Robust Delay Independent Stability Analysis for the Switched Interval Time-Delay Systems with Time-Driven Switching Strategy

Abstract

Some new criteria of delay independent stability for the switched interval time-delay systems are deduced. The switching structure does depend on time-driven switching strategies. The total activation time ratio of the switching law can be determined to guarantee that the switched interval time-delay system is exponentially stable.

1. Introduction

Switched systems constitute an important class of hybrid systems. Such systems can be described by a family of continuous-time subsystems (or discrete-time subsystems) and a rule that orchestrates the switching between them. It is well known that a wide class of physical systems in power systems, chemical process control systems, navigation systems, automobile speed change system, and so forth may be appropriately described by the switched model [1–7]. In the study of switched systems, most works have been centralized on the problem of stability. In the last two decades, there has been increasing interest in the stability analysis for such switched systems; see, for example, [8, 9] and the references cited therein. Two important methods are used to construct the switching law for the stability analysis of the switched systems. One is the state-driven switching strategy [9]; the other is the time-driven switching strategy [8].

The state-driven switching method is that if all subsystems have the common Lyapunov function or the multiple Lyapunov functions, there are many choices of switching strategy to make the whole system stable. However, using these kinds of methods, the system must meet conditions completely. Therefore, the common Lyapunov function or the multi-Lyapunov function is difficult to construct for practical systems; even if we can construct the function, it is more complicated and not easy to implement on practical systems.

The time-driven switching method is based on the concept of dwell time [2] that when all subsystem matrices are Hurwitz stable, then the entire switched system is exponentially stable for any switching signal if the time between consecutive switching (dwell time) is sufficiently large. [10] that switching among stable linear systems results in a stable system provided that switching is slow-on-the-average. But in many applications, unstable subsystems of switched systems cannot be avoided in fact [11]. If the average dwell time is chosen sufficiently large, and the total activation time of unstable subsystems is relatively small compared with that of Hurwitz stable subsystems, then exponential stability of a desire degree is guaranteed.

Furthermore, the time-delay phenomenon also cannot be avoided in practical systems, for instance, chemical process, long distance transmission line, hybrid procedure, electron network, and so forth. The problem of time-delay may cause instability and poor performance of practical systems [12–14]. Therefore, the stability analysis of switched systems with time delay is very worthy to be researched. In a control system, uncertainties may be due to measure errors, modeling errors, linearization approximations, and so forth. There seem to be some alternatives in formulating uncertainties or perturbations. [13, 15] shared the formulation which systems matrices are assumed to be perturbation. However, it does not take the position that the system matrices are expressed in as the sum of the two terms, but that the bounds for them are somehow known. These systems are called interval time-delay systems. The goal of this paper is to derive some robust sufficient stability conditions for the switched interval time-delay system.

Basically, current efforts to achieve stability in time-delay systems can be divided into two categories, namely, delay-independent criteria and delay-dependent criteria. In this paper, in view of delay-independent analysis, we expect to aid in studying stability and designing time-driven switching law to achieve and implement in a practical switched interval time-delay system.

The following notations will be used throughout the paper: 𝜆(𝐴) stands for the eigenvalues of matrix 𝐴, ‖𝐴‖ denotes the norm of matrix 𝐴; that is, ‖𝐴‖=Max[𝜆(𝐴^𝑇𝐴)]^1/2, and 𝜇(𝐴) means the matrix measure of matrix 𝐴; that is, 𝜇(𝐴)=Max[𝜆((𝐴+𝐴^𝑇)/2)]. ‖𝐴‖=Max[𝜆(𝐴^𝑇𝐴)]^1/2.

2. System Description and Problem Statement

First, consider the following switched time-delay system 𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐵_𝜎(𝑡)𝑥(),𝑥𝑡𝑡−𝜏₀=𝑥₀,𝑥(𝑡)=𝜓(𝑡),𝑡∈[−𝜏,0], (1) where 𝑥(𝑡)∈𝐑^𝑛 is state, 𝐴_𝜎(𝑡)∈𝐑^𝑛×𝑛, 𝐵_𝜎(𝑡)∈𝐑^𝑛×𝑛, 𝑡₀≥0 is the initial time, 𝑥₀ is the initial state, and 𝜎(𝑡)∶[𝑡₀,∞)→{1,2,…,𝑁} is a piecewise constant function of time, called a switch signal; that is, the matrix 𝐴_𝜎(𝑡) switches between matrices 𝐴₁,𝐴₂,…,𝐴_𝑁 belonging to the set 𝚨≡{𝐴₁,𝐴₂,…,𝐴_𝑁} and 𝐴_𝑖, 𝑖∈{1,2,…,𝑁}; the matrix 𝐵_𝜎(𝑡) switches between matrices 𝐵₁,𝐵₂,…,𝐵_𝑁 belonging to the set 𝐵≡{𝐵₁,𝐵₂,…,𝐵_𝑁} and 𝐵_𝑖, 𝑖∈{1,2,…,𝑁}. 𝜏>0 is the time-delay duration. 𝜓(𝑡) is a vector-valued initial continuous function defined on the interval [−𝜏,0], and finally 𝜓(𝑡), defined on −𝜏≤𝑡≤0, is the initial condition of the state.

Let us consider the switched interval time-delay system described by 𝑥̇(𝑡)=𝐴^𝐼_𝑖𝑥(𝑡)+𝐵^𝐼_𝑖𝑥(𝑡−𝜏),𝑖∈{1,2,…,𝑁}, (2) where 𝐴^𝐼_𝑖 and 𝐵^𝐼_𝑖 are matrices whose elements vary in prescribed defined as 𝐴^𝐼_𝑖=𝑎_{𝑘𝑙_𝑖},𝐵^𝐼_𝑖=𝑏_{𝑘𝑙_𝑖}, (3) where 𝑘,𝑙=1,2,…,𝑛. 𝑎_{𝑘𝑙_𝑖}≤𝑎_{𝑘𝑙_𝑖}≤𝑎_{𝑘𝑙_𝑖} and 𝑏_{𝑘𝑙_𝑖}≤𝑏_{𝑘𝑙_𝑖}≤𝑏_{𝑘𝑙_𝑖}.

Denote: 𝐴_𝑖=𝑎_{𝑘𝑙_𝑖},𝐴_𝑖=𝑎_{𝑘𝑙_𝑖},𝐵_𝑖=𝑏_{𝑘𝑙_𝑖},𝐵_𝑖=𝑏_{𝑘𝑙_𝑖}, (4) and let 𝐴_𝑖=𝐴_𝑖+𝐴_𝑖2,𝐵_𝑖=𝐵_𝑖+𝐵_𝑖2, (5) where 𝐴_𝑖 and 𝐵_𝑖 are the average matrices between 𝐴_𝑖, 𝐴_𝑖, and 𝐵_𝑖, 𝐵_𝑖, respectively. Furthermore, 𝐴^𝑏_𝑖=𝐴^𝐼_𝑖−𝐴_𝑖,𝐵^𝑏_𝑖=𝐵^𝐼_𝑖−𝐵_𝑖, (6) where 𝐴^𝑏_𝑖 and 𝐵^𝑏_𝑖 are the bias matrices between 𝐴_𝑖, 𝐴_𝑖, and 𝐵_𝑖, 𝐵_𝑖, respectively. Also, 𝐴^𝑚_𝑖=𝐴_𝑖−𝐴_𝑖,𝐵^𝑚_𝑖=𝐵_𝑖−𝐵_𝑖, (7) where 𝐴^𝑚_𝑖 and 𝐵^𝑚_𝑖 are the maximal bias matrices between 𝐴_𝑖, 𝐴_𝑖, and 𝐵_𝑖, 𝐵_𝑖, respectively.

From the properties of matrix norm, we have 𝐴^𝑏_𝑖≤𝐴^𝑚_𝑖=𝛼_𝑖,𝐵^𝑏_𝑖≤𝐵^𝑚_𝑖=𝛽_𝑖, (8) and denote 𝛼_𝐼=max𝛼_𝑖,𝛽_𝐼=max𝛽_𝑖, (9) where 1≤𝑖≤𝑁.

In this paper, we study the robust stability analysis and switching law design for the switched interval time-delay systems.

3. Delay-independent Stability Analysis

Some helpful lemmas and definitions are given below.

Lemma 1 (see [16]). Consider the time-delay system: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏), (10) where 𝑥∈𝑅^𝑛, 𝐴 and 𝐵 are matrices in proper dimensions, and 𝜏 is the delay duration. The stability of the time-delay system implies the stability for the following systems: 𝑤̇(𝑡)=(𝐴+𝑧𝐵)𝑤(𝑡),∀|𝑧|=1, (11) and vice versa.

In the light of Lemma 1, for the switched time-delay system (1), all individual subsystems can be implied 𝑤̇(𝑡)=𝐴_𝑖+𝑧𝐵_𝑖𝑤(𝑡)=𝐴_𝑖(𝑧)𝑤(𝑡),∀|𝑧|=1. (12) Therefore, the system (12) is exponentially stable if and only if the switched time-delay system (1) is exponentially stable.

Lemma 2 (see [17]). For matrices 𝐴∈𝐑^𝑛×𝑛 and 𝐵∈𝐑^𝑛×𝑛, the following relation holds: )𝑡𝜇()𝑡𝜇(𝐴)+‖𝐵‖𝑡|𝑧|exp[(𝐴+𝑧𝐵]≤exp𝐴+𝑧𝐵≤exp,∀=1. (13)

Without loss of generality, we assume that the switched interval time-delay system (2) at least has one individual subsystem whose 𝜇(𝐴_𝑖)+‖𝐵_𝑖‖+𝛼_𝑖+𝛽_𝑖 values are less than zero, the that of remaining individual system are not less than zero; that is,𝜇𝐴_𝑖+𝐵_𝑖+𝛼_𝑖+𝛽_𝑖𝜇𝐴<0,1≤𝑖≤𝑟,_𝑖+𝐵_𝑖+𝛼_𝑖+𝛽_𝑖≥0,𝑟+1≤𝑖≤𝑁. (14a)(14b)

Definition 3. Consider 𝜆⁻_doi=min𝜇𝐴_𝑖+𝐵_𝑖+𝛼_𝑖+𝛽_𝑖𝜆,1≤𝑖≤𝑟,_+doi=max𝜇𝐴_𝑖+𝐵_𝑖+𝛼_𝑖+𝛽_𝑖,𝑟+1≤𝑖≤𝑁. (15a)(15b)
Furthermore, we assume that 𝑇⁺(𝑡) (or 𝑇⁻(𝑡)) is the total activation time of individual subsystems whose 𝜇(𝐴_𝑖)+‖𝐵_𝑖‖+𝛼_𝑖+𝛽_𝑖 values are not less than zero (total activation time of individual subsystems whose 𝜇(𝐴_𝑖)+‖𝐵_𝑖‖+𝛼_𝑖+𝛽_𝑖 values are less than zero). The total activation time ratio between 𝑇⁻(𝑡) and 𝑇⁺(𝑡) can be called a switching law of the switched interval time-delay system (2). Therefore, we will find the ratio for the total activation time such that the switched interval time-delay system (2) is globally and exponentially stable with stability margin 𝜆.

Theorem 4. Suppose that the switched interval time-delay system (2) exists in at least one individual subsystem whose 𝜇(𝐴_𝑖)+‖𝐵_𝑖‖+𝛼_𝑖+𝛽_𝑖 value is less than zero. The switched interval time-delay system (2) is globally and exponentially stable with stability margin 𝜆, if the system (2) satisfies the following switching law: inf_𝑡≥𝑡0𝑇⁻(𝑡)𝑇⁺(𝑡)≥𝜆⁺_𝑑𝑜𝑖+𝜆^∗𝜆⁻_𝑑𝑜𝑖−𝜆^∗, (16) where 𝜆∈(0,𝜆⁻_𝑑𝑜𝑖) and 𝜆^∗∈(𝜆,𝜆⁻_𝑑𝑜𝑖).

Proof. By Lemma 1, the stability of the switched interval time-delay system (2) can be transformed into the following system: 𝑤̇(𝑡)=𝐴^𝐼_𝑖+𝑧𝐵^𝐼_𝑖𝑤(𝑡)=𝐴^𝐼_𝑖𝑤(𝑡). (17) The trajectory response of system (17) is written as follows: 𝑤(𝑡)=𝑒^{𝐴𝐼𝑝𝑖+1(𝑡−𝑡𝑖)}𝑒^{𝐴𝐼𝑝𝑖(𝑡𝑖−𝑡𝑖−1)}⋯𝑒^{𝐴𝐼𝑝1(𝑡1−𝑡0)}𝑤𝑡₀. (18) In view of Lemma 2, we can obtain the inequality ‖𝑤(𝑡)‖≤𝑒^{(𝐴𝐼𝑝𝑖+1+𝑧𝐵𝐼𝑝𝑖+1)(𝑡−𝑡𝑖)}⋅𝑒^{(𝐴𝐼𝑝𝑖+𝑧𝐵𝐼𝑝𝑖)(𝑡𝑖−𝑡𝑖−1)}⋯𝑒^{𝐴𝐼𝑝1+𝑧𝐵𝐼𝑝1(𝑡1−𝑡0)}𝑤𝑡₀≤𝑒^{[𝜇(𝐴𝐼𝑝𝑖+1)+‖𝐵𝐼𝑝𝑖+1‖](𝑡−𝑡𝑖)}⋅𝑒^{[𝜇(𝐴𝐼𝑝𝑖)+‖𝐵𝐼𝑝𝑖‖](𝑡𝑖−𝑡𝑖−1)}⋯𝑒^{[𝜇(𝐴𝐼𝑝1)+‖𝐵𝐼𝑝1‖](𝑡1−𝑡0)}𝑤𝑡₀. (19) From the properties of matrix measure, we have 𝜇𝐴^𝐼_𝑖≤𝜇𝐴^𝑏_𝑖+𝐴_𝑖≤𝜇𝐴^𝑏_𝑖+𝜇𝐴_𝑖≤𝜇𝐴_𝑖+𝛼_𝑖,𝐵^𝐼_𝑖≤𝐵_𝑖+𝛽_𝑖. (20) Hence, the inequality (19) can be written as ‖𝑤(𝑡)‖≤𝑒^{[𝜇(𝐴𝑝𝑖+1)+𝛼𝑝𝑖+1+‖𝐵𝑝𝑖+1‖+𝛽𝑝𝑖+1](𝑡−𝑡𝑖)}⋯𝑒^{[𝜇(𝐴𝑝𝑖)+𝛼𝑝𝑖+‖𝐵𝑝𝑖‖+𝛽𝑝𝑖](𝑡𝑖−𝑡𝑖−1)}⋯𝑒^{[𝜇(𝐴𝑝1)+𝛼𝑝1+‖𝐵𝑝1‖+𝛽𝑝1](𝑡1−𝑡0)}𝑤𝑡₀≤𝑒^{𝜆+doi𝑇+−𝜆−doi𝑇−}𝑤𝑡₀. (21) Furthermore, the switching law (16) means that 𝜆⁺_doi𝑇⁺(𝑡)−𝜆⁻_doi𝑇⁻(𝑡)≤−𝜆^∗𝑇⁺(𝑡)+𝑇⁻(𝑡)=−𝜆^∗𝑡−𝑡₀. (22) Finally, if we choose 𝜆∈(0,𝜆⁻_doi) and 𝜆^∗∈(0,𝜆⁻_doi), the following inequality can be obtained: ‖𝑤(𝑡)‖≤𝑒^{−𝜆∗(𝑡−𝑡0)}𝑤𝑡₀≤𝑒^{−𝜆(𝑡−𝑡0)}𝑤𝑡₀. (23)
From the previous inequality (23), the system (17) is globally and exponentially stable with stability margin 𝜆 and implies that the system (2) is also stable as the systems (17) and (2) have same stability as properties. Hence, the switched interval time-delay system (2) is also globally and exponentially stable with stability margin 𝜆.

Remark 5. By Theorem 4, the stability condition of the switched interval time-delay system (1) is independent of time-delay.

4. Example

Example 1. Consider the switched interval time-delay system with interval matrices.
Subsystem 1. Consider𝐴^𝐼₁=,𝐵−0.3,0.30.8,1.20.8,1.2−0.3,0.3_𝐼1=.−0.1,0.3−0.1,0.1−0.1,0.1−0.1,0.3 (24a)Subsystem 2. Consider 𝐴^𝐼₂=,𝐵−3.1,−2.9−0.2,0.20.9,1.1−3.3,−2.7_𝐼2=.−0.2,0−0.1,0.1−0.4,0−0.4,−0.2 (24b)
From (5) and (7), we obtain the average matrices and maximal bias matrices.
Subsystem 1. Consider 𝐴₁=0110,𝐵₁=,𝐴0.1000.1_𝑚1=0.30.20.20.3,𝐵^𝑚₁=.0.20.10.10.2 (25)Subsystem 2. Consider 𝐴₂=−301−3,𝐵₂=,𝐴−0.10−0.2−0.3_𝑚2=0.10.20.10.3,𝐵^𝑚₂=.0.10.10.20.1 (26)
From (15a) and (15b), we can calculate 𝜆⁺_doi=1.9 and 𝜆⁻_doi=1.4867. Finally, the total activation time ratio for the switching law is (with 𝜆=0.3, 𝜆^∗=0.6) 𝑇⁻(𝑡)𝑇⁺(𝑡)≥𝜆⁺_doi+𝜆^∗𝜆⁻_doi−𝜆^∗=2.8194. (27)

In order to satisfy the switching law (27), we choose the total activation time ratio 3 : 1. The activation time of subsystem 1 is 0.1 sec, and the activation time of subsystem 2 is 0.3 sec, respectively. The trajectory of the switched interval time-delay system (for the average matrices 𝐴₁, 𝐵₁, 𝐴₂, and 𝐵₂) is shown in Figure 1 with initial state 12^𝑇 and time-delay 0.1 sec.

Figure 1 

Trajectory response in Example.

5. Conclusion

We have developed methodologies for the delay-independent stability criteria of switched interval time-delay systems with time-driven switching strategy. On delay-independent stability analysis, the sufficient conditions of the switched laws are presented, and the total activation time ratio under the switching laws is required to be not less than a specified constant, such that the switched interval time-delay system is delay-independent and exponentially stable with stability margin. In addition, the main advantages of our approach showed that we can quantify the region of stability, extend to arbitrary subsystems of switched time-delay systems, and develop the simple time-driven switching rule to stabilize the switched interval time-delay systems.

Acknowledgment

This work is supported by the National Science Council, Taiwan, under Grants no. NSC 102-2221-E-218-017 and NSC100-2632-E-218-001-MY3.