Model Reference Control for Linear Time-Varying Systems: A Direct Parametric Approach

Abstract

This paper investigates model reference control for linear time-varying (LTV) systems. It is shown that the problem can be decomposed into two sub-problems: a state feedback stabilization problem of LTV systems and a feed-forward compensation problem. Firstly, the sufficient condition for the existence of the model reference control is deduced. The condition concerns two coefficient matrices such that two matrix equations are met simultaneously, and the state feedback gain matrix stabilizes LTV systems according to the unified spectral theory. Secondly, generally complete parametrization of the model reference tracking controller including a state feedback controller and a feed-forward compensator is proposed based on the solution to a type of generalized Sylvester matrix equations. Furthermore, a general procedure for solving the model reference control problem is also provided. Finally, a numerical example shows the feasibility and effectiveness of the parametric design approach to the model reference control in LTV systems. INDEX TERMS Linear time-varying systems, model reference control, parametric approach, PD-eigenstructure, Sylvester equations.

Model reference control in quasi-linear systems with a parametric feed-forward compensator and state-feedback stabilization controller

“…The subject of designing model reference control systems has been studied by many researchers in recent years that mainly aim to regulate the output of the system to a desired output of the reference model. Various types of systems have been studied for the model reference control, such as multivariable linear systems (De La Torre et al, 2016;Duan et al (2001), second-order linear system (Duan and Huang, 2008;Tian and Duan, 2020), constrained linear systems (Di Cairano and Borrelli, 2016), multi-agent systems (Liu and Jia, 2012), networked control systems (Sakthivel et al, 2017), switched linear parameter varying systems with parametric uncertainties (Abdullah, 2018), piecewise affine systems with input disturbances (Di Bernardo et al, 2016), markov jump systems (Boukas, 2009), uncertain dynamical systems with performance guarantees (Arabi et al, 2019) and linear timevarying systems (Liu et al, 2020). In the meantime, effective results were successfully achieved in practical application by utilizing model reference control technology, especially in the active damping of driveline vibration in power-split hybrid vehicles (Liu et al, 2019), training recurrent neural networks (Jafari and Hagan, 2018), precise speed tracking of industrial robot (Xie et al, 2019) and so forth.…”