Petersen’s lemma on matrix uncertainty and its generalizations

Khlebnikov, M. V.; Shcherbakov, Pavel

doi:10.1134/s000511790811009x

Cited by 40 publications

(10 citation statements)

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“…Use of Petersen lemma [11] brings us to an equivalent condition, the existence of an ε > 0 such that AP + P A − BB + εF F + 1 ε P H HP 0, or, by Schur lemma [12] AP + P A − BB + εF F P H HP −εI 0.…”

Section: Resultsmentioning

confidence: 99%

Estimates of the attraction domain of linear systems under L 2-bounded control

Khlebnikov

2015

Autom Remote Control

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Section: Resultsmentioning

confidence: 99%

Estimates of the attraction domain of linear systems under L 2-bounded control

Khlebnikov

2015

Autom Remote Control

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“…In particular, [18] deals with nonlinear systems with conic-type nonlinearities and [12] with singular systems with Markovian jumps. Differently from [12], [17], [18] which use norm bounded matrix uncertainties together with the Petersen lemma [19], the uncertainty of the system in this work will be described using a polytopic representation. In comparison with the norm-bounded uncertainty representation, the polytopic representation is more flexible and describes more accurately the uncertainty [20, p. 335].…”

Section: Introductionmentioning

confidence: 99%

Finite time boundedness and stability analysis of discrete time uncertain systems

Kussaba

Ishihara

Borges

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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In this paper the problems of finite time boundedness and stability of uncertain systems in discrete-time are addressed. By using the Finsler lemma, novel parameter dependent linear matrix inequalities analysis conditions are designed. These conditions can be efficiently solved using a homogeneous polynomial relaxation procedure and readily available numerical solvers.In the numerical examples section, it is shown that the proposed finite time conditions are less conservative than others linear matrix inequality based conditions proposed in the literature.

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“…In this section we formally generalize Petersen's sign-definiteness lemma [14], [48], [49] to multiple complex valued uncertainties . The complex-valued version of this lemma for a single uncertainty is proved in [26].…”

Section: Sign-definiteness Lemma With Multiple Uncertaintiesmentioning

confidence: 99%

The Sign-Definiteness Lemma and Its Applications to Robust Transceiver Optimization for Multiuser MIMO Systems

Gharavol

Larsson

2013

IEEE Trans. Signal Process.

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We formally generalize the sign-definiteness lemma to the case of complex-valued matrices and multiple norm-bounded uncertainties. This lemma has found many applications in the study of the stability of control systems, and in the design and optimization of robust transceivers in communications. We then present three different novel applications of this lemma in the area of multi-user multiple-input multiple-output (MIMO) robust transceiver optimization. Specifically, the scenarios of interest are: (i) robust linear beamforming in an interfering adhoc network, (ii) robust design of a general relay network, including the two-way relay channel as a special case, and (iii) a half-duplex one-way relay system with multiple relays. For these networks, we formulate the design problems of minimizing the (sum) MSE of the symbol detection subject to different average power budget constraints. We show that these design problems are non-convex (with bilinear or trilinear constraints) and semiinfinite in multiple independent uncertainty matrix-valued variables. We propose a two-stage solution where in the first step the semi-infinite constraints are converted to linear matrix inequalities using the generalized signdefiniteness lemma, and in the second step, we use an iterative algorithm based on alternating convex search (ACS). Via simulations we evaluate the performance of the proposed scheme.

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Petersen’s lemma on matrix uncertainty and its generalizations

Cited by 40 publications

References 10 publications

Estimates of the attraction domain of linear systems under L 2-bounded control

Estimates of the attraction domain of linear systems under L 2-bounded control

Finite time boundedness and stability analysis of discrete time uncertain systems

The Sign-Definiteness Lemma and Its Applications to Robust Transceiver Optimization for Multiuser MIMO Systems

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