm controllers for linear time-invariant systems with polytopic uncertainties has received considerable attention in the past decades. Many studies devoted to investigating robust stability and This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Improved H2 and H<FONT FACE=Symbol>¥</FONT> conditions for robust analysis and control synthesis of linear systems

Cited by 8 publications

References 7 publications

Parameterized LMIs for robust and state feedback control of continuous‐time polytopic systems

Parameterized LMIs for robust and state feedback control of continuous‐time polytopic systems

Robust stabilization and ℋ<inf>∞</inf> control by means of state-feedback for polytopic linear systems using LMIs and scalar searches

New iterative linear matrix inequality based procedure for H₂ and H_∞ state feedback control of continuous‐time polytopic systems

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Improved H2 and H<FONT FACE=Symbol>¥</FONT> conditions for robust analysis and control synthesis of linear systems

Cited by 8 publications

References 7 publications

Parameterized LMIs for robust and state feedback control of continuous‐time polytopic systems

Parameterized LMIs for robust and state feedback control of continuous‐time polytopic systems

Robust stabilization and &#x210B;<inf>&#x221E;</inf> control by means of state-feedback for polytopic linear systems using LMIs and scalar searches

New iterative linear matrix inequality based procedure for H2 and H∞ state feedback control of continuous‐time polytopic systems

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Robust stabilization and ℋ<inf>∞</inf> control by means of state-feedback for polytopic linear systems using LMIs and scalar searches

New iterative linear matrix inequality based procedure for H₂ and H_∞ state feedback control of continuous‐time polytopic systems