PIProceedings of the American Control Conference Anchorage, AK May 8-3 0,2002

PID Tuning Revisited: Guaranteed Stability and Non-Fragility
Guillermo J. Silva IBM Server Group

11400 Burnet Road Austin, TX 78758 U.S.A. guilsilv@us.ibm.com

Aniruddha Datta and S.P. Bhattacharyya Department of Electrical Engineering Texas A & M University College Station, TX 77843-3128 U.S.A. datta@ee.tamu.edu bhatt@ee.tamu.edu

Abstract In this paper we present a study of some PID tuning techniques that are based on first-order models with time delays. Using the characterization of all stabilizing PID controllers derived in [6], each tuning rule is analyzed t o first determine if the proportional gain value dictated by that rule, lies inside the range of admissible proportional gains. Then, the integral and derivative gain values are examined t o determine conditions under which the tuning rule exhibits resilience with respect t o controller parameter perturbations. Furthermore, a new tuning rule is presented for resilient controller design.

ate in tbe sense of providing PID controller parameters that are resilient in the space of the controller coefficients. A controller for which the closed-loop system is destabilized by small perturbations in the controller coefficients is said t o be fmgile [4]. Any controller that is t o be practically implemented must necessarily be non-fragile or Tesilienl so that: (1) round-off errors during implementation d o not destabilize the closed-loop; and (2) tuning of the parameters about the nominal design values is allowed. Three tuning techniques will be discussed: (1) the classical Ziegler-Nichols step response method; (2) the Cohen-Coon method; and (3) the Internal Model Control design technique. In each case, we will first study if the proposed proportional gain value lies inside the allowable range determined in (61. We will then examine for this fixed proportional gain, the location of the integral and derivative gain values inside the stability region described in (61. This procedure will allow us t o determine conditions under which each tuning technique provides a good l 2 parametric stability margin in the space of the controller coefficients. In this way, we will avoid undesirable scenarios such as PID controller parameters that are dangerously close to instability. This pa.per also presents a new PID tuning rule for resilient controller design. This tuning rule is based on determining first the ultimate gain and the ultimate frequency of the time-delay system. Next, these two parameters are used t o derived a first order model with deadtime of the system. Using the results in 161, the set of stabilizing PID controllers can then be obtained and used to enhance the robustness of the closed loop system t o perturbations in the controller coefficients.

1 Introduction
The PID controller is by far the most commonly used algorithm in process control applications (I]. The Japan Electric Measuring Instrument Manufacturers' Association conducted asurvey of the 7. state of process control systems in 1989 1 1 According t o that survey more than 90% of the control loops were of the PID'type. The popularity of the PID controller can be attributed t o its three distinct features: (i) it provides feedback; (ii) it has the ability t o eliminate steady-state offsets through integral action; and (iii) it can anticipate the future through derivative action. Over the last forty years, numerous methods have been developed for setting the parameters of the PID controller. Some of these methods are based on characterizing the dynamic response of the plant t o be controlled with a first-order model with time delay. Traditionally, this model is obtained by applying a step input to the plant and measuring a t the output the following three parameters: the steady-state gain, the time constant, and the time delay. Although many of these tuning techniques work in practice, not much is known about the robustness or stability of these algorithms beyond what has been observed in empirical studies. Perhaps, the only exception is the Internal Model Control (IMC) algorithm where the stability constraint is built into the PID design method. Recent results on PID stabilization obtained in (61, however, make it possible to revisit these classical tuning rules and t o justify them in terms of stability and robustness. One of the main objectives of this paper is t o do precisely that. In this paper, we will analyze several PID tuning techniques that are based on first-order models with time delay. This analysis will attempt t o describe when each tuning technique is appropri'This work was supported in part by the National Science Foundation under Grant No. ECS-9903488, in part by the Texas Advanced Tecknology Program under Grant No. 000512-00991999, and in part by the National Cancer Institute under Grant No. CA90301.

2 Preliminary Results

The tuning techniques analyzed in this paper are based on characterizing the plant t o be controlled by the following transfer function k e-Ls G ( s )= (1)

1+Ts

where k is the steady-state gain, L is the apparent time delay, and T is the apparent time constant. We will consider the feedback control system shown in Fig. 1 where T is the command signal, y is the output of the plant, G ( s ) given by (1) is the plant t o be controlled, and C ( s ) is the controller. We focus on the case when the controller is of the PID type, Le., the controller has a proportional term, an integral term and a derivative term. There are different ways of representing the PID control algorithm. In our case, we will use the following representation:

C ( S ) kp + - + kds =
S

ki

where k, is the proportional gain, ki is the integral gain and kd is the derivative gain.

0-7803-7298-0/02/$17.00 0 2002 AACC

5000

3 The Ziegler-Nichols Step Response Method
CONTROLLER
PLAN

A simple way t o determine the parameters of a PID controller based on step response data was developed by Ziegler and Nichols in 1942 (81. This method first characterizes the plant by the parameters L and a , where the parameter a is defined as a =k-

Fig. 1: Feedback control system.
The following theorem [6] provides an analytical characterization of the set o f controller parameters (k,,ki,kd) for which the closedloop system in Fig. 1 is stable.

L T

Once these parameters are determined, the PID controller parameters are then given in terms of L and a by the following formulas:

-

1.2

-- 0.6 k. aL -d = -0.6L li . a (5)

Theorem 2.1 The range of k, values for which a given openloop stable plant, with transfer function G(s) as in (l), continues to have closed-loop stability with a PID controller in the loop is given by

-z1 < k, < I, 1''- sin(a1) - cos(ai)] p
T

This tuning rule was developed by empirical simulations of many different systems and is only applicable t o open-loop stable plants. We now define the parameter r as the ratio of the a p parent time delay and the apparent time constant of the plant, i.e.,

(2)

where a1 is the solution of the equation t a n ( a ) = -a (3) T + L in the interval (0, rr). For k , values outside this range, there are no stabilizing PID controllers. The complete stabilizing region is given by: (see Fig. 2)

L r=-. T
First, we consider the proportional gain value given in ( 5 ) and rewrite it as a function of r: - 1.2 kP -

kr

Since - k >

1 . For each k , E the crosslsection of the stabilizing region in the ( k ; ,k d ) space is the trapezoid T. 2. For k , = the cross-section of the stabilizing region in the ( k , , k d ) space is the triangle A.

(-2, i),

0 and r > 0 (the plant is open-loop stable), then k, > 0. From Theorem 2.1, we can rewrite the upper bound on k p as a function of the parameter r:

i,

k,, where a1

=

[:a, sin(a1) - cos(a1)

1

(7)

is now the solution of the equation

3. For each k , E ( ~ , k , , , := [!alsin(al)-cos(al)]), the cross-section of the stabilizang region in the ( k i , kd) space is the quadrilateral Q.

1 tan(a) = - a 1+r in -the interval (0,n).We now compare 5 and kupp by plotting

k,k and k,,k as functions of the parameter r. As can be seen from Fig. 3, the proportional gain value given by the ZieglerNichols step response method is always less than the upper bound kUPP

-05

I

0

.

05

.

1

.

15,=,

.

2

.

25

.

3

.

35

.

i

2

90
2

0 5

I

0

*

05

.

1

.

15

I

2

,

25

I

3

I

35

I

4

l

Fig. 2: The k, < i ; (b) k,

stabilizing region of (ki,kd) for: (a)

=

i ;( c ) 2 < k ,

< k,,,.

-i <
Fig. 3:

In Fig. 2, the parameters as follows: A L2 7 % = 27 bj mj, b j ,

and w j , for j = 1 , 2 , are defined

Comparison of given by the Ziegler-Nichols Method and the upper bound /cup,.

5

2 -k [sin(zj) + -zjcos(zj) L krj w3

e kk, T 2 [sin(sj) + -zj(cos(z3) + 1) kL "

1

(4)

L

where

21, 2 2 , z2

> z1 are the solutions of

I
5001

Thus, this tuning technique always p r o v i d s a feasible proportional gain value k,. We now set k, = k, and consider two cases, requiring different treatment according to the results presented in Section 2. MoEover,for clarity of presentation, let us rewrite the parameters k, and kd in ( 5 ) as:

+ CO+)

T . - -zszn(z) = 0 L

in the interval (0, x ) .

Case 1: T 2 1.2. In this case, we have 0 < 5 Then the stabilizing set is given either by Fig. 2(a) or by Fig. 2(b). is always less than as Notice from (9) t h a t the parameter illustrated in Fig. 4. T h e derivative gain value provided by the

&

i.

parameter & is less than b2 for all rewritten ils follows

T

< 1.2. From (4), bz can be

5 t 1
T

where 22 >; z1 > 0 is the solution of 1.2 T COS(.) - z sin(z) = 0 in the interval (0, T T ) . By sweeping T in the range (0,1.2), it can < b z . Fig. 6 shows the location of & with be shown that respect t o the stabilizing set in the space of (k,,kd).

+

'

T

-1 k .

Fig. 4:

Location of the parameters

(G&)when T 2 1.2.
I

I w2 Ziegler-Nichols method is robust in the sense that it is not close Following the same principle; we to the stability boundary would like to guarantee that the integral gain value is also far away from the stability boundary. k t 2 1 be the ki-coordinate of the point where the h e kd = kd intersects the line k d = mlk, bl. F r o E F i g . 4 , we now find the conditions under which the parameter ki lies in the range (O.221,0.821). Following the same derivation used in (61, 51 can be expressed as follows

E.

Ki

'1

Fig. 6:

Location of the p a r a m e t e r s

(G,z)when 0 < i- <

1.2.
We will now analyze for which values of T, the parameter lies inside the range (0.2z1,0.8~1). As in Case 1, we can plot the versus T. This graph is terms 0 . 2 e 2 1 , 0.8qz1,and &own in d g . 7 for 0 < T < 1.2. From this graph we see that kt lies in the range (O.221,0.821) for 0 < i- < 1.07. For the lies in the range (O.lzl,O.8z1), we relaxed condition where : have 0 e T < 1.2.

+

z1= --z

T k L2

1

[Tsin(zl)

+ zi (cos(z1) + 0.6)]
=
0 [using (6)]

(10)

qF

where z1 is the solution of 1.2

+ TCOS(Z) z sin(z) -

E

in the interval (0, n).
From (8) and ( l o ) , we can plot the terms q z 1 , and versus T. This graph is shown in Fig. 5 for T 2 1.2. As can be

q G

5.
45-

05

2 .

'0

02

04

06

08

1 1.07 1 2

14

Fig. '7: Comparison of 0 . 2 F 2 1 , O.8$21, 0 < 7 < 1.2.
Fig. 5:
Comparison of 0 . 2 q 2 1 , 0 . 8 q z 1 , a n d $%for
7

and

$ for z

2 1.2.

From the previous analysis, we conclude that the Ziegler-Nichols step response method gives a resilient PID controller for 0 < T < 1.07. Resilience is here understood as good parametric stability margin in the space of ( k i , kd).

seen from this graph, does not lie in the range (0.2~1,0.8z1) for anyvalue of T. If we relax our robustness condition and now make ki lie inside the range (O.lz1,0.8~1), see from Fig. 5 that we will this occurs for 1.2 _< T < 3. In this way, for 1.2 5 T < 3, be located 10% of 5 1 away from the kd-axis which corresponds t o a good 12 parametric stability margin. Case 2: 0 < T < 1.2. In this case, we have $ < < kuPp. The stabilizing set is given by Fig. 2(c). We now show that the

G

Ziegler-Nichols d e is applicable if 0.1 < T < 0.6. In this range, the derivative action often gives significant improvement of performance. Comparing this range with the one previously obtained for resilience, we see that the former is included in the latter. Thus, forO.l < T < 0.6, the Ziegler-Nichols step response method not only gives good performance but also is resilient with respect to controller parameter perturbations.

Remark 3.1 It has been determined empirically 111 that the

5002

4 The Cohen-Coon Method
Early papers on PID control introduced a tuning technique called dominant pole design. This technique attempts to position a few closed-loop poles to achieve certain control performance specifi2 cations. T h e Cohen-Coon method 11 is a dominant pole design method based on the first-order model with time delay (1). It attempts t o locate three closed-loop poles, a pair of complex poles and one real pole, such that the amplitude decay ratio for load disturbance response is and the integral error JOm e ( t ) d t is minimized. Based on analytical and numerical computations, Cohen and Coon derived some formulas for the PID controller parameters in terms of the plant parameters k, T , and L . These formulas'are:

9

.

,

,

,

a

3 -

'ita34

i

i

6

6

i

ir

a&

b

IO

k,

= =

1.35 (I+- 0.18b) aL 1- b
E(1 + a Fig. 8: Comparison of 0 . 2 q x 1 , O . 8 q 2 1 , and
( 12.5-22b ___ - 0.396)
(0'37 -0'37b) 1 - 0.81b

q K for

kd where E) b=- (12) (13)

r >_ 1.7834.
Case 2:

T

< 1.7834. In this case

<

< kuPp and the stabi-

kL a=T

L L+T'

lizing - set is given by Fig. 2 ( c ) . First, we compare the parameter kd given by (16) with the parameter b2 given by

We start our robustness analysis by rewriting the parameter as a function of r =

+:

where

22

> z1 > 0 is the solution of
1.35(1+ 0 . 1 8 ~ ) rcos(z) - zsin(r) = 0

-

k, =

1.35

(;1 + 0.18)

+

If we now consider k > 0 and an open-loop stable plant (r > 0), then > 0. As in the previous section, we can plot and the upper bound k,,, in (7) as a function of 7. These plots are similar to the ones shown in Fig-3 and are thus omitted here. I t is clear from this figure that k, < k,,, for all values of 7 . Thus, the Cohen-Coon Method always provides a proportional gain value inside the stabilizing range presented in Theorem 2.1. We - now analyze the location of the controller parameters h and

in the interval (0, T ) . By ploting < 4 for all 0 < 7 be shown that

G

+G and

$b2

versus

T,

it can

< 1.7834.

Next, w e study the location of the parameter to achieve some robustness. Fig. 9 shows the plot of 0.2qz1,O . 8 q s 1 , and versus r. It can be shown using this figure that the parameter lies inside the interval (0.221,0.821) for all 0 < r < 1.7834.
35

,

,

,

,

,

,

,

,

kd in the space of (ki,k d ) . From (12)- (13), we can express these parameters in terms of T as follows

k,

=

T - b . 5 4 (1 + 0.18r)(l + 0 . 6 1 ~ ) k L2
(1

+ 0.27)

We now consider two different scenarios.

Case 1: r 2 1.7834. In this case, we have 0 < 5 5 so the stabilizing set in the space of ( k i , k d ) is given by Figs. 2(a) or < . 2(b). Since 7 > 0 then 1 0 . 1 8 ~ 1 0 . 1 9 ~ Thus, from (16) it < 0.4995:. This implies that is not difficult to see that 0 < the parameter & given by the Cohen-Coon method lies inside the stabilizing range of derivative gain values for T 2 1.7834 (see Figs. 2(a) or 2(b)).

i,

+

+

Fig. 9: Comparison of 0 . 2 q x 1 , O.8$x1, 0 < -r < 1.7834.

and

!zfor $

To study the robustness of the tuning technique, we now compare the parameter ki in (15) with 2 1 defined in Fig. 4. This parameter is given in this case by

Thus, we conclude from Cases 1 and 2 that the Cohen-Coon method gives resilient PID parameters in the sense of the parametric stability margin when the plant under study satisfies the < 8.53. property 0 <

4

where z1 is the solution of 1.35(1

+ 0.187) + TCOS(Z) zsin(z) = 0 -

in the interval (0, T ) . We now find the values of the parameter T for which lies inside the range (0.211,0.821). Fig. 8 shows the as a function of T . From plot of 0 . 2 q 2 1 , O . 8 q z 1 , and - figure we conclude that for 1.7834 5 -r < 8.53 the parameter this k, lies inside the range ( 0 . 2 z 1 , 0 . 8 ~ 1 ) .

5 The Internal Model Controller Design Technique
The internal model controller (IMC) structure has become a popular one in process control applications. In this structure the controller includes an explicit model of the plant. It is particularly appropriate for the design and implementation of controllers

5003

for open-loop stable systems. The fact that many of the plants encountered in process control happen to be open-loop stable possibly accounts for the popularity of IMC among practicing engineers. The IMC principle is a general method that can be applied to the design of PID controllers. The plant t o be controlled is modeled as the first-order system with time delay given by ( l ) , and the dead-time is approximated by a first-order Pad6 approximation, Following the standard IMC procedure 151, the following parameters are obtained for a PID controller:

Case 1: T 2 -. Thus, 0 < 5 and the stabilizing region in the ( k i ,k d ) space is given by Figs. 2(a) or 2(b). Since A/L > 0, it is not difficult t o show from ( 2 2 ) that

i

for any X / L > 0. k Thus, the parameter provided by the IMC design technique lies, in this case, inside the stabilizing set of derivative gain values as illustrated in Fig. 4. 0

<

< 0.5-

T

6

We now t.urn our attentionto the parameter g. We want t o find the values of T for which k , lies inside the range ( 0 . 2 z 1 , 0 . 8 ~ 1 ) , where 11 is given by

kd

=

where X > 0 is a small number. By properly selecting the design variable A, the resultant PID controller can achieve a good compromise between performance and robustness. As discussed in [SI, a suitable choice for X is X > 0.2T and X > 0.25L. We start our robustness analysis by studying the parameter From (17), can be rewriten in terms of r = as follows

in the interval (0, n . By fixing the parameter X / L at the values ) 0 . 1 , 0 . 2 5 , 0 . 5 , 1 ,we can plot kL&, O.2kLz1, and O.8kLzl versus T . Fig. 11 shows these plots. From these figures it is clear that
(a)UL=Ol

5

5.

Since the plant is open-loop s a b l e , then r > 0. Moreover, we know that X / L > 0. Thus, k p > 0. Now, for different values of the parameter AIL, we plot & and k,,, given by ( 7 ) versus T . Fig. 10 shows these plots for X / L = 0 , 0 . 2 5 , 0 . 5 , 1 . I t is clear from this figure that for these values of AIL, the parameter given by the IMC design technique is inside the allowable range of stabilizing proportional gain values. Now, for a fixed value of

5

445i
35

5 ,

Fig. 11:
,

Comparison of 0 . 2 k L z 1 , O . 8 k L z 1 , a n d

: :

7

1+2X/L'

2

...

..

ki lies inside the suggested range for all X/LC[O 1 , 1 ] .

In this case, < k , < k,,, and the stabilizing region in the ( k i , k d ) space is given by Fig. 2 ( c ) . We first analyze the parameter and compare it with bz given by
Case 2: 0

< T < *.

Fig. 10: Comparison of 5 given by the IMC design nique and the upper bound kupp.

tec-

where z~

> z1

is the solution of 2t-7 TCOS(I) 2(1+ AIL)

T , we see from ( 2 0 ) that is a monotonically decreasing function of X / L . We have also seen from Fig. 10 that for X / L = 0, the is less than k,,, for all T > 0. Thus, for any corresponding is less than k,,, for value of X / L > 0, the corresponding all T > 0. We conclude that for any X/L > 0, the IMC design technique provides a stabilizing proportional gain value.

5

in the interval (0,n). By ploting and $ b versus r for different values of the parameter AIL, it can be shown that < b for all 0 < r < and ail X/L > 0.

Now, - we study the parameter

Next, we express the parameters and ( 1 9 ) respectively, a a function of AIL: s

given by ( 1 8 ) and

and find the values of r for which ki lies inside the range ( 0 . 2 1 1 ~ O . 8 s l )Fig. 12 shows the plots of . O.SkL:rl, O.8kL11, and k L 6 versus T for X / L = 0 . 1 , 0 . 2 5 , 0 . 5 , 1 . From theseplots we can find the range of T values for which the parameter ki lies inside the interval (O.211,0.821).For example, for A/L = 0.25, the range of T is (0.37,1.3333). Thus, from the previous analysis we conclude that the robustness but of the controller parameters depends not only on the ratio also on the parameter X / L . The following values of T guarantee a good parametric stability margin:

As in the previous sections, we now consider two different cases.

5004

+

TL ___ 2 k ( L A)

and

z1

+

is the solution o f 24-T 2(1 X / L )

+

T

COS(.)

- z sin(.)

=0

LLZ

for

2
T

2 &,

where

i -

+

- rsin(t) = 0

+z

&

6

-

4

where a is the amplitude of the sinusoidal input t o the relay nonlinearity and d is the height of the relay. T h e conditions for the presence of limit cycle oscillations can be derived by investigating the propagation of a sinusoidal signal around the loop. Since the plant G(s) acts as a low pass filter, the higher harmonics produced by the nonlinear relay will b e attenuated a t the output of the plant. Hence, the condition for oscillation is that the fundamental sine waveform comes back with the same amplitude and phase after traversing through the loop. This means that for sustained oscillations at a frequency of w u , we must have G(jw,)N(a) = -1. Hence,
1-0 0 - 00

(23)

Fig. 12:

Comparison of 0.2kLx1, O.SkLx1, and k L k , for
O < T < * ,

and aTgG(jw,) = -T (25) The quantity k, defined in (24) is called the ultimate gain in the sense that if one were to replace the relay non-linearity in the loop by a static positive gain, then setting the gain value equal to the ultimate gain would cause self-sustained oscillations in the loop. Substituting s = jw, into (l),we obtain the following expressions for the magnitude and phase of G ( j w , ) :

(0.37, M)

(0.41,~)
(0.50, CO)

Remark 5.1 It is commonly recommended [5] that AIL should be &ed at 0.25. A mtio of AIL = 0.25 offers a good compromise between performance and robustness. However, from the previous table, it can be seen that as AIL increases, the lower bound on T for resilience increases. Thus, if the plant under analysis has a small 7,a smaller AIL should be selected.

From (24), (25), (26), (27), using the fact that w, = and solving for T and L , we obtain the following relationships:

2

6 A Tuning Technique for Resilient Controller Design

Tu[x- atan( d ( k k , ) ' - l)] (29) 2r 7 Also, the steady state gain k can be found by applying a unit

L

=

In this section, we propose a new PID tuning technique. The starting point for this technique is the plant description (1) in terms of the parameters k , T and L. Since most PID tuning rules make use of experimentally measured quantities, let us first see how the experimentally measured quantities can be related to k , T and L .

step input t o the plant and observing the amplitude of the steady state output.

6.1 Determining k , T and L from Experimental Data The parameters T and L in the model (1) can be determined experimentally by measuring the ultimate gain and the ultimate period of the plant. This can be done by using the closed-loop configuration [l] shown in Fig. 13. In this figure, the relay is adjusted t o induce a self sustained oscillation in the loop. T h e ultimate gain ( k , ) and the ultimate period (T,) can now be determined by measuring the amplitude and the period of the oscillations.

6.2 Algorithm for computing the largest ball inscribed inside the PID stabilizing region Now that the parameters of the first order model with time delay (1) have been determined, we can proceed to compute the set of stabilizing PID controller parameters for this model. This can be achieved by using Theorem 2.1. We can then choose the s . PID settings a the center of the thme dimenszonal ball of largest mdius inscribed inside the stabilizing region. T h e radius of this ball represents the maximum 12 parametric stability margin in the space of the controller parameters. The problem of finding the largest ball inscribed inside the PID stabilizing region for rational plants has already been solved in [3]. The method developed in [3] is also applicable here since, for a fixed value of the parameter k,, the stabilizing regions of ( k c ,k d ) values are convex polygons (see Fig. 2). Even though the center of the largest ball inscribed inside the stabilizing region cannot be determined in closed form, it can be computed using the following algorithm.
Before presenting the algorithm, we first introduce some concepts. Consider a sphere B ( z , r ) in the three dimensional k,-

Fig. 13: Block diagram for the relay experiment.
We next proceed to derive the relationship between the model parameters T and L and the experimentally observed quantities k , and Tu. The describing function (or the equivalent fundamental frequency gain) of the the relay nonlinearity is given by

k,-kd space with radius T and centered a t I = ( I k p , z k , , i k d ) . Given any angle 0 E [;], let C(Z,T,0) denote the circle with radius T C O S ( ~ ) , centered at ( z k p +rsin(0), z k , , z k d ) and parallel to the k,-kd plane as illustrated in Fig. 14. It is clear that

A

i,

PI:
N ( Q )=

4d "T

Consider C ( z , r , 0 ) with fixed X k , T and 0 so that k, = z k P + r s i n 8 is fixed. Now, the stabifizing set of PID controllers is

kp

t

7 Concluding Remarks
In this paper we have presented an analysis of the robustness of some common PID tuning techniques. This analysis was motivated by the fact that a good PID controller design should exhibit robustness with respect to small perturbations in the controller coefficients. Our criterion was to ensure first that the controller parameters k, and kd were inside the stabilizing set of gain values. Then, the parameter ki was forced t o lie inside a box located 20% of 34 from the boundaries of the stabilizing set in the (kit kd) space. Here, 11 represents the maximum stabilizing integral gain value for t,he fixed proportional and derivative gains provided by the particular tuning rule. As a result of this criterion, the range of values that ensures robustness was determined for each T tuning technique. These values are summarized below: definition of

J ki
Fig. 14: Sphere B ( z , r ) a n d the
C(X,r, 0).

the circle

Ziegler-Nichols Step Response Method Cohen-Coon Method IMC Design Technique (for AIL = 0.25)

:

0
0

< - < 1.07
< - < 8.53
T
L T

:
:

L T L

0.37 <

formed by convex polygons, parallel t o the ki-kd plane, with either three or four sides (see.Fig. 2). Let the convex polygon , associated with this fixed k be given by the set of linear inequalities: P = { z l a r I 5 bej , j = 1,2, .., 4) O where agj E R2, bej E R, and each inequality represents the half plane containing one side of the polygon. Define zC = ( z k i , ~ k , ) ~ .Then from [3], C(z,r , 0) lies inside the stabilizing O region P if and only if ag

In addition to analyzing existing PID tuning rules from the literature, we have presented a new PID design procedure that guarantees maximal robustness with respect t o controller coefficient perturbations. Designing PID controllers t o optimize other performance criteria such as maximum overshoot, settling time, etc. are topics for future research.

+ r cosO((aej11

5

bej, ( j = 1 , 2 , ..,4)

(30)

holds. Let Se denote the set of feasible solutions of (30). From ; , $1, the the geometrical structure, we know that for all 0 E [ centers of the circles C(i,r, 0) have the same ( k i ,lid) coordinates. Since Se is the set of feasible (ki,kd) coordinates of the centers associated with C ( z , r , 0 ) ,it follows that L?(z,r) lies inside the k stabilizing (kp, i , kd) region if and only if

. References Astrom, K., & Hagglund, T . (1995). PID Controllers: Theory, Design, and Tuning, Research Triangle Park: Instrument Society of America.

[I]

(21 Cohen, G. H., & Coon, G. A. (1953). Theoretical Consideration of Retarded Control. Tmnsactions of the American Society of Mechanical Engineers, 76, pp. 827-834. [3] Ho, M. T., Datta, A., & Bhattacharyya, S. P. (2001). Robust and Non-fragile PID Controller Design. International Journal of Robust and Nonlinear Control, 11, pp. 681-708.

We now present the algorithm.

[4] Keel, L. H., & Bhattacharyya, S. P. (1997). Robust, Fragile or Optimal? IEEE Tmnsactions on Automatic Contml, 42((8),pp. 1098-1105.

Step 1: Initialize k = , and step = 2- ( k u p p N+l where N is the desired number of points. Step 2: Increase k p as follows: k = kp , Step 3: If kp < k , the algorithm.

-%

+i),

IS] Morari, M., & Zafiriou, E. (1989). Robust Process Control, Englewood Cliffs: Prentice-Hall.
[6] Silva, G. J . , Datta, A., & Bhattacharyya, S. P. (2002). New Results on the Synthesis of PID Controllers, IEEE Transactions on Automatic Control ( t o appear). [7] Yamamoto, S., & Hashimoto, I. (1991). Present Status and Future Needs: The View from Japanese Industry. Pruceedings of the 4th Internationnl Confeznee on Chemical Process Contml, pp. 512-521. [8] Ziegler, J. G., & Nichols, N. B. (1942). Optimum Settings for Automatic Controllers. Tmnsactzons ofthe American Society of Mechanical Engineers, 64, pp. 759-768.

+ step.
- k) ,

then go t o Step 4. Else, terminate

Step 4: Set T L = 0 and ru = min(k, Step 5: Set r =

w.

+ i, k,,,

Step 6: Sweep over all 0 E and determine the set of all feasible solutions Se for (30) at each stage. Step 7: If set T U = T .

[-;,

$1

Se # 0 then set

r L = r ; otherwise

Step 8: If ITU - r L I 5 specified level, then store r and go t o Step 2; otherwise go t o Step 5.
The above algorithm determines a family of spheres having different radii and centers, with each sphere corresponding t o a particular value of k, used in Step 2. Among these spheres, we pick the one with the largest radius. Setting the (kp,k i , k d ) values a t the center of this sphere will yield the maximum 12 parametric stability margin with respect t o the controller parameters.