A linear dynamic feedback controller for non-linear systems described by matrix differential equations of the second and first orders

Abstract

The paper presents a design scheme of the linear dynamic feedback controller for some non-linear systems. These systems are mathematically described by matrix non-linear differential equations of the first and second orders. A first-order form of the studied systems includes some types of differential-algebraic equations. The stability property of the non-linear systems with the linear controller is assured by an appropriate definition of the system output, and the linear dynamic compensator is an important part of the feedback control system. The order of the dynamic part is equal to the size of the system input and is independent of the size of the system state vector. The asymptotic stability in the Lyapunov sense is analysed and proved by the use of Lyapunov functionals and LaSalle’s invariance principle. Stabilisation in a wide range of controller parameters improves the system’s robustness.

Design of Terminal Sliding Mode Controllers for Disturbed Non-Linear Systems Described by Matrix Differential Equations of the Second and First Orders

“…As shown in the sections below, a TSMC for a first-order matrix system can somehow be considered as a special and simplified form, due to the lower order of the system, of a TSMC for a second-order matrix system. More recently, in [42], it has been shown and formally proven that two different classes of non-linear control systems can be asymptotically stabilized with the help of the same linear dynamic feedback control law. The dynamic behavior of one class is described by non-linear first-order differential equations, while the other class is modeled by non-linear second-order differential equations.…”