Output stabilization of a class of second-order linear time-delay systems: An eigenvalue approach

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Abstract

This paper studies linear time-invariant systems with an input delay and two repeated or distinct real poles. The closed-loop system eigenvalue-loci with respect to the output feedback controller gain are investigated by using the Lambert function and root-locus construction techniques. Output feedback stabilization conditions and stability robustness with respect to the delay time uncertainty are established. Also, the response performance is discussed. Three examples and related simulations are presented to illustrate the analysis results.

Introduction

The retarded type time-delay systems modeled by the combination of ordinary differential equations (ODEs) and difference terms in time, or the differential difference equations (DDEs) [1], are an important research subject in the field of control engineering. This paper studies linear time-invariant systems described by a DDE with a second-order ODE and an input delay. The second-order ODE part is assumed to have two repeated or distinct real poles. Such systems can be encountered in many engineering applications [2].

Recently, the linear matrix inequality (LMI) approach has been widely applied to analyze the stability and to design stabilizing controllers for time-delay systems (see, e.g., [3], [4], [5], [6], [7]). However, the resultant LMI conditions are mostly sufficient only, and more direct and detailed results are often expected, especially for low-order systems with a single delay. For example, the authors of [8] focus on second-order time-delay systems with one unstable open-loop pole, and propose an approach to design the PID-controllers via Nyquist plot. In [9], an eigenvalue approach is proposed for stability analysis and stabilization of dynamical systems subjected to time-delays, and many iterative methods for finding the dominant eigenvalues or the rightmost eigenvalue are presented. Different from the methods in [9], a numerical method using the Lambert function is introduced in [10] to find the spectrum for first-order time-delay systems, and the state feedback stabilization problem is studied thoroughly in [11].

In [12], [13], the authors introduce an auxiliary matrix to combine with 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]. Despite so, the Lambert function is still a useful tool for handling lower-order systems [11]. In this paper, second-order time-delay systems with repeated open-loop poles are treated by extending the method of [11] to discuss the output feedback stabilization problems. Then, the more common systems with two distinct real open-loop poles are considered. By further extending the results for systems with repeated poles and by incorporating the root-locus construction techniques, output feedback stabilization conditions and the robustness with respect to delay time uncertainty are established. Furthermore, with the full view of spectrum, how to select a feedback gain in order to obtain a better response performance is also explored in this paper. Three examples and related simulations are presented to illustrate the analysis results.

Section snippets

System description and the Lambert function

Consider the second-order time-delay systemÿ(t)=p1ẏ(t)p0y(t)+bu(th),t0y(t)=y0,ẏ(t)=y1,t=0where y is the real output signal, u is the real input signal, p1,p0 and b are system parameters, h is the nonnegative constant delay time and y1andy0 are the initial conditions. When an output feedback control u(t)=Ky(t) is applied, (1) becomesÿ(t)=p1ẏ(t)p0y(t)+bKy(th),t0[y(t)ẏ(t)]={[y0y1],t=0[00],t<0

The DDE model (2) has a transcendental characteristic equationΔ(s)=s2+p1s+p0bKesh=0with

Systems with repeated open-loop poles

Consider first the system (2) with two repeated open-loop poles s1=s2=a, so (3) can be rewritten as Δ(s)=(sa)2=bKesh, where p0=a2 and p1=−2a.

Systems with distinct real open-loop poles

Systems with distinct real open-loop poles are more common than those with repeated poles in practice. In this paper, the principal eigenvalue-loci of (2) with two distinct real open-loop poles are plotted with the corresponding loci in Fig. 3 as the basis. Assume the open-loop system (2) has two distinct real poles s1>s2. There are three possibilities:

  • (a)

    both s1 and s2 are negative;

  • (b)

    s1 is positive and s2 is negative;

  • (c)

    both s1 and s2 are positive.

In Theorem 3 to be developed soon, it will be clear

Examples and simulations

Example 1

System (2) with the parameter combination (p1, p0, b)=(−4, 4, 1) cannot be stabilized by any output feedback gain K since a>0 in Theorem 1, no matter what h is. By Theorem 1, Theorem 2 system (2) with the parameter combination (p1, p0, b, h)=(6, 9, 1, 1.8) has the set of stabilizing gain Θh={K|K(10.6722,9)}. For K=2 the state responses with x1=yandx2=ẏ are shown in Fig. 5. It can be checked that for K=1.1>0 or K=−2<0 with |K|<9, the stability of system is kept for all constant h>0, but for K

Conclusions

In this paper, the eigenvalue-loci of a class of second-order LTI time-delay system are sketched. Via the Lambert function and the root-locus construction techniques, the rightmost eigenvalues are identified. Furthermore, theorems are developed for designing stabilizing output feedback controllers. Besides, the response performance is investigated due to the full view of spectrum is sketched. Then three examples and simulations are shown to illustrate the derived results.

This paper just focuses

Acknowledgment

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

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