Relationship on stabilizability of LTI systems by P and PI controllers.

Zhang, Zhiping ; Wang, Qing-Guo ; Zhang, Yong 等

INTRODUCTION

The Proportional-plus-Integral-plus-Derivative (PID) controllers have found wide acceptance and applications in industry for the past few decades. Over 90% of controllers used in chemical process industries are of the PID type (Ho and Edgar, 2004). Although the integral action may eliminate steady-state offset, it is believed to contribute negatively to the stability of the closed-loop systems due to the addition of one open-loop pole at the origin. Thus, control engineers seem to think that integral action is useless for stabilization and PI control cannot do better stabilization than P control. Most people believe that a system that cannot be stabilized by the P controller is not stabilizable by the PI controller. However, this belief is not fully tested or theoretically proven. In fact, a systematic answer to this stabilizability problem is lacking. Let us address it more rigorously, the question to ask is whether there is the equivalence between stabilizability by P and PI in general. In other words, can P stabilize all the systems that PI stabilizes, and conversely, can PI stabilize all the systems that P stabilizes? This paper aims to answer these questions and correct the common perception that the PI controller is poorer than the P controller in stabilization of the system. It is found that PI is no poorer than P in stabilization. PI can stabilize all the systems that P stabilizes but in general the converse is not true. The equivalence holds for all stable systems and for several types of low-order unstable systems. Non-equivalence examples are presented for complementary cases to equivalence ones. The proof for the high-order equivalent cases and search for non-equivalent examples are the most challenging and difficult part of our research. The results of our research can be seen in Table 1.

Please note that uncertainties of a system model in general do not affect the validity of the results in Table 1. The robustness or stability margin may be briefly discussed in two ways. Let us first consider gain margin. The change of gain causes no change to our results. It is readily seen by having kG(s) (k is a positive real number) in Table 1 instead of G(s), then all the conditions hold for any k. Another simple way to consider stability robustness is to keep some common distance, d, of all poles from the stability boundary, the imaginary axis. Let s = s'-d. We require the poles at s-plane to have their real parts less than -d < 0. This is equivalent to make the poles at s'-plane have their real parts less than 0. When s = s'-d is substituted to G(s) to transform s-plane to s'-plane, the derivations and the results of this paper are still applicable, since the numbers of zeros and poles do not change.

This paper is organized as follows. The next section gives the problem formation and preliminaries. The following three sections discuss the nonzero, one zero and two zero plants, respectively. The sixth section discusses other plants followed by the conclusion.

PROBLEM FORMULATION AND PRELIMINARIES

Consider a LTI system described by the transfer function,

G (s) = N (s)/D (s) (1)

where N(s) = [b.sub.m][s.sup.m] + [b.sub.m-1] [s.sup.m-1] + ... + [b.sub.1]s + [b.sub.0] and D(s) = [s.sup.n] + [a.sub.n-1][s.sup.n-1] + ... + [a.sub.1]s + [a.sub.0] are co-prime polynomials with n ??m. In this paper, we assume that G(s) has no zero at s = 0 to avoid any unstable zero-pole cancellation with a PI controller:

N(0) [not equal to] 0. (2)

This assumption is necessary to address a meaningful stabilizability comparison between P and PI control because otherwise PI control can never internally stabilize a system with a zero at the origin.

The system (1) is controlled in the conventional unity negative output feedback configuration, where the controller C(s) can be of P type:

[C.sub.P] (s) = K (3)

or of PI type:

[C.sub.PI] (s) = [K.sub.P] [K.sub.I]/s, [K.sub.I] [not equal to] 0 (4)

where [K.sub.I] [not equal to] 0 is imposed so that the latter always has nonzero integral action and P is never a special case of PI. This is to make two controllers exclusive of each other and thus, stabilizability equivalence study meaningful. Please note that K, [K.sub.P] and [K.sub.I] could be either negative or positive. The resulting closed-loop characteristic equation is

D(s) + KN(s) = 0 (5)

for P-control, and

sD(s) + ([K.sub.P]s + [K.sub.I])N(s) = 0 (6)

for PI-control.

The problem at hand is to find the class of the system, G(s), for which both Equations (5) and (6), can be made stable (having all the roots with negative real parts) by suitable choice of relevant parameters involved. If this is the case, G(s) is called stabilizability-equivalent by P and PI controllers. Thus, stabilizability-equivalent cases are the systems that both P and PI can stabilize or the conditions for the stabilizability by P and PI are the same. The non-equivalent cases are the systems which P cannot stabilize but PI can, or P can stabilize but PI cannot.

For a stable system, by the Root-Locus, it is always stabilizable by P-control as long as the gain K is sufficiently small. For PI-control, let [K.sub.P] = 0 so that it reduces to I-control. A stable system with (2) is also stabilizable by I-control as long as |[K.sub.I]| is sufficiently small and G(0)[K.sub.I] > 0. This establishes Lemma 1 below and leads us to consider the problem for unstable systems only.

Lemma 1. The class of stable systems is stabilizability-equivalent by P and PI controllers.

Lemma 2. If a system given by Equations (1) and (2) is stabilizable by a P controller, so is it by a PI controller.

Proof. If a system given by Equations (1) and (2) is stabilizable by a P controller, then there is some K such that the characteristic Equation (5) is stable. The closed-loop characteristic equation with PI, (6), can be rewritten as s [D(s) + KN(s)] + [([K.sub.P] - K) s + [K.sub.I]]N(s) = 0 or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This can be viewed as the closed-loop characteristic equation with [K.sub.I]/S controlling the plant:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which has a nonzero static gain due to Equation (2) and is stable as its denominator is the same as the left side of Equation (5). It follows from the Root-Locus technique that there is always a nonzero [K.sub.I] such that the closed-loop is stable, that is, there also exits a PI controller stabilizing G(s). This completes the proof.

Lemma 2 states that stabilizability by P implies stabilizability by PI. As a result, one only needs to address the converse case: when does stabilizability by PI imply stabilizability by P. Combined with Lemma 1, this side of problem on stabilizability equivalence for unstable systems will be discussed in terms of the number of zeros associated with the system in the subsequent sections.

PLANTS WITH NO ZERO

The equivalence of stabilizability holds for a plant of up to fourth-order with no zero. Because the proof on the plants of third or lower order are relatively simple, only the proof on the fourth-order plant is presented below for the sake of demonstration. For non-equivalent cases, one example of fifth-order is provided and explained.

Equivalent Case

The transfer function of fourth-order plant with no zero is given by

G (s) = [b.sub.0]/[s.sup.4] + [a.sub.3] [s.sup.3] + [a.sub.2][a.sup.2] + [a.sub.1]s + [a.sub.0], [b.sub.0] [not equal to] 0 (7)

The closed-loop characteristic equation with a P controller is [s.sub.4] + [a.sub.3][s.sup.3] + [a.sub.2][a.sup.2] + [a.sub.1]s + ([a.sub.0] + [Kb.sub.0]) = 0. It follows from the Routh Test that this characteristic equation is stable if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

The closed-loop characteristic equation with a PI controller is [s.sup.5] + [a.sub.3][s.sup.4] + [a.sub.2][a.sup.3] + [a.sub.1] [s.sup.2] + ([a.sub.0] + [K.sub.p][b.sub.0])s + [K.sub.I] [b.sub.0] = 0. Again, it follows from the Routh Test that this characteristic equation is stable if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

It is straightforward to see, by comparing Equation (9) with Equation (8), that Equation (9) will imply Equation (8) if one shows (iv) and (vi) of Equation (8) using Equation (9). Suppose that Equation (9) is true: let K = [K.sub.P] - [K.sub.I]/[a.sub.3]. It follows from (v), (vi) and (vii) of Equation (9) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

One then sees

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where the last equality is due to (viii) of Equation (9). Besides, from (vii) of Equation (9), one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Therefore, Equation (8) is true as well. PI stabilization guarantees P stabilization here. The equivalence of stabilizability holds.

Non-Equivalent Case

It is found that the equivalence of stabilizability by P and PI does not hold when the plant is of fifth-order with no zero. This is due to the increased elements of the Routh array of the closed-loop characteristic equation. With P, one coefficient in the first column of the Routh array contains the square of K. It is thus possible that this coefficient is always non-positive for some specific plants if other coefficients in the first column are kept positive. But with PI, this coefficient could be positive due to the presence of another variable, [K.sub.I] . One example, which cannot be stabilized by any P controller but can be stabilized by PI, is provided here.

Example 1. Let

G (s) = 1/[s.sup.5] + [2s.sub.4] + [2s.sup.3] + [2s.sup.2] + s + 0.5

With a P controller, the closed-loop characteristic equation is [s.sup.5] + [2s.sub.4] + [2s.sup.3] + [3s.sup.2] + s + (0.5 + K) = 0. In order for it to be stable, the Routh Test requires that 2K and - [(K - 0.5).sup.2]/2K are positive simultaneously. Since no K exists such that these two terms are positive simultaneously, P controller cannot stabilize this system. On the other hand, a PI controller

[C.sub.PI] (s) = 0.5 + 0.1/s

is found to be able to stabilize this system. The closed-loop characteristic equation is [s.sup.6] + [2s.sub.5] + [2s.sup.4] + [3s.sup.3] + [s.sup.2] + s + 0.1 = 0. The poles are located at s = -0.0478 [+ or -] 1:0145j, s = -0.0229 [+ or -] 0.7153j, s = -1.7505 and s = -0.1081, which are all stable.

PLANTS WITH ONE ZERO

The equivalence of stabilizability holds for a plant of up to thirdorder with one zero. Again, because the proof on the plants of second- or first-order are relatively simple, only the proof on the third-order plant is mentioned briefly below for the sake of demonstration. For non-equivalent cases, one example of fourthorder is provided and explained.

Equivalent Case

The transfer function of third-order plant with one zero is given by

G (s) = [b.sub.1]s + [b.sub.0]/[s.sup.3] + [a.sub.2][s.sup.2] + [a.sub.1]s + [a.sb.0], [b.sub.1] [not equal to] 0 (10)

The closed-loop characteristic equation with a P controller is [s.sup.3] + [a.sub.2][s.sup.2] + ([a.sub.1] + [Kb.sub.1]) + ([a.sub.0] + [Kb.sub.0]) = 0. The closed-loop characteristic equation with a PI controller is [s.sup.4] + [a.sub.2][s.sup.3] + ([a.sub.1] + [K.sub.P][b.sub.1]) [s.sup.2] + ([a.sub.0] + [K.sub.P][b.sub.0] + [K.sub.I][b.sub.1]) s + [K.sub.I][b.sub.0] = 0. Following the same logic in first section, we prove that PI stabilization ensures P stabilization here. The equivalence holds.

Non-Equivalent Case

The equivalence of stabilizability by P and PI does not hold when the system is of fourth-order. The reason is similar to the case of the fifth-order with no zero. One example, which cannot be stabilized by any P controller but can be stabilized by PI, is provided here.

Example 2. Let

[G.sub.2] (s) = s - 1/[s.sup.4] + [s.sup.3] + [3s.sup.2] + s + 3

With a P controller, the closed-loop characteristic equation is [s.sup.4] + [s.sup.3] + [3s.sup.2] + (K + 1) s+(3 - K) = 0. In order for it to be stable, the Routh Test requires that 2 - K and -[(K - 1).sup.2]/2 - K are positive simultaneously. Since no K exists such that these two terms are positive simultaneously, P controller cannot stabilize this system. On the other hand, a PI controller

[C.sub.PI] (s) = 1 - 0.5/s

is found to be able to stabilize this system. The closed-loop characteristic equation is [s.sup.5] + [s.sup.4] + [3s.sup.3] + [2s.sup.2] + 1.5s + 0.5 = 0. The poles are located at s = -0.1267 [+ or -] 1.4562j, s = -0.1561 [+ or -] 0.7172j, and s = -0.4344, which are all stable.

PLANTS WITH TWO ZEROS

It is found that the equivalence of stabilizabilities by P and PI does not hold in general for a plant with two or more zeros. Based on the Routh Stability criterion, all the parameters in the closed-loop characteristic equation should be positive in order to have stability of the closed-loop system. When the term of s does not exist in both the denominator and numerator of the openloop system transfer function, there would be no way for P to stabilize the system but it is possible for PI to stabilize, since PI has an s term that P does not. One example of second-order is provided here.

Example 3. Let

[G.sub.3] (s) = [s.sup.2] + 1/[s.sup.2] + 2

With a P controller, the closed-loop characteristic equation is (K + 1) [s.sup.2] + (2K + 1) = 0. This equation lacks the term of s, so it always has roots located in the right-half plane or at the imaginary-axis regardless of what K is chosen. P controller cannot stabilize the system. However, a PI controller, such as,

[C.sub.PI] (s) = 1 + 1/s

can stabilize the system. Its closed-loop characteristic equation is [2s.sup.3] + [s.sup.2] + 3s + 1 = 0. The roots of this equation are s = -0.0772 [+ or -] 1.2003j and s = -0.3456, all located in the left-half plane.

OTHER PLANTS

In this section, we discuss plants other than what have been studied for equivalence of stabilizability by P and PI. Plants of Higher Order and With More Zeros Consider Example 3 again with the plant being cascaded with [([beta]s + 1).sup.t]/[([alpha]s + 1).sup.m], where [alpha] and [beta] are some small positive numbers, with l and m as positive integers. The Nyquest curve of such a new open-loop for either P or PI case can be made as close as possible to the counterpart of the original loop and thus causes no change of encirclements with the critical point. Therefore, the conclusion drawn in Example 3 holds for the plant of higher order or with more zeros, that is, PI may stabilize but P cannot. The equivalence of stabilizability by P and PI fails in such cases as well.

Example 4. Let

[G.sub.4] (s) = [s.sup.2] + 1/[s.sup.2] + 2 (0.001s + 1/0.002s + 1)

With a P controller, the closed-loop characteristic equation is (0.002 + 0.001K) [s.sup.3] + (1 + K) [s.sup.2] + (0.004 + 0.001K) s + (2 + K) = 0. According to the Root-Loci of positive and negative gains, this equation always has some of its roots located in the right-half plane or at the imaginary-axis for any value of K. But a PI controller, such as,

[C.sub.PI] (s) = 1 + 1/s

can stabilize the system, since its closed-loop characteristic equation, 0.003[s.sup.4] + 2.001[s.sup.3] +1.005 [s.sup.2] + 3.001s + 1 = 0, has the roots at s = -666.4996, s = -0.0773 [+ or -] 1.2003j and s = -0.3457, all in the left-half plane.

PLANTS WITH TIME DELAY

For the sake of completeness, let us address the problem for time-delay plants (Gu et al., 2003). In Lu (2006), five types of unstable time-delay plants of up to second-order with no zero are studied and the stabilizability results are displayed in Table 2. All the first-order unstable plants are studied there and all the cases of up to second-order except the plants with two unstable poles, which neither P nor PI can stabilize, are also considered. From their results and our Lemma 2 before, one therefore concludes that the equivalence of stabilizability by P and PI holds for time-delay plants of first-order and second-order with no zero.

CONCLUSION

In this paper, the relationship on stabilizability of LTI systems by P and PI controllers is studied. It is shown that the equivalence of stabilizability by P and PI holds for the classes of: (1) stable plants; (2) plants of up to fourth-order with no zero; and (3) plants of up to third-order with one zero. The equivalence fails for other classes of plants in general and non-equivalent examples are provided.

Manuscript received September 7, 2006; revised manuscript received October 30, 2006; accepted for publication March 15, 2007.

REFERENCES

Gu, K., V. Kharitonov and J. Chen, "Stability of Time-Delay Systems," Birkhauser, Boston (2003).

Ho, B. and T. F. Edgar, "PID Control Performance Assessment: The Single-Loop Case," AIChE J. 50(6), 1211-1218 (2004).

Lu, X., "New Design for Control Systems," PhD Thesis, National University of Singapore (2006).

Zhiping Zhang (1), Qing-Guo Wang (1)* and Yong Zhang (2)

(1.) Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260, Singapore

(2.) GE Water and Process Technologies, 1800 Cailun Road, Zhangjiang High-tech Park, Pudong Shanghai 201203, China

* Author to whom correspondence may be addressed.

E-mail address: [email protected]

Table 1. Stabilizability equivalence by P and PI controllers

G (s) =
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Equivalence

stable n < [infinity]
unstable with no zero: [b.sub.m] = [b.sub.m-1], = n [less than or
... [b.sub.1] = 0 equal to] 4
unstable with one zero: [b.sub.m] = [b.sub.m-1], = n [less than or
... [b.sub.2] = 0, [b.sub.1] [not equal to] 0 equal to] 3
unstable with two zeros: [b.sub.2] [not equal to] 0 no

Table 2. Summary of some stabilizability results in Lu (2006)

Plant model P PI

1/s [e.sup.-Ls] [for all]L > 0 [for all]L > 0

1/s (s + 1) [e.sup.-Ls] [for all]L > 0 [for all]L > 0

1/s - 1 [e.sup.-Ls] L < 1 L < 1

1/s (s - 1) [e.sup.-Ls] none none

1/(s - 1)(Ts + 1) [e.sup.-Ls] L < 1 - T L < 1 - T