Delay-based stabilisation and strong stabilisation of LTI systems by nonsmooth constrained optimisation

ABSTRACT

Stabilisation and strong stabilisation (i.e. the problem of stabilising by a stable controller) of linear time-invariant (LTI) multi-input multi-output systems are considered. The considered systems may be finite-dimensional or time-delay systems, which may have multiple discrete time delays. The controllers to be designed may also be either finite-dimensional or time-delay LTI controllers. In the controller design phase, a constrained nonsmooth optimisation-based design method is employed. The effectiveness of the proposed method is illustrated through various benchmark examples which are solved by using the recently developed Matlab-based software called DrStabilization . The examples also show that, in many cases, time-delay controllers may have certain advantages over conventional finite-dimensional controllers.

KEYWORDS:

• Stabilisation
• strong stabilisation
• time-delay systems
• delayed feedback
• optimisation

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The software package is publicly available both in https://github.com/smertozer and in https://mathworks.com/matlabcentral/profile/authors/21374970.

2 Recall that, when l = 0, the controller reduces to the form (12(12) $\begin{array}{r}u(t)=\sum _{i=0}^{\hat{\sigma }}{K}_{i}y(t-{\hat{h}}_i)\end{array}$(12) ), which is $\hat{\mu }$-stable for any $\hat{\mu }\in \mathbb{R}$; in which case, constraint (18c(18c) $\begin{array}{rl}& \hat{c}(\overline{p})\le \hat{\mu }.\end{array}$(18c) ) must be omitted; thus $\mathrm{\nabla }\hat{c}(\overline{p})$ is not needed.

Additional information

Funding

Financial support has been received from the Scientific Research Projects Commission of Eskişehir Technical University [grant number 20DRP067].