New Results on Eigenvalue Distribution and Controller Design for ...

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New Results on Eigenvalue Distribution and Controller Design for Time Delay Systems Honghai Wang, Jianchang Liu, and Yu Zhang

Abstract—This paper considers the eigenvalue distribution of a linear time-invariant (LTI) system with commensurate time delays and its application to proportional-integralderivative (PID) controller design for a delay plant via dominant eigenvalue assignment. A new result on the root distribution of a quasi-polynomial is first produced by applying part of Pontryagin’s conclusions. This result which gives a necessary and sufficient condition can be directly used to judge the number of the right-half plane eigenvalues of the characteristic equation of a time delay system. Based on the proposed result, necessary and sufficient conditions on dominant eigenvalue assignment for PID control of time delay systems are presented and an algorithm is then provided to determine the PID controller gains. The proposed approaches can assign some (one or two) eigenvalues to the desired positions and all the other eigenvalues to the left of a given line to guarantee the dominance of the assigned ones, which enables us to design the controller according to the desired performance indexes for a standard first-order system or a standard second-order system in addition to stability. The method is effective for the closed-loop characteristic equation being retarded type or neutral type. The controller gains to achieve the control objective can be characterized by a straightforward computation. Further, a result on degradation to proportional-integral (PI) control of time delay systems is given. Index Terms—Dominant eigenvalue assignment, eigenvalue distribution, linear time-invariant (LTI) system, proportional-integral-derivative (PID) controller, time delay.

I. INTRODUCTION IME delay is prevalent in many practical problems in science and engineering, especially in control systems [1]– [3]. As the time delay gives rise to a closed-loop characteristic equation with an infinite number of roots, the delayed response of the control system to disturbances appearing in the process results in the generation of oscillations in the closed-loop system,

T

Manuscript received December 2, 2015; revised June 8, 2016; accepted October 12, 2016. Date of publication December 7, 2016; date of current version May 25, 2017. This work was supported in part by the National Natural Science Foundation of China (NSFC) (No. 61374137, No. 61533007, No. 61573087), the IAPI Fundamental Research Funds (2013ZCX02-03). Recommended by Associate Editor C.-Y. Kao. The authors are with the Institute of Automation, College of Information Science and Engineering, State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang Liaoning Province 110819, P. R. China (e-mail: water. [email protected]; [email protected]; [email protected]. edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2016.2637002

and also in the non-stability of this system [4]. Thus, the topic of time delay has attracted substantial interest and the study of delay systems has a long history [5], [6]. Linear time-invariant (LTI) delay systems which are often convenient to apply frequency domain analysis constitute an important part in the field of time delay (see [3] and the references therein). As the eigenvalue distribution of LTI delay systems is a significant problem and plays a fundamental role in the study on the stability analysis and controller design, it is thus unsurprising that this subject has been studied long and well [7]. In earlier years, Pontryagin studied some problems on the zeros of quasi-polynomials [4]. Based on Pontryagin’s results, a suitable extension of the Hermite-Biehler theorem to the quasipolynomial was developed. Such a result can be used to deal with the characteristic functions of time delay systems which are reducible to quasi-polynomials and permit one to obtain the necessary and sufficient condition for the characteristic function being Hurwitz, i.e., the zeros of the characteristic function are all in the open left-half complex plane [8], [9]. In the last decades, according to the root continuity argument, the τ -decomposition idea was provided and largely applied in the analysis of the distribution of the characteristic roots of LTI systems with time delay, see e.g., [11]–[16]. Besides, for a retarded time delay system, some other approaches which can be adopted to compute the eigenvalues in a given region were proposed. In [17], a spectral method was presented for computing all characteristic roots in a given right-half plane. Based on the asymptotic features of infinitely many eigenvalues of a retarded system with time delay proposed by [8], an approach on computing its all eigenvalues located in a large region of the complex plane was presented in [18]. The Lambert W function method described in [19] and [20] can be applied to calculating the eigenvalues of a retarded delay system one by one from right to left in the complex plane. Moreover, the Nyquist criterion [21] and some softwares, such as DDE-BIFTOOL [22], also play an important role in the analysis of the eigenvalue distribution of time delay systems. It has to be stressed that the extension of the Hermite-Biehler theorem to a quasi-polynomial developed by Pontryagin is always a significantly theoretical result which not only can be applied to judging whether a quasi-polynomial is Hurwitz but also plays an important role in stability analysis and controller stabilization of time delay systems [10]. In recent years, using this theorem develops several valuable results on analytical characterization of all stabilizing low-order controllers, including proportional (P), proportional-integral (PI), and proportionalintegral-derivative (PID) controllers, for different models with time delay, see e.g., [5], [9], [23]–[27]. However, when a given quasi-polynomial is not Hurwitz stable, the extension of the Hermite-Biehler theorem cannot provide any information about the root distribution of the quasi-polynomial. On the other hand,

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WANG et al.: NEW RESULTS ON EIGENVALUE DISTRIBUTION AND CONTROLLER DESIGN FOR TIME DELAY SYSTEMS

in control systems, the fundamental requirement to be achieved is to guarantee the stability. However, the final control objective is to improve the performance under the condition of stability. As is known that many performances of a closed-loop system directly depend on the positions of its poles. Hence pole placement, especially dominant pole placement, is one of the mainstream methods in control system design. Nevertheless, it is a challenging work for a time delay system since it has infinite poles. By the method in [28], one can obtain the complete region of PI/PID controller gains which assign some rightmost eigenvalues of a closed-loop system with first-order delay plants to the desired positions. However, the information of the other eigenvalue locations is still indistinct. Thus, the further study on controller gains for guaranteeing the dominance of the assigned ones is very valuable. All the facts above motivate the present paper. By applying part of Pontryagin’s conclusions on quasipolynomial zero location, this paper produces a new result applicable to the case of quasi-polynomials that are not necessarily Hurwitz. This result has the ability to present the number of the right-half plane roots of a class of quasi-polynomials exactly, which also provides a new approach to learn the information of the eigenvalues of LTI time delay systems including to judge their stability. The proposed result is considered to become a fundamental theory for controller design. Based on the presented result, a novel and effective approach on PID controller design for time delay systems via dominant eigenvalue assignment is provided in this paper. Comparing with several significant studies engaged in calculating the complete set of PID controller gains for stabilizing a plant with time delay (see e.g., [5], [24], [26], and so forth), this work gives a theoretical method applicable to directly compute the PID gains according to the desired performance indexes on the premise of stability. Over the last several decades, various strategies have been developed for the PID type controller design via dominant eigenvalue assignment. In earlier years, [29] and [30] proposed some dominant pole assignment methods on PID controllers based on some simple or simplified models of plants. Recently, a guarantee of dominance in the pole placement with PID controllers based on the root locus and Nyquist plot applications was presented in [31]. A method on the parameter determination of more general controllers by shifting the quasi-direct poles as far to the left of the assigned ones as possible was provided in [32]. Moreover, ultimate frequency based dominant pole placement method was proposed in [33], [34]. The problem of dominant three-pole placement for PID controller design was considered in [34]–[36]. By and large, the main contributions of the present paper on controller design lie as follows: 1) the proposed approaches can assign one or two eigenvalues to the desired positions and all the other eigenvalues to the left of a given line to guarantee the dominance of the assigned ones, which enables us to design the controller according to the desired performance indexes for a standard first-order system or a standard secondorder system in addition to stability; 2) the method is effective for the closed-loop characteristic equation being retarded type or neutral type; 3) the controller gains to achieve the control objective can be characterized by a straightforward computation. Moreover, using the presented method, we also provide a result on PI tuning to time delay systems. The rest of this paper is organized as follows. The problem statement is given in Section II. Section III provides some preliminary results. A new result on the root distribution of a class of quasi-polynomials is derived in Section IV. This result is

Fig. 1.

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Unit feedback control system.

applied to studying the problem of controller design via dominant eigenvalue assignment for time delay systems in Section V. Some numerical examples are provided to illustrate the effectiveness of the proposed approaches in Section VI. Finally, a brief summary is given in Section VII. I. Notation: R (R− , R+ ) is the set of real (negative, positive) numbers; Z (Z+ ) is the set of integer (positive) numbers; C is the set of complex numbers. (x) and (x) denote the real part and imaginary part of x, respectively. II. PROBLEM STATEMENT Consider an LTI system with commensurate time delays δ(s) = a0 (s) +

ν 

aj (s)e−θ j s ,

(1)

j =1

where aj (s) are polynomials with real coefficients; θ1 , θ2 , . . ., θν are positive real numbers called delays satisfying 0 < θ1 < θ 2 < · · · < θν . Multiplying δ(s) by eθ ν s gives the following expression H(s)  bν (s)eτ ν s +

ν −1 

bj (s)eτ j s ,

(2)

j =0

where bν (s) = a0 (s), τν = θν , and bj (s) = aν −j (s), τj = θν − θν −j for j = 0, 1, 2, . . . , ν − 1. To mention, since eθ ν s vanishes nowhere in C, δ(s) and H(s) have the same set of zeros. In this paper, we will discuss the root distribution of the quasi-polynomial H(s) and provide a new result which is able to present the number of the right-half plane roots of H(s). Such a result plays a significant role in the analysis of the eigenvalue distribution of an LTI system with a single delay or commensurate delays. In addition, this result can be also applied in controller design for time delay systems. It is found that δ(s) in (1) with ν = 1 is similar to the characteristic function of a unit feedback control system shown in Fig. 1, where G(s) is a general high-order delay plant described as G(s) =

N (s) −θ s e D(s)

(3)

and C(s) is a PID controller whose transfer function is C(s) = kp +

ki + kd s. s

In G(s), θ ∈ R+ is the time delay, N (s) and D(s) are the co-prime polynomials defined as N (s) = dm sm + dm −1 sm −1 + · · · + d1 s + d0 , D(s) = sn + cn −1 sn −1 + · · · + c1 s + c0 ,

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where d0 , d1 , . . . , dm and c0 , c1 , . . . , cn −1 are real numbers and n > m. In C(s), kp , ki , kd ∈ R are proportional gain, integral gain, and derivative gain, respectively. The closed-loop characteristic function is

The polynomial (8) that has a principal term sM ZM (u, v) can be represented in the form  (N ) f (s, u, v) = sM Z∗ (u, v) + sμ Zμ(ν ) (u, v), (9)

(N )

μ< M ν ≤N

δ(s, kp , ki , kd )  sD(s) + (kd s2 + kp s + ki )N (s)e−θ s . (4) where In this paper, according to the new result on root distribution of a quasi-polynomial, we also consider to propose an approach on PID controller design for the delay system above. The control objective is to assign some eigenvalues ρ (dominant poles) to the desired positions, and furthermore, make the other eigenvalues to the left of a given line s = −σ, σ ∈ R+ . Here, we denote the number of dominant eigenvalues by κ. In PID control, when κ = 3, it is difficult to both assign three eigenvalues to the desired positions and guarantee the dominance of them because of the degrees of freedom of the controller gains. Thus, κ is considered as 2 or 1 in this study. The dominant eigenvalues ρ can be given in three cases: 1) ρ1,2 = a ± jb; 2) ρ1 = a1 , ρ2 = a2 ; 3) ρ = a, where a, a1 , a2 ∈ R− and b ∈ R+ . σ can be chosen as −v(ρ), where v is usually 3 ∼ 5 [31]. III. PRELIMINARY RESULTS The form of the function (2) is the same as that of a certain class of quasi-polynomials given by the following definition. Definition 1: [4] Let (s, t) → h(s, t) be a polynomial in two variables with complex coefficients  h(s, t) = αμν sμ tν , αμν ∈ C. (5) μ,ν ≥0

The term αM N sM tN is called the principal term of the polynomial (5) if αM N = 0 and for every other term αμν sμ tν , αμν = 0, we have M ≥ μ, N ≥ ν and at least one of these inequalities is strict. The function C s → h(s, es ) ∈ C is called a quasi-polynomial. The polynomial (5) with the principal term αM N sM tN may be represented in the form  (N ) h(s, t) = sM X∗ (t) + αμν sμ tν , (6) μ< M ν ≤N

where (N )

X∗

(t) =



αM ν tν .

(7)

(N )

Z∗

(u, v) =



(ν )

ZM (u, v).

(10)

ν ≤N

Here, we present two important theorems on the quasipolynomial h(s, es ). The first theorem refers to the zeros of the quasi-polynomial h(s, es ) in a semi-infinite strip of sufficient width. The second theorem discusses the number of zeros or real zeros of a function with the form (8) in a given region. Theorem 1: [4] (i) Let h(s, t) be a polynomial of the form (5) with the principal term αM N sM tN ; (N ) (ii) Let η ∈ R be a number such that X∗ (ex+j η ) = 0, (N ) ∀x ∈ R, and X∗ is the function defined by (7); (iii) The quasi-polynomial s → F (s) = h(s, es ) has no zeros on the imaginary axis. Then   M V (−2lπ + η, 2lπ + η) = 2π 2lN − RF + 2 + l , l −−−→ 0 l→∞

(11)

where V (−2lπ + η, 2lπ + η) denotes the angle described by the vector F (jω) around the origin when ω ranges through the interval −2lπ + η ≤ ω ≤ 2lπ + η; RF is the total number of zeros of the quasi-polynomial F (s) in the semi-infinite strip {s ∈ C : (s) ≥ 0, −2lπ + η ≤ (s) ≤ 2lπ + η}. Theorem 2: [4] If (i) Let f be a polynomial of the form (N ) (8), and sM ZM (u, v) its principal term; (ii) Let η ∈ R be a (N ) (N ) number such that Φ∗ (η + yi) = 0, ∀y ∈ R, where Φ∗ (·) is the function defined by (N )

Φ∗

(N )

(s) = Z∗

[cos(s), sin(s)],

where by Lemma 6.4.1 in [4] such a number always exists. Then in the strip −2lπ + η ≤ (s) ≤ 2lπ + η the function s → F (s) = f [s, cos(s), sin(s)] has exactly 4lN + M zeros, starting from a sufficiently large l ∈ Z+ . Thus, the function F has only real zeros if and only if it has exactly 4lN + M real zeros in the interval [−2lπ + η, ≤ 2lπ + η] ⊂ R, starting from a sufficiently large l ∈ Z+ .

ν ≤N

Next, we introduce a concept of principal term for polynomials in three variables of the form  f (s, u, v) = sμ Zμ(ν ) (u, v), (8) μ,ν ≥0 (ν )

where Zμ (u, v) is a homogeneous polynomial of degree ν; (ν ) Zμ (1, ±i) = 0, ∀μ ≥ 0, ∀ν ≥ 1. (N ) Definition 2: [4] An expression sM ZM (u, v) is called the principal term of polynomial (8), provided that for every other (ν ) term sμ Zμ (u, v) we have M ≥ μ, N ≥ ν and at least one of these inequalities is strict.

IV. A NEW RESULT ON ROOT DISTRIBUTION OF QUASI-POLYNOMIALS In this section, we make use of the theorems in Section III to develop a new result on root distribution of the quasi-polynomial (2). Such a result is applicable to judge the number of the roots of quasi-polynomials as the form (2) in the right-half plane. For the quasi-polynomial (2), we first consider that deg[bj (s)] and τj are nonnegative integers, and define deg[bν (s)] = M and τν = N , where M, N ∈ Z+ . Moreover, denote by α0 and αν the coefficients of the highest order terms of b0 (s) and bν (s), respectively. For deg[bj (s)] in the quasi-polynomial (2), we consider the following two cases: (A) deg[bj (s)] < M for j = 0, 1, 2, . . . , ν − 1;

WANG et al.: NEW RESULTS ON EIGENVALUE DISTRIBUTION AND CONTROLLER DESIGN FOR TIME DELAY SYSTEMS

(B) deg[bj (s)] < M for j = 1, 2, . . . , ν − 1, deg[b0 (s)] = M , and |α0 | < |αν |. Remark 1: The quasi-polynomial (2) with Case (A) belongs a class of quasi-polynomials of retarded type and with Case (B) becomes a class of quasi-polynomials of neutral type. Before stating the main result, we give some definitions. Definition 3: Let H(jω) = Hr (ω) + jHi (ω), where ω ∈ R. Definition 4: For a sufficiently large integer l, define  π for M even, 2lπ + 2N Ωi = 2lπ for M odd,  2lπ for M even, Ωr = π 2lπ + 2N for M odd. Definition 5: Let 0 = ωi,0 < ωi,1 < ωi,2 < · · · < ωi,Q −1 be the real and distinct zeros of Hi (ω) with odd multiplicities in [0, Ωi ) and ωi,Q be the smallest positive real zero of Hi (ω) with ω > Ωi . Then, the imaginary signature of H(s) denoted as γi (H) is defined by

above, we can conclude that Hr (ω) or Hi (ω) has only real roots in the interval [2lπ + η, 2(l + 1)π + η] if and only if Hr (ω) or Hi (ω) has exactly 2N real roots in this interval.  Lemma 2: Consider the quasi-polynomial H(s) in (2). For Case (A) or Case (B), there exists a positive integer l such that Hr (ω) and Hi (ω) have the following properties: (i) Hi (ω) has only 2N roots in the interval [Ωi + 2kπ, Ωi + 2(k + 1)π] and these roots are all real; (ii) Hr (ω) has only 2N roots in the interval [Ωr + 2kπ, Ωr + 2(k + 1)π] and these roots are all real; (iii) The roots of Hi (ω) and those of Hr (ω) interlace in the interval [Ωi , +∞) or [Ωr , +∞); (iv) [∠H(jω)] > 0 in the interval [Ωi , +∞) or [Ωr , +∞). Here, k is a given nonnegative integer arbitrarily. Proof: By substituting s = jω into H(s) in (2), the real part Hr (ω) and the imaginary part Hi (ω) yield

Hr (ω) =

+ · · · + (−1)

2sgn[Hr (ωi,Q −1 )]}

Hi (ω) =

+ and sgn[Hi (ωi,t )] denotes the sign of Hi (ω) soon after the occurrence of the zero sgn[Hi (ωi,t )]. Definition 6: Let 0 < ωr,1 < ωr,2 < · · · < ωr,P be the real and distinct zeros of Hr (ω) with odd multiplicities in [0, Ωr ) and ωi,P +1 be the smallest positive real zero of Hr (ω) with ω > Ωr . Then, the real signature of H(s) denoted as γr (H) is defined by

γr (H)  {2sgn[Hi (ωr,1 )] − 2sgn[Hi (ωr,2 )] + · · · + )], + (−1)P −1 2sgn[Hi (ωr,P )]} · (−1)P sgn[Hr (ωr,P + )] denotes the sign of Hr (ω) soon after the where sgn[Hr (ωr,t occurrence of the zero sgn[Hr (ωr,t )]. As a preliminary to the main result of this section, we need the following lemmas. Lemma 1: Let η be a constant such that the coefficients of terms of highest degree in Hr (ω) and Hi (ω) do not vanish at ω = η. Then, the necessary and sufficient condition under which Hr (ω) or Hi (ω) has only real roots in the interval [2lπ + η, 2(l + 1)π + η] for a sufficiently large integer l is that Hr (ω) or Hi (ω) has exactly 2N real roots in this interval. Proof: According to Theorem 2, Hr (s) or Hi (s) has exactly 4lN + M roots in the strip −2lπ + η ≤ (s) ≤ 2lπ + η for a sufficiently large integer l, where s = ω + yi. Under this fact, one can ascertain that Hr (s) or Hi (s) has only 4(l + 1)N + M roots in the strip −2(l + 1)π + η ≤ (s) ≤ 2(l + 1)π + η. Moreover, it is clear that Hr (ω) is an even function and Hi (ω) is an odd function, which means all the real roots of Hr (ω) and Hi (ω) are symmetrical about the origin. From the discussion

[bj r (ω) cos(τj ω) − bj i (ω) sin(τj ω)],

ν 

[bj i (ω) cos(τj ω) + bj r (ω) sin(τj ω)],

j =0

+ · (−1)Q −1 sgn[Hi (ωi,Q −1 )],

where sgn[·] is the standard signum function given by ⎧ −1 if x < 0, ⎪ ⎨ 0 if x = 0, sgn[x] = ⎪ ⎩ +1 if x > 0,

ν  j =0

γi (H)  {sgn[Hr (ωi,0 )] − 2sgn[Hr (ωi,1 )] + 2sgn[Hr (ωi,2 )] Q −1

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where bj r (ω) and bj i (ω) are the real and imaginary parts of bj (jω) with j = 0, 1, 2, . . . , ν, respectively. Now we consider the root distribution of Hr (ω) and Hi (ω) in Case (A) and Case (B). Case (A): We first consider Properties (i)–(iii). The zeros of Hr (ω) and Hi (ω) are respectively given by ν 

[bj r (ω) cos(τj ω) − bj i (ω) sin(τj ω)] = 0,

(12)

[bj i (ω) cos(τj ω) + bj r (ω) sin(τj ω)] = 0.

(13)

j =0 ν  j =0

For M even, divide the two sides of (12) and (13) by αν ω M , respectively. Then, for a sufficiently large number ω, we have (−1)M /2 cos(N ω) + O1 (ω) = 0, O1 (ω) −−−−→ 0,

(14)

(−1)M /2 sin(N ω) + O2 (ω) = 0, O2 (ω) −−−−→ 0.

(15)

ω →+∞

ω →+∞

From (14) and (15), it is sufficient that Hi (ω) has exactly 2N real zeros in the interval [Ωi + 2kπ, Ωi + 2(k + 1)π] and Hr (ω) has exactly 2N real zeros in the interval [Ωr + 2kπ, Ωr + 2(k + 1)π] for a sufficiently large integer l and any nonnegative integer k. By Lemma 1, Hr (ω) and Hi (ω) have only real zeros in the interval [Ωi + 2kπ, Ωi + 2(k + 1)π] and [Ωr + 2kπ, Ωr + 2(k + 1)π], respectively. Hence, Property (i) and Property (ii) hold. Moreover, Property (iii) is clear according to (14) and (15). Similarly, the results can be verified for M odd.

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Next, we consider Property (iv). The signum of the first order derivative of ∠H(jω) with respect to ω is given by 

  Hi (ω)  sgn {[∠H(jω)] } = sgn arctan Hr (ω)   Hi (ω)Hr (ω) − Hi (ω)Hr (ω) = sgn Hr2 (ω) 

 Hi (ω) . (16) = sgn Hr (ω)

Theorem 3: Let H(s) in (2) be a given quasi-polynomial with no roots on the imaginary axis. Then, H(s) possesses RH roots in the right-half complex plane if and only if  4lN + M + 1 − 2RH for M even, (19) γi (H) = for M odd, 4lN + M − 2RH

For M even, we have

ν  

Hi (ω) j =0 [bj i (ω) cos(τj ω) + bj r (ω) sin(τj ω)] = ν Hr (ω) j =0 [bj r (ω) cos(τj ω) − bj i (ω) sin(τj ω)]

for a sufficiently large integer l. Proof: We first consider the case given by (19). Let η = Ωi − 2lπ. Then, it follows from Theorem 3 that H(s) possesses RH roots in the right-half complex plane if and only if the condition (11) holds, where l → +∞. The term V (−2lπ + η, 2lπ + η) can be rewritten as

≈

N cos2 (N ω)

>0

or

 γr (H) =

4lN + M − 2RH

for M even,

4lN + M + 1 − 2RH

for M odd,

(20)

V (−2lπ + η, 2lπ + η) = −V (−2lπ, −2lπ + η)

with ω ≥ Ωi or ω ≥ Ωr . Similarly, the result can be verified for M odd. Thus, Property (iv) holds. Case (B): First, we consider Properties (i)–(iii). For M even, divide the two sides of (12) and (13) by αν ω M , respectively. For a sufficiently large number ω, we get that α0 (−1)M /2 cos(N ω) + (−1)M /2 + O3 (ω) = 0, αν O3 (ω) −−−−→ 0, ω →+∞

(17)

and (−1)M /2 sin(N ω) + O4 (ω) = 0, O4 (ω) −−−−→ 0. ω →+∞

(18)

It is seen from (17) and (18) that the roots of Hr (ω) tend to those of cos(N ω) = −α0 /αν and the roots of Hi (ω) tend to those of sin(N ω) = 0. It is sufficient that Hi (ω) has only 2N real zeros in the interval [Ωi + 2kπ, Ωi + 2(k + 1)π]. Because 0 < |α0 |/|αν | < 1 with α0 = 0 and |α0 | < |αν |, Hr (ω) also has only 2N real zeros in the interval [Ωr + 2kπ, Ωr + 2(k + 1)π]. According to Lemma 1, the roots of Hi (ω) in [Ωi + 2kπ, Ωi + 2(k + 1)π] and the roots of Hr (ω) in [Ωr + 2kπ, Ωr + 2(k + 1)π] are all real. Moreover, it is not difficult to see from (17) and (18) that the zeros of Hr (ω) and Hi (ω) interlace in the interval [Ωi , +∞) or [Ωr , +∞). Along the same lines as above, such results can be verified for M odd. Therefore, Properties (i)–(iii) hold. Next, we consider the signum of the first order derivative of ∠H(jω) with respect to ω. For M even, the expression (16) becomes   sgn [∠H(jω)]  ν   j =0 [bj i (ω) cos(τj ω) + bj r (ω) sin(τj ω)] ν = sgn j =0 [bj r (ω) cos(τj ω) − bj i (ω) sin(τj ω)]   [1 + αα ν0 sin(N ω)] = sgn N α 0 [ α ν + cos(N ω)]2 with ω ∈ [Ωi , +∞) or ω ∈ [Ωr , +∞). As 0 < |α0 |/|αν | < 1, [∠H(jω)] > 0 with ω ≥ Ωi or ω ≥ Ωr . Similarly, the result is true for M odd. Thus, Property (iv) holds. The analysis above completes the proof of this lemma.  Now, we present the main result of this section.

+ V (−2lπ, 0) + V (0, 2lπ) + V (2lπ, 2lπ + η).

(21)

For M even, a root of Hi (ω) tends to ω = 2lπ and a root of Hr (ω) tends to ω = 2lπ ± η, where η = π/(2N ). Moreover, all the roots of Hi (ω) or Hr (ω) are symmetrical about the origin since Hi (ω) is an odd function and Hr (ω) is an even function. Then, we can obtain the following expressions: π V (−2lπ, −2lπ + η) = + 1 , 1 −−−−→ 0, (22) l→+∞ 2 π (23) V (2lπ, 2lπ + η) = + 2 , 2 −−−−→ 0, l→+∞ 2 V (0, 2lπ) = V (−2lπ, 0) + 3 , 3 −−−−→ 0. l→+∞

(24)

According to (21)–(24), the condition (11) is equivalent to π V (0, Ωi ) = (4lN + M + 1 − 2RH ) + ε1 , ε1 −−−−→ 0. l→+∞ 2 (25) Next, we discuss this case from necessity and sufficiency. Necessity. Define that

 Hi (ω) ϑ(ω)  ∠H(jω) = arctan . Hr (ω) Let Δba ϑ, a, b ∈ R, denote the net change in the argument ϑ(ω) as ω increases from a to b. Similarly to the polynomial case in [10], we have the following conclusions: 1) If ωi,t , ωi,t+1 are both zeros of Hi (ω) in the interval [0, Ωi ), then ω

Δω ii ,, tt + 1 ϑ π + )]; = {sgn[Hr (ωi,t )] − sgn[Hr (ωi,t+1 )]} · sgn[Hi (ωi,t 2 (26) 2) If ωi,t is a zero of Hi (ω) in the interval [0, Ωi ) while ωi,t+1 is not, such a possible situation occurs only when ωi,t+1 is the smallest positive real zero of Hi (ω) with ω > Ωi , then π ω + Δω ii ,, tt + 1 ϑ = sgn[Hr (ωi,t )] · sgn[Hi (ωi,t )]; (27) 2 3) For t = 0, 1, 2, . . . , Q − 2, + + )] = −sgn[Hi (ωi,t )]. sgn[Hi (ωi,t+1

(28)

WANG et al.: NEW RESULTS ON EIGENVALUE DISTRIBUTION AND CONTROLLER DESIGN FOR TIME DELAY SYSTEMS

By using (28) repeatedly, we obtain + sgn[Hi (ωi,t )]

Q −1−t

= (−1)

·

+ sgn[Hi (ωi,Q −1 )]

(29)

for t = 0, 1, 2, . . . , Q − 1. From (26), (27), and (29), we have i ΔΩ 0 ϑ

=

Q −2 

ω

ω

Δω ii ,, tt + 1 ϑ + Δω ii ,, QQ −1 ϑ + ε2

t=0

=

Q −2  t=0

+ · (−1)Q −1−t · sgn[Hi (ωi,Q −1 )] +

· =

for H(s) in (2) being Hurwitz can be also obtained as follows. Corollary 1: Let H(s) in (2) be a given quasi-polynomial with no roots on the imaginary axis. Then, all the roots of H(s) are in the left-half complex plane if and only if  4lN + M + 1 for M even, γi (H) = 4lN + M for M odd, or

π {sgn[Hr (ωi,t )] − sgn[Hr (ωi,t+1 )]} 2

+ sgn[Hi (ωi,Q −1 )]

+ ε2

π γi (H) + ε2 , ε2 −−−−→ 0. l→+∞ 2

i Since ΔΩ 0 ϑ = V (0, Ωi ), we can get that π π (4lN + M + 1 − 2RH ) + ε1 = γi (H) + ε2 . 2 2 Then, we have 2 γi (H) − (4lN + M + 1 − 2RH ) = (ε1 − ε2 ). π It is clear that for a sufficiently large integer l, π |ε1 − ε2 | ≤ |ε1 | + |ε2 |  . 2 Moreover, γi (H) and 4lN + M + 1 − 2RH are both integers. Thus, we have

γi (H) = 4lN + M + 1 − 2RH .

 γr (H) =

π sgn[Hr (ωi,Q −1 )] 2

(30)

Sufficiency. If the condition (30) holds for a sufficiently large integer l, we have γi (H) + 4kN = 4lN + M + 1 − 2RH + 4kN, which means π π γi (H) + 2kN π = (4lN + M + 1 − 2RH ) + 2kN π. 2 2 By Lemma 2, it is sufficient that π i +2k π ΔΩ ϑ = γi (H) + 2kN π + ε3 0 2 i +2k π for ∀k ∈ Z+ , where ε3 → 0 as k → +∞. Since ΔΩ ϑ= 0 V (0, Ωi + 2kπ), we can obtain that π V (0, Ωi + 2kπ) = [4(l + k)N + M + 1 − 2RH ] + ε3 , 2 ε3 −−−−−−→ 0,

l+k →+∞

i.e., the condition (25) holds. Along the same lines as above, we can get the conclusion that a given quasi-polynomial H(s) with M being odd has RH roots in the right-half complex plane if and only if γi (H) = 4lN + M − 2RH for a sufficiently large integer l. Similarly, the conditions (20) for this theorem can be verified. The proof of this theorem is completed.  In view of Theorem 3, a necessary and sufficient condition

2891

4lN + M 4lN + M + 1

for M even, for M odd,

for a sufficiently large integer l. Remark 2: Note that τj for j = 1, 2, . . . , ν, which are the degrees of es , considered above are integer numbers. With τj being non-integers, one could find a positive real number τ by which each τj can be divisible and substitute s = χ/τ into H(s), where χ ∈ C. Then, the degrees of eχ in H(χ/τ ) are integers. It is clear that H(s) in s and H(χ/τ ) in χ have the same numbers of the right-half plane roots, the left-half plane roots, and imaginary roots, respectively. Thus, to ascertain the number of roots of H(s) in the right-half plane, one could also directly substitute s = jz/τ with z ∈ Z+ into H(s) and follow the presented results. Here, we give an example to explain the proposed result and Remark 2. Example 1: Consider a quasi-polynomial H(s) = (s + 1)e1.6s + e0.8s + 3.

(31)

Determine the number of the right-half plane roots of H(s). Let τ = 0.8. We can get χ H = (1.25χ + 1)e2χ + eχ + 3. (32) τ Substituting s = jz/τ into H(s) gives   jz H = Hr (z) + jHi (z), τ where Hi (z) = 1.25z cos(2z)+sin(2z)+sin(z) and Hr (z) = cos(2z) − 1.25z sin(2z) + cos(z) + 3. It is seen that N = 2 and M = 1 in H(χ/τ ) in χ. Let l = 2 which is a sufficiently large integer in this example. By Definition 4, we have Ωi = 4π. The real and distinct zeros of Hi (z) with odd multiplicities in [0, Ωi ) are zi,0 = 0, zi,1 = 1.3207, zi,2 = 2.4105, zi,3 = 3.9545, zi,4 = 5.6125, zi,5 = 7.1671, zi,6 = 8.6534, zi,7 = 10.2213, and zi,8 = 11.8372. Then we have γi (H) = {sgn[Hr (zi,0 )] − 2sgn[Hr (zi,1 )] + · · · + + )] = 13. By Theo(−1)8 2sgn[Hr (zi,8 )]} · (−1)8 sgn[Hi (zi,8 rem 3, RH = [4lN + M − γi (H)]/2 = 2. Therefore, H(s) or H(χ/τ ) has 2 roots in the right-half plane. The corresponding root distributions by using the bifurcation analysis package DDE-BIFTOOL in Matlab [22] are given by Fig. 2.

V. CONTROLLER DESIGN VIA DOMINANT EIGENVALUE ASSIGNMENT In this section, we present a new method on PID controller design for a high-order delay plant via dominant eigenvalue assignment by using the proposed result in Section IV.

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in the left-half λ-plane. Then, (kp , ki , kd ) values in C(s) can be calculated according to the relationship (34). B. Related Definitions For ease of presentation in the sequel, we introduce some related definitions. Multiplying Δ(λ, σ, k˜p , k˜i , k˜d ) by eθ λ , it yields H(λ, σ, k˜p , k˜i , k˜d )  Δ(λ, σ, k˜p , k˜i , k˜d )eθ λ ˜ ˜ (λ, σ). = D(λ, σ)eθ λ + (k˜d λ2 + k˜p λ + k˜i )N

(36)

It is clear that H(λ, σ, k˜p , k˜i , k˜d ) and Δ(λ, σ, k˜p , k˜i , k˜d ) have the same roots in λ since eθ λ vanishes nowhere in C. Let us define ˜ (−λ, σ). H(λ)  H(λ, σ, k˜p , k˜i , k˜d )N (37)

Fig. 2.

The eigenvalue distribution for example 1. (a) H (s), (b) H ( χτ ).

A. An Important Property Taking s = λ − σ into (4), where λ ∈ C is a variable and σ is a constant, yields ˜ σ) Δ(λ, σ, k˜p , k˜i , k˜d )  D(λ, ˜ (λ, σ)e−θ λ , + (k˜d λ2 + k˜p λ + k˜i )N where

⎡

⎤

⎡ k˜p 1 ⎢˜ ⎥ ⎣ ⎣ ki ⎦ = −σ k˜d

0

⎤⎡ ⎤ 0 −2σ kp 1 σ 2 ⎦ ⎣ ki ⎦ kd 0 1

(33)

(34)

and ˜ (λ, σ) = eθ σ N (λ − σ), D(λ, ˜ N σ) = (λ − σ)D(λ − σ). (35) Now, we present an important property for this study. Proposition 1: For a given set of (kp , ki , kd ) values, in the complex plane, the positions of the roots of (4) in s are equivalent to the positions that all the roots of (33) with respect to λ renovate a left shift σ in the horizontal direction. Proof: For a given set of (kp , ki , kd ) values, if s is a root of (4), then λ is clearly a root of (33). Also because s = λ − σ, we can get the result of Proposition 1.  Remark 3: To achieve the control objective of Section II, by Proposition 1, one can first obtain (k˜p , k˜i , k˜d ) values which are able to assign the rightmost roots of (33) exactly at the positions λ = ρ + σ in the right-half λ-plane and place all its other roots

By Remark 2, substituting λ = jz/θ into H(λ), we have  z H j = p˜(z, σ, k˜i , k˜d ) + j q˜(z, σ, k˜p ), θ where   z2 ˜ σ), (38) p˜(z, σ, k˜i , k˜d ) = p˜1 (z, σ) + k˜i − k˜d 2 R(z, θ z˜ q˜(z, σ, k˜p ) = q˜1 (z, σ) + k˜p R(z, σ). (39) θ Here,   ˜ r (z, σ)N ˜r (z, σ) + D ˜ i (z, σ)N ˜i (z, σ) cos(z) p˜1 (z, σ) = D   ˜ i (z, σ)N ˜r (z, σ) − D ˜ r (z, σ)N ˜i (z, σ) sin(z), (40) − D   ˜ r (z, σ)N ˜r (z, σ) + D ˜ i (z, σ)N ˜i (z, σ) sin(z) q˜1 (z, σ) = D   ˜ i (z, σ)N ˜r (z, σ) − D ˜ r (z, σ)N ˜i (z, σ) cos(z), (41) + D ˜ σ) = N ˜r2 (z, σ) + N ˜i2 (z, σ), R(z,

(42)

˜ i (z, σ), N ˜r (z, σ), and N ˜i (z, σ) are the real and ˜ r (z, σ), D and D ˜ ˜ imaginary parts of D(jz/θ, σ) and N (jz/θ, σ), respectively. ˜ From the property of polynomial, for a given σ, D(λ, σ) has exactly n + 1 zeros in complex plane, where one zero is λ = σ and the other zeros are determined by D(λ − σ) = 0, and ˜ (λ, σ) has exactly m zeros which are ascertained by N (λ − N σ) = 0. Let ςn +1 (σ) = σ, ςh (σ) with h = 1, 2, . . . , n be the roots of D(λ − σ), and ζt (σ) with t = 1, 2, . . . , m be the roots of N (λ − σ). Then, we have ˜ D(λ, σ) =

n +1

[λ − ςh (σ)],

(43)

h=1

˜ (−λ, σ) = (−1)m dm eθ σ N

m 

[λ + ζt (σ)].

(44)

t=1

Denote by LD˜ , LN˜ , RD˜ , and RN˜ the numbers of roots of ˜ ˜ (λ, σ) in the left-half plane and in the right-half D(λ, σ) and N plane, respectively. ˜ ˜ (λ, σ) have no roots on the Assumption 1: D(λ, σ) and N imaginary axis.

WANG et al.: NEW RESULTS ON EIGENVALUE DISTRIBUTION AND CONTROLLER DESIGN FOR TIME DELAY SYSTEMS

Note that σ is a chosen number rather than a given number. Thus, this assumption is reasonable. Define ςh = (ςh ) + j(ςh ) and ζt = (ζt ) + j(ζt ). Substituting λ = jz/θ into (43) and (44), by the geometry property of complex variable multiplication, we get that   ˜ j z , σ = ξ˜˜ (z) {cos[ϕ(z)] + j sin[ϕ(z)]} , D D θ  z  ˜ −j , σ = ξ˜ ˜ (z) {cos[ψ(z)] + j sin[ψ(z)]} , N N θ where

 n +1  z/θ − (ςh ) ϕ(z) = arctan , −(ςh ) h=1

ψ(z) =

m 

arctan

t=1

ξ˜D˜ (z) = (−1)R D˜



˜ 2 (z, σ) D r

(−1)L N˜ +m dm ξ˜N˜ (z) = |dm |

open right-half λ-plane and all its other roots are in the open left-half λ-plane if and only if H(λ) has exactly κ + LN˜ roots in the open right-half λ-plane and all its other roots are in the open left-half λ-plane. According to Theorem 3, H(λ) has only κ + LN˜ right-half λ-plane roots and all its other roots are in the open left-half λ-plane if and only if in the interval [0, Z),  4l + m + n − 2LN˜ + 2 − 2κ for m + n odd, γi (HZ ) = 4l + m + n − 2LN˜ + 1 − 2κ for m + n even, (46) where γi (HZ ) = {sgn[˜ p(z0 , σ, k˜i , k˜d )] − 2sgn[˜ p(z1 , σ, k˜i , k˜d )] + · · · + (−1)Q −1 2sgn[˜ p(zQ −1 , σ, k˜i , k˜d )]}



z/θ + (ζt ) , (ζt )



+

+ ˜ · (−1)Q −1 sgn[˜ q (zQ −1 , σ, kp )].

˜ (λ, σ) has no roots on the imaginary, the condition (46) Since N can be rewritten as  4l + n − (LN˜ − RN˜ ) + 2 − 2κ for m + n odd, γi (HZ ) = 4l + n − (LN˜ − RN˜ ) + 1 − 2κ for m + n even.

˜ 2 (z, σ), D i

˜ 2 (z, σ) + N ˜ 2 (z, σ). N r i

˜ = ξ˜˜ (z)/ξ˜ ˜ (z) For representing simply, we indicate that ξ(z) D N and φ(z) = z + ϕ(z) + ψ(z). Definition 7: For a sufficiently large integer l, define  2lπ + π2 for m + n odd, Z= 2lπ for m + n even. Let 0 = z0 < z1 < z2 < · · · < zQ −1 be the real and distinct roots of q˜(z, σ, k˜p ) with odd multiplicities in [0, Z). C. The Admissible Region of k˜p In this subsection, we discuss the admissible region of k˜p to set all the roots of (36) but λ = ρ + σ in the left-half λ-plane. ˜ Definition 8: If the equation d[ξ(z)/z]/dz = 0 has some positive real roots, denote the largest one by 1 , otherwise, let 1 = 0. Then, define l1 , Z1 , and Q1 as follows: (i) l1 is a nonnegative integer which satisfies π φ(1 ) < 2(l1 − ℘)π + , 2

A necessary condition for this to hold is that the number of the totally real and distinct zeros of q˜(z, σ, k˜p ) with odd multiplicities in the interval [0, Z) satisfies that ⎧ n − (LN˜ − RN˜ ) + 3 ⎪ ⎪ − κ for m + n odd, ⎨ 2l + 2 Q≥ ⎪ ⎪ ⎩ 2l + n − (LN˜ − RN˜ ) + 1 − κ for m + n even. 2 (47) When z increases from 0 to Z, φ(z) increases from 0 to 2lπ + [(LD˜ − RD˜ ) − (LN˜ − RN˜ ) + 1]π/2 for m + n odd and to 2lπ + [(LD˜ − RD˜ ) − (LN˜ − RN˜ )]π/2 for m + n even. Moreover, when z increases from 0 to Z1 , φ(z) increases from 0 to 2l1 π + π/2. Thus, we have φ(Z) − φ(Z1 ) = πK1 with

L

where ℘ =  2N˜  + 2 with · being a round up function; (ii) For a fixed l1 , define a positive real number Z1 ascertained by the following equation: π φ(Z1 ) = 2l1 π + ; 2 (iii) Let 0 = z0 < z1 < z2 < · · · < zQ 1 −1 be the real and distinct roots of q˜(z, σ, k˜p ) with odd multiplicities in [0, Z1 ). Theorem 4: A necessary condition for k˜p resulting in the quasi-polynomial (36) possessing only κ roots in the open righthalf λ-plane and all its other roots being in the open left-half λ-plane is to make Q1 ≥ 2l1 + RD˜ + 1 − κ.

2893

(45)

Proof: It is clear that all the roots of the quasi-polynomial H(λ) consist of those of H(λ, σ, k˜p , k˜i , k˜d ) and those of ˜ (−λ, σ). Then, H(λ, σ, k˜p , k˜i , k˜d ) has only κ roots in the N

K1 

(48)

⎧ (LD˜ − RD˜ ) − (LN˜ − RN˜ ) ⎪ ⎪ ⎪ 2(l − l1 ) + ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎨ for m + n odd, ⎪ (L ˜ − RD˜ ) − (LN˜ − RN˜ ) − 1 ⎪ ⎪ 2(l − l1 ) + D ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ for m + n even.

The roots of q˜(z, σ, k˜p ) in (39) can be determined by k˜p z = − sin[φ(z)]. ˜ θξ(z)

(49)

˜ According to Definition 8, we can ascertain that |ξ(z)|/z is a strictly increasing function with z ∈ [ρ1 , +∞). Also because Z1 > ρ1 and the expression (48) holds, the number of the totally real and distinct zeros of q˜(z, σ, k˜p ) with odd multiplicities in [Z1 , Z), i.e., Q − Q1 , satisfies Q − Q1 ≤ K1 .

(50)

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Now, we consider the relationship of the condition (47) and the condition (45). On the one hand, if the condition (47) holds, one can obtain from (47) and (50) that

ρ˜1 = a1 + σ, ρ˜2 = a2 + σ for ρ1 = a1 , ρ2 = a2 . Substituting λ = ρ˜ into H(λ, σ, k˜p , k˜i , k˜d ) = 0, we have

n − 1 LD˜ − RD˜ − +2−κ 2 2 = 2l1 + RD˜ + 1 − κ.

(51)

ˆ the number of the totally real and distinct zeros Denote by Q ˜ of q˜(z, σ, kp ) with odd multiplicities in the interval [0, 2(l1 − ℘)π + π/2). It can be found that ! ! LN˜ RD˜ + LN˜ ˆ Q ≤ 2l1 − 2 − 2 + . 2 2 Then, we have

!

!

RD˜ + LN˜ +3−κ 2 ! ! LN˜ RD˜ + LN˜ − + 3 − κ. = RD˜ + LN˜ − LN˜ + 2 2 2 # " # " R D˜ +L N˜ L , 2 2N˜ ≥ LN˜ , and κ < 3, we Since RD˜ + LN˜ ≥ 2 ˆ ≥ 1. This fact means that there is no less than have Q1 − Q one real root of q˜(z, σ, k˜p ) with odd multiplicities in the inter˜ val [2(l1 − ℘)π + π/2, Z1 ). Also because |ξ(z)|/z is a strictly increasing function with z ∈ [ρ1 , +∞) and ρ1 ≤ 2(l1 − ℘)π + π/2, it is sufficient that ˆ ≥ R ˜ + 2 LN˜ Q1 − Q D 2

−

Q − Q1 = K1 .

(52)

Combining (52) with (51), we can obtain the condition (47). From the discussion above, we can conclude that the condition (47) holds if and only if Q1 ≥ 2l1 + RD˜ + 1 − κ. The proof of this theorem is completed.  Remark 4: It is clear from Theorem 4 that the determination of Q1 with different values of k˜p is very critical in order to determine the admissible range of k˜p . One can see that z0 = 0 is one odd multiple root of q˜(z, σ, k˜p ) and the other roots can be ˜ given by the equation k˜p + θ ξ (z ) sin[φ(z )] = 0. Let c = k˜p and ˜

a ˜ 2 + b2 a ˜ 2 − b2 , ιi2 =  [G(˜ ρ1 )] −  [G(˜ ρ1 )] , 2˜ a 2˜ ab 1 1  [G(˜ ρ1 )] ; ιd1 = − , ιd2 = − 2˜ a 2˜ ab Case 2: for ρ˜1 = a1 + σ, ρ˜2 = a2 + σ, ιi1 = −

On the other hand, if the condition (45) holds, we have n − 1 LD˜ − RD˜ − + 2 − κ. 2 2

(53)

where ιi1 , ιi2 , ιd1 , and ιd2 are given as follows: Case 1: for ρ˜1,2 = a ˜ ± jb,

Q1 ≥ 2l1 +

Q1 ≥ 2l1 +

k˜i = ιi1 k˜p + ιi2 , k˜d = ιd1 k˜p + ιd2 ,

z

)] f (z) = − θ ξ (z ) sin[φ(z . Then, the number of the real, distinct, z and odd multiple zeros of q˜(z, σ, k˜p ) in (0, Z1 ), i.e., Q1 − 1, with different values of k˜p can be ascertained by using the algorithm provided in Appendix A. To mention, this algorithm plays a significant role in determining the controller gains by adopting a straightforward computation.

D. Two Dominant Eigenvalue Assignment In this subsection, we consider to determine the PID gains to assign a pair of eigenvalues of the closed-loop system to the desired positions s = ρ, where ρ1,2 = a ± jb or ρ1 = a1 , ρ2 = a2 , and furthermore, guarantee their dominance, i.e., set the other eigenvalues to the left of a given line s = −σ. ˜ ˜ (λ, σ). It is obDefine ρ˜ = ρ + σ and G(λ) = D(λ, σ)eθ λ /N vious that ρ˜1,2 = a ˜ ± jb for ρ = a ± jb, where a ˜ = a + σ, and

ιi1 = −

ρ˜1 ρ˜2 ρ˜2 ρ˜2 , ιi2 = 2 2 2 [G(˜ ρ1 )] − 2 1 2 [G(˜ ρ2 )] , ρ˜1 + ρ˜2 ρ˜1 − ρ˜2 ρ˜1 − ρ˜2

ιd1 = −

1 1 1 , ιd2 = 2 [G(˜ ρ2 )] − 2 [G(˜ ρ1 )] . ρ˜1 + ρ˜2 ρ˜1 − ρ˜22 ρ˜1 − ρ˜22

Remark 5: A pair of roots of (36) will be exactly at the positions λ = ρ˜ if and only if k˜p , k˜i , k˜d satisfy the condition (53). Remark 6: It is found from (53) that the degrees of freedom of controller gains are reduced if the dominant eigenvalues are determined. Thus, the parameters (k˜p , k˜i , k˜d ) can be ascertained by the effective set of k˜p . According to Theorem 3, for a given value of k˜p , we need calculate all the real and distinct zeros of q˜(z, σ, k˜p ) with odd multiplicities in [0, Z) to ascertain the number of the roots of H(λ) in the right-half plane. However, Z is a sufficiently large number, which is not practically usable and also brings us a great difficulty. The following result allows us to overcome this problem. Definition 9: For a given value of k˜p , define that  2 ξ˜2 (z) − k˜p2 zθ 2 X (z)  2 . ιi1 k˜p + ιi2 − (ιd1 k˜p + ιd2 ) zθ 2 If the equation |X (2 )| = 1 has some positive real roots, denote the largest one by 2 , otherwise, let 2 = 0. Then, define l2 , Z2 as follows: (i) l2 is a nonnegative integer which satisfies π φ(2 ) < 2(l2 − 1)π + ; 2 (ii) For a fixed l2 , define a positive real number Z2 determined by the following equation: π φ(Z2 ) = 2l2 π + . 2 ∗ Definition 10: Let l = max{l1 , l2 }, Z ∗ = max{Z1 , Z2 }, and 0 = z0 < z1 < z2 < · · · < zQ ∗ −1 be the real and distinct roots of q˜(z, σ, k˜p ) with odd multiplicities in [0, Z ∗ ). Lemma 3: When k˜p is a given value and satisfies the condition (45), any root zt of q˜(z, σ, k˜p ) in [Z ∗ , Z) leads to + ˜ p(zt , σ, k˜p )] · (−1)Q −1 sgn[˜ q (zQ (−1)t sgn[˜ −1 , σ, kp )] = 1, (54) where p˜(z, σ, k˜p ) denotes the function with substituting (53) into p˜(z, σ, k˜i , k˜d ).

WANG et al.: NEW RESULTS ON EIGENVALUE DISTRIBUTION AND CONTROLLER DESIGN FOR TIME DELAY SYSTEMS

2895

Proof: According to (40), (42), and the geometry property of complex variable multiplication, we have

Considering the expression (53), (63), and (64), for a fixed value of k˜p , the following result is not difficult to be obtained.

p˜1 (zt , σ) ˜ t ){cos(zt ) cos[ϕ(zt ) + ψ(zt )] = ξ(z ˜ t , σ) R(z

+ ˜ q (zQ sgn[˜ p(zQ −1 , σ, k˜p )] = sgn[˜ −1 , σ, kp )] = 0

− sin(zt ) sin[ϕ(zt ) + ψ(zt )]}.

(55)

(−1)Q −1 sgn[˜ p(zQ −1 , σ, k˜p )]

From q˜(zt , σ, k˜p ) = 0, we can obtain sin(zt ) = −

k˜ p z t θ ξ˜(z t )

+ cos(zt ) sin[ϕ(zt ) + ψ(zt )] cos[ϕ(zt ) + ψ(zt )]

+ ˜ = (−1)Q −1 sgn[˜ q (zQ −1 , σ, kp )] = 0.

.

(56)

Moreover, by the property of trigonometric function, we have cos(zt ) = cos[φ(zt ) − ϕ(zt ) − ψ(zt )] = cos[φ(zt )] cos[ϕ(zt ) + ψ(zt )] + sin[φ(zt )] sin[ϕ(zt ) + ψ(zt )]. From the expression (49), one can get that ⎧ k˜ z ⎪ ⎪ ⎪ sin[φ(zt )] = − p t , ⎪ ⎪ ˜ t) θξ(z ⎨  ⎪ ⎪ ξ˜2 (zt ) − ⎪ ⎪ ⎪ ⎩ cos[φ(zt )] = ± ˜ t) ξ(z

Therefore, we can get that

(65)

According to (65) and (62), the conclusion of this lemma holds. The proof of this lemma is completed.  Theorem 5: For a fixed value of k˜p satisfying the condition (45), the quasi-polynomial (36) possesses only two roots in the open right-half λ-plane and all its other roots are in the open left-half λ-plane if and only if

(57)

γi (HZ ∗ ) = 4l∗ + n − 2 − (LD˜ − RD˜ ),

(66)

where γi (HZ ∗ ) = {sgn[˜ p(z0 , σ, k˜p )] − 2sgn[˜ p(z1 , σ, k˜p )] + · · · k˜ p2 z t2 θ2

(58) .

Combining (55) with (56)–(58), we have $ k˜p2 zt2 p˜1 (zt , σ) = ± ξ˜2 (zt ) − 2 . ˜ t , σ) θ R(z

· (−1)Q

(59)

˜ t , σ) It is seen from (58) and (59) that the sign of p˜1 (zt , σ)/R(z depends on the value of φ(zt ). According to the proof of Theorem 4, for a fixed value of k˜p satisfying the condition (45), it is sufficient that   1 3 φ(zt ) ∈ (2l1 + t − Q1 + )π, (2l1 + t − Q1 + )π 2 2 (60) for a zero zt of q˜(z, σ, k˜p ) in [Z1 , Z). Then, for zt , zt+1 ∈ [Z1 , Z), we get cos[φ(zt )] = − cos[φ(zt+1 )], and increasingly,



 p˜1 (zt , σ) p˜1 (zt+1 , σ) sgn = −sgn = 0. (61) ˜ t , σ) ˜ t+1 , σ) R(z R(z From Definition 9, one can judge that sgn[˜ p(zt , σ, k˜p )] = sgn [˜ p1 (zt , σ)] = 0 ˜ σ) > 0, it is sufficient that for zt ∈ [Z2 , Z). And because R(z, sgn[˜ p(zt , σ, k˜p )] = −sgn[˜ p(zt+1 , σ, k˜p )] = 0

+ (−1)Q

(62)

for zt , zt+1 ∈ [Z ∗ , Z). Furthermore, p˜(z, σ, k˜i , k˜d ) in (38) and q˜(z, σ, k˜p ) in (39) can be also rewritten as  2 ˜ t ) cos[φ(zt )] + (k˜i − k˜d z ) , ˜ σ) ξ(z p˜(z, σ, k˜i , k˜d ) = R(z, θ2 (63) % & ˜ t ) sin[φ(zt )] + k˜p z . ˜ σ) ξ(z q˜(z, σ, k˜p ) = R(z, (64) θ

∗

∗

−1

−1

2sgn[˜ p(zQ ∗ −1 , σ, k˜p )]}

+ ˜ sgn[˜ q (zQ ∗ −1 , σ, kp )].

The two roots in the right-half plane are exactly at the positions λ = ρ˜1,2 , where ρ˜1,2 = a + σ ± jb or ρ˜1 = a1 + σ, ρ˜2 = a2 + σ. Proof: According to Theorem 3, for a fixed value of k˜p , the quasi-polynomial (36) has only two roots in the open right-half λ-plane and all its other roots are in the open left-half λ-plane if and only if  4l + n − 2 − (LN˜ − RN˜ ) for m + n odd, γi (HZ ) = 4l + n − 3 − (LN˜ − RN˜ ) for m + n even (67) with a sufficiently large integer l, where γi (HZ ) = {sgn[˜ p(z0 , σ, k˜p )] − 2sgn[˜ p(z1 , σ, k˜p )] + · · · + (−1)Q −1 2sgn[˜ p(zQ −1 , σ, k˜p )]} + ˜ · (−1)Q −1 sgn[˜ q (zQ −1 , σ, kp )].

Considering Z ∗ ≥ Z1 and the proof of Theorem 4, for a fixed value of k˜p satisfying the condition (45), we have the following two results: 1) the exact number of the real and distinct zeros of q˜(z, σ, k˜p ) with odd multiplicities in [Z ∗ , Z) is K∗ , where ⎧ (L ˜ − RD˜ ) − (LN˜ − RN˜ ) ⎪ ⎪ 2(l − l∗ ) + D ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎨ for m + n odd, K∗  (68) ⎪ (LD˜ − RD˜ ) − (LN˜ − RN˜ ) − 1 ⎪ ∗ ⎪ 2(l − l ) + ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ for m + n even; 2) there is no less than one root of q˜(z, σ, k˜p ) in [ρ1 , Z ∗ ).

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TABLE I THE PID GAINS FOR OBJECTIVE 1) OF EXAMPLE 2

According to Result 1) and Lemma 3, we can ascertain that + ˜ (−1)Q −1 sgn[˜ q (zQ −1 , σ, kp )]

Q −1 

(−1)t 2sgn[˜ p(zt , σ, k˜p )] = 2K∗ .

kp ki kd

(69)

˜ Moreover, since |ξ(z)|/z is a strictly increasing function with z ∈ [ρ1 , +∞), it is sufficient from Result 2) that  π π . (70) φ(zQ ∗ −1 ) ∈ 2l∗ π − , 2l∗ π + 2 2 + ˜ q (zQ (−1)Q −1 sgn[˜ −1 , σ, kp )] ∗

2

3

4

5

6

7.4721 4.7887 1.7027

7.4266 4.7676 1.6324

7.3810 4.7464 1.5622

7.3354 4.7253 1.4919

7.2899 4.7042 1.4217

7.2443 4.6830 1.3514

H(λ, σ, k˜p , k˜i , k˜d ) = 0, the values of k˜p , k˜i , and k˜d satisfy ρ). k˜i = −k˜p ρ˜ − k˜d ρ˜2 − G(˜

Considering (70), (60), and Z ∗ ≥ Z1 , we can get that

= (−1)Q

1

t=Q ∗

−1

+ ˜ sgn[˜ q (zQ ∗ −1 , σ, kp )].

(71)

Then, (67) is equivalent to (66) by applying (69) and (71). On the other hand, the function p˜(z, σ, k˜p ) is obtained by substituting (53) into p˜(z, σ, k˜i , k˜d ). It is obvious that two zeros of the quasi-polynomial (36) are exactly at the positions λ = ρ˜1,2 if and only if the values of k˜p , k˜i , k˜d satisfy the expression (53). Thus, the two roots in the open right-half λ-plane are exactly at the positions λ = ρ˜1,2 . This completes the proof of this theorem.  Now we give the result on the controller gains (kp , ki , kd ). Theorem 6: For PID control of the given delay plant (3), the rightmost eigenvalues of the closed-loop system are assigned to the desired positions s = ρ1,2 , where ρ1,2 = a ± jb or ρ1 = a1 , ρ2 = a2 , and all the other eigenvalues are to the left of the line s = −σ if and only if kp , ki , and kd satisfy ⎡ ⎤ ⎡ ⎤⎡˜ ⎤ kp 1 0 2σ kp ⎥ ⎣ ki ⎦ = ⎣ σ 1 σ 2 ⎦ ⎢ (72) ⎣ k˜i ⎦ , ˜ kd 0 0 1 kd where k˜p satisfies the condition (45) and makes (66) hold, while the values of k˜p and k˜i are determined by (53). Proof: According to Theorem 5, for a fixed value of k˜p satisfying the condition (45), the quasi-polynomial (36) possesses only two roots λ = ρ˜1,2 , where ρ˜1,2 = a + σ ± jb or ρ˜1 = a1 + σ, ρ˜2 = a2 + σ, in the open right-half λ-plane and all its other roots are in the open left-half λ-plane if and only if (66) holds. Moreover, the corresponding values of k˜i and k˜d are given by (53). Then, by Proposition 1 and the expression (34), the rightmost closed-loop eigenvalues are assigned to the desired positions s = ρ1,2 , where ρ1,2 = a ± jb or ρ1 = a1 , ρ2 = a2 , and all the other eigenvalues are to the left of the line s = −σ if and only if kp , ki , and kd satisfy (72). This completes the proof of this theorem.  E. One Dominant Eigenvalue Assignment In this subsection we mainly consider to assign the dominant closed-loop eigenvalue to the desired position s = ρ, where ρ = a, and also set the other eigenvalues to the left of a give line s = −σ by using PID controllers. Remark 7: When a root of (36) is exactly at the position λ = ρ˜, where ρ˜ = ρ + σ, by substituting λ = ρ˜ into

Definition 11: Define that  2 ρ) ξ˜2 (z) − k˜p2 zθ 2 − k˜p ρ˜ − G(˜ Y1 (z)  , 2 ρ˜2 + zθ 2  2 ρ) − ξ˜2 (z) − k˜p2 zθ 2 − k˜p ρ˜ − G(˜ . Y2 (z)  2 ρ˜2 + zθ 2

(73)

(74)

(75)

− Define + 3 and 3 which satisfy the following conditions: 1) + − − 3 , 3 ≥ Z1 ; 2) + 3 , 3 are no less than the largest real roots of dY1 (z)/dz = 0 and dY2 (z)/dz = 0, respectively. − + Let 3 = max{+ 3 , 3 }. Define l3 ∈ Z which satisfies

φ(3 ) < 2(l3 − β)π + where

π , 2

⎡ " R +L # ⎤ 2 D˜ 2 D˜ + 2 − RD˜ ⎥ + 1. β=⎢ ⎢ ⎥ 2 ⎢ ⎥

Then, we define Z3 ∈ R+ determined by the equation φ(Z3 ) = 2l3 π +

π . 2

Let 0 = z0 < z1 < z2 < · · · < zQ 3 −1 be the totally real and distinct roots of q˜(z, σ, k˜p ) with odd multiplicities in [0, Z3 ). Lemma 4: Substitute (73) into p˜(z, σ, k˜i , k˜d ) in (38) and denote the resulting function by p˜(z, σ, k˜p , k˜d ). Then, for a fixed value of k˜p , the necessary condition for k˜d leading to the existence of PID gains for one dominant eigenvalue assignment is that it makes P3 ≥ 2l3 + RD˜ − 1,

(76)

where P3 is the number of the totally real and distinct zeros of p˜(z, σ, k˜p , k˜d ) with odd multiplicities in the interval (0, Z3 ). Proof: Substituting (73) into p˜(z, σ, k˜i , k˜d ) in (38) yields

 z2 ˜ 2 ˜ ˜ ˜ ˜ ˜ p˜(z, σ, kp , kd ) = −kp ρ˜ − G(˜ ρ) − kd ρ˜ − kd 2 R(z, σ) θ + p˜1 (z, σ). By Theorem 3, for a given set (k˜p , k˜d ), the quasi-polynomial (36) has only one root in the open right-half plane and all its

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Fig. 3. The eigenvalue distribution and step response plot with k p = 7.2899, k i = 4.7042, k d = 1.4217 for the plant (81). (a) Eigenvalue distribution, (b) step response plot. TABLE II THE PID GAINS FOR OBJECTIVE 2) OF EXAMPLE 2

kp ki kd

1

2

3

4

5

6

4.1652 2.5110 1.1393

1.5470 2.1065 -1.8385

2.3119 2.2340 -1.2012

3.0767 2.3615 -0.5638

3.8415 2.4889 0.0736

4.6064 2.6164 0.7109

other roots are in the open left-half plane if and only if  4l + n − (LN˜ − RN˜ ) for m + n odd, γi (HZ ) = 4l + n − 1 − (LN˜ − RN˜ ) for m + n even, (77) for a sufficiently large integer l, where γi (HZ ) = {sgn[˜ p(z0 , σ, k˜p , k˜d )] − 2sgn[˜ p(z1 , σ, k˜p , k˜d )] + · · · + (−1)Q −1 2sgn[˜ p(zQ −1 , σ, k˜p , k˜d )]} + ˜ · (−1)Q −1 sgn[˜ q (zQ −1 , σ, kp )].

Moreover, it is obvious from Definition 11 that (74) is a strictly increasing function and (75) is a strictly decreasing function with z ≥ ρ3 . According to the discussion above, there are two necessary conditions on k˜d for the condition (77) to hold as follows: 1) the values of k˜d result in sgn[˜ p(zt , σ, k˜p , k˜d )]=−sgn[˜ p(zt+1 , σ, k˜p , k˜d )] with zt , zt+1 ∈ [Z3 , Z); 2) the values of k˜d lead to P3 ≥ 2l3 + RD˜ − 1. In fact, the condition 1) is equivalent to k˜d ∈ [Y1 (Z3 ), Y2 (Z3 )]. Further, the range of k˜d for the condition 2) to be true is in [Y1 (Z3 ), Y2 (Z3 )]. Then, we have the result of this lemma.  Remark 8: It is found that the roots of p˜(z, σ, k˜p , k˜d ) are the same as those of the function k˜d − Y(z). Let c = k˜d and f (z) = Y(z). Then, the values of P3 with different k˜d can be determined by using the algorithm provided in Appendix A. Theorem 7: For a fixed value of k˜p which satisfies the condition (45) and a fixed value of k˜d which satisfies the condition (76), the quasi-polynomial (36) possesses only one root in the open right-half λ-plane and all its other roots are in the open left-half λ-plane if and only if

It is found that

  sgn[˜ p(z, σ, k˜p , k˜d )] = sgn Y(z) − k˜d ,

γi (HZ 3 ) = 4l3 + n − (LD˜ − RD˜ )

(78)

with

where Y(z) 

p˜1 (z ,σ ) ˜ (z ,σ ) R

− k˜p ρ˜ − G(˜ ρ) ρ˜2 +

z2 θ2

γi (HZ 3 ) = {sgn[˜ p(z0 , σ, k˜p , k˜d )] − 2sgn[˜ p(z1 , σ, k˜p , k˜d )] .

+ · · · + (−1)Q 3 −1 2sgn[˜ p(zQ 3 −1 , σ, k˜p , k˜d )]} + · (−1)Q 3 −1 sgn[˜ q (zQ , σ, k˜p )]. 3 −1

From (59), we have

 z2 ± ξ˜2 (zt ) − k˜p2 θ t2 − k˜p ρ˜ − G(˜ ρ)

Y(zt ) =

ρ˜2

+

z t2 θ2

.

It is clear that (61) holds when z ≥ Z1 . Thus, if  z2 ± ξ˜2 (zt ) − k˜p2 θ t2 − k˜p ρ˜ − G(˜ ρ) Y(zt ) = z t2 2 ρ˜ + θ 2 with zt ≥ Z1 , then Y(zt+1 ) =

 z2 ∓ ξ˜2 (zt+1 ) − k˜p2 tθ+2 1 − k˜p ρ˜ − G(˜ ρ) ρ˜2 +

z t2+ 1 θ2

.

The root in the open right-half λ-plane is exactly at the position λ = ρ˜, where ρ˜ = a + σ. Proof: Following the same lines as the proof of Theorem 5, one can obtain the conclusion of this theorem.  Theorem 8: For PID control of a given delay plant (3), the rightmost eigenvalue of the closed-loop system can be assigned to the desired position s = ρ, where ρ = a, and all the other eigenvalues are to the left of the line s = −σ if and only if the values of kp , ki , and kd are given by (72), where the set (k˜p , k˜d ) satisfies the condition (45) and (76) and results in (78), while the value of k˜i is calculated by (73). Proof: The proof is similar to that of Theorem 6. 

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Fig. 4. The eigenvalue distribution and step response plot with k p = 4.6064, k i = 2.6164, k d = 0.7109 for the plant (81). (a) Eigenvalue distribution, (b) step response plot. TABLE III THE PID GAINS FOR OBJECTIVE 1) OF EXAMPLE 3

kp ki kd

1

2

3

4

5

6

0.4075 0.2946 0.2380

0.3883 0.2879 0.2090

0.3690 0.2812 0.1801

0.3498 0.2745 0.1512

0.3306 0.2678 0.1223

0.3113 0.2611 0.0934

Step 12: Step 13: Step 14:

F. Implementation In view of Subsection V-B∼Subsection V-E, we provide an algorithm to determine the values of (kp , ki , kd ) for dominant eigenvalue assignment. Here, Λ is defined as the set of (kp , ki , kd ) values that meet the design requirement. Step 1: Substitute s = λ − σ into the given N (s) and D(s) ˜ (λ, σ) and D(λ, ˜ in (3) and calculate N σ) by (35). Then, compute the corresponding p˜(z, σ, k˜i , k˜d ) and q˜(z, σ, k˜p ) by (38) and (39), respectively. Step 2: Determine the value of Z1 from Definition 8. According to Theorem 4, ascertain the admissible region of k˜p denoted by [k˜pl , k˜pu ]. Grid this interval of k˜p by N , where N is a desired number of points, and set 1 = N 1+1 (k˜pu − k˜pl ). Step 3: According to the dominant pole positions s = ρ, calculate ρ˜ = ρ + σ. Determine the expressions (53) for κ = 2 or (73) for κ = 1. Then, substitute (53) or (73) into p˜(z, σ, k˜i , k˜d ) in (38) and get p˜(z, σ, k˜p ) or p˜(z, σ, k˜p , k˜d ). Step 4: Initialize k˜p = k˜pl . Step 5: If κ = 2 go to Step 6; if κ = 1 go to Step 11. Step 6: Determine the value of Z ∗ from Definitions 9 and 10. Step 7: Solve for the real and distinct zeros of q˜(z, σ, k˜p ) in (39) with odd multiplicities in the interval [0, Z ∗ ). Step 8: Calculate sgn[˜ p(zt , σ, k˜p )] for all t = 0, 1, . . . , Q∗ − 1 + and sgn[˜ q (zQ ∗ −1 , σ, k˜p )]. Then, compute γi (HZ ∗ ). Step 9: If γi (HZ ∗ ) = 4l∗ + n − 2 − (LD˜ − RD˜ ), select this value of k˜p . Step 10: Ascertain the values of k˜i and k˜d by (53). Determine the values of (kp , ki , kd ) by (72) and put them into the set Λ. Go to Step 18. Step 11: Determine the value of Z3 from Definition 11. Ascertain the allowable range of k˜d by Lemma 4 and

Step 15: Step 16:

Step 17: Step 18:

denote it by [k˜dl , k˜du ]. Grid this interval of k˜d by M, where M is also a desired number of points, and set 1 (k˜du − k˜dl ). 2 = M+1 ˜ Initialize kd = k˜dl . Solve for all the real and distinct zeros of q˜(z, σ, k˜p ) in (39) with odd multiplicities in the interval [0, Z3 ). for all t= Compute sgn[˜ p(zt , σ, k˜p , k˜d )] 0, 1, . . . , Q3 − 1 and sgn[˜ q (zQ + −1 , σ, k˜p )]. Then, 3 calculate γi (HZ 3 ). If γi (HZ 3 ) = 4l3 + n − (LD˜ − RD˜ ), select these values of k˜p and k˜d . Ascertain the value of k˜i by (73). Then, determine the values of (kp , ki , kd ) by (72) and put them into the set Λ. Go to Step 13 with k˜d = k˜d + 2 until k˜d > k˜du . Go to Step 5 with k˜p = k˜p + 1 . If k˜p > k˜pu , terminate the algorithm.

G. Degradation to PI Control The presented results can be also applied in PI control of high-order delay systems via dominant eigenvalue assignment. In PI control, the transfer function of the controller is C(s) = kp + ki /s and the dominant eigenvalue number κ is considered as 1. Below we directly give a theorem for PI control of a given delay plant (3) with the purpose of dominant eigenvalue assignment. The proof of this theorem and the algorithm for determining the gains (kp , ki ) are omitted here since they can be easily completed following the analysis above. It is worth noting that the result plays a significant role in application as PI controllers are widely used in industry. Theorem 9: For PI control of a given delay plant (3), the rightmost eigenvalue of the closed-loop system is assigned to the desired position s = ρ, where ρ = a, and all the other eigenvalues are to the left of the line s = −σ if and only if the values of kp and ki are given by  



1 0 k˜p kp = , (79) σ 1 ki k˜i ρ) and k˜p results in where k˜i = −k˜p ρ˜ − G(˜ γi (HZ 1 ) = 4l1 + n − (LD˜ − RD˜ )

(80)

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Fig. 5. The eigenvalue distribution and step response plot with k p = 0.3690, k i = 0.2812, k d = 0.1801 for the plant (82). (a) Eigenvalue distribution, (b) step response plot. TABLE IV THE PID GAINS FOR OBJECTIVE 2) OF EXAMPLE 3

kp ki kd

1

2

3

4

5

6

0.3476 0.1630 0.2094

0.3951 0.1713 0.2390

0.4425 0.1796 0.2687

0.4900 0.1880 0.2984

0.5375 0.1963 0.3280

0.5849 0.2046 0.3577

with p(z0 , σ, k˜p )] − 2sgn[¯ p(z1 , σ, k˜p )] + · · · γi (HZ 1 ) = {sgn[¯ + (−1)Q 1 −1 2sgn[¯ p(zQ 1 −1 , σ, k˜p )]} + · (−1)Q 1 −1 sgn[˜ q (zQ , σ, k˜p )] 1 −1

˜ σ). Here, in which p¯(z, σ, k˜p ) = p˜1 (z, σ) − [k˜p ρ˜ + G(˜ ρ)]R(z, the necessary condition for k˜p leading to the existence of (80) is given by (45). VI. NUMERICAL EXAMPLES In this section, we give two examples to illustrate the effectiveness of the presented methods, where the characteristic equation of the first example is retarded type and the characteristic equation of the second example is neutral type. Example 2: Consider a plant with time delay s2 − s + 1 e−0.3s . (81) + + 28s3 + 44s2 + 46s + 19 Choose a PID controller to achieve the following control objectives for the unit feedback closed-loop system: 1) assign the dominant poles at ρ1,2 = a ± jb, where a = −0.3242 and b = 0.4424; 2) assign the dominant pole at ρ = a, where a = −0.2. To ensure the dominance of these poles, we assign all the other poles to the left of the line s = −σ = 3a. 1) According to Steps 1–2 in the proposed algorithm in Subsection V-F, we can ascertain that Z1 = 20.4806 and the admissible region of k˜p is given by (2.9756, 16.7333). Here, we set k˜pl = 2.9756, k˜pu = 16.7333, and N = 150. By Step 3, we have k˜i = −0.4751k˜p + 1.1085, k˜d = −0.7711k˜p + 4.9106, and the expression p˜(z, σ, k˜p ) with σ = 0.9726. By Remark 2, the delay degree becomes 1. Then, from Steps 4–10 and Step 18, we have the set Λ shown in Table I. By Section 5–6 in [37], the poles ρ1,2 = −0.3242 ± j0.4424 mean that the overshoot and the settling time ts (2%) of a G(s) =

s5

7s4

standard second-order closed-loop system are 10% and 13s. As a pair of dominant poles, they can make the closed-loop system response of this example close to these indexes. Now, we choose kp = 7.2899, ki = 4.7042, kd = 1.4217 in Λ. The corresponding eigenvalue distribution and step response plot are depicted in Fig. 3. The eigenvalue distribution in this example is obtained by applying the bifurcation analysis package DDE-BIFTOOL in Matlab [22]. 2) According to Steps 1–2 of the algorithm in Subsection V-F and following the results in 1), we can ascertain that the admissible region of k˜p is (−1.0224, 4.7084). Here, we set k˜pl = −1.0224, k˜pu = 4.7084, and N = 5. By Step 3, we have k˜i = −0.4k˜p − 0.16k˜d + 1.7236 and the expression p˜(z, σ, k˜p , k˜d ) with σ = 0.6. Then, we set M = 9. By Steps 4– 5 and Steps 11–18, the set Λ given by Table II can be obtained. From [37], the pole ρ = −0.2 indicates that the settling time ts (2%) of a standard first-order closed-loop system is 20s and no overshoot exists. As a dominant pole, it can make the closedloop system response of this example close to these indexes. Now, we choose kp = 4.6064, ki = 2.6164, kd = 0.7109 in Λ, which are all positive. The corresponding eigenvalue distribution and step response plot are shown in Fig. 4. Example 3: Consider a plant with time delay G(s) =

−0.4s + 1 e−s . (s + 1)(0.8s + 1)

(82)

Design a PID controller by the following control objectives for the unit feedback closed-loop system: 1) assign the dominant poles at ρ1,2 = a ± jb, where a = −0.3326 and b = 0.3488; 2) assign the dominant pole at ρ = a, where a = −0.2. To ensure the dominance of these poles, we assign all the other poles to the left of the line s = −σ = 4a. 1) Here, let N = 50. Following the same lines as 1) in Example 2 and applying the proposed algorithm give the set Λ in Table III. After the dominant pole assignment, the closed-loop system response can be close to the indexes as follows: the overshoot is 5% and the settling time ts (2%) is 13s. Now, we choose kp = 0.3690, ki = 0.2812, kd = 0.1801 in Λ. Fig. 5 depicts the corresponding eigenvalue distribution and step response plot. The eigenvalue distribution in this example is calculated by using the Pad´e approximation in Matlab. 2) Here, choose N = 11 and M = 35. Following the same lines as 2) in Example 2 and applying the presented algorithm, we obtain the set Λ depicted in Table IV.

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Fig. 6. The eigenvalue distribution and step response plot with k p = 0.3951, k i = 0.1713, k d = 0.2390 for the plant (82). (a) Eigenvalue distribution, (b) step response plot.

The dominant pole assignment can make the closed-loop system response close to the indexes as follows: the settling time ts (2%) is 20s and no overshoot exists. Now, we choose kp = 0.3951, ki = 0.1713, kd = 0.2390 in Λ. Fig. 6 shows the corresponding eigenvalue distribution and step response plot. Now, we give some discussions on these examples. Discussion 1: We want the dominant poles to be located in some desired positions and our algorithm can achieve that. Discussion 2: The two examples above illustrate that the proposed methods of this paper enable us to directly calculate the controller gains according to the desired performance indexes for a standard first-order system or a standard second-order system in addition to stability. Discussion 3: The effective region of controller gains will be narrowed down with increasing the value of σ. There always exists a bound such that one cannot obtain effective PID gains if σ is larger than that.

VII. CONCLUSION This paper produces a new result on the root distribution of a class of quasi-polynomials by applying part of Pontryagin’s conclusions. Such a result can be applied to determining the number of the right-half plane eigenvalues of an LTI time delay system. It is also a generalization of the extension of the Hermite-Biehler theorem applicable to the case of quasi-polynomials that are not necessarily Hurwitz. According to the presented result, this paper provides some dominant eigenvalue assignment approaches on PID (including PI) control of high-order delay systems. The procedure is based on first transforming the characteristic equation into another form and defining some new parameters. Then, the sets of these new parameters are ascertained by using the proposed result. Finally, the controller gains are determined by such new parameters. The method is also effective when the characteristic equation of the closed-loop system is neutral type besides retarded type. It has been demonstrated that the controller gains to achieve the control objective can be characterized in a straightforwardly computational way. In addition, we expect to apply the proposed method to higher order controller or state feedback controller design for delay systems. Then, as controller parameters increase in these cases, the numbers of assigned poles and the degrees of freedom of the controller parameters to be considered are both different, which makes the task more complicated. Besides, it is more important to further study the eigenvalue distribution of delay systems and

extend the presented dominant eigenvalue assignment approach to systems with non-commensurate delays in our future work. APPENDIX A In this appendix, we provide a method to determine the number of the odd multiple roots of F[c, f (x)] = c − f (x) for a given value of c with x ∈ (0, X), where f (x) has no poles. Step 1: Solving the equation df (x)/dx = 0 gives all the real, distinct, and odd multiple roots which are denoted by x1 , x2 , . . . , xg −1 in (0, X). Let 0 = x0 < x1 < x2 < · · · < xg −1 < xg = X. Define Ti = f (xi ), i = 0, 1, 2, . . . , g. Step 2: If f  (0) > 0, let Lj  Ti , j = 1, 2, . . . , g + 1, which satisfy L1 ≥ L2 ≥ · · · ≥ Lg +1 ; if f  (0) < 0, let Lj satisfy L1 ≤ L2 ≤ · · · ≤ Lg +1 . Step 3: Define ⎧ 0.5, if i = g is odd, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −0.5, if i = 0 or i = g is even, nL j = ⎪ 1, if 1 ≤ i ≤ g − 1 and i is odd, ⎪ ⎪ ⎪ ⎪ ⎩ −1, if 1 ≤ i ≤ g − 1 and i is even. Step 4: Let −∞ = υ0 < υ1 < · · · < υp−1 < υp = +∞ be constants such that for any j = 1, 2, . . . , g + 1, ∃Lj = − 1. Then, for k = 1, 2, . . . , p − υk with 1 ≤ k ≤ p nL j and Eυ k = nL j , 1, define Nυ k = L j =υ k

L j =υ k

where · is a round up function. In addition, define Eυ 0 = Nυ 0 = 0. Step 5: Denote by U (υk , υk +1 ) and U (υk ) the number of the real and distinct roots of the function F[c, f (x)] with odd multiplicities in (0, X) when min{υk , υk +1 } < c < max{υk , υk +1 } and c = υk , respectively. Thus, for k = 0, 1, 2, . . . , p − 1, we have U (υk , υk +1 ) = 2

k 

Nυ h .

h=0

Then, we can obtain U (υk ) = U (υk , υk +1 ) − Eυ k .

WANG et al.: NEW RESULTS ON EIGENVALUE DISTRIBUTION AND CONTROLLER DESIGN FOR TIME DELAY SYSTEMS

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Honghai Wang was born in Tieling City, Liaoning Province, China, in 1983. He received the B.S. degree in automation, the M.S. degree, and the Ph.D. degree in control theory and control engineering from Northeastern University, Shenyang, China, in 2006, 2008, and 2015, respectively. He is currently a Lecturer at the Collage of Information Science and Engineering, Northeastern University. His current research interest includes eigenvalue distribution of time delay systems, eigenvalue assignment and stabilization for PID control, robust control, and computer-controlled systems.

Jianchang Liu was born in Heishan City, Liaoning Province, China, in 1960. He received the B.S., M.S., and Ph.D. degrees from Northeastern University, Shenyang, China, in 1980, 1989, and 1998, respectively. He is currently a Professor at the Collage of Information Science and Engineering, Northeastern University. His research interest covers modeling, control and optimization for complex process, time delay systems, fault diagnosis, intelligent control theory and application.

Yu Zhang was born in Yingkou City, Liaoning Province, China, in 1983. She received the B.S. degree in automation and the M.S. degree in control theory and control engineering from Northeastern University, Shenyang, China, in 2005 and 2009, respectively. She is an Engineer at the Collage of Information Science and Engineering, Northeastern University. Her research covers process control, time delay systems, PID control, and computer control systems.