Delay Margin of Low-Order Systems Achievable by PID ... - IEEE Xplore

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2018.2853567, IEEE Transactions on Automatic Control IEEE TRANSACTIONS ON AUTOMATIC CONTROL

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Delay Margin of Low-Order Systems Achievable by PID Controllers Dan Ma, Member, IEEE and Jie Chen, Fellow, IEEE

Abstract—This paper concerns the delay margin achievable using PID controllers for linear time-invariant (LTI) systems subject to variable, unknown time delays. The basic issue under investigation addresses the question: What is the largest range of time delay so that there exists a single PID controller to stabilize the delay plants within the entire range? Delay margin is a fundamental measure of robust stabilization against uncertain time delays and poses a fundamental, longstanding problem that remains open except in simple, isolated cases. In this paper we develop explicit expressions of the exact delay margin and its upper bounds achievable by a PID controller for low-order delay systems, notably the first- and second-order unstable systems with unknown constant and possibly time-varying delays. The effect of nonminimum phase zeros is also examined. PID controllers have been extensively used to control and regulate industrial processes which are typically modeled by first- and second-order dynamics. While furnishing the fundamental limits of delay within which a PID controller may robustly stabilize a delay process, our results should also provide useful guidelines in tuning PID parameters and in the analytical design of PID controllers. Index Terms—Delay margin, robust stabilization, uncertain time delay, time-varying delay, PID controller.

I. I NTRODUCTION IME delays arise in the transport of energy, mass, information and such, and are omnipresent in natural and engineered systems. Large-scale distributed and networked systems, such as sensor networks, multi-agent systems, cyberphysical systems, biological and industrial processes, can be particularly affected by long, variable delays. A chronical challenge is thus to cope with the negative impact of time delays, which are likely to degrade a system’s performance and robustness, and in not so rare instances even render a system unstable. For this purpose, it is essential to design feedback control systems to robustly stabilize delay plants, and more broadly, to mitigate the negative effect of the delays on the system’s performance and robustness. There has been a vast volume of published work on the stability and stabilization of time-delay systems; we refer to [25], [7], [18], [5] and [13], [15], [35], [36] for a sample of comprehensive survey articles and research monographes devoted to this subject. Delay margin is a fundamental measure of robust stabilization against uncertain time delays and it addresses a central issue in the study of feedback stabilization of timedelay systems: What is the largest range of delay so that

T

This research was supported in part by the Hong Kong RGC under Project CityU 11201514, CityU 111613, and in part by NSFC under Grants 61603079 and 61773098. Dan Ma is with the College of Information Science and Engineering, State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, China. [email protected]. Jie Chen is with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China. [email protected].

there exists a single controller that can stabilize the delay plant within that entire range? Much of the past work on the delay margin problem can be seen as a tradeoff between the complexity of control laws and the delay margin achievable. When allowed more sophisticated control laws, such as linear periodic controllers [17], nonlinear periodic controllers [6], and nonlinear adaptive controllers [4], [12], the delay margin can be made infinite; in other words, a LTI delay plant can be stabilized by these complex, non-LTI controllers for arbitrarily long uncertain delays. On the other hand, when confined to LTI controllers, the delay margin can only be finite [14] (see also, e.g., [16]), and the achievable margins vary. More specifically, for general LTI systems, upper bounds on the delay margin were obtained in [8], [16], [24], which establish limits beyond which no finite-dimensional LTI output feedback controllers may exist to robustly stabilize the delay plant family within the margin. The bounds, however, are tight generally for systems with no more than one unstable pole and nonminimum phase zero, for which they were found to be exact; otherwise, for systems containing more unstable poles, the bounds tend to be crude and pessimistic. In contrast, lower bounds were derived in [23] for systems with an arbitrary number of unstable poles and nonminimum phase zeros, wherein it was also shown that a LTI controller can be synthesized using H∞ optimal control techniques, which guarantees closed-loop robust stabilization when the delay value falls into the range. Other bounds were also made available under more specialized circumstances by employing finite- or infinite-dimensional LTI controllers based on a predictor feedback mechanism (see, e.g., [14], [31], [33], [37], [38]), and a numerical method was proposed in [14] with the goal to increase the delay margin in an iterative manner. In this paper we study the delay margin achievable by LTI controllers of further restricted structure and complexity, i.e., those of PID type [1], [32]. A time-honored method seemingly of infinite staying power, PID control is favored for its ease of implementation and undoubtedly, has been the most popular means in controlling industrial processes with its unparallel simplicity and unsurpassed effectiveness; a recent expert survey shows the remarkable, widespread approval of PID controllers [26]. Earlier work on the delay margin by PID control includes [14] (see also [15] (pp. 154)), where the exact delay margin was found for first-order systems achievable by proportional static feedback. In the comprehensive developments of [28], [29], likewise, the delay margin problem was addressed using full PID controllers, wherein it was shown that for first-order systems PID controllers can double the delay margin achievable solely by proportional feedback, a margin that was also found to be achievable by a predictor-type observer feedback [14]. The analysis in [28], [29] was carried

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out on the closed-loop quasipolynomial with the delay plant, by a rather elaborate derivation based on an extended HermiteBiehler theorem for quasipolynomials. Similarly, the HermiteBiehler theorem has also been applied to analyze stability and design problems, for systems with a fixed delay (see, e.g., [19]–[22], [27], [30]). Despite the considerable effort and lengthy derivations in [28] and elsewhere, the delay margin was found only for the first-order system. It appears difficult to extend this approach to higher-order plants, and the highly intricate nature of the approach tends to obscure the insight much sought after. From a rather different perspective, in this paper we analyze the open-loop frequency response of a delay system. The idea is to identify the critical frequency where a stable delay-free closed-loop system loses its stability due to time delay. We consider a number of first- and second-order delay plants, with one or two unstable poles, correspondingly. The effect of nonminimum phase zeros is also examined. In each case we derive exact delay margin and its bounds, which in turn avail two sets of results, dependent on the PID coefficients or independent of the controller, respectively. As such, the former set of results provide useful guidelines in tuning PID parameters and in designing analytically a PID controller, while the latter give the intrinsic limits in robustly stabilizing delay plants by a PID controller. More specifically, for systems with one unstable pole, we find analytical expressions of the delay margin, while for systems with more than one unstable pole, we find that the delay margin can be computed by finding the maximum of an explicit function of two real variables. The results consequently reduce considerably the complexity in determining the exact margin, and make it possible to numerically compute the margin; otherwise, the computation can be intensive whether by a brute-force search or using the Hermite-Biehler theorem. Indeed, in the latter vein, even with a fixed delay, finding a set of stabilizing PID coefficients is by no means an easy task. Moreover, the upper bounds provide a priori the limits independent of design, which, other than their accuracy, shed much insight desired; for example, the results show how the delay margin may be constrained by the plant unstable poles and nonminimum phase zeros. Note also that in using PID control, it is prevalent to consider loworder plants, and in fact, in many instances first-order plants, partly because industrial processes are often modeled by firstand second-order systems, partly due to the limitation of PID controllers in controlling high-order dynamics. Indeed, by and large PID control is known to be effective only for first- and second-order systems (see, e.g., [11], [34]); in general it is not guaranteed that PID controllers can stabilize a high-order unstable system free of delay, lest that the system may contain delays. Thus, while theoretically it may be of interest to extend our development to higher-order systems, in practice it appears adequate and moderately non-restrictive to consider systems up to the second order. Our contribution can be summarized as follows. In Section II, we introduce the delay margin problem and its variant defined with PID controllers. In Section III, we consider first-order unstable delay plants, for which we derive exact expressions of the delay margin with fixed PID controller coef-

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ficients and accordingly the maximal delay margin achievable by optimizing the PID controller. The result coincides with that of [28], and interestingly, that of [14], [16] in a more general setting, thus showing that in robustly stabilizing a first-order delay plant, a PID controller is in fact among the optimal; further to this point, our constructive proof reveals that it suffices to employ a PD controller only. The development, which arguably appears much simpler than those based on the Hermite-Biehler theorem, also yields useful insight: it shows that the delay margin is achieved mainly by the action of derivative control, while the integral control does no more to robust stabilization, a conflict seen as the tradeoff between control performance and robustness consistent with longheld design heuristics. Yet further, this insight and technical simplicity carries us forward to Section IV, where we derive exact expressions and bounds for several second-order unstable plants. Systems with two or more unstable poles pose a significant technical difficulty; as we noted above, no exact result has been found for such plants with general LTI controllers or PID controllers. Plants with nonminimum phase zeros are considered in Section V. The results are then extended in Section VI to systems containing time-varying delays. Numerical examples are provided in Section VII, and the paper concludes in Section VIII. II. P RELIMINARIES AND P ROBLEM F ORMULATION We consider the feedback system depicted in Fig.1, in which Pτ (s) denotes a plant to be controlled subject to an unknown delay τ , whose transfer function is given by Pτ (s) = P0 (s)e−τ s , τ ≥ 0,

(1)

where P0 (s) is a finite dimensional delay-free plant. Suppose -g −6

- Pτ (s)

-

K(s) Fig. 1.

Feedback control of a time-delay system

that P0 (s) can be stabilized by a certain finite-dimensional LTI controller K(s). How large may τ be, before the closedloop system loses its stability? This question concerns the computation of the so-called delay margin, defined as τ¯ = sup{µ ≥ 0 : There exists some K(s) that stablizes Pτ (s), ∀τ ∈ [0, µ)}. Stated in words, the problem is to determine the largest delay range within which Pτ (s) can be robustly stabilized by a finitedimensional LTI controller K(s). The delay margin problem has been under scrutiny for some time. In [16], exact delay margin was found for LTI plants containing one unstable pole and one nonminimum phase zero, achievable by general LTI controllers of a possible high order. Of particular interest in this paper is the delay margin achievable by LTI controllers of a specific structure, that is,



3

(v)

those of the PID structure: ki + kd s. KP ID (s) = kp + s In other words, we are concerned with

(

(2)

τ¯P ID = sup{µ ≥ 0 : There exists some KP ID (s) that stablizes Pτ (s), ∀τ ∈ [0, µ)}. This amounts to determining the largest range of delay tolerable by a PID controller. Also of interest is the delay margin achievable by a specific PID controller, with a fixed set of the controller parameters kp , ki , and kd . We define the latter by

−1

tan

ξ + tan

−1

η=

tan−1 ξ − tan−1 η = tan−1

ξ−η . 1 + ξη

III. F IRST- ORDER UNSTABLE SYSTEMS 1 −τ s e , s−p

p>0

(4)

was considered, where it was found that

τ¯P ID = sup{¯ τP ID (KP ID ) : KP ID (s) stabilizes Pτ (s), ∀τ ∈ [0, µ)}. In designing PID controllers, it is rather customary to consider low-order, and in fact first-order systems. This is consistent with control of industrial processes, where the process dynamics is often modeled by a first-order plant. Otherwise, one attempts to control the dominant modes of higher-order systems. It is worth noting that for a stable plant, one can trivially stabilize it robustly independent of delay; that is, the delay margin is infinite1 . For this reason, we follow the suit by considering first- and second-order unstable plants. Consider the open-loop transfer function (3)

Throughout this paper, we impose the following assumption. Assumption 2.1: (i) |L0 (0)| > 1. (ii) |L0 (∞)| < 1. We note that the assumption (i) is necessary for a system to achieve disturbance attenuation, while the assumption (ii) simply means that the open loop gain must roll off at high frequencies. Both are required of any realistic control design; indeed, it is typically required that |L0 (jω)| >> 1 at low frequencies and |L0 (jω)| 1.

In Silva et al. [28], the first-order unstable delay plant

Evidently,

L0 (s) = P0 (s)KP ID (s).

ξ+η 1−ξη , ξ+η 1−ξη +

(vi)

Pτ (s) =

τ¯P ID (KP ID ) = sup{µ ≥ 0 : KP ID (s) stablizes Pτ (s), ∀τ ∈ [0, µ)}.

tan−1 tan−1

πξ p tan−1 ξ ≤ . π − 2 + 2 1 + ξ2

(iv) −1 tan ξ − tan−1 η ≤ |ξ − η| . 1 It should be noted, nonetheless, that the tuning of PID controllers remains a delicate task even for stable plants. The stabilization of stable delay plants, while not a subject of our present study, is extensively discussed in, e.g., [28], [29].

τ¯P ID =

2 . p

(5)

Independently, this bound was also found in [14] as the maximum delay range within which the system (4) can be stabilized by an observer-based delay feedback. Furthermore, in light of [16], it is clear at once that for this system, τ¯ = τ¯P ID . In other words, for a first-order unstable plant, PID controllers are in fact among the optimal to achieve the largest possible delay margin. In this section, we provide an alternative, constructive proof for this result, which, unlike the elaborate analysis in [28] based on the Hermire-Bieler stability criterion, appears conceptually simple. The proof also lends an launching pad toward our investigation of more sophisticated plants of a higher-order. Furthermore, the approach leads to the exact margin τ¯P ID (KP ID ) for any given stabilizing PID controller KP ID (s), and shows how in the limit it may be constructed to achieve asymptotically the delay margin τ¯P ID . Theorem 3.1: Let Pτ (s) be given by (4). Then for KP ID (s) to stabilize P0 (s) and satisfy Assumption 2.1, it is necessary and sufficient that kp > p, ki > 0, and |kd | < 1. Under these conditions, k

tan−1 ωp0

τ¯P ID (KP ID ) =

ω0

tan−1 +

kd ω0 − ωi 0 kp

ω0

,

(6)

where ω0 > 0 is given by

ω02 =

kp 2 −2kd ki −p2 1−kd 2

r +

(

kp 2 −2kd ki −p2 2 ) 1−kd 2

2

ki + 4 1−k 2 d

2

.

Furthermore, τ¯P ID (KP ID ) ≤ τ¯P ID

=

1 kd + , p kp 2 . p

(7) (8)

Proof. We first determine the range of the triplet (kp , ki , kd ) so that KP ID (s) stabilizes P0 (s). This concerns the closedloop characteristic equation (1 + kd )s2 + (kp − p)s + ki = 0.



4

It follows from the Routh-Hurwitz criterion that for this second-order polynomial to be stable, it is both necessary and sufficient that all the coefficients have the same sign, i.e.,    kp > p  kp < p ki > 0 ki < 0 (i) or (ii)   kd > −1, kd < −1. Note, however, that |L0 (∞)| = |kd |. In light of Assumption 2.1, Case (ii) is excluded. Next, we examine the open-loop frequency response ki 1 kp + + jkd ω , (9) L0 (jω) = −p + jω jω which gives rise to 2

|L0 (jω)| =

(kp 2 − 2kd ki ) + kd 2 ω 2 + ω 2 + p2

ki 2 ω2

.

Setting |L0 (jω0 )| = 1 yields the unique solution

ω02 =

kp 2 −2kd ki −p2 1−kd 2

r (

+

kp 2 −2kd ki −p2 2 ) 1−kd 2

2

ki + 4 1−k 2 d

2

.

In other words, there exists a unique ω0 > 0 such that |L0 (jω0 )| = 1. At ω = ω0 , we have ∠L0 (jω) = π + tan−1

kd ω0 − ω0 + tan−1 p kp

ki ω0

.

Since L0 (jω0 ) = e

j∠L0 (jω0 )

= −e

j(tan−1

ω0 p

+tan−1

k kd ω0 − i ω0 kp

)

,

we can match the phase of L0 (jω0 ) with that of the delay, by setting τ0 ω0 = tan−1

ki ω0

kd ω0 − ω0 + tan−1 p kp

(10)

for some τ0 ≥ 0. Evidently, 1 + P0 (jω0 )KP ID (jω0 )e−jτ0 ω0 = 0, and for any τ < τ0 , 1 + P0 (jω)KP ID (jω)e−jτ ω 6= 0 for all ω ≥ 0; that is, the system is stable for all τ < τ0 . Consequently, k

τ¯P ID (KP ID ) = τ0 =

tan−1 ωp0 ω0

kd ω0 − ωi 0 kp

tan−1 +

ω0

.

This proves (6). To establish (7), we note that if kd ω0 ≥ then from Lemma 2.1 (ii), τ0 ≤

k 1 kd − ω0i2 1 kd + ≤ + . p kp p kp

ki ω0 ,

On the other hand, if kd ω0 < ωki0 , then according to Lemma 2.1 (iv), we have ki −1 ω −1 ω0 −kd ω0 0 tan − tan p kp τ0 = ω0 ki ω −k 0 − ω0 d ω0 kp p ≤ ω0 ki 1 1 kd − = + p kp kp ω0 2 1 kd ≤ + . p kp Thus, (7) is proved. Finally, to establish (8), we note that for the ranges of |kd | < 1 and kp > p, 1 1 2 τ¯P ID ≤ + = . (11) p p p The upper bound in (11), however, can be achieved asymptotically by selecting   kd = 1 − ε, kp = p + ε2 , (12)  ki = ε3 , for sufficiently small ε > 0. Indeed, with kd , kp and ki given in (12), we have ω02 = pε + o(ε) → 0 as ε → 0. Consequently, τ¯P ID (KP ID ) → 2/p. This proves (8), and hence completes the proof. Remark 3.1: From the above construction of PID gains (kp , ki , kd ), it is clear that only kd and kp contribute to the robust stabilization of Pτ (s) for τ ∈ [0, τ¯P ID ), while ki is immaterial. On the contrary, it is clear from the proof of Theorem 3.1 that ki or alternatively, the integral control tends to make the delay margin smaller. This is consistent with one’s intuition, since integral control is generally introduced to achieve performance objectives such as asymptotic tracking, which necessarily is in conflict with and is therefore seen as a tradeoff to stabilization. Indeed, it is not difficult to see that the delay margin in (8) can be achieved solely by the use of a PD controller. To this end, we note that with ki = 0, the closed-loop characteristic equation becomes (1 + kd )s + (kp − p) = 0. The crossover frequency ω0 → 0 when kp → p. In light of Lemma 2.1 (i), both tan−1 (ω0 /p) /ω0 and tan−1 (kd ω0 /kp ) /ω0 are monotonically decreasing functions of ω0 , which implies that ! tan−1 kdkωp 0 tan−1 ωp0 2 + = . τ¯P ID = lim kp →p, ω0 ω0 p kd →1

On the other hand, when in the absence of derivative control, the delay margin is reduced. We summarize this observation in the following corollary, which states that with PI control or static feedback alone, the delay margin reduces to 1/p. Note that the case of static feedback was established previously in [10], [14], [15]. Corollary 3.2: Let Pτ (s) be given by (4), and kd = 0. Then 1 τ¯P ID = . (13) p



5

Proof. The proof follows analogously as in that for Theorem 3.1, and hence is omitted. In light of Remark 3.1 and Corollary 3.2, we assert that PD controllers are essential for stabilization. Hence, throughout the rest of the paper, we shall consider PD controllers only. Remark 3.2: In practice, it is typical to implement the derivative control in conjunction with a low-pass filter [1], [32], so that the PID controller possesses the form of kd s ki T + , (14) KP fID (s) = kp + s 1 + Tf s where Tf > 0. By mimicking the above derivation, it is T possible to estimate the delay margin achievable by KP fID (s) as well, which helps demonstrate the tradeoff between the achievable margin and the practical implementation. The following corollary is presented to this effect. The result shows that the inclusion of the filter will reduce the delay margin achievable; intuitively, the filter will reduce the bandwidth of the derivative control and hence the reduction will depend on the time constant Tf relative to the unstable pole p. The effect, however, can be made negligible by selecting Tf such that Tf p p and kd +Tf (kp −p)+1 > 0. Under these conditions, T τ¯P ID (KP fID )

=

tan−1 ωp0

ω0 where ω0 > 0 is given by ω02

=

tan−1 +

kd kp

ω0

k ω02 1+Tf Tf + kd p

ω0

, (15)

(kd + kp Tf )2 − (1 + Tf2 p2 ) 2Tf2 rh i2 (kd + kp Tf )2 − (1 + Tf2 p2 ) + 4Tf2 (kp2 − p2 ) . + 2Tf2

With kp and kd given by (12), where ε → 0, 1 1 Tf τ¯P ID (KP ID ) ≤ 1+ . p 1 + 2Tf p

(16)

Proof. Note that under the condition kp > p and |kd | < 1, the T augmented PD controller KP fID (s) stabilizes P0 (s). Furthermore, Assumption 2.1 is rendered moot. The expressions for T τ¯P ID (KP fID ) is then obtained by straightforward calculation, analogously as in the proof of Theorem 3.1, yielding T

τ¯P ID (KP fID ) =

=

tan−1

ω0 p

ω0

tan−1 ωp0 ω0

tan−1 +

kd kp

ω0 kd 1+Tf Tf + kp ω02

ω0

T

τ¯P ID (KP fID ) ≤ ≤

(kd /kp ) 1 + p 1 + T T + kd ω 2 f f 0 kp 1 (kd /kp ) . + p 1 + Tf2 ω02

Furthermore, when specifying kp and kd by (12), it can be shown that s √ 2p ω0 = + o( ε). Tf This then establishes (16) as ε → 0. IV. S ECOND -O RDER P LANTS In this section we study second-order unstable systems. Delay margin for second-order delay systems generally presents more difficult problems, which has barely been addressed in the previous work. At present, no exact delay margin has been found for second-order plants, either with general LTI controllers or PID controllers. A. Real Poles We first consider plants that contain a pair of distinct real unstable poles p1 and p2 : Pτ (s) =

1 e−τ s , (s − p1 )(s − p2 )

p1 > 0, p2 > 0.

(17)

Both the exact margin and explicit bounds are given in the following theorem. Theorem 4.1: Let Pτ (s) be given by (17), and ki = 0. Then for KP ID (s) to stabilize P0 (s) and satisfy Assumption 2.1, it is necessary and sufficient that kp > p1 p2 and kd > p1 + p2 . Under these conditions, τ¯P ID (KP ID ) =

tan−1 ωp10

+

ω0

tan−1 ωp20 ω0

tan−1 +

kd ω0 kp

ω0

−

π , ω0

(18)

where ω0 > 0 is given by ω0 2 =

kd 2 − (p21 + p22 ) 2q +

2

(kd 2 − (p21 + p22 )) + 4(kp2 − p21 p22 ) 2

Furthermore, τ¯P ID (KP ID ) ≤

kd , kp

. (19)

(20)

τ¯P ID = sup {¯ τP ID (KP ID ) : kp > p1 p2 , kd > p1 + p2 } , (21) and q q 2p2 2p1 tan−1 p1 + p2 √ . (22) τ¯P ID ≤ 2p1 p2

ω0

ω0 tan−1 Tf ω0 − ω0 tan−1

+

kd +kp Tf kp

where the second equality follows by using Lemma 2.1 (vi). In view of Lemma 2.1 (ii), we have

,

Proof. Consider first the delay-free system, i.e., τ = 0. It follows analogously from the Routh-Hurwitz criterion that



6

P0 (s) can be stabilized by the PD controller if and only if kp > −p1 p2 and kd > p1 + p2 . On the other hand, to satisfy Assumption 2.1, it is necessary that |L0 (0)| = |kp |/(p1 p2 ) > 1, i.e., |kp | > p1 p2 . This implies that kp > p1 p2 . Consider now the magnitude of the open-loop frequency response 2

2

|L0 (jω)| =

2

∠L0 (jω0 ) = 2π + tan−1

ω0 kd ω0 ω0 + tan−1 + tan−1 . p1 p2 kp

kd ω0 According to Lemma 2.1 (i), tan−1 ≤ π/2, and kp ω 0 tan−1 ≤ π/2, i = 1, 2. Hence, the smallest τ0 > 0 such pi that 1 + P0 (jω)KP ID (jω)e−jτ0 ω0 = 0 is determined as ω0 kd ω0 ω0 + tan−1 + tan−1 − π, p1 p2 kp

(23)

that is, τ¯P ID (KP ID ) = τ0 . This gives (18). Indeed, invoking Lemma 2.1 (v), we can rewrite (23) as kd (p1 + p2 ) ω0 ω0 − tan−1 2 , kp ω0 − p1 p2

(24)

which yields τ0 ω0 = 0 at kd = p1 + p2 , and τ0 ω0 = π/2 when ω0 → ∞. Furthermore, it can be shown that τ0 ω0 > 0 for all kp > p1 p2 , kd > p1 + p2 . Toward this end, we first note that ω02 > p1 p2 + kp whenever kd > p1 + p2 . It follows that kp (p1 + p2 ) < (ω02 − p1 p2 )(p1 + p2 ), or equivalently, p1 + p2 p1 + p2 kd < < . 2 ω0 − p1 p2 kp kp

We claim that g¯(kd ) is monotonically decreasing with kd > q 2 p1 + p2 . To this end, denote ω1 (kd ) = kd − (p21 + p22 ). It follows that kd ω1 (kd )/kp d¯ g (kd ) ω12 (kd ) = dkd 1 + (kd ω1 (kd )/kp )2 dω1 (kd ) −1 kd ω1 (kd ) − tan kp dkd ω12 (kd )/kp + . 1 + (kd ω1 (kd )/kp )2 Note that dω1 (kd )/dkd = kd /ω1 (kd ). This leads to kd ω1 (kd )/kp kd d¯ g (kd ) = ω12 (kd ) dkd ω1 (kd ) 1 + (kd ω1 (kd )/kp )2 kd ω1 (kd ) − tan−1 kp kd ω1 (kd )/kp ω12 (kd ) + . (26) 1 + (kd ω1 (kd )/kp )2 kd2 By introducing the variable x = kd ω1 (kd )/kp , and noting that ω12 (kd ) ≤ kd2 , we may rewrite (26) as x kd d¯ g (kd ) − tan−1 x = ω12 (kd ) dkd ω1 (kd ) 1 + x2 x ω12 (kd ) + 1 + x2 kd2 2x kd −1 − tan x . ≤ ω1 (kd ) 1 + x2 2x − tan−1 x. (27) 1 + x2 It is an easy exercise to show that√h(x) is a monotonically decreasing function √ for x > 1/ 3, and it achieves the maximum at x0 = 1/ 3: √ √ 3 π h(1/ 3) = − > 0. 2 6 Note, however, that √ √ 3 π h( 3) = − < 0. 2 3 ∗ Hence, √ by √ the continuity ∗of h(x), there exists some ∗x ∈ (1/ 3, 3) such that h(x ) = 0, and that h(x) ≤ h(x ) = 0 for all x ≥ x∗ . Furthermore, for kd > p1 + p2 , we have kd x = ω1 (kd ) k p p1 + p2 p ≥ 2p1 p2 p p √ 1 2 ≥ 2 2, h(x) =

(p1 + p2 ) ω0 kd ω0 > tan−1 2 , kp ω0 − p1 p2

and hence τ0 ω0 > 0. As such, a nontrivial delay margin is determined by (18). Note also from (24) that τ0
p1 + p2 } ,

2

We solve ω0 such that |L0 (jω0 )| = 1, or equivalently, ω0 4 − kd2 − (p1 2 + p2 2 ) ω0 2 − kp 2 − p21 p22 = 0,

τ0 ω0 = tan−1

2.1 (ii) that ∂f (kp , kd )/∂kp ≤ 0. In other words, f (kp , kd ) is monotonically decreasing with kp . Hence,

tan−1 kdkωp 0 ω0

.

By invoking Lemma 2.1 (i), we arrive at (20). Let us then write tan−1 kdkωp 0 , (25) f (kp , kd ) = ω0 with ω0 given by (19). It follows as well that τ¯P ID ≤ sup {f (kp , kd ) : kd > p1 + p2 , kp > p1 p2 } . Fix kd and denote ω0 by ω0 (kp ). Taking the derivative of f (kp , kd ) with kp , we have the equation given at the top of the next page. Since dω0 (kp )/dkp ≥ 0, it follows from Lemma



∂f (kp , kd ) ω02 (kp ) ∂kp

7

  dω0 (kp ) − ω0 (kp ) kp dk kd ω0 (kp ) p   − dω0 (kp ) tan−1 kd ω0 (kp ) = 1 + (kd ω0 (kp )/kp )2 kp2 dkp kp kd ω02 (kp )/kp2 kd ω0 (kp )/kp dω0 (kp ) −1 kd ω0 (kp ) − tan . = − 1 + (kd ω0 (kp )/kp )2 kp dkp 1 + (kd ω0 (kp )/kp )2

√ √ in other words, for all kd > p1 + p2 , x ≥ 2 2 > 3. This allows us to conclude that d¯ g (kd )/dkd < 0 for kd > p1 + p2 , thus establishing the claim. As a result, τ¯P ID ≤ g¯(p1 + p2 ). This gives rise to (22), and hence completes the proof. For a given P0 (s), the exact expression (21) in Theorem 4.1 shows that the delay margin τ¯P ID can be found by a search of maximum over two variables kp and kd , with τ¯P ID (KP ID ) given explicitly as a function of kp and kd in terms of (18) and (19). This result thus simplifies considerably the complexity in determining the exact margin than done by a brute-force method; in the latter vein, one should note that even with a fixed delay, finding a set of stabilizing PID coefficients is a nontrivial task. In contrast, the explicit bound in (22) gives a fundamental limit showing how the delay margin may be constrained by the unstable poles. Remark 4.1: In the earlier work [8], allowing the use of general LTI controllers, it was shown that the delay margin τ¯ satisfies the inequality 2 p11 + p12 τ¯ ≤ . (28) 1 + pp21 + pp21 From Lemma 2.1 (iii), we have tan−1

r

2p2 + p1

r

2p1 p2

≤

≤

Pτ (s) =

1 (s − p)

q

2p2 p1

+

q

q

2p1 p2

τ¯P ID (KP ID ) =

2 tan−1 ωp0

τ¯P ID ≤ 2p1 p2

where the latter inequality follows from the fact that p2 p1 + ≥ 2. It then follows from (22) that p1 p2 π 1 1 τ¯P ID ≤ + , π + 4 p1 p2 which can be shown to improve (28) whenever p1 8 p2 + ≤1+ . p1 p2 π Remark 4.2: It can be shown analogously that the inclusion of the integral term, i.e., ki 6= 0, will lead to the reduction of the delay margin, thus reaffirming the consideration of PD control only. It is also interesting to see from (24) that at kd = p1 + p2 , τ¯P ID (KP ID ) = 0. This implies that the delay margin will never be reached at the boundary of the PD parameter space, and it highlights again the critical role of the derivative control in stabilization.

−τ s

,

p > 0.

(29)

tan−1 +

kd ω0 kp

−

π , ω0

ω0 ω0 where ω0 > 0 is given by q 2 2 2 (kd 2 − 2p2 ) + 4(kp 2 − p4 ) k − 2p + d 2 . ω0 = 2 Furthermore, kd τ¯P ID (KP ID ) ≤ , kp τ¯P ID = sup τ¯P ID (KP ID ) : kp > p2 , kd > 2p ,

2 π − 2 + 2 5 + 2p p1 + q q 2p2 2p1 + π p1 p2 , π+4

2e

The result given below follows as a simple corollary of Theorem 4.1. Corollary 4.2: Let Pτ (s) be given by (29), and ki = 0. Then for KP ID (s) to stabilize P0 (s) and satisfy Assumption 2.1, it is necessary and sufficient that kp > p2 and kd > 2p. Under these conditions,

and π

In a limiting case, it is of interest to consider plants with double unstable poles, that is,

√ ! tan−1 (2 2) 1 √ . p 2

(30)

(31)

(32) (33)

(34)

Remark 4.3: In [8], with general LTI controllers, an explicit upper bound on the delay margin of the plant (29) was found as 4 1 . τ¯ ≤ 3 p The bound in (34) is approximately equal to 0.87/p, which is significantly less than, and certainly consistent with that in [8]. Also of interest is the case when the plant contains an integrator, in addition to an unstable pole. The preceding results (Theorem 3.1, Corollary 3.2) indicate that in the presence of a pure integrator, the delay margin is infinite, seemingly pointing to the contention that an integrator has no effect on the delay margin. The following result, however, shows otherwise, that together with an unstable pole, the presence of an integrator will reduce the delay margin. Corollary 4.3: Let Pτ (s) =

1 e−τ s , s(s − p)

Then, τ¯P ID =

1 . p

p > 0.

(35)



8

Proof. To stabilize the delay-free plant, it is necessary and sufficient that kd > p and kp > 0. When kp → 0, the system approaches that of a first-order plant as given in (4) with a static feedback kd . This suggests that τ¯P ID ≥ 1/p. On the other hand, we may determine the crossover frequency ω0 > 0, given by q (kd2 − p2 ) + (kd2 − p2 )2 + 4kp2 , ω02 = 2 at which τ0 ω0 = tan−1

π kd ω0 + tan−1 ω0 − . p kp 2

By Lemma 2.1 (ii), we assert that τ0 ω0 ≤ tan−1 ωp0 , and hence τ¯P ID ≤ 1/p. The proof is completed. In comparison to Theorem 3.1, Theorem 4.1 and Corollary 4.2 both point to the fact that it is considerably more difficult to robustly stabilize a delay plant with more than one unstable pole, a difficulty that may manifest itself through the technical difficulty as well in obtaining an explicit expression of the exact delay margin. It further exhibits the limitation of LTI controllers in robustly stabilizing systems of a higher-order.

A second-order delay plant with a pair of complex conjugate poles is described by 1 e−τ s , (s − p)(s − p¯)

(36)

where p = α+jβ, Re(p) = α > 0, and p¯ denotes the complex conjugate of p. As stipulated in the preceding section, we consider as well a PD controller given by KP ID (s) = kp + kd s. Likewise, we show below that the exact delay margin can be determined by finding the maximum of a function of two variables, and an explicit upper bound can be established. Theorem 4.4: Let Pτ (s) be given by (36), and ki = 0. Then for KP ID (s) to stabilize P0 (s) and satisfy Assumption 2.1, it is necessary and sufficient that kp > |p|2 and kd > 2α. Under these conditions, tan−1 ω0α−β tan−1 ω0α+β τ¯P ID (KP ID ) = + ω0 ω0 −1 kd ω0 tan π kp − , + ω0 ω0

(37)

where ω0 > 0 is given by ω02 =

kd2 + 2β 2 − 2α2 2 q +

τ¯P ID

√   tan−1 2 |p|2α    √   2|p| ≤ 1 (π/4)  r  √ 2  |p|   2α π  − 2 |p|

if β < α, if β ≥ α.

(41)

Proof. The proof is similar to that of Theorem 4.1; hence, we shall only provide a sketch of it. First, in order for the PD controller to stabilize the delay-free plant and satisfy Assumption 2.1, from the Routh-Hurwitz criterion, it is necessary and sufficient that kd > 2α, 2 kp > |p| . The open-loop frequency response is given by P0 (jω)KP ID (jω) =

kp + jkd ω . (−α + j(ω + β))(−α + j(ω − β))

Solving the equation |L0 (jω0 )| = 1 yields the crossover frequency √ ω0 given in (38). Since, kd > 2α, kp > |p|2 , we find ω0 > 2|p|. We then solve for τ0 such that ω0 − β ω0 + β kd ω0 + tan−1 + tan−1 − π, α α kp (42) which gives rise to (37). Likewise, τ0 ω0 > 0 for all kp > |p|2 , kd > 2α, and τ0 ω0 = 0 at kd = 2α and τ0 ω0 = π/2 when ω0 → ∞. Furthermore, the inequality (39) holds for kp > |p|2 , kd > 2α. It then follows analogously as in the proof of Theorem 4.1 that τ¯P ID ≤ sup f (kp , kd ) : kd > 2α, kp > |p|2 = sup {ˆ g (kd ) : kd > 2α} , τ0 ω0 = tan−1

B. Complex Conjugate Poles

Pτ (s) =

and

4

(kd2 + 2β 2 − 2α2 )2 + 4(kp2 − |p| ) 2

. (38)

Furthermore, τ¯P ID (KP ID ) ≤

kd , kp

(39)

τ¯P ID = sup τ¯P ID (KP ID ) : kp > |p|2 , kd > 2α , (40)

where ω0 is given by (38), f (kp , kd ) is defined by (25), and √ 2 k +2β 2 −2α2 −1 tan kd d |p|2 . gˆ(kd ) = p 2 kd + 2β 2 − 2α2 We claim that similarly, gˆ(kd ) is monotonically decreasing for p kd > 2α whenever β < α. To see this, let ω1 (kd ) = kd2 + 2β 2 − 2α2 and x = kd ω1 (kd )/kp . With h(x) defined by (27), it follows as in the proof of Theorem 4.1 that dˆ g (kd ) kd x ω12 (kd ) = − tan−1 x dkd ω1 (kd ) 1 + x2 x ω12 (kd ) + 1 + x2 kd2 kd β 2 − α2 = h(x) + 2 ω1 (kd ) kd2 kd ≤ h(x). ω1 (kd ) √ 2 2α 2 Note that for kp = |p| and kd > 2α, x > >2 |p| whenever β < α. Thus, as shown in the proof of Theorem



9

4.1, h(2) < 0 and h(x) < 0 for all kd > 2α, which suggests that dˆ g (kd )/dkd < 0 for kd > 2α. Consequently, sup {ˆ g (kd ) : kd > 2α} = gˆ(2α) =

√ tan−1 2 |p|2α

√ 2|p|

.

π(kd /|p|2 ) 2 2 −α2 k k2 1 + |p|d2 |p|d2 + 2 β |p| 2

π(kd /|p|2 )

≤

k2

2

2

−α π − 2 + 2 |p|d2 + 2 β |p| 2

s kd∗ = |p|

2

−

1 √

2α |p|

2 .

C. Oscillatory Poles Systems with imaginary poles exhibit rather different characteristics in robust stabilization against time delays. As demonstrated in [16], such poles generally make the delay margin larger, compared to their real counterparts with the same magnitudes. This is consistent with the intuition that systems with purely imaginary poles are less difficult to control than those with strictly unstable poles. Consider the plant 1 Pτ (s) = 2 e−τ s , ωc ≥ 0, (43) s + ωc2 where P0 (s) contains a pair of imaginary poles at ±jωc . In this case, we give below a more computable expression of the exact delay margin, which reduces the search over two variables to one involving a single variable. Theorem 4.5: Let Pτ (s) be given by (43), and ki = 0. Then for KP ID (s) to stabilize P0 (s) and satisfy Assumption 2.1, it is necessary and sufficient that kp > ωc2 and kd > 0. Under these conditions, τ¯P ID (KP ID ) =

kd ω0 kp

ω0

p (kd /ωc ) (kd /ωc )2 + 2 p , (kd /ωc )2 + 2

(47)

(48)

,

where ω0 > 0 is given by q kd2 + 2ωc2 + (kd2 + 2ωc2 )2 + 4(kp2 − ωc4 ) ω02 = . 2

Proof. The conditions for stabilizing the delay-free plant and meeting Assumption 2.1 are found to be kd > 0 kp > ωc2 . At some ω0 > ωc , we find that kd2 ω02 + kp2 = 1, (ωc2 − ω02 )2 kd ω0 ∠P0 (jω0 )KP ID (jω0 ) = π + tan−1 , kp

This completes the proof.

tan−1

τ¯P ID

|P0 (jω0 )KP ID (jω0 )|

We are then led to π r 4|p| π

−1

tan 1 sup = ωc kd >0

2

s π π 2α2 β 2 − α2 −1+ = |p| − 2. 2 2 |p| 2 |p|

sup {ˆ g (kd ) : kd > 2α} ≤

(46)

Finally, τ¯P ID = ∞ when ωc = 0.

.

The right hand side of the last inequality achieves the maximum (π/4) 1 r √ 2 |p| 2α π − 2 |p| at

kd , kp

! √ √ π tan−1 ( 7/3) 1 1 p √ ≤ τ¯P ID ≤ . ω ω 2 2 (7/3) c c

r π−2+2

τ¯P ID (KP ID ) ≤ and

This establishes the bound in the case β < α. If β ≥ α, we may invoke Lemma 2.1 (iii) to gˆ(kd ), which gives rise to gˆ(kd ) ≤

Furthermore,

(44)

(45)

=

which result in (45) and (44), respectively, such that 1 + P0 (jω0 )KP ID (jω0 )e−jτ0 ω0 = 0. The inequality (46) then follows analogously by using Lemma 2.1 (ii). Since, as shown in the proof of Theorem 4.1, tan−1 (kd ω0 /kp )/ω0 is monotonically decreasing with kp > ωc2 , we have ) ( tan−1 kdkωp 0 2 : kd > 0, kp > ωc τ¯P ID = sup ω0   p  tan−1 ωkd2 kd2 + 2ωc2  pc : k > 0 . = sup d   kd2 + 2ωc2 This proves (47). The upper bound in (48) follows by setting α = 0 and β = ωc > α in (41). To establish the lower bound, we note from Lemma 2.1 (ii) that p tan−1 (kd /ωc ) (kd /ωc )2 + 2 (kd /ωc ) p ≥ 2. 2 (kd /ωc ) + 2 ((kd /ωc )2 + 1) The right hand√side of the inequality achieves its maximum at kd /ωc = 1/ 3. This gives rise to the lower bound in (48), and in turn τ¯P ID = ∞ for ωc = 0. It is clear from Theorem 4.5 that the exact delay margin for systems with one pair of imaginary poles can be computed by employing, e.g., a line search method. The upper bound in (48) is significantly smaller than that found in [16], which concerns the delay margin achievable using general LTI controllers. Theorem 4.5 also corroborates with earlier results in [31], [37], [38], where it was shown that for systems with unstable poles solely at the origin, an arbitrarily large delay margin can be achieved by finite-dimensional state feedback [37], and that for systems with non-zero imaginary poles, only finite delay margin can be attained [31].



D. Further Discussions We conclude this section with a discussion on possible extensions to high-order systems. At the outset, we note that it is generally difficult, if ever possible, to stabilize high-order unstable delay systems using PID control. Indeed, a simple argument based on the root locus or Routh-Hurwitz criterion reveals that a delay-free triple integrator system, or one with three real unstable poles, can in no event be stabilized by PID control, lest that the system may contain delays. This appears to be the main limitation of our results, and indeed, that of PID control in general; a low-order, fixed structure controller such as a PID controller may not be able to cope with sophisticated high-order dynamics. Notwithstanding this fundamental limitation of PID control, it remains possible to stabilize high-order systems by augmented, PID-based controllers. To illustrate, consider, for example, the plant Pm (s) −τ s e , (49) Pτ (s) = s−p where Pm (s) is a stable and minimum phase transfer function. By employing the controller K(s) =

−1 Pm (s) KP ID (s), (1 + δs)n

where n is an integer so that K(s) remains proper, one can readily show that with a sufficiently small δ > 0, 2 τ¯ = . p In other words, using an appropriately augmented PID controller, one can still attain the delay margin, despite that the plant may be of a high order and contain an unstable pole. Evidently, this control scheme extends to high-order systems containing two unstable poles, enabling us to recover the bounds presented in the preceding subsections. Note also that this scheme can be combined with the filtered derivative control in (14) for plants with two unstable poles as well, making it possible to obtain bounds similar to that given in Corollary 3.3. Alternatively, stabilization is also possible for restricted classes of high-order systems using strictly PID control. To illustrate, consider the class of systems Pτ (s) = P0 (s)e−τ s with 1 P0 (s) = n−1 , (50) (s + · · · + a1 s + a0 ) (s − p) where p ≥ 0 and the polynomial a(s) = sn−1 + · · · + a1 s + a0 is assumed to be stable with ai > 0, i = 0, 1, · · · , n − 2. For the PD controller to stabilize P0 (s), we first determine the feasible range of the parameters kp and kd using the RouthHurwitz criterion; for example, it is necessary that kp > a0 p and kd + a0 − a1 p > 0. One may then seek to determine the crossover frequency ω0 by solving the polynomial equation in ω02 , 2 2 R (ω0 ) + ω02 I 2 (ω02 ) (ω02 + p2 ) − kd2 ω02 − kp2 = 0, and accordingly compute the critical delay τ0 such that ω0 kd τ0 ω0 = tan−1 + tan−1 ω0 − ∠a(jω0 ). p kp

10

Here R(ω 2 ) = Re{a(jω)}, and ωI(ω 2 ) = Im{a(jω)}. Note that ∠a(jω) ≥ 0 for all ω ≥ 0. Evidently, if ∠a(jω0 ) ≥ π, then τ0 ≤ 0. For ∠a(jω0 ) ∈ [0, π), one can easily see that  2  tan−1 I(ω02) ω0 if R(ω02 ) ≥ 0 R(ω0 ) ∠a(jω0 ) = 2  π − tan−1 I(ω02) ω0 if R(ω02 ) < 0 |R(ω0 )|

For a given KP ID (s), the margin τ¯P ID (KP ID ) can then be computed readily. The computation of τ¯P ID requires a search in the two-dimensional parameter space of (kp , kd ) with a feasible region determined by the Routh-Hurwitz criterion. Example 7.1 provides a further illustration of this procedure. More generally, several methods developed in the recent years can be employed to study stabilization problems for general LTI delay systems containing an arbitrary number of unstable poles, including the numerical method in [14], the interpolation method in [23], and a series of works based on predictor feedback [31], [33], [37], [38]. These methods generally produce high-order controllers and as expected, avail more degrees of design freedom and therefore are expected to perform better than PID controllers. With their varying degrees of conservatism, however, they appear less advantageous when applied to low-order systems. For example, when applied to the first-order system (4), a lower bound obtained in [23] on the delay margin τ¯ is given as 1.722/p, while Theorem 3.1 provides the exact τ¯ = τ¯P ID = 2/p. For comparisons to predictor feedback method2 , we note that the delay margin bound in [31] (Theorem 7) can be weakened to 1 , n p P 2 2(n + 1) pi i=1

provided that the n-th order delay-free system contains only unstable poles and not all unstable poles pi are strictly on the imaginary axis. For systems with solely imaginary poles pi , the bound can be weakened to 1 s . n P 2 2n |pi | i=1

In other words, the lower bound on the delay margin τ¯ in [31] cannot exceed these quantities in the respective cases. The following table provides for selected cases a comparison of the delay margin bounds obtained in this paper versus those in [31]. Clearly, for each of these low-order systems, the bounds TABLE I L OWER B OUNDS ON τ¯. P0 (s) Predictor State Feedback PID Output Feedback

1 s−p 1 4

2 p

1 p

1 s(s−p)

1 √ 2 6 1 p

1 2 s2 +ωc 1 p

1 1 √ 2 2 ωc √ tan−1 ( 7/3) 1 √ ωc (7/3)

herein outperform those based on predictor state feedback. 2 The results in [37] are restricted to systems with unstable poles at the origin only, while those of [33], [38] require the exact knowledge of the delay in designing the stabilizing feedback controllers and hence are inappropriate to address robust stabilization problems such as the delay margin.



11

Additional comparisons may also be sought after, e.g., for the cases of unstable real poles and unstable complex conjugate poles, by deriving lower bounds on τ¯P ID in these cases. V. E FFECT OF N ONMINIMUM P HASE Z EROS In this section we extend the preceding results to nonminimum phase systems. Consider the plant 1 z−s e−τ s , (51) Pτ (s) = z + s (s − p) where z > 0 is a real nonminimum phase zero of P0 (s), and p > 0 is an unstable real pole of P0 (s). Analogously, we take KP ID (s) = kp + kd s . Theorem 5.1: Let Pτ (s) be given by (51). Assume that z > p. Then for any kp > p, kp /z ≤ kd < 1, and ki = 0, τ¯P ID (KP ID ) =

tan−1

ω0 p

ω0

tan−1 −2 ω0

ω0 z

tan−1 +

kd ω0 kp

ω0

ω0 =

kp 2 − p2 . 1 − kd 2

(53)

Furthermore, 1 2 kd − + , p z kp 1 1 − . = 2 p z

τ¯P ID (KP ID ) ≤ τ¯P ID

The upper bound, however, can be achieved asymptotically by constructing kd = 1 − ε, kp = p + ε2 , for sufficiently small ε such that 0 < ε < ε + (p + ε2 )/z < 1, which leads to r 2pε + ε3 ω0 = . 2−ε This completes the proof. Remark 5.1: It can be readily verified that under the conditions kd < 1, kp > p, the inequality

,

(52)

where ω0 is given by s

where the last inequality follows by using Lemma 2.1 (iv). This proves (54). It follows as well that 1 1 . − τ¯P ID ≤ 2 p z

(54) (55)

Proof. The closed-loop characteristic equation of the delayfree system is given by (1 − kd )s2 + [(1 + kd )z − (kp + p)]s + z(kp − p) = 0. Thus, the necessary and sufficient condition for the PD controller KP ID (s) to stabilize P0 (s) is  1 − kd < 0  kp − p < 0 (56a)  (1 + kd )z < kp + p, or  1 − kd > 0  kp − p > 0 (56b)  (1 + kd )z > kp + p. Together with the requirements |L0 (0)| = |kp /p| > 1, |L0 (∞)| = |kd | < 1, it is evident that the parameters kp and kd must be chosen to satisfy (56b). Calculating next the crossover frequency ω0 such that |L0 (jω0 )| = 1, we obtain (53). We then match the phase of P0 (jω0 )KP ID (jω0 ) to that of e−jτ0 ω0 , which results in (52). Note that under the condition kd > kp /z, it is guaranteed that τ¯P ID (KP ID ) > 0. To establish (54) and (55), we rewrite ω0 ω0 τ0 ω0 = tan−1 − tan−1 p z −1 kd ω0 −1 ω0 + tan − tan kp z ω0 ω0 kd ω0 ω0 ≤ − + − , p z kp z

(1 + kd )z > kp + p and hence (56b) cannot be satisfied whenever z < p; in other words, the delay-free plant P0 (s) cannot be stabilized by a PD controller, lest the delay plant. More generally, one may contend that P0 (s) cannot be stabilized by any PID controller under this circumstance. This can be verified by analyzing the corresponding closed-loop characteristic polynomial, or alternatively observed from a root-locus perspective. The exact margin (55) coincides with that of [16], which was shown to be achievable by a general LTI controller for plants with a single nonminimum phase zero. The result thus reveals, surprisingly, that in the presence of one unstable pole p and one nonminimum phase zero z, a PD controller is among the best in robustly stabilizing the second-order delay plant (51) whenever p < z. In view of [10], which concerns the delay margin achievable by static controllers for first-order unstable plants, it is tempting to ask whether the same can be accomplished by a static controller if the plant contains also a nonminimum phase zero. The following corollary provides a definitive answer, whose proof is similar to that of Theorem 5.1 and hence is omitted. Both Theorem 5.1 and Corollary 5.2 indicate that a pair of closely located real unstable pole and nonminimum phase zero will confine severely the range of the delay over which the delay plant can be robustly stabilized. Corollary 5.2: Let Pτ (s) be given as s − z −τ s Pτ (s) = e , s−p where z > 0 is a real nonminimum phase zero of P0 (s), and p > 0 an unstable real pole of P0 (s). Assume that z > p, and let KP ID (s) = kp . Then for any −1 < kp < −p/z, τ¯P ID (KP ID ) =

tan−1 ω0

ω0 p

−

tan−1 ω0

ω0 z

,

(57)

where ω0 is given by s ω0 =

kp 2 z 2 − p 2 . 1 − kp 2

Furthermore, τ¯P ID =

1 1 − . p z

(58)



12

VI. U NSTABLE PLANTS WITH TIME - VARYING DELAY Robust stabilization against time-varying delays in the spirit of the delay margin has been studied previously in [23], which, as expected, poses a more difficult task and generally does not admit a necessary and sufficient stability condition. In this section we show how PID control may be used to stabilize robustly low-order systems subject to a time-varying delay. Consider the delay system described by the state-space form x(t) ˙ = Ax(t) + Bu(t − τ (t)) y(t) = Cx(t),

(59)

where τ (t) is a time-varying delay. It is customary to restrict the range and variation rate of the delay function τ (t) to 0 ≤ τ (t) ≤ τM , 0 ≤ |τ˙ (t)| ≤ ρ < 1,

(60)

where τM represents the maximal delay range and ρ is the maximal variation rate permissible. We assume that (A, B) is controllable and (C, A) is observable. Let P0 (s) = C(sI − A)−1 B be the transfer function of the delay-free plant. Let also K(s) be a LTI output feedback controller, i.e., u(s) = K(s)y(s), that stabilizes P0 (s). Define the complementary sensitivity function of the delay-free plant by P0 (s)K(s) . 1 + P0 (s)K(s) Then the following lemma, adopted from [9], [23], provides sufficient conditions for the system (59) to be robustly stabilizable. Lemma 6.1: Denote by k · k∞ the H∞ norm of a stable transfer function. Let K(s) stabilize P0 (s). Then the system (59) can be robustly stabilized by K(s) for all τ (t) (i) if 0 ≤ τ (t) ≤ τM , and T0 (s) =

||τM sT0 (s)||∞ < 1. (ii) if 0 ≤ τ (t) ≤ τM , 0 ≤ |τ˙ (t)| ≤ ρ < 1, and r τM s 2−ρ , W < (s)T (s) 0 1 + τM s τM 2 2 ∞

(61)

(62)

where Wτ (s) is some stable and minimum phase rational function such that τ (jω) W (jω) ∀ ω ≥ 0, (63) ≥ φ(ω), τ 1 + τ (jω) 2

with

φ(ω) =

2 sin (τ ω/2) |τ ω| ≤ π, 2 |τ ω| > π.

Remark 6.1: In [23], delay-dependent rational functions of the form τs Wτ (s) 1 + τ2s are constructed to approximate the transcendental function e−τ s − 1, wherein a number of such functions meeting the requirements in Lemma 6.1 (ii) are given (cf. Example 7.3). Since

τs

τs

≤ τ s kWτ (s)k∞

1 + τ s Wτ (s) 1 + 2 2 ∞ ∞ = 2kWτ (s)k∞ ,

the condition (63) mandates that kWτ (s)k∞ ≥ 1. Generally, a higher-order rational function Wτ (s) may provide a better approximation, which in turn yields a kWτ (s)k∞ closer to the value of 1. Note also that for the classes of Wτ (s) under consideration in [23], kWτ (s)k∞ is in fact invariant of τ . Based on the small-gain conditions given in Lemma 6.1, we now derive robust stabilization conditions for systems containing time-varying delays. We consider the first-order unstable plant given by x(t) ˙ = px(t) + u(t − τ (t)),

p > 0.

(64)

For this system, it is known from Corollary 3.2 that in the presence of an uncertain constant delay τ (t) = τM , the delay margin achievable by a static feedback controller is 1/p. Our following result shows that even with a time-varying delay, this delay margin can be achieved by a static controller irrespective of the delay variation rate, thus surprisingly, resulting in a rare necessary and sufficient condition for robust stabilizability in the case of a time-varying delay. Inadvertently, it also shows that with a static controller, the best result is achieved without incorporating the delay variation rate ρ. Theorem 6.1: The system (64) can be robustly stabilized by the proportional controller u(t) = −kp x(t) for all τ (t) satisfying (60) if kp > p and τM
p, we arrive at the condition (66), thus establishing its sufficiency. The next result shows that by using a PD controller, the range of the permissible delay can be judiciously improved by incorporating the delay variation rate. Theorem 6.2: The system (64) can be robustly stabilized by the PD controller KP ID (s) = kp + kd s for all τ (t) satisfying (60) if kp > p, |kd | < 1, and one of the following conditions holds: r 2|kd | 2|kd | 2−ρ , kWτM (s)k∞ < , (i) τM < kp 1 + kd 2 r 2|kd | τM kp 2−ρ (ii) τM ≥ , kWτM (s)k∞ < , kp 1 + kd 2



13

where WτM (s) is a stable, minimum phase rational function satisfying the condition (63). Furthermore, the system can be robustly stabilized by a PD controller if √ 2 2−ρ 1 √ . (67) τM < √ p 2 2kWτM (s)k∞ − 2 − ρ Proof. It follows from the proof of Theorem 3.1 that to stabilize the system (64) for ρ = 0, we require that kp > p and |kd | < 1. Assume that this is the case. In view of Lemma 6.1 (ii), the system (64) can be robustly stabilized by the PD controller KP ID (s) = kp + kd s if r

τM s

2−ρ

, (68) kWτM (s)k∞ T0 (s) < 1 + τM2 s 2 ∞ where T0 (s) is found as T0 (s) =

kp + kd s . kp − p + (1 + kd )s

We then proceed to calculate the H∞ norm of It follows that

τM s

T (s)

1 + τM s 0 2

∞

τM s T0 (s). 1 + τM2 s

Fig. 2.

The effect of filtered PID control

which implies that, unsurprisingly, the condition (67) cannot be as good as that for the case of a constant delay. On the other hand, when WτM (s) is appropriately constructed, specifically when √ 2 2−ρ √ > 1, √ 2 2kWτM (s)k∞ − 2 − ρ

kp + kd s

τM s

≤

1 + τM s kp − p + (1 + kd )s . or equivalently, ∞ 2 ∞

√ 3 2−ρ √ , 2 2 the condition (67) always improves that in (66). This improvement can be significant for a small value of ρ. In the limit and

2|kd | when ρ → 0, the upper bound in (67) may approach 2/p with τM

kp + kd s

=

τM , |kd | > 2 kp a well-constructed WτM (s) such that kWτM (s)k∞ ≈ 1.

1 + τM s kp , |kd | < τ2M kp We conclude by commenting that it is possible to extend 2 ∞ Theorem 6.2 (i)-(ii) then follows by invoking Lemma 6.1 (ii). Theorem 6.1 and Theorem 6.2 to second-order systems, albeit By the monotonicity properties of |kd |/(1+kd ) and 1/(1+kd ), at the expense of increased complexity in the resultant conditions; the complexity may obscure the insight and hence is we find that pursued herein. not 2|kd | τM τM kp τM kp kp inf : |kd | > = inf : |kd | < 1 + kd 2 1 + kd 2 VII. E XAMPLES τM kp , = We now use a number of numerical examples to illustrate 1 + τ2M kp our results. In all cases, we set ki = 0 in the PID controller τM p at kp = p. KP ID (s). which further achieves the minimum 1 + τ2M p Example 7.1: In this first example we examine the effect Consequently, we have of filtered derivative control and that of high-order stable

τM s

dynamics on the delay margin. Consider first the first-order τM p

inf

1 + τM s T0 (s) : kp > p, |kd | < 1 ≤ 1 + τM p . unstable delay plant (4) to be stabilized by the filtered PID 2 2 ∞ controller (14). We choose Tf = 0.001, 0.01, 0.1, 1, 10, Hence, according to Lemma 6.1 (ii), the system can be robustly and allow p to vary in the interval [0.3, 3]. For each T , we f stabilized by a PD controller if Tf compute the delay margin achievable by K (s), namely, P ID r T the maximum of τ¯P ID (KP fID ) by optimizing kp and kd within τM p 2−ρ . T τM kWτM (s)k∞ < their allowable ranges. Fig. 2 exhibits the effect of KP fID (s), 1+ 2 p 2 demonstrating a tradeoff between the achievable delay margin The inequality (67) is then immediate. and the implementation cost. As expected, the higher is the Remark 6.2: Note from Remark 6.1 that if kWτ (s)k∞ bandwidth of the filter, the larger becomes the delay margin. is invariant of τ , then kWτM (s)k∞ is independent of τM . In the limit, when Tf approaches to 0, the delay margin tends Furthermore, under the condition kWτM (s)k∞ ≥ 1, it is to the ideal 2/p, while if Tf → ∞, the delay margin is reduced always true that to 1/p. √ 2 2−ρ Next, we consider a fourth-order unstable delay plant with √ ≤ 2, √ P 2 2kWτM (s)k∞ − 2 − ρ 0 (s) given as in (50), where p > 0 is an unstable pole and By a straightforward calculation, we obtain

τM s τM

kp − p + (1 + kd )s = 1 + kd , ∞

kWτM (s)k∞
0, i = 0, 1, 2. Using the Routh-Hurwitz criterion, we determine the allowable ranges of kp and kd , which yields the conditions a2 > p, a1 > a2 p, a1 p − a0 < kd < (a1 − a2 p)(a2 − p) − (a0 − a1 p), (a1 − a2 p)(a2 − p) − (a0 − a1 p + kd )2 a0 p < kp < + a0 p. (a2 − p)2 It is clear from these conditions that in general a PD controller cannot stabilize such a fourth-order plant even in the absence of delay. It is also clear that the ranges of the PD parameters, or alternatively, the design of the PD controller, are constrained by the stable part of the plant, thus limiting the delay margin achievable. Choose a0 = 170, a1 = 97, and a2 = 18. The polynomial a(s) is stable and has roots at −10, −4 ± j, respectively. For p in [0.2, 2], we then compute the delay margin as outlined in Section IV-D, by searching over the allowable region of (kp , kd ) determined by the conditions given above. Fig. 3 shows that the margin can be found similarly for a fourth-order plant, and that the stable part of the plant does constrain the maximal delay margin achievable. Example 7.2: We next consider the second-order plant 1 Pτ (s) = e−τ s , (s − p1 )(s − p2 ) where p1 and p2 are both unstable poles. Different combinations of these poles are examined in the sequel. Distinct Real Poles: p1 > 0, p2 > 0. In this case, we take three pairs (p1 , p2 ) = (0.6, 0.8), (1, 1.2), and (0.4, 2), taking into account the proximity of the poles from the imaginary axis and their relative locations. The computation gives the maximum of τ¯P ID (KP ID ) as 0.3960, 0.2513 and 0.2595, respectively, while the upper bound in (22) yields 1.2596, 0.7954 and 1.0381. In spite of this seemingly significant discrepancy, Fig. 4, which plots the upper bound in (22) versus the available lower and upper bounds achievable by general LTI controllers by fixing p1 = 0.2 and allowing p2 to vary, demonstrates that for an appreciable range of the unstable poles, the upper bound in (22) remains to offer a sizable improvement. Complex Conjugate Poles: p1 = p = α + jβ, p2 = p¯, α > 0. The computation gives the maximum of τ¯P ID (KP ID ), that

Fig. 5. State responses of (59) with proportional controller for different τM .

is, the delay margin τ¯P ID , as 0.0115, 0.7675 and 0.2329, respectively. The upper bound in (41), instead, is found to be 0.0359, 0.7833 and 0.7351. It is of interest to note that the first two cases correspond to the conditions α >> β and α 0. We choose ωc = 1, 2, 4. In these cases, the delay margin τ¯P ID is computed as 0.6198, 0.3099 and 0.1550, respectively, while the upper bound in (48) is found as 0.6267, 0.3133 and 0.1567, demonstrating the striking accuracy of the bound. Example 7.3: Our final example concerns the system (59), with the time-varying delay τ (t) = a(1 − sin bt). It follows that the maximal delay range and the variation rate are τM = 2a and ρ = ab, respectively. Let p = 2. Theorem 6.1 tells that the system (59) can be stabilized by a proportional controller regardless of the delay variation rate ρ whenever τM < 0.5. Fig. 5 confirms this assertion: For τM = 0.48 < 1/p, the state response decays asymptotically for kp = 2.1 > p, while for τM = 0.64 > 1/p, the state response diverges asymptotically. We next consider the stabilization of the system (59) by a PD controller. In this vein, we note that the following rational



15

Fig. 8. State response of (59) with PD controller: (τM , ρ) outside the stablizable region. Fig. 6.

Stablizable region of (ρ, τM ) with different Wτ (s).

Fig. 7. State responses of (59) with PD controller: (τM , ρ) inside the stablizable region.

functions [23] satisfy the condition (63): Wτ(1) (s) = 1.216, 0.1791(τ s)2 + 0.7093τ s + 1 Wτ(2) (s) = . 0.1791(τ s)2 + 0.5798τ s + 1 0.02952(τ s)4 +0.210172(τ s)3 +0.70763(τ s)2 +1.3188τ s+1 Wτ(3)(s)= 0.02952(τ s)4 +0.191784(τ s)3 +0.64174(τ s)2 +1.195282τ s+1. (1)

The H∞ norms of these functions are found as kWτ (s)k∞ = (3) (2) 1.2160, kWτ (s)k∞ = 1.0908, and kWτ (s)k∞ = 1.0831, respectively. For p = 2, the region of (ρ, τM ) for which the system (59) can be stabilized by a PD controller is shown in Fig. 6, represented by the region below the curve determined by (62). We then choose, at random, a pair (ρ, τM ) = (0.12, 0.6) (3) from the stabilizable region given with the function Wτ (s). This corresponds to a = 0.3 and b = 0.4, and τ (t) = 0.3(1 − sin 0.4t). Select kp = 2.2, kd = 0.4, which together with (ρ, τM ) = (0.12, 0.6) satisfy the condition (ii) in Theorem 6.2. Fig. 7 shows that the corresponding PD controller indeed stabilizes the system. Also shown in the figure is the state response of a closed-loop system with better transient behavior, achieved by the same PD controller for a set of more desirable (ρ, τM ), i.e., (ρ, τM ) = (0.066, 0.44). On the other hand, for (ρ, τM ) = (0.15, 1), which lies outside the stabilizable region, Fig. 8 shows that the plant can no longer be stabilized by this PD controller. VIII. C ONCLUSION In this paper we have studied the delay margin of LTI delay systems achievable using PID controllers. We derived

exact delay margin and its bounds for first- and secondorder delay plants, both dependent on and independent of PID controller parameters. Other than the exact results, the bounds obtained herein are significantly tighter than those known; in other words, when using PID control, the earlier bounds obtained elsewhere will be overly pessimistic compared to the ones derived herein. From both a conceptual and technical standpoint, it appears, arguably, that our analysis is much simpler than those based on the Hermite-Biehler theorem for quasipolynomial analysis, and sheds much light into the limit of robust stabilization of unstable time-delay systems, as well as the limitation of PID controllers in stabilizing such plants. While giving the fundamental limits in the robust stabilization of delay plants by PID controllers, the results should also provide useful guidelines in tuning PID parameters and in the analytical design of PID controllers. Notwithstanding the ability in stabilizing first- and secondorder delay systems, with only three parameters available for design of a stabilizing controller, PID control is inherently limited in controlling high-order systems; a low-order, fixed structure controller such as PID control may simply be unable to cope with sophisticated high-order dynamics beyond its capability. While we have analyzed cases where an extension may be possible, our results remain largely restricted to loworder systems. In contrast, for high-order systems, the more sophisticated design methods, such as that based on the H∞ optimal control in [23], the numerical method in [14], and the predictor feedback method in [31], [38], enjoy higher degrees of design freedom and are more broadly applicable, though each of these methods also suffers from its own limitations and expenses. Finally, while in this paper we have focused exclusively on unstable delay systems, the design of PID controllers for stable delay plants remain to be of practical interest, where likely the tuning of the PID coefficients may override the pursuit of a larger delay margin; indeed, sheer for maximizing the range of stabilization, the delay margin for a stable plant can be made arbitrarily large. ACKNOWLEDGEMENT The authors are grateful to the Associate Editor and the reviewers for their valuable comments. R EFERENCES [1] K. Astr¨om and T. Hagglund, PID Controllers: Theory, Design, and Tuning, 2nd Ed. ISA, 1995.



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Dan Ma (M’17) received the B.S. degree in Automation from Liaoning Institute of Technology, Jinzhou, China in 1999, the M.S. degree from Shenyang University of Technology in 2004, and the Ph.D. degree from Northeastern University in 2007, Shenyang, China, both in Control Theory and Control Engineering. Since 2006, she has been with School of Information Science and Engineering, Northeastern University, Shenyang, where she is appointed an Associate Professor. She was a Postdoctoral Fellow at Northeastern University from 2008 to 2010, a Guest Professor with Department of Electrical Engineering, University of Notre Dame, South Bend, Indiana, USA, in 2012, and a Research Fellow at Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China, in 2017. Her main research interests include network-based control systems, switched systems, and time-delay systems. She is a member of CAA Youth Committee, and a member of TCCT Technical Committee on Nonlinear Systems and Control.

Jie Chen (S’87-M’89-SM’98-F’07) is a Chair Professor in the Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China. He received the B.S. degree in aerospace engineering from Northwestern Polytechnic University, Xian, China in 1982, the M.S.E. degree in electrical engineering, the M.A. degree in mathematics, and the Ph.D. degree in electrical engineering, all from The University of Michigan, Ann Arbor, Michigan, in 1985, 1987, and 1990, respectively. Prior to joining City University, he was University of California, Riverside, California, from 1994 to 2014, where he was a Professor and served as Professor and Chair for the Department of Electrical Engineering from 2001 to 2006. His main research interests are in the areas of linear multivariable systems theory, system identification, robust control, optimization, time-delay systems, networked control, and multi-agent systems. He is a Fellow of IEEE, a Fellow of AAAS, a Fellow of IFAC, a Yangtze Scholar/Chair Professor of China, and a recipient of 1996 US National Science Foundation CAREER Award, 2004 SICE International Award, and 2006 Natural Science Foundation of China Outstanding Overseas Young Scholar Award. He served on a number of journal editorial boards, as an Associate Editor and a Guest Editor. He was also the founding Editor-inChief for Journal of Control Science and Engineering. He currently serves as an Associate Editor for SIAM Journal on Control and Optimization.
