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Quantitative Performance Design for Inverse-Response Processes† Weidong Zhang,*,‡ Xiaoming Xu,‡ and Youxian Sun§ Department of Automation, Shanghai Jiaotong University, Shanghai 200030, People’s Republic of China, and National Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou 310027, People’s Republic of China
In this paper, the Smith predictor, known as the dead-time compensator, is extended to the control of a class of nonminimum phase processes referred to as inverse-response processes. An analytical design procedure is developed based on modern H∞ control theory. The resulting controller can provide not only the suboptimal frequency domain performance but also the quantitative time domain specifications for step response. The performance and robustness of the closed-loop system are discussed. For illustration, numerical examples are given to compare the proposed controller to those previous. 1. Introduction Commonly the inverse-response process means the process showing a step response in the opposite direction initially to where it eventually ends up. Such a dynamic behavior is called inverse response and exhibited by a small number of processing units, such as drum boiler and distillation column (Buckley et al., 1975; Luyben, 1973; Shinskey, 1979). The essential characteristic of the process with inverse response is that the process transfer function has one or an odd number of zeros in the open right half plane. This is a class of nonminimum phase (NMP) processes. Generally, these processes are particularly difficult to control and thus require special attention. The most common inverse-response process is that resulting from the conflict of two first-order systems with opposing effects. Several research papers have discussed the characteristics of an inverse-response process, such as Bernardo (1994), McWilliams and Sain (1989), and Vidyasagar (1985). Though a lot of methods, such as the state space method, can be used for controlling the inverse-response process, there are only two popular ways controlling such a plant in the past in the context of process control (Stephanopoulos, 1984; Tyner and May, 1968; Wang and Zhu, 1991): the first uses a PID controller with Ziegler-Nichols tuning (Waller and Nygardas, 1975), and the second uses an inverse-response compensator (Iinoya and Altpeter, 1962). Unfortunately, both of them are empirical methods and cannot provide satisfactory performance (Scali and Rachid, 1998). In this paper, the well-known Smith predictor used for the control of processes with time delay is extended to the control of processes with inverse response. A new H∞ design method presented by Doyle et al. (1992), Zhang et al. (1999), and Zhang and Sun (1997) is used for designing a controller analytically. The quantitative step response is obtained. It is shown that the new structure can be further simplified to avoid † Part of the paper has been published in ACC97 (Zhang, W. D.; Sun, Y. X.; Xu, X. M. Quantitative Control of Inverse Response Processes. Am. Control Conf. 1997, 3257-3261). This is a modified and extended version. * Corresponding author. Tel: +86.21.62933329. Fax: +86.21.62813329. E-mail:
[email protected]. ‡ Shanghai Jiaotong University. § Zhejiang University.
unnecessary complications. This will result in a PID controller. The controller provides not only the optimal performance but also an adjustable performance and robustness. Some contents of the paper are as follows. In section 2, the inverse-response process resulting from two opposing first-order processes is introduced. In section 3, the previous results are given which are necessary for understanding the main results of this paper. The Smith predictor is extended to the control of inverseresponse processes in section 4. The quantitative step response and system robustness are discussed in section 5. The results of the previous sections are illustrated with selected numerical examples in section 6. In section 7 conclusions are presented. 2. Conditions Yielding Inverse Response Assume that there are two opposing first-order stable processes
G1(s) )
K1 K2 , G2(s) ) τ1s + 1 τss + 1
(1)
where K1, K2, τ1, and τ2 are positive constants. Figure 1 shows a possibility of inverse response. The overall response equals to
y(s) ) G(s) u(s) ) (G1(s) - G2(s))u(s) )
(
)
K1 K2 u(s) τ1s + 1 τ2s + 1
(2)
or
y(s) )
(K1τ2 - K2τ1)s + (K1 - K2) u(s) (τ1s + 1)(τ2s + 1)
(3)
where G(s) is the overall transfer function. We have inverse response when τ1/τ2 > K1/K2 > 1, i.e., initially process 2, which reacts faster than process 1, dominates the response of the overall system, but ultimately process 1 reaches a higher steady-state value than process 2 and forces the response of the overall systems in the opposite direction (Figure 2). Here, the system
10.1021/ie990067z CCC: $19.00 © 2000 American Chemical Society Published on Web 04/19/2000
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Figure 1. Inverse-response process. Figure 4. Compensation control structure.
reflects must exclude the information of inverse response. This is possible if in the open-loop response y(s) we add the quantity ys(s) given by
ys(s) ) C(s) Gc(s) e(s) ) C(s) K
(
)
1 1 e(s) τ2s + 1 τ1s + 1
(6)
Then we can easily find that Figure 2. Inverse-response curve.
y0(s) ) y(s) + ys(s) ) C(s) × [(K1τ2 - K2τ1) + K(τ1 - τ2)]s + (K1 - K2) e(s) (7) (τ1s + 1)(τ2s + 1) and for
Kg Figure 3. Unity feedback control system.
transfer function has a zero in the open right half plane:
z)
K2 - K1 >0 K1τ2 - K2τ1
(4)
This is a typical inverse-response process which has been studied by many researchers (Stephanopoulos, 1984; Zhang, 1998). 3. Traditional Control Methods In the past, there are two popular ways to control the process with inverse response in process control: one is a PID controller with Ziegler-Nichols tuning, and the other is an inverse-response compensator. Waller and Nygardas (1975) demonstrated numerically that the Ziegler-Nichols classical tuning of a PID controller could yield good control for systems with inverse response. Iinoya and Altpeter (1962) utilized the concept of the Smith predictor to cope with the inverse response of a process and proposed a compensation scheme referred as an inverse compensator. Consider the unity feedback control system of Figure 3. The open-loop response of the system is
y(s) ) C(s)
(K1τ2 - K2τ1)s + (K1 - K2) e(s) (τ1s + 1)(τ2s + 1)
K1τ2 - K2τ1 τ2 - τ1
(8)
we find that the zero of the open-loop transfer function is in the open left half plane:
z)
K2 - K1 (K1τ2 - K2τ1) + K(τ1 - τ2)
e0
(9)
Adding the signal ys(s) to the main feedback signal y(s) means the creation of a partial loop Gc(s) with the gain K as shown in Figure 4. The system in this local loop is referred as the inverse-response compensator. It predicts the inverse behavior of the process and provides a corrective signal to eliminate it. Stephanopoulos (1984) and Tyner and May (1968) referred to Iinoya and Altpeter’s work respectively in two of the rare process control textbooks treating inverse-response control and recommended choosing C(s) as the PI controller. They did not, however, do much to improve the scheme. Limited by the level of control theory development, they left the readers several unanswered questions: (1) How can the controller be turned to obtain good performance? (2) Can the controller be designed analytically or not? (3) How does one deal with the plant uncertainties? The following discussion will solve the above problems.
(5)
It has a zero in the open right half plane. To eliminate the inverse response, what the measurement signal
4. Modified Inverse-Response Compensator In this section, we will extend the Smith predictor to the control of inverse-response processes in another way
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and present a new inverse-response compensator. It differs from that given by Iinoya and Altpeter in two aspects: (1) When we add the quantity ys(s) in the open loop response y(s), the controller C(s) of the unity feedback loop is changed to the predictive controller R(s) of the new scheme. Hence, the new structure is equivalent to the unity feedback control structure. The controllers C(s) and R(s) are related through
C(s) )
R(s) 1 + Gc(s) R(s)
(10)
(2) In the Smith predictor, a plant model is incorporated to eliminate the effects resulting from time delay. Instead of the compensator presented by Iinoya and Altpeter, we will use a new plant model to predict the inverse behavior of the process. The new inverseresponse compensator is
Gc(s) ) G2(s) - G1(s) )
K1 K2 τ2s + 1 τ1s + 1
)
K2 K1 S(s) ) 1 + R(s) τ2s + 1 τ1s + 1
|| [
(
)]||
K2 K1 1 1 + R(s) min s τ2s + 1 τ1s + 1
|| [
(
)]||
(
)]||
g |W(z)| (14)
∞
Therefore
min
|| [
K1 K2 1 1 + R(s) s τ2s + 1 τ1s + 1
)
∞
(K1τ2 - K2τ1) + K(τ1 - τ2) (15) K1 - K 2
The optimal R(s) is obtained as follows:
Ropt(s) )
(τ1s + 1)(τ2s + 1) (K1 - K2)
(16)
(12)
In process control, controllers are often designed for step input, so we here define system input as a step. Let the performance specification be min |W(s) S(s)|∞, where W(s) is a weighting function. It should be selected such that the normalized input is 2 norm bounded. For mathematical convenience, W(s) can be selected as 1/s (see, for example, Zhang, 1998; Zhang et al., 1999). This is an H∞ minimum sensitivity optimizing problem (see, for example, Doyle et al., 1992, Zhang and Xu, 1999), which means that the error e(s) caused by setpoint r(s) is minimized or the output y(s) caused by disturbance d(s) is minimized. The definition and characteristics of the ∞ norm has been discussed by many papers (see, for example, Doyle et al., 1992), and it is not repeated here. From the well-known Youla parametrization we can know, under the nominal condition, that the close-loop system is internally stable if and only if R(s) is stable. This can also be seen in Figure 4. Under the nominal condition, y0(s) ) 0, then the system is open loop. Because the plant is stable, the internal stability of the system is equivalent to the stability of R(s). It follows that
min |W(s) S(s)|∞ )
|W(s) S(s)| ) K1 K2 1 1 + R(s) s τ2s + 1 τ1s + 1
(11)
The sensitivity function of the system, i.e., the transfer function from the setpoint r(s) to error e(s) or from the disturbance d(s) to system output y(s), can be described as
(
the given plant. An alternative method to design the H∞ controller analytically will be given here. Theorem 1 (Maximum Modulus Theorem). Suppose that Ω is an unempty, open, and connected set in the complex plane and G(s) is an analytical function in Ω. If G(s) is not constant, then |G(s)| cannot reach its maximum value at the interior point of Ω. Let Ω be the open right half plane, and the mentioned plant has a zero z in Ω; then all R(s)’s satisfy the condition that
Obviously, Ropt(s) is improper and cannot be realized physically. It can be overcome by introducing a low-pass filter described by
J(s) )
1 , λ>0 (λs + 1)2
We obtain the proper but suboptimal controller
R(s) )
(τ1s + 1)(τ2s + 1) (K1 - K2)(λs + 1)2
C(s) ) (τ1s + 1)(τ2s + 1)
∞
where R(s) is stable. For H∞ controller design of singleinput and single-output system, there are two effective ways in the frequency domain: one is Nevanlinna-Pick interpolation, and the other is loop shaping (Doyle et al., 1992). However, both of them are not analytical methods: the former is a numeral method while the latter is an empirical method. They are very tedious for
(18)
When λ tends to be zero, the proper R(s) tends to be improper Ropt(s) and the optimal performance is recovered. In systems with time delay, the Smith predictor cannot be simplified to a unit feedback control loop, because there exists time delay of infinity dimensions. The proposed structure can be simplified to a unit feedback control loop, which is easy to implement. Simple computations give the controller of a unity feedback control loop
2
(13)
(17)
2
λ (K1 - K2)s + [2λ(K1 - K2) + (K2τ1 - K1τ2)]s
(19)
This is in fact a PID controller. Following the discussion above, we see that good control performance of inverseresponse processes can be obtained by adjusting a PID controller. It is worth noticing that the controller resulting from the H∞ design method is similar to but different from that derived by the internal model control method (Rivera et al., 1986; Scali and Rachid, 1998). For the internal model control method, the resulting H2
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optimal controller is
Ropt(s) )
(τ1s + 1)(τ2s + 1) (K2τ1 - K1τ2)s + (K1 - K2)
(20)
It is found that the proposed H∞ controller tends to cancel all poles of the process, while the internal model controller or the H2 controller tends to cancel the poles of the minimum phase part of the process. The filter that makes the improper controller proper can be chosen arbitrarily. The only constraint imposed on it is that it should be a low-pass transfer function with static gain being 1. This implies that the proposed controller can be equivalent to the internal model controller and most of the other frequency domain controllers by selecting different filters (Zhang, 1998; Zhang and Xu, 1999). Extensions to high order systems with one zero on the right half plane, such as the inverse-response process yielded by two second-order processes with opposing effects, are easy and straightforward. Certainly, the resulting controller is more complicated than the PID controller.
In the theory and practice of process control, the system performance is often specified by overshoot (or undershoot) and rise time. The controller presented in this paper can provide not only the suboptimal frequency domain performance but also the quantitative time domain performance. When it is supposed that the setpoint is a step signal, the system setpoint response Hr(s) can be computed as follows:
Hr(s) ) T(s) )
1 -z-1s + 1 1 ) s (λs + 1)2 s
1 1 λ λ s λs + 1 (λs + 1)2 z(λs + 1)2
By inverse Laplace transform we get the time domain setpoint response
1 1 hr(t) ) 1 - e-(1/λ)t - te-(1/λ)t - 2 te-(1/λ)t λ λz
S(s) ) 1 -
K1τ2 - K2τ1 s+1 K1 - K2 (λs + 1)2
-z-1s + 1 )1(λs + 1)2
(21)
K1τ2 - K2τ1 s+1 K1 - K2
T(s) )
(λs + 1)2
-z-1s + 1 ) (λs + 1)2
σu ) 1 - e-1/(1+λz) -
1 e-1/(1+λz) 1 + λz 1 e-1/(1+λz) (26) λz(1 + λz)
There is no overshoot in the system. Define the rise time as the time to reach 90% of the final value. Let hr(t) ) 0.9, and then
] [
]
t 1 + λz t + ) 10 1 + t 2 λ λz λ2z
or equivalently
(22)
|
G∆ - G e |∆(jω)|, ∀ ω G
Then the sufficient and necessary condition of robustness is
||W(s) S(s)| + |∆(s) T(s)||∞ < 1
λ 1 + λz
Substituting this into hr(t) gives the undershoot σu
[
Theorem 2 (Doyle et al., 1992). Suppose that the practical plant is G∆(s) and the maximum multiplicative uncertainty bound of the plant is ∆(s):
|
t)
e(1/λ)t ) 10 1 +
and the complementary sensitivity function is
(25)
Let dhr(t)/dt ) 0, and one can get the time of achieving undershoot
5. Features of the New Compensator Many optimal control methods have their own deficiency in practice resulting from their bad robustness. However, at this point, the controller given by this paper is different from that given by the modern control theory and can provide very good robustness. It is easy to find that the sensitivity function of the nominal system is
(24)
(23)
As we can see, when λ increases, S(s) increases and T(s) decreases. The system can endure larger uncertainty with a worse nominal performance. When λ decreases, the system tends to be optimal with a worse robustness. The relationship is monotonic. Because there always exists uncertainty in practical processes, λ is usually chosen properly larger in order to obtain good robustness. In addition, using the above theorem, we can also get quantitatively the maximum uncertainty bound of the system.
λz 1 + λz λz/(1+λz) t e + λz λ 1 + λz
[
e(t/λ)+[λz/(1+λz)] ) 10
]
(27)
The rise time tr is the solution of the equation. This is a transcendental equation, and a rigorous mathematical treatment is difficult. Only a numerical solution can be obtained. Because the undershoot is usually large, the adjustable parameter λ should provide a suitable compromise among the nominal performance, undershoot, and robustness. 6. Example Consider the following inverse-response process from Iinoya and Altpeter (1962):
G(s) )
K2 K1 τ1s + 1 τ2s + 1
where τ1 ) 2, K1 ) 12, τ2 ) 1, and K2 ) 9. When the Iinoya-Altpeter method is used, the compensator is
(s +1 1 - 2s 1+ 1)
6
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Figure 5. Responses of nominal systems.
Here, the controller parameters are P ) 1/6 and I ) 0.5. When the Ziegler-Nichols method is used to tune the PID controller, the controller parameters are P ) 0.3, I ) 3.0, and D ) 0.75. Now, let us try the control method given in this paper. The compensator is
Figure 6. Responses of nominal systems.
3(2s - 1) (2s + 1)(s + 1) The parameter λ in the controller is adjustable. It should be selected such that a satisfied tradeoff between the performance and robustness of the closed loop is obtained. Here, we take λ ) 1 or 2:
(s + 1)(2s + 1) 3(λs + 1)2 Suppose that the setpoint is a unit step signal at t ) 0 and the disturbance is a step signal with amplitude 0.5 in the controlled variable at t ) 30. Then the responses of the nominal system are shown in Figure 5. Among them, the method given in this paper provides the best control performance. The Iinoya-Altpeter method provides strong oscillation and bad control performance because it uses an approximation of the plant model. Although the PID controller with ZieglerNichols tuning seems to be better than the IinoyaAltpeter method, its inverse undershoot is too large to use. Both the internal model controller and the proposed controller can provide optimal or suboptimal response under their own performance index. Because there is an adjustable parameter in the two controllers respectively, one can always get similar performance and robustness. For example, take the parameter λ to be 0.5 for the internal model controller. One can get a response similar to that of the proposed controller with parameter λ being 1 (Figure 6). As we have discussed, parameter λ has a direct relation with the system robustness. Now assume that there exist uncertainties in the control system. We might as well let ∆(jω) ) (25% by adjusting the process gain and get the system response shown in Figures 7 and 8. The controller given by the Iinoya-Altpeter method is unstable, and the PID controller leads to very large overshoot and undershoot when ∆(jω) ) +25%. Only the new controller provides good control. From Figures 5-8, it can be seen that simulation results are identical with those obtained by theoretical analysis. 7. Conclusions In many textbooks, little attention is paid to compensation of the transient effects caused by inverse-
Figure 7. Responses of perturbed systems (∆(jω) ) -25%).
Figure 8. Responses of perturbed systems (∆(jω) ) +25%).
response processes. Since the studies of Iinoya and Altpeter (1962) and Waller and Nygardas (1975), there are few papers about the control of inverse-response processes (Zhang, 1998). This paper, based on the modern robust control theory, extends the Smith predictor to the control of inverse-response process and develops an analytical design procedure. The proposed method can be extended easily to the control of highorder processes, even if it is unstable (Zhang, 1998; Zhang et al., 1999). In common opinions, the traditional Smith predictor has bad robustness. However, the controller given in this paper overcomes the defect. Its parameters can be used to adjust the system robustness as that of the internal model control parameter. What is more, it provides new insight into the control of inverse-response processes by evaluating the system response quantitatively. Though the new controller can be implemented in a unity feedback control loop, the original control structure can help us understand how to cancel the effects caused by right half plane zeros.
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Acknowledgment This project was supported by National Natural Science Foundation of China (69804007) and Science and Technology Phosphor Program of Shanghai (99QD14012). Literature Cited Bernardo, A. On Undershoot in SISO Systems. IEEE Trans. Autom. Control 1994, 39, 578-581. Buckley, P. S.; Cox, R. K.; Rollins, D. L. Inverse Response in a Distillation Column. Chem. Eng. Prog. 1975, 71, 83-84. Doyle, J. C.; Francis, B. A.; Tannenbaum, A. R. Feedback Control Theory; Macmillan Publishing Company: New York, 1992. Holmberg, U.; Myszkorowski, P.; Piguet, Y.; Longchamp, R. On Compensation of Nonminimum Phase Zeros. Automatica 1995, 31, 1433-1441. Iinoya, K.; Altpeter, R. J. Inverse Response in Process Control. Ind. Eng. Chem. 1962, 54, 39-43. Luyben, W. L. Process Modeling, Simulation and Control for Chemical Engineers; McGraw-Hill Company: New York, 1973. McWilliams, L. H.; Sain,, M. K. Qualitative Step Response Limitations of linear Systems. IEEE Conference on Decision Control, Tampa, FL, 1989; pp 2223-2227. Rivera, D. E.; Morari, M.; Skogestad, S. Internal Model Controls 4: PID Controller Design. Ind. Eng. Chem., Process Des. Dev. 1986, 25, 252-265. Scali, C.; Rachid, A. Analytical Design of Proportional-IntegralDerivation Controllers for Inverse Response Processes. Ind. Eng. Chem. Res. 1998, 37, 1372-1379.
Shinskey, F. G. Process Control System, 2nd ed.; McGraw-Hill Book Company: New York, 1979. Stephanopoulos, G. Chemical Process Control: An Introduction to Theory and Practice; Prentice-Hall: London, 1984. Tyner, M.; May, F. P. Process Engineering Control; Ronald Press: New York, 1968. Vidyasagar, M. On Undershoot and Nonminimum Phase Zeros. IEEE Trans. Autom. Control 1986, 31, 440-441. Waller, K. T. V.; Nygardas, C. G. On Inverse Response in Process Control. Ind. Eng. Chem. Fundam. 1975, 14, 221-223. Wang, J. C.; Zhu, H. Y. Chemical Process Control, 2nd ed.; Chemical Industrial Press: Beijing, 1991 (in Chinese). Zhang, W. D. Analytical Design Methods for Process Control. Postdoctoral Research Report, Shanghai Jiaotong University, Shanghai, 1998. See also Generalized Digital Controller Design for Nonminimum Phase Processes and Unstable Processes. Automatica 1999, submitted for publication. Zhang, W. D.; Sun, Y. X. A Class of Smith predictor and its Robust Tuning. ACTA Autom. Sin. 1997, 23, 660-663 (in Chinese). Zhang, W. D.; Xu, X. M. Analytical Design Formulas for NearOptimal H∞ control of Systems with Time Delay. Automatica 1999, submitted for publication. Zhang, W. D.; Xu, X. M.; Sun Y. X. Quantitative Performance Design for Integrator/Time delay Processes. Automatica 1999, 35, 719-723.
Received for review January 13, 1999 Revised manuscript received November 2, 1999 Accepted November 9, 1999 IE990067Z
