19th Mediterranean Conference on Control and Automation Aquis Corfu Holiday Palace, Corfu, Greece June 20-23, 2011
ThBT5.1
Optimal Tuning of PID Controllers for Integrating Processes via the Symmetrical Optimum Criterion Konstantinos G. Papadopoulos
Konstantina Mermikli, Nikolaos I. Margaris
ABB Ltd. Switzerland Automation Drives and Development Department of Medium Voltage Drives CH 5300, Turgi, Switzerland, tel.+41-5858-93242
[email protected]
Aristotle University of Thessaloniki School of Engineering Department of Electrical & Computer Engineering GR 54124, Thessaloniki, Greece, tel.+30-2310-9-96283
[email protected],
[email protected]
Abstract— An analytical PI, PID type control law for typeII closed loop control systems is proposed. Type-II closed loop control systems are capable of tracking step and ramp reference signals achieving zero steady state position and velocity error respectively. The development of the proposed control law is based on the well known Symmetrical Optimum criterion and focuses on optimizing output disturbance rejection in comparison with the conventional design. For the derivation of the optimal control law, a generalized transfer function of the process model is employed. Therefore, the proposed theory can be applied in any linear SISO stable or integrating process. For verifying the proposed control law, a comparison between the conventional Symmetrical Optimum criterion and the revised theory is performed for several benchmark processes. The proposed control law shows significant improvement regarding output disturbance rejection, of up to 38%, in comparison with the conventional Symmetrical Optimum design.
I. I NTRODUCTION The strong effectiveness of the PID control algorithm has been widely proved throughout its application to many industrial control problems, [1]. Over the literature, the demanding problem of controlling integrating processes drove many researchers at employing various control techniques, either by exploiting the use of well established control schemes, such as the Smith predictor, [2] - [5] and the IMC concept [6]. Careful studying of the former PID tuning techniques [7], [8], tackled with the problem of controlling integrating processes, brings out that simple process models are employed so that the aforementioned proposed control laws are justified. The same status is observed even in cases where analytical tuning PID expressions are defined, [9] - [10]. An in-depth investigation presented in [11] reveals that the PID controller tuning, via the Symmetrical Optimum criterion, can handle efficiently, linear SISO integrating processes regardless of their complexity. The Symmetrical Optimum criterion was initially suggested by C. Kessler, [12]. The name of this criterion comes from the symmetry exhibited by the open loop frequency response of the final closed loop control system. In reality, the Symmetrical Optimum criterion is not something different, but the application of the Magnitude Optimum criterion, [13], [14] in type-II control systems. Let it be noted that the target of the Magnitude Optimum criterion is to design a controller able to render the magnitude of the closed loop frequency response
978-1-4577-0123-8/11/$26.00 ©2011 IEEE
as close as possible to unity, in the widest possible frequency range. When this criterion is fullfilled, then optimized disturbance rejection is achieved as well, at the output of the controlled process. However, according to [15], the design of PID type controllers via the conventional Symmetrical Optimum criterion presents at least two disadvantages. (1) For determining the controller parameters (zeros of the controller transfer function), exact pole-zero cancellation, between process’s poles and controller’s zeros has to take place. (2) This assumption leads to suboptimal results since it restricts the controller parameters to be tuned only with real zeros and whatsmore, further plant parameters for tuning the PID controller, are disregarded. Motivated by these two restrictions, we propose a flexible and more relaxed scheme that allows PID type controllers to be tuned with conjugate complex zeros if needed. Moreover, controller parameters are determined analytically, through specific optimization conditions, as a function of all plant parameters and not as a function of the dominant time constant (pole-zero cancellation). The aforementioned optimization conditions are the result of the application of the magnitude optimum criterion to a general model of a closed loop control system defined in the frequency domain as proved in the Appendix. II. D EFINITIONS AND P RELIMINARIES According to Fig.1 the error e (s) is given by e (s) = r (s) − y (s) = [1 − T (s)] r (s) = S (s) r (s) . Let the closed loop transfer function be defined in general by T (s) =
y(s) sm bm + sm−1 bm−1 + · · · + s2 b2 + sb1 + b0 = n . (1) r(s) s an + sn−1 an−1 + · · · + s2 a2 + sa1 + a0
The resulting error e (s) is defined by n an s + · · · + cm sm + · · · + c1 s + c0 r (s) e (s) = an sn + an−1 sn−1 + · · · + a1 s + a0
(2)
where c j = (a j − b j ), ( j = 0 . . . m). According to the final value theorem, e (∞) is equal to an sn + · · · + c2 s2 + c1 s + c0 e (∞) = lim s r (s) . (3) s→0 an sn + an−1 sn−1 + · · · + a1 s + a0
1289
n r (s)
r(s)
e(s)
+
d i (s)
controller
C(s)
-
do(s) 1.5
u(s) + +
G(s)
kp
+
43.4%
1.25
y(s)
+
10
0.707
Ccex(s)
with Cex(s)
0.5
-1
10
trt = 6.6τ
0.25
Mr
0
+
kh
yo(τ)
trt = 3.1τ
-0.25
S
+
-2
10
-0.5 0
2.5
5
7.5
10
12.5
15
17.5
20
22.5
Fig. 1. Block diagram of the closed loop control system. G(s) is the plant transfer function, C(s) is the controller transfer function, r(s) is the reference signal, do (s) and di (s) are the output and input disturbance signals respectively and nr (s), no (s) are the noise signals at the reference input and process output respectively. k p is the plant’s dc gain and kh is the feedback path.
(a)
s→0
c0 a0
(4)
,
which becomes zero when c0 = 0, or when a0 = b0 . Hence, sensitivity S (s) = dy(s) and closed loop transfer function T (s) o (s) are defined by T (s) =
sm bm + · · · + s2 b2 + sb1 + a0 , sn an + · · · + s2 a2 + sa1 + a0
an sn−1 + an−1 sn−2 + · · · − bm sm−1 S (s) = s
−bm−1 sm−2 + s (a2 − b2 ) + a1 − b1
(5) !
(6)
sn an + sn−1 an−1 + · · · + s2 a2 + sa1 + a0
respectively. If (5), (6) hold by, the closed loop control system is said to be of type-I. In similar fashion, if r (s) = s12 , then the velocity error is equal to, n an s + · · · + cm sm + · · · + c1 s + c0 1 e (∞) = lim (7) s→0 an sn + an−1 sn−1 + · · · + a1 s + a0 s which becomes finite if c0 = 0 or a0 = b0 . As a result the final value of the error is given by c1 a1 − b1 = lim (8) lim evss (t) = lim t→∞ s→0 a0 s→0 a0 and becomes zero when c1 = 0 or when a1 = b1 . In that case, the closed loop control system is said to be of typeII. Sensitivity S and closed loop transfer function T take the following forms respectively, T (s) =
sm bm + sm−1 bm−1 + · · · + sa1 + a0 , sn an + sn−1 an−1 + · · · + sa1 + a0
an sn−2 − · · · − bm sm−2 − bm−1 sm−3 + a2 − b2 sn an + sn−1 an−1 + · · · + s2 a2 + sa1 + a0
(9) !
.
(10)
III. T HE C ONVENTIONAL S YMMETRICAL O PTIMUM C RITERION We consider the closed loop system of Fig.1. Let the integrating process G(s) met in many industry applications be defined by
978-1-4577-0123-8/11/$26.00 ©2011 IEEE
27.5
30
-2
10
-1
0
10
10
1
10
u = ωTΣ
(b)
Fig. 2. Type-II closed loop control system. The effect of the two degree of freedom controller to the (a) step and (b) frequency response of the closed loop control system. (a) Step and (b) frequency response (solid black), filtered step response (dotted black). yo (s) = S (s) do (s) if r(s) = nr (s) = di (s) = no (s) = 0, yr (s) = T (s) r (s) if do (s) = nr (s) = di (s) = no (s) = 0.
If r (s) = 1s , then e (∞) = lim
25
τ = t/TΣ
n o (s)
S (s) = s2
|S(ju)|
|T(ju)|
yr(τ)
1
0.75
y f (s)
without Ccex(s)
0
8.1%
G(s) =
1 , sTm (1 + sTp1 )(1 + sTΣp )
(11)
where Tm is the integrator’s plant time constant, Tp1 the plant’s dominant time constant and TΣp the process parasitic time constant, [9]. For controlling (11), we will apply PID control defined by (1 + sTn )(1 + sTv ) , (12) C (s) = sTi (1 + sTΣc ) where TΣc stands for the controller parasitic dynamics. If Tn = Tv = 0, I control cannot be applied, because it is easily proved that the closed loop transfer function is unstable1 . If the dominant time constant Tp1 is evaluated, we still cannot apply PI control through (12) by setting Tn = Tp1 and Tv = 0, because the closed loop transfer function becomes unstable again for the same reason stated previously. Later on, it is proved that the current process can be controlled optimally under PI control via the proposed method, if no compensation occurs between the process’s pole and the controller’s zero. Assuming again that the dominant time constant Tp1 is accurately measured, we set Tv = Tp1 . The closed loop transfer function becomes then equal to T (s) =
k p Tn s + k p . s3 Ti Tm TΣ + s2 Ti Tm + skh k p Tn + kh k p
(13)
for which TΣ = TΣc + TΣp and TΣc TΣp ≈ 0 has been set. The magnitude of (13) is given by v h i u u k p k p 1 + (ωTn )2 u |T ( jω)| = u !2 . u u t (k p k p − Ti Tp ω 2 )2 + ω 2 k p k p Tn − 1 Ti Tp1 TΣ ω 2 (14) The denominator of (14) is equal to D(ω) = (Ti Tp1 TΣ )2 ω 6 + Ti Tp1 (Ti Tp1 − 2k p kh Tn TΣ ) ω 4 h i + (k p kh Tn )2 − 2k p kh Ti Tp1 ω 2 + k2p kh2
(15)
1 Intermediate terms of s j from the denominator polynomial at the final closed loop transfer function are missing, see Appendix.
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and becomes minimum, see [15] - [17], in the lower frequency range, when
n r (s)
di(s)
controller
e(s)
r ΄ (s) +
Cex(s)
r(s)
u(s) +
C(s)
+
-
kh = 1,
Tn = 4TΣ ,
Ti = 8k p kh
TΣ2 , Tm
Tv = Tp1 .
Using (16) along with (13) results in 1 + 4TΣ s . T (s) = 8TΣ3 s3 + 8TΣ2 s2 + 4TΣ s + 1
y f (s)
IV. T HE R EVISED S YMMETRICAL O PTIMUM C RITERION For the derivation of the proposed control law we assume the integrating process defined in general by
(20)
where n − 1 > m. The proposed PID controller is given by (21)
where parameter Tpn stands for the parasitic controller time constant. The flexible form of numerator Nc (s) = 1 + sX + s2Y allows parameters X, Y to become conjugate complex if possible, since X = Tn + Tv and Y = Tn Tv . In the following analysis, controller’s parameters will be determined analytically, as a function of all plant parameters. In contrast to the Symmetrical Optimum design, zero-pole cancellation will not take place directly, see (16), and X, Y, Ti will be defined as functions X = f1 (bi , a j , Td ), Y = f2 (bi , a j , Td ), Ti = f3 (bi , a j , Td ) of all plant parameters. The product k pC(s)G(s) is defined by " m " k p 1 + sX + s2Y ∑ s j β j j=0 (22) C(s)G(s) = n s2 Ti2 ∑ (si pi ) esTd i=0
978-1-4577-0123-8/11/$26.00 ©2011 IEEE
y(s)
+
n o (s)
(17)
Fig. 3. Block diagram of a two degree of freedom controller. Controller Cex (s) filters the reference input towards the overshoot decrease.
According to Fig.1, T (s) is given by T (s) =
k pC(s)G(s) 1 + k p khC(s)G(s)
(23)
and along with the aid of (22) we obtain m " " k p 1 + sX + s2Y ∑ s j β j j=0 # % . (24) T (s) = 1 + sX n m 2 2 sT i j s Ti e d ∑ (s pi ) + k p kh ∑ (s β j ) +s2Y i=0 j=0 We consider a general purpose time constant c1 and we normalize the time by setting s′ = sc1 . Therefore we obtain x=
rj =
r(s)=0
1 + sX + s2Y C(s) = 2 sTi (1 + sTpn )
+
+
kh S
the overshoot decreases from 43.4% to 8.1%. Let it be noted that the rise time increases from trt = 3.1TΣ to trt = 6.6TΣ . At this point, it should be stressed that in many industry applications, disturbance rejection (careful internal controller design) is of primary concern, while the reference signal varies not that frequently. As a result, the choice for the external filter parameters is not that critical, since it does not participate into . So (s) = dy(s) o (s)
sm βm + sm−1 βm−1 + · · · + sβ1 + 1 −sTd e s (sn−1 an−1 + · · · + s3 a3 + sa1 + 1)
+
G(s)
(16)
Normalizing the time by setting s′ = sTΣ , (17) becomes " 1 + 4s′ T s′ = ′3 . (18) 8s + 8s′2 + 4s′ + 1 The respective step and frequency response of (18) are shown in Fig.2(a), 2(b). It is clear that the step response of the closed loop control system exhibits an undesired overshoot of 43.4%. In order to overcome that obstacle, the reference input is filtered by adding an external controller Cex (s), Fig.3. The great overshoot of the step response in (17) is owed to the numerator of the transfer function, N (s) = 1 + 4TΣ . In that, if we choose an external controller of the form 1 r′ (s) = , (19) Cex (s) = r(s) 1 + 4TΣ s
G(s) =
d o (s)
kp
X , c1
pj , ci1
y=
Y , c21
∀ j = 1, . . . n,
ti =
Ti , c1
zi =
βi , ci1
d=
Td , c1
(25)
∀ i = 1, . . . m. (26)
′
Time delay es d is approximated by the Taylor series 7 " 1 ′ k k s′ d e =∑ s d . k=0 k!
(27)
Substituting (25)-(27) into (24) leads to " m " k p 1 + s′ x + s′2 y ∑ s′ j z j j=0 # % (28) T (s′ ) = 1 + s′ x n m ′ 2 ′ jz ) s′2ti es d ∑ (s′i ri ) + k p kh (s ∑ j +s′2 y i=0 j=0 or in a more compact form T (s′ ) =
N (s′ ) D1 (s′ ) + kh N (s′ )
=
N (s′ ) , D (s′ )
(29)
where " " m " N s′ = k p 1 + s′ x + s′2 y ∑ s′ j z j
(30)
j=0
and " D1 s′ = s′2ti2
#
7
1 " k ∑ k! s′ d k k=0
%
n
∑ i=0
" ′i s ri .
(31)
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If we expand (31), we have
r2 + r1 d+ 2 3 4 D1 (s′ ) = s′ ti2 + s′ ti2 (r1 + d) + s′ ti2 1 2 d 2! 1 2 1 3 . ′5 2 + s ti r3 + r2 d + d r1 + d 2! 3! 1 1 1 6 + s′ ti2 r4 + r3 d + d 2 r2 + d 3 r1 + d 4 + · · · 2! 3! 4! (32) Substituting the constant terms of (32) with, 1 q0 r1 + d q 1 1 r2 + r1 d + d 2 q2 2! 1 1 Q= q = r3 + r2 d + d 2 r1 + d 3 3 2! 3! q4 r 4 + r 3 d + 1 d 2 r2 + 1 d 3 r1 + 1 d 4 .. 2! 3! 4! . .. .
(33)
results in " D1 s′ = · · · + s′8ti2 q6 + s′7ti2 q5 + s′6ti2 q4
+ s′5ti2 q3 + s′4ti2 q2 + s′3ti2 q1 + s′2ti2 q0
.
(34)
From (30) we find that p " " " r N s′ = k p ∑ s′ yz(r−2) + xz(r−1) + z(r)
(35)
r=0
where zr = 0, if r < 0, and z0 = 1. As a result, polynomial D1 (s′ ) of the closed loop transfer function is defined by " " " D s′ = D1 s′ + kh N s′ =
k
∑ j=0
p
" 2 " ′ ( j+2) ti q j s
.
(36)
" r yz(r−2) + xz(r−1) + z(r) + kh k p ∑ s′ r=0
According to (29), (35) and (36), the resulting closed loop transfer function is given by p
" r " k p ∑ s′ yz(r−2) + xz(r−1) + z(r) " r=0 T s′ = yz(r−2) + p " k " " ( j+2) r + k p kh ∑ s′ xz(r−1) + ∑ ti2 q j s′ j=0 r=0 z(r) (37) where q(−2) = q(−1) = 0 and q0 = 1. For the derivation of the proposed control law we will make use of the optimization conditions defined in Appendix. Equations (53)-(56) will be used for the derivation of the proposed control law since for the developed method, PID type controllers are employed. Optimization Condition 1: a0 = b0 The application of (53) to (37) leads to kh = 1
978-1-4577-0123-8/11/$26.00 ©2011 IEEE
(38)
which implies that the final closed loop control system exhibits steady state position and velocity error. From (37) it is apparent that if kh = 1, then the respective terms of s0 , s, N(s′ ) = · · · + s′ k p (x+z1 )+k p and D(s′ ) = · · ·+s′ k p kh (x+z1 )+k p kh become equal. Optimization Condition 2: a21 − 2a2 a0 = 0. By making use of a21 − 2a2 a0 = b21 − 2b2 b0 we end up with ti = 0. For that reason, we set a21 − 2a2 a0 = 0 as another means of optimizing the magnitude of (51), [16], [17]. This results in, " 1 ti2 = k p kh x2 − 2y + z21 − 2z2 . (39) 2 Let it be noted, that in cases where no zeros exist in the plant "transfer function, the integral gain is equal to ti2 = 1 2 2 k p kh x − 2y . Optimization Condition 3: a22 − 2a3 a1 + 2a4 a0 = b22 − 2b3 b1 + 2b4 b0 . The application of (55) to (37) leads to [ti + k p kh (y + z1 x + z2 )]2 − 2k p kh q1 (x + z1 )ti
.
(40)
x2 + 4 (z1 − q1 ) x + 2y + z21 + 2z2 + 4q2 − 4q1 z1 = 0
(41)
+ 2k p kh q2ti = k2p (y + xz1 + z2 )2 By substituting (38), (39) into (40), we find that
and in cases where no zeros exist, (zi = 0, i = 1, . . . , m), (41) becomes equal to x2 − 4q1 x + 2y + 4q2 = 0
(42)
a23
+ 2a1 a5 − 2a6 a0 − 2a4 a2 = Optimization Condition 4: b23 + 2b1 b5 − 2b6 b0 − 2b4 b2 . The application of (56) to (37), along with the use of (38), (39) leads to " 2 q1 − 2q2 x2 + 4 (q1 z2 − q2 z1 + q3 − z3 ) x− " 2 q21 − 2q1 z1 + 2z2 y+ ! . (43) q1 z3 + q3 z1 − " 2 " 2 q1 − 2q2 z1 − 2z2 + 4 =0 q4 − z4 − q2 z2
From (41)-(43), we finally end up with the optimal control law given by, (44) kh = 1, " 1 (45) ti2 = k p kh x2 − 2y + z21 − 2z2 , 2 1 1" y = − x2 + 2 (q1 − z1 ) x − z21 + 2z2 + 4q2 − 4q1 z1 , (46) 2 2 x2 [2 [q1 (q1 − z1 ) − q2 + z2 ]] − % # 3 q1 − 3q21 z1 + 2q1 z21 + q1 z2 + + 4x q2 z1 − q3 + z3 − 2z1 z2 " . " 2 q1 − 2q1 z1 + 2z2 z21 + 2z2 + 4q2 − 4q1 z1 + ! " q1 z3 + q3 z1 − " =0 2 q1 − 2q2 z21 − 2z2 + 4 q4 − z4 − q2 z2
(47)
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yr(τ)
For comparing the conventional Symmetrical Optimum design with the revised theory, three responses of the control loop are investigated. The step, ramp response of the final closed loop control system for the revised PI, PID control law and the corresponding step, ramp response via the Symmetrical Optimum criterion design. Controller unmodelled dynamics have been chosen equal to tsc = 0.1 and normalizing time constant is equal to s′ = sTp1 . A. Integrating process with two dominant time constants In the first example we consider the integrating process given by " G s′ =
1
yo(τ)
r(τ) = τ 20
-0.5
PID
PID
PI: tss = 105τ PID: tss = 71.6τ PID-SO: tss = --
PID-SO
PI
-1
PI
0
-20
-1.5 0
50
100
150
0
200
20
40
(a)
60
80
100
(b)
Fig. 5. Control of an integrating non-minimum phase process. Comparison between the revised control law and the design through the Symmetrical Optimum criterion. (a) step response (b) ramp response of the closed loop control system. 150
1.2
yr(τ)
125
PID-SO
75 0
yo(τ)
50
r(τ) = τ
PID
-0.4
PI: tss = 123τ PID: tss = 81.9τ PID-SO: tss = unstable
PI 0
50
100
150
200
250
300
350
PI
25
PID 0
-0.8
0
400
25
50
(a)
75
100
125
150
(b)
Fig. 6. Control of an integrating process with large delay. Comparison between the revised control law and the design through the Symmetrical Optimum criterion. (a) step response (b) ramp response of the closed loop control system.
25 1
yr(τ)
0.8
20
0.6 15
PI PID
yo(τ)
10
PID
0
r(τ) = τ
PI: tss = 30.2τ PID: tss = 11.1τ PID-SO: tss = 18τ
-0.2
PID-SO
5
PI
PID-SO 0
30
40
50
0
(a)
5
10
15
20
25
(b)
Fig. 4. Control of an integrating process with two dominant time constants. Comparison between the revised control law and the design through the Symmetrical Optimum criterion. (a) step response (b) ramp response of the closed loop control system.
978-1-4577-0123-8/11/$26.00 ©2011 IEEE
Finally, we consider an integrating process with a delay constant, four times greater than the dominant pole. Its transfer function is given by
(49)
The design through the Symmetrical Optimum criterion has led to an almost unstable control loop, Fig.5. The respective PI, PID type controllers according to the proposed control 1+29.9s′ law are given by CrevPI (s′ ) = , CrevPID (s′ ) = 451.2s′ 2 (1+s′ tsc ) ′ ′ (1+16.4s )(1+s ) 1+24.4s′ +65.7s2 , CPIDSO (s′ ) = . The transition 236.7s′ 2 (1+s′ tsc ) 134.48s′ 2 (1+s′ tsc ) from the proposed PI to PID control law has led to faster disturbance rejection, tss = 71.6τ compared to tss = 105τ.
20
PID-SO
C. A pure time delay integrating process
" (1 − 2s′ ) (1 − 1.8s′ ) G s′ = . s (1 + s′ )5
10
40
9.87s (1+s tsc )
Let the integrating non minimum phase plant be defined by
0
60
0
0.4
B. An integrating non-minimum phase process
-0.4
80
0.5
100
" " " s(1 + s′ )2 1 + 10−2 s′ 1 + 10−3 s′ 1 + 10−4 s′
6.84s (1+s tsc )
0.2
1
0.8
(48) consisting of two dominant time constants. From Fig.4 it is apparent that the conventional Symmetrical Optimum criterion fails to tune a stable PI controller, in contrast to the revised theory. Output disturbance rejection has been improved, since settling time is tss = 11.1τ (proposed PID control law) compared to tss = 18τ, (symmetrical optimum criterion). The respective PI, PID type controllers according to the proposed control 1+7.82s′ ′ law are given by, CrevPI (s′ ) = 30.56s ′2 (1+s′ t ) , CrevPID (s ) = sc ′ 1+s′ 1+4.44s (1+3.15s′ )(1+1.88s′ ) )( ) ( , and CPIDSO (s′ ) = . ′2 ′ ′2 ′
0.4
100
1.5
V. S IMULATION R ESULTS
" G s′ =
1 s (1 + s′ )5
′
e−4s .
(50)
In that example, the Symmetrical Optimum criterion fails to tune a stable PID controller, leading finally to an unstable closed loop control system, Fig.6. On the contrary, the proposed control law acts efficiently even in that benchmark case. Let it be noted that in all previous examples, the transfer function of the controller has been chosen equal to Cex (s′ ) = Nc1(s′ ) where Nc (s′ ) is the numerator of the internal controller C(s′ ). Therefore, Cex (s′ ) = 1 1 ′ 1+s′ x and Cex (s ) = 1+s′ x+s′2 y in case of PI and PID control. The resulting revised PI, PID type controllers for that 1+34.8s′ , CPID (s′ ) = example are given by CPI (s′ ) = 606.5s′ 2 (1+s′ tsc ) ′ 1+s′ ′ 2 1+16.4s )( ) ( 1+27.22s +82.3s , CPIDSO (s′ ) = . The proposed 288.22s′ 2 (1+s′ tsc ) 134.48s′ 2 (1+s′ tsc ) control law leads to a control loop able to track a ramp reference signal with satisfactory performance. The addition of the D term is necessary since faster disturbance rejection is achieved, tss = 123τ → tss = 81.9τ, Fig.6. VI. C ONCLUSIONS We have revised the Symmetrical Optimum design criterion by proposing a new PID type control law for controlling
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SISO linear stable or integrating processes regardless of their complexity. In contrast to the conventional Symmetrical Optimum design, the revised control law allows the PID controller’s parameters to be tuned with conjugate complex values if needed. For tuning the zeros of the controller, polezero cancellation does not take place and therefore all plant parameters are considered for the controller’s tuning. This led to a significant improvement regarding output disturbance rejection since |T ( jω)| is satisfied in a wider frequency region. Moreover, in cases where the conventional design fails to tune a stable PID controller, when the complexity of the process increases, the revised theory still returns a feasible PI, PID control law since it determines analytically all controller gains as a function of all process parameters. A PPENDIX A. Proof of Optimization Conditions Let the closed loop transfer function be defined by (51), T (s) =
sm bm + sm−1 bm−1 + · · · + s2 b2 + sb1 + b0 N (s) = sn an + sn−1 an−1 + · · · + s2 a2 + sa1 + a0 D (s) (51)
where m ≤ n. Target of the Symmetrical Optimum criterion is to maintain |T (s)| ≃ 1 in the wider possible frequency range. Thus, by setting s = jω into (51) and squaring |T ( jω)| leads to |N ( jω)|2 . (52) |T ( jω)|2 = |D ( jω)|2 By making equal the terms of ω j ,( j = 1, 2, . . . , n) in polynomials |D( jω)|2 , |N( jω)|2 it is easily proved that conditions a0 = b0
(53)
a21 − 2a2 a0 = b21 − 2b2 b0
(54)
a22 − 2a3 a1 + 2a4 a0 #
a23 + 2a1 a5 − 2a6 a0 −2a4 a2
%
= =
b2 − 2b3 b1 + 2b4 b0 #2 2 b3 + 2b1 b5 − 2b6 b0 −2b4 b2
(55) %
(56)
··· = ··· have to hold by. Since the proposed analysis sticks to the PID control law, further optimization conditions are redundant. B. The Conventional Symmetrical Optimum Criterion Let the integrating process be defined by G(s) =
1 , sTm (1 + sTp1 )(1 + sTΣp )
(57)
where Tm , Tp1 , TΣp have been defined in Section III. If for controlling (57), I control of the form C (s) =
1 , sTi (1 + sTΣc )
(58)
is applied, then the closed loop transfer function is given by T (s) =
kp s2 Ti Tm (1 + sTp1 )(1 + sTΣ ) + kh k p
978-1-4577-0123-8/11/$26.00 ©2011 IEEE
(59)
where TΣp TΣc ≈ 0 and TΣ = TΣp + TΣc . From (59) it is evident kp T (s) = 4 (60) s Ti Tm Tp1 TΣ + s3 Ti Tm (Tp1 + TΣ ) + s2 Ti Tm + kh k p According to (60), it is evident that T (s) is unstable since the term of s is missing. In similar fashion, if PI control of the form 1 + sTn , (61) C (s) = sTi (1 + sTΣc ) is employed, then for determining controller parameter Tn via the conventional Symmetrical Optimum criterion, polezero cancellation must take place, Tn = Tp1 . Therefore, T (s) becomes kp . T (s) = 4 s Ti Tm Tp1 TΣ + s3 Ti Tm (TΣ + Tp1 ) + s2 Ti Tm + kh k p (62) which is unstable again for the same reason as stated for (60). Finally, PID control by cancelling two real or conjugate complex poles of G(s) cannot be applied, since it is proved that T (s) becomes unstable for the same reason as for (60). R EFERENCES [1] Ang K. H., Gregory Chong, Yun Li, “PID Control System Analysis, Design, and Technology”, IEEE Trans. Control Syst. Technol., vol. 13, No 4, pp. 559-576, 2005. [2] Smith O.J.M., “Closed control of loops with dead-time”, Chemical Engineering Progress, vol. 53, pp. 217-219, 1959. ¨ [3] Astrom K.J., Hang C.C., Lim B.C., ‘A New Smith Predictor for controlling a Process with an Integrator and a Long Dead Time’, IEEE Trans. Autom. Control, vol. 39, pp. 343-345, 1994. [4] M. R. Matauˇsek, A. D. Mici´c, “On the Modified Smith Predictor for Controlling a Process with an Integrator and Long Dead-Time”, IEEE Trans. Autom. Control, vol. 44, No 8, pp. 1603-1606, 1999. [5] Watanabe, K. & Ito, M., “A process model control for linear systems with delay”, IEEE Trans. Autom. Control, AC26(6), pp. 12611268, 1981. [6] Kaya I., “Controller design for integrating processes using userspecified gain and phase margin specifications and two degree-offreedom IMC structure”, IEEE Conference on Control Applications, pp. 898 -902, vol.2, June, 2003. [7] Julio E. Normey-Rico and Camacho E. F., “Robust Tuning of DeadTime Compensators for Processes with an Integrator and Long DeadTime”, IEEE Trans. Autom. Control, vol. 44, No 8, pp. 15971603, 1999. [8] Lee Y., Lee J., Park S., “PID controller tuning for integrating and unstable processes with time delay”, Chemical Engineering Science, vol. 55, pp. 3481-3493, 2000. [9] Poulin E., Pomerleau A., “PI Settings for Integrating Processes Based on Ultimate Cycle Information”, IEEE Trans. Control Syst. Technol., vol. 7, No 4, pp. 509-511, 1999. [10] Isaksson A. J., Graebe S. F., “Analytical PID parameter expressions for higher order systems”, Automatica, vol. 35, pp. 1121-1130, 1999. [11] Margaris N.I, ‘Lectures on Applied Control’, Tziolas Ed., Thessaloniki, (in Greek), 2003. [12] C. Kessler, “Das Symmetrische Optimum”, Regelungstechnik, pp. 395400 und 432-426, 1958. [13] Sartorius H., “Die zweckm¨assige Festlegung der frei w¨ahlbaren Regelungskonstanten”, Dissertation, Technische Hochscule, Stuttgart, Germany, 1945. [14] Oldenbourg R. C., Sartorius H., “A Uniform approach to the optimum adjustment of control loops”, Trans. of the ASME, pp.1265-1279, 1954. [15] Voda A. A., Landau I. D., “A Method for the auto-calibration of PID controllers”, Automatica, vol.31, No.1, pp.41-53, 1995. ¨ [16] Astrom K.J., Hagglund T., “Advanced PID Control”, ISA - The Instrumentation, Systems, and Automation Society, 2005. [17] Preitl S., Precup R.E., ‘An extension of tuning relation after symmetrical optimum method for PI and PID controllers’, Automatica, vol. 35, pp. 1731-1736, 1999.
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