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Ind. Eng. Chem. Res. 2006, 45, 4028-4031
Relay Feedback Method under Nonlinearity and Static Disturbance Conditions Su Whan Sung* and Jietae Lee Department of Chemical Engineering, Kyungpook National UniVersity, 1370 Sankyeok-dong, Buk-gu, Daegu 702-701, Korea
We proposed a new relay feedback method to manipulate output nonlinearities and static disturbances. It guarantees the symmetry of the relay output by setting the time length of the lower value of the relay to half that of the previous period. It also rejects the effects of static disturbances and output nonlinearity by changing the input reference value of the relay. We demonstrate with simple examples that the proposed method can be successfully applied to identify the Wiener-type nonlinear process with a static disturbance. 1. Introduction Various relay feedback methods have been proposed and applied to various purposes since the original relay feedback method was proposed.1-5 A modified relay feedback method, using a six-step signal, has been proposed to increase the accuracy of the identified frequency data set by reducing the high-order harmonic terms of the relay output.6 A saturated relay feedback method has been suggested for the same purpose of reducing high-order harmonic terms.7 Several relay feedback methods have been proposed in order to obtain the frequency data set of the process that corresponds to a lower frequency than the ultimate frequency. Some papers have used an artificial time delay,8,9 and others have used a hysteresis.10 A two-channel relay method, with a proportional part and an integral part, has been also used for the same purpose.11 Many relay feedback methods have been proposed to reject static disturbances for which the magnitudes are smaller than the magnitude of the relay.12-15 Their approaches bias the reference value of the relay, as much as the static disturbance, to achieve the same accuracy as in the case of no disturbance. All previous approaches cannot be applied to the design of an approximate sinusoidal test signal for processes with output nonlinearities. Also, the relay feedback methods to reject static disturbances can be applied to only linear processes because they remove the effects of the static disturbances by equalizing the peak value and the valley value of the process output while shifting the output reference value of the relay. In this research, we propose a new relay feedback method in order to design an approximate sinusoidal test signal of the ultimate frequency of the process, which can be applied to nonlinear processes with output nonlinearities and static disturbances. 2. Proposed Relay Feedback Method Figure 1a shows the schematic diagram for the relay feedback methods. u(t) and y(t) are the relay output and the process output, respectively. d represents a static disturbance. Rin and Rout are the input reference value and the output reference value of the relay, respectively. Parts b and c of Figure 1 show typical behaviors of the proposed relay feedback method and the original relay feedback method. Here, the input static disturbance (d) with a magnitude of 0.8 enters at time ) 0. * To whom all correspondence should be addressed. Tel.: +82-53950-6838. Fax: +82-53-950-6615. E-mail:
[email protected].
Figure 1. (a) Schematic diagram of relay feedback methods, (b) typical dynamic behaviors of the proposed relay feedback method for a static disturbance, and (c) the original relay feedback method for a static disturbance.
The proposed method designs the symmetric relay output of the ultimate frequency through the following procedure: (1) Obtain one-period oscillation in the same way as the ordinary relay feedback method. (2) Enter the lower value of the relay and wait for a time equal to half that of the previous period. This enforces the symmetry of the relay output in the cyclic steady state of one frequency. Also, after the time is equal to half that of the previous period, set Rin ) y. This forces the
10.1021/ie050822a CCC: $33.50 © 2006 American Chemical Society Published on Web 12/24/2005
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Figure 2. (a) Wiener-type nonlinear process; (b) Hammerstein-Wienertype nonlinear process.
two crossing points in one period between the process output and the input reference value to converge to the same value. For the case of a severe measurement noise, we recommend Rin ) meanP/2-RPet-toffeP/2+RP[y(t)], which is the mean value of y measurements for the time span P/2 - RP e t - toff e P/2 + RP. P and t - toff are the previous period and time from the relay off. The effects of measurement noises decrease as R increases, but R should be small enough for y to be linear with respect to t within the time span. (3)Enter the upper value of the relay and wait for the process output to cross the input reference value (Rin) for the relay. Repeat steps 2 and 3 until a cyclic steady state is obtained. The output reference value (Rout) of the proposed method is always zero (Rout ) 0). From parts b and c of Figure 1, we realize that the proposed relay feedback method provides a symmetric relay output regardless of the static disturbance, while the original relay feedback method shows a severely asymmetric relay output. As a result, the original relay output cannot be approximated by a sinusoidal signal. We should focus on two remarkable aspects of the proposed method. One is that it always guarantees a symmetric relay output when a cyclic steady state of one period is obtained because the proposed method sets exactly the time length of the lower value of the relay to the half-period. Note that all previous relays determine the on and off status of one cycle on the basis of two zero(reference)-crossing points. Meanwhile, the proposed method determines the on status and the off status on the basis of one reference-crossing point and the previous half-period, respectively. This is the main difference between the proposed approach and the previous approaches. Setting the time length of the lower value of the relay to the half-period in the proposed method guarantees a symmetric relay output in the cyclic steady state of one frequency, because the previous period and the present period are the same in the cyclic steady state. Meanwhile, the previous approaches have no equipment to force the relay output to be symmetric. The other remarkable aspect is that the proposed method updates Rin in order to reject the effects of the static disturbance. These two aspects of the proposed method allow us to generate an approximate sinusoidal signal (equivalently, a symmetric relay output) of the ultimate frequency under circumstances of static disturbances and output nonlinearities. Even though shifting the reference value of the proposed method to reject static disturbances is conceptually similar to the several previous methods, the previous approaches cannot manipulate nonlinearity and their detailed strategies are totally different. For example, consider the Weiner-type nonlinear process with a static disturbance in Figure 2a. Several previous relay feedback methods can reject the effects of static disturbances for linear processes by equalizing the valley and the peak values of the process output while updating the output reference value of the relay. These previous methods, however, cannot be applied to the process in Figure 2a because they equalize the peak and
the valley values of the process output, which are affected by nonlinearity as well as static disturbances. Meanwhile, the proposed method can remove the effects of static disturbances for the Wiener-type nonlinear process because it automatically finds the input reference value of the relay corresponding to the static disturbances, while guaranteeing a symmetric relay output. Also, consider the Hammerstein-Wiener-type nonlinear process in Figure 2b. If the input nonlinear static function is not symmetric, usual relay feedback methods show an asymmetric relay output because the output (w(t)) of the input nonlinear static function is asymmetric for the symmetric relay output (u(t)). So, the previous approaches cannot be used for the Hammerstein-Wiener-type nonlinear process in order to design the approximate sinusoidal test signal, while the proposed method can successfully design the symmetric relay output of the ultimate frequency of the linear dynamic subsystem. The proposed method can also provide a symmetric relay output for other types of nonlinear processes. 3. Case Study We simulated the following Wiener-type nonlinear process with a static input disturbance of 0.2 in order to demonstrate the proposed relay feedback method. Let us estimate the ultimate frequency data of the linear dynamic subsystem and the output nonlinear static function by using the proposed relay feedback method.
w(t) ) u(t) + 0.2
(1)
z(s) exp(-0.3s) ) G(s) ) 3 w(s) s + 3s2 + 3s + 1
(2)
y(t) ) 0.2z(t) + (1 - exp(-3z(t)))
(3)
The activated process output by the proposed relay feedback method is shown in Figure 3. Figure 3a is the case of no measurement noises. Figure 3b is the case of Gaussian measurement noises of 0.001 variance and R ) 0.05. Figure 3c is the case that the ramp disturbance of 0.002t is added to the process output. The measured process outputs and inputs of the last period in Figure 3 are used for the following modeling. Assume that ur is the magnitude of the relay. The fundamental harmonic term of the relay output is
u(t) )
4ur sin(ωt) π
(4)
The high-order harmonic terms of the symmetric relay output are relatively small compared with the fundamental harmonic term of 4ur/π. Also, the process dynamics attenuates much more the high-order harmonic terms than the fundamental harmonic term. So, the signal of z(t) in cyclic steady state can be approximated by the following sinusoidal signal.
z(t) ≈ 0.2G(0) -
4ur |G(jω)| sin(ωt) π
(5)
where G(0) and G(jω) are the frequency responses at zero and the relay (ultimate) frequency, respectively. Now, consider that the inverse model for the output nonlinear static function of eq 3 is
zˆ(t) ) ˆf 1y + ˆf 2y2 + ˆf 3y3 + ‚‚‚ + ˆf nyn
(6)
We can assume that 4ur|G(jω)|/π ) 1 without loss of generality
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Figure 4. Estimated output nonlinear static functions.
which can be solved analytically by the least-squares method. N
min f
(sin(ωti) + ˆf 0 + ˆf 1yi + ˆf 2yi2 + ˆf 3yi3 + ‚‚‚ + ˆf nyin)2 ∑ i)0
(8)
The frequencies of the obtained cyclic steady state in parts a, b, and c of Figure 3 are 1.285, 1.303, and 1.293 (the exact ultimate frequency is 1.304), respectively. The estimated output nonlinear static models are
zˆ(t) ) 1.386y + 0.728y2 + 0.808y3 for the case of no measurement noises (9) zˆ(t) ) 0.991y + 2.581y2 0.953y3 for the case of measurement noises (10) zˆ(t) ) 1.361y + 1.028y2 + 0.556y3 for the case of the ramp disturbance (11) The model performances are acceptable as shown in Figure 4. It is notable that we could successfully identify the nonlinear process with output nonlinearity and static disturbance using only one relay test. Remarks. 1. For the original relay feedback method,1 the following Fourier series expansion is obtained for the Wiener process in Figure 2a:
u(t) ) Figure 3. The dynamic behavior of the proposed relay feedback method for the Wiener-type nonlinear process for the cases of (a) no measurement noises, (b) measurement noises, and (c) ramp disturbance.
because we have one degree of freedom to change the coefficients ˆfi, i ) 1, 2, ‚‚‚, n, in order to compensate for the assumption to guarantee the same overall input-output relation. Then, the output of the linear dynamic subsystem model is
zˆ(t) ≈ - ˆf 0 - sin(ωt)
z(t) )
4ur 4ur 4ur sin(ωt) + sin(3ωt) + sin(5ωt) + ‚‚‚ (12) π 3π 5π 4ur|G(jω)| sin(ωt + ∠G(jω)) + π 4ur|G(j3ω)| sin(3ωt + ∠G(j3ω)) + 3π 4ur|G(j5ω)| sin(5ωt + ∠G(5jω)) + ‚‚‚ (13) 5π
(7)
where zˆ(t) corresponds to the normalized output, z(t)/(4ur|G(jω)|/ π). f0 is -0.2G(0)/(4ur|G(jω)|/π). Now, it is clear from eqs 4 and 7 that the ultimate frequency and the amplitude ratio of the linear dynamic subsystem are the relay frequency and π/4ur, respectively, that is, G ˆ (jω) ) (π/4ur) exp(-jπ). Also, we can estimate the output nonlinear static function by solving the following optimization problem obtained from eqs 6 and 7,
The ratios of the third and fifth harmonic terms to the fundamental term in eq 12 are 0.3333 and 0.2000, respectively. It seems that the high-order harmonic terms are too big compared to the fundamental term to be ignored. Fortunately, the process dynamics further attenuate these high-order terms, because the amplitude ratios of usual process transfer functions decrease dramatically as the frequency increases (that is, |G(jω)| . |G(j3ω)| . |G(j5ω)| . ‚‚‚, ω is the ultimate frequency).
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So, the effects of the high-order harmonic terms are not so serious in the approximation of z(t) ≈ (4ur|G(jω)|/π) sin(ωt + ∠G(jω)). 2. The proposed strategy to design the symmetric relay signal can be straightforwardly extended to other types of relay feedback methods. If we apply the proposed strategy to the previous relay feedback method, which uses a multistep signal,6 we can reduce the high-order harmonic terms sufficiently. In this paper, the two-step relay signal is just exemplified. If a highly accurate model is required, the proposed strategy can be combined with other types of relay methods that suppress the high-order harmonic terms sufficiently. 3. We can determine the order of the polynomials more systematically by minimizing a joint cost function (for example, such as Akaike’s information theoretic criterion), which includes the modeling error term and a penalty term for model complexity.16 4. Conclusions A new relay feedback method has been proposed to manipulate output nonlinearities and static disturbances. It guarantees a symmetric relay output by setting the time length of the lower value of the relay to the half-period. Also, it rejects the effects of static disturbances and output nonlinearity effectively, by changing the input reference value of the relay. Literature Cited (1) A° stro¨m, K. J.; Ha¨gglund, T. Automatic Tuning of Simple Regulators with Specifications on Phase and Amplitude Margins. Automatica 1984, 20, 645. (2) Hang, C. C.; A° stro¨m, K. J.; Wang, Q. G. Relay Feedback Autotuning of Process ControllerssA Tutorial Review. J. Process Control 2002, 12, 143.
(3) Panda R. C.; Yu, C. C. Shape Factor of Relay Response Curves and Its Use in Autotuning. J. Process Control 2005, 15, 893. (4) Lee J.; Kim, D. H. Relay Oscil1ation Method for Estimation of Sorption Kinetics in the Biporous Adsorption. Chem. Eng. Sci. 2001, 56, 6711. (5) Sung, S. W.; Lee, J. Modeling and Control of Wiener-type Processes. Chem. Eng. Sci. 2004, 59, 1515. (6) Sung, S. W.; Park, J. H.; Lee, I. Modified Relay Feedback Method. Ind. Eng. Chem. Res. 1995, 34, 4133. (7) Shen, S.; Yu, H.; Yu, C. Use of Saturation-Relay Feedback for Autotune Identification. Chem. Eng. Sci. 1996, 51, 1187. (8) Kim, Y. H. PI Controller Tuning Using Modified Relay Feedback Method. J. Chem. Eng. Jpn. 1995, 28, 118. (9) Tan, K. K.; Lee, T. H.; Wang, Q. G. Enhanced Automatic Tuning Procedure for Process Control of PI/PID Controllers. AIChE J. 1996, 42, 2555. (10) Chiang, R.; Yu, C. Monitoring Procedure for Intelligent Control: On-Line Identification of Maximum Closed-Loop Log Modulus. Ind. Eng. Chem. Res. 1993, 32, 90. (11) Friman, M.; Waller, K. V. A Two-Channel Relay for Autotuning. Ind. Eng. Chem. Res. 1997, 36, 2662. (12) Hang, C. C.; A° stro¨m, K. J.; Ho, W. K. Relay Auto-tuning in the Present of the Static Load Disturbance. Automatica 1993, 29, 563. (13) Park, J. H.; Sung, S. W.; Lee, I. Improved Relay Auto-tuning with Static Load Disturbance. Automatica 1997, 33, 711. (14) Park, J. H.; Sung, S. W.; Lee, I. A New PID Control Strategy for Unstable Processes. Automatica 1998, 34, 751. (15) Shen, S.; Wu, J.; Yu, C. Autotune Identification under Load Disturbance. Ind. Eng. Chem. Res. 1996, 35, 1642. (16) Ljung L. System identification: Theory for the user; PrenticeHall: Englewood Cliffs, NJ, 1987.
ReceiVed for reView July 13, 2005 ReVised manuscript receiVed November 16, 2005 Accepted November 28, 2005 IE050822A
