How to add drift to a state equation by ssm() (State space function)?

''ssm creates a standard, linear, state-space model object with independent Gaussian state disturbances and observation innovations.'' However, I cannot create a state-space model in which the state equation has a drift. For example:

A = 1;
B = 1;
C = 1;
D = 1;
Mdl = ssm(A,B,C,D)

The codes will create the ssm model as: State equation: x1(t) = x1(t-1) + u1(t)

Observation equation: y1(t) = x1(t) + e1(t)

What should I do if I want to add a drift term to the state equation, like x1(t) = x1(t-1) + u1(t)+c, where c is a constant (drift)?

Answer 1

[Shantanu Dixit]

Hi Wilson,

You can add a drift term 'c' in the state equation when using the 'ssm' function by augmenting the state vector. Since 'ssm' assumes a standard linear state-space model without an explicit drift term, introducing an additional state variable allows the drift to be incorporated naturally.

You can refer to an example script which adds drift to a 'ssm' model

A = [1, 1; 0, 1]; 
B = [1; 0]; 
C = [1, 0];   
D = 1;  

% Define initial state (x1 and drift c)
initial_x1 = 0;  
c = 0.5;   
Mean0 = [initial_x1; c]
Cov0 = [0.1, 0; 0, 0];   

% Create the state-space model
Mdl = ssm(A, B, C, D, 'Mean0', Mean0, 'Cov0', Cov0);

This ensures that the state equation follows:

Hope this helps!