Estimation of state-space vectors using gradient descent can be implemented in MATLAB. This can be done by initializing the state-space matrices randomly and then using gradient descent to adjust them. Here is the step-by-step process to achieve this:
Start by defining the dimensions of your state-space model and initialize the matrices “A”, “B”, “C”, and “D” with random values. This can be done using the “randn” function. For more information, refer to this link - https://www.mathworks.com/help/matlab/ref/randn.html.
n = 4; % Number of states
m = 2; % Number of inputs
p = 3; % Number of outputs
num_samples = 100; % Number of samples
A = randn(n, n);
B = randn(n, m);
C = randn(p, n);
D = randn(p, m);
Then the learning parameters must be defined. Choose the number of iterations and a learning rate, which guides how quickly or slowly the model learns. Once this is done, the cost function can be determined. This could be the mean squared error between the predicted output “yk” and the actual output.
% Define learning parameters
num_iterations = 1000;
learning_rate = 0.001; % Reduced learning rate
Once this is done, the iterative gradient descent algorithm can be used. For each iteration, set the gradients to zero, which will accumulate changes needed for the matrices. Also initialize the state vector to begin the simulation of the state-space model.
% Gradient descent loop
for iter = 1:num_iterations
% Initialize gradients
grad_A = zeros(size(A));
grad_B = zeros(size(B));
grad_C = zeros(size(C));
grad_D = zeros(size(D));
% Initialize state
Xk = zeros(n, 1);
As the code loops through each sample of the data, predict the output using the current matrices and compare it to the desired output. This comparison gives an error, which is used to adjust the matrices. By accumulating these adjustments, the matrices are refined over time.
% Compute gradients
for k = 1:size(U, 2)
% Get current input and desired output
uk = U(:, k);
yk_desired = Y(:, k);
% Predict output
yk = C * Xk + D * uk;
% Compute error
error = yk - yk_desired;
% Accumulate gradients
grad_C = grad_C + 2 * error * Xk';
grad_D = grad_D + 2 * error * uk';
% Update state
Xk_next = A * Xk + B * uk;
% Compute gradients for A and B
grad_A = grad_A + 2 * (C' * error) * Xk';
grad_B = grad_B + 2 * (C' * error) * uk';
% Move to next state
Xk = Xk_next;
end
Finally, after processing all samples, update the matrices using the accumulated gradients, converging to an optimal state.
% Update matrices using gradients
A = A - learning_rate * grad_A;
B = B - learning_rate * grad_B;
C = C - learning_rate * grad_C;
D = D - learning_rate * grad_D;
This process repeats for the number of iterations set, gradually improving the model's accuracy.
Hope this Helps!
