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Ind. Eng. Chem. Res. 2003, 42, 4488-4494
Relay Autotuning of Cascade-Controlled Open-Loop Unstable Reactors Vikas Saraf, Futao Zhao, and B. Wayne Bequette* Howard P. Isermann Department of Chemical Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180-3590
Continuous stirred tank reactors (CSTRs) can present operational problems as a result of complex open-loop behavior such as input and output multiplicities, parametric sensitivities, and nonlinear oscillations. It is sometimes appropriate to operate a chemical reactor at an unstable steady state. Traditionally, autotuning procedures for cascade-control strategies are applied using a sequential, one-loop-at-a-time method. Such a method, however, cannot be used on open-loop unstable systems because the secondary loop cannot stabilize the system when the primary loop is open. As a result, a new simultaneous autotuning procedure is proposed for the cascade controllers of open-loop unstable CSTR processes. This new approach can also be used for the autotuning of cascade-controlled open-loop stable systems, thereby covering a wide operation range for CSTR cascade-control systems. 1. Introduction Relay feedback autotuning approach has received a great deal of attention since the pioneering work done by Astrom and Hagglund.1 This method provides a very useful tool for obtaining appropriate controller parameters for chemical processes whose dynamic models are often difficult to obtain because of process complexity, frequent disturbances, and time variability. A relay test is carried out in a closed-loop, and the process variable is maintained around the set-point value.2 The advantage of a closed-loop test is particularly attractive for chemical processes that demonstrate open-loop instabilities or non-self-regulating characteristics. Open-loop tuning methods have difficulties in tuning controllers for such processes. There has been much recent interest in relay autotuning for open-loop unstable systems. In 1998, Luyben3 showed how to tune a controller through relay feedback testing for an open-loop unstable reactor using the Tyreus and Luyben (TL) tuning procedure.4 Tan et al.5 discussed relay tuning for an unstable time delay process. Shen et al.6 discussed the feasibility of relay feedback testing for some systems with RHP poles and/ or zeros. The use of relay techniques with additional delay has been proposed for low-order open-loop unstable processes.7 All of these techniques focus on singleloop feedback control structures. Cascade control is a multiloop control scheme commonly used in chemical process control.8 Cascade controllers are extremely effective for disturbances that directly affect secondary process measurements. The tight tuning of the secondary controller allows rapid rejection of the disturbance, resulting in little effect on the primary output. The manual tuning of cascade controllers, however, is generally a tedious and timeconsuming task. In view of its widespread applications, automation of the tuning procedure would be very useful. * To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: (518) 276-6683. Fax: (518) 276-4030.
It is appropriate to develop a tuning technique that does not require much a priori knowledge of the process and is able to deal with changes in the operating conditions and unmodeled disturbances. A sequential method of autotuning (Sq-ATV) was proposed by Hang et al.9 for open-loop stable cascade systems, in which the secondary loop is autotuned first with the primary loop open. With the secondary controller in place, the primary loop is then autotuned. Subsequent retuning of the primary and secondary controllers can be done depending on the response. However, this method is inapplicable to cascade-controlled unstable systems for which the primary loop must be closed because the secondary loop alone cannot stabilize the system. However, sometimes it is appropriate to operate a chemical reactor at an unstable steady state because, in such a state, the reaction rate provides good productivity and the reactor temperature is low enough to prevent side reactions or catalyst degradation. To obtain the parameters for both the primary and secondary controllers simultaneously, a “simultaneous” approach of autotuning (St-ATV) is proposed for the control of CSTR systems that demonstrates both openloop stable and unstable characteristics at different operating conditions. 2. Autotuning Procedure The relay feedback autotuning of PI/PID controllers is well-developed and is commercially available in single-loop controllers. If an ideal relay with amplitude h is connected in a feedback loop in which the output lags behind the input by -π rad around an open-loop system, the system will oscillate in a limit cycle with period Pu. The switching occurs once the system output crosses a certain steady-state value from either side. The estimated ultimate frequency from this relay feedback experiment is
ωu ) 2π/Pu
(1)
From the Fourier series expansion, the amplitude a can be considered as the result of the primary harmonic
10.1021/ie0011268 CCC: $25.00 © 2003 American Chemical Society Published on Web 03/14/2003
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4489
of the relay output. Therefore, the ultimate gain can be approximated as
There are advantages in having a relay with hysteresis instead of a pure relay.10 The hysteresis width is often selected to avoid the relay switch because of noise. With hysteresis installed, the relay can generate appropriate oscillation for controller autotuning for openloop unstable processes. The formula for calculating the ultimate gain is thus modified to
where parameter λ can be adjusted for aggressive or sluggish responses. The approach to use relay identification in combination with an integrator plus dead time model is useful when the process time constant is dominant. However, for processes that are dominated by nonminimum phase characteristics (dead time and/or RHP zeros) and not by time constants, the gain plus dead time model is a more useful two-parameter model. For the gain plus dead time model, the dead time and process gain can be found from the relay parameters using the equations (see Appendix B)
Kcu ) 4h/[π(a2 - 2)0.5]
θ ) Pu/2
Kcu ) 4h/πa
(2)
(3)
The ultimate period is still obtained from the period of oscillation as in eq 1. 3. Controller Design Two types of controller designs are investigated in this work. One set of controller parameters is obtained using the Tyreus and Luyben (TL) tuning method.4 The other set involves an IMC-based PID design method. The following two transfer functions can approximate most chemical process dynamics
g1(s) )
Ke-θs τs + 1
(4.1)
g2(s) )
Ke-θs τs - 1
(4.2)
Friman and Waller11 have noted that the dynamics of many chemical processes are dominated by large time constants. If τ is large, the step responses of g1(s) and g2(s) are very close to an integrator plus dead time process at the initial stage. The larger the value of τ, the closer these responses. Thus, when the goal of modeling is the tuning of a controller, the determination of the the ratio k ) K/τ is often adequate, as opposed to the determination of both K and τ in a first order with time delay (FODT) process with large τ (i.e., τ . θ). Therefore, a chemical process with a large time constant can be approximated by an integrator plus dead time model12
g(s) )
ke-θs s
(5)
This model has only two parameters, the dead time, θ, and the initial nonzero slope of the step response, k. For model g(s), the dead time and process gain can be determined from relay tests using the equations (see Appendix A)
θ ) Pu/4 k ) 4a/Puh
(6)
The IMC-based PI tuning parameters can be found from13
kc )
2λ + θ k(λ + θ)2
τI ) 2λ + θ
(7)
k ) a/h
(8)
The IMC-based PI tuning parameters can be found from14
kc )
θ k(2λ + θ)
τI ) θ/2
(9)
In this work, the integrator plus dead time model is used because, in the case of CSTRs, dead time is often small and whatever delay occurs is mainly due to measurement and actuator lags. The IMC-based PI controllers are designed and compared with the TL tuning results. Actually, the proposed simultaneous autotuning approach (St-ATV) is not limited to PI control. When dead time is significant, derivative action might be necessary for the controller to stabilize an open-loop unstable process. In that case, an IMC-based PID controller can be designed instead. 4. Modeling Equations The standard CSTR model describing an exothermic diabatic irreversible first-order reaction (A f B) is a set of three nonlinear ordinary differential equations obtained from dynamic material and energy balances (with assumptions of constant volume, perfect mixing, negligible cooling-jacket dynamics, and constant physical parameters). The process flow diagram for the jacketed exothermic CSTR is given in Figure 1. The equations are written in dimensionless form as8
dx1 ) q(x1f - x1) - φx1κ(x2) dt dx2 ) q(x2f - x2) - δ(x2 - x3) + βφx1κ(x2) dt
(10)
dx3 ) δ1[qc(x3f - x3) + δδ2(x2 - x3)] dt where x1, x2, x3, and qc are the dimensionless concentration, reactor temperature, cooling-jacket temperature and cooling-jacket flow rate, respectively. The dimensionless cooling-jacket flow rate, qc, is the manipulated input, whereas the dimensionless reactor temperature, x2, and the dimensionless cooling-jacket temperature, x3, are the measured outputs. The dimensionless variables and parameters (for example, φ, β, q, etc.) are defined in Table 1. The values of the parameters are given in Table 2. The CSTR exhibits ignition/extinction
4490 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003
Figure 2. Steady-state dimensionless reactor temperature versus cooling-jacket flow rate for the jacketed exothermic CSTR.
The following matrices give the state-space model
Figure 1. Process flow diagram for the jacketed exothermic CSTR.
[
-1.7771 -0.3394 0.0000 A ) 6.2165 1.4155 0.3000 0.0000 3.0000 -10.6795
]
Table 1. Dimensionless Variables and Parameters for the CSTR Model
Ca Caf0
x1 ) q)
Q Q0
x2 )
(T - Tf0) γ Tf0
Q0 τ) t V κ(x2) ) exp
x2f )
( ) x2
x2 1+ γ
qc )
Qc Q0
β)
δ2 )
FCp FcCpc
x3f )
γ)
Ea RTf0
Caf x1f ) Caf0 φ)
V k e-γ Q0 0
δ)
(-∆H)Caf0 γ FCpTf0 (Tcf - Tf0) γ Tf0
UA FCpQ0
(Tc - Tf0) x3 ) γ Tf0 δ1 )
V Vc
(Tf - Tf0) γ Tf0
Table 2. Model Parameter Values parameter
value
φ β δ γ Q δ1 δ2 x1f x2f x3f
0.072 8.000 0.300 20.000 1.000 10.000 1.000 1.000 0.000 -1.000
behavior for this set of parameters. Here, we study the system at two operating points, one unstable (Op1) and one stable (Op2); the steady-state input-output curve is shown in Figure 2. 4.1. Operating Point 1 (Op1), Open-Loop Unstable. The steady-state values of the three states and of the manipulated input are
x1s ) 0.5627, x2s ) 2.7000, x3s ) 0.0394, qcs ) 0.7680
[
0.0000 B ) 0.0000 -10.3937
]
The eigenvalues of A are -0.9009, -10.7521, and 0.6120, indicating that the operating point is unstable. 4.2. Operating Point 2 (Op2), Open-Loop Stable System. The high-temperature steady state (Op2) is open-loop stable with the following steady-state values
x1s ) 0.2028, x2s ) 5.0000, x3s ) 0.4079, qcs ) 0.9785 The following matrices give the state-space model
[
-4.9311 -0.5102 0.0000 A ) 31.4485 2.7817 0.3000 0.0000 3.0000 -12.7851
]
[
0.0000 B ) 0.0000 -14.0789
]
The eigenvalues of A are -1.0492 ( i0.9343 and -12.8361, indicating that the operating point is openloop stable. 5. Simultaneous Autotuning (St-ATV) for Cascade Control
The proposed scheme involves switching both the controllers on the relay and carrying out the autotuning procedure to obtain limit cycles. The output of the primary autorelay loop is the input of the secondary autorelay loop. The secondary loop is forced to oscillate at the frequency of the primary loop with two relays inserted in the feedback loop because of the condition , ωsecondary . Thus, the ultimate gain and ultiωprimary cu cu mate period are first obtained for the primary loop. The primary controller parameters can also be obtained. Then, the secondary loop executes relay feedback with the primary controller in place in order to obtain its own ultimate period, and the secondary controller parameters are obtained. The proposed autotuning procedure is as follows: Step 1. Select the relay amplitude, h, and deadband, , for the two relays.
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4491
Step 2. Execute two relay feedbacks simultaneously and obtain a and Pu of the primary loop. Calculate k and θ from eq 6 and Kcu from eq 3 for the primary loop. Step 3. Select filter parameter λ1 for the primary IMCbased PI controller. Calculate kc and τI from eq 7 for the primary controller. Step 4. Perform the secondary-loop relay feedback with the primary controller in place to obtain a and Pu of the secondary loop. Calculate k and θ from eq 6 and Kcu from eq 3 for the secondary loop. Step 5. Select filter parameter λ2 for the secondary IMC-based PI controller. Calculate kc and τI from eq 7 for the second controller. Step 6. Put the secondary controller in place to realize cascade feedback control. In the above procedure, values for h and for the two relays and λ1 and λ2 for the two IMC-based PI controllers need to be selected. Some guidelines for these selections are provided in the following discussion. The choice of h is dictated by the physical constraints on the manipulated variable, as well as the allowed deviation in the measured output. The choice of depends on the measurement noise level. With a large deadband , the adverse effect of noise for a relay feedback is avoided. However, the detected value of Kcu will decrease and Pu will increase as increases. For most chemical processes, an interesting phenomenon is that the product of Kcu and Pu is essentially a constant for /h ∈ [0.001, 0.1]. A small value of /h can result in aggressive controller parameters and vice versa. In this St-ATV scheme, use of the value /h ) 0.01 is recommended. It should be noted that the values of and h correspond to the scaled values of the controlled variable and control output, respectively, as in a controller. Through relay feedback, the ultimate gain and ultimate period of a process can be approximately detected. An IMC formulation generally results in only one tuning parameter, the time constant of the IMC filter.15 The selection of parameter λ for an IMC-based PI controller is another important issue in this study. For open-loop stable processes, λ provides a tradeoff between performance and robustness. For open-loop unstable processes, λ must be selected within an appropriate range to ensure stability.16 The formula for selecting λ is given as
λ)
θ
x
(11)
π -1 π - 4R
In eq 11, R is a coefficient for selecting λ. It is recommended that R be chosen in the range R ∈ [1/5, 2/3]. A smaller value of R provides a larger value of λ. Depending on the obtained process ultimate gain and period, other tuning methods, such as Ziegler-Nichols and Tyreus-Luyben, can also be used for the tuning of P, PI, and PID controllers. 6. Simulations for a CSTR Cascade Control The exothermic irreversible CSTR was chosen for investigating the effectiveness of the proposed St-ATV tuning method. Both the open-loop unstable and stable cases are studied. Measurement and actuator lags of 0.1 are used in the simulations to mimic unmodeled dynamics. The covariance of the measurement noise is σ ) 0.002 for both the dimensionless reactor temperature (x2) and the dimensionless cooling-jacket temper-
Figure 3. Autorelay response for the primary loop for open-loop unstable case.
Figure 4. Autorelay response of the secondary loop for open-loop unstable case.
ature (x3). The parameters for the relays are set as ) 5σ ) 0.01 and correspondingly h ) 1. The sample time Ts ) 0.01. The controllers tuned by the St-ATV method are tested for a set-point change of 0.1 in the dimensionless reactor temperature (x2) and a step disturbance of 1.0 in the cooling-jacket feed temperature (x3f) at time t ) 1. All resulting performances are compared with the TL tuning results. It should note here that all of the following response curves and data represent deviations from steady states. 6.1. Open-Loop Unstable Operating Point (Op1). First, the two relays are put into use simultaneously to detect the integrator plus dead time model of the primary loop. Oscillations are generated as shown in Figure 3. Using eqs 6, the model of the primary loop can be identified by observing the oscillation amplitude and period of x2. The parameters of the primary controller are then obtained from eqs 11 and 7. The primary controller is hereafter put in place. To generate the oscillation for the secondary loop, the secondary loop remains on the relay feedback. The oscillation of x3 is shown in Figure 4. The amplitude and period of the resulting oscillation are then used to calculate the parameters of the secondary controller. The parameters obtained from the St-ATV method are reported in Table 3.
4492 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 Table 3. Parameters Obtained by the St-ATV Tuning Method for the Open-Loop Unstable Case
a
parameter
primary loop
secondary loop
a Pu Kcu k θ Ra λ kc τI
0.054 1.0 24.0 0.216 0.25 0.4 0.584 9.44 1.42
0.2806 0.111 -4.54 -10.11 0.0278 0.4 0.065 -1.81 0.158
R is selected in the range R∈ [1/5, 2/3].
Figure 7. Autorelay response for the primary loop for open-loop stable case.
Figure 5. Tracking performance of the St-ATV and TL tuning methods.
Figure 8. Autorelay response of the secondary loop for open-loop stable case.
Figure 6. Regulatory performance of the St-ATV and TL tuning methods.
Using the detected Pu and Kcu values, the TL tuning method provides the following results: primary controller, kc ) 7.5 and τI ) 2.2; secondary controller, kc ) -1.42 and τI ) 0.2442. The tracking performances achieved by the St-ATV and TL tuning methods are shown in Figure 5. The integral absolute error (IAE) of the St-ATV method is 7.38, whereas the IAE of the TL method is 11.34. The regulatory performances achieved by the St-ATV and TL tuning methods are shown in Figure 6. The IAE of the St-ATV method is 2.10, whereas the IAE of the TL method is 5.64.
The results shown in Figures 5 and 6 demonstrate that, for this open-loop unstable case, the St-ATV method provides satisfactory performance that is superior to the performance of the TL tuning method. 6.2. Open-Loop Stable Operating Point (Op2). The autotuning procedure is the same as that of the open-loop unstable case. First, oscillation is generated for the primary loop, as shown in Figure 7. Accordingly, the parameters of the primary controller are obtained. The oscillation generated for the secondary loop is shown in Figure 8. The amplitude and period of the resulting oscillation are then used to tune the parameters of the secondary controller. The parameters obtained from the St-ATV tuning method are listed in Table 4. Using the detected Pu and Kcu values, the TL tuning method provides the following results: primary controller, kc ) 6.125 and τI ) 2.002; secondary controller, kc ) -1.29 and τI ) 0.1991; The tracking performances achieved by the St-ATV and TL tuning methods are shown in Figure 9. The IAE of the St-ATV method is 4.90, whereas that of the TL method is 6.65. The regulatory performances generated by the StATV and TL tuning methods are shown in Figure 10.
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4493 Table 4. Parameters Obtained by the St-ATV Tuning Method for Open-Loop Stable Case
a
parameters
primary loop
secondary loop
a Pu Kcu k θ Ra λ kc τI
0.0657 0.91 19.6 0.289 0.228 0.39 0.556 7.54 1.34
0.3093 0.0905 -4.12 -13.67 0.0226 0.37 0.06 -1.53 0.143
R is selected in the range R∈ [1/5, 2/3].
systems because the secondary loop cannot stabilize the system when the primary loop is open. A new approach of simultaneous autotuning of open-loop unstable systems in a cascade-controlled strategy has been proposed in this paper. The method is also applicable to openloop stable systems and is found to give satisfactory results for both set-point changes and disturbance rejections. The St-ATV procedure provides a relay autotuning method that covers a wide operating range of CSTR cascade-control systems. Acknowledgment Acknowledgment is made to Merck Research Laboratories for their partial support during this work. Nomenclature Roman Letters
Figure 9. Tracking performance of the St-ATV and TL tuning methods.
A ) heat transfer area Ca ) concentration of reactant A in the reactor Caf ) concentration of reactant A in the reactor feed Dr ) diameter of the CSTR k ) process gain kc ) controller gain kcu ) ultimate gain q ) dimensionless reactor flow rate Q ) reactor flow rate qc ) dimensionless cooling-jacket flow rate Qc ) cooling-jacket flow rate t ) time Tc ) cooling-jacket temperature Tcf ) cooling-jacket feed temperature Tf ) reactor feed temperature U ) overall heat-transfer coefficient V ) reactor volume Vc ) cooling-jacket volume x1f ) dimensionless reactor feed concentration x2f ) dimensionless reactor feed temperature x3f ) dimensionless cooling-jacket temperature x2sp ) set point of the dimensionless reactor feed temperature x3sp ) set point of dimensionless cooling-jacket temperature Greek Letters
Figure 10. Regulatory performance of the St-ATV and TL tuning methods.
The IAE of the St-ATV method is 2.30, whereas that of the TL method is 4.38. Figures 9 and 10 show that, also in this case, the StATV tuning method achieves satisfactory performance that is better than the performance of the TL tuning method. 7. Conclusions Over a wide operating range, a CSTR cascade-control system often demonstrates both open-loop stable and unstable characteristics. The traditional sequential autotuning method is inapplicable to open-loop unstable
β ) dimensionless heat of reaction γ ) dimensionless activation energy δ ) dimensionless heat-transfer coefficient δ0 ) nominal dimensionless heat-transfer coefficient δ1 ) reactor-to-cooling jacket volume ratio δ2 ) reactor-to-cooling jacket density heat capacity ratio κ(x2) ) dimensionless Arrhenius reaction rate nonlinearity λ ) IMC filter time constant θ ) process dead time τ ) process time constant τI ) integral time φ ) nominal Damkohler number based on the reactor feed ωu ) ultimate period
Appendix A The input to process g(s) ) ke-θs/s is a square wave with frequency ωu and amplitude h. The process output is a triangular wave (up and down ramps) with frequency ωu and amplitude a. Corresponding to the positive part of one period of the input square wave, the output triangular wave moves from a negative peak point to a positive peak point
hk 1 a ) L-1 2 s s t)Pu/2
( )
(A1)
4494 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003
The phase lag for the occurrence of oscillation is
arg g(jωu) ) -θωu -
π ) -π 2
(A2)
Equations A1 and A2 give
k) θ)
4a Puh
Pu π ) 2ωu 4
(6)
Appendix B The input to process g(s) ) ke-θs is a square wave with frequency ωu and amplitude h. Because g(s) is an allpass process, the process output is a square wave with frequency ωu and amplitude a
a ) h|g(jωu)|
(B1)
The phase lag condition for oscillation is
arg g(jωu) ) -θωu ) -π
(B2)
Because |g(jωu)| ) k, eqs B1 and B2 give
k) θ)
a h
Pu π ) ωu 2
(8)
Literature Cited (1) Astrom, K. J.; Hagglund, T. Automatic Tuning of Simple Regulators with Specifications on Phase and Amplitude Margins. Automatica 1984, 20, 645.
(2) Tan, K. K.; Wang, Q. G.; Hang, C. C.; Hagglund, T. Advances in PID Control; Advances in Industrial Control Series; SpringerVerlag: London, 1999. (3) Luyben, W. L. Tuning Temperature Controllers on Open Loop Unstable Reactors. Ind. Eng. Chem. Res. 1998, 37, 4322. (4) Tyreus, B. D.; Luyben, W. L. Tuning of PI Controllers for Integrator/Dead Time Processes. Ind. Eng. Chem. Res. 1992, 31, 2625. (5) Tan, K. K.; Wang, Q. G.; Lee, T. H. Finite Spectrum Assignment Control of Unstable Time Delay Processes with Relay Tuning. Ind. Eng. Chem. Res. 1998, 37, 1351. (6) Shen, S. H.; Yu, H. D.; Yu, C. C. Autotune Identification for Systems with Right-half-plane Poles and Zeros. J. Process Control 1999, 9, 161. (7) Marchetti, G.; Scali, C.; Lewin, D. R. Identification and Control of Open-Loop Unstable Processes by Relay Methods. Automatica 2001, 37, 2049. (8) Russo, L. P.; Bequette, B. W. State-Space versus Input/ Output Representation for Cascade Control of Unstable Systems. Ind. Eng. Chem. Res. 1997, 36, 2271. (9) Hang C. C.; Loh, A. P.; Vasnani, V. U. Relay Feedback AutoTuning of Cascade Controllers. IEEE Trans. Control Syst. Technol. 1994, 2, 42. (10) Astrom, K. J.; Hagglund, T. Automatic Tuning of PID Controllers; Instrument Society of America: Research Triangle Park, NC, 1988. (11) Friman, M.; Waller, K. V. Autotuning of Multiloop Control Systems. Ind. Eng. Chem. Res. 1994, 33, 1708. (12) Luyben, W. L. Getting More Information from Relay Feedback Tests. Ind. Eng. Chem. Res. 2001, 40, 4391. (13) Chien, I. L.; Fruehauf, P. S. Consider IMC Tuning to Improve Controller Performance. Chem. Eng. Prog. 1990, 86, 33. (14) Rivera, D. E.; Morari, M.; Skogestad, S. Internal Model Control. 4. PID Controller Design. Ind. Eng. Chem. Process Des. Dev. 1986, 25, 252. (15) Morari, M.; Zafririou, E. Robust Process Control; Prentice Hall: Englewood Cliffs, NJ, 1989. (16) Rotstein, G. E.; Lewin, D. R. Simple PI and PID Tuning for Open-Loop Unstable Systems. Ind. Eng. Chem. Res. 1991, 30, 1864.
Revised manuscript received January 8, 2003 Resubmitted for review January 15, 2003 Accepted February 10, 2003 IE0011268
