Stabilization of a Wireless Networked Control System with Packet Loss and Time Delay: An ADS Approach

Abstract

The stabilization problem of a wireless networked control system is considered in this paper. Both time delay and packet loss exist simultaneously in the wireless network. The system is modeled as an asynchronous dynamic system (ADS) with unstable subsystems. A sufficient condition for the system to be stable is presented. A numerical example is given to demonstrate the effectiveness of the proposed approach.

1. Introduction

Networked control system (NCS) has received increasing attention in recent years due to its advantages, such as low cost, easy maintenance, and flexibility [1], and it has been applied in many areas, such as automobiles, building supervision, and manufacturing plants [2]. As an alternative for the wired network, the wireless network also has a wide range of applications due to its mobility; see [3–5] and the references therein.

As the network is introduced into the control system, many new problems arise due to the limited transmission capacity of the network, such as network-induced delay and packet loss. Many studies have been done in these areas, and some results are available in the literature, for example, the delay issue was studied in [6–12] and packet loss was studied in [13–18]. However, in all the results above, the time delay and packet loss are considered separately, while in practice they exist simultaneously in the network. Recently, some papers also deals with this problem [19–22]. In [19], a continuous system is considered and the effect of packet loss is considered to be equal to the time delay. A new model is proposed in [20] while the delay is assumed to be less than one sampling period. However, as to the discrete-time system, if the plant to be controlled is unstable intuitively, no method has been addressed, which will be considered here. In this paper, a switched system with time-varying delay model is firstly proposed to describe the wireless NCS, and a sufficient condition is given to stabilize the closed-loop system, and the controller is designed to be dependent on both the packet loss and time delay. A numerical example shows the effectiveness of the proposed approach.

The organization of this paper is as follows. The structure of the wireless network control system is given in Section 2 and a switched system with time-varying delay model is established. New criteria for the switched time-varying delay system with unstable subsystem to be stable are obtained in Section 3. The controller design problem is solved in Section 4, while Section 5 gives an example to illustrate the effectiveness of the proposed approach. Finally, the conclusion is given in Section 6.

2. Modeling of the Wireless NCS

In practise, the sensors may be very far from the controller and actuators, for example [16], where the sensors and the controller are connected by a wireless network. The system structure is as in Figure 1.

Figure 1 

System configuration.

As the wireless network is introduced into the system, the data, from the sensor to the controller, will be lost or delayed due to the disturbance in the environment and the limited transmission capacity of the wireless network. The graph of the system can be described as in Figure 2, where the system to be controlled is 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵𝑢(𝑘),(1)𝑥(𝑘)∈ℝ^𝑛 is the system state, 𝑢(𝑘)∈ℝ^𝑚 is the control input 𝑑_𝑘 is the time-varying delay of the network, and 𝐾_delay,loss is the controller to be designed, which depends on the time delay and packet loss, denoted as 𝐾_𝑑,𝑠 in the graph for simplicity. The data loss in the network can be viewed as the behaviour results of a pair of switches [14]. If the switches are closed, the data is transmitted successfully (3), otherwise, the previous value will be used in the controller (2). Thus, the dynamics of the switches can be modeled as 𝑆∶𝑤(𝑘)=𝑤(𝑘−1),(2)𝑆∶𝑥𝑥(𝑘𝑘)−𝑑_𝑘Notdelayed,Delayed.(3)

Figure 2 

System graph.

Without loss of generality, the following assumptions are made for the description of the system model.(A1)The sensors and controller are time driven.(A2)The data transmitted by the sensors are time stamped, so the controller can obtain the delay value of each data packet.(A3)The maximum delay in the network is 𝑑_𝑀, where 𝑑_𝑀 is a known integer.

Let 𝑧(𝑘)=𝑥^𝑇(𝑘)𝑤^𝑇(𝑘−1)^𝑇.(4) Then the dynamics of the closed-loop system can be described by the following subsystems.

Case 1. When the data is lost, the corresponding controller is denoted as 𝐾_0,0, and the closed-loop system is described as Σ₁∶𝑧(𝑘+1)=𝐴₁𝑧(𝑘),𝐴₁=⎡⎢⎢⎣𝐴𝐵𝐾^0,00𝐼⎤⎥⎥⎦.(5)

Case 2. When the data is transmitted successfully without delay, the corresponding controller is denoted as 𝐾_0,1, and the closed-loop system is described as Σ₂∶𝑧(𝑘+1)=𝐴₂𝑧(𝑘),𝐴₂=⎡⎢⎢⎣𝐴+𝐵𝐾^0,1𝐼00⎤⎥⎥⎦.(6)

Case 3. When the data is transmitted successfully with 𝑑_𝑘-step delay, the corresponding controller is denoted as 𝐾_𝑑𝑘,1, and the closed-loop system is described as Σ₃𝐴∶₃𝑧(𝑘+1)=𝐴₃𝑧(𝑘)+𝐴_𝑑3𝑧𝑘−𝑑_𝑘,=⎡⎢⎢⎣𝐴000⎤⎥⎥⎦,𝐴_𝑑3=⎡⎢⎢⎣𝐵𝐾^𝑑𝑘,1𝐼00⎤⎥⎥⎦.(7) So, the closed-loop system can be described as a system switched within three subsystems: Σ₁, Σ₂, and Σ₃.
If 𝑑_𝑘=0 in Σ₃, then Σ₃ becomes Σ₂, and the difference of Σ₂ and Σ₁ is only the first element value of its system matrix. So, Case 3 includes Cases 1 and 2 by appropriately changing the value of the matrices. So the wireless networked control system can be represented by the following discrete-time switched system with time-varying delay: 𝑧(𝑘+1)=𝐴_𝑖𝑧(𝑘)+𝐴_𝑑𝑖𝑧𝑘−𝑑𝑑_𝑘,𝑖=1,2,3,_𝑘=1,…,𝑑_𝑀,(8) where 𝐴_𝑑1=𝐴_𝑑2=0.
From above, it is easy to see that if the open system is unstable, when the data loss happens, the system will be uncontrollable, and the corresponding subsystem Σ₁ is unstable.
The following lemma plays a crucial role in the proof of our main results.

Lemma 1 (see [23]). For any appropriately dimensioned matrices 𝑅=𝑅^𝑇>0, 𝑌, 𝑋, and two positive integer time-varying 𝑑_𝑘1, 𝑑_𝑘2 satisfying 𝑑_𝑘1+1≤𝑑_𝑘2<𝑑_𝑀, the following equality holds: −^{𝑘−𝑑𝑘1−1}_𝑙=_{𝑘−𝑑𝑘2}𝜂^𝑇(𝑙)𝑅𝜂(𝑙)=+2𝜉^𝑇𝑑(𝑘)𝑌𝑥𝑘−𝑑_k1−𝑥𝑘−𝑑_𝑘2_𝑘2−−𝑑_𝑘1𝜉^𝑇(𝑘)𝑋𝜉(𝑘)_{𝑘−𝑑𝑘1−1𝑙=}_{𝑘−𝑑𝑘2}⎡⎢⎢⎣𝜉𝜂((𝑘𝑙))⎤⎥⎥⎦^𝑇⎡⎢⎢⎣𝑋∗𝑅𝑌⎤⎥⎥⎦⎡⎢⎢⎣𝜉𝜂((𝑘𝑙))⎤⎥⎥⎦.(9)

3. Stability of the Switched System with Time-Varying Delay

As to the discrete-time switched system with time-varying delay, some results are obtained in the literature to test its stability [24–30]. However, all the results above are obtained under the assumption that all the subsystems are stable and they cannot be applied here. So we will give criteria to test the stability of the discrete-time switched time-varying delay system with unstable subsystem in this section.

Consider the following discrete-time switched system with time-varying delay, 𝑧(𝑘+1)=𝐴_𝑖𝑧(𝑘)+𝐴_𝑑𝑖𝑧𝑘−𝑑_𝑘,𝑑𝑖=1,2,…,𝑁._𝑘=1,…,𝑑_𝑀,(10) where 𝑁 is the number of the subsystems.

Choose the following Lyapunov function: 𝑉𝑉(𝑘)=𝑉₁(𝑘)+𝑉₂(𝑘)+𝑉₃(𝑘)+𝑉₄(𝑘),₁𝑉(𝑘)≜𝑧^𝑇(𝑘)𝑃𝑧(𝑘),₂(𝑘)≜^𝑘−1_𝑙=_{𝑘−𝑑𝑘}+𝜆^{−2(𝑙−𝑘)}𝑧^𝑇(𝑙)𝑄₁𝑧(𝑙)₀_{𝑚=−𝑑𝑀+2𝑘−1𝑙=𝑘+𝑚−1}𝑉𝜆^{−2(𝑙−𝑘)}𝑧^𝑇(𝑙)𝑄₁𝑧(𝑙),₃(𝑘)≜^𝑘−1_𝑙=_{𝑘−𝑑𝑀}𝑉𝜆^{−2(𝑙−𝑘)}𝑧^𝑇(𝑙)𝑄₂𝑧(𝑙),₄(𝑘)≜⁻¹_{𝑚=−𝑑𝑀}^𝑘−1_{𝑙=𝑘+𝑚}𝜆^{−2(𝑙−𝑘)}𝜂^𝑇(𝑙)𝑅𝜂(𝑙),(11) where 𝜂(𝑙)=𝜆⁻¹𝑧(𝑙+1)−𝑧(𝑙).

In the following lemma, a criterion is given to guarantee that the Lyapunove function 𝑉(𝑘) exponentially decays or increases along any state trajectory of system (10).

Lemma 2. For a given scalar 𝜆>0 and any delay satisfying 0≤𝑑≤𝑑_𝑀, if there exist symmetric positive definite matrices 𝑃, 𝑄₁, 𝑄₂, 𝑅, symmetric matrix 𝑋=^𝑋1𝑋2_∗𝑋3≥0 and appropriate matrices 𝐺,𝑁=[𝑁^𝑇1𝑁^𝑇2]^𝑇, 𝑀=[𝑀^𝑇1𝑀^𝑇2]^𝑇, such that the following LMIs hold: Ψ𝑑_𝑘=⎡⎢⎢⎢⎢⎢⎢⎣Ψ¹¹∗ΨΨ₁₂₂₂−𝑁₁𝑃+𝐴_𝑖−𝐼𝐺^𝑇−𝐺−𝑁₂𝐴_𝑑𝑖𝐺^𝑇∗∗∗∗−∗𝑄₂𝑃+𝑑_𝑀0𝑅−𝐺−𝐺^𝑇𝑑⎤⎥⎥⎥⎥⎥⎥⎦<0,_𝑘=0,…,𝑑_𝑀⎡⎢⎢⎣𝑋∗𝑁𝑅⎤⎥⎥⎦>0,⎡⎢⎢⎣𝑋∗𝑀𝑅⎤⎥⎥⎦>0,(12) where Ψ₁₁=𝑑_𝑀Ψ𝑄₁+𝑄₂+𝑑_𝑀𝑋₁+Sym𝑀₁+𝐺𝐴^𝑇𝑖−𝐼₁₂,Ψ₂₂=d_𝑀𝑋₂+𝑁₁−𝑀₁+𝑀^𝑇2+𝐺𝐴^𝑇𝑑𝑖,=−𝑄₁+𝑑_𝑀𝑋₃+Sym𝑁₂−𝑀₂,(13) and 𝐴_𝑖=𝜆⁻¹𝐴_𝑖, 𝐴_𝑑𝑖=𝜆^{−1−𝑑𝑘}𝐴_𝑑𝑖, then 𝑉(𝑘)<𝜆^{2(𝑘−𝑘0)}𝑉𝑘₀.(14)

Proof. See the appendix.

Remark 3. The last two terms of Δ𝑉₄ are abandoned to obtain our result, which may introduce some conservatism. In order to reduce the conservatism, a free matrix 𝐺 is introduced.

Suppose that the rates of the corresponding subsystems are 𝑟₁,𝑟₂,…,𝑟_𝑁, which represent the fraction of time that each subsystem occurs, thus ∑^𝑁𝑖=1𝑟_𝑖=1. The following theorem gives a sufficient condition for the system (10) to be stable.

Theorem 4. The switched system with time-varying delay (10) is stable if there exist a Lyapunov function 𝑉(𝑘) and scalars 𝜆₁,…,𝜆_𝑁 corresponding to each subsystem such that 𝑉(𝑘)<𝜆^{2(𝑘−𝑘0)𝑖}𝑉𝑘₀,𝑖=1,…,𝑁_𝑁,_𝑖=1𝜆^𝑟𝑖_𝑖<1.(15)

Proof. Suppose that the transition time of the subsystems is 𝑡₁=0,𝑡₂,…,𝑡_𝑘=𝑘, then 𝑉(𝑘)<<𝜆⋯^{𝑡𝑘−𝑡𝑘−1}_𝑡𝑘<𝑉_𝑁𝑡_𝑘−1<𝜆^{𝑡𝑘−𝑡𝑘−1}_𝑡𝑘𝜆^{𝑡𝑘−1−𝑡𝑘−2}_𝑡𝑘−1𝑉𝑡_𝑘−2_𝑖=1𝜆^𝑇𝑖_𝑖𝑉(0),(16) where 𝑇_𝑖 is the total time that one subsystem occurs and as 𝑘→∞, 𝑟_𝑖𝑘=𝑇_𝑖. Thus, 𝑉(𝑘)<^𝑁_𝑖=1𝜆^𝑇𝑖_𝑖𝑉(0)=^𝑁_𝑖=1𝜆^𝑟𝑖_𝑖^𝑘𝑉(0).(17) It is obvious that the system is stable if ∏^𝑁𝑖=1𝜆^𝑟𝑖_𝑖<1, which completes the proof.

4. Controller Design for the Wireless NCS

In this section, the controller of the wireless networked control system is given using the criteria in Section 3. As the data packets are time stamped, and the packet loss and time delay are known to the controller, the controller is designed to depend on both the packet loss and time delay.

Theorem 5. The system (8) is stable if there exist scalars 𝜆_𝑖, 𝑖=1,2,3, satisfying ∏^3𝑖=1𝜆^𝑟𝑖_𝑖<1 and there exist 2𝑛×2𝑛 symmetric positive definite matrices 𝒫, 𝒬₁, 𝒬₂, ℛ, 𝒳₁, 𝒳₃, matrices ℳ₁, ℳ₂, 𝒩₁, 𝒩₂, 𝒳₂ and 𝑛×𝑛 matrices 𝐺₁, 𝐺₂, 𝑚×𝑛 matrices 𝐾_𝑔0,0, 𝐾_𝑔0,1, 𝐾_{𝑔𝑑𝑘,1},𝑑_𝑘=1,…,𝑑_𝑀 such that the following LMIs hold: ΦΦ₀+Φ₁<0,₀Φ+Φ₂<0,₀⎡⎢⎢⎢⎢⎣𝒳₁+Φ₃𝑑_𝑘<0,𝑑_𝑘=1,…,𝑑_𝑀,𝒳₂𝒩₁∗𝒳₃𝒩₂∗∗ℛ⎤⎥⎥⎥⎥⎦>0⎡⎢⎢⎢⎢⎣𝒳¹𝒳₂ℳ₁∗𝒳₃ℳ₂∗∗ℛ⎤⎥⎥⎥⎥⎦>0,(18) whereΦ₀=⎡⎢⎢⎢⎢⎢⎢⎣Θ¹¹∗−𝒬₁𝑑_𝑀𝒳₂+𝒩₁−ℳ₁+ℳ^𝑇2−𝒩₁𝒫−𝒢^𝑇−𝒢∗+𝑑_𝑀𝒳₃+∗Sym𝒩₂−ℳ₂−−𝒩𝒬₂₂0∗∗∗𝒫+𝑑_𝑀0ℛ−𝒢−𝒢^𝑇⎤⎥⎥⎥⎥⎥⎥⎦,(19)Θ𝒢₁₁==𝑑⎡⎢⎢⎣𝐺_𝑀1𝒬₁+𝒬₂+𝑑_𝑀𝒳₁+Symℳ₁+𝒢,Φ₁0𝐺0₂⎤⎥⎥⎦,=⎡⎢⎢⎢⎢⎢⎢⎣Λ¹¹00Λ₁₄00000000Λ₁₁0000⎤⎥⎥⎥⎥⎥⎥⎦,=⎡⎢⎢⎣𝜆¹⁻¹𝐴𝐺^𝑇1+𝐺₁𝜆¹⁻¹𝐴^𝑇𝐵𝐾_𝑔0,0Λ∗0₁₄⎤⎥⎥⎦,=⎡⎢⎢⎣𝜆¹⁻¹𝐴𝐺^𝑇1𝐵𝐾_𝑔0,0Φ00₂⎤⎥⎥⎦,=⎡⎢⎢⎢⎢⎢⎢⎣Ω¹¹00Ω₁₄00000000Ω₁₁0000⎤⎥⎥⎥⎥⎥⎥⎦,=⎡⎢⎢⎣Sym𝜆²⁻¹Ω𝐴𝐺∗^𝑇1+𝐵𝐾_𝑔0,100₁₄⎤⎥⎥⎦,=⎡⎢⎢⎣𝜆²⁻¹Φ𝐴𝐺^𝑇10+𝐵𝐾_𝑔0,100₃⎤⎥⎥⎦,=⎡⎢⎢⎢⎢⎢⎢⎣Υ¹¹Υ₁₂0Υ₁₄000Υ₂₄0000Υ₁₁0000⎤⎥⎥⎥⎥⎥⎥⎦,=⎡⎢⎢⎣𝜆³⁻¹Υ𝐴0𝐺^𝑇10₁₂0⎤⎥⎥⎦,=⎡⎢⎢⎣𝐾^{𝑔𝑑𝑘,1}𝐵^𝑇𝐺₁Υ00₁₄⎤⎥⎥⎦,=⎡⎢⎢⎣𝜆³⁻¹𝐴0𝐺^𝑇100⎤⎥⎥⎦,Υ₂₄=⎡⎢⎢⎣𝐵𝐾^{𝑔𝑑𝑘,1}𝐺_𝑇100⎤⎥⎥⎦,(20) and the controllers are given by 𝐾𝐾_{0,0𝑑𝑘,1}=𝐾_𝑔0,0𝐺^2−𝑇,𝐾_0,1=𝐾_𝑔0,1𝐺^1−𝑇,=𝐾_{𝑔𝑑𝑘,1}𝐺^1−𝑇,𝑑_𝑘=1,…,𝑑_𝑀.(21)

Proof. Substituting the system matrices of (8) into (12) and letting 𝐾_𝑔0,0=𝐾_0,0𝐺^𝑇2,𝐾_𝑔0,1=𝐾_0,1𝐺^𝑇1,and 𝐾_{𝑔𝑑𝑘,1}=𝐾_𝑑𝑘,1𝐺^𝑇1, it is easy to get the results according to Lemma 2 and Theorem 4.

Remark 6. The computing complexity of Theorem 5: the line number of the LMIs is (36𝑛+8𝑑_𝑀𝑛) and the number of the scalar variables is (32𝑛²+5𝑛+3𝑚𝑛).

5. Numerical Example

In this section, an example is given to show the effectiveness of the proposed approach.

Consider the following discrete-time system: 𝐴=⎡⎢⎢⎢⎢⎣000...891839610218000...285599873694000...568011329884⎤⎥⎥⎥⎥⎦,𝐵=⎡⎢⎢⎢⎢⎣111⎤⎥⎥⎥⎥⎦.(22) The eigenvalues of 𝐴 are 1.8734, 0.3650+0.2369𝑖, and 0.3650−0.2369𝑖 and the system will diverge if there is no control. Suppose the event rates of the packet loss and time delay are 𝑟₁=0.4, and 𝑟₃=0.5 respectively, and the maximum delay is 𝑑_𝑀=3, and choose 𝜆₁=1.9,𝜆₂=0.4, and 𝜆₃=0.7. As 𝜆^𝑟1₁𝜆^𝑟2₂𝜆^𝑟3₃=1.9^0.4×0.4^0.1×0.7^0.5=0.9869<1,(23) using Theorem 5, a feasible solution is given and the corresponding controllers are 𝐾𝐾_0,00,1=−0.00090.3295−0.2700×10⁻³,𝐾=−0.7707−0.6772−0.7465_1,1,𝐾=−0.5713−0.5048−0.5528_2,1,𝐾=−0.3999−0.3533−0.3870_3,1,=−0.2799−0.2473−0.2709.(24) The simulation is shown in Figure 3. The first graph depicts the time delay and packet loss of the wireless network, if the delay value is −1, it means the packet is lost. The second graph is the trajectories of the closed-loop system. We can see that the trajectories of the system diverge at the instance when the packets are lost, while they converge to zeros at last, which proves the effectiveness of the proposed approach.

(a)

(b)

Figure 3 

System simulation.

6. Conclusion

The stabilization problem for a wireless networked control system is studied in this paper. A new model is proposed to describe the system as a discrete-time switched system with time-varying delays. New criteria for the system to be stable with unstable subsystem are obtained, based on which, delay- and packet-loss-dependent controller is given to stabilize the system. A numerical example illustrates the effectiveness of the proposed approach.

Appendix

Proof of Lemma 2.

Proof. Applying the transformation 𝑧(𝑘)=𝜆^{(𝑘−𝑘0)}𝜉(𝑘),𝜆>0, we obtain the following system from (10): 𝜉(𝑘+1)=𝐴_𝑖𝜉(𝑘)+𝐴_𝑑𝑖𝜉𝑘−𝑑_𝑘,𝑑𝑖=1,2,…,𝑁,_𝑘=1,…,𝑑_𝑀,(A.1) where 𝐴_𝑖=𝜆⁻¹𝐴_𝑖 and 𝐴_𝑑𝑖=𝜆^{−1−𝑑𝑘}𝐴_𝑑𝑖.
Choose the following Lyapunov function for system (A.1): 𝑊𝑊(𝑘)=𝑊₁(𝑘)+𝑊₂(𝑘)+𝑊₃(𝑘)+𝑊₄(𝑘),₁𝑊(𝑘)≜𝜉^𝑇(𝑘)𝑃𝜉(𝑘),₂(𝑘)≜^𝑘−1_𝑙=_{𝑘−𝑑𝑘}𝜉^𝑇(𝑙)𝑄₁𝜉(𝑙)+⁰_{𝑚=−𝑑𝑀+2𝑘−1𝑙=𝑘+𝑚−1}𝑊𝜉^𝑇(𝑙)𝑄₁𝜉(𝑙),₃(𝑘)≜^𝑘−1_𝑙=_{𝑘−𝑑𝑀}𝑊𝜉^𝑇(𝑙)𝑄₂𝜉(𝑙),₄(𝑘)≜⁻¹_{𝑚=−𝑑𝑀}^𝑘−1_{𝑙=𝑘+𝑚}𝛿^𝑇(𝑙)𝑅𝛿(𝑙),(A.2) where 𝛿(𝑙)=𝜉(𝑙+1)−𝜉(𝑙). Using Lemma 1, it is easy to obtain that Δ𝑊(𝑘)=+𝜉^𝑇𝜉_𝑇(𝑘+1)𝑃𝜉^𝑇(𝑘+1)−𝜉^𝑇(𝑘)𝑃𝜉(𝑘)(𝑘)𝑄₁𝜉(𝑘)+^𝑘−1_𝑙=𝑘−_𝑑𝑘+1+1−𝜉^𝑇(𝑙)𝑄₁𝜉(𝑙)_{𝑘−1𝑙=𝑘}_{−𝑑𝑘+1}−𝜉_𝑇𝜉^𝑇(𝑙)𝑄₁𝜉(𝑙)+𝜉^𝑇(𝑘)𝑄₂𝜉(𝑘)+𝑑_𝑀𝑘−𝑑_𝑀𝑄₂𝜉𝑘−𝑑_𝑀𝛿^𝑇(𝑘)𝑅𝛿(𝑘)+𝑑_𝑀⎡⎢⎢⎣𝜉𝑘𝜉(−𝑘)𝑑_𝑘⎤⎥⎥⎦^𝑇𝑋⎡⎢⎢⎣𝜉𝑘𝜉(−𝑘)𝑑_𝑘+2⎡⎢⎢⎣𝜉𝑘𝜉(−𝑘)𝑑_𝑘⎤⎥⎥⎦⎤⎥⎥⎦^𝑇⎡⎢⎢⎣𝑁₁𝑁₂⎤⎥⎥⎦𝜉𝑘−𝑑_𝑘+2⎡⎢⎢⎣𝜉𝑘𝜉(−𝑘)𝑑_𝑘−𝜉𝑘−𝑑_𝑀⎤⎥⎥⎦⎡⎢⎢⎣𝑀𝑀¹₂⎤⎥⎥⎦𝜉(𝑘)−𝜉𝑘−𝑑_𝑘−_{𝑘−𝑑𝑘−1𝑙=}_{𝑘−𝑑𝑀}⎡⎢⎢⎢⎢⎣𝜉𝑘𝜉(−𝑘)𝑑_𝑘𝛿(𝑙)⎤⎥⎥⎥⎥⎦^𝑇⎡⎢⎢⎣𝑋∗𝑁𝑅⎤⎥⎥⎦⎡⎢⎢⎢⎢⎣𝜉𝑘𝜉(−𝑘)𝑑_𝑘−𝛿(𝑙)_𝑘−1⎤⎥⎥⎥⎥⎦_𝑙=_{𝑘−𝑑𝑘}⎡⎢⎢⎢⎢⎣𝜉𝑘𝜉(−𝑘)𝑑_𝑘𝛿(𝑙)⎤⎥⎥⎥⎥⎦^𝑇⎡⎢⎢⎣𝑋∗𝑀𝑅⎤⎥⎥⎦⎡⎢⎢⎢⎢⎣𝜉𝑘𝜉(−𝑘)𝑑_𝑘+2𝑥_𝑇𝛿(𝑙)⎤⎥⎥⎥⎥⎦(𝑘)+𝜂^𝑇(𝑘)𝐺+𝐴𝐴^𝑇𝑖−_𝑇𝑑𝑖𝐼𝑥(𝑘)<𝜁_𝑇𝑥𝑘−𝑑_𝑘−𝜂(𝑘)(𝑘)Ψ𝑑_𝑘𝜁(𝑘)<0,(A.3) where 𝜁^𝑇(𝑘)=[𝜉^𝑇(𝑘)𝜉(𝑘−𝑑_𝑘)𝜉^𝑇(𝑘−𝑑_𝑀)𝛿^𝑇(𝑘)], which implies that 𝑊(𝑘)<𝑊(𝑘₀) for any 𝑘≥𝑘₀. Furthermore, 𝑉𝑉₁(𝑘)=𝑧^𝑇(𝑘)𝑃𝑧(𝑘)=𝜆^{2(𝑘−𝑘0)}𝜉^𝑇(𝑘)𝑃𝜉(𝑘)=𝜆^{2(𝑘−𝑘0)}𝑊₁(𝑘),₂(𝑘)=^𝑘−1_𝑙=_{𝑘−𝑑𝑘}+𝜆^{−2(𝑙−𝑘)}𝜆^{2(𝑙−𝑘0)}𝜉^𝑇(𝑙)𝑄₁𝜉(𝑙)₀_{𝑚=−𝑑𝑀+2𝑘−1𝑙=𝑘+𝑚−1}=𝜆_{2(𝑘−𝑘0)}𝜆^{−2(𝑙−𝑘)}𝜆^{2(𝑙−𝑘0)}𝜉^𝑇(𝑙)𝑄₁𝜉(𝑙)𝑉𝑊₂(𝑘),₃(𝑘)=^𝑘−1_𝑙=_{𝑘−𝑑𝑀}=𝜆𝜆^{−2(𝑙−𝑘)}_{2(𝑘−𝑘0)}𝜆^{2(𝑙−𝑘0)}𝜉^𝑇(𝑙)𝑄₂𝜉(𝑙)𝑉𝑊₃(𝑘),₄(𝑘)≜⁻¹_{𝑚=−𝑑𝑀}^𝑘−1_{𝑙=𝑘+𝑚}𝜆^{−2(𝑙−𝑘)}𝜆^{2(𝑙−𝑘0)}𝛿^𝑇(𝑙)𝑅𝛿(𝑙)=𝜆^{2(𝑘−𝑘0)}𝑊₄(𝑘),(A.4) which means that 𝑉(𝑘)=𝜆^{2(𝑘−𝑘0)}𝑊(𝑘).
Noting that 𝑊(𝑘₀)=𝑉(𝑘₀), it is easy to verify that 𝑉(𝑘)<𝜆^{2(𝑘−𝑘0)}𝑉(𝑘₀). This completes the proof.

Acknowledgments

This work is supported by the National Creative Research Groups Science Foundation of China under Grant 60721062, the National Natural Science Foundation of China under Grant 60736021, the National High Technology Research and Development Program of China under Grant 863 Program 2008AA042902, the National Natural Science Foundation of China under Grant 61004034, National Natural Science Foundation of China under Grant 61104102 and China Postdoctoral Science Foundation under Grant 20100480086.