#### Abstract

The reliable and control for a class of Lipschitz nonlinear discrete-time singular systems with time delay is investigated via dynamic feedback control. The main goal of this paper is to design a generalized nonlinear controller such that, for possible actuator failures, the closed-loop system is regular, casual, and stable with a given and disturbance attenuation level being satisfied. Some sufficient conditions are obtained in terms of linear matrix inequalities (LMIs), and the controller design method is also proposed. Finally, a numerical example is included to illustrate the effectiveness of our proposed results.

#### 1. Introduction

Singular systems [1] are also referred to as generalized systems or descriptor systems, differential-algebraic systems, or implicit systems, which arise in many practical physical systems such as electric systems, robotic systems, power systems, networked control systems, and space navigation systems. Considerable efforts have been done to the system analysis [2, 3], engineering applications [4], and control synthesis [5–14] for singular systems. Compared with the stability analysis of normal systems,that of singular systems is much more complicated since regularity and absence of impulse (or casual) are necessary to be considered simultaneously. Meanwhile, nonlinearity is an universal phenomenon existing in practical control systems, which can not be ignored. And It is always a source of instability and poor performance. For nonlinear singular systems, the solutions may not exist even if the systems' linear parts are regular. Currently, for a class of Lipschitz continuous nonlinear singular systems, the robust nonlinear filtering was investigated in [15]. And the authors in [16] considered a class of nonlinear continuous plant presented by a Takagi-Sugeno fuzzy model and designed a nonfragile filter. However, up to date, there are few papers considering the control problem of discrete-time singular systems, especially for the nonlinear discrete-time singular ones. On the other hand, reliable control is an interesting problem in control theory, and has gained considerable attention, and a number of results have been reported in the literature [17, 18]. Up to date, to the best of our knowledge, the research on discrete-time singular nonlinear systems with time delays is still an open problem that deserves further investigation.

Performance analysis has grown, in the past few decades, as one of the most important problems in control theory in addition to stability analysis. Many control problems, to a certain extent, can be equal to designing proper controller such that the closed-loop system is asymptotical stable and its performance satisfies some requirements. The general adopted performance indexes include index, index, index, and index. In particular, in recent years, there are many important results on the problem of stabilization based on and control which have been reported in literature [9, 15, 19–26].

Motivated by the above discussion, in this paper, we focus on a generalized framework for reliable nonlinear and control of a discrete-time Lipschitz singular system subject to time-delay and disturbance uncertainties. The main contributions of this paper are a new criterion for nonlinear discrete-time singular systems is derived. The obtained criterion can ensure the regularity, casuality, and stability of the considered system. With introducing some slack matrices in the derivation, the solution space of the controller parameters is expanded. A novel reliable nonlinear and controller is proposed, which is of more general dynamically framework. In the proposed controller design method, we only solve one strict LMI, but no any semidefinite positive matrix inequality is needed, which causes a simpler design method.

*Notation*

Throughout this paper, and denote, respectively, the -dimensional Euclidean space and the set of all real matrices. Sym(A) denotes for simplicity. The superscripts and indicate the inverse and the transpose of a matrix, respectively. The symbol denotes the symmetric part of a symmetric matrix. is an identity matrix of appropriate dimensions, while is an identity matrix and denotes a block-diagonal matrix. The symmetric matrix (or ) means that is positive definite (or positive semidefinite).

#### 2. Preliminaries and Problem Formulation

Consider the following nonlinear discrete-time singular time-delay system described by where is the state vector, is the measured output, is the controlled output signal, is the control input, is the nonzero exogenous disturbance input that belongs to , and is nonlinear function about the state . Here, we denote . denotes a constant time delay. , , , , , , , , , , and are known real constant matrices of appropriate dimensions. Note that if the matrix is nonsingular, the singular system (1) could be reduced to a conventional state-space system.

In this paper, we assume that is a singular matrix with ; then, there exist nonsingular matrices and , such that Therefore, without loss of generality, we take

We also assume that the system (1) is locally Lipschitz with respect to in a region containing the origin; that is, where is the induced 2-norm and is the Lipschitz real matrix of of appropriate dimensions.

Firstly, we consider the autonomous discrete-time singular time-delay system of (1) and introduce some elementary definitions that will be adopted throughout this paper.

*Definition 1 (see [10]). *The pair () is said to be regular if there exists a scalar such that , the pair is said to be causal if , and the pair () is said to be stable if all the roots of lie in the interior of unit disk. We call the pair admissible if it is regular, casual, and stable, simultaneously. Furthermore, the system (5) is said to be regular, casual, and stable (asymptotically stable) if the pair is admissible.

For system (1), we propose the following dynamic output feedback controller: where and are the state and the output of the controller, respectively. Hence, the matrices , , , , and are the controller gain matrices to be determined.

*Remark 2. *If set , the controller (6) will yield the following normal dynamic structure:

When the actuators experience failures, we use to describe the control input signal sent from the actuator. In general, we consider the actuator failure model [17] with failure parameter where with , for any . It is easy to get

Considering the controller physical implementation convenience in the practical engineering, here, we are interested in a normal controller (6) but not in a singular one. In addition, the design of a controller (6) is simpler than that of a singular controller viewed from the theoretical analysis. Therefore, in this paper, without loss of generality, we assume that the controller is in regular state-space system; that is . Just for convenience, we set .

Define the augmented vector

With the aforementioned operator and system (1), controllers (6) and (8), the closed-loop system is immediate where

The objective of this paper is to design a general nonlinear dynamic output feedback controller of the form (6) such that the resulting closed-loop system (12) is regular, casual, and stable, with a prescribed and performance level being satisfied. More specially, we are dedicated to find the controller gain matrices , , and such that(i)the closed-loop system (12) with is admissible;(ii)under the zero-valued initial state condition, for any nonzero disturbance input , we have where is a known positive scalar and

#### 3. Main Results

Firstly, a generalized stability criterion for discrete-time nonlinear singular system is proposed in this section. And then, based on this obtained result, a sufficient condition for the existence of a desired full-order and controller (6) for system (1) is obtained, which can guarantee that the resulting closed-loop system (12) is admissible (regular, casual, and stable) while satisfying a prescribed and performance . Also, the controller design method is derived.

##### 3.1. and Performance Analysis

In this subsection, we concentrate our attention on the problems of system admissibility containing regularity, casuality, and stability and and performance analysis for the considered system (1). Initially, considering the autonomous nonlinear singular system (5), we have the following.

Theorem 3. *For any nonlinear function satisfying (4), the system (5) is admissible, if there exist positive definite matrices , , matrices , , and a symmetry matrix of the following form:
**
with , such that**where .*

* Proof. *Firstly, we will prove the regularity and casuality of system (5). From (17), we have
Then, if follows from (3) that
It is obvious that
which implies that the matrix is singular. In this case, we can get
when choosing the scalar to be of some value which is not equal to any eigenvalue of the matrix ; then, we have
and therefore
Hence, by Definition 1, the pair is regular; that is, the system (5) is regular. Further, it follows from (21) that
which yields that the pair is casual; that is, the system (5) is casual. Thus, condition (17) in Theorem 3 can guarantee the regularity and casuality of system (5).

Next, in order to show the stability of system (5), we define the following Lyapunov-Krasovskii functional candidate as:
where has been defined in (16). Taking the difference of the Lyapunov functional leads to
It is clear to see that
Then, combining (27) into (26) yields
whereRecalling the state-space model of system (5), it is easy to see that
For any matrices , of appropriate dimension, the following can be obtained:
where
Now, if condition (17) is satisfied, we have
Then,
From (34) and (26), we get ; thus, the system (5) is stable. Summing up the above, we conclude that if condition (17) holds, then, system (5) is regular, casual, and asymptotically stable. Thus, this completes the proof.

In the sequel, we will focus on the and performance analysis of system (12). Based on Theorem 3, we obtain the following.

Theorem 4. *Given a positive scalar , for any consistent initial condition and the nonlinear function in (4), the system (12) is admissible, while satisfying a prescribed and performance , if there exist positive definite matrices , , a symmetry matrix , and matrices , such that the following inequality holds:**where
*

*Proof. *Firstly, it is easy to see that implies that ; in other words, condition (35) can guarantee condition (17). Therefore, according to Theorem 3, if condition (35) is satisfied, the system (1) with is admissible. We denote the following performance index:
Under zero-initial condition, , we have
where
Similar to the prove process in Theorem 3, we can easily get
Setting
yields
Therefore, if condition (35) in Theorem 4 is satisfied, then . We have
Hence, it follows from (39) that
which means
Hence, condition (35) can guarantee ; that is, the performance index is satisfied. It follows from (47) that
Hence, condition (35) can also guarantee ; that is, the performance index is also satisfied. Thus, this completes the proof.

*Remark 5. *By introducing two slack matrix variables and , Theorem 3 presents a novel stability criterion for nonlinear discrete-time singular system, in which the slack matrix variable could provide more free dimensions in the solution space. The condition obtained in Theorem 4 can guarantee not only the disturbance attenuation level but also the disturbance attenuation level. Note that, for the sake of simplicity, we assume that the disturbance attenuation level and the disturbance attenuation level are the same . In practice, if the required two levels are different, we only need to solve two inequalities.

*Remark 6. *Theorem 4 gives a sufficient condition for the existence of a full-order and controller of the forms (6) and (8) for system (1). According to the expression of closed-loop system (12), it can be seen that the condition obtained in Theorem 4 is nonlinear matrix inequality (nLMI) with respect to the parameter matrices , , and and the controller gain matrices , , , and since some crosses of these determined parameters are appearing in (35) in nonlinear fashion. In order to facilitate solving, in the following subsection, some matrix transformations are needed to transform the nLIM (35) to a strict LMI.

##### 3.2. and Controller Design

Theorem 7. *Given scalars , , and , for the discrete-time nonlinear singular system (1), there exists a full-order dynamic feedback controller (6) such that the closed-loop system (12) is admissible with a prescribed and performance , if there exist positive definite matrices , and matrices , , , , and of appropriate form, such that
**
where**
Then, the parameters of the desired and dynamic output feedback controller can be taken as
*

*Proof. *The matrices and in Theorem 4 have been defined in (37). We assume that the matrices , , and in Theorem 4 are of the following form:
where , , , , , and .

Setting

yields
Then, it follows from the above and Schur complement that the inequality (35) is equivalent to
where is defined in (50).

Then, from (10) and (56) and the inequality , for any , we have
where is any unknown arbitrarily positive scalar.

By Schur complement again, we have that in (56) is equivalent to (49). That is to say, the LMI (49) in Theorem 7 can guarantee the inequality (35) in Theorem 4. The matrix variables , , , , and are to be designed. Then, from Theorem 4, if (49) holds, the closed-loop system (12) is admissible with a prescribed and performance . And from (54), the controller gain solution (52) is immediate. Thus, this completes the proof.

*Remark 8. *In this paper, the reliable and dynamic output feedback control problem for discrete-time nonlinear singular systems are studied. Considering the controller physical implementation convenience in the practical engineering, we are interested in a normal state-space controller (6) in this paper. In Theorem 7, the desired full-order dynamic feedback controller (6) can be obtained by solving a strict LMI (49) efficiently. The performance can be obtained and described as
subject to linear matrix inequality (49).

#### 4. Numerical Example

In this section, we give a numerical example to illustrate the effectiveness of the obtained controller design method. Consider the nonlinear singular system (1) with Given the scalars , , the time-delay . We also assume the nonlinear functions as From (4), we can get

By applying the minimization problem in (58), the minimal value of the and performance is . And for a special , the corresponding solutions of the determined matrices are given as follows: Then, from (52), our obtained and dynamic feedback controller parameters can be obtained

#### 5. Conclusion

A generalized and dynamic feedback control problem for nonlinear discrete-time singular systems has been investigated in this paper. By introducing some slack matrix variables, a less conservative condition is obtained, which can ensure that the studied nonlinear discrete-time singular system is regular, casual, and stable. Based on this obtained condition, a sufficient LMI-based condition is obtained such that the resulting closed-loop system is regular, casual, and stable while satisfying a prescribed and performance level . The desired controller parameters can be computed only by solving a strict LMI. Finally, a numerical example shows the validity of our proposed method.

#### 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 (61203143) and Young Teacher Training Program of Shanghai (slg11005).