Impulsive Controller Design for Complex Nonlinear Singular Networked Systems with Packet Dropouts

Globally exponential stability of Complex (with coupling) Nonlinear Singular Impulsive Networked Control Systems (CNSINCS) with packet dropouts and time-delay is investigated. Firstly, the mathematic model of CNSINCS is established.Then, by employing the method of Lyapunov functional, exponential stability criteria are obtained and the impulsive controller design method is given. Finally, some simulation results are provided to demonstrate the effectiveness of the proposed method.


Introduction
At present, singular system is widely used in the control of spacecraft, flexible robot, complex power, large chemical and wireless transmission system [1][2][3][4][5].Many results had been achieved for discrete singular system and time-delay singular system.Such as in [6], the nonlinear discrete singular perturbation model was established and the system condition was given.In [7], chattering free sliding mode control for uncertain discrete time-delay singular system was investigated.The asymptotically stable was established, and the chattering problem that appears in traditional variable structure system was eliminated.As for time-delay singular system, the stability of uncertain time-delay singular systems was researched and the asymptotic stability condition was achieved in [8] by using Jensen integral inequality and feedback control method.
On the other hand, singular system has impulsive behavior in many cases [9][10][11][12][13][14].So it is very important to discuss the problem of impulsive control.For the stability of the impulsive control system, nonlinear impulsive control was put forward and the concept of asymptotic stability condition was provided in [11].Asymptotic stability condition for a class of uncertain impulsive system was established through the comparison theorem in [12].Switch control method was used to research the stability of singular impulsive system, robust stabilization, and  ∞ control problem in [13].Linear approximation and the LMI method were used, respectively, to study the problem of system stability and the sufficient conditions for asymptotic stability in [14].
In network impulsive control system packet dropouts and time-delay exist which will influence the stability of singular system.It is necessary to analyze stability condition and the method of controller design.That is the problem focussed in this study.According to the Lyapunov function theory and comparison theorem, the sufficient conditions for the global exponential stability of the system is obtained.The detailed design process of impulsive controller is given in the paper.System will be stable in accordance with the decay rate to achieve exponential stability.A numerical example is provided to illustrate the correctness of theoretical and the effectiveness of design method.

The Mathematic Model of CNSINCS
The mathematic model of CNSINCS can be described as where   () ∈   is the state vector of the th node. is a constant matrix of  × .  is known scalar. ∈  × is a singular constant matrix, and 0 < rank  =  ≤ , without loss of generality; we hypotheses  = [   0 0 0 ].(⋅) is a nonlinear function.Γ is the internal coupling matrix. =   ∈  × is the coupling matrix of the whole network structure and weights.() is network transmission delay and is assumed to satisfy 0 ≤ () ≤ .
In the process of data transmitting, the buffer's model can be described as: ( The impulsive controller can be designed as where   (  ) ∈   .Substituting (2) and ( 3) into (1), the closed-loop nonlinear singular impulsive networked system model is obtained as follows: where   (  ) = 1 denotes that there are data dropouts and,   (  ) = 0, there are no packet dropouts.
Remark 5.For the case  ≥ 1, we can replace the condition The proof of the above conclusion remains largely the same as Theorem 4, so we omitted it to avoid repetition.

Design Procedure of Impulsive Control for Complex Network
According to Theorem 4, the design process of impulsive control is given as follows.

Numerical Simulation
In this section, a numerical example is presented to illustrate the effectiveness of derived results.
The parameters are given as follows: For simplicity, consider the system with 2 nodes.Assume that the external coupling matrix is  = [ −7 3  3 −4 ] and the internal coupling matrix is Figure 1 shows that the asymptotic stability of the closedloop uncertain system can be guaranteed using the networked impulsive controller designed in this paper.

Conclusion
In this paper, the global exponential stability CNSINCS via impulsive control is investigated.According to the Lyapunov stability theory, the mathematic model of CNSINCS is established.A general model of network consisting of time-delay and packet dropouts has been formulated and the globally exponential stable sufficient conditions have been established.Impulsive controller, which may ensure the system achieves exponential stability with a given decay rate is designed.Therefore our control scheme is efficient and practical in dealing with problems of data transmission with time-delay and packet dropouts.As an application, a numerical simulation is given to demonstrate the usefulness and practicability of proposed theoretical results.

Figure 1 :
Figure 1: The state response of CNSINCS via impulsive control (color online).