Abstract
In this paper, the problem of robust preview control for uncertain discrete singular systems is considered. First of all, by employing the forward difference for uncertain discrete singular systems, the singular augmented error system with the state vector, the input control vector, and the previewable reference signal is derived. Since there is a singular matrix in the system, the existing method cannot be directly applied to this problem. By considering the stability of the transposition system with Linear Matrix Inequality (LMI) method, a new stability criterion for the transposition system is introduced. Then, the robust controller for the augmented error system is obtained, which is regarded as the robust preview controller for the original singular system. At last, the numerical simulation shows the correctness and effectiveness of the results.
1. Introduction
Since the singular system model [1] is proposed by Rosenbrock in the early 1970s, the singular system has been widely studied, and many results are obtained [2, 3]. Especially after the 1980s, singular systems, as an appropriate tool to deal with large-scale complex systems with multiobjective and multidimensional, have been applied to the fields of singularly perturbed theory [4], large-scale systems theory [5], circuit system [6], and so on.
In practical, due to the modeling error, the measurement error, the linear approximation, and the change of working environment, the uncertainty of the system is objective. And they are presented as uncertain model parameters, system perturbation, measurement noise, external interference, etc. There is no doubt that any controller without considering the uncertainties may be difficult to achieve an ideal actual effect or may even cause the collapse of the system. Therefore, robust control is still regarded as one of the hotspot research directions in control theory and the applications. And the study of robust control theory and methods has formed several important branches, such as Lyapunov-Razumikhin method [7], Riccati inequality [8], Linear Matrix Inequality method [9], etc.
For singular systems, uncertain problems also exist [10]. Therefore, the modeling, analysis, and design of uncertain singular systems are particularly important. The related research of uncertain singular systems began in the 1980s. The main purpose of the study is to design a controller to make the corresponding closed-loop system stable and meet certain performance targets, when there are model uncertainty and external interference in the system.
Preview control is a control technique to improve the performance of systems by making full use of known future reference or disturbance signals in advance. Since the preview control model was first proposed in 1960s, the theoretical study and the applications of preview control have never stopped. In recent years, preview control has been extended to multisampling system [11], singular system [12], time-varying system [13], and so on. And it has been applied to many fields, such as automobile control [14], robot control [15], electromechanical valve control system [16], etc.
Based on many results on the robust preview control problem of nonsingular system, the robust preview control problem of singular system is further discussed. Because of the existence of the singular matrix, a proper Lyapunov function cannot be found by the current method to design the robust preview controller. In this paper, a singular augmented error system is constructed by using forward differences. Then, based on the method of [17], a Lyapunov function for the transposition system of the closed-loop system for the singular augmented error system is designed. And by the LMI method, a stability criterion for the transposition system is obtained. Since the two unforced systems before and after transposition have the same stability, the robust controller for the singular augmented error system can be obtained by the transposition system. As for the original singular system, this controller is the robust preview controller.
Notation. denotes the notion that matrix is a positive definite (negative definite) matrix; denotes the unit matrix; denotes a matrix; the symbol denotes the symmetric terms in a symmetric matrix.
2. Problem Statement
Consider the following linear uncertain singular system:where is the state vector, is the input vector, and is the output vector. is the singular matrix with . The matrices , , and are known real constant matrices with appropriate dimensions. The matrices and are uncertain matrices.
First of all, the definition about admissible is needed.
Definition 1 (see [17]). The dynamic systemis said to be admissible if it is regular, causal, and stable.
Then, the following assumption is needed for system (1).
Assumption 2. The matrices pair is stabilizable and the matrix is of full row rank.
By denoting the previewable reference signal , the following assumption is needed.
Assumption 3. The reference signal is previewable. That is, at any time , the future values , are available. When , the reference signal is assumed to be unchanged, namely,where is the preview length [18].
About the uncertain matrices and , the following assumption is needed.
Assumption 4. There exist real constant matrices , with appropriate dimensions and uncertain matrices satisfying
Remark 5. Equation (4) shows that the uncertain matrices and in system (1) satisfy matching conditions, and (5) shows that they are norm bounded.
The purpose in this paper is to design a controller for system (1) with preview compensation to make the output track the reference value accurately when there exist disturbances and in system (1).
3. Construction of the Augmented Error System
The error equation can be defined as follows:Employing the difference operator as to (1) and (6), we can obtainandFurthermore, (8) can be rewritten asPutting (7) and (9) together, we havewhere
Define . Since the future reference signal is known at time , it follows from the Assumption 3 that satisfieswhere
With (10) and (12), the augmented error system can be written as follows:where
According to the properties of and in Assumption 4, we havewhereThe matrices still satisfy ,. Therefore, and are normal bounded.
4. Design of the Robust Preview Controller
In order to obtain the robust preview controller for system (1), the stability problem of the unforced discrete singular nominal system (2) should be considered first and the following lemmas are needed.
Lemma 6 (see [17]). The discrete singular system (2) is admissible if and only if there exists a matrix , a symmetric matrix such thatwhere . is an annihilator basis of the range space of the matrix and .
If the matrices and are replaced with and in system (2), we haveSince is a singular matrix, there always exist nonsingular matrices and , such that and according to the first restricted equivalent form in [19], where with a constant . Substituting into (2) givesPremultiplying on both sides of (20), we have (21) can be rewritten asConcretely, we have that is a stable solution for the second equation of (22). And the stability for the first equation of (22) is decided by the eigenvalues of .
Similarly, substituting into (19), we havePremultiplying on both sides of (20), we haveCorresponding to (22), (24) can be rewritten as is also a stable solution for the second equation of (25). And the stability for the first equation of (25) is decided by the eigenvalues of , which are the same as . Therefore, system (19) is stable if and only if system (2) is stable.
Through the definitions, it is obvious that system (19) is regular and casual if and only if system (2) is regular and casual. As a result, Lemma 7 can be obtained.
Lemma 7. The discrete singular system (2) is admissible if there exists a matrix and a symmetric matrix such thatwhere . is an annihilator basis of the range space of the matrix and .
Proof. According to Lemma 6, if (26) holds, we can get that system (19) is admissible. Because of the equality of the two systems, system (2) is also admissible and vice versa.
Furthermore, Theorem 8 can be obtained.
Theorem 8. The discrete singular system (2) is admissible if there exist a matrix , a symmetric matrix , and matrices such thatwhere. is an annihilator basis of the range space of the matrix and .
Proof. From (27) and (28), it is easy to see thatwhereDefine . It is obvious that . Pre- and postmultiplying (29) by and , from (27) we can getAccording to Lemma 7, it can be concluded that system (2) is admissible. This proves the theorem.
In the following, the robust preview controller will be designed.
Considering the state feedback input of system (14) described bythe closed-loop system iswhere and .
Next, the matrix , which makes the closed-loop system (33) admissible, is to be obtained.
Theorem 9. The closed-loop discrete singular system (33) is admissible, if there exist a matrix , a symmetric matrix , a nonsingular matrix , a matrix , and some given scalars , such thatwhere. is an annihilator basis of the range space of the matrix and . The state feedback controller (32) is .
Proof. Based on the results of Theorem 8, replacing with in (27) and (28), we have (34), where are defined asand . Since there exist some quadratic matrices variables, such as , (34) is not a strict LMI based on the description of (36). To tackle the problem effectively, matrices , and some scalars are introduced just like the condition in [17]. Let and . Substituting into and substituting into in (36), we have (34) and (35). Because is nonsingular, we have and . This proves the theorem.
Since the uncertain terms and are contained in and in (35), the robust controller cannot be directly obtained by computing (34).
Next, the robust preview controller of system (1) is gained. First, some other lemmas are needed.
Lemma 10 (see [20]). Let and be matrices of appropriate dimensions. Let , where are uncertain matrices that satisfy . Then, for arbitrary positive scalars , we havewhere .
Lemma 11 (see [21]). Considering the matrix , where and are symmetric matrices and invertible, the following three conditions are equivalent:(1);(2);(3).
Then, we can get Theorem 12.
Theorem 12. If there exist a matrix , a symmetric matrix , a nonsingular matrix , a matrix , and some given scalars , , such thatwherethe controlleris the robust preview controller for the discrete singular system (1).
Proof. From (34) and (35), we can getDenoteThen (41) can be rewritten asAccording to the Assumption 4, . Applying Lemma 10 to (43), we havewhere .
Therefore, if there exist satisfying , according to Lemma 11, namely, ifwhereThe state feedback controller for system (14) isLet be blocked into , then (47) can be expressed asWith regard to the discrete singular system (1), since , we haveAdd the above equations together on both sides, and move to the right side of the equation. When , we have (50) is the robust preview controller for system (1). This proves the theorem.
Remark 13. Since there exists the term in (50), the future know information about the reference signal is considered in the robust controller.
Remark 14. Unlike the free weighting matrix method for normal system in [22], the difficulty of designing proper Lyapunov function for singular systems is overcome by introducing the method of [17].
5. Numerical Simulation
Consider the following system:where
It is proved that is invertible and satisfy Assumption 4 for any . So the system satisfies the fundamental assumptions.
The reference signal is taken as
The preview length is , , and , respectively. When , , . When , , , . When , , , . And we have Figures 1 and 2.
Figure 1 shows the output response for system (1). Figure 2 shows the tracking error. From the figures we can see that the output response with preview can track the reference signal much faster. And along with the preview length increasing, the overshoots reduced. Meanwhile, the maximum tracking errors decreased.
As a comparison, the “nonrobust” controller is applied to the uncertain system (51). Similarly, the reference signal is also taken asand the preview length are and , respectively. When , , . When , , , . Thus, we have Figures 3 and 4.
Figures 3 and 4 are the output response of the closed-loop system based on the robust controller and the “nonrobust” preview controller when and , respectively. From these two figures, it can be easily seen that the overshoot of the output response based on the robust controller is smaller than that based on the nonrobust controller, which shows the effectiveness of the controller designed in this paper.
6. Conclusion
In this paper, the robust preview control problem for uncertain discrete singular systems is studied. At first, the singular augmented error system is constructed. Then, by considering the stability of the transposition system, some criterions are gained by using LMI method. At last, the robust controller for the singular augmented error system is obtained, which is the robust preview controller for the original uncertain discrete singular system. The numerical example shows the effectiveness of this paper.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant no. 61174209).