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.

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 [

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 [

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 [

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.

Consider the uncertain singular system with randomly occurring time-varying delay:

The parametric uncertainties

To account for the phenomena of ROTD, we introduce the stochastic variable

When the actuators experience failures, we use

Define the following notations:

From notations

The nominal unforced singular systems with randomly occurring time-varying delay of

System

System

System

The uncertain singular system

Let us consider the following controller:

Substituting

For any matrix

Given matrices

Suppose that a positive continuous function

In this section, we first derive a condition to guarantee nominal unforced system

Singular time-delay system

Firstly, the regularity and absence of impulses of system

Then, the problem for the unforced system

Define the infinitesimal operator

Consider

Define

To prove the exponential stability of system

Taking its time derivative yields

On the other hand, we can get from

Applying Lemma

Next, we will present the robust stability criterion for the following system:

Singular time-delay system

Condition

Singular system

By Condition

By Lemma

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

Consider the singular system

The state response of open-loop system (

State response of open-loop system.

In this example, we chose

For the first actuator failure, by Theorem

When actuator failures (

State response of closed-loop system.

Controller output.

For the second actuator failure, by Theorem

When actuator failures (

State response of closed-loop system.

Controller output.

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.

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