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This paper is concerned with the structural controllability analysis for discrete-time linear control systems with time-delay. By adding virtual delay nodes, the linear systems with time-delay are transformed into corresponding linear systems without time-delay, and the structural controllability of them is equivalent. That is to say, the time-delay does not affect or change the controllability of the systems. Several examples are also presented to illustrate the theoretical results.

Controllability has been one of the fundamental concepts in modern control theory and has played an essential role in its development because of the broad applications. Controllability in the classical sense refers to the complete controllability which means that it is possible to steer control systems from an arbitrary initial state to an arbitrary final state using the set of unconstrained admissible controls. In practice, admissible controls are always required to satisfy certain additional constraints. The controllability for linear dynamical systems with constrained controls has also been studied; see [

In this paper, we will consider the problem of structural controllability on unconstrained values of admissible controls. The concept of structural controllability was introduced by Lin in 1974 to study the controllability of linear systems, and it was extended to other systems, such as complex networks and multiagent systems. Roughly speaking, structural controllability generally means that, by adjusting the free parameters of the structured matrix, the control system is completely controllable. Controllability is important in the solution of many control problems, yet the determination of controllability indices, for example, is a particularly ill-posed computational problem as is the problem of checking the controllability of an uncontrollable system. Structural controllability, on the other hand, is a property that is as useful as traditional controllability and can be determined precisely by a computer. The structural controllability is a generalization of traditional controllability concept for linear systems and is purely based on the graphic topologies among state and input vertices. It is now a fundamental tool to study the controllability and enables us to understand the control systems.

The necessary and sufficient conditions of structural controllability were constructed by Lin in [

It is well recognized that time-delay is often encountered in physical and biological systems. Time-delay phenomenon may occur naturally because of the physical characteristics of information transmitting and diversity of signals, as well as the bandwidth of communication channels. Systems with time-delay are more difficult to handle in engineering since the controllability matrices are usually complex.

Studying the linear delayed systems has become an important topic in control theory and many researchers have devoted themselves to the controllability analysis for the delayed systems. For example, a data-based method is used to analyze the controllability of discrete-time linear delayed system by Liu et al. [

In spite of this progress, there is less work concerned with the structural controllability of linear systems with time-delay. This paper is devoted to the structural controllability analysis for discrete-time linear delayed systems. By adding virtual delay nodes, the linear systems with time-delay are transformed into corresponding linear systems without time-delay; the necessary and sufficient conditions with respect to the structural controllability of linear delayed systems are obtained.

This paper is organized as follows. In Section

This section gives some basic definitions and preliminary results.

Consider the following discrete-time linear control system:

The matrix pair

The structured system

The representation graph of structured system (

An alternating sequence of distinct vertices and oriented edges is called a directed path, in which the terminal node of any edge never coincides with its initial node or the initial or the terminal nodes of the former edges. A stem is a directed path in the state vertex set

A vertex (other than the input vertices) is called nonaccessible if and only if there is no possibility of reaching this vertex through any stem of graph

Consider one vertex set

It is well known that for delayed control systems generally two types of controllability are considered: absolute controllability and relative controllability; see paper [

The linear control system (

For the linear system (

The linear control system (

The linear control system (

The following lemma characterizes the structural controllability for the linear structured system (

The linear structured system

there is no nonaccessible vertex in

there is no “dilation” in

Consider the following linear control systems with time-delay in state:

The linear delayed system (

By inserting delays on edges, the linear delayed system is transformed into a corresponding linear system without time-delay.

For every directed edge

(a) A directed graph with 3 state nodes. (b) The directed graph when we add a delay of 4 on the edge

Note

Thus, the linear delayed system (

Consider a very simple example. Without delays we define

As mentioned above, the structural controllability of the linear delayed system (

For example, we construct a linear structured system corresponding to the linear control system described by a directed graph in Figure

In fact, the structure of the representation graph

The following theorems build the equivalence of the structural controllability of the three systems: system (

The linear system (

The necessity is obvious; we then prove the sufficiency. Assume that the linear system (

In the following, we will adjust some parameters of

Firstly, we analyze the characteristics of columns of

Then, from the

Next, assume that the single nonzero element in the

Continue to do elementary transformation according to the rules, multiply the

In this way, matrix

Let

The theorem above reveals that the structural controllability of the time-delayed system (

The linear system (

The proof of the theorem mainly used the results of Lemma

In the first case, there is nonaccessible vertex in

In the second case, there is “dilation” in

It is easy to prove the sufficiency as in the discussion above. Thus, the proof of the theorem is complete.

There are two examples being presented in this section to illustrate the theoretical results.

Consider a linear control system without time-delays with the structured matrices given by

We first consider the case that the message leaving state node

We then construct a linear structured system

Selecting

by simple calculation we obtain

(a) A directed graph with 3 state nodes. (b) The directed graph when we add a delay of 3 on the edge

Consider another linear control system without time-delay with the structured matrices given by

We then construct a linear structured system

Selecting

by simple calculation we obtain

(a) A directed graph with 3 state nodes. (b) The directed graph when we add a delay of 3 on the edge

The structural controllability analysis for discrete-time linear control systems with time-delay is discussed in the paper. We derive necessary and sufficient conditions for the linear delayed systems to be structurally controllable by transforming the delayed systems into a corresponding linear system without time-delay. This method is suitable for the discrete-time linear delayed systems; the structural controllability for the continuous-time linear delayed systems is also an issue we are concern with. We look forward to making some results on continuous-time systems.

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

The authors gratefully acknowledge the support of the Foundation Research Funds for the Central Universities under Grant no. 3122014K008, the Natural Science Foundation of China under Grant no. 61174094, and the Tianjin Natural Science Foundation under Grant no. 14JCYBJC18700.