Model based predictive control using neural network and fuzzy logic.

Balaji, V. ; Vasudevan, N. ; Maheswari, E. 等

Introduction

The Model Predictive Algorithms (MPC) has been widely used in industrial process in recent years. This algorithm is well studied for high performance control of constrained multivariable process because explicit paring of input and output variable is not required and constrains can be incorporated. The general strategy of MPC algorithm is to use a model to predict the output in the future and to minimize the difference between this predicted output and that desired by computing the appropriate control actions.

A Classification of basic classes of MPC algorithms is presented in Fig. 1. It should be treated as a rather simplified one, i.e., many classes of nonlinear optimal MPC algorithms. (with optimization using nonlinear process models) can be further distinguished, but this is beyond the scope of this paper.

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Soft Computing in Nonlinear MPC Algorithms

The presentation of predictive control algorithms using soft computing techniques will be done within the following groups:

* MPC algorithms using fuzzy reasoning:

* Multi-model explicit algorithms in the fuzzy Takagi-Sugeno (TS) structure

* Algorithms with on-line linearization of a fuzzy TS model and QP optimization

* MPC algorithms using artificial neural networks:

* Algorithms with nonlinear optimization and a neural network process model or a neural network prediction model.

* Algorithms with on-line linearization of a neural network model and QP optimization.

* Neural network modeling applied to reduce computational complexity and to approximate the controller.

MPC Algorithms Using Artificial Neural Networks Neural-Network Model of the Plant:

Let the single-input single-output (SISO) process under consideration be described by the following nonlinear discrete-time equation:

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[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

The structure of the neural network is depicted in Fig. 2. The output of the model can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where, [z.sub.ii](k) is the sum of inputs and [v.sub.i](k) is the output of the [i.sup.th] hidden node respectively, [phi] R [right arrow] R is a nonlinear transfer function, k is the number of hidden nodes.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

MPC Algorithms with Nonlinear Optimization (MPC-NO) and Neural Network Models

The gradients of the cost function J(k) are approximated numerically and the nonlinear optimization problem is solved on-line. The cost function is expressed as;

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where, I is the unit matrix and [J.sup.NO] are of dimension [N.sub.u] x [N.sub.u], and the vector [U.sup.NO] is of length [N.sub.u]. The matrix of dimension (N -Nu+1) x Nu, containing partial derivatives of the predicted output w.r.t. future control is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Taking into account equations (2),(3) & (5), the partial derivatives of the predicted output signal w.r.t future controls are calculated from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Obviously,

[partial derivative][z.sub.i](k + p)/[partial derivative]u(k + r\k) = [partial derivative]y(k + p\k)/[partial derivative]u(k + r\k) = 0, r [greater than or equal to] p - [pi] + 1 (7)

It is noted that only some of the output predictions are influenced by future controls. Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Where [I.sub.yp]f(p)=max{min{p-[pi],[n.sub.A]},0} is the number of network input nodes depending on output predictions which are affected by future controls.

Reducing Computational Complexity in MPC with Neural Networks

The MPC-NO algorithm is computationally demanding and the computation time is much longer than that of linearization-based algorithms. To reduce the computational complexity, a few neural-network based alternatives have been suggested. In general, these approaches can be divided into two groups: in the first one, special structures of neural models are used to make the optimization problem simpler (convex), while in the second, explicit approximate algorithms (without on-line optimization) combined with neural networks are used.

Neural Network Based MPC with On-Line Optimization

A structured neural network that implements the gradient projection algorithm is developed to solve the constrained QP problem in a massively parallel fashion. Specifically, the structured network consists of a projection network and a network which implements the gradient projection algorithm. The projection network consists of specially structured linear neurons. A training algorithm is formulated for which the convergence is guaranteed. The networks are trained off-line, whereas the controls are calculated on-line from the networks without any optimization.

In addition to linearization-based MPC schemes which use the QP approach, it is also possible to develop a specially structured neural model to avoid the necessity of nonlinear optimization.

A set of nonlinear affine predictors of the following structures is used in the MPC algorithm:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Where, p = 0, ...., N. The quantities [F.sub.p]([x.bar](k)) and [G.sub.pj] ([x.bar](k)), which depend on the current state of the plant [x.bar](k)] ... y(k - [n.sub.A])u(k - 1) ... [(k - t)].sup.T] (7) are calculated by neural networks. The key idea is that the present and future controls, i.e., the decision variables of the optimization problem occur linearly in the predictor's equation (6). The predictor depends in a nonlinear way only on the past values of input and output signals. Hence, the resulting MPC optimization problem is convex.

Neural Network Based MPC without On-Line Optimization

The key idea is to calculate control signals on-line without any optimization. The main advantage of this approach is its speed. On the other hand, the control law must be precomputed off-line and stored somehow in the controller's memory.

Applications and Exemplary Simulation Results

MPC algorithms with neural network models of different structures have been applied to a wide class of processes, for example, a combustion system (Liu and Daley, 1999), a pneumatic servo system (Norgaard et al., 2000), a mobile robot (Ortega and Camacho, 1996), an industrial packed bed reactor (Temeng et al., 1995), an insulin delivery problem (Trajanoski andWach, 1998), a multivariable chemical reactor (Yu and Gomm, 2003), traffic control on freeways (Parisini and Sacone, 2001), and a biological depolluting treatment of wastewater (Vila and Wagner, 2003).In this simulation examples of a control system with MPC using neural networks is given.

High-Purity High-Pressure Ethylene-Ethane Distillation Column:

The plant under consideration is a high purity, high pressure (1,93MPa) ethyleneethane distillation column shown in Fig. 12 (Lawry 'nczuk, 2003). The feedstream consists of ethylene (approx. 80%), ethane (approx. 20%), and traces of hydrogen, methane and propylene.

[FIGURE 3 OMITTED]

The plant under consideration is a high purity, high pressure (1,93MPa) ethyleneethane distillation column shown in Fig. 12 (Lawrynczuk, 2003). The feedstream consists of ethylene (approx. 80%), ethane (approx. 20%), and traces of hydrogen, methane and propylene. The product of the distillation is ethylene which can contain up to 1000 ppm (parts per million) of ethane. The column has 121trays and the feed stream is delivered to the tray no.37 . Four models of the plant were used. The first one was used as the real process during the simulations. It was based on technological considerations (Lawrynczuk, 2003). An identification procedure was carried out, and as a result two linear models for different operating points and a neural one were obtained. In all the simulations it is assumed that at the sampling instant k = 1 the set-point value is changed from 100 ppm to 350 ppm, 600 ppm and 850 ppm.Because of some technological reasons, the following constraints were imposed on the reflux ratio: [r.sup.min]=4.051, [r.sup.max]=4.4571.

At first, MPC algorithms based on two linear models were developed. The first linear model is valid for a low" impurity level and the resulting control algorithm works well in this region, but exhibits unacceptable oscillatory behavior for medium and big setpoint changes, is shown in Fig. 13. On the contrary, the second linear model captures the process properties for a "high" impurity level and the closed-loop response is fast enough for the biggest setpoint change, but very slow for smaller ones, as is shown in Fig. 14

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Simulation results of MPC-NPL algorithms with a neural network are depicted in Fig. 5. Both algorithms work well for all three setpoint changes. The NPL1 algorithm is slightly slower than NPL2. Simulation results of the MPC-NO algorithm with a neural network are shown in Fig. 6. Compared with suboptimal linearization-based algorithms, nonlinear optimization leads to faster closed loop responses.

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In practice, big changes in the manipulated variable r are not allowed because of technological and safety reasons (high pressure, big production scale). That is why an additional constraint [DELTA][r.sup.max] =0.03 was used. Figure.8 compares simulation results of the MPC-NPL2 and MPC-NO algorithms with a neural network. Although the constraint significantly slows closed-loop responses, it can still be noticed that the MPC-NO algorithm is somewhat faster.

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Conclusions

The subject of the paper was applications of soft computing methods to model-based predictive control techniques. In this paper, emphasis was put on computation efficiency.

A family of MPC algorithms using artificial neural networks (i.e., the most popular multilayer perceptron) was described. In comparison with fuzzy models, neural structures do not suffer from the "curse of dimensionality", which is troublesome in multivariable cases. Algorithms with nonlinear optimization are potentially very precise, but they hinge on the effectiveness of the optimization routine used. Because, in practice, convergence to a global optimum cannot be guaranteed, MPC-NPL algorithms with on-line linearization of a neural network model were presented.

The resulting algorithms make it possible to effectively control highly nonlinear, multidimensional processes, usually subject to constraints, which result from technological, and safety reasons. The algorithms considered can be easily implemented and used on-line.

Acknowledgement

The authors like to acknowledge the management of Sathyabama Institute of Science & Technology for the support and encouragement. I am grateful to my guide Dr.N.Vasudevan for his valuable suggestions and useful discussion while preparing this paper.

References

[1] Lawry'nczuk M. (2003): Nonlinear model predictive control algorithms with neural models.--Ph.D. thesis, Warsaw University of Technology, Warsaw, Poland.

[2] Liu G.P. and Daley S. (1999): Output-model-based predictive control of unstable combustion systems using neural networks.--Contr. Eng. Pract., Vol. 7, No. 5, pp. 591-600.

[3] Norgaard M., Ravn O., Poulsen N.K. and Hansen L.K. (2000): Neural Networks for Modelling and Control of Dynamic Systems.--London: Springer.

[4] Ortega J.G. and Camacho E.F. (1996): Mobile robot navigation in a partially structured static environment, using neural predictive control.--Contr. Eng. Pract., Vol. 4, No. 12, pp. 1669-1679.

[5] Temeng K.O., Schnelle P.D. and McAvoy T.J. (1995): Model predictive control of an industrial packed bed reactor using neural networks.--J. Process Contr., Vol. 5, No. 1, pp. 19-27.

[6] Yu D.L. and Gomm J.B. (2003): Implementation of neural network predictive control to a multivariable chemical reactor.--Contr. Eng. Pract., Vol. 11, No. 11, pp. 1315-1323.

V. Balaji

Research Scholar, Sathyabama University, Chennai, Tamil Nadu, India

N. Vasudevan

Prof & HOD, ECE St Peters' Engg. College, Chennai, Tamil Nadu, India

E. Maheswari

Faculty of Easwari Engineering College, Chennai, Tamil Nadu, India