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

Authors: Shu-An He; I-Kong Fong
Journal: Journal of Control Science and Engineering
 (2011)

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## 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.


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## Body

## 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,(1)ẋ(t)=a0x(t)+bu(t-h),t≥0,x(t)=x0,t=0,
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) becomes(2)ẋ(t)=a0x(t)+a1x(t-h),t≥0,x(t)={x0,t=0,0,t<0,
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 function𝒲(·) is the function that satisfies𝒲(z)e𝒲(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 relations(3)α=eζ(ζcos⁡η-ηsin⁡η),β=eζ(ηcos⁡η+ζsin⁡η).In the next section, it will be clear that onlyz∈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 ofζ=-ηcotη.Based on Figure1 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. (4a)W0(z)={-ηcotη+jη,forη∈(0,π),ζ,forζ≥-1,(4b)W-1(z)={ζ,forζ<-1,-ηcotη+jη,forη∈(-2π,-π)∪(-π,0),(4c)Wk(z)={-ηcot⁡η+jη,forη∈((2k-1)π,2kπ)∪(2kπ,(2k+1)π),k=1,2,…,-ηcot⁡η+jη,forη∈(2kπ,(2k+1)π)∪((2k+1)π,2(k+1)π),k=-2,-3,….In Figure2 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 ofz=α.Lemma 1.
Only parts of𝒲0(z) and 𝒲-1(z) are real-valued, and all other branches of the Lambert function are complex-valued.
In fact, only𝒲0(0)=0, and all other branches are undefined at z=0. Also,(5a)W-1(z)=W0*(z),forz∈(-∞,-e-1),(5b)Wk(z)=W-k*(z),forz∈(0,∞),k=1,2,…,(5c)Wk(z)=W-(k+1)*(z),forz∈(-∞,0),k=1,2,…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 givenz∈ℝ, the real part of 𝒲0(z) is no less than that of 𝒲k(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 anyz∈ℝ and integer k, let 𝒲k(z)=ζk+jηk. Suppose that at z=z0 and z=zk the principal branch and thekth-branch values, respectively, have the same real part ζ0=ζk=ζ. Since in Figure 2, z→-∞ toward right in the branch segments with z∈(-∞,-e-1), the proof will be completed if z0≥zk is always true. Now z0-zk=eζ(ζcosη0-η0sinη0-ζcosηk+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
(6)ζ(cos⁡η0-cos⁡ηk)-η0sin⁡η0+ηksin⁡ηk=-ηksin⁡ηk(η0ηkcos⁡ηkcos⁡η0-cos⁡2ηk+η0ηksin⁡ηksin⁡η0-sin2ηk)=-ηksin⁡ηk[η0ηkcos⁡(ηk-η0)-sin⁡ηksin⁡η0+η0ηksin⁡ηksin⁡η0-1]=-ηksin⁡ηk(cos⁡δk-1)+(ηk-η0)sin⁡η0≥-ηksin⁡ηk(cos⁡δk-1),
where ηk∈(2kπ,2(k+1)π) and sinηk>0. Therefore, z0≥zk. 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 Figure2 it can be seen that ζ0∈[-1,0) and ζ-1∈(-∞,-1]. Thus ζ0≥ζ-1. For k=1,±2,±3,…,η0=0 implies
(7)eζk(ζkcos⁡ηk-ηksin⁡ηk)=ζ0eζ0⟹eζk(-ηkcotηkcos⁡ηk-ηksin⁡ηk)=ζ0eζ0⟹-ηkeζk=ζ0eζ0sin⁡ηk⟹eζ0-ζk=-ηkζ0sin⁡ηk.
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 atz=z0 and z=zk, respectively, the principal branch and thekth-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∈((2k-1)π,2kπ) for k>0 and ηk∈(2kπ,(2k+1)π) for k<0, but with ζk=-ηkcotηk≥0 the ranges of ηk can be limited to ηk∈((2k-1/2)π,2kπ) for k>0 and ηk∈(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≤zk is always true. Now zk-z0=eζ(ζcosηk-ηksinηk)-ζeζ and
(8)ζcos⁡ηk-ηksin⁡ηk-ζ=ζ(cos⁡ηk+sin2ηkcos⁡ηk-1)=ζ(1cos⁡ηk-1).
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](9)λ-a0-a1e-λh=0⟺h(λ-a0)eh(λ-a0)=a1he-a0h⟺W(a1he-a0h)=h(λ-a0)⟺λ=1hW(a1he-a0h)+a0.
In accordance with the Lambert function, λ0=a0+𝒲0(a1he-a0h)/h is called the principal eigenvalue, and λk=a0+𝒲k(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(λk),k=±1,±2,…, for any given set of a0, a1 and h.(ii)
(10)a1he-a0h>-e-1⟺λ0∈R,a1he-a0h=-e-1⟺λ0=-1h+a0,a1he-a0h<-e-1⟺λ0∈C.(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 Section3. 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,(11)|σ|=|a0+a1e-σhcos⁡(-ωh)|≤|a0|+|a1||e-σh|.Ifσ>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 letb 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.
Supposea0≥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.
Supposea0<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 ofK 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
(12)K2bhe-a0h=ζ0eζ0
and K1 makes ζ0=-a0h=-η0cotη0 with η0∈(0,π) and
(13)K1bhe-a0h=eζ0(ζ0cos⁡η0-η0sin⁡η0)=-eζ0(η0cotη0cos⁡η0+η0sin⁡η0)=-eζ0η0sin⁡η0.
Hence, K1=-η0/(bhsinη0) and K2=-a0/b.The result of Theorem5 is known in [18], but here a different derivation is provided.Definition 6.
For a givenh>0, the range of stabilizing gain K is denoted as Θh.


### 5.2. Robust Stability with Respect to Delay Size

The delay timeh 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).
Sinceh is a positive delay time, the condition of Theorem 4 holds for all a0≤0. Select a positive K∈Θh*, 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 Θh is nonpositive for all h>0. Hence the positive K is in Θh 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 negativeK∈Θh* 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 negativeK∈Θh* 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 η0* satisfying a0h*=η0*cotη0*. Therefore Θh*⊆Θhand the feedback system keeps asymptotic stability with the gain K. Next, if h≥h* then Θh⊆Θh* 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 positiveK∉Θh 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 Θh results in a better response performance than that of the uncontrolled system. Actually, for positive a0 all K∈Θh 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), Θh is not empty and a0<0. From Figure 2, the set Θh 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 λk 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.


## 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.
Supposea0≥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.
Supposea0<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 ofK 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
(12)K2bhe-a0h=ζ0eζ0
and K1 makes ζ0=-a0h=-η0cotη0 with η0∈(0,π) and
(13)K1bhe-a0h=eζ0(ζ0cos⁡η0-η0sin⁡η0)=-eζ0(η0cotη0cos⁡η0+η0sin⁡η0)=-eζ0η0sin⁡η0.
Hence, K1=-η0/(bhsinη0) and K2=-a0/b.The result of Theorem5 is known in [18], but here a different derivation is provided.Definition 6.
For a givenh>0, the range of stabilizing gain K is denoted as Θh.


## 5.2. Robust Stability with Respect to Delay Size

The delay timeh 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).
Sinceh is a positive delay time, the condition of Theorem 4 holds for all a0≤0. Select a positive K∈Θh*, 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 Θh is nonpositive for all h>0. Hence the positive K is in Θh 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 negativeK∈Θh* 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 negativeK∈Θh* 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 η0* satisfying a0h*=η0*cotη0*. Therefore Θh*⊆Θhand the feedback system keeps asymptotic stability with the gain K. Next, if h≥h* then Θh⊆Θh* 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 positiveK∉Θh 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 Θh results in a better response performance than that of the uncontrolled system. Actually, for positive a0 all K∈Θh 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), Θh is not empty and a0<0. From Figure 2, the set Θh 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 λk 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 versusK.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 Θh={K∣K∈(-0.7424,-0.3)}. In Figure 4, a feedback gain K∉Θh is used, so the principal eigenvalue locates on RHP and the response of x(t) is divergent. In Figure 5, a feedback gain K∈Θh is used, so the principal eigenvalue locates on LHP and the response of x(t) is convergent.(a) The eigenvalues and (b) the response of system (2) with K=-0.29.
(a)(b)(a) The eigenvalues and (b) the response of system (2) with K=-0.5.
(a)(b)Example 9.
In this example, a stabilizingK 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.The simulations for all cases in Section5.2: (a) Case 1; (b) Case 2 (c); Case 2; (d) Case 3.
(a)(b)(c)(d)Then Figures6(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 Θh={K∣K∈(-5.0488,-0.1976)}. In this example, all K∈Θh 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 Θh={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.The responses of Example10.
(a)(b)


## 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.



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*Source: 719730-2011-12-13.xml*