Design of a Preview Controller for Discrete-Time Systems Based on LMI

Li, Li; Liao, Fucheng

doi:https://doi.org/10.1155/2015/179126

Mathematical Problems in Engineering

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Abstract Introduction Numerical Example Conclusion References Copyright Related Articles

Research Article | Open Access

Volume 2015 | Article ID 179126 | https://doi.org/10.1155/2015/179126

Design of a Preview Controller for Discrete-Time Systems Based on LMI

Li Li¹and Fucheng Liao¹

Academic Editor: Luis J. Yebra

Received26 Sept 2015

Revised02 Dec 2015

Accepted10 Dec 2015

Published29 Dec 2015

Abstract

A preview controller design method for discrete-time systems based on LMI is proposed. First, we use the difference between a system state and its steady-state value, instead of the usual difference between system states, to transform the tracking problem into a regulator problem. Then, based on the Lyapunov stability theory and linear matrix inequality (LMI) approach, the preview controller ensuring asymptotic stability of the closed-loop system for the derived augmented error system is found. And an extended functional observer is designed in this paper which can achieve disturbance attenuation in the estimation process; as a result, the state of the system can be reconstructed rapidly and accurately. The controller gain matrix is obtained by solving an LMI problem. By incorporating the controller obtained into the original system, we obtain the preview controller of the system under consideration. To make sure that the output tracks the reference signal without steady-state error, an integrator is introduced. The numerical simulation example also illustrates the effectiveness of the results in the paper.

1. Introduction

The research problem of preview control theory is as follows: when the reference signal or exogenous disturbance is known or can be previewable, how can we take full advantage of the known future reference signal or disturbance signal to improve the tracking performance of a closed-loop system? In many cases, the future target signal or external disturbance for servo systems is known. For example, in vehicle active suspension [1, 2], wind turbines [3], the robot system [4], and so on, their destination paths are specified in advance. We naturally seek to ascertain how to take full advantage of the known future information to improve the control performance or tracking performances. When considering these applications, preview control theory mentioned here will be used.

Preview control has been around since the 1960s and has been the subject of extensive, in-depth research. Reference [5] discussed the optimal preview control problem for multirate discrete-time systems subject to polynomial previewable desired output and polynomial previewable disturbance signals. Reference [6] studied the optimal preview control for a class of time-varying discrete systems, ingeniously overcoming the difficulty that the differential operator is nonlinear. Using some techniques, [7] combined preview control theory with linear discrete-time descriptor systems to give the preview controller design method. Using and criteria, [8–12] converted the tracking problems of the original systems into standard and control problems by using certain techniques. And the design of preview controllers was given using existing theories. References [13, 14] presented the methods to design preview controllers based on Min-Max control in game theory.

Linear matrix inequality (LMI) is an effective method for robust control theory and has been applied to all aspects of systems and control [15]. The preview control problem for a class of discrete-time systems is presented in this paper. Combining the method of preview control theory with the LMI technique, a preview controller design is proposed. The proposed controller guarantees that the output of the closed-loop system can accurately track the reference signal. This paper adopts the method in [16] to construct an augmented error system, and the sufficient condition of asymptotic stability of the closed-loop system is derived using the second method of Lyapunov. References [17–19] considered the preview control for a polytopic uncertain system by LMI. However, they still adopted the difference method in preview control theory to construct an augmented error system; as a result, this constructive method cannot be extended to general uncertain systems. The paper uses the state transformation method instead of the classical difference method to construct the augmented error system and then considers the preview control problem by combining the second method of Lyapunov with the LMI technique. The benefit is that the results in this paper can easily be extended to uncertain systems.

Notations. and denote the -dimensional real vector space and matrix space, respectively. () denotes the notion that matrix is positive definite (positive semidefinite). () denotes (). is stable; that is, the absolute values of the eigenvalues of are strictly less than 1; denotes the identity matrix, and its dimension can be known from the context of the narrative.

2. Problem Formulation and Basic Assumptions

Consider the discrete-time systemwhere is the state vector, is the control input vector, is the controlled output vector, is the disturbance vector and , and , , , and are known real constant matrices with appropriate dimensions.

Let be the reference signal and define the error signal as

In this paper, our objective is to design a preview controller in such a manner that the output of the closed-loop system tracks the reference signal even in the presence of exogenous disturbances; that is,

Basic assumptions are as follows.

(A1) (full-row rank).

(A1) is the standard assumption of the servomechanism design problems. It can imply that the system has no transmission zeros at .

We assume that the reference signal and the exogenous disturbance are previewable, as more specifically described below.

(A2) Assume the preview length of the reference signal is ; that is, at each time , future values of as well as the present and past values of the reference signal are available. The future values of the reference signal beyond are constant vector :

We assume that the reference signal converges to constant vector as time goes to infinity. That is,

(A3) Assume the preview length of the exogenous disturbance is ; that is, at each time , future values of as well as the present and past values of the exogenous disturbance are available. The future values of the exogenous disturbance beyond are zero:

Remark 1. (A2) and (A3) are standard assumptions for preview control theory. The latter parts of (A2) and (A3) are based on the fact that, according to the characteristic of the control system itself, the previewable signals have a significant effect on the performance of the control system only for a certain time period; therefore, we can ignore the reference signal and the exogenous disturbance when they exceed the preview length. Here, we assume that the reference signal and the exogenous disturbance when they exceed the preview length are any constants. In fact, a regular feedback control system does not take full advantage of the known future reference signal and the exogenous disturbance, or equivalently, the preview length is zero.

The Schur complement lemma will be used in this paper.

Lemma 2 (see [20]). Consider matrix , where and are symmetric matrices and invertible; then the following three conditions are equivalent:(i).(ii), .(iii), .To save space, we directly give the following lemma for stability analysis of the system by Lyapunov stability theory.

Lemma 3 (see [21]). The system is asymptotically stable if and only if there is a positive definite matrix such that

3. Derivation of the Augmented Error System

We derive an augmented error system to transform the tracking problem into a regulator problem and consider the LMI technique to design a preview controller.

If the controlled output of the closed-loop system for system (1) can track the reference signal , there exist constant vectors and satisfying the equation of system (1). On both sides of the system equation and observation equation, letting go to infinity, we obtainThat is,

By (A1), the coefficient matrix of the equation with and and the augmented matrix have the same rank; thus, at least one solution exists. Without loss of generality, we take a fixed solution which is still represented by and .

It should be noted that (A1) implies ; that is, the number of control variables must be greater than or equal to that of the output variables to be controlled. This is quite common in practical control problems. And when , and are existing and unique.

Define new vectorsCombining (1) and (9) givesAnd from (2), we obtain

According to (A2), satisfies the following properties: at each time , , , are known and

The derivation of the augmented error system is given below.

First of all, combining (11) and (12) gives

Then, constructing vectorsand matricesit follows from (A2) and (A3) that we can obtain the equations as follows:where , , , and . By (11), (14), and (17), we obtain the formal systemwhere

Note that the main characteristics of the formal system (18) are that both and with future information are a part of the state vector. Thus, system (18) contains the future information of the reference signal and disturbance signal. It should be pointed out that replaces the difference of in the formal system (18); as a result, the controller obtained of system (18) by the LMI approach does not include the integral of error ; therefore, the final closed-loop system does not contain an integrator that helps to eliminate the static error. For this, we adopt the method in [22] to introduce the discrete integrator defined byNamely, where can be assigned as needed.

Define again, and combining (18) and (20), we obtainwhere

System (22) is the derived augmented error system. Because is a part of the vector , if we can design a state feedbackto make the closed-loop system of system (22) asymptotically stable and we have , then one gets . Thus, the state feedback is applied to system (1) and the output of the corresponding closed-loop system tracks the reference signal without steady-state error.

4. Design of a Preview Controller

As mentioned above, the purpose is accomplished as long as the state feedback obtained of system (22) is in the form as (24) and it makes the closed-loop system of system (22) asymptotically stable. We will give the controller gain matrix in (24) by using the related theory and LMI methods.

Note that when the control input is determined by (24), we see that the closed-loop system of system (22) is given by

Below is the first important theorem in this paper.

Theorem 4. Suppose that (A1)–(A3) are satisfied. If there exist a positive definite symmetric matrix and a matrix such thatthen the closed-loop system (25) is asymptotically stable, where the state feedback gain matrix , and the control law

Proof. From Lemma 3, the closed-loop system (25) is asymptotically stable if and only if there is a positive definite matrix such that Applying the Schur complement and , (28) is equivalent toWe perform the congruent transformation on the left side of (29): premultiplying an invertible symmetric matrix and meanwhile postmultiplying the transposition of this matrix (namely, itself) and then denoting and . It follows that we can obtain the condition of Theorem 4, namely, the matrix of the left side of (26). Since the congruent transformation does not change the positive definiteness, (26) is satisfied if and only if (29) is satisfied. From the above reasoning process, we obtain that (26) and (28) are equivalent. By Lemma 3, Theorem 4 holds.
Synthesis and derivations on Theorem 4 show that if LMI (26) has a feasible solution, there exists state feedback for system (22) such that is a stability matrix. In other words, the pair is stabilizable [23]. In the following, the PBH rank test [24] will be employed to prove the stabilizability of the augmented error system (22).

Lemma 5. The pair is stabilizable if and only if is stabilizable and has full-row rank.

Proof. By the PBH criteria [24], the pair is stabilizable if and only if the matrixhas full-row rank for any complex satisfying .
Note that it follows from the structure of and that both and are nonsingular for any satisfying . From the expression of and , we haveElementary matrix operations do not change the rank of the matrix and is nonsingular for any complex satisfying and . By the elementary transformation of the matrix, we haveThus, the matrix is of full-row rank if and only if is of full-row rank.
When , we have from the elementary transformation of the matrixThus, the matrix is of full-row rank if and only if is of full-row rank.
Lemma 5 holds.

Remark 6. If the condition of Theorem 4 is established, the pair is stabilizable and has full-row rank. And the stabilizability of and guarantee that system (1) is stabilizable via state feedback and has no zeros at , and then these are necessary for the existence of a robustly stabilizing controller for the augmented error system (22). Therefore, the LMI approach is applied to preview control theory and the determination of a preview controller can be converted into a convex optimization problem in terms of linear matrix inequality; as a result, the design of the preview controller becomes a simple matter and some restricted conditions in [24] will be reduced.
We consider the control input of system (1).
When (A1)–(A3) are satisfied, the control input (24) of system (22) is obtained. We decompose the gain matrix then, (24) can be written asSubstituting , , and into the above equation, the following is obtained.

Theorem 7. Suppose that (A1)–(A3) are satisfied. The controller of system (1) can be taken aswhere , , and can be determined by (26). Equation (34) determines the relationship between , and . Under the controller, the output of the closed-loop system of system (1) tracks the reference signal accurately.
We can see from the above equation that the preview controller of system (1) consists of six terms. The first term represents tracking error compensation, the second term represents the state feedback, the third term represents the feedforward or preview action based on the future information of the reference signal, the fourth term represents the feedforward or preview action based on the future information of the exogenous disturbance, the fifth term represents the integral action of the tracking error, and the sixth term represents the compensation by the initial and final values.

5. Design of an Observer-Based Controller

In practical application, the system state information of system (1) often cannot be obtained; thereby, the system state information of system (11) cannot be obtained either. Thus, we need to build an observer which reconstructs the state variables. If we use the method in [23] to design a state observer for system (1), thenLet the estimation error and use (1) and (37) as follows:Obviously, the estimation error will be unavoidably affected by the disturbance ; thus both the rate of convergence for estimating the state variables and the interference rejection will be affected [25–27]. To overcome this disadvantage, we design the extended functional observer to estimate the state and the disturbance simultaneously. Now, for simplicity of presentation, let , , and ; we have Now consider the observation equation of system (1) and the predictability of the disturbance; for the augmented system, we now ought to take the observation equation aswhere . Hence, we obtainThen, the state and the disturbance estimation problem can be converted into a problem of designing a state observer for the augmented system (41). Now for system (41) we consider the state observer given bywhere is the observer state, is an auxiliary variable, and , , , and are constant matrices with appropriate dimensions that should be designed.

Note that (full-column rank). It follows from the verification that that is,Then, we denote , and then matrices and satisfy the equation

Theorem 8. For the augmented system (41), the estimation error converges asymptotically to zero if there exists a matrix such that is stable. Furthermore, matrices , , , and of the functional observer (42) can be determined as

Proof. Multiplying at both sides of the first formula for (41) yieldsand adding to both sides of (47) yieldsSince (45) is , (48) can be rewritten asLetting , system (49) can be rewritten asFor system (50), we consider a state observer as follows:To eliminate the term , we introduce an auxiliary variable , and the observer (51) turns intoComparing (42) with (52), parameter matrices of the functional observer (42) can be designed.
Subtracting (51) from (49) yields the following estimation error equation:Selecting a proper matrix makes be stable; then there must be . Consequently, can be reconstructed, and the estimates of state variables can be determined as .

Lemma 9. There exists the observer gain such that is stable if and only if the pair is detectable.

Proof. If there exists a matrix such that is stable, that is, is stable, or, equivalently, the pair is stabilizable, then we know the pair is detectable based on the duality principle. In the following, we will use PBH criteria to verify the condition of the detectability of the pair .
By the PBH criteria [24], the pair is detectable if and only if the matrixhas full-column rank for any complex satisfying . From the expression for and , we see thatThe pair is detectable if and only if is detectable. Lemma 9 holds.

Remark 10. First we construct the augmented system (39) and reformulate the output equation. Then, the extended functional observer (42) is proposed for the formal system (41). By (53), an appropriate is selected such that is stable [ is a given matrix], and the estimation error will converge asymptotically to zero.

Remark 11. is taken to be a quasi-diagonal matrix and then the coefficient matrix of system (53) is . System (53) can be rewritten aswhere and are the estimation errors of the system state and the disturbance, respectively. According to the estimation error equation (56), is taken to be stable. In fact, selecting can speed up the convergence rate for the disturbance estimation and reduce the interference for system state estimation. Thus, strong interference suppression can be realized. Furthermore, the issue of choosing making stable has already been solved in linear system theory and is not attempted here.
Under the functional observer (42), we can derive the closed-loop system as follows:

6. Numerical Example

In system (1), let , , , and . Through verification, satisfies full-row rank, and is detectable. Therefore, the system satisfies the basic assumptions.

We perform simulations for three situations; that is, the preview lengths of the tracking signal and the exogenous disturbance are , , , , and no preview (namely, ). According to Theorem 4, we use the LMI toolbox of MATLAB to determine matrix variables and in LMI (26); then the state feedback gain matrix is obtained naturally.

When , , we obtain

Let the initial conditions of be and let the initial conditions of be , , and , respectively. In addition, we take the exogenous disturbance asNote that, from the simulation results, it can be seen that choosing properly for different situations can eliminate the oscillation caused by and .

We take the reference signal as the step function and then perform simulations:

Figure 1 shows the output response of the closed-loop system for system (22). Figure 2 shows the control input. It can be seen from three situations (i.e., the preview lengths of the reference signal and the exogenous disturbance are , , , , and no preview (namely, )) that the output can all track the reference signal accurately. However, both the tracking error and the input peak decrease with the increase of the preview length of the reference signal when the preview length of the exogenous disturbance is identical; namely, ; and the output of the closed-loop system can track the reference signal more quickly. Compared with no preview, in addition to the advantages mentioned above, the disturbance attenuation performance is improved by virtue of preview compensation. This is exactly how preview control achieves its goal.

In addition, from (9), the steady-state values of state variables and input variables are given byThe simulations reveal that the steady-state values of and tend toward the theoretical values. Figure 2 shows the curve of the control input changing in time here as an example.

In the following, Figure 1 can be used for further analysis to illustrate the advantages of the preview controller by making use of the dynamic characteristics (rise time, peak time, settling time, etc.). The corresponding results were given out as follows: The rise time: , , and . The peak time: , , and . The maximum overshoot: , , and . The settling time: , , and . The percentage overshoot: , , and .

We perform simulations for three situations; that is, the preview lengths of the reference signal and the exogenous disturbance are , , , , and no preview (namely, ). Figure 3 shows the output response of the closed-loop system. Figure 4 shows the control input. It can be seen from Figures 3 and 4 that the overshoot of the output response decreases, and the settling time can be shortened, by increasing the preview length of the disturbance signal when the preview length of the reference signal is identical; namely, . Although, to weaken the disturbance influence over the system, the control input is increasing with the increase in preview length of the disturbance signal temporarily when the disturbance appears, the input diminishes rapidly and reaches steady state.

From the simulations for various forms of the reference signals and the disturbance signals, we find that the closed-loop system can have fairly good stability, only if the reference signals or only if the disturbance signals are previewable. Now the simulation results about and , respectively, will be given as an example. Figures 5 and 6 show that when the reference signal is not previewable, the overshoot of output response and the input peak decrease and the settling time can be shortened as the preview length of the disturbance signal increases.

From Figures 1, 3, and 5, we can see that the output for the system with has a larger fluctuation when the disturbance appears or disappears. However, the case in which preview action is used can effectively inhibit the fluctuation.

It follows from the simulations of periodic interference signals that the tracking error of the closed-loop system with cannot be eliminated during the survival of periodic interference signals. However, the cases in which preview action is used can eliminate the tracking error. Considering the length of this paper, the figures for these results will no longer be given out.

To design the observer, letand we obtainAs one can easily confirm, all the eigenvalues of matrix lie within the unit circle; that is, is stable. Therefore, we can reconstruct the state of system (1). When the preview lengths are identical, namely, and , Figure 7 shows the output response of the closed loop with the extended functional observer (42) and without the observer.

Let the initial conditions be and . From Figure 7, we can see that the oscillation caused by the deviation of initial value decays rapidly in the reconstruction process of the system state using observer (42) by choosing appropriately and . And through verification, the estimation error is less than after and is not affected by the disturbance (namely, (61)). The observer can give a fast and accurate estimation of the state and restrain the disturbance effectively. The effect of control with observer (42) is ideal.

7. Conclusion

In this paper, the preview control problem based on LMI is proposed for discrete-time systems. First, we adopt the method in [16] to derive an augmented system with previewable reference signal and disturbance signal. Then, by introducing state feedback and applying Lyapunov stability theory, the preview controller existence condition and its design method are presented. Finally, the state feedback gain matrix can be obtained by solving an LMI problem. In addition, an extended functional observer is proposed which can speed up the convergence rate for estimation error and has the ability of anti-interference. The method used to study preview control by LMI in this paper can be extended completely into uncertain systems for preview control. The numerical simulation example also illustrates the effectiveness of the controller in this paper.

Conflict of Interests

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

Acknowledgment

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

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Copyright © 2015 Li Li and Fucheng Liao. 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.

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