#### Abstract

The problem of reliable control is investigated for uncertain continuous singular systems with randomly occurring time-varying delay and actuator faults in this work. The delay occurs in a random way, and such randomly occurring delay obeys certain mutually uncorrelated Bernoulli distributed white noise sequences. The uncertainties under consideration are norm-bounded, and may vary with time. Then, with the constructed Lyapunov function, a sufficient condition is given to ensure the unforced system is mean-square exponentially stable and the corresponding controller can be derived from such condition, and the actuator faults problem is guaranteed. A numerical example is provided to show the effectiveness of the methods.

#### 1. Introduction

During the past three decades, the studies of singular systems have been an active field of research in many scientific and technical disciplines. Dynamic input-output model, electronic network, constrained robots, nuclear reactors, and other noncausal systems all belong to the typical singular systems. Recently, the problems of robust stability analysis and robust stabilization for singular systems have been studied. It is worth noticing that the robust stability problem for singular systems is much more complicated than that for regular systems because it requires considering not only stability robustness, but also regularity and absence of impulses (for continuous singular systems) and causality (for discrete-singular systems) at the same time, and the latter two need not be considered in regular systems. There are lots of papers that have studied these subjects [1–3].

On the other hand, singular systems with time-delays arise in a variety of practical systems such as chemical processes and lossless transmission lines. Since singular systems with time-delays are matrix delay differential equations coupled with matrix difference equations, the study of such systems is much more complicated than that of standard state-space time-delay systems or singular systems. The existence and uniqueness of a solution to a given singular time-delay system is not always guaranteed. In accordance with the advance of robust control theory, a number of robust stabilization methods have been proposed for uncertain time-delay systems [4–9].

For continuous singular systems, a few of the studies have mentioned the robust stabilization of the uncertain time-delay system. In this paper, we address the problems of robust stability and stabilization for uncertain singular systems with randomly occurring time-varying delay (ROTD). Although the randomly occurring delay has appeared in some papers [1, 10, 11], few related results have been established for uncertain singular systems.

Recently, much effort has been devoted to the reliable control with unexpected failures which were often found in the real world. Therefore, designing a controller which could tolerate some actuator failures has been investigated for dynamical systems, discrete-time fuzzy system, networked control system, and so forth. Up to now, the issue of reliable control for uncertain singular systems with randomly occurring time-varying delay and actuator failures has not been fully investigated.

In this paper, we deal with the problem of reliable control and exponential stability analysis for uncertain singular systems with ROTD and actuator faults. A random variable, which obeys Bernoulli distribution, is introduced to account for ROTD. By the LMI approach, a state feedback controller is established to guarantee the resultant closed-loop system is delay-dependent exponentially admissible. Finally, a numerical example is given to show the usefulness of the result derived.

*Notations*. denotes the -dimensional Euclidean space and is the set of all real matrices. The notation , where and are symmetric matrices, means that is positive definite (positive semidefinite). and 0 represent the identity matrix and a zero matrix, respectively; and diag stands for a block-diagonal matrix. The superscript “” represents the transpose and the asterisk “” in a matrix is used to represent the term which is induced by symmetry. means the expectation of the stochastic variable .

#### 2. Problem Formulation and Preliminaries

Consider the uncertain singular system with randomly occurring time-varying delay: where is the state vector and is the control input vector. is a compatible vector valued initial function. The matrix may be singular and it is assumed that . is a time-varying continuous function that satisfies and , where and are the lower and upper bounds of the time-delay . , , and are known real constant matrices with appropriate dimensions. , , and are unknown matrices representing norm-bounded parametric uncertainties and are assumed to be of the form where , , , and are known real constant matrices with appropriate dimensions and is an unknown real and possibly time-varying matrix satisfying

The parametric uncertainties , , and are said to be admissible if both 2 and 3 hold.

To account for the phenomena of ROTD, we introduce the stochastic variable , which is a Bernoulli-distributed white sequence. The natural assumption on is as follows: where the constant .

When the actuators experience failures, we use to describe the control signal sent from actuators. Consider the actuator failure model with failure parameter is the actuator failure matrix with the following property: where

Define the following notations:where

From notations 8a-8b, we have Because actuator failure that might occur is unknown, the matrix is not well known in advance. The matrix satisfies 6 in which the matrix is known in advance.

The nominal unforced singular systems with randomly occurring time-varying delay of 1 can be described as follows:

*Definition 1. *System 11 is said to be regular and impulse-free, if the pair is regular and impulse-free.

*Definition 2. *System 11 is said to be exponentially stable, if there exist scalars and such that .

*Definition 3. *System 11 is said to be exponentially admissible, if it is regular, impulse-free, and exponentially stable.

*Definition 4. *The uncertain singular system 1 is said to be exponentially admissible, if the unforced system is regular, impulse-free, and exponentially stable for all admissible uncertainties and .

Let us consider the following controller: where is the controller gain matrix to be designed.

Substituting for in 1 and considering 5 and 12, the closed-loop system can be described by

Lemma 5 (Lu et al. [12]). *For any matrix , integers and satisfying , and vector function , such that the sums concerned are well defined,
*

Lemma 6 (Wu and Zheng [13]). *Given matrices , and with appropriate dimensions and with symmetrical, then for any satisfying , if and only if there exists a scalar such that .*

Lemma 7 (Mao et al. [9]). *Suppose that a positive continuous function satisfies
**
where , , , , and ; then,
*

#### 3. Main Result

In this section, we first derive a condition to guarantee nominal unforced system 11 to be mean-square exponential stability.

Theorem 8. *Singular time-delay system 11 is exponentially robustly stable, if there exist matrices , , and such that**where
*

*Proof. *Firstly, the regularity and absence of impulses of system 8a, 8b, 8c, and 8d are given. For singular system, we know that there exist nonsingular matrices and such that
Denote
From 17a and using the expressions in 19 and 20, it can be found that . Then, premultiplying and postmultiplying by and , respectively, we have
which implies is nonsingular and thus the pair is regular and impulse-free. Hence, by Definition 1, system 11 is regular and impulse-free.

Then, the problem for the unforced system 11 to be exponentially stable is given. Consider the following Lyapunov function for system 11:
where
Define the infinitesimal operator of as follows:
We obtain
where
Applying Lemma 5, when , we have that
It can be from 17c that 29
which implies
We can get from 27 and 29 that
Then,
So
where
By application of Schur complement and 17b, we have , which guarantees there exists a scalar such that
Set
It is easy to get
where and .

Consider
It can be seen from 17b that
which implies
It is easy to find that matrix satisfying the above inequality is nonsingular. Thus, considering 17a, we can deduce that and .

Define
Then, system 11 is equivalent to
To prove the exponential stability of system 11, we define a function as
where the scalar .

Taking its time derivative yields
Integrating both sides of 39 from 0 to , we get that
By using the similar analysis method of [14], it can be seen from 22, 42, and 44 that if the scalar is chosen small enough, a scalar can be found such that, for any ,
Since , it can be shown from 43 that for any
where . Define
then, from 46, a scalar can be found such that, for any ,
To study the exponential stability of , we construct a function as
By premultiplying the second equation of 41 with , we get that
Adding 50 to 49 yields that
where is any positive scalar.

On the other hand, we can get from 17b that
Premultiplying and postmultiplying 52 by and , respectively, a scalar can be found such that
On the other hand, since can be chosen arbitrarily, can be chosen small enough such that . Then, a scalar can always be found such that
It follows from 48, 49, 51, 53, and 54 that
which infers
where , , and .

Applying Lemma 7 to 56 yields that
We can find from 46 and 57 that system 11 is exponentially stable. This completes the proof.

Next, we will present the robust stability criterion for the following system: via Theorem 8.

Theorem 9. *Singular time-delay system 58 is robustly stable, if there exist a scalar , matrices , , , , and such that
**
where
*

*Proof. *Condition 17b with and replaced by and , respectively, can be written as
where follows the same definition as in Theorem 8 and
By Lemma 6, 61 holds if there exists a scalar such that
By Schur complement, 63 is equivalent to 59. Hence, according to Theorem 8 and Definition 4, system 58 is robustly stable. This completes the proof.

Theorem 10. *Singular system 13 is robustly stable, if there exist a scalar , matrices , , , , and , and matrices with appropriate dimensions such that**where
**
In this case, the desired controller gain is given as .*

*Proof. *By Condition 17b with and replaced by and , respectively, we have
where
Define matrix , , , , , , and . Then, pre- and postmultiplying the left side of 63 by and , respectively, we can find that
where
Noting , we have .

By Lemma 6, 68 holds if there exists a scalar such that
By Condition 70 with 10, we have
whereBy Schur complement, 71 is equivalent to 64a, 64b, and 64c. Hence, according to Theorem 8 and Definition 4, the system controlled by is robustly stable. This completes the proof.

#### 4. Numerical Example

In this section, we give an example to demonstrate the effectiveness of the proposed method.

Consider the singular system 1 with parameters as follows:

The state response of open-loop system () 1 is shown in Figure 1 and it is clear that the open-loop system is unstable.

In this example, we chose , , and . The first actuator failure matrix is , where and and the second actuator failure matrix is , where and .

For the first actuator failure, by Theorem 10, we have the following controller gain:

When actuator failures ( and ) with the designed feedback controller gain 74, the state response of the closed-loop system and the controller output are shown in Figures 2 and 3. In this example, all controllers are partially damaged. Figure 2 shows the reliable feedback control gain can guarantee that the state responses converge to zero asymptotically.

For the second actuator failure, by Theorem 10, we have the following controller gain:

When actuator failures ( and ) with the designed feedback controller gain 75, the state response of the closed-loop system and the controller output are shown in Figures 4 and 5. In this example, not all controllers are partially damaged. Figure 4 shows the reliable feedback control gain can guarantee that the state responses converge to zero asymptotically.

#### 5. Conclusion

The problems of reliable control for uncertain singular systems with randomly occurring time-varying delay and actuator faults have been studied. Based on the LMI approach, a sufficient condition has been established to ensure the considered systems are regular, impulse-free, and exponentially stable. The state feedback controller has been designed to ensure, for all possible actuator failures, the resultant closed-loop system is exponentially admissible. A numerical example has been provided to show the validness and less conservatism of the given result.

#### Conflict of Interests

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