Question

Given a linear dynamic system in state space form x˙=Ax+Bu(t), where A is a nxn

matrix and B is a nx1 matrix, u(t) is 1x1 input vector, which changes with time.
Assume that the initial condition X(0)=0, and the input u(t) is a step input
, i.e. u(t)=1, when t>0. Assume that we expand the x(s) at infinite, i.e. s→[infinity].
Please derive the recursive moment matching formula for response x(s) for this problem.
Hint, assume x(s→[infinity])=m₁s⁻¹+m₂s⁻² +m₃s⁻³+…

Answer 1

To derive the recursive moment matching formula for the response x(s), start with the given linear dynamic system in state space form and use the Laplace transform. Substitute the step input function and solve for X(s). Use the Taylor series expansion to derive the recursive formula for the moments.

Step-by-step explanation:

To derive the recursive moment matching formula for the response x(s), we start with the given linear dynamic system in state space form: x⸗=Ax+Bu(t). Since the input u(t) is a step function, we can express it as u(t) = 1/s (i.e., the Laplace transform of a step function is 1/s).

Substituting this into the state space equation, we get sX(s) = AX(s) + Bu(s), where X(s) is the Laplace transform of x(t). Solving for X(s), we have (sI - A)X(s) = Bu(s). Inverting this equation, we get X(s) = (sI - A)^-1Bu(s).

Now we can use the recursive moment matching formula to find the moments m1, m2, m3, etc. The formula is derived using the Taylor series expansion of X(s) as s approaches infinity. By comparing coefficients of the Taylor series expansion with the moments, we can derive the recursive formula.