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

Abstract

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.

1. Introduction

At present, singular system is widely used in the control of spacecraft, flexible robot, complex power, large chemical and wireless transmission system [1–5]. 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 [9–14]. 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 𝐻_∞ 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.

2. The Mathematic Model of CNSINCS

The mathematic model of CNSINCS can be described as 𝐸𝑥̇_𝑖(𝑡)=𝐴𝑥_𝑖(𝑡)+𝑓𝑥_𝑖(𝑡)+^𝑁_𝑗=1𝐺_𝑖𝑗Γ𝑥_𝑗((𝑡,𝑡𝑡−𝜏))𝑖=1,…,𝑁,𝑡∈_𝑘−1,𝑡_𝑘𝑥_𝑖𝑡⁺_𝑘=𝑐_𝑖𝑥_𝑖𝑡_𝑘+𝑢_𝑖𝑡_𝑘,𝑡=𝑡⁺_𝑘𝑥,𝑘=1,2,…_𝑖(𝑡)=𝜑_𝑖(𝑡),𝑡∈[−𝜏,0], (1) where 𝑥_𝑖(𝑡)∈𝑅^𝑛 is the state vector of the 𝑖th node. 𝐴 is a constant matrix of 𝑛×𝑛. 𝑐_𝑖 is known scalar. 𝐸∈𝑅^𝑛×𝑛 is a singular constant matrix, and 0<rank𝐸=𝑟≤𝑛, without loss of generality; we hypotheses 𝐸=^𝐼𝑟0₀₀. 𝑓(⋅) is a nonlinear function. Γ is the internal coupling matrix. 𝐺=𝐺_𝑖𝑗∈𝑅^𝑛×𝑛 is the coupling matrix of the whole network structure and weights. 𝜏(𝑡) is network transmission delay and is assumed to satisfy 0≤𝜏(𝑡)≤𝜏.

In the process of data transmitting, the buffer’s model can be described as: 𝑢_𝑖𝑡_𝑘=𝑢_𝑖𝑡_𝑘,𝑘=1,2,…,iftransmittedsuccessfully,𝑢_𝑖𝑡_𝑘−1,𝑘=1,2,…,otherwise,𝑥_𝑖𝑡_𝑘=𝑥_𝑖𝑡_𝑘,𝑘=1,2,…,iftransmittedsuccessfully,𝑥_𝑖𝑡_𝑘−1,𝑘=1,2,…,otherwise. (2)

The impulsive controller can be designed as 𝑢_𝑖𝑡_𝑘=𝐾_𝑖𝑥_𝑖𝑡_𝑘, (3) where 𝑢_𝑖(𝑡_𝑘)∈𝑅^𝑚. Substituting (2) and (3) into (1), the closed-loop nonlinear singular impulsive networked system model is obtained as follows: 𝐸𝑥̇_𝑖(𝑡)=𝐴𝑥_𝑖(𝑡)+𝑓𝑥_𝑖(𝑡)+^𝑁_𝑗=1𝐺_𝑖𝑗Γ𝑥_𝑗((𝑡,𝑡𝑡−𝜏))𝑖=1,…,𝑁,𝑡∈_𝑘−1,𝑡_𝑘,𝑥_𝑖𝑡⁺_𝑘=𝑐_𝑖𝑥_𝑖𝑡_𝑘+1−𝜎_𝑖𝑡_𝑘𝐾_𝑖𝑥_𝑖𝑡_𝑘+𝜎_𝑖𝑡_𝑘𝐾_𝑖𝑥_𝑖𝑡_𝑘−1,𝑡=𝑡⁺_𝑘𝑥,𝑘=1,2,…,_𝑖(𝑡)=𝜑_𝑖(𝑡),𝑡∈[−𝜏,0], (4) where 𝜎_𝑖(𝑡_𝑘)=1 denotes that there are data dropouts and, 𝜎_𝑖(𝑡_𝑘)=0, there are no packet dropouts.

Lemma 1. Let 𝑃∈𝑅^𝑛×𝑛 be a symmetric positive definite matrix and 𝑃=𝑄^𝑇𝑄. For any 𝑥,𝑦∈𝑅^𝑛×𝑛 and 𝐴∈𝑅^𝑛×𝑛, then (1)𝑥^𝑇(𝐴^𝑇𝑃𝐸+𝐸^𝑇𝑃^𝑇𝐴)𝑥≤2𝜆_max(𝐴)𝑥^𝑇𝐸^𝑇𝑃𝐸𝑥, (2)𝑥^𝑇(𝐴^𝑇𝑃+𝑃𝐴)𝑥≤2𝜇(𝑄𝐴𝑄⁻¹)𝑥^𝑇𝑃𝑥,(3)|𝑥^𝑇𝑃𝑦|≤𝑥^𝑇𝑃𝑥𝑦^𝑇𝑃𝑦.

Lemma 2. According to the definition of Kronecker product, for a given matrix 𝐴,𝐵, and scalar 𝛼, the following equality can be achieved: (1)(𝛼𝐴)⊗𝐵=𝐴⊗(𝛼𝐵), (2)(𝐴+𝐵)⊗𝐶=(𝐴⊗𝐶)+(𝐵⊗𝐶), (3)(𝐴⊗𝐵)⊗(𝐶⊗𝐷)=(𝐴𝐶)⊗(𝐵𝐷).

If 𝑋(𝑡)=(𝑥^𝑇₁(𝑡),…,𝑥^𝑇_𝑁(𝑡))^𝑇, 𝐶=diag{𝑐₁,…,𝑐_𝑁}, and 𝐹(𝑋(𝑡))=(𝑓^𝑇(𝑥^𝑇₁(𝑡),…,𝑓^𝑇(𝑥^𝑇_𝑁(𝑡))^𝑇, according to Lemma 1, the complex nonlinear singular system can be expressed as 𝐼_𝑁⊗𝐸𝑋̇(𝑡)=𝐼_𝑁𝑋(𝑡)(𝑋(𝑡+()𝑋((𝑡𝑡⊗𝐴+𝐹))𝐺⊗Γ𝑡−𝜏)),𝑡∈_𝑘−1,𝑡_𝑘,𝑋𝑡⁺_𝑘=𝐶𝑋𝑡_𝑘+1−𝜎𝑡_𝑘𝐾𝑋𝑡_𝑘+𝜎𝑡_𝑘𝑋𝑡_𝑘−1,𝑡=𝑡⁺_𝑘. (5) Suppose 𝐸=,𝐸000𝐼_𝑁=𝐼_𝑁0𝑡00,𝑍_𝑘=𝑋𝑡_𝑘𝑋𝑡_𝑘−1,𝐴=.𝐴000 (6) Then (5) is equivalent to the following system: 𝐼_𝑁⊗𝐸𝑧(̇𝑡)=𝐼_𝑁⊗𝐴𝑧(𝑡)+𝐹(𝑥(𝑡0+𝑧((𝑡,))𝐺⊗Γ000𝑡−𝜏))𝐼_𝑁⊗𝐸𝑧𝑡⁺_𝑘=𝐶𝐼_𝑁⊗𝐸𝑧𝑡_𝑘+1−𝜎𝑡_𝑘𝐾𝜎𝑡_𝑘𝐾00𝐼_𝑁⊗𝐸𝑧𝑡_𝑘. (7)

Lemma 3 (see [15]). If 𝐿(𝑡,𝑧(𝑡)) and 𝑈_𝑘(𝑧(𝑡)) satisfy the Lipchitz condition, there exists a uniqueness of solution to nonlinear singular impulsive differential equation which is written as 𝑧(̇𝑡)=𝐿(𝑥,𝑡),𝑡∈𝑡_𝑘−1,𝑡_𝑘,Δ𝑧(𝑡)=𝑈_𝑘(𝑧(𝑡)),𝑡=𝑡⁺_𝑘,𝑘=1,2,…, (8) where 𝑧(𝑡)∈𝑅^𝑛,𝐿∶𝑅₊×𝑅^𝑛→𝑅^𝑛,𝑈_𝑘∶𝑅^𝑛→𝑅^𝑛.

3. The Design of CNSINCS

For the nonlinear singular networked impulsive control system (4), we have the following theorem.

Theorem 4. If there exist 0<𝜌=sup_𝑘∈𝑁{𝑡_𝑘−𝑡_𝑘−1}<∞ and a nonsingular matrix 𝐾∈𝑅^𝑛×𝑛, such that 2ln𝛽𝜌+2𝜆_max𝐴𝜆+2𝐿^max𝐼_𝑁⊗𝐸^𝑇𝑃𝜆_min𝐼_𝑁⊗𝐸^𝑇𝑃+1+𝜆²_max(𝐺⊗Γ)<10<𝛽<1, (9)(10) where 𝛽=𝜆²_max𝑡𝐶+1−𝜎_𝑘𝐾𝜎𝑡_𝑘𝐾00^1/2 (11) then the nonlinear singular networked impulsive control system (4) is asymptotically stable: ‖𝑥(𝑡≤1)‖𝛽𝜆_max(𝑃)𝜆_min(𝑃)sup_{−𝜏≤𝜃≤0}𝜙(𝜃)𝑒^{−(𝜆/2)𝑡}, (12) where 𝜆 is the positive solution of 𝜆+𝑝+𝑞𝑒^𝜆𝜏=0, and 𝑞>0,𝑝+𝑞<0: 𝑝=2ln𝛽𝜌+2𝜆_max𝐴𝜆+2𝐿^max𝐼_𝑁⊗𝐸^𝑇𝑃𝜆_min𝐼_𝑁⊗𝐸^𝑇𝑃+1+𝜆²_max(𝐺⊗Γ). (13)

Proof. From [16], we know that 𝜆+𝑝+𝑞𝑒^𝜆𝜏=0 must have a solution. Set 𝑉(𝑡)=𝑧^𝑇(𝑡)(𝐼_𝑁⊗𝐸)^𝑇𝑃(𝐼_𝑁⊗𝐸)𝑧(𝑡), where (𝐼_𝑁⊗𝐸)^𝑇𝑃(𝐼_𝑁⊗𝐸)≥0.
When 𝑡∈(𝑡_𝑘−1,𝑡_𝑘], the derivative of 𝑉(𝑡) along the trajectories of the CNSINCS (4) is 𝑉̇(𝑡)=𝐼_𝑁⊗𝐸𝑧(̇𝑡)^𝑇𝑃𝐼_𝑁⊗𝐸𝑧(𝑡)+𝐼_𝑁⊗𝐸𝑧(𝑡)^𝑇𝑃^𝑇𝐼_𝑁⊗𝐸𝑧(̇𝑡)=𝐼_𝑁⊗𝐴𝑧(𝑡)+𝐹(𝑥(𝑡0+((𝑡))𝐺⊗Γ000×𝑧𝑡−𝜏))^𝑇𝑃𝐼_𝑁⊗𝐸𝑧(𝑡)+𝐼_𝑁⊗𝐸𝑧(𝑡)^𝑇𝑃𝐼_𝑁⊗𝐴𝑧(𝑡)𝐹(𝑥(𝑡0+𝑧((𝑡))𝐺⊗Γ000𝑡−𝜏))=𝑧^𝑇(𝑡)𝐴^𝑇𝑃𝐼_𝑁⊗𝐸+𝐼_𝑁⊗𝐸^𝑇𝑃^𝑇𝐴×𝑧(𝑡)+𝐹^𝑇(𝑥(𝑡))𝑃𝐼_𝑁⊗𝐸𝑧(𝑡)+𝑧^𝑇(𝑡)𝐼_𝑁⊗𝐸^𝑇𝑃𝐹(𝑥(𝑡))+𝑧^𝑇(𝑡−𝜏(𝑡))𝐺^𝑇𝑃𝐸𝑧(𝑡)+𝑧^𝑇(𝑡)𝐼_𝑁⊗𝐸^𝑇𝑃^𝑇𝐺𝑧(𝑡−𝜏(𝑡)). (14) According to Lemma 3 we have 𝑉̇(𝑡)≤2𝜆_max𝐴𝑧^𝑇(𝑡)𝐼_𝑁⊗𝐸^𝑇𝑃𝐼_𝑁⊗𝐸𝑧(𝑡)+2𝐿𝜆𝐼_𝑁⊗𝐸^𝑇𝑃‖𝑧(𝑡)‖+2𝑧^𝑇(𝑡)𝑃𝐼_𝑁⊗𝐸𝑧(𝑡)×𝑧^𝑇(𝑡−𝜏(𝑡))(𝐺⊗Γ)^𝑇𝐼_𝑁⊗𝐸^𝑇𝑃(𝐺⊗Γ)𝑧(𝑡−𝜏(𝑡))≤𝜆²_max(𝐺⊗Γ)𝑧^𝑇(𝑡−𝜏(𝑡))𝐼_𝑁⊗𝐸^𝑇𝑃𝐼_𝑁⊗𝐸𝑧((𝑡𝜆𝑡−𝜏))+2𝐿^max𝐼_𝑁⊗𝐸^𝑇𝑃𝜆_min𝐼_𝑁⊗𝐸^𝑇𝑃𝑧^𝑇(𝑡)𝐼_𝑁⊗𝐸^𝑇𝑃𝐼_𝑁⊗𝐸𝑧(𝑡)+2𝜆_max𝐴𝑧^𝑇(𝑡)𝐼_𝑁⊗𝐸^𝑇𝑃𝐼_𝑁⊗𝐸𝑧(𝑡)+𝑧^𝑇(𝑡)𝐼_𝑁⊗𝐸^𝑇𝑃𝐼_𝑁⊗𝐸𝑧(𝑡). (15)
On the other hand, when 𝑡=𝑡⁺_𝑘, 𝑉̇𝑡⁺_𝑘=𝑧^𝑇𝑡⁺_𝑘𝐼_𝑁⊗𝐸^𝑇𝑃𝐼_𝑁⊗𝐸𝑧𝑡⁺_𝑘=𝑧^𝑇𝑡⁺_𝑘𝐼_𝑁⊗𝐸^𝑇𝐼_𝑁⊗𝐸^𝑇×𝑃𝐼_𝑁⊗𝐸𝐼_𝑁⊗𝐸𝑧𝑡⁺_𝑘=𝐼_𝑁⊗𝐸𝑧^𝑇𝑡⁺_𝑘^𝑇𝐼_𝑁⊗𝐸^𝑇𝑃𝐼_𝑁⊗𝐸×𝐼_𝑁⊗𝐸𝑧𝑡⁺_𝑘=𝐶𝐼_𝑁⊗𝐸𝑧𝑡_𝑘+1−𝜎𝑡_𝑘𝐾𝜎𝑡_𝑘𝐾00𝐼_𝑁⊗𝐸𝑧𝑡_𝑘^𝑇,𝐼_𝑁⊗𝐸^𝑇𝑃𝐼_𝑁⊗𝐸×𝑡𝐶+1−𝜎_𝑘𝐾𝜎𝑡_𝑘𝐾00𝐼_𝑁⊗𝐸𝑧𝑡_𝑘=𝑧^𝑇𝑡_𝑘𝐼_𝑁⊗𝐸^𝑇×𝑡𝐶+1−𝜎_𝑘𝐾𝜎𝑡_𝑘𝐾00^𝑇×𝐼_𝑁⊗𝐸^𝑇,𝑃𝐼_𝑁⊗𝐸𝑧𝑡_𝑘≤𝜆²_max𝑡𝐶+1−𝜎_𝑘𝐾𝜎𝑡_𝑘𝐾00×𝑧^𝑇𝑡_𝑘𝐼_𝑁⊗𝐸^𝑇𝐼_𝑁⊗𝐸^𝑇𝑃,𝐼_𝑁⊗𝐸𝐼_𝑁⊗𝐸𝑧𝑡_𝑘=𝜆²_max𝑡𝐶+1−𝜎_𝑘𝐾𝜎𝑡_𝑘𝐾00×𝑧^𝑇𝑡_𝑘𝐼_𝑁⊗𝐸^𝑇𝑃𝐼_𝑁⊗𝐸𝑧𝑡_𝑘=𝛽²𝑉𝑡_𝑘, (16) where 𝛽²=𝜆²_max(𝐶+^{(1−𝜎(𝑡𝑘))𝐾𝜎(𝑡𝑘)𝐾}₀₀), supposing that 𝜀>0 is random constant, a comparison system can be established as follows: 𝑣(̇𝑡)=2𝜆_max𝐴𝜆+2𝐿^max𝐼_𝑁⊗𝐸^𝑇𝑃𝜆_min𝐼_𝑁⊗𝐸^𝑇𝑃+1𝑣(𝑡)+𝜆²_max(𝐺⊗Γ)𝑣(𝑡−𝜏(𝑡))+𝜀𝑡≠𝑡_𝑘,𝑣𝑡⁺_𝑘=𝛽𝑣𝑡_𝑘,𝑡=𝑡_𝑘,𝑣(𝜃)=𝜆_max(𝑃)𝜙(𝜃)²,−𝜏≤𝜃≤0. (17) It is clear that 𝑉(𝑡)≤𝑣(𝑡) when −𝜏≤𝜃≤0, according to [16], and we have 𝑉(𝑡)≤𝑣(𝑡) when 𝑡≥0; the trivial solution of the comparison system is 𝑣(𝑡)()𝑣(0)+=𝑊𝑡,0_𝑡0𝑊(𝑡,𝑠)𝜆²_max(𝐺⊗Γ)𝑣(𝑠−𝜏(𝑠)+𝜀)𝑑𝑠,𝑡≥0, (18) where 𝑊(𝑡,𝑠) is Cauchy matrix which satisfies 𝑊(𝑡,𝑠)=𝛽^{2𝜂(𝑡,𝑠)}×𝑒^{{(2𝜆max(𝐴)+2𝐿(𝜆max((𝐼𝑁⊗𝐸)𝑇𝑃)/𝜆min((𝐼𝑁⊗𝐸)𝑇𝑃))+1)(𝑡−𝑠)}}≤𝛽^{2(𝑡−𝑠/𝜌)−1}𝑒^{(𝑝−(2ln𝛽/𝜌))(𝑡−𝑠)}≤𝛽⁻²𝑒^{𝑝(𝑡−𝑠)} (19) in which 𝜂(𝑡,𝑠) is the number of control impulses in the interval (𝑠,𝑡], 0<𝜌=sup_𝑘∈𝑁{𝑡_𝑘−𝑡_𝑘−1}<∞, for 𝑡≥0; we have 𝑣(𝑡)≤𝛽⁻²𝜆_max𝐼_𝑁⊗𝐸^𝑇𝑃𝐼_𝑁⊗𝐸𝜑(0)²+^𝑡₀𝛽⁻²𝑒^{𝑝(𝑡−𝑠)}𝜆²_max(𝐺⊗Γ)𝑣(𝑠−𝜏(𝑠)+𝜀)𝑑𝑠≤𝛾𝑒^𝑝𝑡+𝑒^{𝑝(𝑡−𝑠)}(𝑞𝑣(𝑠−𝜏(𝑠)+𝜀)𝑑𝑠, (20) where 𝛾=𝛽⁻²𝜆_max𝐼_𝑁⊗𝐸^𝑇𝑃𝐼_𝑁⊗𝐸sup_{−𝜏≤𝑠≤0}𝜑(𝑠)²=𝛽⁻²𝜆_max(𝑃)sup_{−𝜏≤𝑠≤0}𝜑(𝑠)². (21)
In the following, we will prove that the following inequality holds: 𝑣(𝑡)≤𝛾𝑒^−𝜆𝑡−𝜀𝛽²𝑝,𝑡≥0 (22) Since 𝜀>0,𝑝<0, so 𝜀/𝛽²𝑝<0. If there exists 𝑡^∗>0 which satisfies 𝑣𝑡^∗≥𝛾𝑒^{−𝜆𝑡∗}−𝜀𝛽²𝑝,𝑣(𝑡)>𝛾𝑒^−𝜆𝑡−𝜀𝛽²𝑝,𝑡<𝑡^∗. (23)(24) From (16) and (24) we have 𝑣𝑡^∗≤𝛾𝑒^{−𝜆𝑡∗}+^𝑡∗₀𝑒^{𝑝(𝑡∗−𝑠)}𝑞𝑣(𝑠−𝜏(𝑠))+𝜀𝑑𝑠<𝑒^𝑝𝑡∗𝜀𝛾−𝛽²+𝑝+𝑞_𝑡∗0𝑒^−𝑝𝑠𝛾𝑞𝑒^{−𝜆(𝑠−𝜏(𝑠))}−𝜀𝑞𝛽²+𝜀𝑝+𝑞𝛽²𝑑𝑠<𝑒^𝑝𝑡∗𝜀𝛾−𝛽²𝑝+𝑞+𝛾𝑞𝑒^𝜆𝜏×^𝑡∗₀𝑒^{−(𝑝+𝜆)𝑠}𝑑𝑠+𝜀𝑝𝛽²𝑝+𝑞^𝑡∗0𝑒^−𝑝𝑠𝑑𝑠<𝑒^𝑝𝑡∗𝜀𝛾−𝛽²𝑒𝑝+𝑞+𝛾^{−(𝑝+𝜆)𝑡∗}−𝜀−1𝛽²𝑒𝑝+𝑞^{−𝑝𝑡∗}−1=𝛾𝑒^{−𝜆𝑡∗}−𝜀𝛽²𝑝+𝑞 (25) which contradicts with (23), and consequently (22) holds. Let 𝜀→0; then 𝑉(𝑡)≤𝑣(𝑡)≤𝛾𝑒^−𝜆𝑡,𝑡≥0. (26) Moreover 𝑉(𝑡)≥𝜆_min(𝑃)‖𝑥(𝑡)‖²,𝑡≥0. (27) Combining the inequality (26) and (27), ‖𝑥(𝑡≤1)‖𝛽𝜆_max(𝑃)𝜆_min(𝑃)sup_{−𝜏≤𝜃≤0}𝜙(𝜃)𝑒^{−(𝜆/2)𝑡} (28)
which implies conclusion (22) and this completes the proof.

Remark 5. For the case 𝛽≥1, we can replace the condition 0<𝜌=sup_𝑘∈𝑁{𝑡_𝑘−𝑡_𝑘−1}<∞ with 0<𝜍=inf_𝑘∈𝑁{𝑡_𝑘−𝑡_𝑘−1}<∞; then the conclusion of Theorem 4 still holds except that now inequality (9) becomes 2ln𝛽𝜍+2𝜆_max𝐴𝜆+2𝐿^max𝐼_𝑁⊗𝐸^𝑇𝑃𝜆_min𝐼_𝑁⊗𝐸^𝑇𝑃+1+𝜆²_max(𝐺⊗Γ)<1. (29) For 𝛽≥1, we have 𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡≤1)𝛽sup_{−𝜏≤𝑡≤0}𝜙(𝑡)𝑒^{−(𝜆/2)𝑡} (30) in which 𝜆=max_𝑘∈𝑁{ln𝛽/(𝑡_𝑘−𝑡_𝑘−1)}.

The proof of the above conclusion remains largely the same as Theorem 4, so we omitted it to avoid repetition.

4. Design Procedure of Impulsive Control for Complex Network

According to Theorem 4, the design process of impulsive control is given as follows.(1)Calculate the parameters 𝐿,𝑚.(2)Choose a matrix 𝑃 which satisfies (𝐼_𝑁⊗𝐸)^𝑇𝑃(𝐼_𝑁⊗𝐸)≥0.(3)For a given parameter 𝜆₀, we can determine the control sequence {𝑡_𝑘}, 𝑡∈𝑁 as follows. If 0<𝛽<1, let Θ∶=(2𝜆_max(𝐴)+2𝐿(𝜆_max((𝐼_𝑁⊗𝐸)^𝑇𝑃)/𝜆_min((𝐼_𝑁⊗𝐸)^𝑇𝑃))+1)+𝜆²_max(𝐺⊗Γ); then the upper bounds of time can be taken as 0<𝜌=sup_𝑘∈𝑁{𝑡_𝑘−𝑡_𝑘−1}=−(ln𝛽)/Θ; if 𝛽≥1, let Θ∶=(2𝜆_max(𝐴)+2𝐿(𝜆_max((𝐼_𝑁⊗𝐸)^𝑇𝑃)/𝜆_min((𝐼_𝑁⊗𝐸)^𝑇𝑃))+1)+𝜆²_max(𝐺⊗Γ); then the lower bounds of control intervals is 0<𝜍=inf_𝑘∈𝑁{𝑡_𝑘−𝑡_𝑘−1}=(ln𝛽)/Θ.

5. Numerical Simulation

In this section, a numerical example is presented to illustrate the effectiveness of derived results.

Example 6. Consider the following complicated nonlinear singular system: 𝐸𝑥̇_𝑖(𝑡)=𝐴𝑥_𝑖(𝑡)+𝑓𝑥_𝑖(𝑡)+^𝑁_𝑗=1𝐺_𝑖𝑗Γ𝑥_𝑗((𝑡,𝑡𝑡−𝜏))𝑖=1,…,𝑁,𝑡∈_𝑘−1,𝑡_𝑘,𝑥_𝑖𝑡⁺_𝑘=𝑐_𝑖𝑥_𝑖𝑡_𝑘+1−𝜎_𝑖𝑡_𝑘𝐾_𝑖𝑥_𝑖𝑡_𝑘+𝜎_𝑖𝑡_𝑘𝐾_𝑖𝑥_𝑖𝑡_𝑘−1,𝑡=𝑡⁺_𝑘𝑥,𝑘=1,2,…,_𝑖(𝑡)=𝜑_𝑖(𝑡),𝑡∈[−𝜏,0]. (31) The parameters are given as follows: 𝐸=1000010000100000,𝑐₁=𝑐₂8=1.2,𝐴=−1010003−10000−830𝑓0021,𝑁=2,𝑡,𝑥_𝑖(𝑡)=0−𝑥_𝑖1(𝑡)𝑥_𝑖3(𝑡)^𝑇𝑥_𝑖1(𝑡)𝑥_𝑖2(𝑡)^𝑇2𝑥_𝑖3(𝑡)^𝑇. (32) For simplicity, consider the system with 2 nodes. Assume that the external coupling matrix is 𝐺=⁻⁷³₃₋₄ and the internal coupling matrix is Γ=0.1−0.2−0.1000.10.20−0.200.100.10.20.1−0.1. (33) Supposing that 𝜏(𝑡)=0.02sint. According to Lemma 1, we can choose 𝑃=𝐼_8×8, 𝑇=𝑡_𝑘+1−𝑡_𝑘=0.005. The region of parameters of chaotic system is 𝐿=80, and the gain of impulsive controller is −1.8653≤𝐾≤−0.5347. The state trajectory diagram of system is depicted in Figure 1. For the case of packet dropouts probability is Pr(𝜎(𝑘)=0∣0.8) and initial condition is 𝑥₁(𝑡)=32−12^𝑇, 𝑥₂(𝑡)=65−48^𝑇, and 𝑡∈(−𝜏,0).

Figure 1 

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

Figure 1 shows that the asymptotic stability of the closed-loop uncertain system can be guaranteed using the networked impulsive controller designed in this paper.

6. 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.

Acknowledgments

This work was supported by the Jiangsu Planned Projects for Postdoctoral Research Funds under Grants 1202005C and the Jiangsu Natural Science Funds for Young Scholar under Grants BK20130696.