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

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.


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 timedelay systems are very abundant [3,4]. In the field of controller design, many literature reviews [5][6][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][11][12][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 2 Journal of Control Science and Engineering 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.

Problem Formulation
Consider the first-order time-delay system, where x ∈ R is the state variable, u ∈ R is the input signal, a 0 ∈ R and b ∈ R are the system parameters, h is a nonnegative constant delay time, and x 0 is the initial condition. When a state feedback controller where a 1 = 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 − a 0 − a 1 e −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.

The Lambert Function
The Lambert function W (·) is the function that satisfies W (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 relations α = e ζ ζ cos η − η sin η , β = e ζ η cos η + ζ sin η . ( 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.

Lemma 2. The largest lower bound of the real parts of the principal values is -1.
Moreover, one has the following less intuitive Lemma.

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 − a 0 − a 1 e −λh = 0 In accordance with the Lambert function, λ 0 = a 0 + W 0 (a 1 he −a0h )/h is called the principal eigenvalue, and Journal of Control Science and Engineering eigenvalue. Note that the parameters a 0 , a 1 , and h in the firstorder 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.

Applications to Some Control Issues
Without loss of generality, subsequently let b be a given positive constant.

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).
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 . (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 a 0 and K, four cases are discussed separately.

Robust Stability with Respect to Delay Size. The delay time h in
Case 1 (a 0 ≤ 0 and K > 0). Since h is a positive delay time, the condition of Theorem 4 holds for all a 0 ≤ 0. Select a positive K ∈ Θ h * , where h * is the assumed delay size. Thus K ≤ −a 0 /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 (a 0 ≤ 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 λ 0 = a 0 + W 0 (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 λ 0 will stay in the LHP. However, when h − h * is large enough, the real part of W 0 (Kbhe −a0h ) will become positive. More specifically, as in the proof of Theorem 5, if a 0 h = η 0 cot η 0 with some η 0 ∈ (π/2, π) and K = −η 0 /(bh sin η 0 ) = −a 0 /(b cos η 0 ), then λ 0 will move to the boundary of LHP. Note that such an η 0 exists if and only if K ≤ a 0 /b, which means if K > a 0 /b then the feedback system  is stable independent of h, and if K ≤ a 0 /b then stability will be lost for h larger than (η 0 cot η 0 )/a 0 .
Case 4 (a 0 > 0 and K > 0). Since positive K / ∈ Θ h for all h when a 0 > 0, this case needs no discussion.

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 a 0 all K ∈ Θ h gives a better performance, but not necessarily so for negative a 0 . Theorem 7. For system (1), only K ∈ (− η 0 /(bh sin η 0 ), 0), where η 0 cot η 0 = 0 and η 0 ∈ (0, π), gives a better response performance than that of the uncontrolled system.

Examples and Simulations
Example 8. Let the parameter combination (a 0 , b, K, h) = (3, 1, K, 0.7) be selected, where a 0 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.
Then consider the parameter combination (a 0 , b, K, h) = (0.3, 1, K, 1.7), where a 0 < 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.
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 (a 0 , 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.
Then Figures 6(b) and 6(c) refer to Case 2 in Section 5.2, with (a 0 , 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 (a 0 , 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].  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.

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.