State space model back-propagation gradient descent

I have a state space model Xk+1 = A xk + B and yk = C Xk + D. I want to start the matrices randomly and adjust them using gradient descent. Is there any way to implement this in Matlab?

Answer 1

[Raghava S N]

Hello @L,

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!

[L]

Thanks @Raghava. I was looking for some built in functions that details such as momentum and gradient clipping anre already addressed. For example, in Python, you can state your model and use loss.backward() . Option like momentum, batch and clipping are parameters to the buil in function. I don’t want to implement these things by hand. It wouldn’t make sense since these are standard procedures. I wanted something close to train() for neural networks were I can choose the optimizer, momentum, clipping, batch etc…

[Raghava S N]

Hey, @L,

In that case, you can explore using the "trainingOptions" function - https://www.mathworks.com/help/deeplearning/ref/trainingoptions.html in tandem with the "trainnet" function - https://www.mathworks.com/help/deeplearning/ref/trainnet.html.

[L]

@Raghava S N, but when using trainnet, I have to specify the layers of a net object. How to specify the state space model?