Complexity

Volume 2018, Article ID 4942906, 10 pages

https://doi.org/10.1155/2018/4942906

## H_{∞} Optimal Performance Design of an Unstable Plant under Bode Integral Constraint

^{1}School of Control Engineering, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China^{2}College of Electrical Engineering, Guizhou University, Guiyang 550025, China

Correspondence should be addressed to Aiping Pang; moc.qq@887425714

Received 6 April 2018; Accepted 11 June 2018; Published 9 August 2018

Academic Editor: Jing Na

Copyright © 2018 Fanwei Meng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper proposed the H_{∞} state feedback and H_{∞} output feedback design methods for unstable plants, which improved the original H_{∞} state feedback and H_{∞} output feedback. For the H_{∞} state feedback design of unstable plants, it presents the complete robustness constraint which is based on solving Riccati equation and Bode integral. For the H_{∞} output feedback design of unstable plants, the medium-frequency band should be considered in particular. Besides, this paper presents the method to select weight function or coefficients in the H_{∞} design, which employs Bode integral to optimize the H_{∞} design. It takes a magnetic levitation system as an example. The simulation results demonstrate that the optimal performance of perturbation suppression is obtained with the design of robustness constraint. The presented method is of benefit to the general H_{∞} design.

#### 1. Introduction

Some constraints are often ignored in theory design so that the designed system could not been achieved [1]. The unstable poles should be considered in the design of unstable plants, which will have an impact on the system running [2]. For example, the performance index of fight aircraft is with a phase margin of 45° and however, the phase margin is 35° at last after lots of money is poured [3]. There is another type of unstable plants such as a magnetic levitation system, which has been built in some universities at home and abroad. But these systems could not run, and there exists a large peak in the data of sensitivity function [4]. And after X-29, the unstable poles are considered under the research on the Fight Aircraft JAS-39 and X-30, which succeeds. This paper solves the problem that it is how to make the control system obtained the optimal disturbance suppression.

There are two types of H_{∞} design, which are cycle formation based on coprime factorization, H_{∞} state feedback and H_{∞} output feedback such as DGKF [5, 6]. Cycle formation applies to control design of the flexible system such as instances in [7]. This paper proposes H_{∞} state feedback and H_{∞} output feedback design together with the magnetic levitation system, which is applicable to unstable plants.

The key to achieve the H_{∞} control design is up to the weight function. The weight function is considered particularly for an unstable plant in H_{∞} control design. There are two different types of unstable plants. The first type is that the frequency band of mathematical model is 10 times larger than unstable mode, for example, in designing autopilot, the unstable mode is less than 1 rad/sec but the bandwidth is larger than 40 rad/sec [2, 8]. These systems will utilize the common H_{∞} design in general. The second type is that the unstable mode and the bandwidth of the closed-loop system are approximate, for example, in magnetic levitation systems in [9, 10], the unstable mode is 60~70 rad/sec and the bandwidth is 100 rad/sec. For the latter, there is obvious feature and it is to be considered in particular when using H_{∞} control design. This paper mainly discusses the second type of unstable plants, and the analysis results will benefit the explanation of the design of the first type.

#### 2. Control Problem of Unstable Plants

In terms of control theory, there may be instability in control design for an unstable plant. Feedback characteristics must be considered in the design of the feedback system. Feedback systems have some performance such as robustness, sensitivity, and disturbance rejection, which can be changed only by feedback. The low sensitivity and disturbance rejection are the reasons why a system needs feedback control, but the robustness is essential performance in the feedback system. Therefore, the purpose of feedback control system design is to achieve low sensitivity and disturbance rejection.

##### 2.1. Control System Performance Description

It is known that the sensitivity function describes the performance of the control system. The schematic of the feedback control system is shown in Figure 1, where is the controller and is the controlled plant. Then, its transfer function is given by