Controllability and observability of time-invariant linear dynamic systems

2021

In this work, we investigate the controllability of singular dynamic systems on time scales. First, we decompose the consider systems into a slow subsystem and a fast subsystem. After that, we use the Laplace transform and convolution theorem to derive the state response of these two subsystems. Finally, we established some necessary and sufficient conditions for the controllability of the slow subsystem and fast subsystem. At last, we provide an example to illustrate the obtained analytical results.

“…x(t 0 ) = x 0 and established the controllability and observability results. Bohner and Nick [20] established the controllability results for the following dynamical system:…”

Section: Introductionmentioning

confidence: 99%

\begin{matrix} alignleft \\ align-1 x^{Δ} (t) & align-2 = A x (t) + B u (t), \\ align-1 x (t_{0}) & align-2 = x_{0} \end{matrix}

and established the controllability and observability results.…”

Section: Introductionmentioning

confidence: 99%

Controllability of singular dynamic systems on time scales

Malik

Sajid²,

2021

“…In the last decades, the problem of controllability for various types of differential equations has been widely studied by many authors; see for instance previous studies [18][19][20][21][22][23]. Further, the controllability results for differential equations on time scales is a new area of research and only few authors have been established these results [24][25][26][27][28][29][30][31]. However, these results cannot be easily extended to the case of switched systems with non-instantaneous impulses.…”

Section: Introductionmentioning

confidence: 99%

Total controllability results for a class of time‐varying switched dynamical systems with impulses on time scales

Djemaï

Defoort

et al. 2020

In this paper, we establish some necessary and sufficient conditions of the controllability results for a class of impulsive switched systems with noninstantaneous jumps on time scales. Firstly, we define the solution of the considered problem by using the parameter variation method. We also define some Gramian matrices to establish the controllability results. Moreover, we give some conditions under which the time invariant impulsive switched system is totally controllable. At the end, we provide one numerical example for different time scales to validate the obtained analytical results.

“…For more details please see [32‐36] and references therein. In addition, controllability and observability results of linear and nonlinear system on time scales is a relatively newer area and only few works have been reported [37‐43]. Particularly, in [37], Davis et al consider the regressive linear system on time scales:

\begin{matrix} x^{Δ} false(t false) & = A false(t false) x false(t false) + B false(t false) u false(t false), t \in double-struckT, \\ y false(t false) & = C false(t false) x false(t false) + D false(t false) u false(t false), \\ x false(t_{0} false) & = x_{0} \in R^{n}, \end{matrix}

where

double-struckT

is a time scale such that

t_{0} \in double-struckT

x false(t false) \in R^{n}

is a state variable.…”

Section: Introductionmentioning

confidence: 99%

Total controllability and observability for dynamic systems with non‐instantaneous impulses on time scales

Malik

2019

In this paper, we establish the total controllability and observability results for a dynamic system with non‐instantaneous impulses on time scales in a finite dimensional space Rn. This paper consists of two segments: the first segment of the work is concerned with total controllability and the other segment with total observability analysis. The tools for study include variation of parameter and Gramian matrix. Some theoretical as well as numerical examples with simulation for two time scales double-struckT=double-struckR and double-struckT=P1,1 are given to illustrate the application of these results.