Sliding Mode Control for Multiagent System with Time-Delay and Uncertainties: An LMI Approach

Abstract

This paper considers the sliding mode control of multiagent systems (MAS) with time-delay and uncertainties in terms of linear matrix inequality (LMI). By constructing virtual feedback control method, the design of control system is simplified for time-delay independent system without uncertainties. For a class of uncertain systems with single time-delay, the essence of SMC design is analyzed in order to acquire a simple method for designing sliding surface. In terms of multiple timedelay system with uncertainties, a sufficient condition for sliding surface with independent time-delay is acquired, while control law is also designed to ensure the robust stability of closed-loop system. Finally, the effectiveness of conclusion is demonstrated by simulation results.

1. Introduction

Research on application of multiagent system (MAS) began in the middle of 1980s with a significant growth trend in recent years. Multiagent technology has been one of the hot issues studied in the field of artificial intelligence presently. Multiagent system, as a frontier discipline in artificial intelligence and an important branch of distributed artificial intelligence research, aims to build large and complicated system into small, mutually communicated, and coordinated system that is easy for management. Research on multipleagent is involving the knowledge, objective, skills, planning as well as how to coordinate between agents, and so forth [1].

However, as the structure of MAS is growing increasingly complicated, more and more problems appear in this field, such as the basic problem of time-delay, random packet loss, uncertainty of system model, and influence of unknown interference on system, all of which can result in interference on the communication of MAS or signal channel congestion. Sliding mode control (SMC) enables a system to switch between two control laws by making use of the discontinuity of control, accordingly generating a new movement having no direct relationship with the original system: sliding mode. Such movement mainly has two features: Firstly, appropriate sliding surface can be selected to achieve dynamic feature of the system so as to meet the demand in performance indicators of closed-loop system. Secondly, closed-loop system reveals insensitivity to uncertainty or random interference. Such invariance proves that sliding mode control is particularly appropriate to be used as robust controller for an uncertain system. Therefore, more and more scholars pay attention to the application of SMC in MAS and relevant research [2–4].

Actually, system usually has uncertainties and varying parameters, while mathematical description is always full of uncertainties and influenced by the disturbance from exterior. In particular, a system usually contains complicated mathematical description which can be viewed as the perturbation of system, whereas these uncertain factors usually fail to completely meet the matching criteria of the system. Usually the original system will be changed to simple form of sliding mode through nonsingular state transformation, then Lyapunov equation of reduced order will be constructed to acquire the sufficient condition of Riccati equation of the designed sliding surface [5–7]. With the development of robust control theory in recent years, particularly the development of linear matrix inequality, many scholars begin to focus on this field [8]. For the uncertain system which meets the matching requirements, Choi designed a new sliding surface, by using quadratic stability theory, acquired a sufficient condition based on LMI, and described the approaching process of SMC [9]. Reference [10] concerned SMC of switched systems with stochastic Perturbation, derived a sufficient condition for the existence of reduced-order sliding mode dynamics. In [11], local stability theory was used to acquire the necessary and sufficient condition for general nonmatching linear SMC system with uncertainties, and then guaranteed the global stability of the system. Reference [6] carried out singular value decomposition of the input matrix and changed the uncertain linear system into simple form through nonsingular state transformation. Moreover, through construction of Lyapunov equation, sliding surface was designed to provide the sufficient condition of system stability in the form of Riccati equation. In [12], through analyzing the features of the uncertain parts in the system after nonsingular transformation, a new system matching condition was provided additionally with a new method for constructing sliding surface, at the same time chattering of the system was well suppressed. References [13, 14] carried out further researches on the design of sliding surface. For a class of singular stochastic system with Markovian switching, [15] designed an integral sliding surface function, by introducing some specified matrices, a new sufficient condition is proposed in terms of strict linear matrix inequality, which guaranteed the stochastic stability of the sliding mode dynamics. Reference [16] designed an observer to estimate the system states, and a sliding-mode control scheme is synthesized for the reaching motion based on the state estimates.

The general communication delay in MAS has always been the focus of scholars. In recent years, with the development of robust control theory, research on the stability of time-delay system and the design of controller has achieved great progress. References [17, 18] presented a common time-delay model, concluded the condition of time-delay system stability, and introduced the design method of state feedback-based controller. Targeting at time-delay dependence model, the literature [19–21] discussed the conditions of stability and analyzed the influence of time-delay transformation model and Lyapunov-Krasovskii function on time-delay system stability. The literature [22] presented a new time-delay transformed model and corresponding Lyapunov-Krasovskii function, yet as the result by Lyapunov-Krasovskii was relatively conservative, the stability condition is conservative and thereby did not fit fast transformed system. In 2001, Fridman put forward a singular time-delay state transformation model and the corresponding Lyapunov-Krasovskii function [23–25], which is less conservative and suitable to fast transformation time-delay systems, while the design method of stabilizing controller was also introduced.

In this paper, multiagent models under three different conditions are studied. First of all, time-delay independent system without uncertainties is investigated, and sliding mode of quadratic stability is constructed by making use of virtual state feedback. Then, uncertain system with single delay is studied, and nonsingular transformation design is applied to make the system reach quadratic stability. In addition, through quadratic stability-based design of robust sliding surface for time-delay independent system with nonmatching uncertainties, we acquire sufficient condition of quadratic stable surface for reduced order system, and controller is designed for the approaching motions of multiple time-delay system. Both approaching and sliding motion show a desirable robust to nonmatching uncertainties and exterior interference. Finally, three illustrative examples show the effectiveness of proposed theorems.

2. Problem Formulation

Concerning a type of MAS, as shown in Figure 1, information flow is transmitted among all agents directly. Let 𝐺={𝑣,𝑒} be a directed graph of order 𝑛 with the set of agents 𝑣={1,2,…,𝑛} and set of edges 𝑒⊆𝑣×𝑣; edge (𝑖,𝑗)∈𝑒 indicates that agent 𝑗 is able to receive information from agent 𝑖. The adjacent matrix 𝐴=[𝑎_𝑖𝑗]∈𝑅^𝑛×𝑛 of directed graph 𝐺 is defined as follows: if information exchange exists between two agents 𝑖 and 𝑗, then 𝑎_𝑖𝑗 denotes the throughput from 𝑖 to 𝑗. At the same time, agent 𝑖 may also receive information from 𝑗, yet 𝑎_𝑖𝑗 is not necessarily equal to 𝑎_𝑗𝑖. Taking into account that if the MAS is static, the connection structure inside the system is not varying with time. In other words, the system is fixed and contains no uncertain items; yet communication delay exists among all agents; then the system model can be summarized as a time-delay independent system without uncertainties, and the model is given as follows: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐴_𝑑𝑥()𝑢(𝑡)(𝑡),𝑥(𝑡)(𝑡)[],𝑡−𝜏+𝐵+𝑓=Ψ𝑡∈−𝜏,0 (1) where 𝑥(𝑡)∈𝑅^𝑛 is system state, 𝑢(𝑡)∈𝑅^𝑚 denotes system control input, 𝑓(𝑡) is disturbance, 𝐴, 𝐴_𝑑(𝑡), and 𝐵 are constant matrices with appropriate dimensions.

Figure 1 

MAS with an fixed structure and time-delay.

However, in practical system, multiagents may be in a dynamic state. Communication connection among agents within communication range may be interrupted as the distance increases; yet agents beyond communication range may also establish connection with each other through movement; this uncertain factor is one of the major reasons resulting in system structure uncertainty. If the state variation of a specific agent in the system is only related to its own state, then the system model is defined as uncertain system with single time-delay, and the model is described as follows: 𝑥̇(𝑡)=(𝐴+Δ𝐴(𝑡))𝑥(𝑡)+𝐴_𝑑+Δ𝐴_𝑑(𝑡)𝑥()𝑢(𝑡)(𝑡),𝑥(𝑡)(𝑡)[],𝑡−𝜏+𝐵+𝑓=Ψ𝑡∈−𝜏,0 (2) where 𝑥(𝑡)∈𝑅^𝑛 denotes system state, 𝑢(𝑡)∈𝑅^𝑚 is system control input, Δ𝐴(𝑡) and Δ𝐴_𝑑(𝑡) are internal parameter perturbation arising from uncertain factors, 𝑓(𝑡) denotes external disturbance, and 𝐴, 𝐴_𝑑(𝑡), and 𝐵 are constant matrices with appropriate dimensions.

If an agent in MAS is influenced by multiple time-delays, then the system model should be a multiple time-delay system with uncertainties, and the model is described as follows: 𝑥̇(𝑡)=(𝐴+Δ𝐴(𝑡))𝑥(𝑡)+^𝑁_𝑖=1𝐴_𝑑+Δ𝐴_𝑑(𝑡)𝑥_𝑖𝑡−𝜏_𝑖𝑢(𝑡)(𝑡),𝑥(𝑡)(𝑡)[],+𝐵+𝑓=Ψ𝑡∈−𝜏,0 (3) where 𝑥(𝑡)∈𝑅^𝑛 denotes system state, 𝑢(𝑡)∈𝑅^𝑚 is system control input, Δ𝐴(𝑡) and Δ𝐴_𝑑(𝑡) are internal parameter perturbation arising from uncertain factors, 𝑓(𝑡) denotes external disturbance, 𝐴, 𝐴_𝑑(𝑡), and 𝐵 are constant matrices with appropriate dimensions, and 𝑁 is the number of agents in the system. In both cases, the system is evolved into one shown in Figure 2. Since the MAS may be in a dynamic state or affected by external connection interference, the communication connection among multiagents features uncertain factors, and the dashed line indicates disconnection or connection occurring at any time.

Figure 2 

MAS with an unfixed structure and time-delay.

For the systems shown in (1), (2), and (3), we will use the following assumptions:

Assumption 1. Supposing that (𝐴,𝐵) could be stabilized, then there is a matrix 𝐾 makes 𝐴=𝐴−𝐵𝐾 stabilizing and rank(𝐵)=𝑚≤𝑛.

Assumption 2. The disturbance has an upper bound, which meets ‖𝑓(𝑡)‖≤𝛿_𝑓.

Assumption 3. Time-delay 𝜏_𝑖 is boundary, and ‖𝜏_𝑖‖_𝑖∈𝑁≤𝜏.

Assumption 4. The perturbation parameter of the system satisfies Δ𝐴Δ𝐴_𝑑=𝐺𝐷(𝑡)𝐻𝐻_𝑑. (4)

Respectively, 𝐺, 𝐻, and 𝐻_𝑑 are known constant matrices, 𝐷(𝑡) is time-delay uncertain matrix, yet Lebesgue-measurable, and 𝐷^𝑇(𝑡)𝐷(𝑡)≤𝐼.

For the design of SMC, usually it consists of two parts. Firstly, design a stable sliding surface which allows time-delay system to maintain a good dynamic performance under internal and external parameter perturbation. Secondly, choose proper control law so as to guarantee the system to meet the sliding mode condition from any states, that is, to reach the sliding surface in finite time.

Besides, a few frequently used and important lemmas are presented here.

Lemma 5. Let 𝑌=𝑌^𝑇, 𝐷, 𝐸, and 𝐹(𝑡) be real matrix of proper dimensions, and 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼, then inequality 𝑌+𝐷𝐹𝐸+(𝐷𝐹𝐸)^𝑇<0 holds if there exists a constant 𝜀, which makes the following equation hold 𝑌+𝜀𝐷𝐷^𝑇+𝜀⁻¹𝐸^𝑇𝐸<0. (5)

Lemma 6 (Schur complement). Given a symmetric matrix 𝑆=^𝑆11𝑆12_𝑆21𝑆22, where 𝑆₁₁ is 𝑟×𝑟 dimensional, the following three conditions are equivalent:(1)𝑆<0;(2)𝑆₁₁<0, 𝑆₂₂−𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂<0;(3)𝑆₂₂<0, 𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<0.

3. Main Results

3.1. Time-Delay Independent System without Uncertainties

In this section, the control law and sliding surface design of MAS as shown in Figure 1 are investigated.

Theorem 7. For system (1) which meets Assumptions 1 and 2, then the system from any initial states will meet the sliding mode reaching condition; that is, it will gradually be driven onto the sliding surface 𝑆(𝑡)=𝐵^𝑇𝑃𝑥(𝑡) within finite time with control law 𝑢(𝑡)=𝑢_𝑒𝑞(𝑡)+𝑢_𝑠𝑤(𝑡), (6) where the equivalent control is 𝑢_𝑒𝑞(𝑡)=−(𝐵^𝑇𝑃𝐵)⁻¹[𝐵^𝑇𝑃𝐴𝑥(𝑡)+𝐵^𝑇𝑃𝐴_𝑑𝑥(𝑡−𝜏)] and nonlinear switch control is 𝑢_𝑠𝑤(𝑡)=−(𝐵^𝑇𝑃𝐵)⁻¹[|𝐵^𝑇𝑃𝐵|𝛿_𝑓+𝜀]sgn(𝑆), in which 𝜀 is a positive constant.

Proof. Choose a Lyapunov function as 1𝑉=2𝑆^𝑇𝑆. (7)
Apparently, for all 𝑆(𝑥,𝑡)≠0, Lyapunov function 𝑉 is positive-definite; it follows that the Lyapunov derivative corresponding to (1), then 𝑉̇=𝑆^𝑇̇̇𝑆,𝑆=𝐵^𝑇𝑃𝑥̇=𝐵^𝑇𝑃𝐴𝑥(𝑡)+𝐵^𝑇𝑃𝐴_𝑑𝑥(𝑡−𝜏)+𝐵^𝑇𝑃𝐵𝑢(𝑡)+𝐵^𝑇𝑃𝐵𝑓(𝑡)=𝐵^𝑇𝑃𝐵𝑢_sw(𝑡)+𝐵^𝑇(𝑡)(𝑆).𝑃𝐵𝑓≤−𝜀sgn (8)
So there is 𝑆̇=𝑆^𝑇𝑆≤−𝜀|𝑆|≤0. (9)
This proves that system from any initial states meets the sliding mode reaching condition, thus the proof of Theorem 7 is complete.

Next, the sliding surface with robust stability will be designed. The control law is modified into the form of virtual feedback, ensuring the sliding mode control system with good robust in time-delay condition.

Theorem 8. The system (1) is quadratically stable if there exists symmetric positive-definite matrices 𝑋 and 𝑉 and general matrix 𝐿 of appropriate dimensions, such that is shown in (10); moreover, the sliding surface of (1) is 𝑆(𝑡)=𝐵^𝑇𝑃𝑥(𝑡), in which 𝑃=𝑋⁻¹𝐴𝑋+𝑋𝐴^𝑇−𝐵𝐿−𝐿^𝑇𝐵^𝑇𝐴_𝑑𝑉𝑋𝑉𝐴^𝑇_𝑑−𝑉0𝑋0−𝑉<0. (10)

Proof. First, we express (6) as a virtual feedback form 𝑢(𝑡)=−𝐾𝑥(𝑡)+𝑣(𝑡), (11) where 𝑣(𝑡)=𝐾𝑥(𝑡)+𝑢_eq(𝑡)+𝑢_sw(𝑡), substituting (1) with (11), then the whole system is represented as 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐴_𝑑𝑥(𝑡−𝜏)+𝐵𝑣(𝑡)+𝑓(𝑡), (12) where 𝐴=𝐴−𝐵𝐾. Choose Lyapunov function as 𝑉(𝑥,𝑡)=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+^𝑡_𝑡−𝜏𝑥^𝑇(𝑠)𝑄𝑥(𝑠)𝑑𝑠. (13)
Its derivative along (12) is 𝑉̇=2𝑥^𝑇𝑃𝑥̇+𝑥^𝑇𝑄𝑥−𝑥^𝑇(𝑡−𝜏)𝑄𝑥(𝑡−𝜏)=2𝑥^𝑇𝑃𝐴𝑥+𝐴_𝑑𝑥(𝑡−𝜏)+𝐵𝑣+𝑓+𝑥^𝑇𝑄𝑥−𝑥^𝑇(𝑡−𝜏)𝑄𝑥(𝑡−𝜏). (14)
From Theorem 7, it has been indicated that with control law (6), the sliding mode control system meets the reaching condition; therefore, the system can reach the sliding surface within limited time, in this case 𝑆=𝐵^𝑇𝑃𝑥=0 holds; in addition, 𝑆^𝑇=𝑥^𝑇𝑃^𝑇𝐵=0 holds; the previous equation can be simplified as 𝑉̇=2𝑥^𝑇𝑃𝐴𝑥+2𝑥^𝑇𝑃𝐴_𝑑𝑥(𝑡−𝜏)+𝑥^𝑇𝑄𝑥−𝑥^𝑇()()=𝑥𝑡−𝜏𝑄𝑥𝑡−𝜏_𝑇(𝑡)𝑥^𝑇()𝑀𝑥(𝑡)𝑥()𝑡−𝜏,𝑡−𝜏 (15) where 𝑀=𝐴^𝑇𝑃+𝑃𝐴+𝑄𝑃𝐴_𝑑𝐴^𝑇_𝑑𝑃−𝑄. (16)
If system (1) is quadratically stable, we have 𝑀<0. Multiply diag𝑃⁻¹𝐼 in both sides of (16), then 𝑃⁻¹𝐴^𝑇+𝐴𝑃⁻¹+𝑃⁻¹𝑄𝑃⁻¹𝐴_𝑑𝐴^𝑇_𝑑−𝑄<0. (17)
Letting 𝑃⁻¹=𝑋 and 𝐿=𝐾𝑥, then 𝑋𝐴^𝑇+𝐴𝑋−𝐵𝐿−𝐿^𝑇𝐵^𝑇+𝑋𝑄𝑋𝐴_𝑑𝐴^𝑇_𝑑−𝑄<0. (18)
From Lemma 6, we have 𝑋𝐴^𝑇+𝐴𝑋−𝐵𝐿−𝐿^𝑇𝐵^𝑇𝐴_𝑑𝑋𝐴^𝑇_𝑑−𝑄0𝑋0−𝑄⁻¹<0. (19)
Multiply diag𝐼𝑄⁻¹𝐼 on both sides of (19), then 𝑋𝐴^𝑇+𝐴𝑋−𝐵𝐿−𝐿^𝑇𝐵^𝑇𝐴_𝑑𝑄⁻¹𝑋𝑄⁻¹𝐴^𝑇_𝑑−𝑄⁻¹0𝑋0−𝑄⁻¹<0. (20)
Letting 𝑄⁻¹=𝑉, so the proof of Theorem 8 is complete.

Remark 9. Based on Theorems 7 and 8, with the effect of robust control (6), system (1) is quadratically stable. From control law (6) and sliding surface (11), the virtual feedback constructed here simplifies the design of sliding surface. In the design process, nonsingular transformation is not required, accordingly simplifies the design of controller.

3.2. Uncertain System with Single Time-Delay

Since the MAS structure may change at any time, which could result in parameters perturbation internally. In this section, uncertain system with single time-delay will be discussed, which is shown in (2).

According to assumption, a nonsingular matrix 𝑇 can be chosen such that 𝑇𝐵=^{0(𝑛−𝑚)×𝑚}_𝐵𝑚, where 𝐵_𝑚 is nonsingular with rank(𝐵_𝑚)=𝑚. According to the definition in [26], let us select 𝑇=^𝑈𝑇2_𝑈𝑇1, where 𝑈₁∈𝑅^𝑛×𝑚 and 𝑈₂∈𝑅^{𝑛×(𝑛−𝑚)} are two unitary matrices resulting from singular value decomposition of matrix 𝐵, then 𝐵=𝑈₁𝑈₂Σ0_{(𝑛−𝑚)×𝑚}𝑉^𝑇, (21) where Σ∈𝑅^𝑚×𝑚 is a symmetric positive-definite matrix, 𝑉∈𝑅^𝑚×𝑚 is a unitary matrix, by the state transformation 𝑦=𝑇𝑥, and (2) have the regular form 𝑦(̇𝑡)=𝐴+Δ𝐴(𝑡)𝑦(𝑡)+𝐴_𝑑+Δ𝐴_𝑑(𝑡𝑦()+0)𝑡−𝜏_{(𝑛−𝑚)×𝑚}𝐵_𝑚𝑢(𝑡)(𝑡,𝑦(𝑡)(𝑡)[],+𝑓)=Φ𝑡∈−𝜏,0 (22) where 𝐴=𝑇𝐴𝑇⁻¹, 𝐴_𝑑=𝑇𝐴_𝑑𝑇⁻¹, Δ𝐴(𝑡)=𝑇Δ𝐴(𝑡)𝑇⁻¹, Δ𝐴_𝑑(𝑡)=𝑇Δ𝐴_𝑑(𝑡)𝑇⁻¹, and Φ(𝑡)=𝑇Ψ(𝑡).

In the following theorem, the design of sliding mode controller is presented. Moreover, the sliding surface can be chosen as 𝑆(𝑡)=𝐺𝑦=𝐶𝑦₁+𝑦₂=0, (23) where 𝐺=𝐶𝐼, 𝑦=^𝑦1_𝑦2, and 𝐶∈𝑅^{𝑚×(𝑛−𝑚)}.

Theorem 10. For uncertain system with single delay such as (2), which meets Assumptions 2 and 4 and sliding surface is given by (23). Then, the system can be driven onto the sliding surface in limit time with control law 𝑢(𝑡)=−𝐵_𝑚⁻¹Φ𝑆+𝐺𝐴+Δ𝐴𝑦(𝑡)+𝐺𝐴_𝑑+Δ𝐴_𝑑𝑦(𝑡−𝜏)+𝐵_𝑚𝛿_𝑓. (24)

Proof. Choose Lyapunov function as 1𝑉=2𝑆^𝑇𝑆. (25)
The Lyapunov derivative corresponding to (22) is given by (26) as follows: 𝑉̇=𝑆^𝑇𝑆̇=𝑆^𝑇𝐺𝑦̇=𝑆^𝑇𝐺𝐴𝑦(𝑡)+𝐺Δ𝐴𝑦(𝑡)+𝐺𝐴_𝑑+Δ𝐴_𝑑𝑦(𝑡−𝜏)+𝐵_𝑚𝑢(𝑡)+𝑓(𝑡). (26)
Taking into account control law (24) and Assumption 2, then 𝑉̇≤−𝑆^𝑇Φ𝑆<0. (27)
The proof of Theorem 10 is complete.

Since the next section provides the design of sliding surface for multiple time-delay cases, (2) can be treated as a special case (𝑁=1). Then, it is easy to acquire the sliding surface in single time-delay condition. For brevity, the result is omitted here.

Remark 11. In accordance with state estimation in [27], when sliding function is approaching the sliding surface, the approaching exponent depends on the value of Φ in the theorem. The larger 𝜆_min(Φ), the faster approaching speed. At the same time, an ideal sliding mode movement is difficult to acquire as it can only be converged maximally within a range surrounding the sliding surface and make back and forth through movement unceasingly.

3.3. Multiple Time-Delay System with Uncertainties

In the last section, the influence of single time-delay on system was discussed in case of internal perturbation. However, in many other cases, states of multiagent may also be influenced by multiple time-delay, and still exist a problem of parameter perturbation, as shown in (3).

By nonsingular transformation mentioned previously, Assumptions 2 to 4 are satisfied at the same time, so (3) is transformed to 𝑦(̇𝑡)=𝐴+Δ𝐴(𝑡)𝑦(𝑡)+^𝑁_𝑖=1𝐴_𝑑+Δ𝐴_𝑑(𝑡)𝑦_𝑖𝑡−𝜏_𝑖+0_{(𝑛−𝑚)×𝑚}𝐵_𝑚𝑢(𝑡)(𝑡),𝑦(𝑡)(𝑡)[],+𝑓=Φ𝑡∈−𝜏,0 (28) where 𝐴=𝑇𝐴𝑇⁻¹, 𝐴_𝑑=𝑇𝐴_𝑑𝑇⁻¹, Δ𝐴(𝑡)=𝑇Δ𝐴(𝑡)𝑇⁻¹, Δ𝐴_𝑑(𝑡)=𝑇Δ𝐴_𝑑(𝑡)𝑇⁻¹, and Φ(𝑡)=𝑇Ψ(𝑡); in addition, we have 𝑦̇₁(𝑡)=𝐴₁₁+Δ𝐴₁₁(𝑡)𝑦₁(𝑡)+^𝑁_𝑖=1𝐴_𝑑11+Δ𝐴_𝑑11(𝑡)×𝑦_𝑖1𝑡−𝜏_𝑖+𝐴₁₁+Δ𝐴₁₂(𝑡)𝑦₂(𝑡)+^𝑁_𝑖=1𝐴_𝑑12+Δ𝐴_𝑑12(𝑡)𝑦_𝑖2𝑡−𝜏_𝑖,𝑦̇₂(𝑡)=𝐴₂₁+Δ𝐴₂₁(𝑡)𝑦₁(𝑡)+^𝑁_𝑖=1𝐴_𝑑21+Δ𝐴_𝑑21(𝑡)×𝑦_𝑖1𝑡−𝜏_𝑖+𝐴₂₂+Δ𝐴₂₂(𝑡)𝑦₂(𝑡)+^𝑁_𝑖=1𝐴_𝑑22+Δ𝐴_𝑑22(𝑡)𝑦_𝑖2𝑡−𝜏_𝑖+𝐵_𝑚𝑢(𝑡)(𝑡),𝑦+𝑓₁(𝑡)=Φ₁(𝑡)[],𝑦𝑡∈−𝜏,0₂(𝑡)=Φ₂(𝑡)𝑡∈[−𝜏,0], (29) where 𝑦₁∈𝑅^{(𝑛−𝑚)}, 𝑦₂∈𝑅^𝑚, 𝐵_𝑚=∑𝑉^𝑇, 𝐴₁₁=𝑈^𝑇₂𝐴𝑈₂, 𝐴₁₂=𝑈^𝑇₂𝐴𝑈₁, 𝐴_𝑑11=𝑈^𝑇₂𝐴_𝑑𝑈₂, 𝐴_𝑑12=𝑈^𝑇₂𝐴_𝑑𝑈₁, Δ𝐴₁₁=𝑈^𝑇₂𝐺𝐷(𝑡)𝐻𝑈₂, Δ𝐴_𝑑11=𝑈^𝑇₂𝐺𝐷(𝑡)𝐻_𝑑𝑈₂, Δ𝐴₁₂=𝑈^𝑇₂𝐺𝐷(𝑡)𝐻_𝑑𝑈₁, Δ𝐴_𝑑12=𝑈^𝑇₂𝐺𝐷(𝑡)𝐻_𝑑𝑈₁, Φ₁(𝑡)∈𝑅^{(𝑛−𝑚)}, and Φ₂(𝑡)∈𝑅^𝑚 are the subblocks of Φ(𝑡).

It is obvious that the first equation of (29) represents the dynamics of sliding motion (28), so the sliding surface is designed as 𝑆=𝐶𝑦₁+𝑦₂=𝐶𝐼𝑦=0, (30) where 𝐶∈𝑅^{𝑚×(𝑛−𝑚)} and 𝑦=^𝑦1_𝑦2, substituting 𝑦₂=−𝐶𝑦₁ to the first equation of (29), then the sliding motion is 𝑦̇₁(𝑡)=𝐴₁₁+Δ𝐴₁₁−𝐴₁₂𝐶−Δ𝐴₁₂𝐶𝑦₁(𝑡)+^𝑁_𝑖=1𝐴_𝑑11+Δ𝐴_𝑑11−𝐴_𝑑12𝐶−Δ𝐴_𝑑12𝐶×𝑦_𝑖1𝑡−𝜏_𝑖,𝑦₁(𝑡)=Φ₁(𝑡)𝑡∈[−𝜏,0]. (31)

Definition 12 (see [28]). The uncertain sliding motion (31) is said to be quadratically stabilizable if there exists symmetric positive-definite matrix 𝑃, 𝑄∈𝑅^{(𝑛−𝑚)×(𝑛−𝑚)}, for any admissible uncertainty the derivation of the Lyapunov functional 𝑉𝑦₁(𝑡),𝑡=𝑦^𝑇₁(𝑡)𝑃𝑦₁(𝑡)+^𝑡_𝑡−𝜏𝑦^𝑇₁(𝑠)𝑄𝑦₁(𝑠)𝑑𝑠 (32) with respect to time 𝑡 satisfies 𝐿𝑦₁(𝑡),𝑡=𝑉̇𝑦₁(𝑡),𝑡≤0 (33) for all pairs (𝑦₁(𝑡),𝑡)∈𝑅^{(𝑛−𝑚)}×𝑅.
For this definition, the purpose is to design a constant gain 𝐶∈𝑅^{𝑚×(𝑛−𝑚)} and a control law 𝑢(𝑡) to make the sliding motion quadratically stable and the system achieve asymptotic stabilization.
The first conclusion is about the design of sliding surface.

Theorem 13. The reduced order system (31) is quadratically stable if there exists symmetric positive-definite matrix 𝑍∈𝑅^{(𝑛−𝑚)×(𝑛−𝑚)}, 𝐿∈𝑅^{(𝑛−𝑚)×𝑛}, 𝑊∈𝑅^{(𝑛−𝑚)×𝑛}, and a general matrix 𝐾∈𝑅^{(𝑛−𝑚)×(𝑛−𝑚)} as well as a positive constant 𝜀 making (34) holds, where Λ=𝑍𝐴^𝑇₁₁−𝐾^𝑇𝐴₁₂+𝐴₁₁𝑍−𝐴₁₂𝐾+𝜀𝑈^𝑇₂𝐺𝐺^𝑇𝑈₂, 𝜂=(1/√𝑁)𝐴_𝑑11𝐿−(1/√𝑁)𝐴_𝑑12𝑊, 𝛿=𝑍𝑈^𝑇₂𝐻^𝑇−𝐾𝑈^𝑇₁𝐻^𝑇, and 𝜃=(1/√𝑁)𝐿𝑈^𝑇₂𝐻^𝑇_𝑑−(1/√𝑁)𝑊𝑈^𝑇₁𝐻^𝑇_𝑑. Moreover, the sliding surface of (29) is 𝑆=𝐶𝑦₁(𝑡)+𝑦₂(𝑡)=𝐶𝐼𝑌=0. Λ𝜂𝛿𝐼∗−𝑉𝜃0∗∗−𝜀𝐼0∗∗∗−𝑉<0, (34) where ∗ is matrix acquired based on matrix symmetry.

Proof. Take symmetric positive-definite matrix 𝑃 and 𝑄∈𝑅^{(𝑛−𝑚)×(𝑛−𝑚)} to construct the following Lyapunov function: 𝑉𝑦₁(𝑡),𝑡=𝑦^𝑇₁(𝑡)𝑃𝑦₁(𝑡)+^𝑁_𝑖=1^𝑡_{𝑡−𝜏𝑖}𝑦^𝑇₁(𝑠)𝑄𝑦₁(𝑠)𝑑𝑠. (35)
For all 𝑡∈[−𝜏,0], 𝑦₁(𝑡)≠0. Taking the derivative of (35) through (31), we have 𝑉̇𝑦₁(𝑡),𝑡=2𝑦^𝑇₁(𝑡)𝑃𝑦₁(𝑡)+𝑁𝑦^𝑇₁𝑄𝑦₁(𝑡)−^𝑁_𝑖=1𝑦^𝑇₁𝑡−𝜏_𝑖𝑄𝑦₁𝑡−𝜏_𝑖. (36)
Letting 𝐴₁₁=𝐴₁₁−𝐴₁₂𝐶, Δ𝐴₁₁=Δ𝐴₁₁−Δ𝐴₁₂𝐶=𝑈^𝑇₂𝐺𝐷(𝑡)𝐻(𝑈₂−𝑈₁𝐶), 𝐴_𝑑11=𝐴_𝑑11−𝐴_𝑑12𝐶, Δ𝐴_𝑑11=Δ𝐴_𝑑11−Δ𝐴_𝑑12𝐶=𝑈^𝑇₂𝐺𝐷(𝑡)𝐻_𝑑(𝑈₂−𝑈₁𝐶), and 𝑌(𝑡−𝜏)=^{𝑦1(𝑡−𝜏1)𝑦2(𝑡−𝜏2)}_{⋮𝑦𝑁(𝑡−𝜏𝑁)}, so there is 𝑉̇𝑦₁(𝑡)=𝑦,𝑡¹(𝑡)𝑦₁𝑡−𝜏₁⋮𝑦_𝑁𝑡−𝜏_𝑁^𝑇𝐴₁₁+Δ𝐴₁₁^𝑇𝑃+𝑃𝐴₁₁+Δ𝐴₁₁+𝑁𝑄𝑃𝐴_𝑑11+Δ𝐴_𝑑11𝐴_𝑑11+Δ𝐴_𝑑11^𝑇×𝑦𝑃−𝑄₁(𝑡)𝑦₁𝑡−𝜏₁⋮𝑦_𝑁𝑡−𝜏_𝑁=𝑦₁(𝑡)𝑌(𝑡−𝜏)^𝑇𝐴₁₁+Δ𝐴₁₁^𝑇𝑃+𝑃𝐴₁₁+Δ𝐴₁₁+𝑁𝑄𝑃𝐴_𝑑11+Δ𝐴_𝑑11𝐴_𝑑11+Δ𝐴_𝑑11^𝑇×𝑦𝑃−𝑄₁(𝑡)𝑌().𝑡−𝜏 (37)
Letting 𝑍=𝑃⁻¹ and 𝐽=𝑍𝑄𝑍, (37) can be written as 𝑉̇𝑦₁(𝑡),𝑡=𝑃𝑦₁(𝑡)𝑃𝑌(𝑡−𝜏)^𝑇𝑀𝑃𝑦₁(𝑡)(),𝑃𝑌𝑡−𝜏 (38) where 𝑍𝐴𝑀=¹¹+Δ𝐴₁₁^𝑇+𝐴₁₁+Δ𝐴₁₁𝑍+𝑁𝐽𝐴_𝑑11+Δ𝐴_𝑑11𝑍𝑍𝐴_𝑑11+Δ𝐴_𝑑11^𝑇−𝐽. (39)
Substituting (4) to (39), it follows from Lemma 5 that 𝑍𝐴𝑀=^𝑇11+𝐴₁₁𝑍+𝑁𝐽𝐴_𝑑11𝑍𝑍𝐴^𝑇_𝑑11+𝑈−𝐽^𝑇2𝐺0×𝐷(𝑡)𝐻𝑈₂−𝑈₁𝐶𝑍𝐻_𝑑𝑈₂−𝑈₁𝐶𝑍+𝐻𝑈₂−𝑈₁𝐶𝑍𝐻_𝑑𝑈₂−𝑈₁𝐶𝑍^𝑇×𝐷^𝑇(𝑡)𝑈^𝑇₂𝐺0^𝑇. (40)
According to Lemma 6, 𝑀<0 is equivalent to 𝑍𝐴^𝑇₁₁+𝐴₁₁𝑍+𝑁𝐽+𝜀𝑈^𝑇₂𝐺𝐺^𝑇𝑈₂𝐴_𝑑11𝑍𝑍𝑈₂−𝑈₁𝐶^𝑇𝐻^𝑇𝑍𝐴^𝑇_𝑑11−𝐽𝑍𝑈₂−𝑈₁𝐶^𝑇𝐻^𝑇_𝑑𝐻𝑈₂−𝑈₁𝐶𝑍𝐻_𝑑𝑈₂−𝑈₁𝐶𝑍−𝜀𝐼<0. (41)
In addition, we have 𝑍𝐴^𝑇₁₁+𝐴₁₁𝑍+𝜀𝑈^𝑇₂𝐺𝐺^𝑇𝑈₂𝐴_𝑑11𝑍𝑍𝑈₂−𝑈₁𝐶^𝑇𝐻^𝑇𝐼𝑍𝐴^𝑇_𝑑11−𝐽𝑍𝑈₂−𝑈₁𝐶^𝑇𝐻^𝑇_𝑑0𝐻𝑈₂−𝑈₁𝐶𝑍𝐻_𝑑𝑈₂−𝑈₁𝐶()𝑍−𝜀𝐼0𝐼00−𝑁𝐽⁻¹<0. (42)
Multiply diag𝐼(1/√𝑁)𝐽⁻¹𝐼𝐼 on both ends of the matrix, then𝑍𝐴^𝑇₁₁+𝐴₁₁𝑍+𝜀𝑈^𝑇₂𝐺𝐺^𝑇𝑈₂1√𝑁𝐴_𝑑11𝑍𝐽⁻¹𝑍𝑈₂−𝑈₁𝐶^𝑇𝐻^𝑇𝐼1√𝑁𝐽⁻¹𝑍𝐴^𝑇_𝑑11−(𝑁𝐽)⁻¹1√𝑁𝐽⁻¹𝑍𝑈₂−𝑈₁𝐶^𝑇𝐻^𝑇_𝑑0𝐻𝑈₂−𝑈₁𝐶𝑍1√𝑁𝐽⁻¹𝐻_𝑑𝑈₂−𝑈₁𝐶()𝑍−𝜀𝐼0𝐼00−𝑁𝐽⁻¹<0. (43)
Letting 𝐶𝑍=𝐾, 𝑄⁻¹𝑍⁻¹=𝐿, 𝐶𝑄⁻¹𝑍⁻¹=𝑊, and (𝑁𝐽)⁻¹=𝑉, then (43) can be expressed as Λ𝜂𝛿𝐼∗−𝑉𝜃0∗∗−𝜀𝐼0∗∗∗−𝑉<0, (44) where Λ=𝑍𝐴^𝑇₁₁−𝐾^𝑇𝐴₁₂+𝐴₁₁𝑍−𝐴₁₂𝐾+𝜀𝑈^𝑇₂𝐺𝐺^𝑇𝑈₂,𝜂=(1/√𝑁)𝐴_𝑑11𝐿−(1/√𝑁)𝐴_𝑑12𝑊, 𝛿=𝑍𝑈^𝑇₂𝐻^𝑇−𝐾𝑈^𝑇₁𝐻^𝑇, and 𝜃=(1/√𝑁)𝐿𝑈^𝑇₂𝐻^𝑇_𝑑−(1/√𝑁)𝑊𝑈^𝑇₁𝐻^𝑇_𝑑. So the proof of Theorem 13 is complete.

The following theorem presents the design of sliding mode controller.

Theorem 14. For a multiagent system (3), which satisfies Assumptions 2 to 4, system can be driven onto sliding surface (45) in limited time with control law (46), which means system from any states will meet the reaching condition (𝑡)(𝑡)𝑢(𝑡)𝑆=𝐶𝐼𝑇𝑥=𝑅𝑥=0,=𝑢_𝑒𝑞+𝑢_𝑠𝑤, (45)(46) where equivalent control 𝑢_𝑒𝑞=−(𝐵^𝑇𝐵)⁻¹[𝐵^𝑇𝐴𝑥(𝑡)+𝐵^𝑇𝐴_𝑑𝑁𝑋] and nonlinear switch control 𝑢_𝑠𝑤=−(𝐵^𝑇𝐵)⁻¹[𝐵^𝑇𝐺(𝐻𝑥(𝑡)+𝐻_𝑑𝑁𝑋)+(𝐵^𝑇𝐵)𝛿_𝑓+𝜀], in which 𝜀 is a positive constant, and ‖𝑥_𝑖(𝑡−𝜏_𝑖)‖_𝑖∈𝑁≤𝑋.

Proof. Conside the following Lyapunov function: 𝑉(𝑡)=12𝑆^𝑇𝑆. (47)
Apparently, for all 𝑆(𝑥,𝑡)≠0, 𝑉 is positive-definite. Taking the derivative of (47) corresponding to (3), then 𝑉̇=𝑆^𝑇𝑆̇=𝑆^𝑇𝑅𝑥̇=𝑆^𝑇𝑅(𝐴+Δ𝐴)𝑥+𝐴_𝑑+Δ𝐴_𝑑×^𝑁_𝑖=1𝑥_𝑖𝑡−𝜏_𝑖+𝐵𝑢+𝑓≤𝑆^𝑇𝑅𝐴𝑥+𝑆^𝑇𝑅Δ𝐴𝑥+𝑆^𝑇𝑅𝐴_𝑑+Δ𝐴_𝑑𝑁𝑋+𝑆^𝑇𝐵_𝑚𝑢+𝑓≤𝑆^𝑇𝑅𝐴𝑥+𝑆^𝑇𝑅𝐺𝐻𝑥+𝑆^𝑇𝑅𝐴_𝑑+𝐺𝐻_𝑑𝑁𝑋+𝑆^𝑇𝐵_𝑚𝑢+𝑓. (48)
Considering control law (46), then 𝑉̇≤𝑆^𝑇−𝐵_𝑚𝛿_𝑓−𝜀sgn(𝑆)+𝐵_𝑚𝛿_𝑓≤−𝜀𝑆^𝑇sgn(𝑆)≤0. (49)
This indicates that system from any states all satisfies the sliding mode reaching condition and the proof of Theorem 14 is complete.

Remark 15. Nonlinear transformation reduces system order, which simplifies the design of sliding mode control system, yet problem arising from multiple time-delay system requires reinforcement of control (46) in order to stabilize the system onto the sliding surface.

4. Numerical Simulations

In this section, 3 simulations are given for testing the theorems developed in this paper. Considering MAS (1), transform the case on [29], where 𝐴=20−11.750.250.8101,𝐴_𝑑=,𝑓(𝑡)(𝑡)001.−100−0.10.250.2−0.20.40.5=0.2sin,𝜏=0.5𝑠,𝐵= (50)

The initial states are 𝑥(𝑡)=10.4−0.8^𝑇 and 𝑡∈−0.50. Based on Theorem 8, we have ,,.𝑋=13.7445−30.166750.9141−30.1667103.4341−49.369050.9141−49.3690364.0445𝑉=23.02170.717434.56850.7174264.0987−55.757934.5685−55.7579350.0517𝐿=−148.8728263.3331782.5076 (51)

Designed to make (𝐴,𝐵) stabilization 𝐾 is 𝐾=−30.3839−2.90189.25, where the poles of closed-loop are configured at 𝐽=−1−2−3.

Thus, the sliding mode coefficient is acquired as 𝑃=𝑋⁻¹=0.97070.2334−0.10410.23340.0665−0.0236−0.1041−0.02360.0141. (52)

The parameter of controller (6) is 𝛿_𝑓=0.2 and 𝜀=0.15, then the simulation result is shown in Figure 3.

(a)

(b)

(c)

(d)

(e)

Figure 3 

States (𝑥1,𝑥2,𝑥3), control input, and sliding mode.

In fact, it is hard to acquire ideal sliding mode. From the figure, we can see that the practical sliding mode is converged within a region around the ideal sliding surface.

Considering MAS (2), where ,𝐷(𝑡)=(𝑡)0(𝑡),𝐻𝐴=−41.542,𝐺=0.10.200.20.5sin00.5sin,𝐻=0.30.20.20.33_𝑑=10,𝑓(𝑡)(𝑡)0.270.170.180.3,𝐵=,𝑇=0110=0.2sin,𝜏=0.5𝑠. (53)

The initial states are 𝑥(𝑡)=1.2−3.42^𝑇 and 𝑡∈−0.50, it follows from Theorem 10 that 𝑐=3.56. The parameters of the controller are Φ=3.72 and 𝛿_𝑓=0.2. The simulation results are shown in Figure 4.

(a)

(b)

(c)

Figure 4 

States (𝑥1,𝑥2), and control input.

From the control law, although the design method is simplified, for systems requiring fast response, control parameters Φ should be enlarged.

Considering MAS (3), where ,𝐷(𝑡)=(𝑡)0(𝑡)𝐴=−5.2252.6,𝐺=0.130.300.280.8sin00.8sin,𝜏=0.5𝑠,𝑁=9,𝐻=0.40.30.262.3,𝐻_𝑑=,𝐴0.350.220.230.4_𝑑=10(𝑡)(𝑡).−2.61.321.56,𝐵=,𝑓=0.2sin (54)

The initial states are 𝑥(𝑡)=2.56−1.42^𝑇 and 𝑡∈−0.50, and the nonlinear transformation is 𝑇=⁰¹₁₀. According to Theorem 13, 𝑍=0.2804, 𝐿=2.5488, 𝑊=7.6719, and 𝐾=0.8441, the controller parameters are 𝛿_𝑓=0.2, 𝜀=2.5784, and 𝑋=4. The simulation results are shown in Figure 5.

(a)

(b)

(c)

(d)

Figure 5 

States (𝑥1,𝑥2), control input, and sliding mode.

5. Conclusion

In this paper, the sliding surface and control law of time-delay independent system without uncertainties, uncertain system with single time-delay and multiple time-delay system with uncertainties are designed. Through virtual feedback control of unstable system, poles of closed-loop are configured within the stable range. LMI is used for acquiring the parameters of sliding mode controller and getting sufficient condition for quadratic stability. Through nonsingular transformation, the equivalent reduced order system is acquired, accordingly reducing the complexity of parameter acquisition. However, it is clear that the system is eventually stabilized within an area and inevitably exists a problem of chattering. For system requiring fast response, control law could only be achieved through increasing the control parameters, yet this is technically difficult in practical engineering, and it is the subject of future work.

Conflict of Interests

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

Acknowledgments

This work has been supported by the National Natural Science Foundation of China (Grant no. 61104109), the Natural Science Foundation of Jiangsu Province of China (Grant no. BK2011703), the Support of Science and Technology and Independent Innovation Foundation of Jiangsu Province of China (Grant no. BE2012178), and the Doctoral Fund of Ministry of Education of China (Grant no. 20113219110027).