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Journal of Control Science and Engineering
Volume 2011 (2011), Article ID 719730, 8 pages
Research Article

Stability and Performance of First-Order Linear Time-Delay Feedback Systems: An Eigenvalue Approach

Department of Electrical Engineering, National Taiwan University, Taipei 10617, Taiwan

Received 25 April 2011; Revised 26 August 2011; Accepted 6 September 2011

Academic Editor: Onur Toker

Copyright © 2011 Shu-An He and I-Kong Fong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Linear time-delay systems with transcendental characteristic equations have infinitely many eigenvalues which are generally hard to compute completely. However, the spectrum of first-order linear time-delay systems can be analyzed with the Lambert function. This paper studies the stability and state feedback stabilization of first-order linear time-delay system in detail via the Lambert function. The main issues concerned are the rightmost eigenvalue locations, stability robustness with respect to delay time, and the response performance of the closed-loop system. Examples and simulations are presented to illustrate the analysis results.

1. Introduction

Systems with delays in internal signal transmissions are quite common in electrical, mechanical, biological, and chemical engineering problems. The delays may be inherent characteristics of system components or part of the control process [1]. The retarded type time-delay systems discussed in this paper belong to those that can be modeled by ordinary differential equations (ODEs) combined with difference terms in time, called the differential difference equations (DDEs) [2].

Recently, research works about the state solution, controllability, observability, and controller design for time-delay systems are very abundant [3, 4]. In the field of controller design, many literature reviews [57] propose linear matrix inequality (LMI) conditions for finding controllers. However, the resultant LMI conditions are mostly sufficient only.

The retarded type time-delay systems are considered infinite dimensional and have transcendental characteristic equations with an infinite spectrum. Over the past decade, many works have been done to find the dominant eigenvalues or even the entire spectrum. For example, in [8] many approaches are introduced to iteratively find the approximate rightmost eigenvalues. Another important numerical method is proposed in [9], which uses the Lambert function to develop an expression for the spectrum. Subsequently, the expression is applied to decide the state solution, to discuss the controllability and observability, and to design state feedback controllers [1013].

Through the Lambert function approach, theoretically the full view of spectrum can be observed. Recently, an auxiliary matrix is introduced to combine with the Lambert function for finding the spectrum of high-order time-delay systems, but it is also pointed out that the existence and uniqueness of such auxiliary matrix are still open problems [14]. Thus, as a basis for complex high-order time-delay systems, this paper discusses the first-order delay systems intensively through the Lambert function.

First-order linear systems with input delay have been focus of study in literature reviews such as [8, 15]. To discuss stability, effects of three factors are of general concern: system parameter, controller gain, and delay size. In [15] P-control is studied, and only the range of stabilizing controller gain is analyzed via Padé approximation and crossing frequencies determination. In [8], a stability region diagram is constructed with respect to the above three factors, but the method suggested is iterative in nature and quite computationally intensive. Through the properties of Lambert function, this paper investigates the stability as well as performance of the first-order linear time delay systems with state feedback. Not only stability conditions are derived, but the entire closed-loop spectrum can be exposed, which facilitates the selection of controller gains to achieve the control objectives. More specifically, the rightmost eigenvalue will be located, and some deeper issues will be emphasized, including stabilization and stability robustness with respect to the delay time. Besides, how to select a feedback gain in order to obtain better response performance is also covered. Finally, examples and simulations are presented to illustrate the analysis results.

2. Problem Formulation

Consider the first-order time-delay system,̇𝑥(𝑡)=𝑎0𝑥𝑥(𝑡)+𝑏𝑢(𝑡),𝑡0,(𝑡)=𝑥0,𝑡=0,(1) where 𝑥 is the state variable, 𝑢 is the input signal, 𝑎0 and 𝑏 are the system parameters, is a nonnegative constant delay time, and 𝑥0 is the initial condition. When a state feedback controller 𝑢(𝑡)=𝐾𝑥(𝑡) is applied, (1) becomeṡ𝑥(𝑡)=𝑎0𝑥(𝑡)+𝑎1𝑥𝑥(𝑡),𝑡0,𝑥(𝑡)=0,𝑡=0,0,𝑡<0,(2) where 𝑎1=𝑏𝐾. The state-space model (2) is a DDE, for which stability and stabilizability are to be studied. The characteristic equation of (2) is Δ(𝑠)=𝑠𝑎0𝑎1𝑒𝑠, a transcendental one that has infinitely many roots. There are some numerical methods to solve this kind of equation, such as an ODE-based approach [16], but utilizing the Lambert function [17] exposes more properties of the eigenvalues.

3. The Lambert Function

The Lambert function 𝒲() is the function that satisfies𝒲(𝑧)𝑒𝒲(𝑧)=𝑧,𝑧. It is a multivalued function, and the function values are classified as the principal branch and the 𝑘th-branch for all non-zero integers 𝑘. Expressing 𝑧=𝛼+𝑗𝛽 and 𝒲(𝑧)=𝜁+𝑗𝜂 in the form of complex numbers, one has the relations𝛼=𝑒𝜁(𝜁cos𝜂𝜂sin𝜂),𝛽=𝑒𝜁(𝜂cos𝜂+𝜁sin𝜂).(3)

In the next section, it will be clear that only 𝑧 needs to be discussed for the purpose of this paper, so either 𝜂=0 or 𝜁=𝜂cot𝜂 with 𝛽=0, and 𝒲(𝑧)=𝜁 or 𝒲(𝑧)=𝜂cot𝜂+𝑗𝜂. A partial plot of the function 𝜁=𝜂cot𝜂 is displayed in Figure 1, and it is easy to infer the rest part of the function.

Figure 1: Plot of 𝜁=𝜂cot𝜂.

Based on Figure 1 and by convention, (4a) below defines the principal branch 𝒲0(z) of the Lambert function, (4b) defines the −1st-branch 𝒲1(𝑧), and (4c) defines the 𝑘th-branch. 𝒲0𝒲(𝑧)=𝜂cot𝜂+𝑗𝜂,for𝜂(0,𝜋),𝜁,for𝜁1,(4a)1𝒲(𝑧)=𝜁,for𝜁<1,𝜂cot𝜂+𝑗𝜂,for𝜂(2𝜋,𝜋)(𝜋,0),(4b)𝑘(𝑧)=𝜂cot𝜂+𝑗𝜂,for𝜂((2𝑘1)𝜋,2𝑘𝜋)(2𝑘𝜋,(2𝑘+1)𝜋),𝑘=1,2,,𝜂cot𝜂+𝑗𝜂,for𝜂(2𝑘𝜋,(2𝑘+1)𝜋)((2𝑘+1)𝜋,2(𝑘+1)𝜋),𝑘=2,3,.(4c)

In Figure 2 a few branches of the Lambert functions are plotted, each with a different color, and the values of 𝑧=𝛼(,) are labeled at corresponding positions with the black font. From Figure 2 many conclusions can be drawn, including the following two Lemmas.

Figure 2: A few branches of the Lambert function parameterized by the values of 𝑧=𝛼.

Lemma 1. Only parts of 𝒲0(𝑧) and 𝒲1(𝑧) are real-valued, and all other branches of the Lambert function are complex-valued.
In fact, only 𝒲0(0)=0, and all other branches are undefined at 𝑧=0. Also,𝒲1(𝑧)=𝒲0(𝑧),for𝑧,𝑒1𝒲,(5a)𝑘(𝑧)=𝒲𝑘𝒲(𝑧),for𝑧(0,),𝑘=1,2,,(5b)𝑘(𝑧)=𝒲(𝑘+1)(𝑧),for𝑧(,0),𝑘=1,2,(5c)and 𝒲0(e1)=𝒲1(e1)=1.

Lemma 2. The largest lower bound of the real parts of the principal values is –1.

Moreover, one has the following less intuitive Lemma.

Lemma 3. For any given z, the real part of 𝒲0(𝑧) is no less than that of 𝒲𝑘(𝑧) for all 𝑘0.

Proof. Consider the three cases: 𝑧(,𝑒1), 𝑧[𝑒1,0), and 𝑧(0,) separately.Case 1 (𝑧(,𝑒1)). In this case, all branches of the Lambert function are complex-valued. Given any 𝑧 and integer 𝑘, let 𝒲𝑘(𝑧)=𝜁𝑘+𝑗𝜂𝑘. Suppose that at 𝑧=𝑧0 and 𝑧=𝑧𝑘 the principal branch and the kth-branch values, respectively, have the same real part 𝜁0=𝜁𝑘=𝜁. Since in Figure 2, 𝑧 toward right in the branch segments with 𝑧(,𝑒1), the proof will be completed if 𝑧0𝑧𝑘 is always true. Now 𝑧0𝑧𝑘=𝑒𝜁(𝜁cos𝜂0𝜂0sin𝜂0𝜁cos𝜂𝑘+𝜂𝑘sin𝜂𝑘)by 𝒲0(𝑧)𝑒𝒲0(𝑧)=𝒲𝑘(𝑧)𝑒𝒲𝑘(𝑧)=𝑧, and 𝜂𝑘=𝜂0+2𝑘𝜋+𝛿𝑘 for all integer 𝑘>0 with some 𝛿𝑘. Since 𝑒𝜁>0 and 𝜁𝑘=𝜂𝑘cot𝜂𝑘, one has 𝜁cos𝜂0cos𝜂𝑘𝜂0sin𝜂0+𝜂𝑘sin𝜂𝑘𝜂=𝑘sin𝜂𝑘cos𝜂𝑘cos𝜂0cos2𝜂𝑘+𝜂0𝜂𝑘sin𝜂𝑘sin𝜂0sin2𝜂𝑘𝜂=𝑘sin𝜂𝑘𝜂cos𝑘𝜂0sin𝜂𝑘sin𝜂0+𝜂0𝜂𝑘sin𝜂𝑘sin𝜂0𝜂1=𝑘sin𝜂𝑘cos𝛿𝑘+𝜂1𝑘𝜂0sin𝜂0𝜂𝑘sin𝜂𝑘cos𝛿𝑘,1(6) where 𝜂𝑘(2𝑘𝜋,2(𝑘+1)𝜋) and sin𝜂𝑘>0. Therefore, 𝑧0𝑧𝑘. For 𝑘=1 and 𝑘2, (5a) and (5c), respectively, can be applied to extend the above result.Case 2 (𝑧[𝑒1,0)). In this case, the principal and the −1st-branch are both real-valued, and from Figure 2 it can be seen that 𝜁0[1,0) and 𝜁1(,1]. Thus 𝜁0𝜁1. For 𝑘=1,±2,±3,,𝜂0=0 implies 𝑒𝜁𝑘𝜁𝑘cos𝜂𝑘𝜂𝑘sin𝜂𝑘=𝜁0𝑒𝜁0𝑒𝜁𝑘𝜂𝑘cot𝜂𝑘cos𝜂𝑘𝜂𝑘sin𝜂𝑘=𝜁0𝑒𝜁0𝜂𝑘𝑒𝜁𝑘=𝜁0𝑒𝜁0sin𝜂𝑘𝑒𝜁0𝜁𝑘𝜂=𝑘𝜁0sin𝜂𝑘.(7) The last ratio of (7) is clearly no less than unity since 𝜂𝑘/sin𝜂𝑘 is so and 𝜁0[1,0).Case 3 (𝑧(0,)). In this case, all but the principal branch of the Lambert function are complex-valued. Again, suppose at 𝑧=𝑧0 and 𝑧=𝑧𝑘, respectively, the principal branch and the kth-branch values have the same real part 𝜁0=𝜁𝑘=𝜁. Note only non-negative 𝜁 needs to be discussed in this case. Consequently, cos𝜂𝑘(0,1), since here 𝜂𝑘((2𝑘1)𝜋,2𝑘𝜋) for 𝑘>0 and 𝜂𝑘(2𝑘𝜋,(2𝑘+1)𝜋) for 𝑘<0, but with 𝜁𝑘=𝜂𝑘cot𝜂𝑘0 the ranges of 𝜂𝑘 can be limited to 𝜂𝑘((2𝑘1/2)𝜋,2𝑘𝜋) for 𝑘>0 and 𝜂𝑘(2𝑘𝜋,(2𝑘+1/2)𝜋) for 𝑘<0. Because in Figure 2𝑧 toward right in the branch segments with 𝑧(0,), the proof will be completed if 𝑧0𝑧𝑘 is always true. Now 𝑧𝑘𝑧0=𝑒𝜁(𝜁cos𝜂𝑘𝜂𝑘sin𝜂𝑘)𝜁𝑒𝜁 and 𝜁cos𝜂𝑘𝜂𝑘sin𝜂𝑘𝜁=𝜁cos𝜂𝑘+sin2𝜂𝑘cos𝜂𝑘11=𝜁cos𝜂𝑘.1(8) Hence the proof is completed.

4. The Spectrum of First-Order Feedback Time-Delay Systems

Consider the first-order time-delay system (2). Let 𝜆 be an eigenvalue of (2). Then [9]𝜆𝑎0𝑎1𝑒𝜆=0𝜆𝑎0𝑒(𝜆𝑎0)=𝑎1𝑒𝑎0𝑎𝒲1𝑒𝑎0=𝜆𝑎01𝜆=𝒲𝑎1𝑒𝑎0+𝑎0.(9) In accordance with the Lambert function, 𝜆0=𝑎0+𝒲0(𝑎1𝑒𝑎0)/ is called the principal eigenvalue, and 𝜆𝑘=𝑎0+𝒲𝑘(𝑎1𝑒𝑎0)/,𝑘=±1,±2,, is called the kth-branch eigenvalue. Note that the parameters 𝑎0, 𝑎1, and in the first-order system are all real, so only β = 0 is discussed for (3). By the above results there are some properties of the eigenvalues of (2) that can be obtained instantly.(i)Re(𝜆0)Re(𝜆k),𝑘=±1,±2,, for any given set of 𝑎0, 𝑎1 and .(ii)𝑎1𝑒𝑎0>𝑒1𝜆0𝑎,1𝑒𝑎0=𝑒1𝜆01=+𝑎0,𝑎1𝑒𝑎0<𝑒1𝜆0.(10)(iii)1/+𝑎0Re(𝜆0)<|𝑎0|+|𝑎1|.(iv)0Im(𝜆0)<𝜋/.

Most of these properties are immediately implied by the results in Section 3. For the upper bound in property (iii), let 𝜆=𝜎+𝑗𝜔, and from 𝜆=𝑎0+𝑎1𝑒𝜆 one has 𝜎=𝑎0+𝑎1𝑒𝜎cos(𝜔). Thus,||𝑎|𝜎|=0+𝑎1𝑒𝜎||||𝑎cos(𝜔)0||+||𝑎1||||𝑒𝜎||.(11)

If 𝜎>0 then Re(𝜆0)<|𝑎0|+|𝑎1|, and if 𝜎0 then Re(𝜆0)0.

5. Applications to Some Control Issues

Without loss of generality, subsequently let b be a given positive constant.

5.1. State Feedback for System Stabilization

For system (1) with the state feedback 𝑢(𝑡)=𝐾𝑥(𝑡), property (ii) of Section 4 indicates that the principal eigenvalue of (2) is real-valued for 𝐾𝑒1/(𝑏𝑒𝑎0), and complex-valued for 𝐾<𝑒1/(𝑏𝑒𝑎0). Moreover, by property (i) of Section 4 the principal eigenvalue has the largest real part among all eigenvalues. To stabilize system (2), one has to move the principal eigenvalue to the left half plane (LHP).

Theorem 4. There exists a state feedback gain K such that system (2) can be asymptotically stabilized if and only if 𝑎0<1/.

Proof. Suppose 𝑎01/. Then property (iii) of Section 4 shows that 0Re(𝜆0) no matter what 𝐾 is. On the other hand, if 𝑎0<1/, or 𝑎0+𝜀=1 for some 𝜀>0, then in Figure 2 it can be seen that there is a 𝐾 such that 𝒲0(𝐾𝑏𝑒𝑎0)=1+𝜀/2>1, which makes 𝜆0=𝜀/(2)<0.

Theorem 5. Suppose 𝑎0<1/ in (1). The range of K for the asymptotic stability of (1) is (𝜂0/(𝑏sin𝜂0),𝑎0/𝑏), where 𝑎0=𝜂0cot𝜂0 and 𝜂0(0,𝜋).

Proof. The values of 𝐾 for the asymptotic stability of (1) are those rendering the real part of 𝜆0=𝑎0+𝒲0(𝐾𝑏𝑒𝑎0)/ smaller than 0, or that of 𝒲0(𝐾𝑏𝑒𝑎0) smaller than 𝑎0. From Figure 2 and the assumption of 𝑎0<1/ or 1<𝑎0, a non-empty range of stabilizing gain 𝐾 exists, and the stabilizing gain 𝐾 must lie in the range of (𝐾1,𝐾2), where 𝐾2 makes 𝜁0=𝑎0 with 𝐾2𝑏𝑒𝑎0=𝜁0𝑒𝜁0(12) and 𝐾1 makes 𝜁0=𝑎0=𝜂0cot𝜂0 with 𝜂0(0,𝜋) and 𝐾1𝑏𝑒𝑎0=𝑒𝜁0𝜁0cos𝜂0𝜂0sin𝜂0=𝑒𝜁0𝜂0cot𝜂0cos𝜂0+𝜂0sin𝜂0𝑒=𝜁0𝜂0sin𝜂0.(13) Hence, 𝐾1=𝜂0/(𝑏sin𝜂0) and 𝐾2=𝑎0/𝑏.

The result of Theorem 5 is known in [18], but here a different derivation is provided.

Definition 6. For a given >0, the range of stabilizing gain K is denoted as Θ.

5.2. Robust Stability with Respect to Delay Size

The delay time in (1) is assumed to be a constant, but may be uncertain. In the practical applications, the delay time may not be estimated accurately. Therefore, it is important to analyze how robust the system (2) is with respect to the delay time under a selected stabilizing state feedback gain 𝐾0. Based on the signs of 𝑎0 and 𝐾, four cases are discussed separately.

Case 1 (𝑎00 and 𝐾>0). Since is a positive delay time, the condition of Theorem 4 holds for all 𝑎00. Select a positive 𝐾Θ, where is the assumed delay size. Thus 𝐾𝑎0/𝑏 and the upper bound is independent of . Also, by Theorem 5 the lower bound in Θ is nonpositive for all >0. Hence the positive 𝐾 is in Θ for all >0. In short, the feedback system is asymptotically stable independent of the delay time h.

Case 2 (𝑎00 and 𝐾<0). Suppose a negative 𝐾Θ is selected, where is the assumed delay size. Note that the principal eigenvalue is 𝜆0=𝑎0+𝒲0(𝐾𝑏𝑒𝑎0)/, so if < then 𝐾𝑏e𝑎0 is less negative, and by observing the parameter values in the principal branch of Figure 2, it can be deduced that 𝜆0 will stay in the LHP. However, when is large enough, the real part of 𝒲0(𝐾𝑏𝑒𝑎0) will become positive. More specifically, as in the proof of Theorem 5, if 𝑎0=𝜂0cot𝜂0 with some 𝜂0(𝜋/2,𝜋) and 𝐾=𝜂0/(𝑏sin𝜂0)=𝑎0/(𝑏cos𝜂0), then 𝜆0 will move to the boundary of LHP. Note that such an 𝜂0 exists if and only if 𝐾𝑎0/𝑏, which means if 𝐾>𝑎0/𝑏 then the feedback system is stable independent of , and if 𝐾𝑎0/𝑏 then stability will be lost for larger than (𝜂0cot𝜂0)/𝑎0.

Case 3 (𝑎0>0 and 𝐾<0). Again, suppose a negative 𝐾Θ is selected, where is the assumed delay size. Note that if <, then by Figure 1 the 𝜂0 satisfying 𝑎0=𝜂0cot𝜂0 is closer to 𝜋/2 than the 𝜂0 satisfying 𝑎0=𝜂0cot𝜂0. Therefore ΘΘand the feedback system keeps asymptotic stability with the gain 𝐾. Next, if then ΘΘ and the feedback system keeps asymptotic stability if and only if is smaller than (𝜂0cot𝜂0)/𝑎0, where 𝜂0=cos1(𝑎0/(𝑏𝐾))(0,𝜋/2).

Case 4 (𝑎0>0 and 𝐾>0). Since positive 𝐾Θ for all when 𝑎0>0, this case needs no discussion.

5.3. The Response Performance

Unlike non-delay systems, for which the system spectrum before and after state feedback stabilization have been thoroughly studied [19, 20], the corresponding problem for time-delay systems has not been probed deeply. Suppose it is desired to control the system (1) such that the response has a fast decay rate but is not oscillatory. In Theorem 5, the range of 𝐾 for the asymptotic stability is decided, but not every 𝐾 in Θ results in a better response performance than that of the uncontrolled system. Actually, for positive 𝑎0 all 𝐾Θ gives a better performance, but not necessarily so for negative 𝑎0.

Theorem 7. For system (1), only 𝐾(̃𝜂0/(𝑏siñ𝜂0),0), where ̃𝜂0cot̃𝜂0=0 and ̃𝜂0(0,𝜋), gives a better response performance than that of the uncontrolled system.

Proof. Suppose for (1), Θ is not empty and 𝑎0<0. From Figure 2, the set Θ consists of three subsets (𝜂0/(𝑏sin𝜂0),̃𝜂0/(𝑏siñ𝜂0)], (̃𝜂0/(𝑏siñ𝜂0),0), and [0,𝑎0/𝑏), where 𝑎0=𝜂0cot𝜂0, ̃𝜂0cot̃𝜂0=0, and 𝜂0,̃𝜂0(0,𝜋). However, only the 𝜆0 corresponding to 𝐾(̃𝜂0/(𝑏siñ𝜂0),0) has a more negative real part than 𝑎0, which means such feedback gain results in a larger decay rate. Next, the set (̃𝜂0/(𝑏siñ𝜂0),0) can be divided into (̃𝜂0/(𝑏siñ𝜂0),𝑒1/(𝑏𝑒𝑎0)) and [𝑒1/(𝑏𝑒𝑎0),0), where ̃𝜂0cot̃𝜂0=0 and ̃𝜂0(0,𝜋). By property (ii), the 𝜆0 from 𝐾(̃𝜂0/(𝑏siñ𝜂0),𝑒1/(𝑏𝑒𝑎0)) implies an oscillatory response. Finally, if 𝐾[𝑒1/(𝑏𝑒𝑎0),0), then 𝜆0 is a real-valued. Although other 𝜆𝑘 are complex-valued, 𝜆0 is the rightmost eigenvalue and far from others. Thus, oscillation only appears in the transient response. In fact, 𝐾=𝑒1/(𝑏𝑒𝑎0) gives the largest decay rate and least oscillation.

6. Examples and Simulations

Example 8. Let the parameter combination (𝑎0,𝑏,𝐾,)=(3,1,𝐾,0.7) be selected, where 𝑎0 does not satisfy the condition of Theorem 4, so there is no stabilizing gain 𝐾. The principal eigenvalue locus versus 𝐾(,) is shown in Figure 3, and it can be seen that the principal eigenvalue always lies on the RHP no matter what 𝐾 is.

Figure 3: The principal eigenvalue locus versus 𝐾.

Then consider the parameter combination (𝑎0,𝑏,𝐾,)=(0.3,1,𝐾,1.7), where 𝑎0<1/ and there exists a feasible set Θ={𝐾𝐾(0.7424,0.3)}. In Figure 4, a feedback gain 𝐾Θ is used, so the principal eigenvalue locates on RHP and the response of 𝑥(𝑡) is divergent. In Figure 5, a feedback gain 𝐾Θ is used, so the principal eigenvalue locates on LHP and the response of 𝑥(𝑡) is convergent.

Figure 4: (a) The eigenvalues and (b) the response of system (2) with 𝐾=0.29.
Figure 5: (a) The eigenvalues and (b) the response of system (2) with 𝐾=0.5.

Example 9. In this example, a stabilizing 𝐾 is fixed to test the robustness of system (2) with respect to delay time . There are four simulations. The first one refers to Case 1 in Section 5.2, where (𝑎0,𝑏,𝐾,)=(3,1,2,), and the principal eigenvalue versus is shown as Figure 6(a). The stability of this system in this simulation is independent on .

Figure 6: The simulations for all cases in Section 5.2: (a) Case 1; (b) Case 2 (c); Case 2; (d) Case 3.

Then Figures 6(b) and 6(c) refer to Case 2 in Section 5.2, with (𝑎0,𝑏,𝐾,)=(3,1,4,) and (3,1,1,), respectively. In Figure 6(b), the stability of this system is dependent on h, and the stability is maintained for [0,0.9142]. In Figure 6(c), stability of this system is independent of h. The last simulation refers to Case 3 in Section 5.2 with (𝑎0,𝑏,𝐾,)=(0.2,1,4,). It is seen that the crossing frequency is 3.995, and the stability is maintained for =[0,0.3807].

Example 10. Let the parameter combination (𝑎0,𝑏,𝐾,)=(0.01,0.0506,𝐾,6) [21] be selected. The parameter 𝑎0 is positive and satisfies Theorem 4, and Θ={𝐾𝐾(5.0488,0.1976)}. In this example, all 𝐾Θ give the better closed-loop system response performance. For 𝐾=1.2867 and 𝐾=1.079, which correspond to 𝜆0=0.1567 and 𝜆0=0.0763, respectively, the responses are shown in Figure 7(a), where the blue line (𝐾=𝑒1/(𝑏e𝑎0)) is better than green line in terms of decay rate. Consider another parameter combination (𝑎0,𝑏,𝐾,)=(1,2,𝐾,0.7). The parameter 𝑎0 is negative and satisfies Theorem 4, and Θ={𝐾𝐾(1.4600,0.5000)}. For this case only 𝐾(1.122,0) produces a better closed-loop response performance. For three different gains K = – 0.37461, 𝐾=0.13049, and 𝐾=0.075062, corresponding to 𝜆0=1.4±1.9558𝑗,2.4286 and −1.4, respectively, the responses are shown in Figure 7(b), in which the blue and red lines have the same decay rate, but the blue line is oscillatory. Also, the green line is the best, since 𝐾=𝑒1/(𝑏𝑒𝑎0)=0.13049.

Figure 7: The responses of Example 10.

7. Conclusions

Low-order time-delay systems are common in chemical engineering applications, and this paper studies the spectrum of first-order linear time-delay systems via the Lambert function. Stability and stabilization of such systems are discussed through the eigenvalue approach. By focusing on the principal eigenvalue, the intervals of stabilizing gains are obtained, and for a fixed stabilizing gain, the stability robustness of the system with respect to delay-time is explored. Moreover, through the full understanding of the spectrum, how to decide the feedback gain to obtain a better performance response is discussed. Three examples and simulations are shown to demonstrate the derived results.

Although this paper just focuses on the first-order system, the higher order systems can be discussed based on the analysis of this paper by using the partial fraction expansion approach. This will be the investigation goal of future works.


This research is supported by the National Science Council of the Republic of China under Grant NSC 98-2221-E-002-148-MY3.


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