Algebraic Riccati equation based Q and R matrices selection algorithm for optimal LQR applied to tracking control of 3rd order magnetic levitation system

Abstract

This paper presents an analytical approach for solving the weighting matrices selection problem of a linear quadratic regulator (LQR) for the trajectory tracking application of a magnetic levitation system. One of the challenging problems in the design of LQR for tracking applications is the choice of Q and R matrices. Conventionally, the weights of a LQR controller are chosen based on a trial and error approach to determine the optimum state feedback controller gains. However, it is often time consuming and tedious to tune the controller gains via a trial and error method. To address this problem, by utilizing the relation between the algebraic Riccati equation (ARE) and the Lagrangian optimization principle, an analytical methodology for selecting the elements of Q and R matrices has been formulated. The novelty of the methodology is the emphasis on the synthesis of time domain design specifications for the formulation of the cost function of LQR, which directly translates the system requirement into a cost function so that the optimal performance can be obtained via a systematic approach. The efficacy of the proposed methodology is tested on the benchmark Quanser magnetic levitation system and a detailed simulation and experimental results are presented. Experimental results prove that the proposed methodology not only provides a systematic way of selecting the weighting matrices but also significantly improves the tracking performance of the system.

Seleção das matrizes da saída controlada por realimentação de estados com norma H2 baseada na solução algébrica de Riccati

“…Em Kumare and Jerome (2016),é sugerido um método analítico para encontrar as matrizes de um regulador linear quadrático usando a equação algébrica de Riccati. Considerando um sistema linear e invariante no tempȯ…”

“…Porém, nadaé dito sobre quais valores tais matrizes da saída controlada devem assumir. O presente trabalho parte do conceito mostrado por Kumare and Jerome (2016), que propôs um algoritmo para a seleção das matrizes Q e R de um controlador LQR baseado na solução algébrica da equação de Riccati. Uma característica importante da sua propostaé que a solução algébricaé dependente do modelo matemático do sistema utilizado.…”

Optimal Design of Fault‐Tolerant Controller for an Electric Power Steering System with Sensor Failures Using Genetic Algorithm

“…where P is the positive definite solution of the Riccati Equation (15). Substituting (23) and (24) into (27) yields…”

Passivity-Based Control Design for Magnetic Levitation System

“…Magnetic levitation system is highly nonlinear unstable system, it is a strong need to control the suspension gap. In recent years, various research methods have been proposed, such as Proportional-Integral-Derivative (PID) control [5,6], Linear Quadratic Regulator (LQR) control [7], and sliding mode controller [8], fuzzy control is presented in [9]. When PID controller is designed, the system has to be linearized, sliding mode control has chattering effect.…”