Is there a way to solve this problem with a neural network?

Hello,

I want to preface this by stating that my knowledge of neural networks is not that large yet. I have looked through a number of resources on this topic, but I am still confused. Currently, I am trying to use a neural network to implement an optimal trajectory planner for a simple 2D linear point-mass 2nd-order system given initial and terminal conditions of the state, . This point-mass system is guided by a control law, , that guides the system to its terminal state, . The code below shows this simulated action in effect:

clc, clear all, close all

%% Initialization

% Initial State

xo1 = [0;0]; % Initial Position

xo2 = [1;1]; % Initial Velocity

xo = [xo1;xo2];

% Terminal State

z = [2;1]; % Terminal Position

vd = [2;2]; % Terminal Velocity

xf = [z;vd];

[t, x] = ode45(@(t,x) EqnMtn(x,xf,t), [0 1], xo);

function dx = EqnMtn(x, xf, t) % Equations of Motion

%% Parsing State Vector

x1 = x(1:2);

x2 = x(3:4);

%% Parsing Terminal State Vector

z = xf(1:2);

vd = xf(3:4);

%% Deriving Time-to-Go

tgo = 1.0001 - t;

%% Gain Tuning

n = 0; % Trajectory Tuning Parameter

k1 = (n+2) * (n+3);

k2 = -(n+1) * (n+2);

%% Output of Acceleration Command

control_term1 = (k1/tgo^2) * (z - x1 - tgo*x2);

control_term2 = (k2/tgo) * (vd - x2);

uu = control_term1 + control_term2; % u(t)

%% Derivatives

dx1 = x2;

dx2 = uu;

dx = [

dx1

dx2

];

return;

end

For the implementation of a neural network, I would hope to input both a given input, (1x4 vector), and terminal state, (1x4 vector), to the neural network and have it output a state trajectory for the point mass to get to its terminal state, x(1:2), and its velocity trajectory, x(3:4). I theorized that the neural net should be trained on five seperate optimal trajectories with different terminal positions and use a seperate trajectory for validation, . I believe that the training and validation data for the network should look something like this:

%% Parsing Training and Validation Data

% Inputs
xo_train = [x1(1,:),x2(1,:),x4(1,:),x5(1,:),x6(1,:)]; % Initial State Vectors
xf_train = [xf1,xf2,xf4,xf5,xf6]; % Terminal State Vectors

% Outputs
x_train = [x1(:,1:2);x2(:,1:2);x4(:,1:2);x5(:,1:2);x6(:,1:2)]; % Position State Trajectories
dx_train = [x1(:,3:4);x2(:,3:4);x4(:,3:4);x5(:,3:4);x6(:,3:4)]; % Velocity State Trajectories

% Validation Data
xo_valid = x3(1,:); 
xf_valid = xf3;
x_valid = x3(:,1:2);
dx_valid = x3(:,3:4);

Also, I would like to train the data on a custom loss function which is described by:

where, are estimates of the optimal guidance trajectory at each time step.

At this point, this is where I am stuck since I do not know which neural network architecture to use to solve this problem. I believe that a feedforward network of some kind would help, but I am not even sure how to go about constructing it. Any advice on where to go from here would be greatly appreciated. Thanks in advance.

[Walter Roberson]

https://www.mathworks.com/help/deeplearning/function-approximation-and-nonlinear-regression.html