Controllability and observability of time-invariant linear dynamic systems. (English) Zbl 1265.34334

Bohner, Martin; Wintz, Nick

Math. Bohem. 137, No. 2, 149-163 (2012).

The paper is concerned with linear dynamic time-invariant control systems of the forms \[ x^{\Delta }(t)=-Ax^{\sigma }(t)+Bu(t),\;\;\;y(t)=Cx^{\sigma }(t),\tag{1} \] where \(x\), \(y\), \(u\) denote the state, input, and output of the system, respectively. It is assumed that \(u\) is an rd-continuous function, and \(A\) is a regressive matrix.
The authors derive necessary and sufficient conditions for the controllability and reachability of system (1), which generalize the usual ones which are well known in the classical control theory.
The final result is a duality theorem, which says that the controllability of (1) is equivalent to the observability of \[ x^{\Delta }(t)=A^T x(t)+C^T u(t),\;\;\;y(t)=B^T x(t). \]

Reviewer: Antonín Slavík (Praha)

MSC:

34N05	Dynamic equations on time scales or measure chains
93B05	Controllability
93B07	Observability
34H05	Control problems involving ordinary differential equations
34A30	Linear ordinary differential equations and systems

Keywords:

time scale; dynamic equation; controllability; reachability; observability; duality principle