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

Abstract

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 [5–7] 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 [10–13].

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,𝑥̇(𝑡)=𝑎𝑥₀𝑥(𝑡()𝑡)=+𝑥𝑏₀𝑢(𝑡−ℎ),𝑡≥0,,𝑡=0,(1) where 𝑥∈ℝ is the state variable, 𝑢∈ℝ is the input signal, 𝑎₀∈ℝ and 𝑏∈ℝ are the system parameters, ℎ is a nonnegative constant delay time, and 𝑥₀ is the initial condition. When a state feedback controller 𝑢(𝑡)=𝐾𝑥(𝑡) is applied, (1) becomes𝑥̇(𝑡)=𝑎𝑥₀(𝑥𝑡)(𝑡=)+𝑎𝑥₁₀𝑥(𝑡−ℎ),𝑡≥0,0,,𝑡𝑡=<00,,(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)=𝑠−𝑎₀−𝑎₁𝑒^−𝑠ℎ, 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)𝑒^𝒲(𝑧)=𝑧,𝑧∈ℂ. 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 relations𝛼𝛽==𝑒𝑒^𝜁_𝜁(𝜁cos𝜂−𝜂sin𝜂),(𝜂cos𝜂+𝜁sin𝜂).(3)

In the next section, it will be clear that only 𝑧∈ℝ 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 𝒲₋₁(𝑧), and (4c) defines the 𝑘th-branch. 𝒲𝒲₀(𝑧)=−𝜁,𝜂cot𝜂+𝑗𝜂,ffoorr𝜁𝜂∈≥(−0,1𝜋,),(4a)₋₁𝒲(𝑧)=𝜁−,𝜂cot𝜂+𝑗𝜂,ffoorr𝜂𝜁∈<−(−12,𝜋,−𝜋)∪(−𝜋,0),(4b)_𝑘(𝑧)=⎧⎪⎪⎪⎨⎪⎪⎪⎩−f−foo𝜂𝜂rrcc𝜂𝜂oott∈∈𝜂𝜂++(((2𝑗𝑗2𝑘𝜂𝜂𝑘𝜋,,−,(21𝑘)𝜋+,21𝑘)𝜋𝜋))𝑘𝑘∪∪==(((221−𝑘𝑘,𝜋22,+,,−(…213𝑘),,𝜋…+,21.)(𝑘𝜋)+,1)𝜋),(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 𝒲₋₁(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,𝒲𝒲₋₁(𝑧)=𝒲^∗0(𝑧),for𝑧∈−∞,−𝑒⁻¹,(5a)_𝑘𝒲(𝑧)=𝒲^∗−𝑘(𝑧),for𝑧∈(0,∞),𝑘=1,2,…,(5b)_𝑘(𝑧)=𝒲^{∗−(𝑘+1)}(𝑧),for𝑧∈(−∞,0),𝑘=1,2,…(5c)and 𝒲₀(−e⁻¹)=𝒲₋₁(−e⁻¹)=−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∈ℝ, the real part of 𝒲₀(𝑧) is no less than that of 𝒲_𝑘(𝑧) for all 𝑘≠0.

Proof. Consider the three cases: 𝑧∈(−∞,−𝑒⁻¹), 𝑧∈[−𝑒⁻¹,0), and 𝑧∈(0,∞) separately.Case 1 (𝑧∈(−∞,−𝑒⁻¹)). In this case, all branches of the Lambert function are complex-valued. Given any 𝑧∈ℝ and integer 𝑘, let 𝒲_𝑘(𝑧)=𝜁_𝑘+𝑗𝜂_𝑘. Suppose that at 𝑧=𝑧₀ and 𝑧=𝑧_𝑘 the principal branch and the kth-branch values, respectively, have the same real part 𝜁₀=𝜁_𝑘=𝜁. Since in Figure 2, 𝑧→−∞ toward right in the branch segments with 𝑧∈(−∞,−𝑒⁻¹), the proof will be completed if 𝑧₀≥𝑧_𝑘 is always true. Now 𝑧₀−𝑧_𝑘=𝑒^𝜁(𝜁cos𝜂₀−𝜂₀sin𝜂₀−𝜁cos𝜂_𝑘+𝜂_𝑘sin𝜂_𝑘)by 𝒲₀(𝑧)𝑒^𝒲0(𝑧)=𝒲_𝑘(𝑧)𝑒^{𝒲𝑘(𝑧)}=𝑧, and 𝜂_𝑘=𝜂₀+2𝑘𝜋+𝛿_𝑘 for all integer 𝑘>0 with some 𝛿_𝑘. Since 𝑒^𝜁>0 and 𝜁_𝑘=−𝜂_𝑘cot𝜂_𝑘, one has 𝜁cos𝜂₀−cos𝜂_𝑘−𝜂₀sin𝜂₀+𝜂_𝑘sin𝜂_𝑘=−𝜂^𝑘sin𝜂_𝑘cos𝜂^𝑘cos𝜂₀−cos²𝜂_𝑘+𝜂⁰𝜂_𝑘sin𝜂_𝑘sin𝜂₀−sin²𝜂_𝑘=−𝜂_𝑘sin𝜂_𝑘cos𝜂^𝑘−𝜂₀−sin𝜂_𝑘sin𝜂₀+𝜂⁰𝜂_𝑘=−𝜂_𝑘sin𝜂_𝑘sin𝜂₀−1sin𝜂_𝑘cos𝛿_𝑘−1+𝜂_𝑘−𝜂₀sin𝜂₀≥−𝜂^𝑘sin𝜂_𝑘cos𝛿_𝑘−1,(6) where 𝜂_𝑘∈(2𝑘𝜋,2(𝑘+1)𝜋) and sin𝜂_𝑘>0. Therefore, 𝑧₀≥𝑧_𝑘. For 𝑘=−1 and 𝑘≤−2, (5a) and (5c), respectively, can be applied to extend the above result.Case 2 (𝑧∈[−𝑒⁻¹,0)). 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 𝜁₋₁∈(−∞,−1]. Thus 𝜁₀≥𝜁₋₁. For 𝑘=1,±2,±3,…,𝜂₀=0 implies 𝑒^𝜁𝑘𝜁_𝑘cos𝜂_𝑘−𝜂_𝑘sin𝜂_𝑘=𝜁₀𝑒^𝜁0⟹𝑒^𝜁𝑘−𝜂_𝑘cot𝜂_𝑘cos𝜂_𝑘−𝜂_𝑘sin𝜂_𝑘=𝜁₀𝑒^𝜁0⟹−𝜂_𝑘𝑒^𝜁𝑘=𝜁₀𝑒^𝜁0sin𝜂_𝑘⟹𝑒^{𝜁0−𝜁𝑘}=−𝜂^𝑘𝜁₀sin𝜂_𝑘.(7) The last ratio of (7) is clearly no less than unity since 𝜂_𝑘/sin𝜂_𝑘 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=𝑧_𝑘, respectively, the principal branch and the kth-branch values have the same real part 𝜁₀=𝜁_𝑘=𝜁. Note only non-negative 𝜁 needs to be discussed in this case. Consequently, cos𝜂_𝑘∈(0,1), since here 𝜂_𝑘∈((2𝑘−1)𝜋,2𝑘𝜋) for 𝑘>0 and 𝜂_𝑘∈(2𝑘𝜋,(2𝑘+1)𝜋) for 𝑘<0, but with 𝜁_𝑘=−𝜂_𝑘cot𝜂_𝑘≥0 the ranges of 𝜂_𝑘 can be limited to 𝜂_𝑘∈((2𝑘−1/2)𝜋,2𝑘𝜋) for 𝑘>0 and 𝜂_𝑘∈(2𝑘𝜋,(2𝑘+1/2)𝜋) for 𝑘<0. Because in Figure 2 𝑧→∞ toward right in the branch segments with 𝑧∈(0,∞), the proof will be completed if 𝑧₀≤𝑧_𝑘 is always true. Now 𝑧_𝑘−𝑧₀=𝑒^𝜁(𝜁cos𝜂_𝑘−𝜂_𝑘sin𝜂_𝑘)−𝜁𝑒^𝜁 and 𝜁cos𝜂_𝑘−𝜂_𝑘sin𝜂_𝑘−𝜁=𝜁cos𝜂_𝑘+sin²𝜂_𝑘cos𝜂_𝑘=𝜁1−1cos𝜂_𝑘−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]𝜆⟺−𝑎ℎ₀−𝜆𝑎−₁𝑒𝑎^−𝜆ℎ₀=0𝑒^{ℎ(𝜆−𝑎0)}=𝑎₁ℎ𝑒^−𝑎0ℎ⟺⟺𝜆𝒲=𝑎1₁ℎ𝑒^−𝑎0ℎ=ℎ𝜆−𝑎₀ℎ𝒲𝑎¹ℎ𝑒^−𝑎0ℎ+𝑎₀.(9) In accordance with the Lambert function, 𝜆₀=𝑎₀+𝒲₀(𝑎₁ℎ𝑒^−𝑎0ℎ)/ℎ is called the principal eigenvalue, and 𝜆_𝑘=𝑎₀+𝒲_𝑘(𝑎₁ℎ𝑒^−𝑎0ℎ)/ℎ,𝑘=±1,±2,…, is called the kth-branch eigenvalue. Note that the parameters 𝑎₀, 𝑎₁, and ℎ 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(𝜆_k),𝑘=±1,±2,…, for any given set of 𝑎₀, 𝑎₁ and ℎ.(ii)𝑎𝑎₁ℎ𝑒^−𝑎0ℎ>−𝑒⁻¹⟺𝜆₀∈ℝ,₁ℎ𝑒^−𝑎0ℎ=−𝑒⁻¹⟺𝜆₀=−1ℎ+𝑎⁰𝑎₁,ℎ𝑒^−𝑎0ℎ<−𝑒⁻¹⟺𝜆₀∈ℂ.(10)(iii)−1/ℎ+𝑎₀≤Re(𝜆₀)<|𝑎₀|+|𝑎₁|.(iv)0≤Im(𝜆₀)<𝜋/ℎ.

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+𝑎₁𝑒^−𝜆ℎ one has 𝜎=𝑎₀+𝑎₁𝑒^−𝜎ℎcos(−𝜔ℎ). Thus,|𝜎|=||𝑎₀+𝑎₁𝑒^−𝜎ℎcos(−𝜔ℎ)||≤||𝑎₀||+||𝑎₁||||𝑒^−𝜎ℎ||.(11)

If 𝜎>0 then Re(𝜆₀)<|𝑎₀|+|𝑎₁|, and if 𝜎≤0 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 𝐾≥−𝑒⁻¹/(𝑏ℎ𝑒^−𝑎0ℎ), and complex-valued for 𝐾<−𝑒⁻¹/(𝑏ℎ𝑒^−𝑎0ℎ). 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/ℎ.

Proof. Suppose 𝑎₀≥1/ℎ. Then property (iii) of Section 4 shows that 0≤Re(𝜆₀) no matter what 𝐾 is. On the other hand, if 𝑎₀<1/ℎ, or 𝑎₀ℎ+𝜀=1 for some 𝜀>0, then in Figure 2 it can be seen that there is a 𝐾 such that 𝒲₀(𝐾𝑏ℎ𝑒^−𝑎0ℎ)=−1+𝜀/2>−1, which makes 𝜆₀=−𝜀/(2ℎ)<0.

Theorem 5. Suppose 𝑎₀<1/ℎ in (1). The range of K for the asymptotic stability of (1) is (−𝜂₀/(𝑏ℎsin𝜂₀),−𝑎₀/𝑏), where 𝑎₀ℎ=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(𝐾𝑏ℎ𝑒^−𝑎0ℎ)/ℎ smaller than 0, or that of 𝒲₀(𝐾𝑏ℎ𝑒^−𝑎0ℎ) smaller than −𝑎₀ℎ. From Figure 2 and the assumption of 𝑎₀<1/ℎ or −1<−𝑎₀ℎ, 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=−𝑎₀ℎ with 𝐾₂𝑏ℎ𝑒^−𝑎0ℎ=𝜁₀𝑒^𝜁0(12) and 𝐾₁ makes 𝜁₀=−𝑎₀ℎ=−𝜂₀cot𝜂₀ with 𝜂₀∈(0,𝜋) and 𝐾₁𝑏ℎ𝑒^−𝑎0ℎ==𝑒−𝑒^𝜁0_𝜁0𝜁₀cos𝜂₀−𝜂₀sin𝜂₀=−𝑒_𝜁0𝜂₀cot𝜂₀cos𝜂₀+𝜂₀sin𝜂₀𝜂₀sin𝜂₀.(13) Hence, 𝐾₁=−𝜂₀/(𝑏ℎsin𝜂₀) 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 ℎ>0, the range of stabilizing gain K is denoted as Θ_ℎ.

5.2. Robust Stability with Respect to Delay Size

The delay time ℎ in (1) is assumed to be a constant, but may be uncertain. In the practical applications, the delay time ℎ may not be estimated accurately. Therefore, it is important to analyze how robust the system (2) is with respect to the delay time ℎ under a selected stabilizing state feedback gain 𝐾≠0. Based on the signs of 𝑎₀ and 𝐾, four cases are discussed separately.

Case 1 (𝑎₀≤0 and 𝐾>0). Since ℎ is a positive delay time, the condition of Theorem 4 holds for all 𝑎₀≤0. Select a positive 𝐾∈Θ_ℎ^∗, where ℎ^∗ is the assumed delay size. Thus 𝐾≤−𝑎₀/𝑏 and the upper bound is independent of ℎ. Also, by Theorem 5 the lower bound in Θ_ℎ is nonpositive for all ℎ>0. Hence the positive 𝐾 is in Θ_ℎ for all ℎ>0. In short, the feedback system is asymptotically stable independent of the delay time h.

Case 2 (𝑎₀≤0 and 𝐾<0). Suppose a negative 𝐾∈Θ_ℎ^∗ is selected, where ℎ^∗ is the assumed delay size. Note that the principal eigenvalue is 𝜆₀=𝑎₀+𝒲₀(𝐾𝑏ℎ𝑒^−𝑎0ℎ)/ℎ, so if ℎ<ℎ^∗ then 𝐾𝑏ℎe^−𝑎0ℎ 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 ℎ−ℎ^∗ is large enough, the real part of 𝒲₀(𝐾𝑏ℎ𝑒^−𝑎0ℎ) will become positive. More specifically, as in the proof of Theorem 5, if 𝑎₀ℎ=𝜂₀cot𝜂₀ with some 𝜂₀∈(𝜋/2,𝜋) and 𝐾=−𝜂₀/(𝑏ℎsin𝜂₀)=−𝑎₀/(𝑏cos𝜂₀), then 𝜆₀ will move to the boundary of LHP. Note that such an 𝜂₀ exists if and only if 𝐾≤𝑎₀/𝑏, which means if 𝐾>𝑎₀/𝑏 then the feedback system is stable independent of ℎ, and if 𝐾≤𝑎₀/𝑏 then stability will be lost for ℎ larger than (𝜂₀cot𝜂₀)/𝑎₀.

Case 3 (𝑎₀>0 and 𝐾<0). Again, suppose a negative 𝐾∈Θ_ℎ^∗ is selected, where ℎ^∗ is the assumed delay size. Note that if ℎ<ℎ^∗, then by Figure 1 the 𝜂₀ satisfying 𝑎₀ℎ=𝜂₀cot𝜂₀ is closer to 𝜋/2 than the 𝜂^∗0 satisfying 𝑎₀ℎ^∗=𝜂^∗0cot𝜂^∗0. Therefore Θ_ℎ^∗⊆Θ_ℎand the feedback system keeps asymptotic stability with the gain 𝐾. Next, if ℎ≥ℎ^∗ then Θ_ℎ⊆Θ_ℎ^∗ and the feedback system keeps asymptotic stability if and only if ℎ is smaller than (𝜂₀cot𝜂₀)/𝑎₀, where 𝜂₀=cos⁻¹(−𝑎₀/(𝑏𝐾))∈(0,𝜋/2).

Case 4 (𝑎₀>0 and 𝐾>0). Since positive 𝐾∉Θ_ℎ for all ℎ 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_ℎ results in a better response performance than that of the uncontrolled system. Actually, for positive 𝑎₀ all 𝐾∈Θ_ℎ gives a better performance, but not necessarily so for negative 𝑎₀.

Theorem 7. For system (1), only 𝐾∈(−𝜂̃₀/(𝑏ℎsin𝜂̃₀),0), where 𝜂̃₀cot𝜂̃₀=0 and 𝜂̃₀∈(0,𝜋), gives a better response performance than that of the uncontrolled system.

Proof. Suppose for (1), Θ_ℎ is not empty and 𝑎₀<0. From Figure 2, the set Θ_ℎ consists of three subsets (−𝜂₀/(𝑏ℎsin𝜂₀),−𝜂̃₀/(𝑏ℎsin𝜂̃₀)], (−𝜂̃₀/(𝑏ℎsin𝜂̃₀),0), and [0,−𝑎₀/𝑏), where 𝑎₀ℎ=𝜂₀cot𝜂₀, 𝜂̃₀cot𝜂̃₀=0, and 𝜂₀,𝜂̃₀∈(0,𝜋). However, only the 𝜆₀ corresponding to 𝐾∈(−𝜂̃₀/(𝑏ℎsin𝜂̃₀),0) has a more negative real part than 𝑎₀, which means such feedback gain results in a larger decay rate. Next, the set (−𝜂̃₀/(𝑏ℎsin𝜂̃₀),0) can be divided into (−𝜂̃₀/(𝑏ℎsin𝜂̃₀),−𝑒⁻¹/(𝑏ℎ𝑒^−𝑎0ℎ)) and [−𝑒⁻¹/(𝑏ℎ𝑒^−𝑎0ℎ),0), where 𝜂̃₀cot𝜂̃₀=0 and 𝜂̃₀∈(0,𝜋). By property (ii), the 𝜆₀ from 𝐾∈(−𝜂̃₀/(𝑏ℎsin𝜂̃₀),−𝑒⁻¹/(𝑏ℎ𝑒^−𝑎0ℎ)) implies an oscillatory response. Finally, if 𝐾∈[−𝑒⁻¹/(𝑏ℎ𝑒^−𝑎0ℎ),0), then 𝜆₀ is a real-valued. Although other 𝜆_𝑘 are complex-valued, 𝜆₀ is the rightmost eigenvalue and far from others. Thus, oscillation only appears in the transient response. In fact, 𝐾=−𝑒⁻¹/(𝑏ℎ𝑒^−𝑎0ℎ) gives the largest decay rate and least oscillation.

6. Examples and Simulations

Example 8. Let the parameter combination (𝑎₀,𝑏,𝐾,ℎ)=(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,ℎ)=(0.3,1,𝐾,1.7), where 𝑎₀<1/ℎ and there exists a feasible set Θ_ℎ={𝐾∣𝐾∈(−0.7424,−0.3)}. In Figure 4, a feedback gain 𝐾∉Θ_ℎ is used, so the principal eigenvalue locates on RHP and the response of 𝑥(𝑡) is divergent. In Figure 5, a feedback gain 𝐾∈Θ_ℎ is used, so the principal eigenvalue locates on LHP and the response of 𝑥(𝑡) is convergent.

(a)

(b)

Figure 4 

(a) The eigenvalues and (b) the response of system (2) with 𝐾=−0.29.

(a)

(b)

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 ℎ. There are four simulations. The first one refers to Case 1 in Section 5.2, where (𝑎₀,𝑏,𝐾,ℎ)=(−3,1,2,ℎ), and the principal eigenvalue versus ℎ is shown as Figure 6(a). The stability of this system in this simulation is independent on ℎ.

(a)

(b)

(c)

(d)

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,ℎ)=(−3,1,−4,ℎ) and (−3,1,−1,ℎ), respectively. In Figure 6(b), the stability of this system is dependent on h, and the stability is maintained for ℎ∈[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,ℎ)=(0.2,1,−4,ℎ). It is seen that the crossing frequency is 3.995, and the stability is maintained for ℎ=[0,0.3807].

Example 10. Let the parameter combination (𝑎₀,𝑏,𝐾,ℎ)=(0.01,0.0506,𝐾,6) [21] be selected. The parameter 𝑎₀ is positive and satisfies Theorem 4, and Θ_ℎ={𝐾∣𝐾∈(−5.0488,−0.1976)}. In this example, all 𝐾∈Θ_ℎ 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=−𝑒⁻¹/(𝑏ℎe^−𝑎0ℎ)) is better than green line in terms of decay rate. Consider another parameter combination (𝑎₀,𝑏,𝐾,ℎ)=(−1,2,𝐾,0.7). The parameter 𝑎₀ is negative and satisfies Theorem 4, and Θ_ℎ={𝐾∣𝐾∈(−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=−𝑒⁻¹/(𝑏ℎ𝑒^−𝑎0ℎ)=−0.13049.

(a)

(b)

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.

Acknowledgment

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