Extending the Symmetrical Optimum criterion to the design of PID type-p control loops

Journal of Process Control 22 (2012) 11–25

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Extending the Symmetrical Optimum criterion to the design of PID type-p control loops Konstantinos G. Papadopoulos a,∗ , Nikolaos I. Margaris b,1 a b

ABB Switzerland Ltd., Department of Power Electronics & Medium Voltage Drives, CH-5300 Turgi, Switzerland Aristotle University of Thessaloniki, Department of Electrical & Computer Engineering, GR-54124 Thessaloniki, Greece

a r t i c l e

i n f o

Article history: Received 17 July 2011 Received in revised form 23 October 2011 Accepted 23 October 2011 Available online 23 November 2011 Keywords: PID control Design Tuning Optimization Process control Control engineering Industrial control

a b s t r a c t An extension of the Symmetrical Optimum criterion for the design of PID type-p closed- loop control systems is proposed. Type-p control loops are characterized by the presence of p integrators in the openloop transfer function. For designing a PID type-p control loop there should exist an PIp D, or PI(p−1) D, or PID and so on, if the controlled process is of type-0 or type-1 or type-(p − 1) respectively. A type-II control loop achieves zero steady state position and velocity error, a type-III control loop achieves zero steady state position, velocity and acceleration error and therefore a type-p control loop is expected to track both faster reference signals and eliminate higher order errors at steady state. For deriving the proposed control law, a transfer function containing dominant time constants and the plant’s unmodelled dynamics has been considered in the frequency domain. The final control law consists of analytical expressions that involve both dominant dynamics and model uncertainty of the controlled process. For justifying the potential of the proposed theory, simulation results for representative processes met in many industry applications are presented. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction The need to design higher order type control loops has always been challenging and critical over the academic and industry society [1,2]. From the theory, it is known that higher order type control loops have the advantage of tracking fast reference signals [1]. In order to achieve this target, and if the design of the control loop is based on the frequency domain, the design of a type-p control loop involves the existence of p integrators in the open-loop transfer function. Hence, if for the design of a type-p control loop, PID type controllers are adopted, then the sum of the pure free integrators of both the process and the controller should be equal to p. Throughout the literature, it is evident that the PID control law offers the simplest, feasible and yet most efficient solution to many real-world control problems, see [5–22]. More than 90% of industrial controllers still implemented, are based around PID algorithms, see [17–23]. However, the demanding problem of designing higher order type control loops has been treated by many researchers after employing or modifying well established control schemes such as the IMC2 principle [1,25,26] or the Smith predictor

∗ Corresponding author. Tel.: +41 58 5 893242; fax: +41 58 5 892580. E-mail addresses: [email protected], [email protected] (K.G. Papadopoulos), [email protected] (N.I. Margaris). 1 Tel.: +30 2310 9 96283; fax: +30 2310 9 96447. 2 Internal Model Control. 0959-1524/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2011.10.014

[27]. Main drawback of the above approaches is that the proposed control laws are restricted to the design of type-II control loops and whatsmore, for verifying their control laws potential, simple process models are employed [9,24,32] (first order plus dead time process, first order reduced integrating process). Over the literature, a first attempt of designing type-III control loops can be found in [3,4]. There, it is shown that by applying the principle of Symmetrical Optimum criterion along with the use of PID type controllers, robust type-III control loops can be designed. In similar fashion and since the aim of this work is to present a feasible control design method that can be applied in many industry applications, once more the simplicity and widespread application of the PID control law will be exploited. Therefore, the approach proposed in that work, is faced with the following problem. Tune a PID type controller, such that the output of the final closed-loop control system eliminates higher order steady state errors, position, velocity, acceleration etc. For developing the proposed theory, the principle of Symmetrical Optimum criterion will be adopted. In the sequel, it will be shown that for achieving the design of type-p control loops along with the aid of the Symmetrical Optimum criterion, the conventional design principle for tuning PID type controllers has to be adopted. This conventional principle implies that exact pole-zero cancellation has to be achieved between the process poles and the controller’s zeros. Therefore, for applying the proposed theory all dominant time constants of the process have to be measured accurately. The Symmetrical Optimum criterion is an extension of

12

K.G. Papadopoulos, N.I. Margaris / Journal of Process Control 22 (2012) 11–25

the Magnitude Optimum criterion, introduced by Oldenbourg and Sartorius [28], and is based on the idea of designing a controller which will render the magnitude of the closed-loop frequency response as close as possible to unity in the widest possible frequency range. In succession, Kessler suggested the Symmetrical Optimum criterion [29], which in reality is the application of the Magnitude Optimum criterion in type-II control systems. The name of the Symmetrical Optimum criterion comes from the symmetry exhibited by the open-loop frequency response [29,30]. In reality, the Symmetrical Optimum criterion is not something different, but the application of the Magnitude Optimum criterion to type-II control systems. The design of control systems both with the Magnitude Optimum and Symmetrical Optimum criteria of Oldenbourg-Sartorius and Kessler respectively presents at least two important advantages: (1) they do not require the complete plant model [30] and (2) the setpoint response of the closed-loop system is satisfactory [2]. However, excluding the German bibliography [33–36], the Magnitude Optimum criterion is rarely referred today [30]. For the development of the proposed theory and for the sake of a clear presentation of this work, the conventional tuning of PID type controllers via the Symmetrical Optimum criterion is presented in Section 3. Based on this principle, in Section 4 the design of PID type-III control loops is presented. Finally, in Section 5 the proposed theory is extended to the design of type-p control loops. Preliminary knowledge regarding the definition of the type of the control loops is provided in Section 2. For testing the proposed theory (see Section 6), representative processes met over the literature and industry are employed, processes with time delay, processes with equivalent dominant time constants and non-minimum phase processes. 2. Definitions and preliminaries 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) . If the closed-loop transfer function T(s) = y(s)/r(s) is defined by bm sm + bm−1 sm−1 + · · · + b1 s + b0 an sn + an−1 sn−1 + · · · + a1 s + a0

T (s) =

(1)

then the resulting error e(s) is defined by



e(s) =

an sn + · · · + cm sm + · · · + c1 s + c0 an sn + an−1 sn−1 + · · · + a1 s + a0

 r(s)

(2)

where cj = aj − bj (j = 0, . . ., m). According to the final value theorem and if e(s) is stable, e(∞) is equal to



e(∞) = lims s→0

an sn + · · · + c2 s2 + c1 s + c0 an sn + an−1 sn−1 + · · · + a1 s + a0



r(s).

(3)

If r(s) = 1/s then e(∞) = lim

s→0

c  0

a0

,

(4)

which becomes zero when c0 = 0, or when a0 = b0 . Hence, sensitivity S(s) = y(s)/do (s)3 and closed-loop transfer function T(s) are defined by T (s) =

sm bm + · · · + s2 b2 + sb1 + a0 , sn an + · · · + s2 a2 + sa1 + a0

S(s) = s

(5)

respectively. If (5) and (6) hold by, the closed-loop control system is said to be of type-I. In similar fashion, if, then the velocity error is equal to,



e(∞) = lim

s→0

an sn + · · · + cm sm + · · · + c1 s + c0 an sn + an−1 sn−1 + · · · + a1 s + a0

1 s

(7)

which becomes finite if c0 = 0 or a0 = b0 . As a result the final value of the error is given by lim evss (t) = lim

t→∞

s→0

c  1

a0

= lim

a − b  1 1

s→0

(8)

a0

and becomes zero when c1 = 0 or when a1 = b1 . In that case, the closed-loop control system is said to be of type-II. 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

S(s) = s2

an sn−2 − · · · − bm sm−2 − bm−1 sm−3 + a2 − b2 . sn an + sn−1 an−1 + · · · + s2 a2 + sa1 + a0

(9)

(10)

According to the above analysis, a closed-loop control system is said to be of type-p when sensitivity S and complementary sensitivity T have the following form S(s) = sp

(an sn−p + an−1 sn−1−p − · · · − bm sm−p − bm−1 sm−1−p + ap − bp ) sn an + sn−1 an−1 + · · · + s2 a2 + sa1 + a0

(11)

and T (s) =

bm sm + · · · + ap sp + ap−1 sp−1 + · · · + a1 s + a0 an sn + · · · + ap sp + ap−1 sp−1 + · · · + a1 s + a0

(12)

respectively. Also, one could argue that type-p control loops are characterized by the order of zeros at s = 0 in the sensitivity function S, see (6), (10) and (11). According to the above, type-p control loops have the advantage of tracking fast reference signals, since they eliminate higher order errors. 3. The conventional Symmetrical Optimum design criterion Let us now consider the closed-loop system of Fig. 1, where r(s), e(s), u(s), y(s), do (s) and di (s) are the reference input, the control error, the input and output of the plant, the output and the input disturbances respectively. An integrating process met in many industry applications can be defined by (13) G(s) =

1 , Tm s(1 + Tp1 s)(1 + Tp s)

(13)

where Tm is the integrator’s plant time constant, Tp1 the plant’s dominant time constant and Tp the process parasitic time constant [26]. Let it be noted that such type of modelling is frequently used in vector controlled induction motor drives. More specifically, time constant Tm stands for the mechanical subsystem of the motor which is the mechanism that involves the electromagnetic and load torque, the difference of which, makes the shaft rotating. Furthermore, time constant Tp1 is involved in the inner current control loop of the electrical drive and represents the stator winding time constant. Finally, Tp stands for the motor’s unmodelled dynamics. If vector control4 is to be followed (control of induction motor drives), kp stands for the pulse width modulator’s gain (kPWM ) which

[an sn−1 + an−1 sn−2 + · · · + (am − bm )sm−1 + (am−1 − bm−1 )sm−2 + s(a2 − b2 ) + a1 − b1 ] sn an + sn−1 an−1 + · · · + s2 a2 + sa1 + a0

3 S(s) stands for the sensitivity of the closed-loop control system and is defined by S(s) = y(s)/do (s) when r(s) = nr (s) = di (s) = nr (s) = 0, Fig. 1.



4

(6)

SFOC: stator field oriented control; RFOC: rotor field oriented control.

K.G. Papadopoulos, N.I. Margaris / Journal of Process Control 22 (2012) 11–25

13

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) and no (s) are the noise signals at the reference input and process output respectively. kp stands for the plant’s dc gain and kh is the feedback path.

is supposed to remain constant all over the whole operating range (0 → 1 p.u.) regarding output frequency.5 Parameter kh is the feedback path of the output measurement and as it will be proved in the sequel, kh should satisfy condition kh = 1. Back to Fig. 1, for controlling (13), the PID controller defined by C(s) =

Symmetrical Optimum design). The closed-loop transfer function becomes then equal to T (s) =

kp Tn s + kp Ti Tm T s3 + Ti Tm s2 + kh kp Tn s + kh kp



(14)

1 C(s) = Ti s(1 + Tc s)

(15)

|T (jω)| =

kp kp [1 + (ωTn )2 ] 2

(kp kh − Ti Tm ω2 ) + ω2 (kp kh Tn − Ti Tm T ω2 )

kp Ti Tm s2 (1 + Tp1 s)(1 + T s) + kh kp

(16)

kp Ti Tm Tp1 T s4 + Ti Tm (Tp1 + T )s3 + Ti Tm s2 + kh kp

.

(17)

From (17), it is clear that T(s) is unstable since the term of s is missing, see Appendix B. In similar fashion, if PI control of the form 1 + Tn s C(s) = Ti s(1 + Tc s)

(18)

is employed, then for determining controller parameter Tn via the conventional Symmetrical Optimum criterion, pole-zero cancellation must take place, Tn = Tp1 . Therefore, the dominant time constant Tp1 has to be evaluated and in that case, T(s) becomes kp Ti Tm T s4 + Ti Tm T s3 + Ti Tm s2 + kh kp

,

(19)

which is unstable again for the same reason as for (17), see Appendix B. Assuming again that the dominant time constant Tp1 is accurately measured and considering a PID controller as that described by (14), Tv = Tp1 is set (pole-zero cancellation, conventional

5

.

(21)

D(ω) = (Ti Tp1 T )2 ω6 + Ti Tp1 (Ti Tp1 − 2kp kh Tn T )ω4 2

+[(kp kh Tn ) − 2kp kh Ti Tp1 ]ω2 + kp2 kh2

(22)

and becomes minimum, see [30,31], in the lower frequency range when kh = 1,

Tn = 4T ,

Ti = 8kp kh

2 T

Tm

,

Tv = Tp1 .

(23)

Using (23) along with (20) results in

where Tp Tc ≈ 0 and T = Tp + Tc . From (16) it is evident that

T (s) =

2

The denominator of (21) is equal to

is applied, then the closed-loop transfer function is given by

T (s) =

(20)

The magnitude of (20) is given by

(1 + Tn s)(1 + Tv s) Ti s(1 + Tc s)

is adopted. For its tuning, the conventional Symmetrical Optimum design method is employed. Time constant Tc stands for the controller’s parasitic dynamics. If Tn = Tv = 0, I control cannot be applied, because the closed-loop transfer function becomes unstable. This is justified as follows. If for controlling (13), I control of the form

T (s) =

.

In many industry applications kp stands for the plant’s dc gain at steady state.

T (s) =

1 + 4T s

(24)

3 s + 8T 2 s + 4T s + 1 8T  

or finally after normalizing the frequency by substituting s = T s results in T (s ) =

8s3

1 + 4s . + 8s2 + 4s + 1

(25)

The respective step and frequency response of (25) are shown in Fig. 2(a) and (b). From there, it is clear that the step response of the closed-loop control system exhibits an undesired overshoot of 43.4% in the time domain Fig. 2(a), and a peak overshoot in the frequency domain Fig. 2(b). This is also justified by the open-loop frequency response Fig. 3 where the phase margin in the crossover ◦ ◦ frequency ωc = 1/(2T ) is ϕm ≈ 35  < 45 .Note also the symmetry of the critical frequencies

1 , 1 4T T

exhibited by |Fol (jω)|

where its slope is equal to −1/deg around the crossover frequency ωc = 1/(2T ), Fig. 3. The open loop transfer function is given by Fol (s ) =

1 + 4s 8s 2 (1 + s )

.

(26)

In order to overcome the obstacle of 43.4% overshoot, the reference input is filtered by adding an external controller Cex (s), Fig. 4. The great overshoot of the step response in (24) is owed to the zero of the transfer function, N(s ) = 1 + 4s . This can be removed by

14

K.G. Papadopoulos, N.I. Margaris / Journal of Process Control 22 (2012) 11–25

Fig. 2. Type-II closed-loop control system. (a) The effect of the two degrees of freedom controller to the step response of the closed-loop control system. Step response (solid black) and filtered step response (dotted black). (b) The effect of the two degrees of freedom controller to the frequency response of the closed-loop control system.

integrating process of the form (13) is assumed, where Tp1 stands for the dominant time constant of the process and Tm , Tp stand for the integrator’s time constant and the unmodelled plant dynamics respectively. Supposing that the dominant time constant Tp1 is evaluated, the proposed I-PID controller is defined by, C(s) =

(1 + Tn s)(1 + Tv s)(1 + Tx s) Ti s2 (1 + Tc s)(1 + Tc s) 1

(28)

2

where Tc , Tc are known and sufficiently small time constants 1 2 compared to Tp1 . By setting Tx = Tp1 (pole-zero cancellation) and assuming that Tc = Tc + Tc , Tc Tc ≈ 0, the transfer func1 2 1 2 tion of the closed-loop control system is equal to T (s) = Fig. 3. Type-II closed-loop control system. Open loop frequency response.

including that zero as a pole in the reference filter. In that, if an external filter of the form 1 r  (s ) = Cex (s ) = 1 + 4s r(s ) 

(27)

is chosen, the overshoot decreases from 43.4% to 8.1%. Let it be noted that the rise time increases from trt = 3.1T to trt = 6.6T . Such dynamics, can for sure be improved by adding additional dynamics in the reference filter. 4. Extending the Symmetrical Optimum design criterion to type-III control loops According to the design of type-II closed-loop control systems, a similar methodology for the design of type-III closed-loop control systems will be proposed. For the following analysis, an

kp Tn Tv s2 + kp (Tn + Tv )s + kp Ti Tm T

s4

+ Ti Tm s3 + kp kh Tn Tv s2 + kp kh (Tn + Tv )s + kp kh

, (29)

where T = Tc + Tp . The magnitude of (29) is given by

 |T (jω)| =

2

kp2 (1 − Tn Tv ω2 ) + kp2 (Tn + Tv )2 ω2 2

[Ti Tm T ω4 + kp kh (1 − Tn Tv ω2 )] + A0 ω2

(30)

where A0 = [kp kh (Tn + Tv ) − Ti Tm ω2 ]2 . The denominator of (30) is defined by D(ω) = (Ti Tm T )2 ω8 + Ti Tm (Ti Tm − 2kp kh Tn Tv T )ω6 2 + kp kh [2Ti Tm T − 2(Tn + Tv )Ti Tm + kp kh Ti2 Tm ]ω4

+ (kp kh )2 (Tn2 + Tv2 )ω2 + (kp kh )2 .

(31)

One way to optimize the magnitude of (30) is to set the terms of ωj , j = 2, 4, 6, . . ., in (31), equal to zero, starting again from the lower

Fig. 4. Two degrees of freedom controller. Controller Cex (s) filters the reference input so that the undesired overshoot at the output y(s) is diminished. Controller Cex (s) affects the closed-loop transfer function T(s) and not the output and input disturbance transfer functions So (s) = y(s)/do (s), Si (s) = y(s)/di (s).

K.G. Papadopoulos, N.I. Margaris / Journal of Process Control 22 (2012) 11–25

15

Fig. 5. (a) Step response and disturbance rejection of type-III closed-loop control system. (b) Frequency response of type-III closed-loop control system.

in Fig. 6 the open-loop frequency response is shown. Its transfer function is given by Fol (s ) =

4n(n − 1)s 2 + (n2 − 4)s + n − 4 8n(n − 1)s 3 (1 + s )

.

(37)

From Fig. 6 it is concluded that the magnitude of the complementary sensitivity |T(ju)| is practically independent of the parameter n. Moreover, sensitivity |S(ju)| becomes maximum if n = 4.1 and minimum, if n = 7.46. In that case (n = 7.46), Tn = Tv holds by. For every other value of parameter n, the shape of the open-loop frequency response is preserved exactly as presented in Fig. 5(a), (≈50%). Since the phase margin is ϕm = 35◦ < 45◦ , an undesired overshoot in the step response of the closed-loop system is expected, Fig. 5(a), which can be decreased along with the aid of an external filter Cex (s) as mentioned in Section 3. Fig. 6. Open-loop frequency response of type-III closed-loop control system.

frequency range. Setting kh = 1 and the term of ω6 equal to zero leads to Ti =

2kp kh Tn Tv T . Tm

(32)

In similar fashion, setting the term of ω4 equal to zero along with the aid of (32), leads to 2 − 4(Tn + Tv )T + Tn Tv = 0. 4T

According to the analysis presented in Section 4 a similar analysis for tuning the PID type controller’s parameters will be presented regarding the design of type-p control loops. Note that parameter p stands for the free integrators of the open-loop transfer function. Therefore, let the process be defined by G(s) =

(33)

If Tv = nT is chosen, then (33) becomes Tn =

5. Extending the Symmetrical Optimum design criterion to type-p control loops

4(n − 1) T . n−4 

(34)

Proper selection of parameter n (n > 4 must hold by) leads to a feasible I-PID control law. Substituting Eqs. (32) and (34) into the closed-loop transfer function results in

Tm sq

2 2 s + (n2 − 4)T s + n − 4 4n(n − 1)T 3 3 4 4 2 2 8n(n − 1)T s + 8n(n − 1)T s + 4n(n − 1)T s + (n2 − 4)T s + (n − 4)

.

(35) Normalizing again the time by setting s = sT where s = ju, (s = jω) which finally leads to u = ωT , (35) becomes equal to T (s) =

2



4n(n − 1)s + (n − 4)s + (n − 4) 2

8n(n − 1)s 4 + 8n(n − 1)s 3 + 4n(n − 1)s 2 + (n2 − 4)s + (n − 4)

.

(36)

Note that the control loop defined in (36) is of type-III, since the  terms of s j , j = 0, 1, 2, are equal, a0 = b0 , a1 = b1 , a2 = b2 , see Section 2. The respective step and frequency responses of (36) for two different values of parameter n, are presented in Fig. 5. In addition,

ns

(1 + Tmj s) j=1

k=1

(1 + Tsk s)

,

(38)

consisting of q integrators and Tm one of the integrator’s time constant. Assuming that the plant’s dominant time constants are defined by Tmj (j = 1, 2, . . ., nm ) and the process unmodelled dynamics by Tsk (k = 1, 2, . . ., ns ) we can substitute in (38), without loss of generality with the approximation ns 

(1 + Tsk s) = 1 + Ts s

k=1

T (s) =

1

nm

(39)

n

s T stands for the process’ small unmodelled where Ts = k=1 sk time constants. Since the target of the design is the final closed-loop control system to be of type-p, according to the analysis presented in Section 4, the proposed PID type controller is given by

nm

C(s) =

p−1

j=1

(1 + Tmj s)

Ti s

nc p−q

r=1

(1 + Tnr s)

(1 + Tcz s) z=1

.

(40)

Thus, according the design of type-II, III control loops, the PID type controller has to contain nm zeros equal to the Tmj dominant time constants (j = 1, 2, . . ., nm ) so that exact pole-zero cancellation is achieved. Moreover, it is proved after some calculus in T(s), that in order the denominator of the final closed-loop transfer function is a full polynomial in terms of the sj coefficients, p − 1 zeros

16

K.G. Papadopoulos, N.I. Margaris / Journal of Process Control 22 (2012) 11–25

must exist. Furthermore, the controller must introduce p–q integrators, so that the final closed-loop is of type-p. Finally, in order the controller transfer function is strictly causal, denominator’s order must be greater or equal to p − 1 + nm . The unmodelled controller’s dynamics are represented by

loop is of type-II p = 2, the PID type controller (according to the Symmetrical Optimum criterion) is given by

nc 

for which Tn2 = Tp1 and (1 + sTp )(1 + sTc ) ≈ 1 + sT have been set. In that, the open-loop transfer function is given by

(1 + Tcz s) = 1 + Tc s

(41)

C(s) =

(1 + Tn1 s)(1 + Tn2 s) , Ti s(1 + Tc s)

z=1

Fol (s) = kp kh

where

nc

Tc =

Tcz .

(42)

z=1

Fol (s) = kp kh G(s)C(s) = kp kh

Ti Tm s

ns p

p−1 r=1

k=1

(1 + Tnr s)

nc

(1 + Tsk s)

z=1

(1 + Tcz s) (43)

or by substituting (39), (40), (41) and (42) results in

p−1

(1 + Tnr s) r=1

Fol (s) = kp kh

(44)

Ti Tm sp (1 + T s)

where T = Ts + Tc and Ts Tc ≈ 0. Finally, the closed-loop transfer function is equal to T (s) =

kp Ti Tm T

sp+1

p−1 r=1

+ Ti Tm

sp

(1 + Tnr s) + kp kh

p−1 r=1

(1 + Tnr s)

.

(45)

T (s) =

ap+1

sp+1

+ ap

sp

+ ap−1

sp−1

+ · · · + a3

s3

+ a2

s2

where bp−1 =

p−1 

Tpj = Tp1 Tp2 . . . Tpp−1 ,

Tni Tnj Tnk ,

(47)





b1 = kp

Tni Tnj ,

i= / j=1

Tni ,

b0 = kp ,

(48)

i=1

and ap+1 = Ti Tm T ,

ap = Ti Tm ,



(49)

p−1

a3 = kp kh

a2 = kp kh

i= / j= / k=1

a1 = kp kh

p−1



is set, as another means of optimizing the magnitude of (56), [30,31]. After some calculus it is obtained Ti = 2kp kh

(50)

i= / j=1

a0 = kp kh .

Tni ,

Tni Tnj ,

2 /T ), see Section 3. and if Tn1 = 4T then Ti = 8kp kh (T m According to Section 4, for a process of one dominant time constant defined again by (13) where (q = 1) then in order the final control loop is of type-III p = 3, the PID type controller is given by

(1 + Tn1 s)(1 + Tn2 s)(1 + Tn3 s) Ti s2 (1 + Tc s)

Tm

Tnr .

(59)

Assuming again pole-zero cancellation, Tn3 = Tp1 and (1 + sTp )(1 + sTc ) ≈ 1 + sT the open-loop transfer function Fol (s) becomes Fol (s) = kp kh

(1 + Tn1 s)(1 + Tn2 s) Ti Tm s3 (1 + T s)

.

(60)

T (s) =

kp (1 + Tn1 s)(1 + Tn2 s) Ti Tm s3 (1 + T s) + kp kh (1 + Tn1 s)(1 + sTn2 )

.

(61)

According to (57) and since n = 2, the integrator’s time constant is calculated via (62)

Finally, after some calculus it was shown that the integrator time constant is equal to T Tn Tn . Tm 1 2

(52)

In similar fashion, for a process of one dominant time constant defined by (13) and if n = k − 1, in order the final control loop is of type-p, the PID type controller is given by

Since the aim is to determine parameters Ti , Tnr (r = 1, . . . , p − 1) the magnitude of (46) will be optimized according to Appendix A. For every order p, the optimal integral gain is given by p−1 T 

.

Ti = 2kp kh

According to (A.9), if a0 = b0 then kh = 1.

(58)

(51)

i=1

Ti = 2kp kh

T Tn Tm 1

a23 = 2a2 a4 .

p−1

Tni Tnj Tnk ,

(56)

Therefore the closed-loop transfer function is equal to

p−1

p−1

.

(57)

i= / j= / k=1

j=1

b2 = kp

p−1

b3 = kp

kp (1 + Tn1 s) Ti Tm s2 (1 + T s) + kp kh (1 + Tn1 s)

a22 = 2a1 a3

C(s) =

+ a1 s + a0 (46)

(55)

According to the analysis presented in Section 3, the integrator’s time constant is calculated if

After some calculus in (45) it is concluded that bp−1 sp−1 + bp−2 sp−2 + · · · + b3 s3 + b2 s2 + b1 s + b0

(1 + Tn1 s) Ti Tm s2 (1 + T s)

and the closed-loop transfer function is then given by T (s) =

In that case, the open-loop transfer function becomes

(54)

(53)

C(s) =

(1 + Tn1 s)(1 + Tn2 s) . . . (1 + Tnk s)

.

(64)

According to the analysis presented previously, it can be claimed regarding the integrator’s time constant Tik−1 , that T  Tnj . Tm k−1

r=1

This can be proved as follows. For a process of one dominant time constant defined by (13) where (q = 1), then in order the final control

Ti sk−1 (1 + Tc s)

(63)

Tik−1 = 2kp kh

j=1

(65)

K.G. Papadopoulos, N.I. Margaris / Journal of Process Control 22 (2012) 11–25

Therefore, for n = k, it has to be proved that

In similar fashion, in type-III control loops for determining parameters Tn1 , Tn2 we make use of a22 − 2a3 a1 + 2a4 a0 = 0, see (A.11). This results in

T  Tnj Tm k

= 2kp kh

Tik

2 4T Tn1 Tn2 − 4T Tn1 Tn2 (Tn1 + Tn2 ) + Tn21 Tn22 = 0

j=1

⎡ ⎤ . k−1  T = 2kp kh  ⎣ Tnj ⎦ Tnk = Tik−1 Tnk Tm

(66)

(1 + Tn1 s)(1 + Tn2 s) . . . (1 + Tnk s)(1 + Tnk+1 s)

(67)

Ti sk (1 + Tc s)

for which Tnk+1 = Tp1 is set, assuming design via pole-zero cancellation. Since again (1 + sTp )(1 + sTc ) ≈ 1 + sT , the open and closed-loop transfer functions are given by

k

Fol (s) = kp kh

T (s) =

(1 + Tnj s) j=1 , k+1 Tik Tm s (1 + T s) kp

k

j=1

(68)

(1 + Tnj s)

Tik Tm sk+1 (1 + T s) + kp kh

k j=1

(1 + Tnj s)

,

(69)

(70)

Tik Tm T sk+2 + Tik Tm sk+1 + kp kh (rk sk + rk−1 sk−1 + · · ·r2 s2 + r1 s + 1)

(Tik Tm )2 = 2kp kh Tik Tm T rk

T (s) =

kp (rk−1 sk−1 + rk−2 sk−2 + · · · + r2 s2 + r1 s + 1) Ti Tm T sk+1 + Ti Tm sk + kp kh (rk−1 sk−1 + rk−2 sk−2 + · · · + r2 s2 + r1 s + 1)

2 rk−1 − 4T rk−2 + 4T rk−3 = 0.

(73)

r=1

Tnr sp+1 + 2kp kh T

(1 + Tnr s)

p−1 r=1

Tnr sp + kp kh

p−1 r=1

(1 + Tnr s)

.

(74)

(85)

p−2



p−1 

i=1

i=1

i=1

(86)

(87)

or p−2



p−1



p 

i=1

i=1

i=1

Tni − 4T

Tni +

Tni = 0.

(88)

The above equation is rewritten in the form of 2 Tnp 4T

p−3



p−2



p−1 

i=1

i=1

i=1

Tni − 4T Tnp

Tni + Tnp

or finally

For determining now parameters Tnr , it will be shown that in order the magnitude of (74) satisfies condition |T(jω)  1|, controller time constants Tnr must satisfy condition p−3



(83)

or after some calculus

2 4T

which is equal to (66). In that case, if (73) is substituted into (69), results in

p−1

(82)

(84)

kp2 rk2 − 2rk−1 kp (2kp T rk ) + 2kp rk−2 (2kp T rk )T = 0

j=1

kp

.

In (72) it was shown that Ti Tm = 2kp kh T rk . By applying (80) to (82) we obtain

2 4T rk−2 − 4T rk−1 + rk = 0

j=1 ⎡ ⎤ k−1  T = 2kp kh  ⎣ Tnj ⎦ Tnk = Tik−1 Tnk

r=1

the ones that satisfy condition |T(jω)  1| in a wide range of frequencies. Therefore, if n = k − 1 then controller C(s) is defined by (64) and the closed-loop transfer function is given by

(72)

T T  r = 2kp kh  Tnj Tm k Tm

2kp kh T

(81)

= 2ak+1 ak−1

If n = k then the closed-loop transfer function is given by (70). Since Ti Tm = 2kp kh T rk then by applying a2k = 2ak−1 ak+1 − 2ak−2 ak+2 to (70) we obtain

k

p−1 2

(80)

a2k

2 rk − 4T rk−1 + 4T rk−2 = 0.

or Tik Tm = 2kp kh T rk . Finally, along with the aid of (70), it is obtained

T (s) =

a2k−1 = 2ak−2 ak − 2ak−3 ak+1 ,

(71)

or

Tm

According to the above, and based on (65) if the closed-loop control system is of type-p, then for determining parameters Tnj (j = 1, 2, . . ., k), the following optimization conditions are claimed to be,

If n = k, then we are going to show that

a2k+1 = 2ak+2 ak

= 2kp kh

(79)

from which after some calculus results in

kp (rk sk + rk−1 sk−1 + · · · + r2 s2 + r1 s + 1)

respectively. Then, according to (57), Ti is calculated by

Tik

or finally,

2 kp2 rk−1 − 2rk−2 kp (2kp T rk−1 ) + 2kp rk−3 (2kp T rk−1 )T = 0,

or T (s) =

(78)

2 Tn1 Tn2 − 4T (Tn1 + Tn2 ) + 4T = 0.

j=1

According to the design of type-p control loops, the PID type controller is given by C(s) =

17



2 4T

p−3



p−2



p−1 

i=1

i=1

i=1

Tni − 4T

Tni +

Tni

Tni = 0

(89)

 Tnp = 0

(90)

This is justified as follows. In type-II control loops for determining parameter Tn1 we make use of a21 − 2a2 a0 = 0 (see (A.10)). This results in

which is true, since (83) holds by. Obviously, the number of combinations of the Tni optimal parameters that satisfy (90) is infinite. More specifically, by applying condition (90) for the design of up to type-V control loops results in Type-V control loops:

kp2 Tn21 = 2kp (2kp Tn1 T )

2 4(Tn1 Tn2 + Tn1 Tn3 + Tn1 Tn4 + Tn2 Tn3 + Tn2 Tn4 + Tn3 Tn4 )T

2 4T

Tni − 4T

Tni +

Tni = 0.

(75)

(76)

− 4(Tn1 Tn2 Tn3 + Tn1 Tn2 Tn4 + Tn2 Tn3 Tn4 + Tn1 Tn3 Tn4 )T

or finally Tn1 − 4T = 0.

(77)

+ Tn1 Tn2 Tn3 Tn4 = 0.

(91)

18

K.G. Papadopoulos, N.I. Margaris / Journal of Process Control 22 (2012) 11–25

Fig. 7. Type-IV control loop. (a) Step and (b) frequency response of the final closed-loop control system for various values of parameter n.

Type-IV control loops: 2 − 4(Tn1 Tn2 + Tn2 Tn3 + Tn1 Tn3 )T + Tn1 Tn2 Tn3 4(Tn1 + Tn2 + Tn3 )T

= 0.

Fol (s) =

3 3 2 2 [4n3 (n − 2)T s + n2 (n2 − 12)T s 2n2 (n − 6)T s + (n − 0.536)(n − 7.464)] 5 5 4 4 8n3 (n − 2)T s + 8n3 (n − 2)T s

(92)

(103)

Type-III control loops: 2 − 4(Tn1 + Tn2 )T + Tn1 Tn2 = 0. 4T

(93)

Type-II control loops: (94)

Note that (93) and (94) are equal to (18) and (33) respectively. In similar fashion with type-III control loops and for the sake of simplicity of the analysis, if we choose Tn1 = Tn2 = · · · = Tnp−1 = nT

(95)

the respective open Fol (s) and closed-loop T(s) transfer functions are given by Fol (s) ≈

p−1

p

2np−1 T sp+1 + 2np−1 T sp + (1 + nT s)p−1

.

(97)

The optimal value of parameter n depends on the type of the control loop we want to design. If we substitute (95) into (92)-(94), we have consequently, Type-V control loops: 4 = 0 ⇒ nopt = 14.32. n2 (n2 − 16n + 24)T

(98)

Type-IV control loops: 2

3 = 0 ⇒ nopt = 10.89. nn − 12n + 12)T

(99)

Type-III control loops: 2

2 = 0 ⇒ nopt = 7.46. (n − 8n + 4)T

(100)

With respect to the above, for the design of a type-IV control loop, a PID type controller of three zeros in its transfer function is required. Therefore, if we chose Tn1 = Tn2 = nT 

(101)

according to (95), then from (92) it is obtained Tn3 =

s5

b3 s3 + b2 s2 + b1 s + b0 + a4 s4 + a3 s3 + a2 s2 + a1 s + a0

3 , b3 = 4n3 (n − 2)T

b1 = 2n2

(104)

4n(n − 2) 4n(n − 2) T . T = (n − 0.536)(n − 7.464)  n2 − 8n + 4

(102)

Based on the above, the corresponding Fol (s) and T(s) transfer functions are given by

b2 = n2

(n − 6) T , n−2 

b0 =

(n2 − 12) 2 T n−2

(n − 0.536)(n − 7.464) n−2

(105) (106)

and 5 a5 = 8n3 (n − 2)T , 3 a3 = 4n3 T ,

a1 = 2n2

(1 + nT s)p−1 p+1

a5

(96)

p

2np−1 T sp (1 + T s)

and T (s) ≈

T (s) = where

Tn1 = 4T .

(1 + nT s)

.

4 a4 = 8n3 T

(n2 − 12) 2 T n−2

a2 = n2

(n − 6) T , n−2 

a0 =

(n − 0.536)(n − 7.464) . n−2

(107) (108) (109)

According to (104), the closed-loop control system is of type-IV since, aj = bj , j = 0, 1, 2, 3, see Section 2. If n < 7.464 the closed-loop control system is unstable. As a result, for having a feasible PID type control law, n > 7.464 has to hold by, see (105). In Fig. 7(b) the frequency response of sensitivity S and complementary sensitivity T of the type-IV closed-loop is presented for several variations of parameter n, n ∈ [7.5, ∞). From there, it is apparent that variations of parameter n do not lead to critical variations of both functions T, S in the frequency domain. Sensitivity S is affected only in the lower frequency region. Note that, in similar fashion with type-III control loops, sensitivity S becomes minimum when all controller zeros are equal, Tn1 = Tn2 = Tn3 , n = 10.89, Fig. 10. There, it is shown how the controller’s zeros are affected in case of variations in design parameter n. Similar results are also observed in the time domain, Fig. 7(a). The step response of the type-IV closed-loop control system exhibits an overshoot of 50%, which is justified by the phase margin ( = 32◦ < 45◦ ) of the open-loop Fol (s) frequency response, Fig. 8. For decreasing the overshoot of the final closed-loop control system, the two degrees of freedom controller structure will again be exploited. If n = 10.89, then the closed-loop transfer function in terms of time constants form is given by T (s) =

N1 (s) D1 (s)D2 (s)D3 (s)

(110)

where N1 (s) = (1 + 10.89T s)3 ,

D1 (s) = (1 + 2.3T s)

(111)

K.G. Papadopoulos, N.I. Margaris / Journal of Process Control 22 (2012) 11–25

19

6. Simulation results For justifying the control’s law potential simulation examples of type-II, III, IV, V control loops are presented. According to the control law presented in Section 4 the I-I-PID type controller for controlling a type-0 process is given by C(s) =

(1 + Tn1 s)(1 + Tn2 s)(1 + Tn3 s) Ti s3 (1 + Tc s)(1 + Tc s) 1

.

(115)

2

In all three examples, it is assumed that the sum T of all time constants of the controlled process is accurately measured. Time

k

Fig. 8. Open-loop frequency response of a type-IV control loop for various values of parameter n.

T + Tc and Tc = Tc1 + Tc2 includes both constant T = j=1 pj plant’s and controller’s unmodelled dynamics. Since type-III control loops are designed Tn1 = Tp1 , Tn2 = 4(n−1) T , Tn3 = nT  . Parameter n−4 n has been chosen equal to n = 7.46. The integrator’s time constant T p−1 is calculated through Ti = 2kp kh T T = 2kp kh Tn2 Tn3 T . In all r=1 nr m three cases Tm = 1 has been set. 6.1. Process with dominant time constants The process described by G(s ) =

2 (1 + s )(1 + 0.84s )(1 + 0.78s )(1 + 0.57s )(1 + 0.28s ) (116)

is considered. From Fig. 11(a) it is apparent that the type-III closedloop control system exhibits an undesired overshoot of 87.4% which is decreased by filtering the reference with an external controller Cex1 (s), Fig. 11. Settling time remains almost unaltered, tss = 143. Note that disturbance rejection has remained the same since the external controller Cex1 (s) acts only at the reference signal outside of the control loop. For manipulating the overshoot of the output, 1 if Cex2 (s) = reference filter is to be used, then 2 Fig. 9. The effect of the two degrees of freedom controller structure to the step response of the type-IV closed-loop control system.

(tn2 tn3 )s +(tn2 +tn3 )s+1

the overshoot is decreased to 6.2%. Since the closed-loop control system is of type-III, the output of the process can track perfectly both ramp and parabolic reference signals, Fig. 12. 6.2. Process with time delay A delay process of the form G(s ) =

Fig. 10. Variations of parameters Tn1 , Tn2 , Tn3 according to variations of parameter n. 2 2 D2 (s) = (2.274)2 T s + 0.99(2.274)T s + 1

D3 (s) =

2 2 (14.75)2 T s

+ 1.9(14.75)T s + 1.

(112) (113)

Thus, by choosing an external controller of the form Cex (s) =

2 s2 + 1.9(14.75)T s + 1] (1 + 2.3T s)[(14.75)2 T 

(1 + 10.89T s)3 (1 + T s)

overshoot is reduced to 14.75%, Fig. 9.

(114)

2  e−s (1 + s )(1 + 0.99s )(1 + 0.57s )(1 + 0.28s )(1 + 0.1s ) (117)

is assumed in this example. Note that the proposed control law does not take into account the effect of the time delay and therefore in this example the robustness of the method to model uncertainties is also tested. If no external filter is used for reference tracking, the control loop exhibits an overshoot of 100.4%, Fig. 13(a). The use of both Cex1 (s), Cex2 (s) eliminates the overshoot to 9.4% and 0% respectively, Fig. 13(a). Disturbance rejection remains unaltered. Cex2 (s) is of the same form as in the previous example. Note that control signal u() is improved in case the reference is filtered, Figs. 11(b) and 13(b). 6.3. A non-minimum phase process Although the proposed theory does not take into account the existence of zeros in the process model, a non-minimum phase process of the form G(s ) =

1.34(1 − 0.771s ) (1 + s )(1 + 0.33s )(1 + 0.12s )(1 + 0.056s )(1 + 0.038s ) (118)

20

K.G. Papadopoulos, N.I. Margaris / Journal of Process Control 22 (2012) 11–25

Fig. 11. Type-III closed-loop control system. (a) Step response of the output of the control system and (b) control signal u(). Output disturbance rejection is applied at 1 is used for decreasing the overshoot of the output. Input disturbance di (s) = 0.1r(s) is applied at  = 250  = 250. External filter of the form Cex (s) = 2 (0.45tn2 tn3 )s +(tn2 +0.45tn3 )s+1

and output disturbance di (s) = 0.1r(s) is applied at  = 500.

Fig. 12. Type-III closed-loop control system. (a) Ramp response of the closed loop control system and (b) parabolic response of the closed-loop control system.

Fig. 13. Type-III closed-loop control system. (a) Step response of the output of the control system and (b) control signal u(). Output disturbance rejection is applied at 1 is used for decreasing the overshoot of the output. Input disturbance di (s) = 0.1r(s) is applied at  = 250  = 250. External filter of the form Cex1 (s) = 2 (0.45tn2 tn3 )s +(tn2 +0.45tn3 )s+1

and output disturbance di (s) = 0.1r(s) is applied at  = 500.

is adopted for testing the robustness of the proposed control law. The step response of (118) is presented in Fig. 14. In addition, in Fig. 15(a) and (b) the step response of the output y() and the control signal u() are presented respectively. If no external filter is used, the overshoot of the step response is 59.9%. Since this is undesirable, if r(s) is filtered by Cex1 (s), Cex2 (s) then the overshoot is reduced to 0% in both cases. Output and input disturbance rejection remain unaltered since the external filter does not participate into Si (s) = y(s)/di (s), So (s) = y(s)/di (s) respectively.

where the PID controller does not achieve pole-zero cancellation. Therefore, parameter Tn1 is determined by Tn1 = (1 + a)Tp1 where a is the error when measuring Tp1 . The process is given by

6.4. Controller tuning without pole zero cancellation

From Fig. 16(a) and (b) it is apparent that if an error of 30% when measuring Tp1 occurs, a small change is observed in the overshoot of the closed loop control system. In addition, both input and output disturbance rejection remain almost unaltered.

For testing the robustness of the proposed control law to parameter uncertainties, a type-III closed loop control system is designed

G(s ) =

1.23 . (1 + s )(1 + 0.872s )(1 + 0.367s )(1 + 0.287s )(1 + 0.11s ) (119)

K.G. Papadopoulos, N.I. Margaris / Journal of Process Control 22 (2012) 11–25

21

that the type-I control loop fails to track both the ramp and the parabolic reference signal achieving constant non-zero steady state velocity and acceleration error. 6.6. A type-IV and a type-V control loop From the Laplace transformation it is known that if r(t) = tn then L{y(t)} = n!/sn+1 . Specifically, if n = 1 then L{r(t)} = 1/s2 and the system is of type-II, or if n = 2 then L{r(t)} = 2/s3 and the system is of type-III. For a type-IV and type-V control loop the Laplace transformation of the reference signal is given if n = 3 and n = 4 for which we have L{r(t)} = 3 !/s3+1 and L{r(t)} = 4 !/s4+1 respectively. According to the proposed theory for a type-IV, V control loop the proposed PID type controllers are given by C(s) =

(1 + Tn1 s)(1 + Tn2 s)(1 + Tn3 s)(1 + Tn4 s) Ti s4 (1 + Tc s)(1 + Tc s) 1

Fig. 14. Step response of the non-minimum phase process defined by (118).

C(s) =

(1 + Tn1 s)(1 + Tn2 s)(1 + Tn3 s)(1 + Tn4 s)(1 + Tn5 s) Ti s5 (1 + Tc s)(1 + Tc s) 1

6.5. Comparison between a type-I and a type-III control loop For showing the advantages of designing a higher order faster control loop, the following process G(s ) =

1.23 (1 + s )(1 + 0.992s )(1 + 0.692s )(1 + 0.139s )(1 + 0.107s ) (120)

is adopted. For this process, a type-I, III closed control loop will be designed. For designing the PID type-I control loop the conventional Magnitude Optimum criterion (see Appendix D) is employed. Note that for determining controller’s zeros, exact pole zero cancellation has to take place (see Appendix D) [28]. From Fig. 17 it is apparent

.

(121)

2

(122)

2

respectively. For determining parameters Tn1 , Tn2 , Tn3 , Tn4 , Ti in (121) according to the proposed theory, we set Tn4 = Tp1 and Tn1 = Tn2 = nT according to (95). For that reason, (92) becomes 4(2nT + Tn3 ) − 4(n2 T + 2nTn3 ) + n2 Tn3 = 0

(123)

or finally Tn3 =

4n(n − 2) T . (n2 − 8n + 4)

(124)

Integrator’s time constant for the type-IV control loop is equal to Ti = 2kp kh Tn1 Tn2 Tn3 T .

(125)

Fig. 15. Type-III closed-loop control system for a non-minimum phase process. (a) Step response of the output of the control system and (b) control signal u(). Output 1 is used for decreasing the overshoot of the output. Input disturbance disturbance rejection is applied at  = 200. External filter of the form Cex1 (s) = 2 (0.45tn2 tn3 )s +(tn2 +0.45tn3 )s+1

di (s) = 0.1r(s) is applied at  = 200 and output disturbance di (s) = 0.1r(s) is applied at  = 300.

Fig. 16. Type-III closed-loop control system. The PID controller is tuned without pole zero cancellation: a = 0.3 and a = −0.3. The PID controller is tuned via exact pole-zero cancellation a = 0.

22

K.G. Papadopoulos, N.I. Margaris / Journal of Process Control 22 (2012) 11–25

Fig. 17. Comparison between a type-I, III PID control loop. The type-I control loop fails to track the ramp r() =  and the parabolic r() =  2 reference signal since constant steady state velocity and acceleration error is observed.

Fig. 18. (a) Response of the type-IV control loop to reference signal r(t) = t3 ; parameter n has been chosen equal to n = 10.89 according to (99). (b) Response of the type-V control loop to reference signal r(t) = t4 ; parameter n has been chosen equal to n = 14.32 according to (99).

In similar fashion, for the (122) PID type controller and since the control loop is of type-V, we set Tn5 = Tp1 and Tn1 = Tn2 = Tn3 = nT . Accordingly, (91) becomes 2 3 2 2 + 3nT Tn4 )T − 4(n3 T + 3n2 T Tn4 ) + n3 T Tn4 = 0 4(3n2 T

(126) and after some calculus results in Tn4 =

4n2 (n − 3) 4n(n − 3) T = 2 T . n(n2 − 12n + 12) n − 12n + 12

(127)

Integrator’s time constant for the type-V control loop is equal to Ti = 2kp kh Tn1 Tn2 Tn3 Tn4 T .

(128)

The controlled process in this example is defined by (120). The respective response to r(t) = t3 and r(t) = t4 reference signals for the type-IV and the type-V control loop are presented in Fig. 18. 6.7. Effect of the process unmodelled dynamics to the control performance

Fig. 19. Step response of the PID type-III control loop when a = 0.15 and a = 0.6 for a process defined by (129).

parasitic time constant of the process is comparable to its dominant time constant. Since

The effect of the process unmodelled dynamics is discussed in this example. The process defined by G(s) =

1 (1 + s )(1 + as )(1 + a2 s )(1 + a3 s )(1 + a4 s )

(129)

is adopted. As proved in Sections 4 and 5 the proposed control law depends on pole-zero cancellation and time constant T which models the process’ unmodelled dynamics (poles of the process far from the origin), see (32) and (34) where T = Tc + Tp and Tp is the process parasitic time constant and Tc  Tp . In Fig. 19 the process is modelled by a = 0.15 containing a relatively large dominant time constant and in the next case where a = 0.6 the

a = 0.15 then Tp = Tp1

4

4

Tp Tp1

j=1

=

4

j=1

aj , it is apparent that when

aj = 0.1764Tp1 and when a = 0.6 then

aj

Tp = Tp1 j=1 = 1.3056Tp1 . The conclusion according to Fig. 18 is that the less accurate the model of the process in terms of zeros, time delay, poles compared to the dominant time constant (T ≈ Tpj ), the poorer the performance becomes (see settling time of the output and input disturbance rejection Fig. 19). 7. Conclusions and discussion The Symmetrical Optimum criterion has been extended for the design of type-p control loop. Based on the conventional tuning

K.G. Papadopoulos, N.I. Margaris / Journal of Process Control 22 (2012) 11–25

for PID type controllers via the Symmetrical Optimum principle, a similar design technique for type-III control loops was proposed. It was shown that type-III control loop achieve zero steady state position, velocity and acceleration error and therefore they are able to track faster reference signals than type-I or II control loops. Based on this technique, the proposed control law was extended for tuning PID type-p control loops so that tracking of faster reference signals is achieved. The development of the proposed control is carried out in the frequency domain where the transfer function of the process involves the dominant time constants and the plant’s unmodelled dynamics. Future work deals with introducing the time delay constant6 as one more parameter in the proposed control law, since nowadays the time delay is straightforward to be measured for most industrial processes. The proposed theory has been evaluated for the control of representative plants met in many industry applications. The robustness of the proposed control law achieves promising results also for the control of processes with parameters the control law disregards, such as non-minimum phase processes and processes with time delay.

23

and |N(jω)|2  (b28 )ω16 + (b27 − b8 b6 )ω14 + (b26 + 2b4 b8 − 2b5 b7 )ω12 + (b25 + 2b3 b7 − 2b2 b8 − 2b4 b6 )ω10 + (b24 + 2b0 b8 + 2b2 b6 − 2b1 b7 − 2b3 b5 )ω8 + (b23 + 2b1 b5 − 2b6 b0 − 2b2 b4 )ω6 + (b22 + 2b0 b4 − 2b1 b3 )ω4 + (b21 + 2b0 b2 )ω2 + (b0 )ω0 . Finally, |T(jω)|2 is equal to |T (jω)|2 =

|N(jω)| |D(jω)|

2

2

=

· · · + B4 ω8 + B3 ω6 + B2 ω4 + B1 ω2 + B0 · · · + A4 ω8 + A3 ω6 + A2 ω4 + A1 ω2 + A0

The authors would like to express their greatful thanks to the three anonymous reviewers for their valuable feedback during the peer review process.

. (A.8)

ωj

(j = 1, 2, . . ., n) in polynomials By making equal the terms of |D(jω)|2 , |N(jω)|2 so that |T(s)|  1 in the wider possible frequency range results in a0 = b0

(A.9)

a21 − 2a2 a0 = b21 − 2b2 b0

(A.10)

a22 − 2a3 a1 + 2a4 a0 = b22 − 2b3 b1 + 2b4 b0

Acknowledgements

(A.7)

a23

+ 2a1 a5 − 2a6 a0 − 2a4 a2 =

b23

(A.11)

+ 2b1 b5 − 2b6 b0 − 2b4 b2 (A.12)

a24 + 2a0 a8 + 2a6 a2 − 2a1 a7 − 2a3 a5 = b24 + 2b0 b8 + 2b6 b2 − 2b1 b7 − 2b3 b5

(A.13)

Appendix A. Optimization conditions ··· = ···

Let the closed-loop transfer function be defined by (A.1), bm sm + bm−1 sm−1 + · · · + b2 s2 + b1 s + b0 N(s) = D(s) an sn + an−1 sn−1 + · · · + a2 s2 + a1 s + a0

T (s) =

(A.1)

where m ≤ n. By applying the Symmetrical Optimum criterion to (A.1) we will force |T(s)|  1 in the wider possible frequency range. Thus, by setting s = jω into (A.1) and squaring |T(jω)| leads to 2

|T (jω)| =

|N(jω)| |D(jω)|

Let the integrating process be defined by G(s) =

2

(A.2)

2

1 , sT m (1 + sT p1 )(1 + sT p )

m

2

N(jω) (jω) bm + · · · + (jω) b2 + (jω)b1 + b0 = . n 2 D(jω) (jω) an + · · · + (jω) a2 + (jω)a1 + a0

(A.3)

C(s) =

1 , sTi (1 + sTc )

T (s) =

N(jω)  · · · + b8 ω8 − b6 ω6 + b4 ω4 − b2 ω2 + b0 + j(· · · − b7 ω7 + b5 ω5 − b3 ω3 + b1 ω)

(A.4)

and D(jω)  · · · + a8 ω8 − a6 ω6 + a4 ω4 − a2 ω2 + a0 + j(· · · − a7 ω7 (A.5)

|D(jω)|2  (a28 )ω16 + (a27 − a8 a6 )ω14 + (a26 + 2a4 a8 − 2a5 a7 )ω12 + 2a3 a7 − 2a2 a8 − 2a4 a6 )ω

− 2a1 a7 − 2a3 a5 )ω

8

+ (a23

10

+ (a24

+ 2a0 a8 + 2a2 a6

+ 2a1 a5 − 2a6 a0 − 2a2 a4 )ω6

+ (a22 + 2a0 a4 − 2a1 a3 )ω4 + (a21 + 2a0 a2 )ω2 + (a0 )ω0

6

(A.6)

Time delay constant  canbe introduced in the process model (38) by the Taylor 

series es d =

∞

k=0

1 k k s d k!

.

(B.3)

kp s4 Ti Tm Tp1 T + s3 Ti Tm (Tp1 + T ) + s2 Ti Tm + kh kp

.

(B.4)

According to (B.4), it is evident that T(s) is unstable since the term of s is missing. In similar fashion, if PI control of the form C(s) =

or

kp s2 Ti Tm (1 + sTp1 )(1 + sT ) + kh kp

where Tp Tc ≈ 0 and T = Tp + Tc . From (B.3) it is evident T (s) =

+ a5 ω5 − a3 ω3 + a1 ω)

(B.2)

is applied, then the closed loop transfer function is given by

Polynomials N(jω) and D(jω) are rewritten as follows

+ (a25

(B.1)

where Tm , Tp1 , Tp have been defined in Section 3. If for controlling (B.1), I control of the form

or T (jω) =

Appendix B. Instability of the PI control – conventional tuning via the Symmetrical Optimum criterion

1 + sT n sTi (1 + sTc )

(B.5)

is employed, then for determining controller parameter Tn via the conventional Symmetrical Optimum criterion, pole-zero cancellation must take place, Tn = Tp1 . Therefore, T(s) becomes T (s) =

kp s4 Ti Tm Tp1 T + s3 Ti Tm (T + Tp1 ) + s2 Ti Tm + kh kp

,

(B.6)

which is unstable again for the same reason as stated for (B.4). 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 (B.4). This is

24

K.G. Papadopoulos, N.I. Margaris / Journal of Process Control 22 (2012) 11–25

justified by the Routh theorem. For a polynomial of the form D(s) = an sn + an−1 sn−1 + · · · + a1 s + a0 , necessary and sufficient condition for D(s) to be stable is aj > 0, j = 0, 1, 2, . . .. Since both in (19), (16) and (B.6), a3 = 0 and a1 = 0 then according the Routh theorem, D(s) is unstable.

By substituting (D.4) into (D.3) and calculating |T(jω)| results in



|T (jω)| ≈

kp2 Ti2 T2 ω4

+ (Ti − 2kp kh T2 )Ti ω2 + kp2 kh2

.

(D.5)

Therefore, condition |T(jω)|  1 is satisfied when Appendix C. Proof of the parameter n

kh = 1,

In Section 5 it was shown that zeros of the controller for type-V, IV, III control loops are given by

(C.1)

− 4(Tn1 Tn2 + Tn2 Tn3 + Tn1 Tn3 )T

+Tn1 Tn2 Tn3 = 0. 2 4T − 4(Tn1 + Tn2 )T + Tn1 Tn2 = 0.

3 3 3 4 + n3 T + n3 T + n3 T )T + n4 T = 0,

(C.3)

References

(C.4)

(C.5) 2 2 − 4(nT + nT )T + n2 T 4T =0

(C.6)

or 4 4 4 24n2 T − 16n3 T + n4 T = 0,

(C.7)

3 3 12nT 3 − 12n2 T + n3 T = 0,

(C.8)

= 0.

(C.9)

/ 0, from (C.6)–(C.9) we obtain (98), (99) and (100) Since T = respectively. Appendix D. The conventional Magnitude Optimum criterion For controlling the process defined by G(s) =

1 , (1 + sT p1 )(1 + sT p2 )(1 + sT 2 )

where T2 = p

n

T i=3 pi

(D.1)

p

stands for the process unmodelled dynam-

ics, the PID controller of the form C(s) =

(1 + Tn1 s)(1 + Tn2 s) Ti s(1 + Tc s)

(D.2)

is adopted. According to the conventional Magnitude Optimum criterion parameters Tn1 , Tn2 , Ti will be determined so that |T(jω)|  1 in the widest possible frequency range. Assuming that Tc  T2 and p

T2 = T2 + Tc , the transfer function of the closed loop control p

system according to Fig. 1 is equal to T (s) =

kp (1 + sTn1 )(1 + sTn2 ) . sTi (1 + sTp1 )(1 + sTp2 )(1 + sT2 ) + kh kp (1 + sTn1 )(1 + sTn2 ) (D.3)

By forcing pole zero cancellation according to Tn1 = Tp1 ,

Tn2 = Tp2 .

1 . 2s2 + 2s + 1

T (s ) =

2 2 2 2 3 − 4(n2 T + n2 T + n2 T )T + n3 T = 0, 4(nT + nT + nT )T

2 + n2 T

1 2 s2 + 2T 2T 2 s + 1

(C.2)

2 2 2 2 2 2 2 3 + n2 T 4(n2 T + n2 T + n2 T + n2 T + n2 T )T − 4(n3 T

− 8nT 2

T (s) =

(D.7)

and normalizing the time by setting s = sT2 leads to

By substituting (95) into (C.1)–(C.3) results in

2 4T

(D.6)

2

+Tn1 Tn2 Tn3 Tn4 = 0 2 4(Tn1 + Tn2 + Tn3 )T

= 2kp kh (T − Tn1 − Tn2 ). By substituting (D.4) and (D.6) into (D.3) results finally

2 4(Tn1 Tn2 + Tn1 Tn3 + Tn1 Tn4 + Tn2 Tn3 + Tn2 Tn4 + Tn3 Tn4 )T

−4(Tn1 Tn2 Tn3 + Tn1 Tn2 Tn4 + Tn2 Tn3 Tn4 + Tn1 Tn3 Tn4 )T

Ti = 2kp kh T2 = 2kp kh (T − Tp1 − Tp2 )

(D.4)

(D.8)

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