Icici014p - Widjiantoro, Predictive Control

International Conference on Instrumentation, Communication and Information Technology (ICICI) 2005 Proc., August 3 -5, 2005, Bandung, Indonesia

Nonlinear Model Predictive Control Containing Neural Model and Controller for MIMO Process Bambang L. Widjiantoro1), The Houw Liong2), Yul Y. N3), and Bambang SPA3) 1) Jurusan Teknik Fisika FTI – ITS Surabaya Kampus ITS Sukolilo Surabaya – Jatim Indonesia E-mail: [email protected] 2) Departemen Fisika ITB Bandung 3) Departemen Teknik Fisika ITB Bandung Jl. Ganesha 10 Bandung

Abstract – Most of industrial processes are characterized as multi input multi output (MIMO) and nonlinear processes as well as influenced by the disturbances that are detrimental to it. These conditions give difficulties in the design of a proper control system, which may overcome the above characteristics. One of model based control strategy that can be applied to improve the control system performance is Nonlinear Model Predictive Control (NMPC), which requires nonlinear model allowing analog characteristics between the model and its respective process In view of nonlinear model predictive control’s restriction, it relies on the nonlinear model derivation, which often requires simultaneous mathematical equations, complex and somewhat difficult, in particular for MIMO and nonlinear processes. Apparently other restriction deals with solution of nonlinear optimization, as consequence of nonlinear model application, which often yields local minima in its solution or even divergent along with iterative time consuming computation. The research proposes a new structure and algorithm of nonlinear model predictive control by integrating neural networks as nonlinear model and controller. Nonlinear model can be developed using neural networks without involving the complex mathematical equations and detail information of the process. The neural networks controller will be able to eliminate a nonlinear optimization solution. The current control signal is given by the output neural networks controller, instead as the solution of the optimization problems and no nonlinear optimization must be solved at every sampling time. The neural networks controller is also able to generate multivariable control signals directly without iterative computation. Therefore, the computational time for determining control signal can be reduced. The type of neural networks in the research is multilayer perceptron (MLP), which comprises input layer, hidden layer with tangent

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hyperbolic activation function and output layer with linear activation function. The proposed Nonlinear Model Predictive Control (NMPC) is applied on cascaded tanks process to control its level. The results show that the proposed NMPC is able to yield good control system performance and to overcome process characteristics mentioned above. The process output can follow the desired set point adequately well and generate the smooth control signals profile. Moreover, with the new NMPC scheme, the requirements of nonlinear model to represent the MIMO process can be simplified and computation time to obtain the control signals can be reduced.

Keywords – Nonlinear Model Predictive Control, optimization, neural networks, multilayer perceptron (MLP), MIMO process.

I. INTRODUCTION Most of industrial processes are characterized as multi input multi output (MIMO) and nonlinear processes as well as influenced by the disturbances that are detrimental to it. These conditions give difficulties in the design of a proper control system, which may overcome the above characteristics. Application of conventional control system often produces worse control performances since it depends on the assumptions such as a linearization and restricted of operation range. Attempts for increasing the control performances have been done by developing model based control system, where a model is used explicitly in determining the control signal. One of model based control strategy that

International Conference on Instrumentation, Communication and Information Technology (ICICI) 2005 Proc., August 3 -5, 2005, Bandung, Indonesia

out to reach the extreme point. As consequence, it worthwhile developing nonlinear model predictive control strategy that require less computational effort.

can be applied to improve the control system performance is Nonlinear Model Predictive Control (NMPC) which requires nonlinear model allowing analog characteristics between the model and its respective process. The nonlinear model predictive control algorithm offers the potential for improved process control performance (Henson, 1998).

Some researchers (Rohani et.al, 1999 and Fernholz et.al, 2001) have used neural networks based nonlinear model predictive control with MISO (Multi Input Single Output) structure for MIMO process. This structure, of course, requires the numerous models to represent the entire process and also need large memory.

The neural networks offer many advantages to build up the nonlinear model of the process. Their ability in nonlinear mapping between input – output is the potential benefits in the development of nonlinear model. The nonlinear model can be built without complicated mathematical equations and detail information about the process.

The research proposes a new structure and algorithm of nonlinear model predictive control by integrating neural networks as nonlinear model and controller. Nonlinear model can be developed using neural networks without involving the complex mathematical equations and detail information of the process. The neural networks controller will be able to eliminate a nonlinear optimization solution. The current control signal is given by the output neural networks controller, instead as the solution of the optimization problems and no nonlinear optimization must be solved at every sampling time. The neural networks controller is also able to generate multivariable control signals directly without iterative computation. Therefore, the computational time for determining control signal can be reduced.

When the predictive control strategy is based on nonlinear model, independently of the nature of model, the prediction equation cannot be solved explicitly, as in the linear case, and an iterative solution of the performance function evaluating the future behaviour of the system is required. At every sampling time, the current manipulated variable is calculated using an optimization procedure, which determines the optimal profile control actions that minimize the objective function. Hence, the use of nonlinear model allows the development of predictive control strategy for nonlinear dynamic process, but it also implies that at every sampling time a nonlinear optimization problem should be solved. This is a potential drawback of those control strategy because the solution of optimization problem are usually computational laborious, particularly in large processes. Most nonlinear optimization algorithms use some form of search technique to scan the feasible space of the objective function until the extreme point is located. This search is generally guided by calculation of the objective function and/or its derivatives and it implies a large computational effort because several iterations should be carried

The paper is organized as follows. Section two describes the concept of nonlinear model predictive control. Section three discusses about the development of nonlinear process model using neural networks. Section four describes about the development of neural networks controller in nonlinear model predictive control algorithm. Section five will illustrate the simulation of the proposed control strategy to MIMO process.

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International Conference on Instrumentation, Communication and Information Technology (ICICI) 2005 Proc., August 3 -5, 2005, Bandung, Indonesia

yˆ is the prediction of future process output from the model. n and m denote the number of output process and signal control, respectively.

II. NONLINEAR MODEL PREDICTIVE CONTROL (NMPC) Model predictive control (MPC) refers to a class of control algorithms in which a dynamics process model is used to predict and optimize process performance. In the beginning, the MPC algorithm is based on linear dynamic models and therefore is referenced by term linear model predictive control. Although often unjustified, the assumption of process linearity greatly simplifies model development and controller. Many processes are sufficiently nonlinear to preclude the successful application of linear model predictive control algorithm. This has led to the development of nonlinear model predictive control in which a more accurate nonlinear model is used for process prediction and optimization.

III. THE DEVELOPMENT OF NONLINEAR PROCESS MODEL USING NEURAL NETWORKS A multi layer perceptron (MLP) has been chosen for modelling purposes of MIMO process. The basic MLP networks is constructed by ordering the neurons in layers, letting each neuron in a layer take as input only the outputs of units in the previous layer or external inputs. If the networks have two such layers of neurons, it refers to as a two layer networks, if it has three layers it is called a three layer networks and so on. Figure 2 illustrates the example of three layer networks including input layer, hidden layer and output layer. The mathematical formula expressing what is going on in the MLP networks takes the form:

The concept of Nonlinear Model Predictive Control (NMPC) is depicted in figure 1 (Garcia et.al, 1989).

⎡nh ⎤ ⎞ ⎛ nϕ yi = Fi ⎢ ∑ Wi, j. f j ⎜ ∑ w j,l ϕl + w j,0 ⎟ + Wi,0 ⎥ ⎟ ⎜ l =1 ⎢⎣ j=1 ⎥⎦ ⎠ ⎝

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Figure 1. The concept of NMPC The idea behind model predictive control (MPC) is at each iteration to minimize the objective function of the following type: m

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International Conference on Instrumentation, Communication and Information Technology (ICICI) 2005 Proc., August 3 -5, 2005, Bandung, Indonesia

In order to determine the weight values, a set of examples of how the outputs, yˆ i , should relate to the inputs, ϕ i must be available. The task of determining the weights from these examples is called training or learning. The aim of training procedure is an adjustment of the weights to minimize the error between the neural networks output and the process output (also called by target). A learning algorithm is associated with any change in the memory as represented by the weights; learning does not in this sense imply a change in the structure of the memory. Therefore, learning can be regarded as a parametric adaptation algorithm. The learning algorithm is Levenberg-Marquardt method. This algorithm requires the information of gradient and Hessian matrices. The convergence will generally be faster than for the back-propagation algorithm. The detail derivation of Levenberg-Marquardt method can be seen in Norgaard et.al (2000).

where g is a mapping function that determines the solution of optimization problem given by the objective function in equation (1) and yˆ is the model output. If the function g was known, the expression given by equation (3) would provide at every time k the control signal that must be applied to the system to reach the desired control objective, when a predictive control strategy is employed. The problem of finding an expression for the function g can now be interpreted as functional approximation problem. Based on approximation capabilities of neural networks, one can decide to approximate the functional g by neural networks controller. The learning of the predictive neural controller consists in determining the weight parameter set (Wc) such that the control law, given by equation (3), provides a predictive performance controller. The training procedure is carry out in on line approach or called by specialized training. In specialized training, the neural networks controller is trained to minimize the objective function so the system output will follow the reference signal closely. The learning algorithm used in this research was back-propagation algorithm.

IV. THE DEVELOPMENT OF NEURAL NETWORKS CONTROLLER In the nonlinear model predictive control (NMPC), the controller has a task to replace the optimization problem for generating the control signal. Thus, there is no nonlinear optimization when the neural networks used in NMPC strategies. The following section will derive the algorithm of the neural networks controller in NMPC. The algorithm was derived and modified from Galvan et.al (1997) and Norgaard et.al (2000).

Suppose Wc be a weight parameter of the neural networks controller and set point remains constant along the prediction horizon, the gradient of the error function given by equation (1) with respect to the parameter can be written as follows:

The derivation of control algorithm using neural networks controller can be viewed as the analog of equation (2). Hence, the control signal u(k) minimizing the objective function in equation (1) can be written as follows :

m N2 n Nu ∂yˆ j ( t + i) ∂U l, k ∂J ( t ) = − 2 ∑ ∑ r j ( t + i) − yˆ j ( t + i) + 2ρ ∑ ∑ U l, k ∂Wc ∂Wc ∂Wc j =1 i = N1 l =1 k =1

(

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(4) with: U l, k = u l ( t + k )

(

sp ˆ ,L, Y ˆ , U ,L, U ) U = g Y1sp , L, Ym , Y, Y 1 m 1 n

Based on the above description, the neural predictive controller parameters can be

(3)

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International Conference on Instrumentation, Communication and Information Technology (ICICI) 2005 Proc., August 3 -5, 2005, Bandung, Indonesia

updated at every sampling time using the following learning rule: ∂J Wcnew = Wcold + α (5) ∂Wc

The system identification with neural networks algorithm was done in NARX (Nonlinear Auto Regressive with eXogenous input) or series parallel model structure. The history length for both the input signal and the output signal was 4. It means that the input spaces consist the present and three past values of the inputoutput signal. Thus, the input layer has 24 variables as the input of regressor. The hidden layer consists of 8 neurons with hyperbolic tangent function. The result of neural networks model in identi-fication of the process is shown in figure 5. From this figure, it can be seen that the neural model can identify and anticipate behaviour of the process satisfactorily.

V. SIMULATION In the following section, the containing neural model and controller will be applied to the MIMO process. The proposed control system is applied to the cascade tanks, which adopted from Babuska (1998). The cascade tanks is depicted in figure 3. X

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The goal of control system is to maintain level in tanks 1 and 2 by adjusting the flow rate q1 and q2. The first step in designing of NMPC is to develop the process model by system identification. Figure 4 illustrates the inputoutput data.

Figure 5. Identification of neural model After developing the nonlinear process model, the next step is to determine the neural networks controller to generate the optimal control signal of the process. The structure of the neural networks controller was also multi layer perceptron (MLP), with input, hidden and output layer, and used the back-propagation learning algorithm to up date the weights of neural controller.

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International Conference on Instrumentation, Communication and Information Technology (ICICI) 2005 Proc., August 3 -5, 2005, Bandung, Indonesia

VI. CONCLUSION

controller to generate the control signal for the process. Simulation of NMPC containing neural model and controller is illustrated in figure 6, while figure 7 shows the control signals that generated by neural controller.

The algorithm of NMPC containing neural model and controller for MIMO process has been presented in the paper. With this structure, the requirements of nonlinear model to represent the entire process can be simplified using single model and solution of nonlinear optimization procedure to generate the control signals can be eliminated.

Performance of NMPC Containing Neural Model and Controller Output #1

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REFERENCES

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[1]. Babuska R (1998), Fuzzy Modelling and Identification Toolbox User’s Guide, http://lcewww.et.tudelft.nl/~babuska. [2]. Carlos E. Garcia, David M. Prett, Manfred Morari, (1989), Model Predictive Control: Theory and Practice a Survey, Automatica, 25, pp. 335-348. [3]. Cybenco G (1989), Approximation by Superpositions of A Sigmoidal Function, Mathematics of Control, Signals and Systems, 2(4), 303 – 314. [4]. Fernholz G, Rossman V, Engell S, Bredehoeft J.P (2001), System Identification Using Radial Basis Function Nets for Nonlinear Model Predictive Control of A Semibatch Reactive Distillation Column, Internal Report, Laboratory of Process Control, University of Dortmund. [5]. Galvan I.M, Zaldivar J.M (1998), Application of Recurrent Neural Networks in Batch Reactors. Part II: Nonlinear Inverse and Predictive Control of The Heat Transfer Fluid Temperature, Chemical Engineering and Processing, 37, 149 – 161. [6]. Henson,M.A (1998), Nonlinear Model Predictive Control: Current Status and Future Directions, Computers and Chemical Engineering, 23, pp. 187-202. [7]. Norgaard M, Ravn O, Poulsen N.K, Hansen L.K (2000), Neural Networks for Modelling and Control of Dynamic Systems, Springer Verlag. [8]. Rohani S, Haeri M, Wood H.C (1999), Modelling and Control of A Continuous Crystallization Process Part II: Model Predictive Control, Computers and Chemical Engineering, 23, 279 – 286.

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Figure 6. Performance of the control system The good set point tracking was achieved in figure 8. The process output can track the set point without offset. Moreover, the neural networks controller can also generate the smooth control signal for the process. Control signal 0.4

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