I have a dynamic Linear system defined using the state-space equations dx/dt = Ax(t) + Bu(t).
1- What are the conditions for the system to be ergodic?
If the system is ergodic and we assume that A is a function of random variable A(g) and g has Gaussian Distribution. We want to find the distribution (mean/std) of x(t) at each time point t.
One way is to simulate the system for different values of g using Monte-Carlo in order to find the distribution of x(t) , t \in [0,T]
Can we use the ergodicity theorem and assume that g is function of time g(t) and has Gaussian distribution in time. We then simulate the system for very long time [ 0 k*T] and divide the waveform x(t) into k parts and find the mean/std along the time interval?
Thanks
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