Journal of Control Science and Engineering

Volume 2015 (2015), Article ID 689767, 5 pages

http://dx.doi.org/10.1155/2015/689767

## Design of Internal Model Control Based on an Optimal Control for a Servo System

Meiji University School of Science and Technology, 1-1-1, Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan

Received 16 August 2014; Revised 10 February 2015; Accepted 10 February 2015

Academic Editor: Ai-Guo Wu

Copyright © 2015 Hiromitsu Ogawa 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 describes a design of internal model control based on an optimal control for a servo system. The control system has the feedback based on the proposed disturbance compensator in the disturbance response. The compensator is designed to become the denominator of the transfer function without a dead time in the disturbance responses. The disturbance response of the proposed method is faster than that of the previous method.

#### 1. Introduction

A PID control is one of the attractive controls for the designers and the researchers because the PID controller is simple and easy to understand sensuously. It is widely studied in the control study and used in the industrial area widely [1–5]. Although the PID control provides the benefits as described above, the parameters tuning is difficult especially for the plant with the dead time. To overcome it, Smith predictor and internal model control (IMC) have been proposed and have been studied widely to control a plant with the dead time [6–11]. One of the key features is to be able to design each parameter easily compared with the PID control for the plant with the dead time. IMC especially possess many advantages over the PID control, particularly in the presence of the significant process dead time. IMC simplify the control system design in a large class of industrial applications for the disturbance response. One of the simple control system design approaches utilizes an optimal control theory [12]. It is not necessary to design some controller parameters also for experienced designers and there is no steady-state error in the disturbance response. However, the parameters of its disturbance compensator are designed by trial and error in consideration of dead time. It is one of the serious issues. This is because there are various plants with the dead time in the industrial areas, such as a DC motors control, a pneumatic actuators control, and a temperature control. Generally, these plants have large dead time. In especially the DC motor control area when the system has some gears, the system has large dead time.

In this paper, we propose the design of IMC based on an optimal control for a servo system. The proposed disturbance compensator is designed to become the denominator of the closed loop transfer function without the dead time in the disturbance responses. As a result, the design of the controller is easy and its stability is improved. For this proposed method, it is not necessary to design the parameters of its disturbance compensator by trial and error like the previous method [12]. We confirm that the disturbance response of the proposed method is improved as compared with the previous method in the simulation result and the actual DC motor control result.

#### 2. Previous IMC Design Method

The design method of IMC is a superior control method for the plants with the dead time. This previous method is one of the approaches to improve the steady-state error in the disturbance response [12]. Let us consider the following second-order plant with the dead time: where is the dead time. The following equation is discretized by the zero-order hold: where is the dead time and is a nominal plant. Then the relation between the plant output and the disturbance in the previous method is given as follows: where is the plant output, is the disturbance at the plant input, is a plant model, is the estimated dead time, and is obtained as follows: Here, to design it simple, the controller is defined as follows: The is a PID controller as Here, the parameters , , and are designed by an optimal control method in this paper. Then, let us consider the following controllable and observable plant model; where is the state variable, is the plant output, and is the plant input. Then, to design the parameters , , and easily, let us consider the following augmented plant: where , , and are described as follows: The performance index is used in practice for quadratic optimal control formulation as follows: where is a positive semidefinite real symmetric matrix and is a positive definite real symmetric matrix. The first term accounts for the relative importance of the state of the plant and the second term accounts for the expenditure of the energy of the control signal. Then, the following Riccati equation is solved to determine the such that this performance index is minimized: Then, the is obtained as follows: The represents the input as follows: Then, the plant input is as follows: The optimal feedback gains and are obtained as PID parameters of . Then, the controller is designed by using these PID parameters. Then, the disturbance response is expressed as follows: To consider it easily, we assume that Then the following disturbance response is obtained as follows: In this previous method [12], the is a PID controller as follows: These parameters, , , and , are designed on a trial basis.

#### 3. Proposed IMC Design Method

Although the previous method does not have the steady-state error in the disturbance response at the plant input, the denominator of the disturbance response contains the dead time. In addition, is designed by trial and error. However, it is difficult to design . To overcome this problem, we propose the design of the disturbance compensator as follows: Then let us substitute (19) into (18) as follows:Since the denominator of the disturbance response does not contain the dead time, the design of is easier than the previous method. The proposed method can control the plant as shown in Figure 1.