High-order fully-actuated system approaches: Part IX. Generalised PID control and model reference tracking

Abstract

Tracking control of general dynamical systems in high-order fully actuated (HOFA) system representation is solved. For the case of tracking a constant or slow time-varying signal in the presence of constant or slow time-varying disturbances, a generalised PID control scheme is proposed, which realises asymptotical tracking to a prescribed signal and also guarantees that the state derivatives of certain orders converge to the origin. For the case that the signal to be tracked is generated by a reference model, a model reference tracking (MRT) controller is presented, which relies on the solution to a type of generalised Sylvester matrix equations and guarantees the desired asymptotical tracking requirement. Due to the full-actuation property of the HOFA models, closed-loop systems under both control schemes are constant and linear. Furthermore, based on a general parametric solution to the type of Sylvester matrix equations and a general parametric eigenstructure assignment result, simple and complete parameterisation of the two types of control designs are provided, and feasibility conditions in terms of the system initial values for sub-fully actuated systems are also derived. An illustrative example is presented to demonstrate the application of the proposed approaches and their effects.

High-order fully actuated system approaches: Part X. Basics of discrete-time systems

“…Control systems design Thanks to the full-actuation property of the proposed discrete-time HOFA models, important control features are revealed as shown in Theorems 5.1 and 5.2. As in the continuous-time system case, these basic results allow us to extend the results to many other design problems, such as robust control (Duan, 2021g, 2021a, adaptive control (Duan, 2021i, 2021a, disturbance rejection (Duan, 2021b), optimal control (Duan, 2021d), and signal tracking control (Duan, 2021e). All such control problems, as done in the continuous-time case, can be converted into corresponding ones for linear systems, hence can be effectively solved by adopting existing methods for linear systems.…”

Discrete-time delay systems: part 1. Global fully actuated case

“…The results proposed in the paper are very fundamental and are also vitally important since they lay a solid basis for the FAS approaches to discrete-time delay systems. These basic results allow us to further address, as in the continuous-time FAS case, many design problems associated with discrete-time delay systems, such as robust control [22,24], adaptive control [23,24], disturbance rejection [25], optimal control [27], and signal tracking control [28]. Furthermore, the results can also be generalized into the time-varying input delay case.…”

“…Very recently, a different approach is proposed in parallel to the state-space one for dynamical control system design, especially for nonlinear control systems. The approach is termed as the fully actuated system (FAS) approach, and was firstly proposed for continuous-time dynamical systems (see [17][18][19][20][21][22][23][24][25][26][27][28]), and then generalized into the case of discrete-time systems [29].…”

“…Hence it allows one to cancel the nonlinearities in the system, and eventually produces a constant linear system with an arbitrarily assignable eigenstructure. The FAS approach is convenient for control problems, and has been sufficiently shown to be extremely effective and simple in dealing with the control problems of general dynamical nonlinear systems, such as robust control [22,24], adaptive control [23,24], disturbance rejection [25], optimal control [27], and signal tracking control [28].…”

Optimal Fuzzy Proportional‐Integral‐Derivative Control for a Class of Fourth‐Order Nonlinear Systems using Imperialist Competitive Algorithms

“…In fact, this control approach is renowned for its simple structure. However, tuning its parameters is a challenge that requires carrying out some strategies such as classical tuning rules, adaptation laws, optimal techniques, and fuzzy systems [6][7][8][9][10][11].…”