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The preview control problem of a class of linear discrete-time descriptor systems is studied. Firstly, the descriptor system is decomposed into a normal system and an algebraic equation by the method of the constrained equivalent transformation. Secondly, by applying the first-order forward difference operator to the state equation, combined with the error equation, the error system is obtained. The tracking problem is transformed into the optimal preview control problem of the error system. Finally, the optimal controller of the error system is obtained by using the related results and the optimal preview controller of the original system is gained. In this paper, we propose a numerical simulation method for descriptor systems. The method does not depend on the restricted equivalent transformation.

Preview control theory takes full advantage of future known reference signals or disturbance signals information to improve the dynamic response, to inhibit the disturbance, and to increase the tracking performance of the systems. The traditional method is to construct an auxiliary system (called error system) by combining the error equation and the difference equation. As a result, the tracking problem can be transformed into a regulation problem. By using known results [

Descriptor systems, also known as singular systems, are a class of dynamic systems. It contains not only the normal differential equations, but also algebraic equations. The study of descriptor system theory begins in the 1970s. After more than 40 years’ development, it has gradually formed a complete theoretical system and method [

The research of preview control theory for descriptor systems begins in 2012. In [

In this paper, the optimal preview control problem for discrete-time descriptor systems, with both reference signal and disturbance signal known, is studied. First, the system is decoupled into a normal equation and an algebraic equation. Then, by applying the first-order forward difference operator to the normal equation, a difference equation is obtained. Thus, the error system is constructed by combing the difference equation and the error equation. Finally, the optimal regulator for the error system is obtained, and as a result, the optimal preview controller for the original descriptor system is also gained. However, because there is a singular matrix in the original system, the closed-loop system cannot be directly simulated. The previous simulation work was figured out by the system after the limited equivalent transformation, rather than by the original system. Therefore, the main contribution in this paper is to design a more general simulation method for descriptor systems.

Consider the discrete-time descriptor system:

In this paper, only causal systems are discussed. First, it is assumed that system (

Assume that

Assume that

Assume that the preview length of the reference signal

Assume that the preview length of the disturbance signal

The error vector is defined as follows:

There are two benefits when introducing

In order to make full use of the conclusions of optimal preview theory in normal system, system (

The transformation above is called the limited equivalent transformation [

The so-called “system (

Since

Due to system (

Since the limited equivalent transformation does not change the dynamic characteristics, we only need to design a preview controller for (

Taking the first-order forward difference operator

Since the reference signal

Expressing the performance index (

Assume that the following conditions are satisfied:

According to Lemma

Firstly, the conditions that ensure that

Because the limited equivalence transformation keeps the stabilizability of the system [

Then, according to the PBH criterion,

Matrix

Since

Since

According to (

Secondly, the conditions that ensure that

Because the limited equivalence transformation keeps the detectability of a system unchanged [

According to the PBH criterion,

Then, we can get the following theorem.

If Assumptions

If Assumptions

Since

Because

This completes the proof.

By observing (

In the following, we need to deal with the numerical simulation problem with the state equation in system (

A new method needs to be designed to solve this problem.

Firstly, an appropriate matrix

Secondly, adding the identical equation

In this way, the simulation can be carried out.

It is obvious that the above iterative method is equal to adding the term

Meanwhile, if the iterative method (

Thirdly, the convergent condition for the iterative method (

Since

In conclusion, if there exists an appropriate matrix

The conclusions of this section can be applied to the numerical simulation of all discrete-time descriptor systems.

Consider system (

Let the initial state vector be

Let

The output response to step function with preview.

It can be seen from Figure

In this paper, the optimal preview controller for linear discrete-time descriptor systems is designed. Firstly, the descriptor system is transformed into a normal system by introducing the limited equivalent transformation. Then, by using a difference operator, the error system is constructed. The optimal preview controller is obtained according to the known conclusions of preview control theory. At the same time, the existence of the optimal preview controller with the basic assumptions is also proved strictly. More importantly, we solved the simulation problem and the simulation method is very effective.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (Grant no. 61174209).

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