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                   OVERVIEW OF FILES RELATED TO SIMULATION OF
                      NEURAL NETWORK BASED OPTIMAL CONTROL
                   
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Matlab tools for training a neural network to act as an optimal controller
and for simulating the closed-loop system with the trained controller network
in the loop.


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DESCRIPTION
Assuming the system to be controlled can be described by
     ^      ^
     y(t) = f(y(t-1),...,y(t-n),u(t-1),...,u(t-m))

then the present files train a neural network controller, g: 

     u(t) = g(r(t+1),y(t),...,y(t-n+1),u(t-1),...,u(t-m+1))
 
to minimize the criterion (theta is a vector containing the weights)

                                              
                  ---                  2                     2
      J(theta) =  >   [ (r(t+1)-y(t+1))   +  rho*(u(t|theta))  ]
                  ---                               
                   t                            

The network is trained with a recursive Gauss-Newton method. This
method is derived for least squares problems but has here been modified
here simply by adding an additional term to the gradient compared to the
implementation of specialized training in 'special2'. The covariance matrix
update is unchanged but that doesn't seem to of be a problem -- the algorithm
still converges quite rapidly. Three different methods for updating the
covariance matrix (the inverse Hessian) have been implemented: exponential
forgetting, constant trace, and the exponential forgetting and resetting
algorithm (EFRA).

The criterion differs from the one considered in "specialized" training of
inverse models in that the squared control actions are penalized by
introduction of an additional term in the criterion. If rho=0 the training
procedure will obviously perform exactly as 'special2' and the controller
simply becomes the inverse model.

The process to be controlled can be a SIMULINK model or a model described by a
set of differential equations specified in a MATLAB-function. If the training
data is obtained through a "real world" experiment (i.e., a MATLAB/SIMULINK
model is not available), the closed-loop system can also be simulated assuming
that the network model is a perfect description of the system to be controlled.


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FILE OVERVIEW:
Optim    - This file
opttrain - Train a network as a "detuned" inverse model using
           indirect/specialized training. The network is trained with a
           recursive Gauss-Newton method.
optrinit - File containing initializations for 'opttrain'.
optcon   - Program for simulating the closed-loop system.
optinit  - File containing the design parameters for 'optcon'.
opttest  - A test example.


The following functions must be available as well:
NNSYSID toolbox        - Neural Network Based System Identification
shift, siggener        - Functions used by opttrain and optcon
general, invsim
smout, springm, spm1   - Files used for demonstration


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SUGGESTED WORKING PROCEDURE FOR TESTING THE CONCEPT ON A KNOWN PROCESS

1: BUILD THE PROCESS
Build the process in SIMULINK or write the differential equations in a
MATLAB-function. If SIMULINK is used, specify input to and output from the
model by an inport and an outport, respectively. If SIMULINK is not available, 
write the differential equations in a MATLAB-function in the format required
by the ODE45 differential equation solver.

2: MAKE AN EXPERIMENT
Use the function 'experim' to generate a data set to be used
for inferring a neural network model of the process.

3. BUILD A MODEL OF THE PROCESS
Identify a "forward" model of the process with the 'nnarx' function and
evaluate the obtained network with 'nnvalid'. This network is used for providing
estimates of the Jacobian of the process which is required for training the
controller network.

4: TRAIN THE CONTROLLER NETWORK
'opttrain' is responsible for training the controller network. The user
specifications necessary for running this function must be written in the file
'optrinit'. One of the things that the user must specify in this file is an
initial controller network. This can for example be an inverse model trained
with 'general'. This will in most cases be more safe than just initializing
the weights at random and in addition it will often result in a more rapid
convergence.  

5: SIMULATION OF THE CLOSED-LOOP SYSTEM
The closed-loop system is simulated with the function 'optcon'. The design
parameters necessary for running this program must be defined in the file
'optinit'. For example, the user must specify the name of the
MATLAB/SIMULINK model, sampling period,, reference signal, and the name of the
file containing architecture and weights of the controller network.