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Ind. Eng. Chem. Res. 2006, 45, 4726-4731
Parameter Estimation for Integrating Processes Using Relay Feedback Control under Static Load Disturbances I4 brahim Kaya* Inonu UniVersity, Engineering Faculty, Department of Electrical and Electronics Engineering, 44280, Malatya, Turkey
Identifying the parameters of an assumed transfer function using the relay feedback control has become an accepted practical procedure. In process control problems, the transfer function may involve an integrator. Therefore, in this paper, a modified relay feedback control for parameter estimation of an integrating plus first-order plus dead time (IFOPDT) plant transfer function has been suggested, using the A-Function method, which is an exact method for investigating a limit cycle to occur. The effect of static load disturbances has been included in the derived expressions for a limit cycle to occur, so that better estimations can be performed under static load disturbances. 1. Introduction Many advanced control strategies incorporate various aspects of the internal model principle, which requires a model of the system. Some proportional-integral-derivative (PID) controllers also include an implicit process model in their design. For some controller design approaches, such as a Smith predictor scheme, a process model must be used. Therefore, being able to obtain an accurate process model is an important task. The relay auto-tuning has become an accepted practical procedure for identifying parameters of an assumed plant transfer function. The method was first suggested in 1984 by Åstro¨m and Ha¨gglund1 to determine the frequency (ωc) at which the plant has a 180° phase shift, as well as the corresponding plant gain (1/Kc). Later, it was also suggested for use in parameter estimation of a plant transfer function,2 where the expressions from the describing function (DF) analysis were used for the identification procedure. However, because the DF method is an approximate method, the estimates that are obtained will have some error. The use of the A-Function method, which is an exact method for investigation of the limit cycle frequency and amplitude that exist in a relay controlled feedback loop, was suggested by Kaya3 and Kaya and Atherton4 for parameter estimation of stable and unstable first-order plus dead time (FOPDT) and second-order plus dead time (SOPDT) plant transfer functions. The fast Fourier transform (FFT) algorithm, which also results in accurate estimates, has been suggested for parameter estimation of stable transfer functions as well.5,6 On the other hand, the use of a relay feedback method for parameter estimation of integrating processes is uncommon. Ho et al.7 used the approximate DF method and a differentiator in front of the process to estimate parameters of an integrating plus first-order plus dead time (IFOPDT) plant transfer function. Kaya8 used the A-Function method for parameter estimation of the IFOPDT; however, it was assumed that the process gain is known or can be obtained by another test. This shortcoming was later erased by introducing an improved relay auto-tuning procedure.8 In the literature, it is, generally, assumed that there are no load disturbances in the relay feedback control system during the parameter estimation procedure. However, load disturbances * To whom correspondence should be addressed. Tel: +90 422 3410010, ext. 4499. Fax: +90 422 3410046. E-mail address: ikaya@ inonu.edu.tr.
Figure 1. Relay feedback system under static load disturbances.
are encountered quite frequently in practical situations. Therefore, the use of expressions obtained for load-disturbance-free situations may lead to significant errors in the estimates under static load disturbances. There are only a few works9-11 that consider relay auto-tuning under static load disturbances. However, all consider calculation of the ultimate gain and frequency by first estimating the disturbance and then injecting a signal to make the limit cycle odd symmetrical. Kaya3 obtained exact expressions to estimate parameters of the stable FOPDT plant transfer function under static load disturbances. The procedure was later extended to stable and unstable FOPDT and SOPDT plant transfer functions.12 The aim of this paper is 2-fold. First, the modified relay feedback method of Kaya8 is extended so that exact parameter estimations also can be performed under static load disturbances. For this, results given by Kaya3 and Kaya and Atherton12 are adopted to obtain parameters of the IFOPDT plant transfer function. Second, additional simulation examples are provided to illustrate that, if the static load disturbance is not considered in the expressions and that the disturbance exist during parameter estimation procedure, then large errors in estimates must be expected. Further simulation results under assumed different measurement noise levels are supplied to illustrate the effectiveness of the method for practical considerations. 2. The A-Function Method When the nonlinearity in Figure 1 is a relay, then exact solutions for the limit cycle frequency and amplitude are possible, and Tyspkin’s approach is one procedure that can be applied to accomplish this task. The method was developed many years ago by Tyspkin and further developed by Atherton,13 who introduced the A-Function. The A-Function of a linear transfer function is a complex function of both time and frequency and, for a transfer function, G(s), and a specific time t, or phase θ (θ ) ωt), and frequency ω, is given by
10.1021/ie060270b CCC: $33.50 © 2006 American Chemical Society Published on Web 05/28/2006
Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006 4727
AG(θ,ω) ) ReAG(θ,ω) + jImAG(θ,ω)
(1)
where ∞
ReAG(θ,ω) )
∑ [VG(nω)(sin nθ) + UG(nω)(cos nθ)]
(2)
n)1
and ∞
ImAG(θ,ω) )
1
∑ [VG(nω)(cos nθ) - UG(nω)(sin nθ)] n)1 n
(3)
Here, Re and Im denote the real and imaginary parts of the A-locus and UG and VG are the real and imaginary parts of the transfer function G(jnω). The real and imaginary parts of the A-Function can be determined analytically from known summations for simple transfer functions or computationally by taking a reasonable number of terms for complex transfer functions. The plant output and its derivative in a relay feedback system, assuming a biased relay and disturbance d, can easily be found3,12 as
c(t) ) dG(0) +
G(0)(h1∆t1 - h2∆t2) + P
(h1 + h2) [ImAG(-ωt,ω) - Im AG(-ωt + ω∆t1,ω)] (4) π c˘ (t) )
(h1 + h2)ω [ReAG(-ωt,ω) - ReAG(-ωt + ω∆t1,ω)] π (5)
Figure 2. Relay input and output.
Figure 3. Modified relay feedback control system under static load disturbances.
for the corrections when this condition is not valid. Equations 6 and 8 give the value of the limit cycle frequency ω and pulse duration ∆t1, and satisfaction can be checked using eqs 7 and 9. 3. Parameter Estimation for Integrating Processes One of the most widely used models for integrating processes is
where G(0) is the steady-state gain, ∆t1 and ∆t2 are the positive and negative pulse durations of the relay output, h1 and h2 are the relay heights, and P is the period (such that P ) ∆t1 + ∆t2), as shown in Figure 2. When the relay has hysteresis (∆), the limit cycle conditions can easily be obtained, by imposing the switching requirements at time t ) 0 and ∆t1, as follows:
x(0) ) -c(0) - dG(0) ) ∆ x˘ (0) ) -c˘ (0) > 0 x(∆t1) ) -c(∆t1) - dG(0) ) -∆ x˘ (∆t1) ) -c˘ (∆t1) < 0 Substituting eqs 4 and 5 into these expressions yields
ImAG(0,ω) - ImAG(ω∆t1,ω) )
[
]
G(0)(h1∆t1 + h2∆t2) -π dG(0) + ∆ + (6) h1 - h2 P ReAG(0,ω) - ReAG(ω∆t1,ω) < 0 ImAG(0,ω) - ImAG(-ω∆t1,ω) )
[
(7)
]
G(0)(h1∆t1 + h2∆t2) π dG(0) - ∆ + (8) h1 - h2 P ReAG(0,ω) - ReAG(-ω∆t1,ω) < 0
(9)
These expressions are valid provided that limsf∞sG(s) ) 0; otherwise, some corrections should be made to the right-hand side of eqs 6-9. The reader may refer to the work of Atherton14
G(s) )
Ke-Ls s(Ts + 1)
(10)
Therefore, the number of unknowns to be determined is four. Three of them are coming from the assumed model plant transfer function parameterssnamely, K, T, and Lsand the other parameter is the disturbance magnitude (d). If a relay with symmetric characteristics is used, the number of expressions to be obtained is two.3 Therefore, to find all unknowns, a second relay test with a different hysteresis value must be conducted, which is a time-consuming procedure. Using a relay with asymmetric characteristics will result in four expressions.3 However, the integrating process model that is given by eq 10 has an infinite value of G(0). Hence, eqs 6 and 8 cannot be used in parameter estimations. To determine the parameters of the integrating processes, Ho et al.7 made a suggestion to place a differentiator in front of the process in the standard relay feedback control system, given in Figure 1. However, this will cause impulses at zero crossings. This shortcoming can be overcome by placing the differentiator at the output of the process8 (see Figure 3). It is assumed that the disturbance enters the system through the same transfer function as the manipulated variable, which is a common procedure that has been used in the literature.9-11 Another disadvantage of the identification method suggested by Ho et al.7 is that the approximate DF analysis was used with a biased relay in the relay-controlled feedback loop. Because of the approximations in the DF method, the results obtained will have some errors. The error in the estimates will get worse as the asymmetry in the oscillations and the hysteresis used in the relay become greater.3,4 Kaya8 used the
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Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006
A-Function method, together with the configuration given in Figure 3, to remove the shortcomings that are related with the identification procedure proposed by Ho et al.7 However, both Ho et al.7 and Kaya8 assumed that no disturbance enters into the relay feedback control system during the identification procedure. Here, the expressions given in ref 8 are extended so that the effect of static load disturbances are taken into consideration. In Figure 3, the plant is assumed to be the IFOPDT plant transfer function, which is given by eq 10. Therefore, the limit cycle measured at c(t) in Figure 3 will be the same as the limit cycle measured at c(t) in the standard relay feedback control system, given in Figure 1, for a stable FOPDT plant transfer function, Ke-Ls/(Ts + 1). As a result, the modeling procedure can be done using the expressions obtained for the FOPDT plant transfer function, instead of the IFOPDT plant transfer function. To obtain the expressions for a limit cycle to exist, one must determine the imaginary part of the FOPDT plant transfer function, which is given as3
K Kπe(θ+ωL)/λ (11) ImAG(θ,ω) ) (θ + ωL + λ - π) - 2π/λ 2 e -1 where λ ) ωT. Substituting eq 11 into eqs 6 and 8, the following two equations can easily be obtained:3
K
K
(
)
-ω∆t1 πeL/T(e∆t1/T - 1) + ) 2 (e2π/λ - 1)
[
[
]
G(0)(h1∆t1 + h2∆t2) -π dG(0) + ∆ + (12) P (h1 - h2)
]
ω∆t1 - 2π πeL/T(e(-ω∆t1+2π)/λ - 1) + ) 2 (e2π/λ - 1)
[
]
G(0)(h1∆t1 + h2∆t2) π dG(0) - ∆ + (13) h1 - h2 P
Substituting eq 11 into eq 4, the plant output can easily be obtained as follows:3
G(0)(h1∆t1 + h2∆t2) + P (h1 - h2)K -ω∆t1 πe(-t+L)/T(e∆t1/T - 1) (14) + π 2 e2π/λ - 1
c(t) ) dG(0) +
(
)
The minimum and maximum of c(t) occurs at t ) L and t ) ∆t1 + L - 2π/ω, respectively. Substituting these values into eq 14, the minimum and maximum of the plant output are determined to be3
G(0)(h1∆t1 + h2∆t2) + amin ) dG(0) + P (h1 - h2)K -ω∆t1 π(e∆t1/T - 1) + 2π/λ (15) π 2 (e - 1)
[
]
G(0)(h1∆t1 + h2∆t2) + P (h1 - h2)K -ω∆t1 πe2π/λ(1 - e-∆t1/T) + (16) π 2 (e2π/λ - 1)
amax ) dG(0) +
[
]
Using the above four equationssthat is, eqs 12, 13, 15, and 16sdetermination of the three unknown parameters of the
IFOPDT plant transfer function and the disturbance magnitude is possible. However, because the equations are nonlinear, initial guesses are required. Thus, to reduce the number of unknowns and make the solution easier to find, Fourier analysis can be used to identify the steady-state gain, K, and disturbance magnitude, d. It is assumed that the steady-state gain can be calculated from
∫t t+P c(t) dt K ) G(0) ) t+P ∫t y′(t) dt
(17)
where c(t) and y′(t) are the plant output and input, respectively, and P is the period of the limit cycle. After steady-state operation occurs, the disturbance magnitude d can be calculated from the relation
d)
1 G(0)P
∫t t+P c(t) dt -
(h1∆t1 + h2∆t2) P
(18)
where ∆t1 and ∆t2 are the respective positive and negative pulse durations of the relay output, and h1 and h2 are the relay heights when the relay is on and off, respectively. Therefore, with the plant transfer function gain K and the disturbance magnitude d determined (from eqs 17 and 18, respectively), the time constant T can be calculated from eq 15 if the amin parameter is measured or eq 16 if the amax parameter is measured. Because of the fact that the measurements are generally not free of error, both amin and amax can be measured and the time constant T can be calculated from eqs 15 and 16 separately; the average value of these two results then can be used as a final value to ensure better accuracy. Finally, with K and T known, the dead time L can be computed using either eq 12 or eq 13. Again, to ensure better accuracy, the average of these two equations can be used for the time delay estimation. Remark 1: It can be claimed that a standard relay feedback control can be performed for parameter estimation if one waits until the dynamics of disturbance disappear. This may not be possible due to two reasons. First, if the system is under seVere static load disturbances, it may not be practical to wait for the dynamics of disturbances to disappear. Second, because a relay with asymmetric characteristics is used, it cannot be predicted whether the asymmetry in the oscillations is due to a disturbance or due to the relay characteristic. Remark 2: Because an exact differentiator is not physically realizable, an approximate differentiator, such as, s/(0.01s + 1) can be used. This approximate differentiator is used in all of the simulations. 4. Simulation Examples In this section, simulation examples are provided to illustrate the use of the proposed identification procedure. In the first example, an IFOPDT plant transfer function is considered. For different static load disturbance magnitudes, the identification method suggested in this paper, as well as those devised by Ho et al.7 and described in ref 8 are used for comparison. Simulations also were performed for different relay heights and hysteresis values. Hence, it is shown that, if the effect of load disturbance is not taken into consideration, quite unsatisfactory solutionssand, in some cases, unacceptable resultsscan be obtained. In the second example, some typical integrating plant transfer functions under static load disturbances are modeled using the proposed identification procedure to show that the models obtained are satisfactory in the sense of controller design
Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006 4729 Table 1. Estimated Parameters for Example 1, Using the Proposed Identification Method h1 ) 0.7, h2 ) -0.5 d
K
T
h1 ) 0.7, h2 ) -0.3 L
K
0.00 0.10 0.15
1.0000 1.0000 1.0000
20.0015 19.9995 20.0040
∆ ) 0.05 10.0096 1.0000 10.0105 1.0000 10.0097 1.0000
0.00 0.10 0.15
1.0000 1.0000 1.0000
20.0047 20.0041 19.9985
∆ ) 0.1 10.0097 1.0000 10.0094 1.0000 10.0108 1.0000
G(s) )
T
L
19.9983 20.0017 20.0029
10.0106 10.0097 10.0098
20.0006 20.0032 20.0025
10.0102 10.0095 10.0099
Table 2. Estimated Parameters for Example 1, Using the Method Described in ref 8 h1 ) 0.7, h2 ) -0.5 d
K
T
0.00 0.10 0.15
1.0000 1.0000 1.0000
20.0173 18.6134 17.2841
0.00 0.10 0.15
1.0000 1.0000 1.0000
20.0157 18.4469 16.9427
h1 ) 0.7, h2 ) -0.3 L
K
T
L
∆ ) 0.05 10.0069 1.0000 9.0608 1.0000 8.3832 1.0000
20.0135 16.5904 13.9341
10.0085 8.4705 7.9532
∆ ) 0.1 10.0065 1.0000 9.0387 1.0000 8.4074 1.0000
20.0132 16.1388 13.4785
10.0069 8.6846 8.5074
Table 3. Estimated Parameters for Example 1, Using the Method Given by Ho et al.7 h1 ) 0.7, h2 ) -0.5 d
K
T
h1 ) 0.7, h2 ) -0.3 L
K
0.00 0.10 0.15
1.0000 1.0000 1.0000
16.4684 16.8888 17.2672
∆ ) 0.05 10.7347 1.0000 11.2752 1.0000 11.7266 1.0000
0.00 0.10 0.15
1.0000 1.0000 1.0000
16.5713 17.2055 17.7775
∆ ) 0.1 11.4189 1.0000 12.0442 1.0000 12.5821 1.0000
T
L
17.2694 19.1331 21.2137
11.7785 13.6369 15.3971
17.8465 21.0356 25.3058
12.8416 15.4010 18.4444
in most cases. As in practice, measurements are usually not noise free; the modeling is performed under assumed measurement noises to illustrate the accuracy of estimates as well. Throughout the paper, the following definition, which is known as the noiseto-signal ratio,15 will be used for the noise level in the measured signals.
N)
Example 1. The IFOPDT plant transfer function,
mean(abs(noise)) mean(abs(signal))
(19)
In eq 19, abs(‚‚‚) denotes the absolute value and mean(‚‚‚) represents the mean value. To avoid the undesirable effects of measurement noise, a relay with hysteresis, where the width of hysteresis is larger than the noise band, can be used. Alternatively, a filter can be used to reduce the unwanted effects of noise.
e-10s s(20s + 1)
which was considered by Ho et al.,7 is considered. For different load disturbance magnitudes, relay heights, and hysteresis values, relay feedback tests were conducted. The parameter estimation method that has been proposed in this paper (which uses exact analysis and considers the effect of load disturbance magnitude), as well as the method given in ref 8 (which uses exact analysis but does not consider the effect of load disturbance magnitude) and the method given in the work of Ho et al.7 (which uses an approximate DF method and does not take the effect of load disturbance into consideration), are performed to determine the unknowns, namely K, T, and L. The estimated parameters are given in Table 1 for the proposed method, in Table 2 for the method given in ref 8, and in Table 3 for the method given by Ho et al.7 Several conclusions can be derived from these tables. Of course, using the expressions given in this paper resulted in exact solutions for all cases, from Table 1. The identification procedure used in ref 8, which uses exact expressions for a limit cycle to occur but does not consider the effect of load disturbances, gives slightly better estimates, when compared to the identification method given by Ho et al.,7 which uses approximate DF analysis. Some common weaknesses of the identification methods suggested in ref 8 and by Ho et al.7 can be derived from Tables 2 and 3. First, as the disturbance magnitude gets larger, the error in the estimates also becomes larger. Second, increasing the bias in the relay also increases the errors in the estimates. In the tables, exact solutions are obtained for the gain K, for all cases and identification methods, as eq 17 that has been given in this paper is used for the identification. Otherwise, completely incorrect solutions may be obtained for the identification method proposed in ref 8 and by Ho et al.7 Example 2. In this example, several typical integrating process transfer functions are considered to show the effectiveness of the method in modeling higher-order process transfer functions for controller design. Table 4 lists some typical integrating process transfer functions. Relay parameters for the simulation were selected to be h1 ) 1, h2 ) -0.8, and ∆ ) 0. The disturbance magnitude for all cases was assumed to be d ) 0.1. The corresponding models found using the IFOPDT model with different noise levels are also provided in Table 4. The Nyquist curves of the actual plant transfer functions and the IFOPDT model transfer functions obtained under different assumed measurement noise levels are shown in Figures 4-6, which illustrates good matching, where the phase change is ∼180°, hence showing that the method can give reasonable models for higher-order plant transfer functions with the IFOPDT plant transfer functions for controller design.
Table 4. Actual and Estimated Model Plant Transfer Functions for Some Typical Integrating Processes IFOPDT Model case
process -0.2s
N ) 0% -0.299s
N ) 1% -0.298s
N ) 5%
a
e s(0.1s + 1)(s + 1.2)
0.843e s(1.072s + 1)
0.862e s(1.082s + 1)
0.881e-0.299s s(1.037s + 1)
b
e-0.5s s(s + 1)(0.5s + 1)(0.2s + 1)(0.1s + 1)
1.000e-1.123s s(1.756s + 1)
0.960e-1.138s s(1.673s + 1)
0.940e-1.144s s(1.632s + 1)
c
e-10s s(s + 1)(0.5s + 1)(0.2s + 1)(0.1s + 1)
1.000e-10.683s 1.234s + 1
1.002e-10.677s 1.251s + 1
1.024e-10.577s 1.461s + 1
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Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006 Table 5. Controller Parameters Gc(s)
Figure 4. Nyquist plots for case a.
Gd(s)
case
N (%)
Kp
Td
Kd
Tf
a b c
5 5 5
1.988 0.487 0.0483
1.037 1.632 1.461
1.898 0.465 0.0462
1.037 1.632 1.461
gested, is used to determine the controller settings. Table 5 gives the tuning parameters for different cases. The controller parameters are given only for the worst case, that is, N ) 5%. In the table, Gc(s) is the controller responsible for set-point tracking (Gc(s) ) Kp(1 + Tds)) and Gd(s) is the controller for a satisfactory disturbance rejection (Gd(s) ) Kd(1 + Tfs)). Closedloop system responses to a unity step set-point change and disturbance with magnitude of -0.5 introduced into the system at time t ) 30 s are shown in Figures 7 and 8, for cases a and b, respectively. The disturbance magnitude for case c was assumed to be -0.1, which was introduced at time t ) 100 s. For cases a and b, very satisfactory closed-loop responses are achieved. For case c, the disturbance rejection is not very satisfactory (see Figure 9), because the controller design used was suggested for processes with small time delays,16 whereas the time delay used for case c is quite large. Actually, a Smith predictor scheme17 must be adopted for processes with large time delays. Because the main objective of this paper is identification, this point will not be considered further here.
Figure 5. Nyquist plots for case b.
Figure 7. Closed-loop responses for case a with N ) 5%.
Figure 6. Nyquist plots for case c.
Example 3. In this example, the accuracy of the IFOPDT models, which were obtained in Example 2 for some higherorder plant transfer functions, is tested for tracking set-point changes and disturbance rejections in a closed-loop system. The controller design approach given in ref 16, where a two-degreesof-freedom internal model control (IMC) structure was sug-
Figure 8. Closed-loop responses for case b with N ) 5%.
Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006 4731
Figure 9. Closed-loop responses for case c with N ) 5%.
5. Conclusions In this paper, a modified relay feedback control under static load disturbances is given to identify parameters of the integrating plus first-order plus dead time (IFOPDT) plant transfer function using exact expressions for a limit cycle to occur. It is shown, by examples, that large errors in estimates can result if the effect of load disturbances is not taken into account. Using the expressions provided in this paper, exact solutions are achieved when the actual plant transfer function matches the IFOPDT perfectly and error-free measurements are assumed. On the other hand, if the process has high-order transfer functions, reasonable models can be obtained with the IFOPDT plant transfer function for controller design. Simulation results for assumed noise levels are provided to show that the method also gives rational estimates when measurement noise is taken into consideration. Literature Cited (1) Åstro¨m, K. J.; Ha¨gglund, T. Automatic Tuning of Simple Regulators with Specifications on Phase and Gain Margins. Automatica 1984, 20 (5), 645-651.
(2) Luyben, W. L. Derivation of Transfer Functions for Highly Nonlinear Distillation Columns. Ind. Eng. Chem. Res. 1987, 26, 2490-2495. (3) Kaya, I˙ . Relay Feedback Identification and Model Based Controller Design. D.Philos. Thesis, University of Sussex, Brighton, U.K., 1999. (4) Kaya, I˙ .; Atherton, D. P. Parameter Estimation from Relay Autotuning with Asymmetric Limit Cycle Data. J. Process Control 2001, 11 (4), 429-439. (5) Bi, Q.; Wang, Q. G.; Hang, C. C. Relay-Based Estimation of Multiple Points on Process Frequency Response. Automatica 1997, 33, 17531757. (6) Wang, Q. G.; Hang, C. C.; Bi, Q. Process Frequency Response Estimation from Relay Feedback. Control Eng. Practice 1997, 5, 12931302. (7) Ho, W. K.; Feng, E. B.; Gan, O. P. A Novel Relay Auto-Tuning Technique for Processes with Integration. Control Eng. Practice 1996, 4 (7), 923-928. (8) Kaya, I˙ . Parameter Estimation for Integrating Processes Using Relay Feedback Control. (In Turk.) In Proceedings of the TOK Conference, Istanbul, Turkey, 2005; pp 173-174. (9) Hang, C. C.; Åstro¨m, K. J.; Ho, W. K. Relay autotuning in the presence of static load disturbance. Automatica 1993, 29, 563-564. (10) Shen, S. H.; Wu, J. S.; Yu, C. C. Autotune identification under load disturbance. Ind. Eng. Chem. Res. 1996, 35, 1642-1651. (11) Park, J. H.; Sung, S. W.; Lee, I. B. Improved relay autotuning with static load disturbance. Automatica 1997, 33, 711-715. (12) Kaya, I˙ .; Atherton, D. P. Exact Parameter Estimation from Relay Autotuning under Static Load Disturbances. Proc. Am. Control Conf. 2001, 3274-3279. (13) Atherton, D. P. Conditions for periodicity in control systems containing several relays. Presented at the 3rd IFAC Congress, 1966, Paper 28E. (14) Atherton, D. P. Nonlinear Control Engineering, Student Edition; Van Nonstrand-Reinhold: London, 1981. (15) Haykin, S. An Introduction to Analogue and Digital Communications. Wiley: New York, 1989. (16) Kaya, I˙ . Two degree-of-freedom IMC structure and controller design for integrating processes based on gain and phase margin specifications. IEE Proc.sControl, Theory Appl. 2004, 151 (3), 481-487. (17) Smith, O. J. A controller to overcome dead time. ISA J. 1959, 6 (2), 28-33.
ReceiVed for reView March 6, 2006 ReVised manuscript receiVed May 1, 2006 Accepted May 3, 2006 IE060270B
