Ind. Eng. Chem. Res. 2004, 43, 735-740
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PROCESS DESIGN AND CONTROL Relay Feedback Approaches for the Identification of Hammerstein-Type Nonlinear Processes Ho Cheol Park, Doe Gyoon Koo, Jung Hoon Youn, and Jietae Lee* Department of Chemical Engineering, Kyungpook National University, Taegu 702-701, Korea
Su Whan Sung Department of Chemical & Biomolecular Engineering and Center for Ultramicrochemical Process Systems, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong, Yuseong-gu, Daejeon, 305-701, Korea
For identification of a Hammerstein-type nonlinear process that consists of a nonlinear static function and a linear dynamic subsystem, the use of a relay feedback test and subsequent triangular input test is investigated. The proposed approach can separate the identification problem of the nonlinear static function from that of the linear dynamic subsystem, which makes it possible to estimate the model parameters without any iterative procedure. From the relay feedback test, the frequency response data of the linear dynamic subsystem are estimated. Then, the model parameters of the nonlinear static function are estimated from the subsequent triangular-type input test. The advantages of the proposed experimental approach are demonstrated with two numerical examples and a simple linearizing control system based on the estimated Hammerstein model. Introduction Linear models have inherent limitations in describing the nonlinear dynamics of industrial chemical processes. For better process models, nonlinear block oriented models that consist of linear dynamic subsystem and memoryless nonlinear static function such as Wiener, Hammerstein, and Hammerstein-Wiener models1 are often used. To obtain these nonlinear models empirically, many methods have been available. A cubic spline function,2 a multilayered feedforward neural,3 and a piecewise linear mapping4 have been used to describe static nonlinear functions. Pottmann et al.5 suggested an identification method based on a multi-model approach with Kolmogorov-Gabor polynomials. Bai6 proposed an optimal two-stage identification algorithm for the Hammerstein-Wiener system. Sung7 proposed a new estimation method for nonlinear static element using a random binary signal to deactivate effects of the nonlinear elements. A˙ stro¨m and Ha¨gglund8 identified ultimate information from a relay feedback test to tune the PID controller automatically. Shen et al.9 suggested the use of a biased relay to estimate the steady-state gain and the ultimate information of the process simultaneously. Some investigators used relay feedback tests and block oriented nonlinear models to identify and tune controllers automatically. To estimate nonlinear models in a very simple manner, Luyben and Eskinat10,11 used two autotune
tests with different relay heights and different dynamic elements inserted in the feedback loop. Huang et al.12 suggested a method to classify and identify nonlinear processes using two relay feedback tests. However, only a few points of the nonlinear function can be estimated in their approach because the relay test signals contain only several levels. Balestrino et al.13 inserted time delay in the feedback loop and then adjusted the relay height so that the output of process oscillates symmetrically in order to estimate nonlinear models. In this research, we investigate the identification problem for a block oriented Hammerstein nonlinear process that consists of a series of nonlinear static functions and a linear subsystem. The proposed method separates the identification problem of the linear dynamic subsystem from that of the nonlinear static function. It enables model parameter identification to be done without any iterative procedure. The Hammerstein process is activated by a relay feedback test signal followed by a periodic triangular test signal. The frequency response data of the linear dynamic subsystem are estimated from the relay feedback test, and the model parameters of the nonlinear static function are analytically obtained from a subsequent triangular input. To demonstrate the advantages of the proposed experimental approach, two numerical examples and a linearizing control system based on the estimated Hammerstein model have been simulated. Hammerstein-Type Nonlinear Processes
* To whom correspondence should be addressed. Tel: +8253-950-5620. Fax: +82-53-950-6615. E-mail:
[email protected].
Haber and Unbehauen1 have discussed various empirical nonlinear process models and summarized model
10.1021/ie030382s CCC: $27.50 © 2004 American Chemical Society Published on Web 01/08/2004
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Figure 1. (a) Simple Hammerstein-type nonlinear process, (b) schematic diagram of the process activation by the biased relay, (c) typical responses of Hammerstein process (solid line, output of the nonlinear function; dashed line, output of the relay; dotted line, process output), and (d) equivalent Hammerstein process.
structure selection methods. Among various types of nonlinear models with a combination of static nonlinear elements and linear dynamics subsystems, a Hammerstein-type nonlinear model is considered in this research. If a nonlinear static element precedes a linear dynamic subsystem, the model is called a Hammersteintype nonlinear process. Consider the following block-oriented Hammersteintype nonlinear process with a series of a nonlinear static function and a linear dynamic subsystem as shown in Figure 1(a):
vo(t) ) Fo(u(t)) Go(s) )
(1)
Y(s) ) Vo(s)
(bmsm + bm-1sm-1 + ‚‚‚ + b1s + b0)exp(-θs) (aksk + ak-1sk-1 + ‚‚‚ + a1s + a0)
(2)
where u(t), vo(t), and y(t) denote the process input, the output of the nonlinear static function, and the process output, respectively. Fo(‚) is a nonlinear static function and assumed to be monotonic. Go(s) is a transfer function of linear dynamic subsystem. The input, u(t), and the output, y(t), are known or measurable, whereas
vo(t) is not measurable. The main issue of this research is to estimate the nonlinear static function and the frequency response data of the linear dynamic subsystem from measurements of u(t) and y(t). Proposed Identification Method We suggest a two-step approach. First, a biased relay feedback method is applied to activate the Hammerstein nonlinear process, and frequency response data of the linear dynamic subsystem are calculated. Then, a triangular-type subsequent input is forced to the Hammerstein process to identify the nonlinear static function. Estimating Frequency Responses of the Linear Dynamic Subsystem. Consider the biased relay feedback experiment of Figure 1(b) and the corresponding outputs of the nonlinear static function and the Hammerstein process of Figure 1(c). It should be noted that the output of the relay is a binary signal and so is the output of the nonlinear static function. As shown in Figure 1(c), their periods are the same, although the mean value and the oscillation magnitude are different. Hence, the following linear equation is valid between the two binary signals:
vr(t) ) Rur(t) + β
(3)
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Figure 2. Schematic diagram to activate the process using a triangular-type test signal: (a) open-loop activation and (b) closed-loop activation.
where R and β are constants, ur(t) and vr(t) represent the relay output and the output of the nonlinear static function, respectively. Keeping this relationship in mind, let us consider the following transfer function for nonzero frequencies:
∫t+P yr(t) exp(-jnωrt)dt G(jnωr) ) for n ) 1, 2, 3, ... ∫t+P ur(t) exp(-jnωrt)dt
A0 )
r
where ωr ) 2π/Pr, and Pr is the relay period. The output yr(t) is the process output for the relay feedback test and G(jnωr) is the frequency response of process at the frequency jnωr. It should be noted that the bias term of β does not affect the estimates of G(jnωr) for n ) 1, 2, 3, ... (For details, refer to Sung and Lee14). Therefore, eq 4 can be rewritten with respect to a transfer function of the linear dynamic subsystem as follows:
∫t+P yr(t) exp(-jnωrt)dt ) ∫t+P vr(t) exp(-jnωrt)dt
R G(jnωr) )
r
r
RGo(jnωr) for n ) 1, 2, 3, ... (5)
where Go(jnωr) is the frequency response of the linear dynamic subsystem in Figure 1(b) at the frequency jnωr. Without loss of generality, G(s) can be used as the transfer function of the linear dynamic subsystem instead of Go(s) because the nonlinear element can be scaled as F(‚) ) Fo(‚)/R. It does not change the inputoutput relationship as shown in Figure 1(d). Frequency responses of the linear dynamic subsystem for the frequencies jnωr, n ) 1, 2, 3, ‚‚‚ can be estimated by eq 4 from the relay output and the corresponding process output. Estimating Nonlinear Static Function. Now we estimate the nonlinear static function. For this, any test periodic test signal whose period is the same as that of the relay feedback test can be used. We use a triangulartype periodic test signal as shown in Figure 2. Two types of open-loop and closed-loop activation strategies are considered. Figure 2(a) shows the open-loop activation method. The process input is a triangular signal whose period is the same as that of the relay feedback test. Hence, the process output can be described by the following Fourier series expansion. ∞
yt(t) ) A0 +
∑ (An cos(nωrt) + Bn sin(nωrt)) n)1
(6)
∫t+P yt(t)dt
(7)
r
An )
2 Pr
dt ∫t+P yt(t) cos2nπt Pr
(8)
Bn )
2 Pr
dt ∫t+P yt(t) sin2nπt Pr
(9)
r
(4)
1 Pr
r
r
where yt(t) denotes the activated process output by the triangular-type test signal. Figure 2(b) describes the closed-loop activation method. The set point of the control system is a triangular type signal whose period is the same as that of the relay feedback system. The controller is tuned using the frequency information of the linear dynamic subsystem using the Ziegler-Nichols tuning method or the margin specification tuning rule.8 Because the process output is oscillating with the same period of the relay feedback system, the process output yt(t) can also be described by the Fourier series expansion as eq 6. To identify the nonlinear static function, the signal vt(t) is required. We reconstruct vt(t) from yt(t). Consider the Hammerstein nonlinear process of Figure 1(d) and the following transfer function representation of the process for the triangular-type test signal.
Vt(s) ) G(s)-1Yt(s) ) Q(s)Yt(s)
(10)
where G(s)-1 ≡ Q(s), and vt(t) and yt(t) are the output of the nonlinear static function and the process output for the triangular-type test signal, respectively. From eqs 6 and 10, vt(t) can be reconstructed from the measured process output yt(t).
vt(t) ) A0Q(0) + z(t)
(11)
∞
z(t) )
∑ An|Q(jnωr)|sin(nωrt + ∠Q(jnωr)) +
n)1
∞
∑ Bn|Q(jnωr)|cos(nωrt + ∠Q(jnωr))
(12)
n)1
where Q(jnωr) ) G-1(jnωr), n ) 1, 2, 3, ‚‚‚, which are estimated from the relay feedback test (eq 4). The coefficients of An and Bn are calculated from equations 8 and 9 with numerical integration of the process output. From eq 12 a data set of ut(t) versus z(t) can be obtained. Without loss of generality, u(t), v(t), and y(t) can be assumed to be deviation variables. This means that vt(t) corresponding to ut(t) ) 0 is also zero. Here we set -A0Q(0) in eq 11 to the value of z(t) correspond-
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Figure 3. (a) Linearizing control structure and (b) equivalent control system.
ing to ut(t) ) 0. The nonlinear static function is identified by relating the data set of the calculated vt(t) ) A0Q(0) + z(t) and ut(t). If it is needed to estimate the inverse model of the nonlinear static function in a polynomial form, we can analytically solve the following optimization problem using the least-squares method. n
∑
min (ut(ti) - uˆ t(ti))2 ξˆ 1,ξˆ 2,‚‚‚ξˆ ni)1
(13)
uˆ (t) ) ξˆ 1v(t) + ξˆ 2v2(t) + ‚‚‚ + ξˆ nvn(t)
(14)
subject to
The proposed estimation procedure can be summarized as follows. Step 1. First, we obtain the frequency response data set of the linear dynamic subsystem (G(jnωr) for n ) 1, 2, 3, ‚‚‚) using eq 4 from the relay feedback test. Step 2. From the triangular-type test, the coefficients of An and Bn are calculated by eqs 8 and 9 with numerical integration. The z(t) value of eq 12 corresponding to ut(t) ) 0 is set to -A0Q(0). Step 3. From the data sets of ut(t) and vt(t), the nonlinear static function is estimated. The inverse of the nonlinear static function, if necessary, can be obtained by solving the least squares problem of eq 13. Linearizing Control Strategy Once model parameters for the nonlinear Hammerstein process are obtained, a simple linearizing control system can be designed. A simple example is shown in Figure 3(a). The inverse of the nonlinear static function linearizes the nonlinearity of the Hammerstein process, resulting in the equivalent linear control system as shown in Figure 3(b). Control performances like those of linear systems can be achieved. On the basis of the dynamics of the linear dynamic subsystem, the PID controller can be tuned regardless of the nonlinear static function. Case Study We simulated two Hammerstein nonlinear processes to confirm the performances of the proposed identification method. Example 1. Consider the following Hammerstein process which has a third-order plus time delay model as the linear dynamic subsystem and a square root
Figure 4. (a) Process activation by the biased relay feedback in Example 1 (solid line, process output; dotted line, relay output), and (b) process activation by the triangular signal test (solid line, process output; dotted line, controller output; dashed line, set point).
function as the nonlinear static function.
v(t) )
{
-x|u(t)| for u(t) < 0
xu(t) for u(t) g 0
Y(s) ) G(s)V(s) )
exp(-s) (s + 1)3
V(s)
(15)
(16)
Figure 4(a) shows the process output activated by the
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Figure 5. Identification results for the nonlinear static function in Example 1 (solid line, real nonlinear static function; dotted line (not distinguishable from the solid line), estimated nonlinear static function; dashed line, polynomial fitting).
Figure 6. Control performances of the proposed linearizing control strategy and the conventional PID controller (solid line, proposed; dotted line, conventional PID controller).
relay feedback. Figure 4(b) shows the process output activated by a triangular-type set point change for the closed-loop test. Using eqs 8-9 and 11-12, we can obtain data sets of the controller output and the output of the nonlinear static function as shown in Figure 5. The estimated nonlinear static function is multiplied by R ) 0.6086 (calculated from eq 5) to compare it with the real static function. We can see that the proposed identification method provides an excellent model. To design the proposed linearizing controller, we fitted the inverse of the nonlinear static function by the following third-order polynomial using the least-squares method of eq 13.
u(t) ) 0.2720v(t) + 0.0033v(t)2 + 0.1159v(t)3
(17)
Figure 6 compares the control performances of the proposed linearizing PID controller and the conventional PID controller. The linearizing controller shows almost the same responses regardless of set points because it removes the nonlinearity, whereas the conventional PID controller shows a strong dependency on the set point values and poor control performance.
Figure 7. (a) Process activation by the biased relay feedback in Example 2 (solid line, process output; dotted line, relay output), and (b) process activation by the triangular signal test (solid line, process output; dotted line, process input; dashed line, output of the nonlinear static function).
Example 2. We simulated the following Hammerstein process that has a third-order plus time delay as the linear subsystem and a nonlinear static function in the presence of a white noise of 0.001 variance.
{
-0.4 for u2(t) < -0.4 v(t) ) u(t) for -0.4 e u(t) e 1.2 1.2 for u(t) g 1.2
(18)
exp(-3s) V(s) (s + 1)3
(19)
Y(s) ) G(s)V(s) )
To reduce the effect of measurement noise, a low-pass filter of 1/(s + 1) is inserted into the feedback loop and numerical integrations in eqs 4 and 6-9 are performed over a several relay period. Figure 7(a) shows the activated process output by a biased relay feedback test and Figure 7(b) shows the activated process output by a triangular-type process input of Figure 2(a). Data sets of the process input and the output of the nonlinear static function computed using eqs 8-9 and 11-12 are shown in Figure 8. A simulation result without measurement noise is also shown in Figure 8. Here, R is
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Hammerstein processes including the saturation nonlinearity. Literature Cited
Figure 8. Identification results for the nonlinear static function in Example 2 (solid line, real nonlinear static function; dotted line, estimated nonlinear static function in the presence of measurement noise; dashed line (not distinguishable from the solid line), estimated nonlinear static function in the absence of measure noise).
calculated as 0.9000 from eq 5. We can see that the proposed method can describe the saturation function well. Conclusions In this research, a simple identification method for the Hammerstein-type nonlinear processes has been proposed. The process is activated by a relay feedback and a triangular test signal in sequence. From these activations, it is possible that the nonlinear static function and the linear dynamic subsystem are identified separately. From the relay feedback test, we estimate the frequency responses of the linear dynamic subsystem, and then the nonlinear static function of the Hammerstein-type nonlinear process is estimated from the triangular signal test. Simulations show that the proposed identification provides excellent models for
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Received for review May 5, 2003 Revised manuscript received October 22, 2003 Accepted November 26, 2003 IE030382S
