I want to create a discrete system from a continuous system. The continuous system is a Linear Time-Invariant (LTI) system with white noise input matrix Q. Computing the Transition Matrix (phi) is well documented. However, computing the discrete system noise Covariance matrix is less straightforward.
I have used the differential equation:
d Qk/dt = F Qk + Qk FT +Q for computing the Qk.
What I expected was that the state covariance matrix in steady state would be the same for both the continuous and discrete systems.
Consider a 1st-order LTI system: xDot = -x/tau + Q
The steady state variance of x is P = Q*tau / 2
The transition matrix (phi) is e- DT / tau
The Qk in steady state is Qk=Q*tau / 2
The variance of the discrete state is given by: P(i+1) = Phi * P(i) * Phi + Qk
The steady state for the discrete P(i) is given by: P = Q * tau / (2 * (1-e-2DT/tau)
So the only time the steady-state variances are equal is when the DT is vey small.
I have a higher-order LTI transfer function that has a white noise input. I set the Q value for the system using the steady-state value. Then I convert the continuous time system to a discrete system by computing phi and Qk. Is there an easy way to make the discrete system have the exact same Covariance time history as the continuous system at the discrete sample instances?