Controllability of linear time-varying systems with delay in control†

Abstract

This paper is concerned with tho study of controllability of linear systems with delay in the control function. It has been illustrated that many of the techniques which proved to be useful in the study of linear systems with no delay (Kalman et al. ] 962, Kroindlcr and Sarachik 1064) call be generalized when dealing with systems having delay in the control.An explicit expression is given for transferring a given state to RIlY desired state using minimum control energy.Tho corresponding conditions for linear time-invariant systems (Sebakhy and Bayoumi 1971) are obtained as a special case. Extensions to mult.iple delays systems are also included.A new degree of controllability is introduced and the corresponding criteria are obtained.

Algebraic conditions on the controllability for a type of discrete‐continuous systems with delays

“…On the other hand, if there is no impulse in system (i.e., E k = F k = 0, k = 1,2, … ), then V 1 , V 2 , … , V M in vanish and D ( s + τ )][ X ( t 0 , s ) B ( s ) + X ( t 0 , s + τ ) D ( s + τ )] ⊤ d s . Simple calculations imply that results in Theorems and include the results in . Therefore, our results generalize the existing results to more general cases.…”

“…It is necessary to study the controllability of time‐varying impulsive systems with time delay in the input. An algebraic approach was utilized to derive conditions for controllability of linear (time‐varying) system with time delay in control in . In , the criteria for controllability and observability of linear systems with state delay were presented using the solution in terms of the matrix Lambert W function.…”

“…By explicitly characterizing solutions of such systems, we derive the controllability criteria expressed in terms of matrix rank conditions, which are easy to check. Specializing in the cases of linear time‐varying impulsive systems or linear delay systems, the obtained criteria include some existing results . Furthermore, it is interesting to find that the time delay in the control is useful in achieving the controllability of impulsive system even if the original system without a delay term is not controllable.…”

A Constructive Method of Solving Local Boundary Value Problems for Nonlinear Systems with Perturbations and Control Delays

“…The study of BVPs over finite time intervals for such types of systems includes finding necessary and sufficient conditions of complete, local, constrained, relative, or approximate controllability for linear [14][15][16][17][18][19][20][21][22], bilinear [23], semilinear [24][25][26], and nonlinear [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45] systems of differential equations; studying and estimating the attainability domain (see [20,46,47]); and developing methods to construct controls under which the trajectory connects the given points in the phase space (see [20,45]). For linear stationary systems, there exist criteria of complete controllability in terms of the matrices of the right-hand side, which take into account a time delay in control [14,15] or several delays in the system state [31].…”

Controllability and null controllability of linear systems

“…It is known in Sebakhy and Bayoumi (1973) that, in the study of economics, biology and physiological systems as well as electromagnetic systems composed of such subsystems interconnected by hydraulic, mechanical and various other linkages, one encounters phenomena which cannot be readily modeled unless relations involving time delays are admitted. Models for such systems can be controlled.…”