Ind. Eng. Chem. Res. 2002, 41, 2941-2947
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Modeling and Control of a Nonlinear Process Based on the Extended Self-Organizing Map Network Hualiang Zhuang,† Wei-Jen Ang,† Masahiro Ohshima,‡ and Min-Sen Chiu*,† Department of Chemical and Environmental Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore, and Department of Chemical Engineering, Kyoto University, Kyoto 606-8501, Japan
An extended self-organizing map (ESOM) network, which consists of a self-organization phase and an optimization phase, was recently developed to construct a local model network (LMN) automatically using the plant data. However, this previous result suffers two drawbacks: (1) increased computation time in the self-organization phase as the number of local models increases, (2) lack of checking stability conditions for both local models and LMN. To overcome these problems, an improved algorithm for the ESOM network is developed in this paper by employing a competitive learning algorithm for data clustering in the self-organization phase and parametric constraints are formulated in the optimization phase to handle the stability of local models. In addition, the global stability of LMN is addressed. With LMN constructed by the ESOM network, it serves as a basis for building a nonlinear controller that combines several local controllers through the weighting functions obtained by the ESOM algorithm. Literature examples are used to illustrate the proposed ESOM-based modeling and controller design method. 1. Introduction The technique of multiple models or a local model network (LMN), which is constructed as the weighted combination of a set of simple local models across the operating space of interest, has received considerable attention in recent years.1-18 Despite the recent advances of LMN, prior knowledge of the process, which may not be readily accessible in most practical applications, has to be exploited for the determination of the LMN structure. To circumvent this problem, Ge et al.7 developed an extended self-organizing map (ESOM) network to partition the operating space of the nonlinear process automatically using the plant data. An ESOM network employs a two-phase algorithm consisting of a self-organization phase and an optimization phase. Because the stability of local models is not taken into account, some of the ESOM local models obtained may be unstable even when the ESOM network is trained with the plant data obtained from a stable process. This feature is not desirable, especially when these local models are to be employed in the controller design. In addition, this previous method requires a long computation time in the self-organization phase as the number of local model increases, and the global stability of LMN was not addressed adequately. In this paper, the previous ESOM algorithm is improved by incorporating the stability of the local models explicitly in its formulation. More specifically, the local stability constraints are formulated and imposed on the parameters of the local models in the optimization phase of the proposed ESOM algorithm. The global stability of LMN is checked by the determination of a common positive-definite matrix for all of the subsystems.17 In addition, a competitive learning * To whom all correspondence should be addressed. Telephone: (65) 68742223. Fax: (65) 67791936. E-mail: checms@ nus.edu.sg. † National University of Singapore. ‡ Kyoto University.
method19 for cluster-center searching is used in the selforganization phase of the proposed ESOM algorithm to improve the computational efficiency. Based on the LMN obtained, a nonlinear controller, termed the local controller network (LCN), can be constructed by combining various local controllers that are designed with respect to the local models of LMN.10,18 Two literature examples are used to illustrate the LCN design based on the LMN obtained form the proposed ESOM algorithm. This paper is organized as follows: in the next section, the formulation of LMN is described and the proposed ESOM algorithm and its utility in the LCN design are discussed. Section 3 presents simulation results to demonstrate the proposed ESOM-based design methods. Finally, the conclusion will be drawn. 2. ESOM Network The idea of LMN is to approximate a nonlinear system with a set of relatively simple local models valid in certain operating regimes. To this end, a weighting function is assigned to each local model to indicate its respective degree of validity in the operating space, and LMN is formed as the weighted combination of these local models. The property that LMN can approximate nonlinear systems has been investigated in great detail recently.11 Moreover, LMN is more applicable for controller design because it gives a higher accuracy than linear models but less complexity as compared to conventional nonlinear models. To construct LMN, the operating space Φ of the process is decomposed into nΦ operating regimes Φj. Define β(k-1) as an operating point vector, which characterizes the process dynamics, as given by
β(k-1) ) [y(k-1), ..., y(k-ny), u(k-nd), ..., u(k-nd- nu+1)]T (1) where u and y are the system input and output, k
10.1021/ie0102605 CCC: $22.00 © 2002 American Chemical Society Published on Web 05/14/2002
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1 and the ith node of layer 2 is defined as Wi ) (wi,1, wi,2, ..., wi,K1)T, where wi,j denotes the connected weight between the jth node in layer 1 and the ith node in layer 2, and Kn is the node number of layer n. The following summarizes the proposed ESOM algorithm. Input Layer. Given the input-output training data v and y, the input of this layer includes y as well as v, i.e., vj ) (v, y), where
v(k) ) [y(k-1), ..., y(k-ny), u(k-nd), ..., u(k-nd-nu+1)]T (7) Figure 1. ESOM architecture.
denotes discrete time samples, ny and nu are integers related to the system’s order, and nd is the time delay. Recently, Ge et al.7 developed an ESOM network to construct LMN automatically using the input-output data of the process. In their work, it was shown that the output of the ESOM network is equivalent to that of a LMN formulated based on the autoregressive exogenous (ARX) local model as represented by nΦ
yˆ (k) )
∑ j)1
nΦ
Fj(β(k-1))yj(k) )
Fj(β(k-1))ψT(k-1) θj ∑ j)1 (2)
where yˆ (k) is the output of LMN, ψ(k-1) is the regression vector, and θj is the local parameter vector for operating regime Φj as defined by
||W vs (l) - v(l)|| ) min||W vi (l) - v(l)||
ψ(k-1) ) [y(k-1), ..., y(k-ny), u(k-nd), ...,
i
u(k-nd-nu+1), 1]T (3) θj ) [θj,1, ..., θj,ny, θj,ny+1, ..., θj,ny+nu, θj,ny+nu+1]
T
(4)
and Fj(β) is the weighting function assigned to each local model and is constrained between 0 and 1. In addition, it satisfies nΦ
Fj(β) ) 1 ∑ j)1
(5)
Note that the steady-state behavior of LMN formulated in eq 2 is obtained by setting y(k) ) ys and u(k) ) us for all k: nΦ
ys )
∑ j)1
ny
Fj(ys,us)(
∑ i)1
nu
θj,iys +
θj,n +ius + θj,n +n +1) ) ∑ i)1 y
y
Kohonen Layer. The weight vector of the Kohonen nodes is formatted as Wi ) (W vi , W yi ), where W vi is called the master weight vector and W yi is the slave weight vector.7 The following gives the self-organizing algorithm to determine a set of cluster centers Ωj, j ) 1, ..., nΦ, which characterize the dynamics of the nonlinear process. Step 1. Initialize Wi, i ) 1, 2, ..., K2, and Ωj, j ) 1, 2, ..., nΦ. Note that the initial Kohonen weight vectors (Wi) are chosen randomly to have values with uniform distribution over the operating space, whereas the cluster centers (Ωj) are initialized by the center data vector of the operating space. Step 2. At each learning step l, only the master weight vector W vi , i ) 1, 2, ..., K2, is used to determine the winner node whose weight vector W vs best matches the input v, i.e.,
u
N(ys,us) (6) where N(‚) is a nonlinear function. Consequently, the LMN given in eq 2 is capable of modeling the dynamic system with input multiplicity or output multiplicity. Despite the advantages of the ESOM method, it suffers two limitations: (1) increased computation time in the self-organization phase as the number of local models increases and (2) lack of checking of the stability conditions for both local models and LMN. To overcome these problems, an improved algorithm for the ESOM network is developed in the remainder of this section. As shown in Figure 1, the ESOM network consists of three layers: an input layer, a Kohonen layer, and an output layer. The connected weight vector between layer
(8)
where ||‚|| denotes the Euclidean norm. Step 3. Update every weight vector Wi ) (W vi , W yi ) in the Kohonen layer as follows:
Wi(l+1) ) Wi(l) + γ(i,l) (vj (l) - Wi(l))
(9)
where γ(i,l) ) 1/[(1 + l)e||pi-ps||2]1/2 and pi and ps are the respective positions of the node i and winner node. Step 4. Check the convergence of Wi, i ) 1, 2, ..., K2, by the criterion ||Wi(l) - Wi(l-1)|| < 1||Wi(l-1)||, where 1 is the tolerance parameter. If all of the inequalities are satisfied, reset learning step l ) 1 and go to step 5; otherwise, l ) l + 1 and go to step 2. To reduce the computational burden in the previous ESOM algorithm, a competitive learning rule19 is employed in steps 5-7 to determine a fixed number of neurons Ωj, j ) 1, 2, ..., nΦ, which are considered as the centers of the clusters of the nodes in the Kohonen layer. Step 5. m ) m + 1. If m > K2, m ) 1. Determine the neuron whose weight vector Ωs(l) best matches the weight vector of W vm(l), i.e.,
||Ωs(l) - W vm(l)|| )
min ||Ωj(l) - W vm(l)||
j)1,2,...,nΦ
(10)
Step 6. Update the weight vector of the neuron Ωs as follows:
Ωs(l+1) ) Ωs(l) + χ(l) (W vm(l) - Ωs(l))
(11)
where χ(l) ) 1/[(1 + l)e||pm-ps||2]1/2 and pm and ps are the respective positions of W vm(l) and Ωs.
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Step 7. Check the convergence of Ωj(l), j ) 1, 2, ..., nΦ, by the criterion ||Ωj(l) - Ωj(l-1)|| < 2||Ωj(l-1)||, where 2 is the tolerance parameter. If the above inequalities are satisfied, stop; otherwise, l ) l + 1 and go to step 5. In relation to the LMN, the cluster centers form the local models. In this paper, the weighting functions Fj are chosen to be the normalized Gaussian functions as follows:
Fj(k) )
AjTPAj - P < 0, j ) 1, 2, ..., nΦ
(20)
[ ][ ] [ ] 1 1 θj,1 1 < -1 1 θj,2 1
(16)
-1 < θj,2 < 1
(17)
where Aj is the system matrix of the jth local model and “< 0” denotes a negative-definite matrix. One remark about implementing the ESOM network is the determination of nΦ and K2. For a specific problem, a large nΦ will normally lead to a LMN with higher accuracy but with more local models. Likewise, a large K2 will normally increase the approximation accuracy of the ESOM network but at the expense of increased computation time because the computation effort (such as the distance measure used in eqs 8 and 9) is linked with the number of Wi, i.e. K2. Thus, there is a tradeoff between complexity and accuracy in choosing the values of nΦ and K2. Unfortunately, no systematic guidelines exist for the determination of these two parameters. In this paper, the following iterative procedure is adopted: starting with an initial value of K2 and the plant input-output data is partitioned into a training data set and a validation data set. The training data is used to train the simplest ESOM network with nΦ ) 2 and those networks with more complexity by gradually increasing the values of nΦ (i.e., more local models), with the one resulting in the smallest modeling error for the validation data being chosen as the bestfit LMN. If the approximation accuracy of LMNs obtained does not meet the modeling requirement, increase the value of K2 and the above procedure repeats. This iterative procedure terminates when at least one LMN obtained can meet or exceed the required modeling accuracy. After LMN is obtained, it can be incorporated into the design of LCN.10 The following summarizes the design procedure of LCN by the ESOM method: Step 1. Generation and acquisition of the process input-output data. Step 2. Construction of the LMN using the proposed ESOM algorithm. Step 3. Local controller design based on each local model. Step 4. The output of LCN is obtained as the weighted sum of the local controller outputs via the weighting functions determined by the ESOM algorithm. In the next section, two literature examples are used to illustrate the proposed ESOM-based modeling and controller design method.
(18)
3. Examples Example 1. The dynamic equations of a first-order exothermic reaction in a continuous stirred tank reactor (CSTR) can be described by the following differential equations:21
(19)
x2 dx1 + d2 (21) ) -x1 + Da(1 - x1) exp dt 1 + x2/µ
exp(-||v(k) - Ωj||2/2σj2) nΦ
, j ) 1, 2, ..., nΦ
exp(-||v(k) - Ωj||2/2σj2) ∑ j)1
(12)
where σj2 is a constant variance. Output Layer. Set vˆ ) (v, 1); the output of this layer is computed as the sum of all signals from the winner nodes: nΦ
y˜ (k) )
Fjvˆ Tθj ) Fˆ θ ∑ j)1
(13)
where Fˆ ) [F1vˆ T, ..., FnΦvˆ T]T, θ ) [θ1T, ..., θnΦT]T, and θj is defined in eq 4. Given the input-output data {v(k), y(k), k ) 1, 2, ..., Kt}, Fˆ is fixed after the self-organizing process, meaning that the output y˜ is a linear combination of the elements of θ. To guarantee the stability of each local model, local parametric constraints are incorporated into the following optimization problem:
min f ) ||Ψθ - Y|| θ
(14)
subject to a set of parametric constraints {Ci(θ) < 0, i ) 1, 2, ...} formulated based on Jury’s stability criteria,20 Ψ ) [Fˆ (1), ..., Fˆ (Kt)]T, and Y ) [y(1), ..., y(Kt)]T. For notational convenience, only the stability constraints up to the third-order model are given as follows:
First-order local model: -1 < θj,1 < 1
(15)
Second-order local model:
[
guarantee the global stability of LMN. Tanaka and Sugeno17 discussed this issue and showed that the system would be globally asymptotically stable if there exists a common positive-definite matrix P for all local models such that
Third-order local model: 1 -1 θj,3 -θj,3
1 1 1 -1
][ ] [ ]
1 1 θj,1 -1 θj,2 < 1 θj,3 1 θj,3 θj,3 1
-1 < θj,3 < 1
By solving the above optimization problem, a set of local models with guaranteed local stability is obtained. However, in the construction of LMN for a dynamic system which is stable over the entire operating space, the stability of local models, i.e., local stability, does not
(
(
)
)
dx2 x2 ) -x2 + BDa(1 - x1) exp + dt 1 + x2/µ R(u - x2) + d1 (22) y ) x1
(23)
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Table 1. ESOM Network for Various Cases in the CSTR Example nΦ ) 3
training data (MSE) validation data (MSE)
nΦ ) 4
nΦ ) 2: 1 stable model, 1 unstable model
case a: 1 stable model, 2 unstable models
case b: 2 stable models, 1 unstable model
case a: 1 stable model, 3 unstable models
case b: 2 stable models, 2 unstable models
case c: 3 stable models, 1 unstable model
1.63 × 10-4
1.45 × 10-4
4.52 × 10-5
2.02 × 10-4
9.14 × 10-5
3.81 × 10-5
3.04 × 10-4
2.35 × 10-4
1.47 × 10-4
4.62 × 10-4
1.98 × 10-4
1.84 × 10-4
Table 3. PID Tuning for ESOM Local Models Kcj 1 2 3
21.59 50.73 24.80
τIj
τDj
3.86 6.68
0.232 0.336 0.930
results obtained by the ESOM algorithm (K2 ) 36, 1 ) 0.01, and 2 ) 0.0066). It is clear that case b of nΦ ) 3 gives the best-fit LMN because it yields the smallest MSE value for the validation test. To obtain this LMN by the proposed ESOM algorithm, the following stability and instability constraints are incorporated into the optimization problem equation (14):
Stability constraint:
[ ][ ] [ ]
Figure 2. Training and validation of the ESOM network with nΦ ) 3 (case b). Solid: ESOM. Dashed: plant. Table 2. LMN for the CSTR Example (case b of nΦ ) 3) 3
y(k) )
∑F Ψ (k-1) θ T
j
j
j)1
θ1 ) [1.9561, -0.9900, 0.0085, 0.0150, 0.0019]T θ2 ) [1.9791, -0.9900, 0.0062, 0.0045, 0.0084]T θ3 ) [0.1373, 1.1373, 0.0319, 0.0336, -0.0222]T
where x1 and x2 are the dimensionless concentration and reactor temperature, respectively. The input u is the temperature of the cooling jacket surrounding the reactor. The constants are Da ) 0.072, µ ) 20, B ) 8, and R ) 0.3, while d1 and d2 represent disturbances to the system. The plant data are sampled every 0.5 units of time, and the operating space of CSTR is x1 ∈ [0.1, 0.8], in which both stable and unstable regions exist. As a result, the LMN to be constructed consists of at least one unstable local model. Because there exists an unstable region in the operating space of CSTR, it is necessary that CSTR is under proportional-integral (PI) control initially to enable the collection of the plant data in the unstable region. Therefore, with a stabilizing PI controller (Kc ) 8 and τI ) 1), a plant test similar to that in work by Banerjee and Arkun3 is used to simulate 1200 input-output plant data, 600 data each for training and validation purpose, as shown in Figure 2. To train the ESOM network, v(k) is chosen as follows:
v(k) ) [y(k-1), y(k-2), u(k-1), u(k-2)]T
(24)
to yield the second-order local models. To determine the best number of winner nodes for the ESOM network, we compare the approximation accuracy of the ESOM network as a function of nΦ and various local stability requirements. Six cases are considered with nΦ ) 2-4. Table 1 summarizes mean square errors (MSEs) for both training and validation
1 1 θj,1 1 < , j ) 1, 2 -1 1 θj,2 1
(25)
-1 < θj,2 < 1, j ) 1, 2
(26)
Instability constraint:
[
][ ] [ ]
-1 -1 θ3,1 -1 < 1 -1 θ3,2 -1
(27)
-|θ3,2| < -1
(28)
In relation to the ESOM network, eqs 25-28 mean that, among three winner nodes (nΦ ) 3) in the ESOM network, two stable winner nodes are denoted by Ω1 and Ω2, respectively. With the aforementioned constraints, Table 2 summarizes the resulting model parameters of LMN. Figure 2 illustrates the predictive performance of this LMN by comparing its predicted output with the plant output. As can be seen, an ESOM-based LMN gives a reasonably good prediction for this reactor. To illustrate the reduction of computation time achieved by the proposed ESOM algorithm, a LMN is obtained by Ge’s ESOM algorithm7 by using the same network structure and training data, and the comparison result shows that the previous ESOM algorithm requires 30.7% more computation time in the self-organization phase. Next, a LCN controller is designed based on the LMN obtained. To this end, three local PID controllers employed are summarized in Table 3 and the LCN controller is constructed as the weighted sum of these local controllers as follows: 3
LCN )
(
1
Fj(β) Kcj 1 + ∑ τ j)1
Ijs
)
+ τDjs
(29)
where Fj(β) is the weighting function obtained by the ESOM algorithm. The servo response of the LCN controller is evaluated for the successive set-point changes in y that varies between 0.1 and 0.8, as illustrated by the solid line in
Ind. Eng. Chem. Res., Vol. 41, No. 12, 2002 2945
Figure 3. Servo response. Solid: LCN. Dashed: stablizing PI. Dash-dotted: set point.
Figure 5. Disturbance rejection with d2 ) -0.1. Solid: LCN. Dashed: stablizing PI.
Figure 4. Disturbance rejection with d1 ) 1/3. Solid: LCN. Dashed: stablizing PI.
Figure 6. Servo response under 20% error in Da. Solid: LCN. Dashed: stablizing PI. Dash-dotted: set point.
Figure 3. For comparison purposes, a simulation result of the stabilizing PI controller is also presented, as given by the dashed line in Figure 3. It is evident that the performance of the LCN controller is superior to that of the stabilizing PI controller over the entire operating space. The disturbance rejection performance of the two controllers is compared in Figures 4 and 5. Again, the LCN controller outperforms the PI controller at the unstable operating point. Similar results are also obtained at the other two stable operating points. Last, the robustness of two controllers is evaluated by assuming that there is a 20% modeling error in the parameter Da. The simulation result in Figure 6 illustrates that the LCN controller has a more consistent and better performance than the PI controller. When the responses given in Figures 3 and 6 are compared, it is obvious that the LCN controller is robust with respect to the parametric uncertainty in Da. Example 2. The operation of the pulse jet fabric filter can be briefly described as follows: during the filtration cycle T, influent gas is passed through a filter bag at a pressure drop ∆P and dust cake is built up at the upstream side of the bag surface. At time T, the bag is subject to a pulse jet of air of high pressure, removing a certain fraction of the cake. Ju et al.22 presented a first principle model of the filtration process. The
average flow rate Q of the exhausted gas during cycle time T is the output of the process, and T is the manipulated variable. The input-output data vector for ESOM training are chosen as
v(k) ) [Q(k-1), Q(k-2), Q(k-3), T(k-1)]T (30) y(k) ) Q(k)
(31)
The operating space under consideration is that Q varies between 408.3 × 10-4 and 304.1 × 10-4 m/s. The input-output data used for the ESOM network are generated by 1000 independent random input signals with uniform distribution ranging from 30 to 150 s, and the corresponding plant outputs are simulated as shown in Figure 7. Because the filtration process is stable in the operating space, all of the local models are constrained to be stable and hence the following parametric constraints hold.
[
1 -1 θj,3 -θj,3
1 1 1 -1
][ ] [ ]
1 1 θj,1 -1 θj,2 < 1 , j ) 1, 2, ..., nΦ θj,3 1 θj,3 θj,3 1
(32)
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Table 4. ESOM Network for Various Cases in the Filter Example nΦ ) 2
nΦ ) 3
10-7
7.72 × 7.42 × 10-7
training data (MSE) validation data (MSE)
nΦ ) 4
10-7
nΦ ) 5
10-7
6.65 × 6.56 × 10-7
2.79 × 4.24 × 10-7
nΦ ) 6
10-7
nΦ ) 7
10-7
2.84 × 4.25 × 10-7
2.30 × 4.32 × 10-7
2.35 × 10-7 4.35 × 10-7
Table 5. LMN for the Filter Example (nΦ ) 4)a 4
y(k) )
∑F Ψ (k-1) θ T
j
j
j)1
θ1 ) [0.1661, 0.05832, 0.1860, -0.5577, 254.5]T θ2 ) [-0.4010, -0.04273, 0.03007, -1.7675, 622.91]T θ3 ) [-0.6733, 0.003756, -0.005957, -1.5788, 743.19 ]T θ4 ) [-1.1996, -0.8011, -0.2014, -3.4386, 1413.9]T a The engineering value of the output can be obtained by scaling this model output by 10-4.
For the purpose of comparison, Ge’s ESOM algorithm is again applied by using the same network structure and training data to obtain a LMN with the following local model parameters:
θ1 ) [0.2166, 0.1391, 0.009628, -1.2453, 300.67]T Figure 7. Training and validation of the ESOM network with nΦ ) 4. Solid: ESOM. Dashed: plant.
θ2 ) [-2.5304, -1.2257, -0.4162, -4.7583, 2216.9]T
- 1 < θj,3 < 1, j ) 1, 2, ..., nΦ
θ3 ) [0.6357, 0.4334, 0.2006, -0.1911, -65.863 ]T
(33)
The proposed ESOM algorithm (K2 ) 36, 1 ) 0.01, and 2 ) 0.012) is then applied to the input-output data vj ) (v(k), y(k)) to solve the optimization problem equation (14) subject to eqs 32 and 33. Table 4 summarizes the training and validation results for the ESOM network with six cases of nΦ. As can be seen, no significant improvement of the modeling results is achieved as the number of winner nodes in the ESOM network is increased above 4. Hence, the LMN with nΦ ) 4 is selected for this example, and its local models are given in Table 5. To check the global stability of this LMN, the local system matrices are obtained from Table 5 as given by
[
]
0.1661 0.05832 0.1860 0 0 , A1 ) 1 0 1 0 -0.4010 -0.04273 0.03007 A2 ) 1 0 0 0 1 0
[
[
]
-0.6733 0.003756 -0.005957 0 0 A3 ) 1 , 0 1 0 -1.1996 -0.8011 -0.2014 0 0 A4 ) 1 0 1 0
[
[
It can be verified that this LMN has two unstable local models, namely, the second and third models. Clearly, this result contradicts the actual plant dynamics, which are stable over the operating space. Furthermore, the computation time required is 38.9% more than that of the proposed ESOM algorithm. To proceed with the LCN design, an ARX model-based predictive control strategy, generalized predictive control (GPC),23 is employed to design the local controller. The following control objective is considered in the design of the jth local GPC controller:
J)
min j
∆Tj(k),∆Tj(k+1),...,∆Tj(k+N u-1) j
] ]
A common positive-definite matrix
1.5139 0.7434 0.3123 P ) 0.7434 0.9687 0.3012 0.3123 0.3012 0.3295
θ4 ) [-0.5497, -0.03607, 0.05545, -1.3651, 670.07]T
]
is found to fulfill eq 20. Therefore, the global stability of this LMN is guaranteed. The simulation result in Figure 7 confirms that this LMN can predict the plant dynamics with good accuracy.
j
Np
[Qj(k+i) - Q ∑ i)1
set
Nu
2
(k+i)] +
λj[∆Tj(k+i-1)]2 ∑ i)1
(34)
where Qj(k+i) is the future process output predicted by the jth local model, Qset(k+i) is the set point, ∆Tj(k+i1) is the future input to be determined, N jp is the prediction horizon, N ju is the control horizon, and λj is a weighting factor. In the local GPC design, a set of future inputs {∆Tj(k+i-1), i ) 1, ..., N ju} which minimize J is determined at each sampling time instant. However, only the first input is implemented, i.e., Tj(k) ) Tj(k1) + ∆Tj(k), and the optimization procedure repeats at the next sampling time instant. The output of the LCN controller is given by 4
T(k) )
Fj(β) Tj(k) ∑ j)1
(35)
For comparison purposes, both the PI controller and conventional GPC controller are designed around the middle operating point Q ) 356.2 × 10-4 m/s. The controller parameters used for the PI controller are Kc
Ind. Eng. Chem. Res., Vol. 41, No. 12, 2002 2947
in controlling the process that is subject to multiple operating regimes. Acknowledgment Support from the research grant received from the National University of Singapore is gratefully acknowledged. We also appreciate very useful comments from the anonymous reviewers. Literature Cited
Figure 8. Servo response: (a) PI controller; (b) conventional GPC; (c) LCN. Dashed: set point. Table 6. MSEs for the Servo Response in Figure 8 PI 3.97 ×
conventional GPC 10-6
3.78 ×
10-7
LCN 2.36 × 10-7
) 0.12 and τI ) 1 and those for the GPC controller are Np ) 1, Nu ) 1, and λ ) 0.8. In the LCN design, Np ) 1 and Nu ) 1 are used for all local GPC controllers, and the weight factors employed are λ1 ) 0.8, λ2 ) 1, λ3 ) 0.3, and λ4 ) 1.2. The set-point response of the three controllers is compared in Figure 8. It is apparent that the LCN controller has a better performance than the other two controllers. To measure the controller performance quantitatively, Table 6 shows that the LCN controller indeed gives the smallest MSE value for servo response, as shown in Figure 8. 4. Conclusion This paper proposes a methodology for the modeling and control of a nonlinear process based on the plant data. In doing so, an improved ESOM algorithm is developed to automatically determine the structure and the weighting function of the LMN, which provides a basis for the design of the LCN controller that is composed of several local controllers. Simulation results illustrate the effectiveness of the proposed ESOM algorithm and show that the resulting LCN design has a better performance than its conventional counterparts
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Received for review March 22, 2001 Revised manuscript received January 15, 2002 Accepted March 13, 2002 IE0102605
