A New Method for Determining the Degree of Controllability of State Variables for the LQR Problem Using the Duality Theorem

Lifei Zhang,Konstantin Avenirovich Neusypin,Maria Sergeevna Selezneva

doi:10.3390/app10155234

Lifei Zhang, Konstantin Avenirovich Neusypin + Show 1 more

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https://doi.org/10.3390/app10155234

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Journal: Applied sciences	Publication Date: Jul 29, 2020
Citations: 4	License type: CC BY 4.0

Affiliation: Bauman Moscow State Technical University

Abstract

The control performance of a dynamic system can be checked by the degree of controllability. In this work, we present a new method for determining the degree of observability of state variables for the linear quadratic optimal estimation (LQE) problem. We carried out the calculation of the degree of controllability for the linear quadratic optimal control (LQR) problem using a duality theorem. Compared with the traditional measures of controllability such as determinant, trace, and maximal eigenvalue of the inverse controllability Gramian, the proposed degree of controllability was developed for each state variable and takes into account both the controllability Gramian and the cost function. The new method is convenient to apply to LQR problem. In the numerical simulation, we determined the influence of the model parameters on the degree of controllability. Besides that, we analyzed the degree of controllability, which gives an insight into the relationship between the system model design and the control performance.

Highlights

In this work, we present a new method for determining the degree of observability of state variables for the linear quadratic optimal estimation (LQE) problem
We proposed a new method for determining the degree of controllability of specific state variables using the duality theorem
We studied the value of the degree of controllability and the control performance of the system under different system model parameters

Summary

Introduction

We present a new method for determining the degree of observability of state variables for the linear quadratic optimal estimation (LQE) problem. “Degree of controllability” is a scalar measure of controllability to check how controllable a given system is [1,2,3]. It reflects the efficiency of the control effort. The most widely-known definition is related to the minimum control input energy required to transfer any initial state x0 to the origin [1,2,3,4,7]. If a system requires smaller input energy than others to regulate the system, it can be considered more controllable [3]

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