Representation of solutions of linear discrete systems with constant coefficients and pure delay

Diblı́k, Josef; Khusainov, D. Ya.

doi:10.1155/ade/2006/80825

Cited by 51 publications

(33 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this paper, the relative controllability of (1.1) is analyzed through a suitable representation formula for its solutions, describing a solution in time t in terms of its initial condition, the control input, and some matrix-valued coefficients computed recursively (see Proposition 2.7). Such coefficients generalize the discrete delayed matrix exponentials introduced in [12] for (1.3) to the case of several delays and matrices. A similar formula has been used in [3] to analyze the stability of a system of transport equations on a network under intermittent damping and in [4] to obtain stability criteria for (1.1) under no control and with time-varying matrices A j , which in particular provide generalizations of classical stability results for difference equations such as the Hale-Silkowski criterion from [31] (cf.…”

Section: Introductionmentioning

confidence: 97%

Relative Controllability of Linear Difference Equations

Mazanti¹

2017

SIAM J. Control Optim.

View full text Add to dashboard Cite

In this paper, we study the relative controllability of linear difference equations with multiple delays in the state by using a suitable formula for the solutions of such systems in terms of their initial conditions, their control inputs, and some matrix-valued coefficients obtained recursively from the matrices defining the system. Thanks to such formula, we characterize relative controllability in time T in terms of an algebraic property of the matrix-valued coefficients, which reduces to the usual Kalman controllability criterion in the case of a single delay. Relative controllability is studied for solutions in the set of all functions and in the function spaces L p and C k . We also compare the relative controllability of the system for different delays in terms of their rational dependence structure, proving that relative controllability for some delays implies relative controllability for all delays that are "less rationally dependent" than the original ones, in a sense that we make precise. Finally, we provide an upper bound on the minimal controllability time for a system depending only on its dimension and on its largest delay.Notations In this paper, we denote by N and N * the sets of nonnegative and positive integers, respectively. For a, b ∈ R, we write the set of all integers between a and b as a,The cardinality of a set N is denoted by #N. For ξ ∈ R N , we use ξ min and ξ max to denote the smallest and the largest components of ξ , respectively. For ξ ∈ R, the symbol ⌊ξ ⌋ is used to the denote the integer part of ξ , i.e., the unique integer such that ξ − 1 < ⌊ξ ⌋ ≤ ξ .The set of d × m matrices with coefficients in

show abstract

Section: Introductionmentioning

confidence: 97%

Relative Controllability of Linear Difference Equations

Mazanti¹

2017

SIAM J. Control Optim.

View full text Add to dashboard Cite

show abstract

“…e.g., [31], [32]) to preform our studies. In discrete-time, we model our algorithm as a delay difference equation whose solution characterization can be found, for example, in [33], [34]. For both of the continuous-and the discrete-time algorithms that we study, we carefully characterize the admissible delay bound over strongly connected and weight-balanced (SCWB) interaction topologies.…”

Section: Introductionmentioning

confidence: 99%

On Robustness Analysis of a Dynamic Average Consensus Algorithm to Communication Delay

Moradian

Kia

2019

IEEE Trans. Control Netw. Syst.

View full text Add to dashboard Cite

This paper studies the robustness of a dynamic average consensus algorithm to communication delay over strongly connected and weight-balanced (SCWB) digraphs. Under delayfree communication, the algorithm of interest achieves a practical asymptotic tracking of the dynamic average of the timevarying agents' reference signals. For this algorithm, in both its continuous-time and discrete-time implementations, we characterize the admissible communication delay range and study the effect of the delay on the rate of convergence and the tracking error bound. Our study also includes establishing a relationship between the admissible delay bound and the maximum degree of the SCWB digraphs. We also show that for delays in the admissible bound, for static signals the algorithms achieve perfect tracking. Moreover, when the interaction topology is a connected undirected graph, we show that the discrete-time implementation is guaranteed to tolerate at least one step delay. Simulations demonstrate our results.

show abstract

“…Motivated by delayed exponential representing a solution of a system of differential or difference equations with one or multiple fixed or variable delays [1][2][3][4][5][6], which has many applications in theory of controllability, asymptotic properties, boundary-value problems, and so forth [3][4][5][7][8][9][10][11][12][13][14][15], we extended representation of a solution of a system of differential equations of second order with delay [1] ( ) = − 2 ( − ) (1) to the case of two delays…”

Section: Introductionmentioning

confidence: 99%