1468 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 9, SEPTEMBER 2005

Comments on “New Results on the Synthesisof PID Controllers”

Gianpasquale Martelli

Abstract—Silva, Datta, and Bhattacharyya (in the above paper) havestudied the feedback control system, provided with a proportional-inte-gral-derivative (PID) controller and relative to a first-order plant with atime delay, and have determined the range of the PID controller parame-ters suitable for the stability by means of a version of the Hermite–Biehlertheorem. The purpose of this note is to find the same results in a simplerway by means of the classical Nyquist’s stability criterion.

Index Terms—Closed-loop systems, delay systems, proportional-inte-gral-derivative (PID) control, reduced-order systems, stability.

The stability zones in the coordinates system (kd; ki) for a suitablevalue of kp are quadrilateral defined in [1] as follows.

1) Fig. 3, ([1]): T > 0; ki > 0;�1 < kkp < kku

�r(z1) < 0 and �r(z2) > 0

andkkdT

< 1 and ki > 0; (1)

2) Fig. 4, ([1]): T < �0:5L; ki < 0; kkl < kkp < �1

�r(z1) > 0 and �r(z2) < 0

andkkdT

< 1 and ki < 0; (2)

where �r(z) and �i(z), which are the real and the imaginary compo-nents of the characteristic equation multiplied by esL and calculatedfor s = jz=L, are given by

�r(z) = kki � kkdz2

L2�

z

Lsin(z)� T

z2

L2cos(z) (3)

�i(z) =z

Lkkp + cos(z)� T

z

Lsin(z) (4)

and z1 and z2 are the first two positive roots of �i(z) = 0.The open-loop transfer function Fol(s) and its real component

Rol(z), imaginary component Iol(z), modulusMol(z), calculated fors = jz=L, are given by

Fol(s) =ke�Ls(ki + kps+ kds

2)

s+ Ts2

Rol(z) = k(A cos(z) +B sin(z)) (5)

Iol(z) = k(B cos(z)� A sin(z)) (6)

Mol(z) = kk2iL

4 + k2p � 2kikd L2z2 + k2dz4 1=2

(L2z2 + T 2z4)1=2(7)

where

A =T (kdz

2� kiL

2) + kpL2

L2 + T 2z2

B =L

z

kdz2� kiL

2� kpTz

2

L2 + T 2z2:

Manuscript received June 28, 2004; revised October 20, 2004 and April 19,2005. Recommended by Associate Editor Hong Wang.

The author is at Via Domenico de Vespolate 8, 28079Vespolate, Italy (e-mail:gianpasqualemartelli@libero.it).

Digital Object Identifier 10.1109/TAC.2005.854644

Before applying Nyquist’s stability criterion, it is necessary to makethe following two observations, which are the fundamental elements ofthis note.

1) Remark 1Differentiating (7) with respect to z, we obtain

dMol(z)

dz=

k2L2

Mol(z)

�m0 � 2m2z2 +m4z

4

(L2 + T 2z2)2z3(8)

where

m0 = k2iL4

m2 = k2i T2L2

m4 = k2d � k2p � 2kikd T 2:

Since (8) has none real root form4 < 0 and one positive rootzm form4 > 0 it can be stated that, when z increases fromzero to infinity, Mol(z) decreases always if m4 < 0 or, incontrary case, decreases for z lower than zm and increases,for z higher than zm, up toMol(+1), given by

Mol(+1) =k kdT

: (9)

Therefore the point of the first part of the Nyquist diagram,lying on the negative real axis and farthest from the coor-dinates origin, is the first intersection or the last one (z =+1). Moreover, Mol(+1) < 1 is one of the conditions,indicated in (1) and (2), which define the stability zones.

2) Remark 2Making equal to zero the imaginary component Iol(z) weobtain from (6)

Im1(z) = Im2(z) (10)

where

Im1(z) = kdz � kiL2

z(11)

Im2(z) = kpLTz cos(z) + L sin(z)

L cos(z)� Tz sin(z): (12)

Replacing in (5) the second member of (11) with the secondmember of (12), the coordinate R0(z) of the points lying onthe real axis can be expressed as

R0(z) =kkpL

L cos(z)� Tz sin(z): (13)

It is easy to verify that R0(z) = �1 and �i(z) = 0 arethe same comparing (13) with (4) and also that (10) maybe obtained eliminating k from �r(z) = 0 and �i(z) = 0;therefore, the system of the equations �r(z) = 0 and �i(z) =0 is equivalent to Im1(z) = Im2(z) and R0(z) = �1 one.Moreover, the lower and the upper bounds of kp, named in [1]kl (T < 0) and ku (T > 0) are, respectively, the minimumand the maximum of kp, deduced from R0(z) = �1, in theinterval of z (0; �).

Now, we can plot the Nyquist diagrams (see Fig. 1 for T > 0and Fig. 2 for T < 0) and the curves relative to (13) and to bothmembers of (10) (see Fig. 3 for T > 0 and kp > 0 and Fig. 4 forT < �L). We consider only the positive frequencies (j! > 0) inthe contour Q; the plot of Fol(s) for negative frequencies is obvi-ously symmetrical to the plot for positive frequencies, with the realaxis as the axis of simmetry. The first part of the Nyquist diagram,from z = 0 to z = +1, is a line, starting from the point Rol(0) =

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 9, SEPTEMBER 2005 1469

Fig. 1. Nyquist plot for T > 0. (a) Contour Q. (b) F (s).

Fig. 2. Nyquist plot for T < 0. (a) Contour Q. (b) F (s).

k(kp � (T + L)ki); Iol(0) = 1 sgn(�Lkki) and spiralling towardthe circle of radius equal toMol(+1) = jkkd=T j.

Considering Nyquist’s stability criterion (see, for example, [2, para.10.3.6, p. 181]) we note the following.

1) Open-loop stable plants (T > 0 and ki > 0): Figs. 1 and 3Since there are no poles of Fol(s) in the right-half plane, forthe stability it is necessary to have none rotation of Fol(s)about �1 + j0 point i.e., R0(z) > �1 for all intersectionswith the negative real axis. For kp > 0 since, according toRemark 1, the farthest of these intersections from the coor-dinates origin is the first one (z = za) or the last one (z =+1), the inequalities z1 < za < z2 and Mol(+1) < 1must be satisfied (see Fig. 3, where z1 and z2 are indicatedas the coordinates of the crossing points R0 = �1 and zaof the crossing point Im1 = Im2). For kp < 0 the plotsof Im2(z) and R0(z) are symmetrical to the shown ones inFig. 3, with the z axis as the axis of symmetry, and the in-equalities za < z1 and z2 < zb replace the previous ones(zb is the second intersection with the negative real axis). Inboth cases, the stability zone is therefore defined by

Im1(z1) > Im2(z1) and Im1(z2) < Im2(z2)

and

Mol(+1) < 1 and ki > 0: (14)

2) Open-loop unstable plants (T < 0 and ki < 0): Figs. 2 and4Since there is one pole of Fol(s) in the right-half plane, forthe stability it is necessary to have one counterclockwise ro-tation of Fol(s) about �1 + j0 point, i.e., R0(z) < �1 forthe first intersection with the negative real axis (z = za) and

Fig. 3. Plots of I ; I , and R for T > 0 and k > 0.

Fig. 4. Plots of I ; I , and R for T < �L.

R0(z) > �1 for all the remaining ones. Since, accordingto Remark 1, the farthest of these remaining intersectionsfrom the coordinates origin is the second (z = zb) or thelast one (z = +1), the inequalities za < z1; zb < z2,and Mol(+1) < 1 must be satisfied (see Fig. 4, where z1and z2 are indicated as the coordinates of the crossing pointsR0 = �1 and za and zb of the crossing points Im1 = Im2).For T > �L the first part of the plot of Im2(z) is alwaysdecreasing, but the proof previously enunciated for T < �Lis fully applicable. The stability zone is therefore defined by

Im1(z1) < Im2(z1) and Im1(z2) > Im2(z2)

and

Mol(+1) < 1 and ki < 0: (15)

Finally since, according to Remark 2, (14) coincides with (1) and(15) with (2), we conclude that the results of [1] have been obtained ina simpler way.

REFERENCES

[1] G. J. Silva, A. Datta, and S. P. Bhattacharyya, “New results on the syn-thesis of PID controllers,” IEEE Trans. Autom. Control, vol. 47, no. 2,pp. 241–252, Feb. 2002.

[2] The Control Handbook,W. S. Levine, Ed., IEEE Press, NewYork, 1996.