My Radial basis network fits 99% to the validation data . Please give me some citations on this topic.I have seen MPC being used on state space models but not much on models obtained by RBF networks
The MPC technique can be used with any ANN model types such as MLP, RBF or any recurrent networks. Since the ANN models are nonlinear, this will lead to non-convex and non-quadratic optimization problem like been argued in [1, 2]. The gradient based optimization used to solve the MPC cost function at every time step can be quite computational demanding. You will need to do the recursion of the RBF model predictor to get the k-step ahead prediction of the system's output, and this would made the control calculation even more difficult to obtain. Depending on your system, the nonlinear MPC can still be implemented as long as your dynamic is reasonable slow and fast convergence is not a requirement.
If the non-linearities in your dynamic system is reasonably smooth and fast convergence solution as priority, you can also consider to extract linear models from the ANN at each time instance [see 3,4]. Basically what you get at each time step is a linear model of the process (either state space or transfer function model), and this will be used in the MPC's prediction and optimization process. In term of performance, the controller is also found to produce satisfactory results as reported in [5,6, 7].
[1] Lawrynczuk, M., A family of model predictive control algorithms with artificial neural networks International Journal of Applied Mathematics and Computer Science, 2007, 17, 217-232.
[2] Norgaard, M., Neural networks for modelling and control of dynamic systems : a practitioner's handbook Springer, 2000, 24.
[3] Kuure-Kinsey, M.; Cutright, R. & Bequette, B. Computationally efficient neural predictive control based on a feedforward architecture Industrial & Engineering Chemistry Research, 2006, 45, 8575-8582
[4] Kuure-Kinsey, M. & Bequette, B. Improved nonlinear predictive control performance using recurrent neural networks American Control Conference, 2008, 2008, 4197-4202
[5] Witt, J.; Boonto, S. & Werner, H. Approximate model predictive control of a 3-DOF helicopter Decision and Control, 2007 46th IEEE Conference on, 2007, 4501-4506
[6] Norgaard, M.; Sorensen, P. H.; Poulsen, N. K.; Ravn, O. & Hansen, L. K. Intelligent predictive control of nonlinear processes using neural networks Intelligent Control, 1996., Proceedings of the 1996 IEEE International Symposium on, 1996, 301-306
[7] Mu, J. & Rees, D. Approximate model predictive control for gas turbine engines American Control Conference, 2004. Proceedings of the 2004, 2004, 6, 5704-5709
MPC will work as long as there is a model, but the challenge in using a nonlinear model as it is in the MPC is the solution to the optimization problem. The answer to your question is yes, it can be done with sequential quadratic programming, but it is challenging to code line by line. But again, the question is whether you need to kill the controller by solving a nonlinear optimization problem at every time step? In many cases, a linear model that changes itself in real-time works just as well. You can do this through the recursive least squares algorithm which will identify the model parameters in real time.
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