Consistent systems and pole assignment: I

Зайцев, В.А.

doi:10.1134/s001226611110120

Cited by 16 publications

(18 citation statements)

References 27 publications

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“…Assertion 4 with condition (f) and Assertion 5 are similar to a sufficient condition for the consistency in Assertion 7 in [3] for systems with continuous time.…”

Section: Assertionmentioning

confidence: 72%

“…, γ n ) ∈ K n , then we say that the spectrum of the system Σ (respectively, Ω) is arbitrarily assignable. It is well known that, for the system Σ = (A, B, C) with complete feedback, i.e., with C = I, the following assertion (see the bibliography in [3]) holds both for K = C and K = R and both for continuous-time and discrete-time systems. 3.…”

Section: Eigenvalue Assignment Problemmentioning

confidence: 85%

“…For continuous-time systems, Theorems 5 and 6 were proved in [3] (see Theorems 1 and 2 there). Next, the assertions 2 ⇐⇒ 3 of Theorems 5 and 6 were proved in [6,7] for continuous-time systems with K = R. The proofs presented there remain valid both for continuous-time systems and for discrete-time systems in the cases of K = R and K = C. The assertions 1 =⇒ 2 in Theorems 5 and 6 follow from Assertion 3, because the matrix A in Hessenberg form is cyclic.…”

Section: Eigenvalue Assignment Problemmentioning

confidence: 96%

“…This fact is illustrated by Example 4 below. The counterexamples 3 and 4 constructed below differ from the corresponding counterexamples (see [3,11]) for continuoustime systems.…”

Section: Eigenvalue Assignment Problemmentioning

confidence: 98%

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Consistency and eigenvalue assignment for discrete-time bilinear systems: II

Зайцев

2015

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We obtain new necessary and sufficient conditions for the consistency of discretetime time-independent linear control systems with incomplete feedback and bilinear systems. We find a relationship between the consistency of time-independent systems and the eigenvalue assignment problem. We show that consistency is sufficient and, in certain cases, necessary condition for the arbitrary assignability of the eigenvalues for discrete-time systems with coefficients of special form.The present paper is the second part of [1], and here we continue the numbering of formulas, assertions, lemmas, theorems, corollaries, and remarks. The notation, definitions, results, and formulas in [1] required in forthcoming considerations are used without additional comments. By J we denote the first unit superdiagonal matrix; i.e., NECESSARY CONDITIONS AND SUFFICIENT CONDITIONS FOR THE CONSISTENCY OF A TIME-INDEPENDENT SYSTEMLet i 1 (λ), . . . , i s (λ) be nontrivial invariant polynomials [2, p. 134] of the matrix A of degrees n 1 ≥ · · · ≥ n s > 0, respectively (n 1 + · · · + n s = n), and let n 0 := n 1 + 3n 2 + · · · + (2s − 1)n s . Assertion 2. If the system Σ (respectively, Ω) is consistent, then mk ≥ n 0 (respectively, r ≥ n 0 ).Proof. If the system Σ (respectively, Ω) is ϑ-consistent for ϑ = 1, then m = k = n (respectively, r = n 2 ); consequently, mk = n 2 . Obviously, n 0 ≤ n 2 . Consider the case in which ϑ > 1. In this case, det A = 0 by virtue of Assertion 1. It is well known [2, p. 192] that the set of matrices commuting with the matrix A forms a linear subspace M ⊂ M n of dimension n 0 . Take a basis P 1 , . . . , P n0 in M. Set H := α 1 P 1 + · · · + α n0 P n0 , where the α i ∈ K are found from the conditionRelation (33) is a system of mk equations with n 0 unknown coefficients α i . If mk < n 0 , then system (33) always has a nontrivial solution α = (α 1 , . . . , α n0 ). In this case, we have H = 0 and C * HB = 0. Since H commutes with A, we have C * A i HA −i B = C * HB = 0 for all i = 1, . . . , n 2 −1.Consequently, relations (31) hold for the matrix H = 0. By Theorem 4, the system Σ is not consistent.510

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“…Assertion 4 with condition (f) and Assertion 5 are similar to a sufficient condition for the consistency in Assertion 7 in [3] for systems with continuous time.…”

Section: Assertionmentioning

confidence: 72%

Section: Eigenvalue Assignment Problemmentioning

confidence: 85%

Section: Eigenvalue Assignment Problemmentioning

confidence: 96%

“…This fact is illustrated by Example 4 below. The counterexamples 3 and 4 constructed below differ from the corresponding counterexamples (see [3,11]) for continuoustime systems.…”

Section: Eigenvalue Assignment Problemmentioning

confidence: 98%

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Consistency and eigenvalue assignment for discrete-time bilinear systems: II

Зайцев

2015

Diff Equat

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“…Therefore conditions (4) imply that the function V 0 (t, y) satisfies conditions 1), 2) of Theorem 1 for the system (6). Consider the set E(V 0 ) = {(t, y) ∈ R + × R n :V 0 (t, y) = 0}.…”

Section: Consider the Periodic System (1) Leṫmentioning

confidence: 99%

Global asymptotic stabilization of bilinear control systems with periodic coefficients

Зайцев¹

2012

Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki

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MSC: 34D23, 34H15, 93D15c V. A. Zaitsev GLOBAL ASYMPTOTIC STABILIZATION OF BILINEAR CONTROL SYSTEMS WITH PERIODIC COEFFICIENTS 1Sufficient conditions for uniform global asymptotic stabilization of the origin are obtained for bilinear control systems with periodic coefficients. The proof is based on the use of the Krasovsky theorem on global asymptotic stability of the origin for periodic systems. The stabilizing control function is feedback control constructed as the quadratic form of the phase variables and depends on time periodically.

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Global asymptotic stabilization of autonomous bilinear complex systems

Зайцев

2022

European Journal of Control

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Consistent systems and pole assignment: I

Cited by 16 publications

References 27 publications

Consistency and eigenvalue assignment for discrete-time bilinear systems: II

Consistency and eigenvalue assignment for discrete-time bilinear systems: II

Global asymptotic stabilization of bilinear control systems with periodic coefficients

Global asymptotic stabilization of autonomous bilinear complex systems

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