다입력-다출력 Hammerstein 비선형 공정을 위한 고급 공정확인방법의 개발

Theories and Applications of Chem. Eng., 2002, Vol. 8, No. 1

화학공학으 이론과 응용 제8권 제1호 2002년

이용준, 성수환, 박선원

한국과학기술원, 생명화학공학과, 초미세 화학공정 연구센터

화학공학의 이론과 응용 제8권 제1호 2002년

Development of Advanced Process Identification Method for Multi-Input-Multi-Output Hammerstein Nonlinear Processes.

Yong Joon LEE, Su Whan Sung, and Sunwon Park

Department of Chemical & Biomolecular Engineering, KAIST

The Center for Ultramicrochemical Process Systems.

1. Introduction

The purpose of this study is to design a special test signal to identify MIMO Hammerstein processes and to develop an advanced process identification method.

The Hammerstein nonlinear process is shown in the following figure.

Figure 1 Simple Hammerstein-type nonlinear process

Where[eqn: u(k)] and [eqn: y(k)] denote the process input and process output, respectively. This nonlinear model is useful to represent chemical processes. All chemical processes always involve the nonlinearity. Even though linear models can represent the process dynamics in the narrow operating range, we must use the nonlinear models when we operate the process in the wide range such as batch reactors. The existing well-established linear identification methods have a limit of expressing the dynamics of an actual process. Therefore the identification method must be developed so as to identify the process dynamics more exactly. In the chemical processes, almost actuators are nonlinear and there are many cases that the actuator is nonlinear to the process output. Because the process gain at steady state is usually more dominant than the time constant, nonlinear model of Hammerstein type can describe the dynamics of the chemical processes very well (Eskinat et al. (1991), Luyben and Eskinat (1994)). Many identification methods for the nonlinear model have been proposed. But, they have the following problems technically. Eskinet et al. (1991) calculate the parameters of Hammerstein models by using the Narendra_Gallman algorithm, they demonstrate the usefulness of the Hammerstein model by applying it to model simulated distillation columns and an experimental heat exchanger process.   However, it has problems of initialization and numerical stability. Because they cannot remove completely the interaction between the two identification problems of the linear subsystem and the nonlinear function, the algorithm needs an iterative procedure to estimate the process parameters. Luyben and Eskinat (1994) estimate the Hammerstein model by using the two relays with both different height and period each other. The autotune approach can identify the nonlinear model in a very simple way. But only several points of the nonlinear function can be estimated because only several data points of height and period of process output is available from the test. All above-mentioned identification methods can be applied only to SISO(Single-Input- Single-Output) process.

In this research, we propose a new process identification method to overcome the disadvantages of previous identification methods. By separating completely the two identification problems of the linear dynamics subsystem and nonlinear function, we can directly use the previous well-established linear system identification methods such as subspace methods, prediction error method in identifying the linear subsystem of Hammerstein process, and a simple non-iterative least squares method to identify the nonlinear function. And, it can manipulate SISO as well as MIMO processes without any difficulty.

2. Development of the Proposed Identification Method

Consider the following MIMO Hammerstein nonliner process of [eqn: m] inputs.

[eqn: { v}_{j }(k)={ p}_{j,1 }{ u}_{1 }(k)+ CDOT CDOT CDOT+{ p}_{j,m }{ u}_{m }(k)+ { p}_{j,m+1 } { u}`_{1 } ^{2 }(k) +{ p}_{j,m+2 } { u}_{1 }(k){ u}_{2 }(k)+ CDOT CDOT CDOT]

[eqn: +{ p}_{j,2m }{ u}_{1 }(k){ u}_{m }(k)+ CDOT CDOT CDOT, j=1,2,CDOT CDOT CDOT,m]	(1)

[eqn: x(k)=Ax(k-1)+Bv(k-1)+Ke(k-1)]	(2)

[eqn: y(k)=Cx(k)+e(k)]	(3)

Where,[eqn: y(k) IN { R}^{l }], [eqn: u(k) IN { R}^{m }] and [eqn: v(k) IN { R}^{m }] denote the measured process output, the given process input, and the nonlinear output, respectively. [eqn: x(k) IN { R}^{n }] is the n-dimensional state. [eqn: e(k) IN { R}^{l }] is a Gaussian noise. (1) represents the nonlinear function. (2) and (3) describe the linear subsystem of Hammerstein model. The problem of this study is to estimate the system matrices [eqn: A, B, C, K] and [eqn: { p}_{k }, k=1,2, CDOTCDOT CDOT] from the discrete-sampled-output and the given process input. It is very important to separate completely the two identification problems of the linear subsystem and the nonlinear function in obtaining of MIMO Hammerstein models. The previous methods cannot separate the two identification problems, so they cannot avoid complexity.

In this research, we use the below test signal to separate the two identification problems. Consider a following test signal that satisfies two conditions:  1. A binary signal composed of two steps, where one must be zero and the other is [eqn: d](which is nonzero constant).2. At any time [eqn: k], if one of [eqn: m] input signals is [eqn: d], the others must be zero. For example, if the first input, [eqn: {u}_{1}(k)] is [eqn: d], the others, [eqn: {u}_{j}(k), j=2, CDOT CDOT CDOT ,m] must be zero. The signals satisfying the above conditions can be easily constructed by generating a random number at each time and set it as one input signal sequentially. It should be noted that all elements of [eqn: v(k)] are also binary for the above designed binary signal, [eqn: u(k)] and we can choose freely any value of [eqn: B]. Then, we can choose the element [eqn: B] such that they satisfy [eqn: v(k)=f(u(k))]. Conclusively, we can identify directly the linear subsystem from the input of the linear subsystem [eqn: v(k)=u(k)] and the corresponding process output, measured [eqn: y(k)] by using the previous well-established linear process identification methods. In this study, subspace process identification methods are used.

After determining the linear subsystem through the above procedure, we identify the nonlinear function. To activate the nonlinear function, we insert the multistep random sequence with the various heights after the above-designed-binary signal. Then, we can solve the model parameters[eqn: { p}_{k }, k=1,2, CDOTCDOT CDOT]by solving the following optimization problem.

[eqn: P= arg LEFT [ min LEFT [ V={ 0.5} over {N } SUM from { {k}=1} to {N} LEFT { y(k)-{y}_{m}(k)RIGHT }^{T} LAMBDA LEFT { y(k)-{y}_{m}(k)RIGHT } RIGHT ] RIGHT ]]	(4)

subject to (1), (5) and (6).

[eqn: {x}_{m}(k)={A}_{m}{x}_{m}(k-1)+{B}_{m}v(k-1)+{K}_{m}e(k-1)]	(5)

[eqn: {y}_{m}(k)={C}_{m}{x}_{m}(k)+e(k)]	(6)

where, (5) and (6) present the optimal prediction of the nonlinear Hammerstein process. [eqn: {A}_{m}, {B}_{m}, {C}_{m}] and [eqn: {K}_{m}] are the model parameters of linear subsystem determined by using the previous linear process identification methods. Note that the optimization of (4) can be solved analytically. To solve analytically, the follows should be considered.

[eqn: { PARTIAL V} over {PARTIALP }= -{1 } over {N } SUM from { { k}=1} to{N} LEFT { y(k)-{y}_{m}(k) RIGHT }^{T} { PARTIAL{y}_{m}(k) } over {PARTIALP }]	(7)

[eqn: { PARTIAL{y}_{m}(k) } over {PARTIALP }={C}_{m}{ PARTIAL{x}_{m}(k) } over {PARTIALP }]	(8)

[eqn: { PARTIAL{x}_{m}(k) } over {PARTIALP }={A}_{m}{ PARTIAL{x}_{m}(k-1) } over {PARTIALP }+{B}_{m}{ PARTIALv(k-1) } over {PARTIALP }-{K}_{m}{ PARTIAL{y}_{m}(k-1) } over {PARTIALP }]	(9)

[eqn: { PARTIAL{v}_{1}(k) } over {PARTIALP }= LEFT [ {u}_{1}(k)~CDOT CDOT CDOT ~{u}_{m}(k)~{ u}`_{1 } ^{2 } (k)~ {u}_{1}(k){u}_{2}(k)~CDOTCDOTCDOTRIGHT ]]	(10)

(10) is the first partial differential equation of  the first element [eqn: { v}_{1 }(k)] of [eqn: v(k)] with respect to variable [eqn: P]. We can simply derive all equations about the other elements of vector [eqn: v(k)] in a similar way. Note that the first partial differentiation of [eqn: v(k)] in (10) is independent of variable . So, all the second order differentiation of [eqn: v(k)] are zero. Then, the second order partial differentiation of [eqn: { x}_{m }(k)] and [eqn: { y}_{m }(k)] become zero. We recognize that the partial diffrentiation of the cost function more than 2 are all zero as follows:

[eqn: { {PARTIAL}^{2} V} over {PARTIAL{P}^{2} }= -{1 } over {N } SUM from { { k}=1} to{N} LEFT ( { PARTIAL{y}_{m}(k) } over {PARTIALP }RIGHT ) {LEFT ( { PARTIAL{y}_{m}(k) } over {PARTIALP }RIGHT ) }^{T}] (11)	[eqn: { {PARTIAL}^{k} V} over {PARTIAL{P}^{k} }= 0, ~~k >= 3]	(12)

From (11) and (12), it is clear that the cost function is quadratic. Therefore, the optimization problem can be solved analytically like the following:

[eqn: P=- {LEFT [ { {{ PARTIAL V} over PARTIALP }right|}}_{P=0}RIGHT ]^-1 } {LEFT [ { {{ {PARTIAL}^{2} V} over {PARTIAL{P}}^2 }right|}}_{P=0}RIGHT ] }]	(13)

Remarks: The previous methods need an iterative-optimization method since it cannot separate completely the two identification problems of the linear subsystem and the nonlinear function. Contrarily, the proposed method did succeed in separating two problems completely and then, it can identify the nonlinear function analytically.

3. Simulation Study

We simulated the following Hammerstein-type nonlinear process of which static nonlinear function shows the input multiplicity and the saturation:

[eqn: v(t)=5.3u(t) - 2.8u^2 (t) + 0.43u^3 (t) ~for~ u(t) <= 4.5]	[eqn: v(t)=6.33375~for~u(t)>4.5]	(14)

[eqn: y(s) over v(s) = {exp(-0.3s) } over {s^4 + 4s^3 + 6s^2 + 4s +1} + e(s)]	(15)

Where [eqn: e(t)] is a white noise of 0.05 variance. The process is activated by the RBS signal(composed of 0.0 and 4.0306201) followed by the uniformly distributed random noise(between 0.0 and 7.0) as shown in Figure 2. Here, the minimum switching time is 10 for both signals.

Figure 2. Process activation using the proposed test signal; activated process output (a), test signal (b)

We obtained the following linear model from the input-output data sets from [eqn: t=0] to [eqn: t=50] using the continuous-time prediction error identification method.

[eqn: {{y}_{m} (s)} over {{v}_{m} (s)} = {1.005exp(-0.577s) } over {2.714s^3 + 4.892s^2 + 3.819s +1.000}]	(16)

Next, we identified the polynomial nonlinear static function of (1) using the analytical solution of (13) from the input-output data sets from [eqn: t=0] to [eqn: t=100].

[eqn: v(t)=5.257u(t) - 2.783u^2 (t) + 0.429u^3 (t) ~for~ u(t) <= 4.5]	[eqn: v(t)=6.328~for~u(t)>4.5]	(17)

The estimated linear dynamic subsystem of (16) and the nonlinear static function of (17) are almost exact for the whole activated region as shown in Figure 3, confirming that the proposed method can effectively incorporate input multiplicity, saturation nonlinear function and measurement noises.

Figure 3. Identification result for the nonlinear static function.

4. Concluding remarks

We designed a special test signal to separate completely the two identification problems of the linear subsystem and the nonlinear function in the MIMO nonlinear Hammerstein model. We can identify the linear subsystem by using the previous well-established linear subsystem identification methods such as the subspace method or prediction error method. And the nonlinear function is estimated analytically. And the simulation of the SISO Hammerstein nonlinear process has been executed. We will simulate the MIMO Hammerstein nonlinear process with the proposed method.

Acknowledgements

This work is partially supported by the Brain Korea 21 project and the Center for Ultramicrochemical Process Systems.

References

Eskinat, E., Johnson, S. H. and Luyben, W. L. (1991)  Use of Hammerstein Models in Identification of Nonlinear Systems,  AIChE Journal, 37, 255

Luyben, W. L. and Eskinat, E. (1994)  Nonlinear Auto-tune Identification,  Int. J. Control, 59, 595