Modified Algebraic Riccati Equation Closed-Form Stabilizing Solution

Alejandro J. Rojas

doi:10.1109/access.2021.3119592

Abstract

A modified discrete-time algebraic Riccati equation (MARE) is a discrete-time algebraic Riccati equation (DARE) for which the quadratic term is weighted by a modifying parameter $\alpha $ . The MARE is known to arise for example when conducting estimation or stabilization of single-input single-output (SISO) systems subject to packet losses in networked control systems (NCSs). In the present paper we characterize the solution to the MARE in closed form that, to the best of the author’s knowledge, is a completely novel result. We then verify the already known critical value of the modifying parameter $\alpha _{c}$ for which the MARE is solvable and propose closed-form expressions for the optimal state feedback gain matrix. We finally present examples to illustrate the obtained contributions.

Highlights

Riccati equations are a recurrent and important feature in many theoretical control design results and have been the subject of study for a long time, including [1] and more recently [2]
The motivation to study the modified algebraic Riccati equation (MARE) and its solution in this work is because it has been observed that the MARE plays a similar key role in networked control systems (NCSs) problems, as the discrete-time algebraic Riccati equation (DARE) does in classic control
The remainder of this paper is organized as follows: In Section II, we introduce the standing assumptions for the present paper and discuss the closed-loop pole locations induced by the closed-form solution to the MARE

Summary

INTRODUCTION

Riccati equations are a recurrent and important feature in many theoretical control design results and have been the subject of study for a long time, including [1] and more recently [2]. The motivation to study the MARE and its solution in this work is because it has been observed that the MARE plays a similar key role in NCS problems, as the discrete-time algebraic Riccati equation (DARE) does in classic control. In order to introduce a Linear Matrix Inequality (LMI) approach, the authors of [34] further relax the identified ARE constraint from an equality to an inequality, which is potentially a second source of approximation on the resulting MARE solution. These are some drawbacks inherent to a numerical approach, such as the LMI approach.

ASSUMPTIONS

Findings

CONCLUSION