Parametric control to a type of descriptor quasi‐linear systems based on dynamic compensator and multi‐objective optimisation

Abstract

This study investigates the parametric approach for a type of descriptor quasi-linear systems by utilising dynamic compensator and multi-objective optimisation. Based on the solutions of generalised Sylvester matrix equation, the generally parameterised expressions of dynamic compensator and the left and right eigenvector matrices are both established, meanwhile, a group of arbitrary parameters are obtained. With the parametric approach, the closed-loop system can be transformed into a linear time-invariant one with an expected eigenstructure by using a group of canonical matrix pairs. Simultaneously, it also presents a novel technique to design multi-objective optimisation for descriptor quasi-linear systems.Multiple performance indexes such as low sensitivity, disturbance attenuation, robustness degree, and low gains are formulated by arbitrary parameters. Based on the above indexes, robustness criteria and low gain criteria can be expressed by a synthetic objective function which includes each performance index weighted. By utilising the degrees of freedom in arbitrary parameters, a dynamic compensator can be obtained by solving a multi-objective optimisation problem. Finally, two examples are proposed to prove that the parametric approach is effective.

Parametric method to design dynamic compensator for descriptor high‐order quasi‐linear systems

“…Lemma 4: Given the DHQ systems (1a) and (1b) satisfying Assumptions 1-4, let N(θ, y, s) and D(θ, y, s) be a pair of polynomial matrices satisfying RCF (21). Based on the results in [19,26], there also exists a pair of polynomial matrices N (θ, y, s)…”

“…Figs. [17][18][19] show that the parametric method leads to the better transient performance than that of Liu and Fei [13], such that the frequency of oscillation in the dynamical process has been reduced effectively and the convergence rate has been improved obviously, which fully demonstrates the benefits of the parametric method.…”

“…Based on the results in [31–33], a general solution to GHSE (20) is given by where is an arbitrary parameter matrix. Lemma Given the DHQ systems (1a) and (1b) satisfying Assumptions 1–4, let and be a pair of polynomial matrices satisfying RCF (21). Based on the results in [19, 26], there also exists a pair of polynomial matrices and for the ADHQ systems (4a) and (4b) such that …”

“…Our team has paid more attention to the quasi-linear system and obtained a series of research results, such as descriptor quasi-linear systems [19][20][21] and dynamic compensators with multi-objective optimisation [19,[22][23][24][25]. Notably, we also present an effective approach to design a static output feedback and a dynamic compensator for high-order quasi-linear systems (see [26,27]), which provides the basis to deal with the parametric design of a dynamic compensator for DHQ systems.…”

Parametric control to permanent magnet synchronous motor via proportional plus integral feedback

“…The parametric approach, creatively proposed by Duan (2014b, 2014c), solved the basic control problem and realized the performance optimization, which opens up the whole new field of research. Further, Gu et al developed its applications in various systems, such as linear time-varying systems (Gu et al 2019a; Gu and Zhang 2020a) and quasi-linear systems (Gu et al 2019b, 2019c; Gu and Zhang 2020b, 2020c, 2020d). The major benefits of the parametric approach are summarized in the following aspects.…”