Quadratic optimal controller to stabilise symmetrical systems

Abstract

It is considered a controller design in order to stabilize a control system. The technique that is used for designing the controller includes a linear regulator and an asymptotical estimator which form the controller. The linear regulator is designed by state estimators feedback for symmetrical systems minimizing a quadratic performance index. Computing the gain matrix of optimal feedback is solved by Riccati's equation, whilst the gain estimator matrix is computed by making use of symmetrical systems properties. The symmetrical systems properties allow us to find the solution for Riccati's dual equation without needing to solve two equations, one for the feedback and other for the estimator, using the controllability and observability matrices can be found a symmetrical and invertible matrix which can be used for computing the Riccati's dual solution. Having the regulator and the estimator gain matrices, we proceed to compute the transfer function of the controller based on estimator. Then, we use this method to stabilize the magnetic levitation system.

Optimal guaranteed cost control for discrete‐time LPV systems with bounded rates of variation

“…Since 1960s, the search for optimal controllers has been an essential issue in linear systems theory and control applications (see other works [1][2][3][4][5][6][7] and the references therein). Among all potential approaches, the one termed linear quadratic regulator is undoubtedly a relevant technique applied by industry practitioners in many engineering problems.…”

Stabilization, estimation and control of linear dynamical systems with positivity and symmetry constraints

“…Furthermore, if the system represented by G(s) is externally passive, then it admits an internally passive minimal realization satisfying (3.67). To see the broad aspect and diverse applications of symmetric systems one can refer to [33][34][35][36][37][38]. The symmetric characteristic of transfer function as explored in Lemma 3.2.2 can be illustrated through the model of a space structure with collocated sensors and actuators [51] described by…”

“…Frequently, such systems admit a positivity constraint as well which makes their stability and control problem even more challenging. A great deal of effort was devoted at early stage of system development to understand the concepts of symmetry and passivity [33][34][35][36][37][38]. Although both positive and symmetric systems have been tackled separately and employed in system theory, the combined presence of them in control application has not been thoroughly investigated.…”