Ind. Eng. Chem. Res. 2001, 40, 5929-5934
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PROCESS DESIGN AND CONTROL Adaptive Slow/Fast Control Systems for Some Interacting Multivariable Processes Jietae Lee* Department of Chemical Engineering, Kyungpook National University, Taegu 702-701, Korea
Byung Su Ko and Thomas F. Edgar Department of Chemical Engineering, University of Texas, Austin, Texas 78712
It is hard to design robust control systems for interacting multivariable processes. When processes are ill-conditioned, difficulties become more serious due to sensitivities on plant variations. For robust control systems, we decompose systems to slow subsystems and fast subsystems. When the slow subsystem is detuned and maintained to be slow, the fast subsystems can be made to be robust to plant variations. On the other hand, the slow loop is sensitive to plant variations. Adaptive control technique is introduced to the slow loop to maintain performances of the slow loop. Because the slow subsystem is chosen to be single-input single-output and controlled to be slow, the adaptive control scheme is very simple. Simulations show that the proposed control system has nominal performances similar to usual multivariable control systems while maintaining better robustness. Introduction Because models for designing control systems usually have errors and plants can be changing from time to time, it is very important that control systems should have robustness for plant variations. Recent robust control methods1 consider plant variations explicitly and design control systems which maintain performance and stability against plant variations. When plant variations are known precisely and can be well described in a mathematical form, control systems having robust performance and stability can be designed with the robust control methods. However, the resulting control systems are usually high orders.2 Furthermore, when plant variations are not known well and so general mathematical descriptions which are not adequate to represent plant variations precisely are used, tight control systems cannot be obtained with robust control methods. Control performances can also be very poor for plant uncertainties not considered in the design stage. Adaptive control methods3,4 consider plant variations explicitly. Adaptive control systems can track plant variations and maintain their control performances. Because multivariable processes have large numbers of parameters to be identified for adaptation, adaptive control systems are somewhat complex to apply to multivariable processes. They also have their own problems, such as estimator wind-up,5 and additional features to solve these problems make them more complicated. When processes are ill-conditioned, difficulties to design robust control systems are increased due to their * To whom correspondence should be addressed. Tel: +8253-950-5620. Fax: +82-53-950-6615. E-mail:
[email protected].
sensitivities on plant variations.6 To control interacting multivariable processes well, including ill-conditioned ones, without sacrificing control performances for robustness, an adaptive control method is applied. Full multivariable adaptive control systems that identify the entire multivariable models and update multivariable controllers suffer from the same sensitivity problems. To avoid this, a different adaptive scheme is devised. If one critical loop is detuned to be a little slow, the robustness of the other loops may be enhanced considerably. However, as shown later, the slower loop becomes sensitive to model parameter uncertainties. An adaptive technique is applied to the slow loop. The adaptation can maintain the speed of the slow loop for wide plant changes and robustness of the whole control system can be enhanced considerably. Design techniques used for this study are very similar to those for the adaptive twotime-scale system.7 However, the proposed control system does not rely on the natural time-scale separation. Simulations show that the achievable control performances of the proposed control system are better than those of other control systems although a loop is adaptively maintained to be slow. Effects of Detuning a Loop. Consider a 2 × 2 process
G(s) )
[
g11(s) g12(s) g21(s) g22(s)
]
(1)
which is controlled with a multiloop controller (Figure 1)
C(s) )
[
c1(s) 0 c2(s) 0
10.1021/ie000357b CCC: $20.00 © 2001 American Chemical Society Published on Web 11/10/2001
]
(2)
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h2(jω) ≈ 1 for the design frequencies ω of the slow first loop. Effective processes for design frequencies of loops become
g˜11(s) ≈ g11(s) -
g12(s)g21(s)
(6)
g22(s)
and
Figure 1. Multiloop control system.
g˜22(s) ≈ g22(s)
(7)
Equations 6 and 7 show that two loops are not affected each other when they are tuned to be slow and fast, respectively. That is, the second loop is not much affected by how the first loop is designed and vice versa. We can design each loop independently. Now effects of plant variations are investigated. Assume that each element of transfer function matrix has a relative error as, at each frequency ω
gij(jω) ) gj ij(jω)(1 + ∆ij(jω))
where gj ij(jω) is a true value and ∆ij(jω) is a relative error. Then, for small relative errors such that h1(s) and h2(s) are not altered much
Figure 2. Decomposition of a multiloop control system.
g˜ 11 ≈ g11 -
g12g21 g22
) gj 11(1 + ∆11) ≈ gj 11 Figure 3. Typical amplitude ratios of hi(jω) and h2(jω) and design frequencies for loops.
g12(s)g21(s) h2(s) g22(s)
[
h2(s) )
(4)
1 + g11(s)c1(s) 1 + g22(s)c2(s)
gj 12gj 21 gj 11gj 22
and
g11(s)c1(s)
g22(s)c2(s)
1 ζh ∆11 (∆ + ∆21 1 - ζh 1 - ζh 12
]
(3)
Here, h1(s) and h2(s) are the closed-loop transfer functions in case that each diagonal system is controlled by each controller
h1(s) )
gj 22(1 + ∆22)
∆22) (9) ζh )
and another control system in which the controller c2(s) is connected to the process
g12(s)g21(s) g˜22(s) ) g22(s) h1(s) g11(s)
gj 12gj 21(1 + ∆12)(1 + ∆21)
gj 12gj 21 gj 12gj 21 + gj 11∆11 (∆12 + ∆21 - ∆22) gj 22 gj 22
) gj 11(1 - ζh) 1 +
The control system can be considered as two subsystems (Figure 2); a control system in which the controller c1(s) is connected to the process
g˜11(s) ) g11(s) -
(8)
(5)
Assume that the first loop is detuned to be sufficiently slow, whereas the second loop is fast. Under this assumption, as shown in Figure 3, we have h1(jω) ≈ 0 for the design frequencies ω of the fast second loop and
g˜ 22 ≈ g22 ) gj 22(1 + ∆22)
(10)
Here, the argument jω is omitted for brevity. We can see that relative error in g˜ 11(jω) is increased, whereas relative error in g˜ 22(jω) is the same as in g22(s). Furthermore, for ill-conditioned processes, relative error in g˜ 11(jω) becomes very large because
|
|1 - ζh(jω)| ) 1 -
|
gj 12(jω)gj 21(jω) gj 11(jω)gj 22(jω)
,1
(11)
for some frequencies. Hence, the first loop can be very sensitive to plant variations, whereas the second fast loop is not. If both loops have similar speeds, both loops will be sensitive to the plant uncertainties. If each uncertainty ∆ij(jω) is small and the first slow loop can maintain its performance, a robust control system can be designed. For this, adaptive control technique is applied to the slow loop. Because the first loop is single-input single-output and it is tuned to be slow, adaptive control system can be designed easily.
Ind. Eng. Chem. Res., Vol. 40, No. 25, 2001 5931
Figure 4. Proposed control system. Figure 5. Parameter identification model.
Adaptive Slow/Fast Control System. The proposed control systems for interacting multivariable processes are as shown in Figure 4. Here, to decompose the system to a slow subsystem and a fast subsystem more effectively, a partial decoupler is introduced as
p1(s) ) - g21(s)/g22(s)
[
g11(s) G12(s) G21(s) G22(s)
]
where G12(s), G21(s), and G22(s) are 1 × (n - 1), (n - 1) × 1 and (n - 1) × (n - 1) matrixes, respectively. The fast controller c2(s) is then designed for G22(s), and the partial decoupler p1(s) is designed to approximate -G22-1(s)G21(s). The process for the slow loop system becomes g˜ 11(s) ) g11(s) - G12(s)G22-1(s)G21(s). Indirect adaptive control method is used. For this, because the first loop is controlled to be slow, simple first-order process
g˜11(s) ≈
b s+a
q1 q2 b b ) + s + a λs + 1 λs + 1 s + a
(12)
This partial decoupler makes automatically h1(s) and h2(s) can be designed from eqs 9 and 10, respectively. It can be easily shown that the partial decoupler does not disturb the robustness nature of the second fast loop whenever the first loop is maintained to be slow. Because there is no addition or subtraction between transfer functions in 12, the partial decoupler is not very sensitive to plant changes. If the partial decoupler is perfect, it decomposes the process into two effective transfer functions of eqs 9 and 10. It will promote the separation of the slow subsystem and the fast subsystem. For some processes, the partial decoupler may not be proper or may have a time lead element. We add a low pass filter to p1(s) when it is not proper and a time delay element when it requires a time lead element. In this case, perfect decoupling cannot be possible but still the following adaptive control technique can be applied because analysis in the previous section is effective in case of no partial decoupler. This decomposition can be easily extended to general n-dimensional processes of
G(s) )
Filtered output is fitted on-line as shown in Figure 5. When estimation of the filtered output yˆ f(t) is perfectly track the filtered output yf(t), we have
(13)
is assumed and its parameters are identified. Among many schemes for continuous time identification, a scheme with the state variable filter4,8 is used. Before applying the state variable filter, a first-order filter gf(s) is introduced to cut the high-frequency data which will excite the ignored high order terms in g˜ 11(s). Hagglund and Astrom9 used several filters to find frequency response data without identifying higher order transfer function. Filter is known to be helpful for robust identification.
(14)
Hence
a ) (1 - q1)/λ b ) q2/λ
(15)
Here, λ is the time constant of state variable filter. The model parameters which approximate the equation
yf(t) ) q0 + q1z1(t) + q2z2(t) ) (1 z1(t) z2(t))(q0 q1 q2)T
(16)
are identified. Standard exponentially weighted leastsquares method is applied as
θ(tk) ) θ(tk-1) + L(tk)(yf(tk) - φT(tk)θ(tk-1)) L(tk) )
P(tk-1)φ(tk) ξ + φT(tk)P(tk-1)φ(tk)
[
P(tk) ) P(tk-1) -
]
P(tk-1)φ(tk)φT(tk)P(tk-1) λ + φT(tk)P(tk-1)φ(tk)
/ξ
(17)
θ(tk) ) [q0 q1 q2]T φ(tk) ) [1 z1(tk) z2(tk)]T The forgetting factor ξ is to discard old data and is introduced to track time varying parameters. It is usually set to near 1 or made to vary as contents of information in input data.10 When parameters are estimated, controller can be designed. Because the process is assumed to be first order, proportional and integral (PI) controller is sufficient. For the closed-loop time constant to be τc, PI control system is designed as
kc )
1 τcbˆ
τI )
1 aˆ
(18)
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Figure 6. (a) Responses of the proposed control system with and without adaptation for example 1. (b) Trajectories of model parameter estimates.
where aˆ and bˆ are estimates of a and b in the first-order process model 13. Examples. Example 1. Consider the high purity distillation column of Skogestad et al.6
G(s) )
[
0.878 -0.864 1 75s + 1 1.082 -1.096
]
PI controller for the fast controller is designed for the transfer function g22(s) by the pole assignment technique as
(
c2(s) ) -136 1 +
1 1.987s
)
This controller is such that h2(s) ) (1.987s + 1)/(s + 1)2, that is, the closed-loop poles of the fast loop are all at -1. The partial decoupler is designed as
p1(s) ) -g21(s)/g22(s) ) 0.987 The time constant for the first slow loop τc is set to 40, which is slower by approximately 20 times than the fast loop. The time constants for the state variable filter and the low pass filter gf(s) are chosen to be faster a little than the closed-loop time constant τc. Here both are set to 20. Until the time 200, pulse input is added to the input u1 and model parameters are identified with 200 data of z1 and z2. Least-squares method is applied to these data and initial values for identification are calculated. Then initial controller is designed from eq 18 and recursive identification of eq 17 is started. Control responses and identified parameter trajectories are shown in Figure 6. At the time 600, the process is changed to
G(s) )
[
]
0.878 -0.864 1 × 75s + 1 1.082 × 1.1 -1.096 1.1 exp(-0.1s) 0 0.9 exp(-0.1s) 0
[
]
where the gain of g21(s) is increased by 10% and input variables have 10% gain errors and time delays of 0.1. Uncertainties in input variables were considered in
Figure 7. (a) Responses of the proposed control system for example 1 when random noises between -0.25 and 0.25 are added to the output y1. (b) Trajectories of model parameter estimates.
many robust control applications.11,12 However, their effects on the control performances are not as large as expected. On the other hand, the gain uncertainty in g21(s) affects the control performances very much. This type of uncertainty is our major concern of this study. We can see that the parameter estimation tracks plant changes. When the process change is occurred at the time 600, the proposed control system fluctuates but settles down rapidly, showing good control performances. As shown by the dotted line in Figure 6, the control system without adaptation becomes unstable for the above process change. Figure 7 shows control performances when random noises between -0.25 and 0.25 are added to the output y1. It shows that the proposed adaptive control system is not much influenced by the process noises. There are many drawbacks to be solved for practical applications in the adaptive control method.13 Among them, the estimator wind-up problem14 is critical. That is, the estimates of parameters can be drift when the process is controlled well and is not sufficiently perturbed. Many methods are suggested to resolve this problem. These practical aspects of the adaptive control system5 are not investigated further here because aim of this paper is to show that the adaptive control method can be used to control interacting multivariable processes robustly. Example 2. Consider the Wood and Berry column15
[
19.4 exp(-3s) 6.6 exp(-7s) 14.4s + 1 10.9s + 1 GWB(s) ) 18.9 exp(-3s) 12.8 exp(-s) 21s + 1 16.7s + 1 -
]
This process has a condition number of 8 at the steady state and is not considered as an ill-conditioned process. This process will show that the proposed adaptive control method can also be applied to general interacting multivariable processes which are not highly illconditioned. Applying the internal model control technique16 to the transfer function g22(s), PI controller for the fast loop can be designed as
(
c2(s) ) 0.691 1 +
1 17.7s
)
Ind. Eng. Chem. Res., Vol. 40, No. 25, 2001 5933
multivariable subsystem. The fast subsystem is found to be robust for plant variations whenever the slow subsystem is maintained to be slow. However, the slow subsystem is very sensitive to plant variations. That is, effects of plant variations are concentrated on the slow subsystem. We apply the adaptation technique to the slow subsystem. Because the slow subsystem is singleinput single-output and it is controlled to be slow, the adaptive scheme become very simple. Simulations show that this adaptive control system is effective to control interacting multivariable processes including ill-conditioned processes. Nomenclature
Figure 8. (a) Responses of the proposed control system and multiloop PI control system designed by the biggest log-modulus tuning method for example 2. (b) Trajectories of model parameter estimates.
This controller makes the approximate time constant for the fast loop be 2. The partial decoupler is designed as
p1(s) ) -1.477
16.7s + 1 exp(-2s) 21s + 1
The time constant for the first slow loop τc is set to 20, which is slower by approximately 10 times than that of the fast loop. The time constants for the state variable filter and the low pass filter gf(s) are set to both 10. Control performances and model parameter trajectories are shown in Figure 8. Until the time 200, the slow loop is open and pulse input is introduced to u1 to gather data for initial values of adaptive control. At the time 400, time delays of the diagonal transfer functions are increased by 40%, and the process gains of the offdiagonal elements are increased by 40%. The dotted line shows the responses when fixed multiloop control system designed by the biggest log-modulus tuning method15 is used. We can see that far better performances can be obtained with the proposed control system. Conclusion It is hard to design robust control systems for interacting multivariable processes. When processes are illconditioned, difficulties are more involved due to their sensitivities on plant variations. Lee et al.17 have shown that it is impossible to design fixed control systems which have both features of performance and robustness at the same time for all possible plant uncertainties and all possible disturbances. That is, tight control systems for ill-conditioned processes suffer from sensitivity problems for some plant uncertainties and disturbances, and robust control systems are hard to obtain high control performances. To resolve this problem, we propose an adaptive control system for interacting multivariable processes. Multivariable processes have large number of parameters to be identified for adaptation of control systems and this classical multivariable adaptive control systems can have the same sensitivity problem. Here, we decompose the control systems into two subsystems of a scalar slow subsystem and fast
a, b ) Parameters for the approximate first-order process. aˆ , bˆ ) Estimates of a and b. C(s), ci(s) ) Controller transfer function matrix and its element. G(s), gij(s) ) Plant transfer function matrix and its element. gf(s) ) Filter transfer function. g˜ ii(s) ) Effective transfer functions when loops are connected. gj ij(s) ) Uncertainty free element of process transfer function matrix. hi(s) ) Closed-loop transfer function for the diagonal process. kc ) Controller gain for the slow loop. p1(s) ) Partial decoupler. q0, q1, q2 ) Quantities to be identified. s ) Laplace variable. u ) Process input vector. y ) Process output vector. yf ) Filtered output. z1, z2 ) Outputs of the state variable filter. R ) Constant for the effective gain estimation. ∆ij(s) ) Relative model uncertainties. τc ) Closed-loop time constant for the slow loop. τI ) Integral time for the slow loop. λ ) Time constant for the state variable filter. Literature Cited (1) Green, M.; Limebeer, D. J. N. Linear Robust Control; Prentice Hall: New Jersey, 1995. (2) Liu, Y.; Anderson, B. D. O. Frequency Weighted Controller Reduction Methods and Loop Transfer Recovery. Automatica 1990, 26, 487. (3) Astrom, K. J.; Wittenmark, B. Adaptive Control, 2nd ed.; Addison-Wesley: New York, 1995. (4) Narendra, K. S.; Annaswamy, A. M. Stable Adaptive Systems; Prentice Hall: New Jersey, 1989. (5) Hagglund, T.; Astrom, K. J. Supervision of Adaptive Control Algorithms. Automatica 2000, 36, 1171. (6) Skogestad, S.; Morari, M.; Doyle, J. C. Robust Control of Ill-Conditioned Plants: High Purity Distillation. IEEE Trans. Automatic Control 1988, AC-33, 1092. (7) Al-Ashoor, R. A.; Khorasani, K. A Decentralized Indirect Adaptive Control for a Class of Two-Time-Scale Nonlinear Systems with Application to Flexible-Joint Manipulators. IEEE Trans. Ind. Electron. 1999, 46, 1019. (8) Elliott, H. Hybrid Adaptive Control of Continuous Time Systems. IEEE Trans. Automatic Control 1982, AC-27, 419. (9) Hagglund, T.; Astrom, K. J. Industrial Adaptive Controllers Based on Frequency Response Techniques. Automatica 1991, 27, 599. (10) Fortescue, T. R.; Kershenbaum, L. S.; Ydstie, B. E. Implementation of Self-Tuning Regulators with Variable Forgetting Factors. Automatica 1981, 17, 831.
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(11) Limebeer, D. J. N.; Kasenally, E. M.; Perkins, J. D. On the Design of Robust Two Degree of Freedom Controllers. Automatica 1993, 29, 157. (12) Whidborne, J. F.; Postlethwaite, I.; Gu, D. W. Robust Controller Design Using H-infinity Loop-Shaping and the Method of Inequalities. IEEE Trans. Control Systems Technology 1994, 2, 455. (13) Seborg, D. E.; Edgar, T. F.; Shah, S. L. Adaptive Control Strategies for Process Control: a Survey. AIChE J. 1986, 32, 881. (14) Anderson, B. D. O. Adaptive Systems, Lack of Persistency of Excitation and Bursting Phenomena. Automatica 1985, 21, 247.
(15) Luyben, W. L. Simple Method for Tuning SISO Controllers in Multivariable Systems. Ind. Eng. Chem. Process Des. Dev. 1986, 25, 654. (16) Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: New Jersey, 1989. (17) Lee, J.; Cho, W.; Edgar, T. F. Effects of Diagonal Input Uncertainties and Element Uncertainties in Ill-conditioned Processes. Ind. Eng. Chem. Research 1998, 37, 1009.
Received for review March 30, 2000 Accepted September 14, 2001 IE000357B
