#### Abstract

This paper explicates a new method of designing a preview controller for a class of continuous-time systems. The augmented error system is constructed by the error system with the derivative of the tracking error signal, the state equation, and an identical equation of the derivative of the control input, which transforms a tracking problem into a regulation problem. Therefore, in the paper, the performance index contains the derivative of the control input. Based on the theory of optimal control, the regulator problem of the augmented error system is solved. Thus, the controller with preview compensation for the original system is deduced. The response speed of the closed-loop system is accelerated by the previewed demand output. A final numerical example is given to illustrate the validity of the proposed method.

#### 1. Introduction

The ability to take the known future demand output or disturbance signal into consideration to design a controller with preview compensation is the greatest distinction of the preview control methods [1–3]. It has been proved that better output tracking can be obtained and the performance of the closed-loop system can be improved when the preview control method is applied; this has been demonstrated in many laboratory simulations such as electromechanical valve actuators in [4], networked control systems in [5], and vehicle active suspension systems in [6]. Recently, preview control has drawn greater attention [2, 7]. It has been integrated with other control theories to propose some new conceptions such as preview control in [8, 9], FI (full-information) preview control in [9], and stochastic optimal preview control in [10, 11].

In the preview control method, the most important thing is that the known future demand output or disturbance signal affects the operation of the control system. As a typical method, the augmented error system plays a vital role in the design of the controller in discrete-time systems. Since the relationship between the system and the future signal was established by the difference operator, discrete-time systems have become very popular in many research fields [12, 13].

In continuous-time systems, differential operation is used instead of difference operation. Therefore, it is difficult to construct augmented error systems. In [14], reasonable assumptions about the known future signals are made. Derivatives are taken of both the tracking error signal and the state equation. Thus, the tracking error signal and the derivative of the state vector are put together as the state vector of the augmented error system. In [14], the optimal preview control problem of continuous-time constant systems is solved. In this paper, the augmented error system is constructed by the error system with the derivative of the tracking error signal, the state equation, and an identical equation of the derivative of the control input. Thus, the augmented state vector contains the tracking error signal, the state vector, and the control input. Based on optimal control theory, the optimal controller of the augmented error system is derived. A differential equation relevant to the control input of the original system is solved, and an optimal preview controller is deduced.

This paper is organized as follows. In Section 2, the continuous-time system, desired output, performance index, and basic assumptions needed for the proof of the main results are given. In Section 3, the augmented error system is constructed, in which the control input is taken as a part of the augmented state vector. This is the greatest distinction from [14]. Then, an optimal controller for the augmented error system is deduced, based on the theory of optimal control. Furthermore, the optimal preview controller for the original system is obtained through integration. In Section 4, a numerical example is presented to illustrate the effectiveness of the proposed method. The paper ends with conclusions and cited references.

#### 2. Problem Statement

We consider a class of linear continuous-time systems described by where , , and are, respectively, the state vector, control input, and output vector. , , and are all constant matrices. The positive real number denotes the initial time and represents the terminal time. The vector is the initial state of system (1). When , both the state vector and the controlled input are zero.

Let be the desired output or the demand vector. The tracking error signal is defined as the subtraction of the output vector and the desired output ; that is,

Concerning the desired output , the following assumptions will be needed throughout the paper.

*Assumption 1. *The desired output is piecewise differentiable. At some nondifferential points , we take the left derivative or the right derivative instead of .

*Assumption 2. *The desired output is previewable in the sense that the future value of () is available at each instant of time . The value of is presented as the preview length of the desired output in [14].

*Remark 3. *In Assumption 1, is in low requirement; hence our method can be used as widely as possible for more desired output. In fact, if is continuously differentiable, the most commonly used step signal will be excluded.

The basic design problem considered in this paper is to find a controller such that the output vector can track the desired output without static error. Therefore, we wish to obtain the optimal controller with preview compensation such that the performance index
is minimized, where and are both positive definite matrices and and are nonnegative definite matrices.

*Remark 4. *It is well known in [15] that introducing the derivative of the control input into the quadratic performance index can allow the controller to contain integration terms, which may help the system reduce static errors. So is contained in here. Additionally, it is realistic to bring into the quadratic performance index, which can limit .

We wish to design an optimal preview controller for system (1) under the performance index (3).

#### 3. Design of the Optimal Preview Controller

In order to obtain the optimal preview controller for system (1), we construct an augmented error system that includes the future information on the desired output as well as the tracking error signal , the state vector , and the controlled input .

Since the desired output is contained in the tracking error signal defined by (2), differentiate (2) and we can get Combining (4) with the first formula of system (1) yields

Equation (5) is the error system of (1). Due to the derivative of the control input being contained in performance index (3), we consider taking as the control input of the augmented error system. Now define the augmented state vector Putting (5) and together gives The corresponding output equation is where

Noting the tracking error signal of system (1) as being a part of the augmented state vector in (7), performance index (3) can be expressed as where

Therefore, the optimal controller can be derived by solving the control problem that minimizes the performance index of (10),subject to the dynamic constraint of (7).

Equation (7) is the augmented error system that we need. It is obvious that the tracking error signal is a part of the augmented state vector in (7). So it is natural to conclude that the output vector of (1) can track the desired output with less static error if the controller of (7) makes the state vector stable to zero.

*Remark 5. *Concerning the construction of the augmented error system, the innovative features of this paper mainly lie in the following two aspects, which are also the primary differences from [14]. First, we only take the derivative of the tracking error signal , whereas [14] took the derivatives of both the tracking error signal and the two sides of the state equation. Second, the control input is put into the augmented state vector. And then and are both introduced into the performance index to restrict themselves within an allowable range. The method described in this paper can also ensure that the closed-loop system of (1) contains integration terms to reduce any static errors.

Next, the optimal preview controller of system (1) will be deduced by solving the controlled input of (7) under performance index (10).

For system (7) and performance index (10), the Hamilton function is chosen as
According to the theory of optimal control [16], if we want to enforce optimal control on system (7), the Hamilton function must satisfy the canonical equations:
that is,
and the boundary conditions

As the constraint condition, Equation (7) needs to be met for . Combining (14a) and (14b) with (7) holds as follows:
Eliminating from the first three equations of (15), we can get the following differential equations relevant to and :

Then from the above differential equations and the conclusions in [14], we can apprehend that the relationship between and can be expressed as
where the matrix and the vector function , respectively, satisfy the following Riccati equation and linear differential equation:
with the boundary conditions , .

Substituting (17) into the third formula of (15) obtains Theorem 6.

Theorem 6. *Under performance index (10), the optimal control input of augmented error system (7) is
**
where is the nonnegative definite solution of the Riccati equation:
**
and is the solution of the following time-varying differential equation:
*

*Remark 7. *Theoretically, we can obtain in (21) after obtaining the nonnegative definite solution of (20). Then substituting into (19), we can deduce the expression of . So, through integration, the optimal control input with preview compensation of system (1) can be derived easily.

*Remark 8. *Practically, all variables mentioned in Remark 7 can be given in the form of numerical solutions. Consequently, the control law in Theorem 6 can be realized. But we still hope to get further results.

First, we solve from (21). Let . Equation (21) can be written as follows:

Suppose is a fundamental matrix solution of the homogeneous equation ; then based on the theory of differential equations [16], the solution of (22) is given by the formula
Thus the following formula holds:
Using , we get

Considering the characteristics of the desired output and , we realize that can be given as

Further, we want to deduce the relationship among the derivative of the controlled input , the tracking error signal , the state vector , and the desired output of system (1). In view of being nonnegative definite matrices, partitioning gives
where , , and . Then, based on the structure of and , substituting (26) and (27) into (19) gives
that is,

Let ; we have . So the above formula can be rewritten as

Noticing , , and when , so we have
Therefore, the subsequent theorem to describe the controlled input of continuous-time system (1) follows immediately.

Theorem 9. *Under performance index (3), the optimal preview controller of continuous-time system (1) is
**
where , () is shown as (27), is the nonnegative definite solution of the Riccati equation (20) [14], and is the fundamental matrix solution of the homogeneous equation of Equation (22).*

As in Theorem 9, it is easy to find that the controller of continuous-time system (1) is the combination of integral actions related to the tracking error signal, the state vector, and the desired output. Compared with [14], the controller described in this paper does not contain the state feedback but the integral of the state vector.

#### 4. Numerical Example

*Example 1. *Consider a continuous-time system with single input and single output described by
The coefficient matrices of system (1) are, respectively,
The weight matrices in performance index (3) are
The solution of Riccati equation
is
And the matrix of is

Let the desired output be When there is no preview compensation, that is, in the controller, the output response of system (33) is shown in Figure 1 through Matlab simulation. When the preview length is chosen as , the corresponding output responses of system (33) are shown, respectively, in Figure 2. Comparing Figures 1 and 2, we find that preview control for continuous-time systems can decrease the static error.

#### 5. Conclusions

Based on the desired output being piecewise differentiable, we take only the derivative of the tracking error equation, the state equation, and an identical equation of the derivative of the control input as the new state equation of the augmented error system. Then, according to optimal control theory, a differential equation relevant to the control input is given. So the optimal preview controller for the original continuous-time system is deduced. The controller in this paper does not contain the state feedback, but rather, the desired output and the integrals of the tracking error signal and the state vector. The preview controller can also reduce static errors effectively.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 61174209) and the Oriented Award Foundation for Science and Technological Innovation, Inner Mongolia Autonomous Region, China (no. 2012).