Nonlinear Model Predictive Control of Chemical Processes with a Wiener Identification Approach

MohammadMehdi Arefi1, Allahyar Montazeri1, Javad Poshtan1, and MohammadReza Jahed-Motlagh2

1Electrical Engineering Department, 2Computer Engineering Department Iran University of Science and Technology, Tehran, Iran

Email: mmarefi@ee.iust.ac.ir

Abstract- Some chemical plants such as pH neutralization process have highly nonlinear behavior. Such processes demand a powerful wiener identification approach based on neural networks for identification of the nonlinear part. In this paper, the pH neutralization process is identified with NN-based wiener identification method and two linear and nonlinear model predictive controllers with the ability of rejecting slowly varying unmeasured disturbances are applied. Simulation results show that the obtained wiener model has good capability to predict the step response of the process. Parameters of both linear and nonlinear model predictive controllers are tuned and the best obtained results are compared. For this purpose, different operating points are selected to have a wide range of operation for the nonlinear process. Simulation results show that the nonlinear controller has better performance without any overshoot in comparison with linear MPC and also less steady-state error in tracking the set -points.

I. INTRODUCTION

There are very few design techniques that can be proved to stabilize processes in the presence of nonlinearities and constraints. Model Predictive Control (MPC) -an optimal control, model based method- has been one of the successful controllers in manufacturing industries for the past two decades. MPC refers to a class of computer control algorithms that control the future behavior of a plant through the use of an explicit process model. At each control interval the MPC algorithm computes an open-loop sequence of manipulated variable adjustments in order to optimize future plant behavior. The first input in the optimal sequence is injected into the plant, and the entire optimization is repeated at subsequent control intervals [1]. By now, the application of MPC controllers based on linear dynamic models cover a wide range of applications and linear MPC theory can be considered quiet mature. Nevertheless, many manufacturing processes are inherently nonlinear and there are cases where nonlinear effects are significant and can not be ignored. These include at least two broad categories of applications [1]:

1- Regulator control problems where the process is highly nonlinear and subject to large frequent disturbances (pH control, etc.)

2- Servo control problems where the operating points change frequently and span a wide range of nonlinear process dynamics (polymer manufacturing, ammonia synthesis, etc.)

In fact higher product quality specifications and increasing productivity demands, tighter environmental regulations and demanding economical considerations require to operate systems over a wide range of operating conditions and often near the boundary of admissible region [2]. Besides the operating point in some batch processes is not in steady-state and all of the operations are performed in transient conditions [3]. Under these conditions linear models are often not sufficient to describe the process dynamics adequately and nonlinear models must be used.

In recent years several nonlinear model predictive control (NMPC) techniques from identification as well as control points of view are addressed for different processes in literatures. Among these techniques, neural networks play important role especially in identification phase. For example in [4], neural networks are used to develop a model for highly nonlinear CSTR and pH neutralization processes. A nonlinear internal model controller is designed based on these models and results are compared with PID controller. In [5], an RBF neural network is used to model and control of an unstable CSTR process. The use of neural network for modeling of a CSTR process has also reported in [6]. The controller is designed using classical optimization methods. The main problem with neural network as a model is that it performs well in the range of data used for training neural network but has poor extrapolation property in other regions. To cope with this problem, dynamic nonlinear models are proposed [2], [7]. The use of wiener models where a linear dynamic model is followed by a static nonlinearity is one of the solutions. A nonlinear model predictive controller for pH neutralization and CSTR processes using wiener model method is addressed in [8], [9]. In these methods a static nonlinear term is used to model the inverse of the nonlinearity of the plant and selected as polynomial with proper degree or piece-wise linear terms. In this paper a nonlinear model predictive control based on classic optimization methods with nonlinear identification using wiener model for a highly nonlinear chemical process is proposed. The main difference between this method and those in [8], [9] is that here, the static nonlinear term is neural

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network. The use of polynomials for direct identification of these highly nonlinear plants will result in poor validation. For this purpose after this introduction, the theory of wiener identification using neural network as the static nonlinear term is presented. Also selection of the test signal to achieve the best result for identification is studied. In section 3 the design of nonlinear model predictive controller based on the identified wiener model is presented. The simulation results for identification and control of pH neutralization process are given in section 4 and finally conclusions come.

II. NN-BASED WIENER IDENTIFICATION

A. Wiener Identification A Wiener model consists of a dynamic linear block (H1) in

cascade with a static nonlinearity at the output (H2), as shown in Fig. 1. Here lRkz ∈)( is an intermediate signal which has not necessarily a physical meaning.

Fig. 1. The Wiener model.

State-space representation of a wiener model can be stated as

follows:

)())(()()()()()()()1(

kvkzfkykukxkzkukxkx

+=+=+=+

DCBA

(1)

where )(kx is an 1×n state vector at time k , )(ku an 1×m vector of control input, )(ky an 1×l vector of measured output and )(kv is a measurement noise with zero mean which is independent of )(ku for all k’s. The system matrices

DCBA ,,, are real with proper dimension and (.)f is a nonlinear vector function defined on ll RR → . The sequences of input and output data used for identification of (1) are available. Besides, it is assumed the input sequences )}({ ku are persistent exciting [10] and statistically independent of noise sequences )}({ kv . The systematic approach for identification of the above problem is stated completely in [11]. The first step is identification of linear part using state-space methods. So assuming the nonlinear mapping as an identity, the linear dynamic characterized by quadruple ),,,( DCBA will be identified. Then using the identified matrices ),,,( DCBA , the sequences of the output of this LTI system N

kkz 1)}(ˆ{ = will be computed. With this sequence a primary identification of the nonlinear part of the wiener model can be estimated. Here, this static nonlinear term is identified with a single layer neural network with the following structure:

)()1,(),()(),,(),())((1 1

ksbisbkzjisiskzf si

l

jjs ευβφα

υ

+++

+= ∑ ∑

= =

(2)

In the above equation (.)sf and )(kzs are used to characterize the sth input and output of this nonlinear term. Besides, ),( isα ، ),,( jisβ ، ),( isb and )1,( +υsb are unknown real coefficients stacked in the vector parameter

)1)2(( ++∈ υllRθ and must be estimated using nonlinear least square methods. υ is the number of neurons in the hidden layer and the last term )(kε shows the estimation error. The cost function which minimized for estimation of the vector parameters θ is:

2

1

1 1

1 11

)1,(),()(ˆ),,(),()(

)1,1(),1()(ˆ),,1(),1()(min ∑

∑ ∑

∑ ∑

=

= =

= =

++

+−

++

+−

N

k

i

l

jjl

i

l

jj

lbilbkzjililky

bibkzjiiky

υ

υ

υβφα

υβφα

θ

(3)

Finally the best parameters for the linear and nonlinear parts are identified with an optimization algorithm. For this purpose, the system matrices identified for the linear part and the vector parameter estimated for the nonlinear part are used as initial conditions for the calculation of the final parameters. In spite of the vector parameter defined for the nonlinear part a full parameterization of the wiener model in (1) requires that the system matrices ),,,( DCBA and also the vector of initial condition )1(x be included in the vector parameter. To have minimum parameter for the matrices the pair ),( CA must be transformed with similar transformations to what is called output normal form. Definition: The pair ),( CA of the system

matrices ),,,( DCBA is output normal form if nTT ICCAA =+ ,

where nnn R ×∈I is an identity matrix. The above definition

explicitly shows that the matrix A must be asymptotically stable. In order the state space description of the system be unique, the matrices A and C are transformed such that

AC

be lower triangular with positive elements in the diagonal.

After these transformations, the parameterization can be performed using nl parameters. More details about this method of parameterization can be found in [12, 13]. All parameters of the system matrices after this parameterization are stacked in the vector onθ . The estimation of the all parameters of the parameterized wiener system can be obtained by minimizing this performance index:

2

1,),1(),),1(,(ˆ)(min ∑

=

−=N

kon

onxxkyky θθ

θθ (4)

where N is the number of samples used for identification. To obtain all parameters of the system, the above least square minimization must be solved. The method used here is Levenberg-Marquardt which tries to find local minima of the performance index iteratively. If Θ is the vector of the all parameters by defining )(ˆ)( ΘΘ yye −= , where )(Θe is the error between the target and output vector. The parameters of Θ can be updated in each iteration. Suppose the value of these parameters at t iteration of Levenberg-Marquardt algorithm is shown by )(tΘ , then this algorithm can be stated as follows:

)()()1( ttt ΔΘΘΘ +=+ (5) where ΔΘ is obtained by solving this set of nonlinear

equations: ))(())()(( tett TT ΘJΘIJJ −=∆+ µ (6)

H1 H2 u(k) y(k) z(k)

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In the above equation, )(tJ is the Jacobian matrix with these derivatives:

).(:1,:1,))((ˆ

: ΘΘΘ

J lengthjNity

j

iij ==

∂∂

= (7)

The tuning parameter ),0( ∞∈µ is called the Levenberg factor and is necessary for convergence of the algorithm. The derivatives in (7) are calculated analytically for the nonlinear part and approximated using forward difference method for the linear part [13]. All identification procedures are performed using SLICOT toolbox [13]. B. Test Design

Some important factors which must be considered in designing the identification test for nonlinear systems are: the duration of the test signal, its amplitude and shape, its spectrum (the average switching time), the correlation of the test signal in each channel, and the number of manipulated variables in each test.

Since in nonlinear systems the test time depends mainly on the number of parameters in the model and the level of noise and unmeasured disturbances, it is recommended longer test time in comparison with linear systems. This is typically considered about 16-25 times of the settling time of the process. The other factors may be included by choosing one of the following excitation signals [14]:

1- Staircase test: In this type of test the width of the pulses and their numbers must be selected properly. 2- Generalized multiple-level noise (GMN): This type of test which is also used here is a multi-level extension of generalized binary noise. In this test the amplitude and the number of pulses must be selected suitably. The number of levels on this test is equal or greater than the degree of nonlinear polynomial which must be identified. Moreover, the average switching time of the test can be obtained from Tsw=T/3, where T is the %98 of the settling time of the process. 3- Filtered white uniform noise: The flexibility in shaping the spectrum of this type of signal is its main advantage. Each spectrum may be realized with a proper filter. A first order low-pass filter is often suitable for this purpose.

III. NONLINEAR MODEL PREDICTIVE CONTROLLER

If at time k, the future state and behavior of the plant is assumed to be known, they can be written in vector form in MIMO case as follows:

[ ]TTTT Pkzkzkzk )()2()1()( +++= Lz (8)

[ ]TTTT Mkukukuk )()2()1()( +++= Lu (9)

[ ]TTTT Pkykykyk )()2()1()( +++= Ly (10)

[ ]TTTT Pkrkrkrk )()2()1()( +++= Lr (11)

where )(kz is the vector of outputs of linear model, )(ku the vector of manipulating variables, )(ky the vector of the outputs of the wiener model shown in Fig. 2, and )(kr the vector consisting set points.

Fig. 2. The Wiener model for NMPC.

Also M , P are the control and prediction horizon

respectively. The predicted output of the linear model can be written as:

)()()(ˆ kxkk ξβ += uz (12) where β , ξ and )(ku are:

+−

=

∑=

−−−−− )(321

2

P

Mi

iPMPPPP BCABCADBCABCABCA

0DCBCABBCA00DCBCAB000DCB

L

MMMMMβ

(13)

=

PT

T

T

T

AC

ACACAC

M

3

2

ξ (14)

[ ]TTT kkuk )()()( uu = (15) Besides, the predicted output of the wiener model is as

follows:

))(ˆ(

))(ˆ(

))3(ˆ())2(ˆ())1(ˆ(

)(ˆ kf

Pkzf

kzfkzfkzf

k zy =

+

+++

=M

(16)

Finally by solving and minimizing this optimization problem the control signal applied to the process can be obtained:

∑∑∑===

++++++∆+−+=

M

j

M

j

P

jMkukuku

kjkukjkukrkjkyJ1

2

1

2

1

2

)(...,),2(),1()|()|()()|(ˆmin RSQ

(17) where

)1()()( kjkukjkukjku −+−+=+∆ (18)

In the above equations, Q , S and R are the weighting matrices for the output and rate of change of the control input respectively. In addition, it is assumed that:

PtoMjforMujku 1)()( +==+ (19) In this optimization problem the eligible limits for the

control input and its rate of change and also for output signal may be considered with these inequalities:

kkkdkkdkk

∀≤≤∀≤−−≤∀≤≤

)( )1()( )(

maxmin

maxmin

maxmin

yyyuuuu

uuu (20)

x(k+1)=Ax(k)+Bu(k) z(k)=Cx(k)+Du(k)

z(k) NN(.) y(k) u(k)

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The optimization problem stated above can be solved with Successive Quadratic Programming (SQP) method. It is important to notice that the optimization time must be less than the sampling time of the process since the obtained control signal can be applied to the process.

To compensate the eventual mismatch between the process and its model and in order to consider unmeasured disturbances in the process, a term like this must be added to the predicted output of the plant [6]:

)()()( kkk myyd −= (21) where )(ky is the output of the real process and )(kmy is the

model output. The modified predicted output will be: Piforkikik mpred K1),()()( =++=+ dyy (22)

IV. SIMULATION RESULTS

A. IDENTIFICATION RESULTS

The considered pH neutralization process consists of an acid (HNO3) stream, a base (NaOH) stream, and a buffer (NaHCO3) stream that are mixed in a constant volume (V) stirring tank. The process is schematically depicted in Fig. 3.

Fig. 3. Schematic representation of pH neutralization process.

The inputs of the system are the base flow rate (u1), buffer flow rate (u2), and the acid flow rate (u3), while the pH level of the effluent solution is considered as the output (y). Usually the acid flow rate and the volume of the tank are assumed to be constant and the pH level of the effluent solution is controlled by changing the base flow rate. The buffer flow rate is assumed as an unmeasured disturbance. The governed nonlinear equations which are highly nonlinear and their corresponding parameters are described in [8]. The nominal condition of y and u1 are 7 and 15.55 mls-1 respectively. Fig. 4 shows the nonlinear behavior of the open-loop response of the process for

%10/−+ change in the base flow rate. As it can be seen the gain for %10+ change is about 250% greater than the %10− change.

0 2 4 6 8 10 12 14 16 18 206

6.5

7

7.5

8

8.5

9

9.5

Time (Min)

pH

+10%

-10%

Fig. 4. Open-loop step response of the pH neutralization process for changing the flow rate of the input signal.

To identify this process a GMN signal with six levels 13,

15.55, 17, 18, 20, and 25 is selected to cover the spanned range of the input signal. Switching time between these levels is assumed to be 6 samples. 1280 sample of the input-output data with sampling time of 0.25min is used for identification. Fig. 5 shows the input and output signals of the process (base flow rate and the pH of the solution respectively). A measurement noise with the SNR of 20dB is added to the output signal.

0 200 400 600 800 1000 1200 140010

15

20

25

base

flow

rate

(mls-1

)

0 200 400 600 800 1000 1200 14006

7

8

9

10

11

No of sample

pH

Fig. 5. GMN input (Up) and output (Down) signals for identification of pH process.

For identification, a wiener model with 5 neurons in the hidden layer is chosen. The linear part of this model is a second order state-space. The output data for better training of the neural network are normalized between -1 and +1. 800 samples are used for identification and the rest of the signal is used for validation purpose. Fig. 6 shows the validation results. Also the step response of the model and plant for %10+ change in the input signal are compared in Fig. 7. These plots validate the nonlinear identified model.

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0 50 100 150 200 250 300 350 400 450 5006

6.5

7

7.5

8

8.5

9

9.5

10

10.5

11

samples

pH

Wiener ModelValidation Data (Clean)Validation data (noisy)

Fig. 6. Validation of the wiener model of pH process.

0 2 4 6 8 10 12 14 16 18 206.5

7

7.5

8

8.5

9

9.5

Time (Min)

pH

Real Process Wiener Model

Fig. 7. Step response of the wiener model for %10+ change in the input signal of pH process.

B. CONTROL RESULTS

The behavior of the mentioned regulating points with NMPC is shown in Fig. 8. As it can be seen, the performance of NMPC is good without any overshoot for all operating points. The prediction and control horizons for NMPC are tuned as 5 and 3 respectively. The weighting matrices of NMPC are selected as 500=Q and 70=S . Notice that, in this case, the criterion function is as in (17), unless it does not include the term penalizes the control effort (i.e. R in (17)). Also, saturation constraints in the manipulated variable are imposed to take into account the minimum and the maximum aperture of the valve regulating the base flow rate. For both cases (NMPC and linear MPC) a lower limit of 13 ml/s and an upper limit of 25 ml/s were chosen for this variable.

The results are also compared with linear MPC in Fig. 9. As it is clear from this figure, the NMPC has better performance without any overshoot, while in linear MPC especially when the operating point goes far from the point where linear model is identified the performance is poor.

The control efforts for both NMPC and MPC (i.e. the base flow rate) are shown in Fig. 10. As it can be seen this signal for NMPC is relatively smoother than for Linear MPC and has not large step changes.

0 5 10 15 20 255

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

Time (Min)

pH

SetpointNMPC

Fig. 8. The performance of set-point tracking of pH process using NMPC.

0 5 10 15 20 255

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

Time (Min)

pH

SetpointNMPCLMPC

Fig. 9. Comparison of NMPC (solid line) with MPC (dashed line) in command following for pH process.

0 5 10 15 20 250

5

10

15

20

25

30

Time (Min)

u1 (m

ls-1)

control Signal

NMPCLMPC

Fig. 10. Comparison of NMPC (solid line) with MPC (dashed line) in he base flow rate for pH process.

To see the computation time required for generating the

control signal, in all of the simulation it is shown in Fig. 11. As it can be seen in this figure the maximum computation time for optimization is 1.05sec which is below enough the chosen sampling time of 15sec for the process and hence the control signal is feasible. The SQP optimization is performed using fmincon function of the MATLAB software.

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0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (Min)

Tim

e (s

ec)

CPU Time

Fig. 11. Computation time of the CPU for SQP optimization for pH process.

The performance of the NMPC controller for rejecting the

unmeasured disturbance with the strategy described in section III is shown in Fig. 12. Here the unmeasured disturbance is the buffer flow rate which is reduced from 0.55 to 0.2 ml/s.

0 5 10 15 20 255

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

Time (Min)

pH

SetpointNMPC

Fig. 12. The performance of the NMPC in rejecting unmeasured disturbance for pH process.

V. CONCLUSIONS

In this paper a nonlinear model predictive controller for control of a highly nonlinear chemical process is applied and simulated. The selected process is pH neutralization which has strong nonlinearity and wide range of operating points. These properties cause the linear MPC techniques does not show good results and demands a more complex identification and controller design procedure.

The model which is an important factor for prediction in MPC, is selected a wiener model with neural network as the static nonlinear term. Simulation results from the identification phase approve the validation of the identified model. Besides, the step responses of the plant and this identified model are in good agreement and very close to each other. This shows the ability of this type of model structure for modeling highly nonlinear processes (for example pH neutralization). Simulation of the NMPC controller for a wide range of operating point shows superior performance of the NMPC with

respect to linear MPC. This is especially true when the operating condition of the process is far from the point where the model for linear MPC is identified. Results show that in such conditions the linear MPC for pH neutralization process follows the set point with overshoot, while the nonlinear MPC exhibits a desirable fast response with smoother changes in the control effort. Simulations also confirm that the designed controllers have the capability to reject a slowly varying unmeasured disturbance which are often happens in chemical processes.

ACKNOWLEDGMENT

The authors would like to acknowledge Dr. Vasile Sima from National Institute for Research & Development in Informatics for his useful helps and comments on working with SLICOT software.

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