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This paper presents a method for designing a type one servomechanism for a discrete-time linear system with input delay subject to a previewable desired output and a nonmeasurable constant disturbance. The tracking problem of a delay system is transformed into a regulation problem of a delay-free system via constructing an augmented error system and a variable substitution. A controller is obtained with delay compensation and preview compensation based on preview control theory and the predictor method. When the state vector is not directly measurable, a full-dimensional observer is offered. The effectiveness of the design method is demonstrated by numerical simulations.

Preview control is one of the approaches available for producing a good performance by utilizing future information of the reference signal in the controller. Lots of work on the preview control problem have already been done. One of the early contributions was delineated in [

During the past several decades, there have been very rich research achievements in the area of control systems with time delay. An early control method for systems with time delay is the Smith predictor [

For discrete-time linear systems, if the input vector has a time delay, then it is necessary to reconsider the design of the preview controller. The input delay system was transformed into a delay-free system by using the discrete lifting technique in [

This paper is organized as follows. An introduction is given in Section

Consider a discrete-time system with input delay as follows:

Let

First, we give the following two basic assumptions:

Let the pairs

Let the reference signal

Furthermore, let

The purpose of this paper is to design a controller with preview compensation such that the output

The optimal control method is applied to achieve the goal. The performance index of (

Notice that, as used in [

The basic method of designing a preview controller for (

Using

Using

Combining (

Equation (

Correspondingly, in terms of the augmented state vector

If a controller

Let us introduce a new input vector

Obviously, (

If (A1) and (A2) hold and

The future reference signal value

The preview controller for the augmented error system (

Combining (

In (

The future value

Substituting (

Let us derive a preview controller of (

It is assumed that the initial values of system (

Thus, a preview control theory for system (

Let (A1) and (A2) hold, let

At each time

It is easy to see that the preview controller (

According to the character of the reference signal considered here, the abovementioned design method for the preview controller is applicable to some irregular reference signals which cannot be modeled by the dynamic system’s outputs.

Now, let us apply present theory to an air slider linear motor.

The dynamic equation of the motor is described as follows:

Let

Taking sampling period

The reference signal

Let the preview length of the reference signal be

The output responses to the step signal with different delays.

The output responses to the fading signal with different delays.

The output responses of the closed-loop system tracking to the step signal are shown in Figure

When the preview length is equal to the input delay, the future value of the reference signal is only used to compensate the delay. In order to get a better performance, let us take a larger preview length. When the input delay

The output responses to the step signal with different preview lengths.

The output responses to the fading signal with different preview lengths.

It can be seen from Figures

If the state vector

Since

Let

Consider delay system (

According to the conclusions in [

Since the matrix

From Lemma

If (A1) and (A2) hold,

Consider the control system in Example

The output responses with state observer to the step signal.

The output responses with state observer to the fading signal.

Compared with the responses in Figures

In this paper, a class of preview controller problems of discrete-time linear systems with input delay has been solved. A preview controller with delay compensation and preview compensation is derived. The difficulties of input delay are successfully overcome by predictor feedback. A design method of the full state observer is given when the state vector of the system cannot be measured directly. Numerical results show that the present methods are effective.

Further studies are needed on selecting a proper preview length according to the characteristics of the system, which is a complex problem but a significant one.

The authors declare that they have no competing interests.

The authors would like to thank the National Natural Science Foundation of China for their financial support under Grant no. 61174209.