Characterization of flat outputs of switched linear discrete-time systems: Algebraic condition and algorithm

Abstract

The problem of flat output characterization for switched linear discrete-time systems is addressed. First, an algebraic condition for an output to be flat is provided. It applies for I-flat outputs with the integer I potentially strictly greater than 1. Next, it is proved that such a characterization is decidable. Finally, an efficient algorithm which allows to decide in polynomial time whether a given output is flat is given. The algorithm is built from the observation that the flat output characterization can be expressed in terms of dead-beat stability of an associated constrained switched system. Thus, the notions of joint spectral radius and graph theory play relevant roles in this context.

Linear switched system modelling and analysis for the servo system

“…Assuming that the system (1) is left invertible with left inherent delay r, the input sequence of the system can be recovered from its output sequence [21].…”

“…Assuming that the system () is left invertible with left inherent delay r , the input sequence of the system can be recovered from its output sequence [21]. $$$\left\{\begin{array}{@{}l@{}}{\hat{x}}_{k+r+1}={P}_{\sigma (k:k+r)}{\hat{x}}_{k+r}\hfill \\ +{B}_{\sigma (k)}{I}_{m\times r}{\left({M}_{\sigma (k:k+r)}\right)}^{{\dagger}}{y}_{k:k+r}\hfill \\ {\hat{u}}_{k+r}=-{I}_{m\times r}{\left({M}_{\sigma (k:k+r)}\right)}^{{\dagger}}{\mathcal{O}}_{\sigma (k:k+r)}{\hat{x}}_{k+r}\hfill \\ +{I}_{m\times r}{\left({M}_{\sigma (k:k+r)}\right)}^{{\dagger}}{y}_{k:k+r}\hfill \end{array}\right.$$$ with …”

“…In refs. [21,22], the characterisation of the switched linear discrete-time system is studied with the algebraic condition and algorithm.…”

“…From the aspects of controllability and observability, the flatness property [20] approves for a whole parameterisation of all variables in terms of a finite set of independent variables referred to as flat output [21]. In refs.…”