Finite-Time Boundedness and Stabilization of Networked Control Systems with Time Delay

Abstract

The finite-time control problem of a class of networked control systems (NCSs) with time delay is investigated. The main results provided in the paper are sufficient conditions for finite-time stability via state feedback. An augmentation approach is proposed to model NCSs with time delay as linear systems. Based on finite time stability theory, the sufficient conditions for finite-time boundedness and stabilization of the underlying systems are derived via linear matrix inequalities (LMIs) formulation. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed results.

1. Introduction

Networked control systems (NCSs) are feedback control systems with control loops closed via digital communication channels. Compared with the traditional point-to-point wiring, the use of the communication channels can reduce the costs of cables and power, simplify the installation and maintenance of the whole system, and increase the reliability. NCSs have many industrial applications in automobiles, manufacturing plants, aircrafts, and HVAC systems [1]. However, the insertion of communication networks in feedback control loops makes the NCSs analysis and synthesis complex; see [2–8] and the references therein.

One issue inherent to NCSs, however, is the network-induced delay that occurs while exchanging date among devices connected to the shared medium. This delay, either constant or time varying, can degrade the performance of control systems designed without considering it and even destabilize the system. Thus the issues of stability analysis for NCSs have received considerable attention for decades [9–15]. In [9, 10], NCSs with random delays are modelled as jump linear systems with two modes; the necessary and sufficient conditions on the existence of stabilizing controllers are given. By introducing indicator functions, mean-square asymptotic stability is derived for the closed-loop networked control system in [11]. Based on a discrete system model with time-varying input delays, stability analysis and control design are carried out in [12, 13]. In [14], an observer-based stabilizing controller has been designed for networked systems involving both random measurement and actuation delays. In [15], a novel state feedback 𝐻_∞ control with the compensator for the effects of network delays in both forward and feedback channels is proposed by introducing an augmented state variable.

On the other hand, finite-time boundedness and stability can be used in all those applications where large values of the state should not be attained, for instance, in the presence of saturations. However, most of the results in the literature are focused on Lyapunov stability. Some early results on finite-time stability (FTS) can be found in [16], more recently the concept of FTS has been revisited in the light of recent results coming from linear matrix inequalities (LMIs) theory, which has made it possible to find less conservative conditions for guaranteeing FTS and finite time stabilization of discrete-time and continuous-time systems [17–26]. In [27, 28], sufficient conditions for finite-time stability of networked control systems with packet dropout are provided; however, controller design methods are not given.

To the best of our knowledge, the finite-time stabilization problems for NCSs with delay have not been fully investigated to date. Especially for the case where the plant subjects to external interference, very few results related to NCSs are available in the existing literature, which motivates the study of this paper. The main contributions of this paper are definitions of finite-time boundedness and stabilization are extended to NCSs. Furthermore, sufficient conditions for finite-time boundedness and stabilization linear matrix inequalities formulation are given.

In this paper, the finite-time stabilization and boundedness problems of a class of NCSs with time delay are studied. The sufficient conditions for finite-time stabilization and boundedness of the underlying systems are derived via LMIs formulation. Lastly, an illustrative example is given to demonstrate the effectiveness of the proposed methods.

This paper is organized as follows. An augmentation approach is proposed to model NCSs with time delay as linear system in Section 2. The finite-time stabilization and boundedness conditions for NCSs with time delay are derived via LMIs in Section 3. Section 4 provides a numerical example to illustrate the effectiveness of our results. Finally, Section 5 gives some concluding remarks.

2. Problem Formulation and Preliminaries

Consider NCS depicted in Figure 1 consists of three components: a plant to be controlled, a network such as the Internet, and a controller.

Figure 1 

Illustration of NCSs over communication network.

In this paper, it is assumed that the plant is described by 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡)+𝐺𝑤(𝑡) (2.1) and time-invariant controller 𝑢(𝑘ℎ)=−𝐾𝑥(𝐾ℎ),𝑘=0,1,2,…, (2.2) where 𝑥(𝑡)∈ℝ^𝑛 is the state, 𝑢(𝑡)∈ℝ^𝑚 is the control input, and 𝑤(𝑡)∈ℝ^𝑞 is the exogenous input. 𝐴, 𝐵, and 𝐺 are known real constant matrices with appropriate dimensions. The sampling period ℎ is fixed and known. There are two sources of delays from the network: the sensor-to-controller delay 𝜏_sc and the controller-to-actuator delay 𝜏_ca. For the fixed control law, the sensor-to-controller delay and the controller-to-actuator delay can be lumped together as 𝜏=𝜏_sc+𝜏_ca for analysis purpose. We make the following assumptions about NCSs.

Assumption 2.1. The sensors are clock-driven sensors, and controllers and actuators are event-driven.

Assumption 2.2. The network-induced delay is constant and less than one sampling period.

Assumption 2.3. During the finite time 𝑇, there exists a positive constant 𝑑, such that the exogenous input 𝑤(𝑡) satisfies ^𝑇₀𝑤^𝑇(𝑡)𝑤(𝑡)𝑑𝑡≤𝑑². (2.3)

Then the system equation can be written as (𝑡)(𝑡)(𝑡)(𝑡)[()),𝑦(𝑡)(𝑡),𝑢𝑡𝑥̇=𝐴𝑥+𝐵𝑢+𝐺𝑤,𝑡∈𝑘ℎ+𝜏,𝑘+1ℎ+𝜏=𝐶𝑥₊=−𝐾𝑥(𝑡−𝜏),𝑡∈{𝑘ℎ+𝜏,𝑘=1,2,…}. (2.4) Sampling the system with period ℎ, we obtain 𝑥(𝑘+1)=Φ𝑥(𝑘)+Γ₀(𝜏)𝑢(𝑘)+Γ₁(𝜏)𝑢()(𝑘),𝑦(𝑡)(𝑡),𝑘−1+Ψ𝑤=𝐶𝑥 (2.5) where Φ=𝑒^𝐴ℎ,Ψ=^ℎ₀𝑒^𝐴𝑠𝐺𝑑𝑠. (2.6)Γ₀(𝜏) and Γ₁(𝜏) are defined as follows: Γ₀(𝜏)=₀^ℎ−𝜏𝑒^𝐴𝑠𝐵𝑑𝑠,Γ₁(𝜏)=^ℎ_ℎ−𝜏𝑒^𝐴𝑠𝐵𝑑𝑠. (2.7) Define the augmented state vector 𝑥(𝑘) as follows: 𝑥(𝑘)=[𝑥(𝑘),𝑢(𝑘−1)]^𝑇 (2.8) and the augmented exogenous input vector 𝑤(𝑘) as 𝑤(𝑘)=[𝑤(𝑘),0]^𝑇. (2.9) Then we have the augmented closed-loop system 𝑥(𝑘+1)=𝐴+𝐵𝐾𝑥(𝑘)+𝐺𝑤(𝑘), (2.10) where 𝐴=ΦΓ₁(𝜏),00𝐵=−Γ₀(𝜏),−𝐼𝐺=Ψ000 (2.11) and 𝐾 is defined as follows 𝐾=𝐾0. (2.12)

Remark 2.4. According to Assumption 2.3, we can derive that there exists a positive constant 𝑑, such that the condition ^𝑁_𝑘=1𝑤^𝑇(𝑘)𝑤(𝑘)≤𝑑² (2.13) is satisfied, for finite positive integer 𝑁.

Remark 2.5. When the delay is longer than one sampling period, that is to say, ℎ<𝜏<𝑙ℎ, where 𝑙>1, the augmented state vector 𝑥(𝑘) is defined as 𝑥(𝑘)=[𝑥(𝑘),𝑢(𝑘−𝑙),…,𝑢(𝑘−1)]^𝑇 (2.14) and the corresponding augmented closed-loop system can be derived.

The main aim of this paper is to find some sufficient conditions which guarantee that the system given by (2.10) is bounded over a finite-time interval. The general idea of finite-time stability concerns the boundedness of the state of a system over a finite time interval for given initial conditions; this concept can be formalized through the following definitions.

Definition 2.6. System (2.10) with (2.13) is said to be finite-time bounded with respect to (𝛼,𝑑,𝛽,𝑅,𝑁), where 𝑅 is a positive-definite matrix, 0<𝛼<𝛽, if 𝑥^𝑇(0)𝑅𝑥(0)≤𝛼²⟹𝑥^𝑇(𝑘)𝑅𝑥(𝑘)≤𝛽²,𝑘∈{1,…,𝑁}. (2.15)

Definition 2.7. System (2.10) with 𝑤(𝑘)=0 is said to be finite-time stable with respect to (𝛼,𝛽,𝑅,𝑁), where 𝑅 is a positive-definite matrix, 0<𝛼<𝛽, if 𝑥^𝑇(0)𝑅𝑥(0)≤𝛼²⟹𝑥^𝑇(𝑘)𝑅𝑥(𝑘)≤𝛽²,𝑘∈{1,…,𝑁}. (2.16)

To this end, the following lemma will be essential for the proofs in the next section and its proof can be found in the cited references.

Lemma 2.8 (Schur complement lemma, see [29]). For a given symmetric matrix 𝑊=^𝑊11𝑊12_{𝑊𝑇12𝑊22}, where 𝑊₁₁∈ℝ^𝑝×𝑝,𝑊₂₂∈ℝ^𝑞×𝑞, and 𝑊₁₂∈ℝ^𝑝×𝑞, the following three conditions are mutually equivalent:(1)𝑊<0,(2)𝑊₁₁<0, 𝑊₂₂−𝑊^𝑇₁₂𝑊⁻¹₁₁𝑊₁₂<0,(3)𝑊₂₂<0, 𝑊₁₁−𝑊₁₂𝑊⁻¹₂₂𝑊^𝑇₁₂<0.

3. Main Results

In this section, we will find a state feedback control matrix 𝐾, such that system (2.10) is finite-time bounded with respect to (𝛼,𝑑,𝛽,𝑅,𝑁). In order to solve the problem, the following theorem will be essential.

Theorem 3.1. For given state feedback control matrix 𝐾, system (2.10) is finite-time bounded with respect to (𝛼,𝑑,𝛽,𝑅,𝑁), if there exist symmetric positive definite matrices 𝑃₁ and 𝑃₂ and a scalar 𝛾≥1, such that the following conditions hold: 𝐴+𝐵𝐾^𝑇𝑃₁𝐴+𝐵𝐾−𝛾𝑃₁𝐴+𝐵𝐾^𝑇𝑃₁𝐺𝐺^𝑇𝑃₁𝐴+𝐵𝐾𝐺^𝑇𝑃₁𝐺−𝛾𝑃₂𝜆<0,₂𝜆₁𝛾^𝑁𝛼²+𝜆₃𝜆₁𝛾^𝑁𝑑²<𝛽², (3.1)(3.2) where 𝜆₁=𝜆_min𝑃₁,𝜆₂=𝜆_max𝑃₁,𝜆₃=𝜆_max𝑃₂,𝑃₁=𝑅^−1/2𝑃₁𝑅^1/2. (3.3)

Proof. Choose the Lyapunov function as 𝑉(𝑥(𝑘))=𝑥^𝑇(𝑘)𝑃₁𝑥(𝑘). (3.4) Then we have 𝑉(𝑥(𝑘+1))=𝑥^𝑇(𝑘+1)𝑃₁()=𝐵𝐾(𝑘)+(𝑘)𝑥𝑘+1𝐴+𝑥𝐺𝑤^𝑇𝑃₁𝐵𝐾(𝑘)+(𝑘)=(𝑘)𝑤(𝑘)𝐴+𝑥𝐺𝑤𝑥^𝑇𝐴+𝐵𝐾^𝑇𝑃₁𝐴+𝐵𝐾𝐴+𝐵𝐾^𝑇𝑃₁𝐺𝐺^𝑇𝑃₁𝐴+𝐵𝐾𝐺^𝑇𝑃₁𝐺(𝑘)𝑤(𝑘).𝑥 (3.5) It follows from (3.1) that 𝑉(𝑥(𝑘+1))≤𝛾𝑉(𝑥(𝑘))+𝛾𝑤^𝑇(𝑘)𝑃₂𝑤(𝑘). (3.6) Applying iteratively (3.6), we obtain 𝑉(𝑥(𝑘))≤𝛾^𝑘𝑉(𝑥(0))+^𝑘_𝑗=1𝛾^𝑗𝑤^𝑇𝑘−𝑗𝑃₂𝑤𝑘−𝑗=𝛾^𝑘𝑉(𝑥(0))+^𝑘_𝑗=1𝛾^𝑗−𝑘𝑤^𝑇𝑘−𝑗𝑃₂𝑤𝑘−𝑗≤𝛾^𝑘𝑉(𝑥(0))+𝜆₃^𝑘_𝑗=1𝛾^𝑗−𝑘𝑤^𝑇𝑘−𝑗𝑤𝑘−𝑗. (3.7) Using the fact that 𝛾≥1, we have 𝑉(𝑥(𝑘))≤𝛾^𝑘𝑉(𝑥(0))+𝜆₃^𝑘_𝑗=1𝑤^𝑇𝑘−𝑗𝑤𝑘−𝑗≤𝛾^𝑁𝜆₂𝛼²+𝜆₃𝑑². (3.8) On the other hand, 𝑉(𝑥(𝑘))=𝑥^𝑇(𝑘)𝑃₁𝑥(𝑘)≥𝜆₁𝑥^𝑇(𝑘)𝑅𝑥(𝑘). (3.9) From (3.8) and (3.19), it can be seen that 𝑥^𝑇(𝑘)(𝑘)≤𝜆𝑅𝑥²𝜆₁𝛾^𝑁𝛼²+𝜆₃𝜆₁𝛾^𝑁𝑑²<𝛽² (3.10) which means that 𝑥^𝑇(𝑘)𝑅𝑥(𝑘)≤𝛽²,𝑘=1,…,𝑁. (3.11) This completes the proof.

Corollary 3.2 3.2. For given state feedback control matrix 𝐾, system (2.10) with the disturbance 𝑤(𝑘)=0 is finite-time stable with respect to (𝛼,𝛽,𝑅,𝑁), if there exist symmetric positive definite matrix 𝑃 and a scalar 𝛾≥1, such that the following conditions hold: 𝐴+𝐵𝐾^𝑇𝑃𝐵𝐾𝑃<1𝐴+−𝛾𝑃<0,cond𝛾^𝑁𝛽²𝛼², (3.12) where 𝑃=𝑅^−1/2𝑃𝑅^1/2𝑃=𝜆,cond^max𝑃𝜆_min𝑃. (3.13)

Now we turn back to our original problem, that is, to find sufficient conditions which guarantee that the system (2.4) with the controller (2.2) is finite-time bounded with respect to (𝛼,𝑑,𝛽,𝑅,𝑁). The solution of this problem is given by the following theorem.

Theorem 3.3. System (2.10) is finite-time bounded with respect to (𝛼,𝑑,𝛽,𝑅,𝑁) if there exist symmetric positive definite matrices 𝑄₁₁, 𝑄₁₂, and 𝑄₂, a matrix 𝐿, and a scalar 𝛾≥1, such that the following conditions hold: −𝛾𝑄₁0𝐴𝑄₁+𝐵𝐿𝑆^𝑇0−𝛾𝑄₂𝐺^𝑇𝐴𝑄₁+𝐵𝐿𝑆𝐺−𝑄₁𝜆<0,₅𝜆₄𝛾^𝑁𝛼²+𝜆₅𝜆₆𝛾^𝑁𝑑²<𝛽², (3.14)(3.15) where 𝜆₄=𝜆_min𝑄₁,𝜆₅=𝜆_max𝑄₁,𝜆₆=𝜆_max𝑄₂,𝑄₁=𝑅^−1/2𝑄₁𝑅^1/2 (3.16)𝑆 and 𝑄₁ are defined as follows 𝑆=𝐼000,𝑄₁=𝑄₁₁00𝑄₁₂. (3.17) Then the controller 𝐾 is given by the first 𝑝 columns of 𝐾=𝐿𝑆𝑄₁⁻¹, which is in the form (2.12).

Proof. Let us consider Theorem 3.1 with 𝑄₁=𝑃₁⁻¹ and 𝑄₂=𝑃₂. Condition (3.2) can be rewritten as in (3.15) recalling that for a positive definite matrix 𝑄𝜆_max(𝑄)=1𝜆_min𝑄⁻¹. (3.18) Denote 𝐴=𝐴+𝐵𝐾. Then condition (3.1) can be rewritten as 𝐴^𝑇𝑄₁⁻¹𝐴−𝛾𝑄₁⁻¹𝐴^𝑇𝑄₁⁻¹𝐺𝐺^𝑇𝑄₁⁻¹𝐴𝐺^𝑇𝑄₁⁻¹𝐺−𝛾𝑄₂<0. (3.19) Pre- and postmultiplying (3.19) by the symmetric matrix 𝑄₁00𝐼, (3.20) the following equivalent condition is obtained 𝑄₁𝐴^𝑇𝑄₁⁻¹𝐴𝑄₁−𝛾𝑄₁𝑄₁𝐴^𝑇𝑄₁⁻¹𝐺𝐺^𝑇𝑄₁⁻¹𝐴𝑄₁𝐺^𝑇𝑄₁⁻¹𝐺−𝛾𝑄₂<0. (3.21) By using Lemma 2.8, (3.21) is equivalent to the following: 𝑄₁𝐴^𝑇𝑄₁⁻¹𝐴𝑄₁−𝛾𝑄₁𝑄₁𝐴^𝑇𝑄₁⁻¹𝐺𝐺0_𝑇𝑄₁⁻¹𝐴𝑄₁−𝛾𝑄₂𝐺^𝑇0𝐺−𝑄₁<0. (3.22) Premultiply (3.22) by 𝐼0−𝑄₁𝐴^𝑇𝑄₁⁻¹0𝐼000𝐼 (3.23) and postmultiply it by the transpose of (3.23). In this way, we obtain the following equivalent condition: −𝛾𝑄₁0𝑄₁𝐴^𝑇0−𝛾𝑄₂𝐺^𝑇𝐴𝑄₁𝐺−𝑄₁<0. (3.24) Recalling that 𝐴=𝐴+𝐵𝐾 and letting 𝐾𝑄₁=𝐿𝑆, we obtain that condition (3.1) is equivalent to (3.14). This completes the proof.

Remark 3.4. The chosen structures for matrices 𝑆 and 𝑄₁ guarantee that 𝐾 is in the form (2.12). In fact 𝐾=𝐿𝑆𝑄₁⁻¹𝑄=𝐿𝐼000¹¹00𝑄₁₂𝑄=𝐿⁻¹¹¹0=00𝐾0. (3.25)

Remark 3.5. Once we have fixed 𝛾, the feasibility of the conditions stated in (3.14) can be turned into LMI feasibility problems. On the other hand, for 𝜃₁>0,𝜃₂>0, it is easy to check that condition (3.15) can be guaranteed by 𝜃₁𝑅⁻¹<𝑄₁<𝑅⁻¹,0<𝑄₂<𝜃₂𝛽𝐼,₂−𝜃₂𝑑²𝛾^𝑁𝛼𝛾^𝑁𝛼𝛾^𝑁𝜃₁>0. (3.26)

Corollary 3.6. System (2.10) with the disturbance 𝑤(𝑘)=0 is finite-time stable with respect to (𝛼,𝛽,𝑅,𝑁), if there exist symmetric positive definite matrices 𝑄₁, 𝑄₂, a matrix 𝐿, and a scalar 𝛾≥1, such that the following conditions hold: −𝛾𝑄𝐴𝑄+𝐵𝐿𝑆^𝑇𝑅𝐴𝑄+𝐵𝐿𝑆−𝑄<0,₋₁1<𝑄<𝛾^𝑁𝛽²𝛼²𝑅⁻¹, (3.27) where 𝑄𝑆=𝐼000,𝑄=¹00𝑄₂. (3.28) Then the controller 𝐾 is given by the first 𝑝 columns of 𝐾=𝐿𝑆𝑄⁻¹.

4. Numerical Example

Consider the following system: (𝑡)=𝑥(𝑡)+0𝑢(𝑡)+𝑤(𝑡),𝑦(𝑡)=𝑥(𝑡).𝑥̇010−0.10.110010.10.5 (4.1) Choose the sampling ℎ=0.3𝑠. Suppose 𝜏=0.1𝑠. The corresponding matrices are given by ΓΦ=1.00000.295500.9704,Ψ=0.30000.044600.2955₀,(𝜏)=0.00200.0198,Γ₁(𝜏)=0.00250.0098 (4.2)(4.3) which yields 𝐴=,,1.00000.29550.002500.97040.0098000𝐵=−0.0020−0.0198−1𝐺=0.30000.0446000.29550000. (4.4) It is assumed that 𝛼=1, 𝑑=3, 𝛽=20, 𝑅=𝐼, 𝑁=10. Applying Theorem 3.3 with 𝛾=1.5, it is found that 𝑄₁=,.0.94720.031800.03180.79470000.8616𝐿=0.00300.01870 (4.5) Therefore, the desired controller gain is given by 𝐾=𝐿𝑆𝑄₁⁻¹=𝐾0=0.00240.02350. (4.6)

5. Conclusions

In this paper, we have considered the finite-time boundedness problems of a class of networked control systems (NCSs) subject to disturbances. Based on the augmentation approach, the NCSs with time delay as linear systems. The sufficient conditions for finite-time boundedness of the underlying systems are derived via linear matrix inequalities (LMIs) formulation. Lastly, an illustrative example is given to demonstrate the effectiveness of the proposed results.

Acknowledgment

This work was partly supported by the program of science and technology of Huainan no. 2011A08005.