Ind. Eng. Chem. Res. 2002, 41, 2679-2688
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PROCESS DESIGN AND CONTROL Enhanced Control with a General Cascade Control Structure Yongho Lee* and Sunggook Oh Department of Process Engineering, Samsung Engineering Company Ltd., 946-1 Glass Tower, Taechi-dong, Kangnam-Gu, Seoul 135-708, Korea
Sunwon Park Department of Chemical Engineering, KAIST 373-1, Gusong-Dong, Usong-gu, Taejon 305-701, Korea
In this paper, a general cascade control structure is proposed to handle stable, integrating, and unstable processes for cascade control systems. With the structure, a IMC controller is derived to get a desired closed-loop response and converted to a conventional PID controller. Several examples are given to illustrate the superiority of the structure and the proposed controller. The new structure is more robust to measurement noises than a conventional cascade control structure. 1. Introduction Cascade control is often used to improve single-loop control performance when the disturbances are associated with the manipulated variable or when the final control element exhibits nonlinear behavior. The performance enhanced by cascade control heavily depends on the tuning of controllers in inner and outer loops. However, previous tuning methods for cascade control appear to be rather limited to the frequency response methods1-3 that are tedious to apply because of the need for trial and error graphical calculations. Krishnaswamy and Rangaiah4 provided tuning charts that predict the primary controller settings for minimizing ITAE criterion due to load disturbances on the secondary loop in cascade control systems. However, the method is of limited use for the PI/P configuration and the first-order plus dead time (FOPDT) model over a limited range of model parameters. Lee et al.5 proposed analytical tuning rules for general stable processes for cascade control systems. The foregoing approaches involve two steps for designing controllers for cascade control systems: first, the secondary controller is tuned on the basis of the dynamic model of the inner process; second, the primary controller is tuned on the basis of the dynamic model of the outer process including the secondary loop. However, the tuning of inner and outer loop controllers by their method can be done simultaneously, and no identification is required even when the secondary controller is retuned because the tuning method is based on the model parameters of the process. Although the method proposed by Lee et al.5 eliminated foregoing tedious trial and error graphical calculations, it still cannot handle unstable or integrating cascade processes. * To whom correspondence should be addressed. E-mail:
[email protected]. Tel: 82-31-270-3772. Fax: 82-31-2704019.
In this paper, a general structure for the cascade control system is proposed. With the general structure for the cascade system, the method by Lee et al.5 is extended to integrating and unstable processes. For a more efficient performance, a setpoint filter is added to each loop. As a result, a general PID tuning rule for cascade control systems is developed. This study first enables one to design the practical IMC-based PID controllers for the unstable cascade control system. 2. Theory As shown in Figure 1, the classical cascade control system consists of two feedback loops. Lee et al.5 proposed a PID tuning method for the classical cascade control system for general stable processes. In the method, they used the IMC6 approach to design classical feedback controllers. The IMC structure of Lee et al.5 is shown in Figure 2. With p˜ 2 ) p2 and p˜ p ) q2p2p1, the closed-loop transfer functions for the inner and outer loops are
y2 ) q2p2r2 + (1 - q2p2)pd2d2
(1)
y1 ) p2q2p1q1r1 + (1 - p2q2)p1(1 - p2q2p1q1)pd2d2 + (1 - p2q2p1q1)pd1d1 (2) In the structure with the setpoint filter, controllers can be designed with characteristics:7,8 efficient performance of disturbance rejection; elimination of overshoots for unstable processes. Further, unstable processes are often controlled with a cascade structure. Here, we generalize the cascade structure with a new structure as shown in Figure 3 to eliminate overshoots, which often happen for unstable processes, and to get a better performance of disturbance rejection. A generalized feedback control structure corresponding to the
10.1021/ie010157f CCC: $22.00 © 2002 American Chemical Society Published on Web 05/01/2002
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Figure 1. Classical cascade control system.
Figure 2. IMC structure for cascade control of Lee et al.5
Figure 3. Generalized IMC structure for cascade control.
generalized IMC structure is in Figure 4. With p˜ 2 ) p2 and p˜ p ) qf2q2p2p1, the closed-loop transfer functions for the inner and outer loops become
y2 ) q2p2qf2r2 + (1 - q2p2)pd2d2
(3)
y1 ) p2q2qf2p1q1qf1r1 + (1 - p2q2)p1(1 p2q2qf2p1q1)pd2d2 + (1 - p2q2qf2p1q1)pd1d1 (4) 2.1. Design of the Secondary Controller. The controller in the inner loop has to be designed to reject
Ind. Eng. Chem. Res., Vol. 41, No. 11, 2002 2681
Figure 4. Generalized feedback control structure for cascade control.
the disturbances into the inner loop quickly. For this, the secondary variable should follow its sepoint as good as possible. For general stable processes, the method by Lee et al.5 can give a stable overdamping response. For integrating and unstable processes, the method by Lee et al.8 is used to get an enhanced performance of disturbance rejection and to handle unstable processes in the inner loop. Consider a general process model of the inner loop
p2(s) ) p2m(s) p2a(s)
m
f2 )
(λ2s + 1)2m
m
p2a(
|
Risi + 1) ∑ i)1
(λ2s + 1)2m
) 0 (7)
s)dup12,...,dupm2
Thus, the IMC controller is m
Risi + 1 ∑ i)1
(5)
where the process model of the inner loop can be divided into two parts: p2m, the portion of the model inverted by the controller, and p2a, the portion of the model not inverted by the controller. p2a is the portion of the model with right-half-plane zeros and time delays. Unstable Processes or Stable Processes with Poles near Zero. To get a good response for unstable processes or stable processes with poles near zero, the IMC controller for the secondary loop should satisfy following conditions. (i) If the process model p2 has unstable poles, up12, up22, ..., then q2 should have zeros at up12, up22, .... (ii) If the process model pd2 has unstable poles dup12, dup22, ... or has poles near zero, then 1 - p2q2 should have zeros at dup12, dup22, ... or at the poles near zero. IMC controller q2 is set as q2 ) p2m-1f2. Then, the IMC controller q2 satisfies the first condition automatically because p2m-1 is the inverse of the model portion with the unstable poles. Through filter f2 design, the second condition is satisfied. For this case, the filter is set as
Risi + 1 ∑ i)1
1 - p2q2|s)dup12,...,dupm2 ) |1 -
q2 ) p2m-1 (λ2s + 1)2m
(8)
Then, we get m
y2 r2
y2 d2
p2a( ) p2q2qf2 )
[
Risi + 1) ∑ i)1
(λ2s + 1)2m
qf2
m
]
Risi + 1) ∑ i)1
p2a(
) (1 - p2q2)p2 ) 1 -
(9)
(λ2s + 1)2m
p2 (10)
The lead term in eq 9 causes an overshoot in the closedloop response. To eliminate the overshoot, we set a setpoint filter qf2 as
qf2 )
1 m
(11)
Risi + 1 ∑ i)1
Then
(6)
Here, Ri values are determined to cancel the unstable poles or poles near zero of pd2 in eq 7 and m is the number of poles canceled. Equation 6 functions as a filter with adjustable time constant λ.
p2a y2 ) p2q2qf2 ) r2 (λ2s + 1)2m
(12)
Normal Stable Processes. Process response with cancellation of the poles near zero is faster than the one without cancellation as the poles close to zero because the poles near zero cause a long time lag. Whether a secondary process is considered as a stable process with
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poles near zero or as a normal stable process depends on controller design sense. In the case that the process p2 does not have unstable poles or poles near zero, the IMC and setpoint filters become
f2 )
1 ; qf2 ) 1 (λ2s + 1)m
1 - ppq1|s)dup11,...,dupn1 ) |1 -
|
n
βisi + 1) ∑ i)1
ppa(
(λ1s + 1)2n
(13)
s)dup11,dup21,...,dupn1
Thus, the IMC controller is
where m is the order of stable processes. With this setting, the IMC controller is
q2 )
p2m-1 (λ2s + 1)m
) 0 (20)
n
( q1 ) ppm-1
(14)
βisi + 1) ∑ i)1
(21)
(λ1s + 1)2n
Then, we get
Then
p2a y2 ) p2q2qf2 ) r2 (λ2s + 1)m
n
(15)
y1 r1
Note that eq 15 is the same form as eq 12. 2.2. Design of the Primary Controller. The transfer function between setpoint and primary output is
y1 ) p2q2qf2p1q1qf1 r1
(16)
d1
To design the IMC controller q1 for a primary loop, consider a primary process as follows:
pp ) p2q2qf2p1
y1
(17)
ppa( ) ppq1qf1 )
(
qf1
n
ppa(
)
βisi + 1) ∑ i)1
(λ1s + 1)2n
(22)
p1
(23)
The lead term in eq 22 causes an overshoot in the closedloop response. To eliminate the overshoot, we set a setpoint filter qf1 as
q f1 )
(18)
where ppm the portion of the model inverted by the controller, and ppa, the portion of the model not inverted by the controller. ppa is the portion of the model with right half plane zeros and time delays. Unstable Processes or Stable Processes with Poles near Zero. To get a good response for unstable processes or stable processes with poles near zero, the IMC controller for a primary loop should satisfy following conditions. (i) If the process model pp has unstable poles, up11, up21, ..., then q1 should have zeros at up11, up21, .... (ii) If the process model pd1 has unstable poles dup11, dup21, ..., or has poles near zero, then 1 - ppq1 should have zeros at dup11, dup21, ... or at the poles near zero. The IMC controller q1 is set as q1 ) ppm-1f1. Then, the IMC controller q1 satisfies the first condition automatically because ppm-1 is the inverse of the model portion with the unstable poles. Through filter f1 design, the second condition is satisfied. For this case, again the filter is set as
(λ1s + 1)2n
) (1 - ppq1)pd1 ) 1 -
This process is decomposed as two parts:
pp ) ppappm
βisi + 1) ∑ i)1
1
(24)
n
βis ∑ i)1
i
+1
Then
ppa y1 ) ppq1qf1 ) r1 (λ1s + 1)2n
(25)
Normal Stable Processes. As referred to in the section of normal stable processes for a secondary controller, whether a primary process is considered as a stable process with poles near zero or as a normal stable process depends on the controller design sense. In the case that the process does not have unstable poles or poles near zero, the IMC and setpoint filters become
f1 )
1 ; qf 1 ) 1 (λ1s + 1)n
(26)
where n is the order of stable processes. With this setting, the IMC controller is
n
f1 )
βis i + 1 ∑ i)1 (λ1s + 1)2n
q1 )
(19)
Here, βi values are determined to cancel the unstable poles or poles near zero of pd1 in eq 20 and n is the number of poles canceled. Equation 19 functions as a filter with adjustable time constant λ1.
ppm-1 (λ1s + 1)n
(27)
Then
ppa y1 ) ppq1qf1 ) r1 (λ1s + 1)n
(28)
Ind. Eng. Chem. Res., Vol. 41, No. 11, 2002 2683
2.3. PID Controller Settings. In practical control problems, PID controllers are preferred to IMC controllers. So, here, a conversion procedure from IMC controllers to PID controllers is presented. This procedure was developed by Lee et al.7 for general process models. Then, the controller transfer function is represented by
Gc )
q 1 - pq
m
Risi + 1) ∑ i)1
(λ2s + 1)2m
Gcs(s) )
m
p2a( 1-
R is ∑ i)1
i
(30)
+ 1)
(λ2s + 1)2m
With eqs 5 and 14, the secondary controller for normal stable processes becomes
p2m-1
Gcs(s) )
(λ2s + 1)m - p2a
(31)
Primary Controller. With a relation between the IMC and feedback controllers, Gcp ) q1/(1 - ppq1), and eqs 18 and 21, the primary controller for processes with unstable poles or with poles near zero becomes
βisi + 1)
(λ1s + 1)2n
Gcp(s) )
n
ppa( 1-
∑ i)1
(λ1s + 1)n - ppa
)
1 + τDs + ... τIs
(36)
Kc ) f ′(0); τI ) f ′(0)/f(0); τD ) f ′′(0)/2f ′(0)
(37)
This approximation method is validated with several research results.5,7,8 The integral and derivative time constants (τI and τD) can have negative values for some complicated process models independent of the selection of a filter time constant. In this case, the simple PID controller cascaded with a first-order lag of the form 1/(as + 1) or a second-order lag of the form 1/(b2s2 + b1s + 1) is recommended. To obtain a PID controller cascaded by a first-order lag [i.e., Kc(1 + 1/τIs + τDs)/(as + 1)], we rewrite Gc(s) as
1 1 f(s) h(s) Gc(s) ≡ f(s) ) s s h(s)
(38)
where h(s) ≡ 1 + as. Now, we expand the quantity f(s) h(s) in a Maclaurin series about the origin and choose the parameter a so that the third-order term in the expansion becomes zero. The expansion of eq 38 then becomes
Gc(s) ) {f(0) + [f ′(0) + af(0)]s + [f ′′(0) + 2af ′(0)]s2/2 + [f ′′′(0) + 3af ′′(0)]s3 + ...}/s(as + 1) (39)
f ′′′(0) 3f ′′(0)
(40)
and the PID parameters are
βisi + 1)
ppm-1
(35)
where
(32)
Kc ) f ′(0) + af(0); τI ) Kc/f(0); τD ) [f ′′(0) + 2af ′(0)]/2Kc (41)
(λ1s + 1)2n
(33)
Conversion Procedure. Here, derived feedback controllers are converted into classical PID controllers. The subscripts s and p representing secondary and primary are abbreviated in the conversion procedure. Because pa(0) is 1, the controller Gc can be expressed with an integral term as
Gc ≡ f(s)/s
(
Gc(s) ) Kc 1 +
a)-
With eqs 18 and 27, the primary controller for normal stable processes becomes
Gcp(s) )
]
Selecting the lag parameter a to drop the third-order term gives
n
∑ i)1
ppm-1(
[
f ′′(0) 2 1 f(0) + f ′(0) s + s + ... s 2
The first three terms of the above expansion can be interpreted as the standard PID controller given by
(29)
where q is the IMC controller. Secondary Controller. With a relation between the IMC and feedback controllers, Gcs ) q2/(1 - p2q2), and eqs 5 and 8, the secondary controller for processes with unstable poles or with poles near zero becomes
p2m-1(
Gc(s) )
(34)
Now, the controller is approximated as a PID controller by expanding Gc in a Maclaurin series in s, and it gives
Again, the PID lag controller is Kc(1 + 1/τIs + τDs)/(as + 1). To obtain a PID controller cascaded with a secondorder lag, we write Gc(s) as
Gc(s) = N(s)/sD(s)
(42)
where N(s) and D(s) are polynomials obtained by substituting high-order (higher than third) Pade approximations for the exponential terms in pm(s) and pa(s). This gives n
Risi + 1) ∑ i)1
k(λ) ( Gc(s) )
n-1
s(
bi(λ) si + 1) ∑ i)1
(43)
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where 0 e ai and bi(λ) and bi(λ) and k(λ) are functions of λ. Dropping terms higher than second order in the numerator and higher than third order in the denominator gives
Gc(s) =
a2s2 + a1s + 1
k(λ) s b (λ) s2 + b (λ) s + 1 2 1
1-
K2e τ2s + 1
K2 e-θ2s τ2s + 1
Gcs(s) )
(λ2s + 1)m - p2a
Kcs )
τDs )
ppm (
p2a(
R is ∑ i)1
2(λ2 + θ2)
θ22 6(λ2 + θ2)
(
3-
)
θ2 τIs
Gcp(s) )
)
n
βis ∑ i)1
i
+ 1)
(λ1s + 1)2n (λ2s + 1)(τ1s + 1)(βs + 1) K1[(λ1s + 1)2 - e-(θ2+θ1)s(βs + 1)]
+ 1)
(53)
βisi + 1) ∑ i)1
(54)
Expanding Gcp in a Maclaurin series in s gives
(λs + 1)2m (τ2s + 1)(Rs + 1) K2[(λ2s + 1)2 - e-θ2s(Rs + 1)]
(48)
τIs K2(2λ2 + θ2 - R)
τIp K1(2λ1 + θ1 + θ2 - β)
1 1 (θ1 + θ2)3 - (θ1 + θ2)2β 6 2 λ2τ1 + λ2β + τ1β 2λ1 + θ1 + θ2 - β τDp ) τIp 1 λ12 + (θ1 + θ2)β - (θ1 + θ2)2 2 (55) 2λ1 + θ1 + θ2 - β
1 λ22 + θ2R - θ22 2 τIs ) τ2 + R 2λ2 + θ2 - R 1 3 1 2 θ - θ2 R 6 2 2 1 2 2 τ2R 2λ2 + θ2 - R λ2 + θ2R - 2θ2 τDs ) τIs 2λ2 + θ2 - R
Kcp )
1 λ12 + (θ1 + θ2)β - (θ1 + θ2)2 2 τIp ) λ2 + τ1 + β 2λ1 + θ1 + θ2 - β
Expanding Gcs in a Maclaurin series in s gives
Kcs )
θ22
(λ1s + 1)2n
1)
i
τIs K2(λ2 + θ2)
n
ppa(
m
1-
-1
m
(λ2s + 1)
Gcs(s) )
(52)
Design of the Primary Controller for Unstable Processes or Stable Processes with Poles near Zero. Equation 32 becomes
(47)
2m
K2(λ2s + 1 - e-θ2s)
Expanding Gcs in a Maclaurin series in s gives
(46)
Risi + 1) ∑ i)1
λ2 2 -θ2/τ2 e τ2
τ2s + 1
)
(45)
Design of the Secondary Controller for Unstable Processes or Stable Processes with Poles near Zero. Equation 30 becomes
p2m-1(
(51)
τIs ) τ2 +
Then, the model of the secondary loop is decomposed as
p2(s) ) p2mp2a )
[ ( ) ]
p2m-1
-θ1s
K1e p1(s) ) τ1s + 1
(50)
s)-1/τ2
Design of the Secondary Controller for Normal Stable Processes. Equation 31 becomes
-θ2s
p2(s) )
)0
(λ2s + 1)2
R ) τ2 1 - 1 -
(44)
The controller given by eq 44 can be viewed as an ideal PID controller cascaded with a second-order lag or as a floating integral controller cascaded with a second-order lead-lag transfer function. The controller parameters are Kc ) k(λ) τI, τI ) a1, and τD ) a2/τI. The second-order lag is given by 1/(b2s2 + b1s + 1). All of the parameters except possibly Kc are positive. See the examples ref 7. 2.4. FOPDT Model. Because the most commonly used approximate model for chemical processes is the FOPDT model, a cascade system with a FOPDT model for both secondary and primary loops is considered as an example as follows:
|
e-θ2s(Rs + 1)
(49)
To cancel the unstable pole or stable pole near zero,
Ind. Eng. Chem. Res., Vol. 41, No. 11, 2002 2685 Table 1. Resulting Tuning Rules for the FOPDT Modela inner loop
outer loop
Stable Process with a Pole or Unstable Process process model
K2e-θ2s p2(s) ) τ2s + 1
p1(s) )
reference trajectory
C2 e-θ2s ) R2 λ2s + 1
C1 e-(θ1+θ2)s ) R1 λ1s + 1
Kc
Kcs )
τI
τD
setpoint filter
τIs
Kcp )
K2(2λ2 + θ2 - R)
K1e-θ1s τ1 s + 1
τIp K1(2λ1 + θ1 + θ2 - β)
1 λ22 + θ2R - θ22 2 τIs ) τ2 + R 2λ2 + θ2 - R
1 λ12 + (θ1 + θ2)β - (θ1 + θ2)2 2 τIp ) λ2 + τ1 + β 2λ1 + θ1 + θ2 - β
1 3 1 2 θ - θ2 R 6 2 2 1 2 2 τ2R 2λ2 + θ2 - R λ2 + θ2R - 2θ2 τDs ) τIs 2λ2 + θ2 - R
1 1 (θ + θ2)3 - (θ1 + θ2)2β 6 1 2 λ2τ1 + λ2β + τ1β 2λ1 + θ1 + θ2 - β τDp ) τIp 1 λ12 + (θ1 + θ2)β - (θ1 + θ2)2 2 2λ1 + θ1 + θ2 - β
qf2 ) 1/(Rs + 1)
qf1 ) 1/(βs + 1) Normal Stable Process
K2e-θ2s τ2s + 1
process model
p2(s) )
reference trajectory
C2 e-θ2s ) R2 λ2s + 1
Kc
Kcs )
τI
τIs ) τ2 +
τD
τDs )
a
p1(s) )
K1e-θ1s τ1 s + 1
C1 e-(θ1+θ2)s ) R1 λ1s + 1
τIs
Kcp )
K2(λ2 + θ2) θ22
6(λ2 + θ2)
( ) 3-
K1(λ1 + θ1 + θ2)
τIp ) λ2 + τ1 +
2(λ2 + θ2)
θ22
τIp
θ2 τIs
λ2τ1 τDp )
(θ1 + θ2)2 2(λ1 + θ1 + θ2) (θ1 + θ2)3
(θ1 + θ2)2 6(λ1 + θ1 + θ2) + τIp 2(λ1 + θ1 + θ2)
R ) τ2[1 - (1 - λ2/τ2)2e-θ2/τ2] and β ) τ1[1 - (1 - λ1/τ1)2e-(θ1+θ2)/τ1].
the following condition should be satisfied:
e-(θ1+θ2)s(βs + 1) 1|s)-1/τ1 ) 0 (λ1s + 1)2
τIp ) λ2 + τ1 + (56) λ2τ1 -
This becomes
[ ( )
β ) τ1 1 - 1 -
λ1 2 -(θ1+θ2)/τ1 e τ1
]
τDp ) (57)
Design of the Primary Controller for Normal Stable Processes. Equation 33 becomes
Gcp(s) )
ppm-1 (λ1s + 1)n - ppa
)
(λ2s + 1)(τ1s + 1) K1(λ1s + 1 - e-(θ2+θ1)s) (58)
Expanding Gcp in a Maclaurin series in s gives
Kcp )
τIp K1(λ1 + θ1 + θ2)
(θ1 + θ2)2 2(λ1 + θ1 + θ2)
(θ1 + θ2)3 (θ1 + θ2)2 6(λ1 + θ1 + θ2) + τIp 2(λ1 + θ1 + θ2)
(59)
The resulting tuning rules are given in Table 1. Guideline for Closed-Loop Time Constants λ1 and λ2. In the proposed approach, λ1 and λ2 are adjustable tuning parameters for the speed of process response. λ1 and λ2 should be chosen to give a good performance and robustness. The larger values of the closed-loop time constants λ1 and λ2 are selected to sacrifice the performance for robustness as the uncertainty between the plant and model increases. The closed-loop bandwidth is usually chosen such that it does not exceed 10 times the open-loop bandwidth.6 Therefore, as a rough guideline, it is recommended that at least λ1 and λ2 are 10 times less than those of the corresponding open-loop time constants. The optimal value of λ is also a function of the process dead time.
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Table 2. Tuning Values by the Proposed Method and Other Methods for Example 1 ITAE method Lee et al. proposed method
inner loop controller
outer loop controller
Kcs ) 2.97 Kcs ) 3.44, τIs ) 20.66, τDs ) 0.64 Kcs ) 6.92, τIs ) 4.60, τDs ) 0.79 qf2 ) 1/(3.66s + 1)
Kcp ) 7.3, τIp ) 200 Kcp ) 5.83, τIp ) 105, τDp ) 4.8 Kcp ) 10.11, τIp ) 31.05, τDp ) 5.24 qf1 ) 1/(24.93s + 1)
For open-loop stable processes, λ1 and λ2 were proposed as λ1/(θ1 + θ2) ) 0.5 and λ2/θ2 ) 0.5.5 An extensive simulation indicates that λ2/θ2 ) 0.5-1 and λ1/(θ1 + θ2) ) 0.5-1 for stable processes with poles near zero and λ2/θ2 ) 1-2 and λ1/(θ1 + θ2) ) 1-2 for unstable processes with ISE criterion. 2.5. Integrating Process Model with Time Delay.
G(s) ) Ke-θs/s
(60)
We cannot use directly eq 48 or eq 54 for integrating processes as the terms including R and β disappear at s ) 0. However, we can use eq 48 or eq 54 for integrating processes by approximating the integrator as an unstable pole near zero. Usually, the controller based on the model with an unstable pole near zero can give more robust closed-loop responses than that based on the model with an integrator. How to choose an unstable pole near zero depends on the control design sense. To choose an unstable pole a little bit far from zero means to get a more robust response.
Figure 5. Comparison of the closed-loop responses due to a load change (d2) for example 1.
3. Simulation Study To illustrate this method for cascade control structure, simulation studies for typical examples were performed using MATLAB (control system design and simulation software).9 3.1. Examples To Test the Robustness of the Proposed Method. We choose examples 1 and 2 to test the robustness of the proposed method. These examples are normal processes without integrators or unstable poles. To show the superiority of the proposed method, the simulation results of the proposed method are compared with those of other methods (ITAE by Krishnaswamy and Rangaiah4 and the method by Lee et al.5). Example 1. First, as a basic example, the following open-loop stable process model4 was considered.
GP1 ) GD1 )
e-10s 2e-2s ; GP2 ) GD2 ) 100s + 1 20s + 1
(61)
Here, we consider the process model by eq 61 as a process which has poles near zero. The PID controllers were tuned by the proposed method with λ2 ) 1 and λ1 ) 8. The values of setpoint filter parameters, R and β, were chosen as 3.667 and 24.930, respectively. The values of simulation results were compared with those by the ITAE method and the method of Lee et al.5 (Here, the process by eq 61 is considered as a normal stable process.) In Table 2, the PID tuning values used in the simulation are presented. Figures 5 and 6 show the closed-loop responses tuned by the ITAE method, the method by Lee et al.,5 and the proposed method for unit step changes in d2 and d1, respectively. The results shown in the figures illustrate the superior performance of the proposed method. Example 2. To evaluate the robustness of the proposed method against structural mismatch in plant-
Figure 6. Comparison of the closed-loop responses due to a load change (d1) for example 1.
model, the following complicated process5 was tested:
GP1 ) GD1 )
10(-5s + 1)e-5s
; (30s + 1)3(10s + 1)2 GP2 ) GD2 )
3e-3s (62) 13.3s + 1
White noises are added to y2 and y1 to represent real process measurements. The variances of the noises are 1 × 10-4 and 1 × 10-4 in measurement, respectively. The processes in the inner and outer loops are identified with FOPDT models. The models were obtained by minimization of the squared error between the process output data and model data. The following process models were obtained:5
Gm1 ) GD1 )
10.2e-61.71s 2.98e-3.66s ; Gm2 ) GD2 ) 66.49s + 1 13.28s + 1 (63)
Ind. Eng. Chem. Res., Vol. 41, No. 11, 2002 2687 Table 3. Tuning Values by the Proposed Method and Other Methods for Example 2 ITAE method Lee et al. proposed method
inner loop controller
outer loop controller
Kcs ) 0.66 Kcs ) 0.88, τIs ) 14.5, τDs ) 1.11 Kcs ) 1.62, τIs ) 7.44, τDs ) 1.37 qf2 ) 1/(5.78s + 1)
Kcp ) 0.13, τIp ) 485.37 Kcp ) 0.09, τIp ) 90.53, τDp ) 18.2 Kcp ) 0.12, τIp ) 87.88, τDp ) 21.21 qf1 ) 1/(59.34s + 1)
Figure 7. Comparison of the closed-loop responses due to a load change (d2) for example 2.
Figure 9. Closed-loop response due to a load change (d2) for example 3. Table 4. Resulting Tuning Values for Example 3 inner loop outer loop
Kc
τI
τD
setpoint filter
6.92 -3.31
4.60 36.22
0.79 3.08
qf2 ) 1/(3.66s + 1) qf1 ) 1/(32.91s + 1)
earities due to the heat generation term in the energy balance. The simulation results of these examples show a good perfomance of the proposed method for the unstable or integrating cascade processes. Example 3. To test the capability of the proposed method for unstable processes, the following process model is considered. The process of the outer loop is an unstable process, while the process of the inner loop is a stable process. This is a typical example of the CSTR system which includes highly exothermic reactions.10
GP1 ) GD1 ) Figure 8. Comparison of the closed-loop responses due to a load change (d2) for example 2.
The closed-loop time constants are chosen as λ1 ) 30.85 and λ2 ) 1.83. The tuning parameters are in Table 3. Figure 7 shows the closed-loop responses tuned by the method of Lee et al.,5 the ITAE method, and the proposed method for the unit step change in d2. Figure 8 shows the closed-loop responses for the noises with the increased variances, 2 × 10-2, 2 × 10-2. The increased noises deteriorate the performance by the method which includes the derivative mode. However, because the proposed structure itself contains a lowpass filter, even though the proposed method use derivative mode in controllers, the performance by the proposed structure is less sensitive to the noises than the one by the conventional structure. 3.2. Challenging Examples for Unstable and Integrating Cascade Processes. Example 3 is an unstable cascade process, and example 4 is an integrating cascade process which cannot be handled with other methods. These types of processes are typical examples of batch chemical reactors, which have strong nonlin-
e-4s 2e-2s ; GP2 ) GD2 ) -20s + 1 20s + 1
(64)
The closed-loop time constants are chosen as λ2 ) 1 and λ1 ) 8. The tuning parameters are given in Table 4. Because any other methods cannot handle these kinds of unstable cascade processes, the simulation results are not compared with those of other methods. Figure 9 shows the closed-loop response of the unstable process given by eq 64 to a unit step change in d2. Example 4. Finally, the following process model was considered. The process for the inner loop can often be an integrator with time delay.
GP1 ) GD1 )
e-4s 2e-2s ; GP2 ) GD2 ) -20s + 1 s
(65)
To design a feedback controller for the integrating process with time delay, the integrator is approximated into an unstable pole near zero. The integrating model can be changed to following form:
2e-2s -200e-2s 2e-2s ) ) s s - 0.01 -100s + 1
(66)
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process models. General processes are divided into two parts. One part consists of integrating unstable processes and stable prcesses with poles near zero. The other part consists of normal stable processes. For these processes, IMC controllers to give the desired closedloop responses are obtained and converted to PID controllers. The PID controllers tuned by the proposed method give superior performances to those by other design methods. Literature Cited
Figure 10. Closed-loop response due to a load change (d2) for example 4. Table 5. Resulting Tuning Values for Example 4 inner loop outer loop
Kc
τI
τD
setpoint filter
0.35 -3.31
5.02 36.22
0.82 3.08
qf2 ) 1/(4.07s + 1) qf1 ) 1/(32.91s + 1)
With the approximated model (eq 66), a PID controller for the integrating process of the inner loop is designed. The closed-loop time constants are chosen as λ2 ) 1 and λ1 ) 8. The tuning parameters are given in Table 5. Figure 10 shows the closed-loop response of the unstable process given by eq 65 to a unit step change in d2. 4. Conclusion We generalize the IMC approach for the cascade control system with a general cascade control structure and show how to obtain PID tuning values for general
(1) Jury, F. D. Fundamentals of three-mode controllers; Technical Monograph No. 28; Fisher Controls Company: 1973. (2) Hougen, J. O. Measurement and control applications. Ph.D. Thesis, Pittsburgh, PA, 1979. (3) Edgar, T. F.; Heeb, R. C.; Hougen, J. O. Computer-aided process control system design using interactive graphics. Comput. Chem. Eng. 1982, 5 (4), 225. (4) Krishnaswamy, P. R.; Rangaiah, G. P. When to use cascaded control. Ind. Eng. Chem. Res. 1990, 29, 2163. (5) Lee, Y.; Lee, M.; Park, S. PID controller tuning to obtain desired closed loop responses for cascade control systems. Ind. Eng. Chem. Res. 1998, 37, 7 (5), 1859. (6) Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: Englewood Cliffs, NJ, 1989. (7) Lee, Y.; Lee, M.; Park, S.; Brosilow, C. PID controller tuning for desired closed loop responses for SI/SO systems. AIChE J. 1998, 44 (1), 106. (8) Lee, Y.; Lee, J.; Park, S. PID controller tuning for integrating and unstable processes with time delay. Chem. Eng. Sci. 2000, 55 (17), 3481. (9) Shahian, B.; Hassul, M. Control System Design Using Matlab; Prentice Hall: Englewood Cliffs, NJ, 1993. (10) Russo, L. P.; Bequette, B. W. Effect of process design on the open loop behavior of a jacketed exothermic CSTR. Comput. Chem. Eng. 1996, 20 (4), 417.
Received for review February 19, 2001 Revised manuscript received February 6, 2002 Accepted March 13, 2002 IE010157F
