2012 IEEE International Conference on Control Applications (CCA) Part of 2012 IEEE Multi-Conference on Systems and Control October 3-5, 2012. Dubrovnik, Croatia
Digital PID Type-III Control Loop Design via the Symmetrical Optimum Criterion Konstantinos G. Papadopoulos1 , Nikolaos D. Tselepis2 , Nikolaos I. Margaris2 Abstract— An analytical digital PID control law for the design of type-III control loops is developed. The proposed control law involves both dominant time constants of the controlled process, its unmodelled dynamics plus the sampling time of the controller. Basis of the proposed theory is the well known Symmetrical Optimum criterion. The development of the control law takes place in the frequency domain. For applying the proposed theory an accurate estimation of the plant’s dominant time constants is required. The final control loop has the advantage of achieving zero steady state position, velocity and acceleration error. The potential of the proposed theory is justified via simulation examples for benchmark process models met frequently in many industry applications.
I. I NTRODUCTION The need for designing higher order type (type-I, II, III . . .) control loops arises from the fact that the higher the order is, the faster reference signals can be perfectly tracked by the output of the controlled process. By perfect tracking, it is meant that the control loop is able to eliminate higher order errors at steady state. Specifically, if a type-I control loop achieves zero steady state position error, then a typeII control loop achieves both zero steady state position and velocity error. On a theoretical basis, the type of the control loop is determined by the number of the free pure integrators in the open loop transfer function if modelling in the frequency domain is to be followed, see [1], [2]. In industry applications, type-II control loops are frequently designed when the controlled process has an integrating behaviour and PID control is to be applied. A common industrial example of this case is met on the vector control of induction motor drives when designing the inner current control loop and the outer speed control loop. In addition to the aforementioned definitions, a typeIII control loop is expected to achieve zero steady state position, velocity and acceleration error. In this case, the existence of three free pure integrators is required in the open loop transfer function. Since scope of this work is to present a feasible control law that can be applied in many industry applications, the simplicity and widespread application of the PID control law will be exploited, [3]. To this end, if the process is of type-0, the proposed control *This work was supported by Aristotle University of Thessaloniki and is not a statement by ABB Switzerland Ltd. 1 K.G Papadopoulos is with ABB Switzerland Ltd., Department of Medium Voltage Drives, Turgi, CH-5300, Switzerland, email:
[email protected]. 2 N.D Tselepis and N.I. Margaris are with the Aristotle University of Thessaloniki, Department of Electrical & Computer Engineering, GR-54124, Greece, email:
[email protected],
[email protected].
978-1-4673-4504-0/12/$31.00 ©2012 IEEE
law has to be of PI3 D (type-III control loop) and if the plant is of type-I (integrating process) then consequently, the proposed control law has to of be PI2 D, (type-III control loop). Recently, a promising systematic PID tuning method for the design of type-III control loop has been proposed, see [1], [2]. The method presented in [1] proposes a PID control law which can be applied for the design of type-p control loops. For the development of this theory, the principle of the well known Symmetrical Optimum criterion [4]–[7] has been adopted. The Symmetrical Optimum criterion implies that the controller is designed in such a way so that the magnitude of the closed loop transfer function is rendered around the unity in the widest possible frequency range, |T ( jω)| ≃ 1. Basic principle for tuning the PID controller when the Symmetrical Optimum criterion is to be followed, is that an accurate estimation of the plant’s dominant time constants is required. If this is possible, then pole-zero cancellation takes place between the process’ poles and the controller’s zeros and the integrator’s time constant (I) is determined after setting |T ( jω)| ≃ 1. In similar fashion with the aforementioned method, scope of the current work is to follow the same principle presented in [1] (conventional symmetrical optimum criterion tuning) and to further extend the proposed control law to the design of digital control loops. Since nowadays, most of PID control loops are digitally implemented, purpose of this work is to involve the sampling time Ts of the controller in the final control law. This will allow control engineers to 1) design a type-III control loop and 2) perform off-line accurate investigations regarding the affect of the sampling time of the controller on the control loop performance. Such investigations will be presented at the end of this work in section IV. For the sake of a clear presentation of the proposed method, this work is organized as follows. In section II the analog PID control law is presented for type-III control loops. Based on this theory, in section III the sampling time of the controller is introduced in the control loop. The determination of the controller gains takes place in the frequency domain and their final tuning formulas involve also the sampling time Ts apart from the plant’s dominant time constants and unmodelled dynamics. For justifying the proposed theory’s potential, the control of two benchmark processes met frequently in many industry applications is investigated in section IV. II. T YPE -III C ONTROL L OOPS -A NALOG D ESIGN Let us consider the closed loop control system of Fig.1 where r(s), e(s), u(s), y(s), do (s) and di (s) are the reference
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input, the control error, the input and output of the plant, the output and the input disturbances respectively. For extracting the suboptimal control law, the integrating process 1 G (s) = (1) sTm (1 + sTp1 )(1 + sTΣp ) met in many industry applications is adopted. Tp1 represents the process’s dominant time constant and Tm , TΣp stand for the integrator’s time constant and the unmodelled plant dynamics respectively. For controlling (1) the proposed PID type controller C (s) =
(1 + sTn ) (1 + sTv ) (1 + sTx ) . s2 Ti (1 + sTΣc1 )(1 + sTΣc2 )
(2)
is employed, see [1], [2]. I and PI control cannot be applied since the final control loop is unstable, see Appendix B. Note that the product kh k pC(s)G(s) (open-loop transfer function) contains three free integrators so that the final closed-loop control system is of type-III. For determining controller parameters according to the Symmetrical Optimum criterion, zero-pole cancellation has to be achieved. In that, by setting Tx = Tp1 and assuming that TΣc = TΣc1 + TΣc2 , TΣc1 TΣc2 ≈ 0, the transfer function of the control loop is equal to
one way to optimize the magnitude of (4) is to set the terms of ω j , j = 2, 4, 6, . . ., in (4), equal to zero, starting from the lower frequency range, [1], [2], [4]. Setting kh = 1 and 2k k T T T the term of ω 6 equal to zero leads to Ti = p hTmn v Σ . In similar fashion, setting the term of ω 4 equal to zero along with the aid of Ti , drives to 4TΣ2 − 4 Tn + Tv TΣ + Tn Tv = 0. By choosing Tv = nTΣ then Tn becomes Tn = 4(n−1) n−4 TΣ . Note that n > 4 must hold by so that a feasible PI2 D control law is extracted. By substituting the definitions of the gains Ti , Tn , Tv into the closed loop transfer function results in 4n (n − 1) TΣ2 s2 + n2 − 4 TΣ s + n − 4 # . (8) T (s) = " 8n (n − 1) TΣ4 s4 + 8n (n − 1) TΣ3 s3 + 4n (n − 1) TΣ2 s2 + n2 − 4 TΣ s + (n − 4)
Normalizing the time by setting s′ = sTΣ , (8) becomes 4n (n − 1) s′ 2 + n2 − 4 s′ + (n − 4) ′ #. T s =" 8n (n − 1) s′ 4 + 8n (n − 1) s′ 3 + 4n (n − 1) s′ 2 + n2 − 4 s′ + (n − 4) (9) The control loop defined in (9) is of type-III. This is justified by the equality of the terms of s′ j , j = 0, 1, 2, a0 = 2 s k p Tn Tv + sk p (Tn + Tv ) + k p b , a = b1 , a2 = b2 of the closed-loop transfer function. The #, 0 1 T (s) = " 4 3 2 respective step and frequency response of (9) are presented s Ti Tm TΣ + s Ti Tm + s k p kh Tn Tv + sk p kh (Tn + Tv ) for two different values of parameter n in Fig.4(a), Fig.4(b) +k p kh respectively. In addition, in Fig.2 the open loop frequency (3) where TΣ = TΣc + TΣ p and TΣc TΣ p ≈ 0. Since the magnitude of (3) is given by 10 |FOL(ju)| v n = 7.46 u 2 (1 − T T ω 2 )2 + k2 (T + T )2 ω 2 u k n v n v p p |T ( jω)| = u #2 " #2 . u" 10 u Ti Tm TΣ ω 4 + (k k (T + T ) p n v h n = 4.1 t 2 + ω uc = 0.5 uc = 1 k p kh (1 − Tn Tv ω 2 ) −Ti Tm ω 2 ) (4) |φOL(ju)| = and the denominator of (4) is defined by D(ω) √ A0 + A1 + A2 where -150 5
0
= (Ti Tm TΣ )2 ω 8 + Ti Tm (Ti Tm − 2k p kh Tn Tv TΣ ) ω 6 (5) # " 2Ti Tm TΣ − 2 (Tn + Tv ) Ti Tm ω4 (6) = k p kh +k p kh Tn2 Tv2 = (k p kh )2 Tn2 + Tv2 ω 2 + (k p kh )2 (7)
A0 A1 A2
n r (s)
r(s)
e(s)
+
d i (s)
controller
C(s)
-
u(s) +
+
do(s)
kp
G(s)
+
n = 7.46
n = 4.1 -180 -2 10
Fig. 2. system.
o
φm = 35 -1
2
10
10
u = ωΤΣ
Open loop frequency response of type-III closed loop control
n r (s)
y(s)
di(s)
controller
Cex(s)
e(s)
r ΄ (s) +
C(s)
-
d o (s)
u(s) + +
kp
G(s)
+ +
y(s)
+
kh S
1
10
+
r(s)
y f (s)
0
10
+
y f (s)
+
kh S
n o (s)
+
n o (s)
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 stands for the plant’s dc gain and kh is the feedback path.
Fig. 3. 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 disturbance transfer function So (s) = dy(s) . o (s)
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2
n = 7.46
1.5
|S(ju)|
|T(ju)|
0
10
0.707
yr(τ) 1
-1
10
n = 4.1
n = 4.1
0.5
yd(τ)
un = 0.28
-2
10
0
n = 7.46
un = 0.85
-0.5
-3
10
n = 7.46
-1 0
5
10
15
20
25
30
-4
-3
10
(a) Step response (|T (s)| = | y(s) r(s) |) and disturbance rejection
-2
10
-1
10
0
10
1
10
2
10
10
(b) Frequency response of type-III closed loop control system. Sensitivity |S( ju)| = | dy(o (ju) ju) | and complementary sensitivity |T ( ju)| =
(|S(s)| = | dy(s) |) of type-III closed loop control system. o (s)
ju) | y( r( ju) |.
Fig. 4. Step and frequency response of the type-III closed loop control system for various values of the parameter n. The shape of the step response remains practically unaltered regardless of the value of the parameter n.
response is shown, from which we conclude that the magni ju) tude of the closed-loop transfer function |T ( ju)| = y( r( ju) is practically independent of the parameter n. Moreover, sensitivity |S ( ju)| = dy(o (ju) 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.2, (≈ 50%). Since the phase margin is ϕm = 35◦ < 45◦ , an undesired overshoot in the step response of the closed loop system is expected, Fig.4(a). This can be decreased along with the aid of an external filter Cex (s) = 1 1+0.1(Tn +Tv )s for example, Fig.5. III. T YPE -III C ONTROL L OOPS -D IGITAL D ESIGN Let the integrating process in series with its constant gain k p 1 at steady state in Fig.5 be defined by (1). The proposed PID type controller is given by C(s) = C∗ (s)CZOH (s) = (1 + sTn ) (1 + sTv ) (1 + sTx ) ∗ 1 − e−sTs = s2 Ti (1 + sTΣc1 ) (1 + sTΣc2 ) sTs
1 slope
of the step response.
Ti Ts , tΣc1
= TTms , t p1 = is given by Tx Ts , tm
=
TΣc 1 Ts
, tΣc2 =
Tp1 Ts , tΣP
=
TΣ p Ts .
TΣc 2 Ts
, tn =
Tn Ts , tv
The product k p
(1 + s′tn ) (1 + s′tv ) (1 + s′tx ) k pC s G s = Ts ′2 s ti (1 + s′tΣc1 ) (1 + s′tΣc2 ) kp s′tm (1 + s′t p1 )(1 + s′tΣ p ) ′
′
Tv Ts , tx = C(s′ )G(s′ )
=
∗
′
(1 − e−s ) s′
(13)
For determining the proposed control law, the same line presented in section II will be followed. In that, zero-pole cancellation will take place by setting tx = t p1 . To this end, (13) becomes ∗ k p Ts (1 + s′tn ) (1 + s′tv ) k pC s′ G s′ = ′ s tm (1 + s′tΣ p ) s′2ti (1 + s′tΣc1 ) (1 + s′tΣc2 ) ′
(1 − e−s ) s′
(10)
where CZOH (s) stands for the zero order hold transfer function and Ts stands for the controller sampling period. All time constants in the control loop are normalized in the frequency domain with the sampling period Ts . In that by substituting s′ = sTs (1) and (10) become 1 (11) G(s′ ) = ′ s tm (1 + s′t p1 )(1 + s′tΣ p ) and C(s′ ) = C∗ (s′ )CZOH (s′ ) = ′ (1 + s′tn ) (1 + s′tv ) (1 + s′tx ) ∗ (1 − e−s ) = Ts ′2 s ti (1 + s′tΣc1 ) (1 + s′tΣc2 ) s′
where ti =
(14)
The transition from the Laplace domain to the z domain takes ′ s′ and 1′ 2 = place by utilizing the relation s1′ = z′z−1 = se′ Ts z′ (z′ −1)2
′
Ts es
=
2
′
(es −1)
. Since
n r (s)
r(s)
-
*
Ts
C (s)
C ZOH (s)
e −1
s ′ k pC (s ) G (s′ )
=
′ es ,
di(s)
d o (s)
u(s)
+
+
controller
e(s)
+
z′
+
the product
kp
G(s)
y(s)
+
kh
(12)
+
+
n o (s)
Fig. 5. Closed loop control system including the digital PID type controller.
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it is concluded that (19) becomes equal to
takes the form "
k p Ts (1 + s′tn )(1 + s′tv ) k pC(s )G(s ) = ′ ′ s tm (1 + s tΣ p ) ti (1 + s′tΣc1 )(1 + s′tΣc2 ) ′
′
′
′
(1 − e−s )es Ts es ′
′
#
T (s′ ) =
′
=
2
(es − 1)(es − 1)
′
′
k p es (1 + s′tn ) (1 + s′tv ) 2 ′ ′ ′ s′tmti (1 + s′tΣ ) es − 1 + k p kh es (1 + s′tn ) (1 + s′tv ) h ′ i ′ ′ s′2 k ptntv + s′ k p (tn + tv ) + k p (D1 (s′ ) + 1)
(15)
or finally after some calculus,
(s′2titmtΣ + s′tmti )D2 (s′ )+ h ′ i ′ ′ s′2 k p khtntv + s′ k p kh (tn + tv ) + k p kh (D1 (s′ ) + 1) (22)
For simplifying the determination of the optimal control law, the following substitutions take place. In the ′ ′ numerator of (22) it is set z1 = k p + k p (tn + tv ), z2 = ′ ′ ′ ′ 1 ′ 1 ′ 1 ′ es 2! k p + k p (tn + tv ) + k p tn tv , z3 = 3! k p + 2! k p (tn + tv ) + k p tn tv ′ ′ ′ 2 ′ and z4 = 4!1 k p + 3!1 k p (tn + tv ) + 2!1 k ptntv . In the denominator (es − 1) (16) of (22) it is set r1 = kh z1 , r2 = kh z2 , r3 = kh z3 + tmti , r4 = kh z4 +tmti +tmtitΣ , r5 = kh z5 + 0.5833tmti +tmtitΣ , r6 = kh z6 + ′ 0.25t 2 m ti +0.5833tm ti tΣ and r7 = kh z7 +0.0861tm ti +0.25tm ti tΣ . The substitution k p = k p Ts results in For determining the proposed analytical control law, the ′ ′ optimization conditions (34) to (37) (Symmetrical Optimum k p es (1 + s′tn )(1 + s′tv ) ′ ′ k pC(s )G(s ) = . criterion) presented in Appendix B will be applied to (22). 2 ′ s′tmti (1 + s′tΣ p )(1 + s′tΣc1 )(1 + s′tΣc2 )(es − 1) Optimization Condition 1: a0 = b0 (17) By applying the first optimization condition to the closed loop transfer function results in By setting tΣc1 tΣc2 ≈ 0 and tΣc = tΣc1 +tΣc2 it is found that (1+ kh = 1. (23) s′tΣ p )(1 + s′tΣc1 )(1 + s′tΣc2 ) = (1 + s′tΣ p )(1 + s′tΣc ). In similar fashion, if tΣc tΣ p ≈ 0 and tΣ = tΣc + tΣ p then (17) becomes which implies that the final closed loop control system equal to exhibits steady state position, velocity error. From (22) it is apparent that if kh = 1, then N(s′ ) = k ptntv s′2 + s′ k p (tn + ′ s′ ′ t )(1 + s′ t ) ′ ′2 ′ e (1 + s k n v p (18) tv ) + k p and D(s ) = · · · + k p khtntv s + s k p kh (tn + tv ) + k p kh k pC(s′ )G(s′ ) = 2 ′ ′ ′ s respectively. s tmti (1 + s tΣ )(e − 1) Optimization Condition 2: a21 − 2a0 a2 = 0 According again to Fig.1, the closed loop transfer function The application of (35) to (22) results after some calculus in T (s′ ) = y(s) r(s) is equal to tn2 + tv2 = 0. (24) " # k p Ts2 (1 + s′tn )(1 + s′tv ) k pC(s )G(s ) = ′ s tm (1 + s′tΣ p ) ti (1 + s′tΣc1 )(1 + s′tΣc2 ) ′
′
′
′
Optimization Condition 3: a22 + 2a0 a4 − 2a1 a3 = 0
k p es (1 + s′tn )(1 + s′tv )
2 ′ In similar fashion, the application of (36) to (22) results in s′tmti (1 + s′tΣ )(es − 1) = ′ ′ s′ ′ ′ (25) k p khtn2tv2 − 2tmti (tn + tv − tΣ ) = 0 k p e (1 + s tn )(1 + s tv ) 1 + kh 2 ′ s′tmti (1 + s′tΣ )(es − 1) Optimization Condition 4: a23 + 2a1 a5 − 2a6 a0 − 2a2 a4 = 0 ′ s′ ′ ′ k p e (1 + s tn )(1 + s tv ) The integrator’s time constant is calculated after the appli= 2 ′ ′ ′ cation of (37) to (22). This leads to s′tmti (1 + s′tΣ )(es − 1) + kh k p es (1 + s′tn )(1 + s′tv ) ′ (19) (26) titm − k p kh [2tΣtntv + 0.166(tΣ − tn − tv )] = 0
T (s′ ) =
k pC(s′ )G(s′ ) = 1 + k p khC(s′ )G(s′ )
or finally
′
Approximating the delay time constant es by the series ′
D1 (s′ ) = es −1 = s′ +
′
1 ′2 1 ′3 1 ′4 1 ′5 1 ′6 s + s + s + s + s +· · · 2! 3! 4! 5! 6! (20)
k p kh [2tΣtntv + 0.166(tΣ − tn − tv )] tm Substituting (27) into (25) results in ti =
and ′
s′
4tΣtntv (tn + tv − tΣ ) − 0.3332(tn + tv − tΣ )2 = tn2tv2 2
D2 (s ) = (e − 1) 1 2 1 3 1 4 1 5 1 6 = (s′ + s′ + s′ + s′ + s′ + s′ + · · · )2 2! 3! 4! 5! 6! 6 5 4 3 2 = · · · + 0.0861s′ + 0.25s′ + 0.5833s′ + s′ + s′ (21)
(27)
(28)
If the same line (section II) of choosing parameter tv is to be followed, it is set tv = ntΣ . To this end, if tv = ntΣ is substituted into (28) results in √ −[(n − 1)tΣ (4ntΣ2 − 0.6664)] ± ∆ tn1,2 = (29) 2[ntΣ2 (4 − n) − 0.3332]
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2
800
79.4%
PID control 700
1.5
y(τ)
with Cex(s)
600 500
1
400
400 350
0.5
300
do(τ) = 0.25r(τ)
300
y(τ)
250 200
200 150
0 100
r( τ ) = τ
100
2
r( τ ) = τ
50
2
0
PID control
0
-0.5
50
100
150
200
250
300
350
400
0 0
500
1000
1500
2000
0
(a) Step response and disturbance rejection of type-III closed loop control system. Fig. 6.
100
200
300
400
500
600
700
800
(b) Ramp response of the type-III closed loop control system. Reference signal is s12 .
Step and ramp response of the type-III closed loop control system.
(a) Step response and disturbance rejection of type-III closed loop control system. Fig. 7.
(b) Parabolic response of the type-III closed loop control system. Reference signal is s23 .
Step and parabolic response of the type-III closed loop control system.
where ∆ = n(n − 1)2tΣ4 [16n − 1.33(n − 3)]. Since tΣ = TTΣs , Ts is a design parameter, parameter tn is calculated out of (29). Parameter n must be chosen so that n(n − 1)2tΣ4 [16n − 1.33(n − 3)] > 0. To this end, n > 0 and n > 1. If n is chosen such so that n > 1 then is is easily shown that ∀n > 1, 16n − 1.33(n − 3) > 0.
presented. Controller unmodelled dynamics have been set equal to tsc = 0.1 and normalizing time constant is equal to s′ = sTs . A. Process with dominant time constants In this example the process
IV. E VALUATION R ESULTS For testing the proposed control law, two representative processes met in many industry applications are investigated. In the following examples the control of both a process with two dominant time constants and a non-minimum phase process is considered. In the second example the robustness of the proposed control law is investigated, since no zeros are considered in the process defined by (11). In each example we plot the step response (r(s) = 1s ) of the final control loop. The response to a ramp (r(s) = 1s , section IV-A) and a parabolic reference signal (r(s) = s22 , section IV-B) is also
G s′ = "
1 (1 + 10s′ ) (1 + 8.7s′ ) (1 + 3.7s′ ) (1 + 2.6s′ ) (1 + 2.1s′ )
#
(30) consisting of five equal dominant time constants is considered. In Fig.6(a) the step response of the type-III control loop when r(s) = 1s and do (s) = 0.25r(s) is presented. The transfer function of the external controller (reference filter) has been chosen equal to Cex (s) = 1+s(t + t1v )+t tv s2 . The time instant of n 4 n 4 the disturbance do (s) is equal to τ = 1000. The overshoot of the control loop without filtering the reference r(s) is equal
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to 79.4% while the overshoot after filtering the reference is decreased to 10.87%. In Fig.6(b) the ramp response of the type-III control loop is presented. The output of the control loop achieves zero steady state position 6(a) and velocity error Fig.6(b).
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 2 a1 − 2a2 a0
a22 − 2a3 a1 + 2a4 a0
B. A non-minimum phase process
a23 + 2a1 a5 − 2a6 a0
The non-minimum phase process G s′ = "
−2a4 a2
1 − 5s′
(1 + 10s′ ) (1 + 8.8s′ ) (1 + 4.7s′ ) (1 + 4.4s′ ) (1 + 2.05s′ )
v
n
= =
−2b4 b2
(35) (36) !
(37)
have to hold by.
v
V. C ONCLUSIONS AND O UTLOOK An analytical PID control for digital controllers has been developed regarding the design of type-III control loops. The advantage of type-III control loops is their ability to eliminate higher order errors at steady state. Such loops are able to track a step, ramp and parabolic reference signal achieving zero steady state position, velocity and acceleration error respectively. The proposed control law is based on the well known Symmetrical Optimum criterion and consists of analytical expressions that involve dominant time constants of the process, unmodelled dynamics plus the sampling time of the controller. To this end, control engineers are able to design 1) type-III control loops 2) perform extensive investigations on the robustness of the control loop in terms of input di (s), output do (s) disturbances and plant’s parameters variations 3) investigate the affect of the sampling time on the control loop performance.
B. The Conventional Symmetrical Optimum Criterion Let the integrating process be defined by 1 , (38) G(s) = sTm (1 + sTp1 )(1 + sTΣp ) where Tm , Tp1 , TΣp have been defined in section II. If for 1 , is controlling (38), I control of the form C (s) = s2 T (1+sT i Σc ) applied, then the closed loop transfer function is given by kp T (s) = 3 (39) s Ti Tm (1 + sTp1 )(1 + sTΣ ) + kh k p where TΣp TΣc ≈ 0 and TΣ = TΣp + TΣc . From (39) it is evident T (s) =
kp 5 4 s Ti Tm Tp1 TΣ + s Ti Tm (Tp1
+ TΣ ) + s3 Ti Tm + kh k p (40) According to (40), it is evident that T (s) is unstable since the terms of s, s2 are missing. In similar fashion, if PI n is employed, then control of the form C (s) = s2 T1+sT i (1+sTΣc ) 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 . 4 s Ti Tm TΣ + s3 Ti Tm + kh k p
(41)
which is unstable again for the same reason as stated for (40). Finally, PID control by canceling 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 (41).
A PPENDIX A. Proof of Optimization Conditions Let the closed loop transfer function be defined by (32), T (s) =
=
(34)
b21 − 2b2 b0 b22 − 2b3 b1 + 2b4 b0 b23 + 2b1 b5 − 2b6 b0
··· = ···
#
(31) is considered in this example. The response of the control loop to a step reference input r(s) = 1s and a step output disturbance do (s) = 0.25r(s), is presented in Fig.7(a). From there, it is evident that the use of the external controller Cex (s) has increased the rise time tr of the step response but the overshoot has been decreased from 104.61% to 0%. Output disturbance rejection is unaffected since Cex (s) is , (r(s) = di (s) = not involved in the inner loop So (s) = dy(s) o (s) nr (s) = no (s) = 0). The output of the control loop achieves zero steady state position 7(a) and acceleration error 7(b). The external controller for filtering the reference input has been chosen equal to Cex (s) = 1+s(t +0.55t1 )+t 0.55t s2 . n
!
= b0
sm bm + sm−1 bm−1 + · · · + s2 b2 + sb1 + b0 N (s) = sn an + sn−1 an−1 + · · · + s2 a2 + sa1 + a0 D (s) (32)
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 (32) and squaring |T ( jω)| leads to |N ( jω)|2 . (33) |T ( jω)|2 = |D ( jω)|2 1608
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