Necessary and Sufficient Conditions for Assignability of the Lyapunov Spectrum of Discrete Linear Time-Varying Systems

Babiarz, Artur; Banshchikova, I.N.; Czornik, Adam; Makarov, E. K.; Niezabitowski, Michał; Popova, S. N.

doi:10.1109/tac.2018.2823086

Cited by 26 publications

(14 citation statements)

References 18 publications

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“…The proof of Lemma 7 is similar to the proof of Lemma 6. By ( [49], Theorem 4.1.7), system (21) is exactly controllable on [0, ϑ] if and only if for some γ > 0 and all x ∈ X:…”

Section: Resultsmentioning

confidence: 99%

“…In a series of studies [13][14][15][16][17], the results on arbitrary assignability of Lyapunov exponents and other Lyapunov invariants for system (2) in finite-dimensional spaces were proved, based on the property of uniform complete controllability in the sense of Kalman. In recent studies [18][19][20][21][22][23], these results have been partially extended to discrete-time systems. In finite-dimensional spaces, the Lyapunov exponents, the Bohl exponents, and other Lyapunov invariants were studied, for example in [24][25][26] for continuous-time systems and in [27][28][29][30][31][32][33] for discrete-time systems.…”

Section: Introductionmentioning

confidence: 99%

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On Assignment of the Upper Bohl Exponent for Linear Time-Invariant Control Systems in a Hilbert Space by State Feedback

Зайцев

Zhuravleva

2020

Mathematics

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We consider a linear continuous-time control system with time-invariant linear bounded operator coefficients in a Hilbert space. The controller in the system has the form of linear state feedback with a time-varying linear bounded gain operator function. We study the problem of arbitrary assignment for the upper Bohl exponent by state feedback control. We prove that if the open-loop system is exactly controllable then one can shift the upper Bohl exponent of the closed-loop system by any pregiven number with respect to the upper Bohl exponent of the free system. This implies arbitrary assignability of the upper Bohl exponent by linear state feedback. Finally, an illustrative example is presented.

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“…The proof of Lemma 7 is similar to the proof of Lemma 6. By ( [49], Theorem 4.1.7), system (21) is exactly controllable on [0, ϑ] if and only if for some γ > 0 and all x ∈ X:…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Assignment of the Upper Bohl Exponent for Linear Time-Invariant Control Systems in a Hilbert Space by State Feedback

Зайцев

Zhuravleva

2020

Mathematics

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“…From ( 27), (31), and (32), it follows that there exist α 5 = α 5 (ϑ) > 0 and α 6 = α 6 (ϑ) > 0 such that…”

Section: ) Necessary Conditions and Sufficient Conditionsmentioning

confidence: 99%

Lyapunov Spectrum Local Assignability of Linear Discrete Time-Varying Systems by Static Output Feedback

et al. 2021

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We consider a linear discrete time-varying input-output system. Our goal is to study the problem of local assignability of the Lyapunov spectrum by static output feedback control. To this end we introduce the notion of uniform consistency for discrete-time linear systems which is the extension of the notion of uniform complete controllability to input-output systems. The property of uniform consistency is investigated, some necessary and sufficient conditions for this property are obtained. The notion of uniform local attainability is introduced for the closed-loop system. We prove that uniform consistency implies uniform local attainability of the closed-loop system. The property of local Lyapunov reducibility is introduced for the closed-loop system. We prove that uniform local attainability implies local Lyapunov reducibility. We prove that, for a locally Lyapunov reducible system, the Lyapunov spectrum is locally assignable, if the free system is diagonalizable or regular (in the Lyapunov sence) or has the stable Lyapunov spectrum.INDEX TERMS linear discrete time-varying input-output systems, local assignability, Lyapunov spectrum, pole assignment problem, static output feedback, uniform consistency.

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“…On the other hand, for the DLTV systems of the form x(k + 1) = A(k)x(k), there exist some general requirements, i.e., require ||A(k)|| and the rate of variation of the system matrix A(k) = A(k + 1) − A(k) are bounded [25]- [28]. These requirements restrict the practical applications and theoretical research of DLTV systems.…”

Section: Introductionmentioning

confidence: 99%

Stability Analysis of Discrete-Time Switched Linear Time-Varying Systems Based on Function-Dependent LMIs

Chen

Yang

2020

IEEE Access

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This paper considers the problem of uniformly asymptotic stability (UAS) for discrete-time switched linear time-varying (DSLTV) systems. Starting with discrete-time linear time-varying (DLTV) systems, some stability conditions are given by using function-dependent linear matrix inequalities (LMIs). Comparing with the existing results, the conditions obtained allow the norm and rate of variation of system matrix are unbounded. Furthermore, the obtained results are extended to study the UAS of DSLTV systems, the stability condition is given by combining the methods of function-dependent LMIs with average dwell time. Finally, some further discussions about the relaxation of conditions obtained are also proposed. Three numerical examples are given to illustrate the theoretical results. INDEX TERMS Uniformly asymptotic stability, discrete-time systems, switched systems, linear timevarying systems, function-dependent linear matrix inequalities, average dwell time.

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Necessary and Sufficient Conditions for Assignability of the Lyapunov Spectrum of Discrete Linear Time-Varying Systems

Cited by 26 publications

References 18 publications

On Assignment of the Upper Bohl Exponent for Linear Time-Invariant Control Systems in a Hilbert Space by State Feedback

On Assignment of the Upper Bohl Exponent for Linear Time-Invariant Control Systems in a Hilbert Space by State Feedback

Lyapunov Spectrum Local Assignability of Linear Discrete Time-Varying Systems by Static Output Feedback

Stability Analysis of Discrete-Time Switched Linear Time-Varying Systems Based on Function-Dependent LMIs

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