Ind. Eng. Chem. Res. 2006, 45, 4671-4678
4671
Closed-Loop Process Identification under PI Control: A Time Domain Approach Rodrigo Silva,† Daniel Sba´ rbaro,*†,§ and Bernardo A. Leo´ n de la Barra‡ Departments of Electrical Engineering and Metallurgical Engineering, UniVersidad de Concepcio´ n, Concepcio´ n, Chile, and School of Engineering, UniVersity of Tasmania, Tasmania, Australia
Closed-loop identification plays an important role in improving existing underperforming closed-loop controllers, also enabling the identification of systems that, either by technical or economical reasons, cannot operate in open-loop fashion. A simple and noise-immune closed-loop identification approach based on step responses is presented. This method is entirely conceived in the time domain, and it is flexible enough to identify a set of simple low-order models that can be obtained from either reference or disturbance closedloop step responses. This latter feature makes the proposed approach an ideal option as a building block of an adaptive controller. A comparative study considering other closed-loop identification methods available in the literature highlights the effectiveness of the new approach. Finally, a set of experimental examples demonstrates its ability to deal with real industrial processes. 1. Introduction Closed-loop identification is very important in industrial settings, where normally there exists a large number of underperforming closed-loop controllers. Some knowledge of the process dynamics is required to tune these controllers. This knowledge can be obtained from open-loop or closed-loop experiments carried out on the process. From a practical viewpoint, when closed-loop and open-loop identification procedures are compared, the former tend to be less troublesome to the operators, usually take less time, and are the only alternative if the open-loop system is unstable or if, for some operational reason, it is not possible to perform open-loop identification tests. In addition, recent work1,2 has shown that closed-loop identification also yields modeling errors (i.e., bias and variance) that are ideally tuned for the design of a new controller with better performance. A considerable number of identification methods to carry out closed-loop identification have been reported in the literature. Those methods based on the identification of a closed-loop transfer function (between an exogenous signalssetpoint or disturbancesand an internal signal of the loopsprocess or controller input or output) from which a process model can be obtained using knowledge of the operating controller constitute what has been termed indirect closed-loop identification. In this framework, there are two main paradigms: the first one is based on the prediction-error approach and considers discrete time representations,3,4 and the second one is based on continuous time descriptions. The work presented in this paper is based on the latter paradigm. In this context, Yuwana and Seborg5 presented a simple method based on a few data points of a closed-loop transient step response. The underlying open-loop process was assumed to be operating under proportional control, and it was modeled as a first-order plus delay time (FOPDT) system. This method offers significant advantages over open-loop tests such as the reaction-curve method. Jutan and Rodrı´guez6 proposed an * To whom correspondence should be addressed. E-mail: dsbarbar@ udec.cl. † Department of Electrical Engineering, Universidad de Concepcio´n. ‡ School of Engineering, University of Tasmania. § Department of Metallurgical Engineering, Universidad de Concepcio´n.
improved version of Yuwana and Seborg’s method by using a higher-order approximation of the delay in the closed-loop transfer function denominator, and Chen7 extended it further by determining the process ultimate data directly from the closed-loop step response. However, as argued by Suganda et al.,8 a second-order plus delay time (SOPDT) model can represent some process dynamics more accurately than an FOPDT model, especially if the process is underdamped or if it has high order. The same authors proposed an on-line process identification method based on information gathered from a closed-loop test under proportionalintegral (PI) control. Their method uses specific points of the system response depending on the sort of closed-loop step response obtained. Furthermore, it considers a high-order Pade´ approximation of the delay in the closed-loop transfer function denominator. Rangaiah and Krishnaswamy9 presented an identification method to estimate the parameters of an SOPDT model, by using two or three specific points from an underdamped process transient response. Most of the above methods can be quite sensitive to noise because they depend on the values of only a few points of the closed-loop response. An alternative method has been proposed by Wang et al.,10 where the open-loop process step response is reconstructed from data obtained in a closed-loop test. The frequency response of the process is calculated using the fast Fourier transform (FFT), and then, by using the inverse fast Fourier transform (IFFT), the open-loop step response is reconstructed. Finally, a second-order plus delay time with zero (SOPDTZ) model is obtained by using a regression method on the reconstructed open-loop step response. Note that, to deal with noisy measurements, it is necessary to use an instrumental variable method in order to have consistent estimates, and as pointed out in ref 10, there are no general guidelines for selecting the instrumental variable matrix. The use of Laguerre series expansions, first proposed by Wang and Cluett,11 offers an alternative method for dealing with noisy measurements in the identification of models from step responses. The same authors carried out an analysis of their method when applied to noisy open-loop step response data. The bias and variance of the estimated model parameters were analyzed, and a frequency domain bound on the modeling error was estimated. In the context of closed-loop identification, Park et al.12 proposed to carry out the approximation of the reference
10.1021/ie051229w CCC: $33.50 © 2006 American Chemical Society Published on Web 05/27/2006
4672
Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006
closed-loop step response by means of Laguerre series expansions and least-squares techniques to obtain a high-order openloop transfer function. An open-loop SOPDT model was identified in the frequency domain by using again a least-squares (LS) approach, so that the error between the module of this model and the high-order model was minimized. This approach cannot be easily extended to systems with zeros, since the minimization is carried out using solely the transfer function modulus. To overcome the limitations of the previous methods, we propose a closed-loop identification technique based entirely on Laguerre series expansions, which can also identify a process model from both reference and disturbance step responses. To the best of our knowledge, this is the first continuous-time identification method that makes full use of the information obtained from a disturbance step response to identify the process model. The proposed approach carries out all the calculations in the time domain, and it requires a single experimental closedloop test. Because of the filtering properties of the Laguerre filters, there is no need for the introduction of instrumental variables. By using constrained regression methods in the time domain, it is possible to consider models with and without zeros. The paper is organized as follows. In Section 2, a brief introduction to Laguerre series is provided and then the different steps of the proposed identification method are presented. In Section 3, the main features of the proposed approach are illustrated through simulation examples. The new approach is also compared with some other identification methods available in the process control literature. Section 4 presents some results of its application to a real industrial process. Conclusions and final remarks are provided in Section 5. 2. Identification Method We consider set-point or disturbance step changes since they are normally found in practice, even though it is well-known that step signals are suboptimal regarding modeling errors when compared to excitation signals with flat spectra or with a high signal-to-noise ratio (SNR) around the system’s crossover frequency (see ref 1). The proposed method requires data collected from a set-point or disturbance step change. First, the recorded closed-loop response is approximated by a Laguerre series expansion. Second, this expansion can then be used to build the open-loop process step response. Finally, a low-order model is identified from the open-loop step response by solving a constrained regression problem. We will also assume the system to be linear and time-invariant during the identification test. Before providing details about the identification method, we will briefly review the use of Laguerre functions in the context of system modeling. 2.1. Laguerre Function. Laguerre functions have been extensively used for process modeling and control.13,14 These functions are a complete orthonormal set in the space L2[0, +∞] and can be used to represent any stable impulse response. The nth order Laguerre function has the following time-domain expression
e-pt dn-1 n-1 -2pt ln(t) ) x2p (t e ) (n - 1)! dtn-1
(s - p)n-1 (s + p)n
Each Laplace transform represents a low-pass filter in cascade with an (n - 1)th order all-pass filter. The impulse response of a stable linear system H(s) in the space L2[0, +∞] can be modeled using a finite dimensional Laguerre model of order N, namely, N
H(s) )
∑anLn(s)
(2)
(3)
n)1
where an is the coefficient of the nth order Laguerre function. Since the expression (eq 3) is linear in the parameters an, a recursive algorithm can be applied to model the system dynamics. Moreover, the representation using Laguerre functions has several advantages,13 namely, • Laguerre functions have a similar transient response to the usual time delay process. • A good approximation can be obtained with a small number of terms in the Laguerre series because of its orthogonal property. • The representation does not require exact a priori information such as process order and time delay. • The representation quantifies the errors due to nonmodeled dynamics and disturbances. 2.2. Modeling the Closed-Loop Step Response. Let us consider a single-input single-output control system, as illustrated in Figure 1, where P(s) and C(s) denote the transfer functions of the process and the controller, respectively. It is assumed that the feedback system is internally stable. We will consider system responses, Y(s), as a result of step changes either in the reference signal or in the disturbance signal. To obtain an estimate of the plant model, the closed-loop response y(t) is approximated using Laguerre functions by minimizing the following cost function, M
min
∑ (y(tm) - yˆ (tm))2
(4)
a1,...,aNm)1
where M is the number of samples, tm is the time of the mth measurement, and yˆ (tm) is the estimate of y(tm), i.e., N
yˆ (tm) ) a0 +
∑anln(tm)
(5)
n)1
where ln(tm) is defined as in eq 1 and constant a0 will represent the steady-state value of the output. The output can be expressed in the Laplace domain as
(1)
where n ) 1, ‚‚‚, N, and p is a time scale factor. Their corresponding Laplace transform is
Ln(s) ) x2p
Figure 1. Typical feedback control system.
Yˆ (s) )
a0 s
+ x2p‚
N
∑ i)1
(
)
(s - p)i-1
ai
(s + p)i
(6)
Let us introduce the following vectors Y(M × 1), θ((N + 1) × 1), and m((N + 1) × 1), together with the matrix M(M × (N + 1)),
[ ] [ ] [ ] [] mT(t1) mT(t2) M) l mT(tM)
y(t1) y(t2) Y) l y(tM)
a0 a θ) 1 l aN
1 l (t) m(t) ) 1 l lN(t)
where M denotes the number of measurements and N is the number of terms in the Laguerre series, respectively. Then, eq 4 can be rewritten in a compact form, by using an L2 norm, as
min||Y - Mθ||2 where the estimate θˆ is found to be
) a0 + sx2p‚
N
∑ i)1
(
(s + p)i
)
H ˆ (s) (1 - H ˆ (s))C(s)
(9)
a0 + sx2p‚ P(s) )
(
( ∑(
∑ i)1
1 - a0 - sx2p‚
(13)
1-H ˆ D(s)C(s)
Yˆ (s) D(s)
) a0 + sx2p‚
N
∑ i)1
(
)
ai(s - p)i-1 (s + p)i
h s
Y(s) ) P(s)
Y(s) )
) ))
(s + p)
N
ai(s - p)i-1
i)1
(s + p)i
s + a1s + a2
e-LsU(s)
(15)
or in the time domain by
y¨(t) + a1y˘ (t) + a2y(t) ) b1u˘ (t - L) + b2u(t - L) (16) Under zero initial conditions, with a time delay L g 0, integrating both sides of eq 16 gives
∫0ty(τ) dτ + a2∫0t∫0τy(τ1) dτ1 dτ ) t t τ b1∫0 u(τ - L) dτ + b2∫0 ∫0 u(τ1 - L) dτ1 dτ
y(t) + a1
(17)
Assuming that the input is a step with magnitude h, the righthand side of the above expression can be written for any time t g L as
∫0tu(τ - L) dτ ) (t - L)h ∫0t∫0τu(τ1 - L) dτ1 dτ ) 21(t - L)2h
(18)
Substituting eq 18 into eq 17 gives
∫0ty(τ) dτ - a2∫0t ∫0τy(τ1) dτ1 dτ +
(-b L + 21b L )h + (b - b L)th + 21b t h (19) 2
1
(11)
C(s)
(12)
2
2
1
2
2
If the following vectors are defined
Z ) [y(t1) y(t2) ‚‚‚ y(tn) ]T Γ ) [Φ(t1) Φ(t2) ‚‚‚ Φ(tn) ]T
For a step disturbance, we have the following closed-loop transfer function,
Yˆ (s) P(s) H ˆ D(s) ) ) D(s) 1 + P(s)C(s)
b1s + b2 2
y(t) ) -a1
ai(s - p)i-1 i
(14)
where h represents the step magnitude. 2.4. Estimation of the Process Transfer Function. To estimate the open-loop transfer function, it is necessary to have available some appropriate data. To this end, eqs 11 and 13 are used to generate the open-loop step response (eq 14). First, we consider systems with relative degree 1. To illustrate the procedure proposed in Wang et al.,16 we estimate a SOPDTZ model, which can be represented in the Laplace domain by
(10)
Hence, the plant can be expressed in terms of the Laguerre series and the controller transfer function as N
H ˆ D(s) )
ai(s - p)i-1
The plant transfer function can be obtained from the closedloop transfer function of the system
P(s) )
H ˆ D(s)
where HD(s) is obtained from a disturbance step and can also be written in terms of Laguerre functions as follows:
(8)
The estimation of the closed-loop system response depends on the sample time (tM), the number of terms in the Laguerre series (N), and the time scale factor (p). The following guidelines provide good starting points for the selection of the above parameters: • Sample and sampling times: it is recommended that the time of the final measurement should be between 1.2ts and 1.5ts, where ts is the step response settling time,12 and that the sampling time should be around ts/10. • Number of terms in the Laguerre series: if the step response is quite oscillatory, a large number of terms in the Laguerre series are recommended to guarantee an acceptable accuracy. It has been suggested that 5-10 terms are needed for a loworder process with large time delay, and that 10-15 terms are needed for a high-order underdamped process with time delay.12 N • Time scale factor: p must be chosen to maximize ∑i)1 ai2, where ai are the coefficients in the Laguerre series. Wang and Cluett15 described a procedure to determine the time scaling factor which maximizes the above summation. 2.3. Modeling the Open-Loop Step Response. The Laplace transform of a step response is given by eq 6. Thus, the closedloop transfer function can be approximated in terms of the Laguerre series as follows:
Ysp(s)
P(s) )
Finally, the open-loop step response can be obtained as
θˆ ) (MTM)-1MTY
Yˆ (s)
and the plant model can be estimated from
(7)
θ
H ˆ (s) )
Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006 4673
1 ϑT ) a1 a2 -b1L + b2L2 b1 - b2L b2 2
[
where
(20)
]
(21)
4674
Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006
[
ΦT(t) ) -
∫0ty(τ) dτ -∫0t∫0τy(τ1) dτ1 dτ
12 th 2
h th
]
then the least squares algorithm can be applied to find the set of parameters ϑ* that minimizes the sum of squared errors between the step response of the model and the estimated openloop step response; i.e.,
ϑ* ) (ΓTΓ)-1ΓTZ
(22)
Since the number of unknown parameters is equal to the dimension of ϑ, then eq 22 can be solved for a1, a2, b1, b2, and L. This result can also be extended to other models with a relative degree >1. In this case, the number of unknown parameters is smaller than the dimension of the estimate ϑ, and therefore, eq 22 will be overdetermined. We will illustrate a simple procedure to deal with these types of models by considering a second-order plus delay model represented by
Y(s) )
b1 s + a1s + a2 2
e-LsU(s)
(23)
Following the same procedure described for the SOPDTZ model, we notice that we have four unknown parameters and the dimension of the estimated vector is five; this means that there must exist a relationship between some of the estimated parameters. In this example, the parameters are
1 ϑ4 ) -b1L ϑ3 ) b1L2 ϑ5 ) b1 2
(24)
and the relationship between them is
2ϑ5ϑ3 - ϑ42 ) 0
(25)
Equation 25 can be readily added to the cost function as follows,
min θ,λ
{
}
where λ is a positive real number representing a Lagrange multiplier. Thus, the optimal estimated parameters ϑ* and λ* must satisfy the following necessary conditions,
(26)
[ ]
where the matrix W is defined as
0 0 W) 0 0 0
0 0 0 0 0
0 0 0 0 2
0 0 0 -2 0
C(s) ) Kc +
Ti s
Table 1 provides the estimated models obtained by three different methods; i.e., the proposed method, Suganda et al.’s method,8 and Wang et al.’s method.10 To obtain the parameters of a SOPDT model by Wang et al.’s method,8 the time delay is directly calculated from a set of overdetermined equations.17 Different sets of PI parameters were tested for these examples, producing different closed-loop dynamics. For the first two examples, we considered underdamped responses, while for the third and four examples, we considered mildly underdamped and overdamped responses, respectively. A reference unit step change was applied to the system under closed-loop PI control. Thirteen terms of the Laguerre series were considered for approximating the closed-loop response, and the values of their coefficients are given in Table 2. In this case, the proposed method provides a good model of the openloop step response, with low IAE, which was almost identical to that of the actual process. From a comparative analysis, we can conclude that the proposed method outperforms Suganda et al.’s method and provides slightly better results than Wang et al.’s method.10 The results of the approximation of the process with time delay (process 1) using Laguerre functions are presented in Figure 2a, and the comparison between the real process output and the estimated one is presented in Figure 2b. To illustrate the effect of using constraint 25 in the estimation of the parameters of a system with relative degree >1, we consider again system 1, but this time we calculate the parameters by using eq 22, obtaining
ϑ* ) [0.6635 0.0849 0.1356 -0.0892 0.0849 ]
(Z - Γϑ)T(Z - Γϑ) + λ(2ϑ5ϑ3 - ϑ42) 2
ϑ* ) (ΓTΓ - λ*W)-1ΓTY 2ϑ5*ϑ3* - ϑ4*2 ) 0
that illustrate different processes have been adopted from the literature on the subject. All these processes were controlled by a PI controller with the following structure:
0 0 2 0 0
The solution to the problem posed by eq 26 can be obtained by doing a one-dimensional search for the Lagrange multiplier.
Using only ϑ5 ) b1 and ϑ4 ) -b1L, we obtain the following model,
e-1.05s 11.77s2 + 7.81s + 1 which is clearly less accurate than the one obtained with the proposed method; see Table 1. The next two examples correspond to a higher-order process with large dead time and a nonminimum phase process, respectively. The results obtained by the proposed method and by Wang et al.’s method10 are summarized in Table 3. The estimated models give smaller IAE values when compared to the ones obtained by Wang et al.’s method.10 In both examples, the identified model is almost identical to the real process. The following example illustrates how accurate are the estimates when the following long delay process is considered:
e-10s 4s + 4 + 1 2
3. Simulation Examples To evaluate the performance of the proposed method, several examples found in the literature were simulated. As a performance criterion, the IAE (integral of the absolute value of the error) was calculated for each test. 3.1. Identification of a Model from a Reference Step Response. In this section, four examples identified in Table 1
The estimated model was
e-10.03s 4.2836s + 4.0645s + 1 2
from which we can see that the time delay was estimated with
Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006 4675 Table 1. Identification Results for Examples 1-4 Suganda et al.’s method8
proposed method process Pˆ (s)
controller Kc,Ti
process Pˆ (s)
IAE
process Pˆ (s)
IAE
Wang et al.’s method10 process Pˆ (s)
IAE
1
e-s 12s2 + 8s + 1
0.4, 2
e-1.03s 12.03s2 + 7.98s + 1
0.0362
e-0.99s 12.04s2 + 7.98s + 1
0.0532
e-1.02s 12.06s2 + 8.01s + 1
0.0412
2
1 (s + 1)5
0.35, 2
e-1.45s 4.28s + 3.63s + 1
0.1985
0.999e-1.81s 2.98s2 + 3.312s + 1
0.2590
e-1.37s 4.30s + 3.72s + 1
0.2229
e-3s (s + 1)2(2s + 1)
0.55, 3
e-3.21s 4.21s + 3.76s + 1
0.1338
e-3.30s 3.72s + 3.59s + 1
0.1814
e-3.42s 3.12s + 3.49s + 1
0.1305
e-2.93s 2.86s + 3.35s + 1
0.4191
3
4
(0.5s + 1) e-3s (s + 1) (2s + 1) 2
2
2
0.2, 2.4
(0.0312s + 0.3373) e-3.16s
2
0.1111
s + 1.1339s + 0.0.3375 2
Table 2. Terms of the Laguerre Series for Example 1 a0 ) 1.0003 a1 ) -2.3174 a2 ) 0.2612 a3 ) 1.0294 a4 ) -0.3534
a5 ) -0.4149 a6 ) 0.2547 a7 ) 0.1351 a8 ) -0.1525 a9 ) 0.0327
2
a10 ) 0.0732 a11 ) 0.0059 a12 ) -0.0527 a13 ) 0.0269
an accuracy of 0.3 (%) and the coefficients of the denominator were estimated with 7.0 (%) and 1.6 (%) accuracies, respectively. 3.2. Identification of a SOPD Model from a Disturbance Step Response. The following examples illustrate the identification of two different processes, given in Table 5, from a closedloop disturbance step response. The result of the approximation of the closed-loop response of the second process, using Laguerre series, is presented in Figure 3a. The comparison between the real process output and the estimated one is presented in Figure 3b, where it can be seen that the final approximation is very good.
2
2
0.0967
s2 + 1.222s + 0.3657
Table 4 summarizes the coefficients of the Laguerre series expansion for the closed-loop response, and Table 5 summarizes the final identified models. It is worth noting that these models are almost identical to the ones estimated with the step reference responses; see Table 1. 3.3. Measurement Noise. Measurement noise is normally present in many industrial settings. To demonstrate the noise immunity of the proposed method, we have synthesized a set of data corrupted with zero mean Gaussian noise with different standard deviations; to present the results, we use the noise-tosignal ratio (NSR) defined as by Wang et al.10:
NSR )
mean(abs(noise)) mean(abs(signal))
We considered the underdamped closed-loop response of the system
P(s) )
Figure 2. Approximation using Laguerre series and estimated open-loop process response.
(0.0038s + 0.3657) e-3.16s
2.4(s + 5) e-s (s + 1)3(s + 3)(s + 4)
under PI control with Kc ) 0.6 and Ti ) 2. The results for a SOPDTZ model, considering an NSR of 20.98, are presented in Figure 4. Figure 4a shows the approximation of the closedloop response by a Laguerre expansion. Figure 4b shows a comparison between the open-loop step response of the real process and the estimated one. Table 6 lists the IAE for different noise intensities, where we can see that a 200 (%) increment in the noise level produces just a 20 (%) deterioration in the performance index. The results presented in Table 6 and Figure 4 show that the proposed method works well, even in the presence of significant noise levels in the measurement. To illustrate the effect of the measurement noise on the estimates accuracy, we considered system 6, and we estimated its parameters, as defined in eq 15, under different NSR conditions. The results, summarized in Table 7, show the moderate impact of the high noise level on the estimated parameters. 3.4. Comments about Indirect Closed-Loop Identification using Prediction Error Methods. Indirect closed-loop identification using prediction error methods has a strong theoretical background and different variants according to the excitation source and the measured variable. It is indeed possible to identify a closed-loop transfer function from any of the following pairs of signals: (ysp, y), (ysp, u), (d, y), and (d, u). A model for P(s) can then be found using knowledge of C(s) and mappings like eq 10. It has been shown in refs 2 and18 that, depending on the singularities of the controller, several of these variants should
4676
Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006
Table 3. Identification Results for Processes 5 and 6 Wang et al.’s method10
proposed method controller Kc,Ti
process P(s)
1.08 e-10s (s + 1)2(2s + 1)2
5
(-4s + 1) e-1s
6
0.3, 6.2
0.11, 1.4
process Pˆ (s)
1.079 e-10.67s 12.8518s2 + 6.1877s + 1
0.4222
(-0.4479s + 0.1112) e-1.025s
9s + 2.4s + 1
0.1231
s + 0.2651s + 0.1112
2
a5 ) 0.3603 a6 ) -0.1094 a7 ) 0.0407 a8 ) -0.0154
NSR (%) 0
(-0.4327s + 0.1110) e-1.065s
0.5818
0.1319
process Pˆ (s)
IAE
(0.2271s + 0.4322) e-2.11s
0.3015
s2 + 1.2213s + 0.4323 9.45
(0.2589s + 0.4419) e-2.15s
0.4101
s2 + 1.2321s + 0.4415
proposed method controller Kc,Ti
process Pˆ (s)
IAE
e-s 12s + 8s + 1
0.22, 4.1
e-1.02s 12.04s + 7.97s + 1
0.0301
e-3s (s + 1)2(2s + 1)
0.55, 3
e-3.33s 4.001s2 + 3.601s + 1
0.1355
2
1.08 e-11.68s 13.351s2 + 6.332s + 1
IAE
Table 6. Identification Results for Noisy Measurements
a9 ) 0.0016 a10 ) -0.0075 a11 ) -0.0007 a12 ) -0.0068
Table 5. Identification Results Using Disturbance Step Responses
process P(s)
process Pˆ (s)
s2 + 0.2619s + 0.1110
2
Table 4. Terms of the Laguerre Series for Process 1 a0 ) -0.0010 a1 ) 0.8878 a2 ) -2.2897 a3 ) 1.9091 a4 ) -0.7526
IAE
2
16.28
(0.2629s + 0.4529) e-2.18s
0.5148
s2 + 1.2401s + 0.4528 20.98
(0.2699s + 0.4621) e-2.19s
0.6221
s2 + 1.2601s + 0.4622
not be used, because they would deliver a process model that is completely flawed for control design. For example, when the controller is unstable, like a PI controller, the only valid variant is the one based on the (ysp, u) pair. All other variants would produce a process model that is destabilized by the existing controller and inappropriate for control design. Although the authors of refs 2 and 18 have presented their results in a prediction-error framework, these results are based on SNR
Figure 3. Approximation using Laguerre series and estimated process.
considerations only and are, as such, theoretically independent of the identification method. The results of Subsections 3.2 and 3.3 show, however, that the proposed method is not subject to this problem, since a good model could be obtained in the (ysp, y) case, despite the presence of an integrator in the controller. Hence, the proposed method shows some robustness properties not shared by the standard prediction-error method.
Figure 4. Approximation using Laguerre series and estimated process.
Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006 4677 Table 7. Identification Results in Terms of Parameters Accuracy for Noisy Measurements process Pˆ (s)
IAE
∆L%
∆b1%
∆b2%
∆a1%
∆a2%
(- 0.4479s + 0.1112) e-1.025s
0.1231
2.5
0.78
0.09
0.56
0.09
0.2921
2.8
1.21
1.89
0.56
1.8
0.7885
5.8
1.53
4.5
2.1
4.5
3.9
9.0
6.9
8.9
NSR (%) 0
s2 + 0.2651s + 0.1112 (- 0.4498s + 0.1132) e-1.028s
9.45
s2 + 0.2681s + 0.1131 (- 0.4512s + 0.1161) e-1.058s
16.28
s2 + 0.2722s + 0.1161 (- 0.4617s + 0.1211) e-1.15s
20.98
1.3122
15
s2 + 0.2851s + 0.1210
4. Experimental Results In this section, we apply the proposed method to a real heat transfer process. In this process, air is drawn from the atmosphere by a centrifugal blower, driven to a heater, and delivered through a tube to the downstream process. In addition, there is a valve that can only be manipulated by hand, which fixes the amount of air to be heated. The temperature is measured downstream in the tube, as depicted in Figure 5. The control system measures the air temperature, compares it with a value set by the operator, and generates a control signal, which determines the amount of electrical power supplied to the heater. From a closed-loop reference step response, the open-loop process was identified as
Pˆ (s) )
0.843 e-0.045s ) 0.957s + 2.702s + 1 0.843 e-0.045s (27) (s + 2.38)(s + 0.43) 2
position of the manual opening of the heater. The model identified as a result of this experiment is
Pˆ (s) )
0.848 e-0.01s ) 1.410s + 2.935s + 1 0.848 e-0.01s (28) (s + 1.6)(s + 0.43) 2
The step response of the identified model is very close to the true open-loop system step response, as seen in Figure 8. By comparing both transfer functions, i.e., eqs 27 and 28, we can see that there is a small difference in the fastest pole, with the gain factor and the slowest pole being almost identical. 5. Conclusions In this paper, a new and simple method based on Laguerre series expansions has been developed for noise-immune process identification using data from both reference and disturbance
This estimated model is a very good approximation to the real process, as seen in Figure 6. The following experiment illustrates that the proposed method can also identify a model using the data obtained after a disturbance step is applied. The closed-loop response is provided in Figure 7. This disturbance corresponds to a change in the
Figure 5. Experimental setup.
Figure 6. Comparison between the real and estimated process outputs.
Figure 7. Closed-loop response in the presence of a disturbance.
Figure 8. Comparison between the real process and the estimated one.
4678
Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006
step process responses. This method has the advantage that the identification is carried out in the time domain, avoiding the problem of singular points existing in some frequency-domainbased methods (as by Park et al.12) or the need for instrumental variables (as by Wang et al.10). Furthermore, the Laguerre series works as a filter in the presence of noise, providing immunity against high-frequency noise. The accuracy and reliability of the proposed method has been demonstrated, in the presence of noise, by a comparative study with two other identification methods. The experimental results have also confirmed its potential to deal with the identification of a real process based on data collected during both reference and disturbance step changes. The simplicity of this method makes it very attractive as a key component of an adaptive PI controller, which not only will use information from reference step changes but also from disturbance step changes, as the successful Exact commercial controller.19 Literature Cited (1) Gevers, M.; Anderson, B. D. O.; Codrons, B. Issues in modeling for control. In Proceedings of the 1998 American Control Conference, Philadelphia, PA, 1998; American Automatic Control Council: Dayton, OH, 1988; pp 1615-1619. (2) Codrons, B. Process Modelling for Control: A Unified Framework Using Standard Black-box Techniques; Springer: U.K., 2005. (3) Ljung, L. System Identification: Theory for the User, 2nd ed.; Prentice-Hall: Upper Saddle River, NJ, 1999. (4) So¨derstro¨m, T.; Stoica, P. System Identification; Prentice-Hall International: Hemel Hempstead, Hertfordshire, U.K., 1989. (5) Yuwana, M.; Seborg, D. E. A new method for on-line controller tuning. AIChE J. 1982, 28, 434-440. (6) Jutan, A.; Rodrı´guez, E. S., II. Extension of a new method for online controller tuning. Can. J. Chem. Eng. 1984, 62, 802-807.
(7) Chen, C. L. A simple method for on-line identification and controller tuning. AlChE J. 1989, 12, 2037-2039. (8) Suganda, T.; Krishnaswamy, P. R.; Rangaiah, G. P. On-line process identification from closed-loop test under PI control. Trans. Inst. Chem. Eng. 1998, 76 (Part AS), 451-457. (9) Rangaiah, G. P.; Krishnaswamy, P. R. Estimating second-order dead time parameters from underdamped process transients. Chem. Eng. Sci. 1995, 51, 1149-115. (10) Wang, Q. G.; Zhang, Y.; Guo, X. Robust closed-loop identification with application to auto-tuning. J. Process Control 2001, 11, 519-530. (11) Wang, L.; Cluett, W. R. Building transfer function models from noisy step response data using the Laguerre network. Chem. Eng. Sci. 1995, 50, 149-161. (12) Park, H. I.; Sung, S. W.; Lee, I. B. On-Line Process Identification Using the Laguerre Series for Automatic Tuning of the ProportionalIntegral-Derivative Controller. Ind. Eng. Chem. Res. 1997, 36, 101-111. (13) Dumont, G. A.; Zervos, C. C. Deterministic Adaptive Control Based on orthonormal series representation. Int. J. Control 1988, 48, 2333-2359. (14) Wahlberg, B. System Identification using Laguerre Models. IEEE Trans. Autom. Control 1991, 36, 551-562. (15) Wang, L.; Cluett, W. R. Optimal Choice of Time-scaling Factor for Linear Systems Approximations Using Laguerre Models. IEEE Trans. Autom. Control 1994, 39, 1463-1467. (16) Wang, Q. G.; Guo, X.; Zhang, Y. Direct identification of continuous time delay systems from step responses. J. Process Control 2001, 11, 531542. (17) Wang, Q. G.; Zhang, Y. Robust identification of continuous systems with dead-time from step responses. Automatica 2001, 37, 377-390. (18) Codrons, B.; Anderson, B. D. O.; Gevers, M. Closed-loop identification with an unstable or nonminimum phase controller. Automatica 2002, 38, 2127-2137. (19) Kraus, T. W. Pattern-recognizing self-tuning controller. U.S. Patent 4602326, July 22, 1986.
ReceiVed for reView November 7, 2005 ReVised manuscript receiVed April 24, 2006 Accepted April 24, 2006 IE051229W
