Applying Partial Least Squares Based Decomposition Structure to

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Ind. Eng. Chem. Res. 2004, 43, 5888-5898

Applying Partial Least Squares Based Decomposition Structure to Multiloop Adaptive Proportional-Integral-Derivative Controllers in Nonlinear Processes Junghui Chen* and Yui-Chun Cheng Department of Chemical Engineering, Chung-Yuan Christian University, Chung-Li, Taiwan 320, Republic of China

An online tuning method based on a partial least squares (PLS) decomposition structure for multiloop proportional-integral-derivative controllers in nonlinear processes is proposed. Unlike traditional decoupling design, the proposed dynamic PLS (DynPLS) model derived from PLS and the linear dynamic model can decompose a multi-input multi-output (MIMO) process into a multiloop control system in a reduced subspace. To update DynPLS in the nonlinear process, the instantaneous linearized neural network model at each sampling time is used to extract the linear dynamic part of DynPLS for quickly updating the current model. Because of the decomposition of a complex MIMO process, this scheme takes advantage of the simplicity to enhance feasibility. Simulation case studies are used to demonstrate the effectiveness of the control design procedures of nonlinear MIMO processes. 1. Introduction In nearly all chemical industries, the control processes often inherently encounter more than one variable to be controlled. The control of multi-input multi-output (MIMO) systems is not always an easy task because of its complex and interactive nature. Thus, most of the automatic tuning methods are intended for single-input single-output (SISO) processes, and few of them are intended for MIMO processes.1 A critical step in the MIMO dynamic control design is the development of a suitable model that can pair the control loops. The biggest log modulus tuning method2 based on the tradeoff between the stability and performance of the system was designed for each loop. First, all interactions are ignored; then the controller parameters are properly adjusted by the detuning factor to maintain stability. Recently, a sequential design was used for multiloop proportional-integral-derivative (PID) controller systems.3,4 Because it took a lot of time to conduct the identification of the multiloop system and the design of the multiple single-loop controller in a sequential procedure, the partial least squares (PLS) for the decomposition of MIMO system control design was proposed.5,6 First, the dynamic transformation was incorporated into the standard PLS model. Then the synthesis method of controller design was used to tune the controller parameters on each control loop. Similar concepts for nonlinear MIMO processes were proposed.7,8 The PLS outer model was first constructed. The nonlinear dynamic relationship between the input and output scores was built upon the inner model. In the control design based on the PLS-based models, the existing model-predictive control algorithm can be applied. Although the control performances of the above methods were satisfactory, the development of the adaptive control design based on the decomposition of PLS-related models for the multiloop control problem had not been addressed. * To whom correspondence should be addressed. Fax: 8863-265-4199. E-mail: [email protected].

The control design for nonlinear systems is still underdeveloped. In the past years, neural networks constituted a very large research interest. They were able to solve complex mathematical problems because they had been proven to be capable of capturing the characteristics of system patterns and performing any continuous function approximation accurately.9 The benefit of adopting the neural network is especially shown when incorporating the neural network model into the existing model-based predictive control design.10-12 Although the control performances of the above methods achieved satisfactory results, the major difficulties encountered in the development and application of the neural network model were the requirements of the extensive computation effort and the model stability. They might make the implementation strategy realistic only for control of slow dynamic systems. Linearization of nonlinear models is one of the ways often used in the control field to alleviate the design of control problems for nonlinear systems. Although the model extracted by the linearization technique seems to be a crude description system, the rich and well-understood linear control design and process modeling techniques can be integrated to compensate for the performance of the closed loop. It is commonly believed that a model estimated through linearization based on the operating point can be considered to be valid only in a certain region around this point. Because of the nonlinear characteristics and the size of the operating region, it is necessary to consider whether it is appropriate to use a single linear model or to obtain more linearization models around different operating regions. The multiple-model control has been developed to design and control the nonlinear process.13 The divide-and-conquer strategy was proposed to decompose the global model into several local models.14 The model-predictive control and adaptive control using the integration of the local models was then applied. Besides, several combinational methods based on neural networks and traditional linear controller designs have been developed. Ahmed and Tasadduq (1994) mentioned a three-stage procedure for designing

10.1021/ie040013b CCC: $27.50 © 2004 American Chemical Society Published on Web 08/07/2004

Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5889

controllers by linearization through a built neural network.15 Sorensen (1996) extracted the local linearized neural network model at each sampling point to design a pole placement controller for the SISO process problem.16 Wang and Wu (1996) used neuron computation of feedback gain matrices for the pole assignment problem.17 In this paper, a multiloop PID controller design based on PLS and neural networks is developed for the nonlinear MIMO system. The motivation of the control loops based on PID controllers is, despite the advent of the above complicated control theories and techniques, that more than 95% of the control loops based on PID controllers are still being used in the majority of industrial processes. PID controllers have provided many favorable characteristics, such as user-friendly manipulation, low-cost maintenance, and robust performance, to process dynamic variations. Nevertheless, the PID algorithm might be difficult to deal with in highly nonlinear and interactive chemical processes. To improve the control performance, an online multiloop self-tuning PID controller is proposed. With the instantaneous linearization of the nonlinear neural models and the multivariable decomposition of the PLS structure, the traditional adaptive PID control design of the single loop can be implemented directly and separately on each control loop without any modification. This methodology is good for controlling nonlinear MIMO processes without highly demanding computation because the controller design is based on the linear model instead of the nonlinear model. 2. Dynamic Decomposed PLS Structure of a Nonlinear MIMO Process The development in this paper will not be restricted to a squared MIMO process, which has an unequal number of inputs and outputs. There may often exist a coupling in this process; that is, a large disturbance from other loops occurs whenever the manipulated variable of one loop changes. This interaction may cause oscillation and even instability. Without decouplers, the PLS algorithm is used to decompose a multivariable process into a multiloop control system in a reduced subspace. For the dynamic system, the static PLS will be extended to dynamic PLS (DynPLS) to handle the process measurements with dynamic behavior. For the nonlinear processes, the dynamic ARX neural network is constructed based on the input and output measurements to appropriately update the DynPLS model. Because DynPLS is derived from PLS and the neural network, an overview of these two techniques is present first. PLS. PLS regression derived from the classical linear regression is often used to predict properties of processes based on variables only indirectly related to the properties. For the purpose of multiloop design, the PLS model will be applied in order to decompose a MIMO control design system into several SISO loop control design problems. The process data with two blocks is collected, including a dependent block (Y) and an independent block (U). The Y block with a two-way array (I × M) summarizes the I samples and the M controlled variables. The U block with a two-way array (I × N) organizes operating N manipulated variables. In PLS, the data matrices U and Y are expressed as a sum of rank-one outer products:

R

U)

∑trwTr + E ) TWT + E r)1 R

Y)

trcTr + F ) TCT + F ∑ r)1

(1)

where R is the number of principal components retained in the PLS model. The matrix U is decomposed into the summation of the product of score vectors t and loading vectors w plus residual matrix E. Similarly, the matrix Y is the summation of t and c plus F. Assuming familiarity with PLS,18 a recursive fashion to decompose Y and U can be described as follows: (i) Initially, let U0 ) U and Y0 ) Y, with r ) 0. (ii) Compute the direction within the correlation of Ur and Yr that has the largest eigenvalue, wr ) arg maxwr (UrTYr|||wr|| ) 1). (iii) Calculate the directions of maximum covariance between Ur and Yr as pr ) UrTtr/trTtr and qr ) YrTsr/srTsr, where the scores tr ) Urwr, cr ) YrTtr/trTtr, and sr ) YrTcr/crTcr are the projections of Ur and Yr onto the directions wr and cr. (iv) Remove the variance associated with the previously calculated rth direction of wr and cr in the variance of manipulated variables and controlled variables, Ur ) Ur-1 - trwrT and YrT ) Yr-1T - brtrcrT, respectively. PLS is used to extract latent variables. The latent variables explain the best correlation between the controlled variable block (Y) and the manipulated variable block (U). The above procedures are repeated until the description of Y properly reaches convergence. Because of its simplicity and easy interpretation, the applications of this approach can be found in an abundant literature. However, the method is only good for linear or closed-linear processes. It has limited use in a chemical production atmosphere because it does not comply with multivariable dynamic or nonlinear processes. Dynamic Neural Networks. In general, neural networks with MIMO can be modeled for nonlinear multivariable time series dynamic systems, but such models are too complicated in practice. For simplicity and easy utilization, the multi-input single-output models are often used.19-21 Each output ym(k) can be inferred from a time series NNARX (neural network with ARX) model fitted to the observations of the dynamic characteristics of a complex system:

ym(k) ) fm[φ(k)]

(2)

where

φ(k) ) [yT1 (k) ‚‚‚ yTm(k) ‚‚‚ yTM(k) uT1 (k) ‚‚‚ uTn (k) ‚‚‚ uTN(k)] (3) is the regression vector consisting of the past output vectors ym, m ) 1, ..., M, and the past input vectors un, n ) 1, ..., N, respectively.

ym(k) ) [ym(k-1) ym(k-2) ‚ ‚ ‚ ym(k-nym) ]T (4) un(k) ) [un(k-dn-1) un(k-dn-2) ‚ ‚ ‚ un(k-dn-nun) ]T (5)

5890 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004

nym and nun are the orders of output m and input n, respectively, and dn is the time delay for input n. The NNARX model used here is a three-layer feedforward neural network with a linear activation function in the output neuron and with a hyperbolic tangent activation function in the hidden neurons. The mathematical expression of this model is depicted as follows: Nhidden

yˆ m(k) )

o o wm,c zc[hm(k)] + wm,b ∑ c)1 nym

ny1

hm(k) )

∑ i)1

(6)

h wm,iy y1(k-i)

+ ... +

iy)i

∑ i)1

h wm,iy ym(k-i) +

m-1

iy ) ∑ nys+1 s)1

nu1

... +

∑ i)1

nun

h wm,iu u1(k-i) ... +

M

∑ i)1

h wm,iu un(k-i) +

n-1

iu) ∑ nus+1

iu) ∑ nus+1

s)1

s)1

h ... + wm,b

where zc is the transfer function for the hidden neuron o o and wm,b are the weights and the bias of the c, wm,c h h hidden-to-output layer, wm,i and wm,b are the weights and the bias of the input-to-hidden layer, and hm is the summation of all products between inputs and inputto-hidden weights in the input layer. The training procedures adjust the weights of the neural network model such that the network replicates the measured phenomenon. A wide variety of subjects of the training neural networks can be found.22 For the entire coverage, see our previous work.23 Note that, even if the simplicity of the nonlinear blackbox approach is most often preferred, it is computationally demanding when the iterative minimization algorithm is needed for the predictive control strategies. DynPLS Model. As we know, PLS can decompose the multivariable process into several univariate processes in the latent subspace. The neural network is a good approximator for capturing the characteristics of any continuous function. It is worth combining these two merits. The idea is used to decouple the multivariable system with decomposition of the PLS structure and to extract a linear model with instantaneous linearization of the nonlinear neural network model at each sampling point. The structure of the DynPLS identification scheme is shown in Figure 1. (i) ARX Model. On the basis of the NNARX model structure, the approximated linear model at time k ) τ can be obtained by linearizing NNARX around the current state φ(k)τ). The instantaneous linearization (k) can be written as model of the output m, yˆ INST m nym

yˆ INST (k) ) biasm + m

am,iym(k-i) + yhm(k) ∑ i)1

(7)

where biasm ) fm[φ(τ)], depending on the current nym am,iym(k-i) represents the weightoperating point. ∑i)1 ed sum of the past self-output m values. yhm(k) collects the contributions of all outputs except yj, j * m, and inputs. To simplify the explanation, the dynamic relationships between yj, j * m, and ym(k) are neglected in

Figure 1. Structure of the MIMO DynPLS model based on the instantaneous linearization of the neural network models. Su and Sy are the factors that scale the input and output variables, respectively.

this paper. Of course, it is straightforward to include this dynamic linear term of the past other outputs and inputs without any difficulty. am,i is the linear model coefficient of the output ym(k) and the corresponding past output ym(k-i); it is the derivative of the output with respect to the past output ym(k-i)

am,i )

∂fm[φ(k)] ∂ym(k-i) Nhidden

)

φ(k))φ(τ)

∂zc(k) ∂hm(k)

o ∑ wm,c ∂h c)1

Nhidden

)

|

m(k)

∂y(k-i)

i ) 1,2,‚‚‚,nym

(8)

∂zc(k)

o wm,c ∑ ∂h c)1

whm,i m(k)

(ii) PLS Model. Because coupling or interaction exists in the multivariable control system, it is important to know which manipulated (or input) variables should be connected to which controlled (or output) variables. The modeling equation yhm(k), which is the so-called steady-state gain matrix, is defined as

yh(k) ) Mu(k) [yh1 (k),

(9)

yhM(k)]T.

..., where yh(k) ) On the basis of this linearization procedure, a DynPLS model consisting of a series connection of a memoryless PLS model and linear dynamic models is constructed. To identify this model is a nonlinear optimization problem. A two-stage optimization algorithm based on an alternative procedure is used here to solve PLS and then the dynamic models separately rather than solving the nonlinear optimization directly. First, the parameters of the linear dynamic model are obtained from the instantaneous linearized neural networks. Then the parameters of PLS can be estimated. The whole procedure is successively updated at each sampling point. Thus, the static PLS model followed by linear dynamic ARX models can be expressed as

yh(k) ) PLS[u(k)] ny-1

yˆ (k+1) ) bias +

Aiy(k-i) + yh(k) ∑ i)0

(10)


where PLS is the static PLS model. y(k) ) [y1(k), ..., yM(k)]T is the output vector. yˆ (k+1) is the optimal onestep predictor. Ai ) diag(am,i)m)1,2,...,M, i ) 1, 2, ..., na, is obtained from the instantaneous linearized neural networks. bias ) [bias1, bias2, ..., biasm]T is the bias vector obtained from linearizing NNARXs. ny indicates the number of output lag terms. This means the current output value of the time series is expressed as a weighted sum of the past output values and the output from the PLS model. When compared with Kaspar and Ray’s work,6 the proposed method also has the same feature in conducting PLS without increasing the dimensions of inputs or outputs, but the structure of the dynamic model in the proposed method is more flexible to meet the process dynamic behavior. Rather than solving eq 10 by the nonlinear least-squares method, the sequential procedure utilizes a linearized neural network model to extract ARX and identify PLS separately. It can not only decrease the dimension of the search space but also update the DynPLS model directly. Furthermore, the linearization of the nonlinear model can significantly alleviate the difficulty of controller design. The instantaneous linearization of the neural network model can be changed quickly in response to process changes. The control design based on the linearized updated model is similar to an adaptive controller or a gain schedule controller whose model is chosen from a set of predefined linearized models. This methodology has been implemented in the PID controller design of a single loop.24 More importantly, the neural network model is trained offline, and an extra computational load is not required when constructing the current model in the online identification for the current control design. 3. Multiloop PID Controller Design DynPLS can decompose the MIMO process into a multiloop control system in a reduced subspace. The control loop design is then applied to each pair to form a multiloop control system. In each loop, each new controlled variable and the corresponding new manipulated variables can be designed in the latent space. A method of incorporating the adaptive PID control into each independent control loop is developed. The block diagram of the control system to be considered is shown in Figure 2. Conventional PID Controller. The PID controller from the process variable y(t) to the control variable u(t) is

[

u(t) ) us + kc e(t) +

1 τi

∫e(t) dt + τd

]

de(t) dt

(11)

where us is bias value. e(t) ) yset(t) - y(t) is the output error deviated from the setpoint. kc, τi, and τd are known as the proportional gain, the integral time constant, and the derivative time constant, respectively. The discrete PID controller whose integral action is computed using the trapezoidal approximation can be written as T

∆x(k) ) k0e(k) + k1e(k-1) + k2e(k-2) ) e (k) k(k) (12) where

e(k) ) [e(k) e(k-1) e(k-2) ]T

(13)

Figure 2. Scheme of the adaptive multiloop PID control design under the DynPLS structure. Su-1 and Sy-1 are the factors that rescale the input and output variables, respectively.

and

k(k) ) [k0 k1 k2 ]T ) τd ∆t ∆t 2τd kcτd kc 1 + + -kc 1 + 2τi ∆t 2τi ∆t ∆t

[(

) (

) ]

T

(14)

∆t is the sampling period. Multiloop PID Controllers Based on DynPLS Decomposition Structure. The goal of the controller design for the MIMO system is to seek control actions u(k) that can minimize the difference of the process outputs y(k) and the desired outputs yset(k) at the next time step; i.e., the process outputs can reach the desired output at the next time. Besides, from the operational point of view, the variance controller output should be minimized in order to avoid excessive control effort. The objective function of MIMO system is expressed as

J)

1 min [||e(k+1)||2 + µ||∆u(k)||2] 2 kc,r,τi,r,τd,r

(15)

r)1,2,...,R

where µ is the weighting penalty parameter. Here assume that kc,r, τi,r, and τd,r are the PID control parameters of the loop r and R is the number of control loops in the reduced subspace. Because e(k+1) ) yset(k+1) - y(k+1), the objective function involves a term in the future of the next time step; namely, y(k+1) is not available at time k. Using eq 10, a model based on the DynPLS can be used; that is, y(k+1) = yˆ (k+1).

J)

1 2

na-1 set

min [||y (k+1) - [bias + kc,r,τi,r,τd,r r)1,2,...,R

Aiy(k-i) + ∑ i)0

yh(k)]||2 + µ||∆u(k)||2] (16) na-1 Let yh,set(k+1) ≡ yset(k+1) - bias - ∑i)0 Aiy(k-i), h,set h (k+1), and y (k) can be decomposed into the lower y R dimensional space yh,set(k+1) ) ∑r)1 tset r (k+1) cr and R h y (k) ) ∑r)1tr(k) cr. The above equation can be represented as


J)

1 2

Substituting eqs 19 and 20 into the objective function Jr gives

R

2 ∑[tset r (k+1) - tr(k)]cr|| + r)1

min [|| kc,r,τi,r,τd,r r)1,2,...,R

R

µ||

∑∆tr(k) prT||2]

r)1

eJ)

R

1 2

2 2 ∑[tset r (k+1) - tr(k)] ||cr|| + r)1

where T Ar(k) ) (1 + λr)esub,r(k) esub,r (k)

min [ kc,r,τi,r,τd,r r)1,2,...,R

1 Jr ) ∆kTr (k) Ar(k) ∆kr(k) + dTr (k) ∆kr(k) + cr (21) 2

R

[∆tr(k)]2||prT||2] ∑ r)1

T dTr (k) ) {-[tset r (k) - tr(k)]esub,r(k) + T (k)} (22) (1 + λr)kTr (k-1) esub,resub,r

µ ) min [J1 + J2 + ... + JR] kc,r,τi,r,τd,r r)1,2,...,R

) [ min

kc,1,τi,1,τd,1

J1 + min

kc,2,τi,2,τd,2

J2 + ... +

min

kc,R,τi,R,τd,R

JR] (17)

2 2 2 where Jr ≡ (1/2){[tset r (k+1) - tr(k)] ||cr|| + µ[∆tr(k)] || T 2 set pr || }. tr and tr are the current score and the desired score at the inner pair loop r, respectively. The above equation is an easy consequence of the Schwarz inequality. Now the objective function is decomposed into R subobjective functions in the lower dimensional subspace. Only R score variables (tr, with r ) 1, 2, ..., R) need to be designed separately, as compared with M process variables to be lumped together without the decomposition. These multiloop controllers, like decentralized controllers, have a simpler structure, and accordingly tuning parameters are less than the fully cross-coupled ones. This decomposition of the structure from the multidimensional control problem is a key component of the decoupling method. Tuning Each PID Control Loop Based on General Minimum Variance. After the objective function is decoupled into R objective functions, the interactions that exist between control loops are eliminated. The similar SISO design techniques can be directly applied to each SISO structure of the decoupling system. The only difference is that the process variables are decomposed into several score variables in the subspace. Each subobjective (Jr, with r ) 1, 2, ..., R) is arranged into

Jr )

1 min {[tset(k+1) - tr(k)]2 + λr[∆tr(k)]2} 2 kc,r,τi,r,τd,r r

(18)

where ||cr||2 and ||prT||2 of eq 17 with the penalty factor are lumped into a coefficient λr. Here the incremental from of the PID controller (eq 12) is used in each loop T (k) kr(k) tr(k) ) tr(k-1) + esub,r

(19)

where esub,r(k) ) [esub,r(k), esub,r(k-1), esub,r(k-2)]T and esub,r(k) ) tset r (k) - tr(k). Let the updated control parameters of the loop r become

kr(k) ) kr(k-1) + ∆kr(k)

(20)

where ∆kr(k) is the change of the tuning parameters at the sampling point k and kr(k-1) is the old control parameter vector computed at the sampling point k - 1.

1 (k) - tr(k)]2 cr ) [tset 2 r T [tset r (k) - tr(k)][(eset,r(k) kr(k-1)] + 1 T [e (k) kr(k-1)]2 + 2 set,r λr T T k (k-1) esub,r(k) esub,r (k) kr(k-1) 2 r When Jr is minimized with respect to ∆kr(k), the required changes of the control parameters can be computed as

∆kr(k) ) -Ar-1(k) dr(k)

(23)

Note that ∆kr(k) is calculated without the noninvertible problem because the coefficient λr is added to the diagonal elements of Ar(k). When the process variables are close to the desired score variables, esub,r(k) is close to zero. ∆kr(k) would be close to the zero vector because dr(k) approaches a zero vector. Using eq 14, the corresponding PID control parameters are

kc,r(k) ) -[k1,r(k) + 2k2,r(k)] τi,r(k) )

-[k1,r(k) + 2k2,r(k)]∆t k0,r(k) + k1,r(k) + k2,r(k)

τd,r(k) )

(24)

-k2,r(k)∆t k1,r(k) + 2k2,r(k)

The optimimization of the above objective function is not a problem, but physically it is not suitable. Typically, the constraints should be placed on the input and output variables at their minimum and maximum. However, these constraint relationships are coupled in the latent space. To implement the decomposition strategy, a quadratic function (eq 21) is still solved for each control loop, but the manipulated variables mapped back from the latent space and the corresponding controlled variables predicted from the neural network model should satisfy the constraints. If the estimated manipulated variables exceed the bounds, the bound values of the manipulated variables would be applied. In this way, the manipulated variables and the controlled variables will locate in the feasible input constraint regions and the output constraint regions, respectively. This strategy prevents the problem of any setpoints that cannot be reached if the constraints are projected onto the subspace. Remark. Although the control design in the adaptive control structure continues online calculation of the adaptive parameters of the controller, the computation


of the new control action is redundant when the controlled output is close to the desired setpoint. The updated criterion, by a cumulative sum of the past error with the fixed window size, is designed to detect the deterministic shift in the desired setpoint

Sr(k) )

1

k

2 ∑ [tset r (k) - tr(k)] hi)k-h

(25)

where Sr(k) is the current control performance of the loop r. h is the size of a moving window that contains h - 1 past outputs until now. Whenever the current performance Sr(k) is below its control limit Sr(k) e σ2, the current controller parameters are assumed to be fixed until Sr(k) > σ2 and σ2 is the threshold. It can be estimated from the steady-state process data when there is no change in the control action, or it is based on the prior knowledge of the operating process. 4. Procedures of the PLS-Based Multiloop PID Controller Design Two phases are needed to conduct the online tuning of PID controllers in the MIMO processes. Phase I involves identifying the relationship of the dynamic process between the input process variables and the output process variables. NNARXs are trained to derive these relationships in order to accurately predict the output behavior of the possible operating condition. In phase II, based on the PLS model and the linear dynamic model extracted from instantaneous linearized NNARX models, the DynPLS linear model is used to construct the pairs of input-output variables in the latent space. Each controller tuner is directly computed on the basis of the DynPLS model so as to solve the PID control parameter design problem. The detailed procedure is summarized as follows: Phase I: Identify the relationship between the inputs and the outputs to predict the outputs at the next step. Step 1: Train NNARX models based on the observation data. Once trained, each NNARX model represents a nonlinear or complex function for the output that it learned. Set k ) 1. Phase II: Determine the current optimal PID controller parameters of each loop at each time instant. Step 2: At the new sampling time k, sample the process outputs and inputs to compute the performance of the current data window (eq 25) and determine if the control parameters should be updated. If the current performance S(t) is below its control limit, keep the same values of the PID controller parameters and go to step 5. Otherwise, carry out the update control parameters and go to step 3. Step 3: Extract the linearized model through the linearization of NNARX models around the current input-output pair and update the PLS model based on the current measurements to construct the DynPLS model (eq 10). Step 4: With the current DynPLS model and the difference between the predicted output and the process output, calculate the updated control parameters using eqs 23 and 24. Step 5: Implement the new control action on the process based on the calculated PID control parameters. Wait for the next sampling time (k ) k + 1), keep

Figure 3. Setpoint change in both the pH (from 7 and 8.5) and level (from 14 to 12) in example 1: (a) pH and level; (b) acid and base flow rates. Table 1. Simulation Parameters in Example 1 A ) 207 cm2 Cv ) 8.75 mL cm-1 s-1 pK1 ) 6.35 pK2 ) 10.25 Wa1 ) 3 × 10-3 M Wa2 ) -3 × 10-2 M Wa3 ) -3.05 × 10-3 M

Wb3 ) 5 × 10-5 M q1 ) 16.6 mL s-1 q2 ) 0.55 mL-1 q3 ) 15.6 mL s-1 [acid] ) 0.003 M HNO3 [buffer] ) 0.03 M NaHCO3 [base] ) 0.003 M NaOH

collecting the process input-output, and repeat the procedure from step 2 to 5 for the next time step. 5. Illustration Examples In this section, two simulation examples involving processes of different nonlinear MIMO dynamics are discussed to demonstrate the applicability of the proposed decomposition structure to online updated multiloop PID control design. Example 1: pH Neutralization System. A pH neutralization process11 that has three input streams and one outlet stream is considered. The input streams include acid (HNO3), buffer (NaHCO3), and base (NaOH). The process model consists of two reaction invariants, three nonlinear ordinary equations, and one nonlinear algebraic equation. The input streams are subject to


Figure 5. Series of setpoint changes in both the pH and level in example 1: (a) pH and level; (b) acid and base flow rates.

Figure 4. The setpoint change in both the pH (from 7 to 6.5) and level (14 to 16) in Example 1: (a) pH and level; (b) acid and base flow rates.

constraints.

dh 1 ) (q + q2 + q3 - Cvh0.5) dt A 1

(26)

dWa4 1 ) [(Wa1 - Wa4)q1 + (Wa2 - Wa4)q2 + (Wa3 dt Ah Wa4)q3] (27) dWb4 1 ) [(Wb1 - Wb4)q1 + (Wb2 - Wb4)q2 + (Wb3 dt Ah Wb4)q3] (28) 1 + 2 × 10pH-pK2 Wa4 + 1014-pH + Wb4 1 + 10pK1-pH + 10pH-pK2 10-pH ) 0 (29) subject to

0 e q1(k) e 30, 0 e q2(k) e 30

(30)

where Wa4 and Wb4 are the reaction invariants of the effluent streams. Wa ≡ [H+] - [OH-] - [HCO3-] 2[CO32-] and Wb ≡ [H2CO3] + [HCO-] + [CO32-] are carbonate ion and charge balances. h is the liquid level.

q1, q2, and q3 are the acid, buffer, and base flow rates, respectively. The other parameter definitions and the nominal operation conditions are listed in Table 1. The objective is to control the pH value and level h in the tank by manipulating the base (q3) and acid (q1) flow rates. To update the DynPLS model online, two NNARX models from the open-loop simulation data of the pH system need to be established first. In this case, the training data set is generated from pseudorandom variation of inputs q1 and q3. The duration of each variation interval is set at 20 min. The excitation signal is designed at the expected range of the process dynamics. Another data set generated in a similar manner is used to verify the prediction capability of the neural network. In the first test condition, the control strategy shows the setpoint changes in both the level and the pH value. Figures 3a and 4a demonstrate two setpoint changes in both the level and pH. The setpoint of pH changes from 7 to 8.5 and from 7 to 6.5, while the setpoint of h changes from 14 to 12 and from 12 to 16. The two input flows for these two control loops are separately shown in Figures 3b and 4b. Compared with our previous control design based on NNMPC,23 the performance of the proposed control algorithm is a little inferior to that of the NNMPC, but the proposed decomposition structure can significantly reduce the computational load and more feasibly apply to the real world process. Furthermore, the different output setpoint changes used to examine the setpoint tracking ability of the proposed method are shown in Figure 5. Figure 6 shows the process gain variations of the system for pH - q1(k11),


Figure 6. Process gain variations of the pH neutralization system in example 1.

h - q1(k21), pH - q3(k12), and h - q3(k22) in these control regions. It is obvious that there is strong nonlinearity and interaction in these environments, particularly in pH - q3 and h - q3. The traditional decoupling strategy that is used to reduce the interaction between loops cannot be applied because the nonlinearity and interaction may differ at different operation regions. The recursive linear ARX model directly collected from the process data to update the controller parameters online has been used, but the performance is also unsatisfactory because of the inappropriate pairing of inputoutput of the linear model. Besides, the disturbance of the buffer flow rate is also an important control object in the pH neutralization system. Figure 7 shows the rejection of a buffer disturbance at the changes from 0.6 to 0.2 mL/s and from 0.2 to 1.5 mL/s. Using the online adaptive strategy, buffer disturbances can be rejected even if the buffer flow rates cause a large variation in this highly nonlinear system. Example 2: Nonsquare System. Nonsquare systems with an unequal number of inputs and outputs occur frequently in the chemical process industry. In the past control design, multivariable control design mostly dealt with the square system only in order to pair one output with one input and to implement feedback control on each independent loop pair. By doing so, the systems were often squared by adding or deleting the appropriate number of inputs or the output from the process to be controlled. Although there were quite a few research papers on nonsquare system design,25 such research results were not applicable directly to nonlinear nonsquare systems. Here a series chemical reaction processes26 are used to demonstrate the proposed technique. The reaction system involves two series reactions:

A+BfC A+DfE The concentrations of C and E in the product stream are controlled by manipulating three inlet concentra-

Figure 7. Responses of the pH and level in two step changes in the buffer flow rate: (a) from 0.6 to 0.2 mL/s; (b) from 0.6 to 1.5 mL/s.

tions of A, B, and D (u1, u2, and u3). The system is formulated as follows

[]

dx ) f(x) + g(x)Tu dt

where u ) f1 f2 f ) f3 ) f4 f5

[

[u1, u2, u3]T,

and

[]

x y ) x3 5

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]

F -k1(x1x2 + x1cBS + x2cAS) -k2(x1x4 + x1cDS + x4cAS) - x1 V F -k1(x1x2 + x1cBS + x2cAS) - x2 V F k1(x1x2 + x1cBS + x2cAS) - x3 V F -k2(x1x4 + x1cDS + x4cAS) - x4 V F k2(x1x4 + x1cDS + x4cAS) - x5 V

and

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[]

Figure 8. Output responses in example 2 with different numbers of components: (a) the first component; (b) the first two components; (c) all components.

F 0 0 V F 0 0 V g) 0 0 0 F 0 0 V 0 0 0

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The system parameters are k1 ) 1, k2 ) 2, and F ) V ) 3. The desired operating steady state is x ) [0.89, 0.53, 0.47, 0.36, 0.64]T. The steady-state input values are u ) [2, 1, 1]T. Like the previous training procedures in example 1, two NNARX models are trained. After NNARX is trained, the predicted result exactly follows the actual process behavior. With the control design based on the instantaneous linearization of NNARX and the decomposition of the PLS structure, the proposed updated multiloop algorithm is used to track the different setpoint changes shown in Figure 8. The process gain variations for the system with three inputs (u1, u2, u3) and two outputs (y3, y5) shown in Figure 9 also indicate that there is strong nonlinearity and interaction in these environments. The average percentages of variance captured by each PLS component for the training data are listed in Table 2. It is observed that two principal components capture over 97% of the variance in the

Figure 9. Process gain variations in example 2 where kij represents the process gain of output i and input j.

relationships of the MIMO process. This suggests that the process variables are fairly well correlated between inputs and outputs. Here different numbers of components are selected to verify the control performance (Figure 8). Because the first component accounts for only 92% of all of the total input variations and 88% of all of the total output variations, the control loop based


Figure 10. Projection of the controlled variables in Figure 8a onto the latent space: (dashed line) controlled score variable; (solid line) setpoint score variable. Table 2. Average Percentages of Variance Captured by Each PLS Component in Example 2 percent variance captured by each PLS component component

Xblock

total

Yblock

total

1 2 3

91.60 5.54 2.85

97.15 97.15 100.0

88.10 10.21 0.41

88.10 98.31 98.72

on the first component constitutes the minimum control performance that barely meets our expectation. Figure 8a shows that the offset occurs as a result of the model error of the PLS model with only one component even if the response of the controlled score variable is close to the desired setpoint score variable (Figure 10). With the increased number of control loops, the corresponding control performance has been improved. However, the control performance becomes deteriorated when the third component is added, because the first two components already account for 97% of all of the total input variations and 98% of all of the total output variations; this minor contribution of the third component is very sensitive to the process variation. Therefore, the control loops based on the contributions of only a few components in the subspace can be used in the nonsquare multivariable nonlinear system without a substantial loss of control performance. 6. Conclusion A new method that deals with MIMO nonlinear processes using the decomposition of the DynPLS structure is developed. The proposed DynPLS model is built based on a memoryless PLS model and a linear dynamic model extracted from instantaneous linearzation of neural networks. It can decompose the multivariable control problem into the multiloop control problems. Thus, the conventional SISO design strategy can be directly used to determine each loop control parameter. Of additional importance to multivariable process is the PLS applicability to nonsquare matrices. In the multiloop control design, the PID controller parameters of each loop are estimated based on the instantaneous DynPLS model and the generalized minimum variance performance. Because of the linearity of the model, the control design is free from the

convergence problem so that a unique solution can be found. Besides, the implementation is much simpler and less computationally demanding than the predictive control based on the neural network model. This hybrid control strategy integrates the conventional and wellestablished linear control design methods with the powerful PLS and the nonlinear neural network modeling tools. This strategy is particularly suitable for the chemical process that is characterized by high nonlinearity and analytical difficulty. The proposed method is demonstrated through two examples. The results of the examples show that the proposed method is effective in following the setpoint and reducing the variance of the output caused by unknown disturbance. So far, in this paper the control strategy is developed only based on the traditional feedback control. Some advanced control strategies, like model-predictive control with prior information of the change of the setpoint, need to be further considered in future work. For practical consideration, the measurements with noise under the operating process always exist. Even if mapping of neural networks is an efficient approach for noise reduction, tests with different noise-signal ratios for the real experiments will be studied in the future in order to verify the performance of the proposed method. Acknowledgment This work is supported by the National Science Council of Republic of China. Literature Cited (1) Palmor, Z. J.; Halevi, Y.; Karsney, N. Automatic Tuning of Decentralized PID Controller for TITO Processes. Automatica 1995, 31 (7), 1001. (2) Luyben, W. L. Simple Method for Tuning SISO Controllers in Multivariable Systems. Ind. Eng. Chem. 1986, 25, 654. (3) Shiu, S. L.; Huang, S. H. Sequential Design Method for Multivariable Decoupling and Multiloop PID Controllers. Ind. Eng. Chem. Res. 1998, 37, 107. (4) Shen, S. H.; Yu, C. C. Use of Relay-Feedback Test for Automatic Tuning of Multivariable Systems. AIChE J. 1994, 40, 627. (5) Kaspar, M. H.; Ray, W. H. Dynamic PLS Modelling for Process Control. Chem. Eng. Sci. 1993, 48 (20), 3447. (6) Kaspar, M. H.; Ray, W. H. Chemometric Methods for Process Monitoring and High Performance Controller Design AIChE J. 1992, 38, 1593. (7) Lakshminarayanan, S.; Shah, S. L.; Nandakumar, K. Modeling and Control of Multivariable Processes: Dynamic PLS Approach. AIChE J. 1997, 43 (9), 2307. (8) Baffi, G.; Morris, J.; Martin, E. Non-linear Model Based Predictive Control through Dynamic Non-linear Partial Least Squares. Trans. Inst. Chem. Eng. 2002, 80, 75. (9) Hornik, K.; Stinchcombe, M.; White, H. Mutliplayer Feedforward Neural Networks are Universal Approximators. Neural Networks 1989, 2, 359. (10) Nikolaou, M.; Hanagandi, V. Control of Nonlinear Dynamical Systems Modeled by Recurrent Neural Networks. AIChE J. 1993, 39, 1890. (11) Nahas, E. P.; Henson, M. A.; Seborg, D. E. Nonlinear Internal Model Control Strategy for Neural Network Models. Comput. Chem. Eng. 1992, 16, 1039. (12) Psichogios, D. C.; Ungar, L. H. Direct and Indirect Model Based Control Using Artificial Neural Networks. Ind. Eng. Chem. Res. 1991, 30, 2564.

5898 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 (13) Narendra, K. S.; Balakrishnan, J. Adaptive Control Using Multiple Models. IEEE Trans. Autom. Control 1997, 42, 171. (14) Johansen, T. A.; Foss, B. A. Operating Regime Based Process Modeling and Identification. Comput. Chem. Eng. 1997, 21, 159. (15) Wang, J.; Wu, G. A Multilayer Recurrent Neural Network for On-Line Synthesis of Minimum-Norm Linear Feedback Control Systems via Pole Assignment. Automatica 1996, 32, 435. (16) Sorensen, O. Non-linear Pole-placement Control with a Neural Network. Eur. J. Control 1996, 2, 36. (17) Fuli, W.; Mingzhong, L.; Yinghua, Y. Neural Network Pole Placement Controller for Nonlinear Systems through Linearsation. Proc. Am. Control Conf. 1997, 1984. (18) Ho¨skuldsson, A. PLS Regression Methods. J. Chemom. 1988, 2, 211. (19) Wang, S.-D.; Lee, C.-H. Fuzzy System Modeling Using Linear Distance Rules. Fuzzy Sets Syst. 1999, 108 (2), 179. (20) Yu, D. L.; Gomm, J. B.; Williams, D. On-line predictive control of a chemical process using neural network models. IFAC 14th Triennial World Congress, Beijing, China, 1999; Elsevier Science Ltd.: Oxford, U.K., 1999; p 121.

(21) Ljung, L.; Glad, T. Modeling of Dynamic Systems; Prentice Hall: Englewood Cliffs, NJ, 1994. (22) Hagan, M. T.; Demuth, H. B.; Beal, M. Neural Network Design; PWS Publishing Co.: Boston, 1996. (23) Chen, J.; Yea, Y. Neural Network-Based Predictive Control for Multivariable Processes. Chem. Eng. Commun. 2002, 189 (7), 865. (24) Chen, J.; Huang, T.-C. Applying Neural Networks to Online Updated PID Controllers for Nonlinear Process Control. J. Process Control 2004, 14 (2), 209. (25) Reeves, D. E.; Arkun, Y. Interaction Measures for Nonsquare Decentralized Control Structures. AIChE J. 1989, 35 (4), 603. (26) Kolavennu, S.; Palanki, S.; Cockburn, J. C. Nonlinear Control of Nonsquare Multivariable Systems. Chem. Eng. Sci. 2001, 56, 2103.

Received for review January 7, 2004 Revised manuscript received March 26, 2004 Accepted May 7, 2004 IE040013B

Applying Partial Least Squares Based Decomposition Structure to

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