How could we calculate the zero dynamics of a MIMO nonlinear system with non-zero initial condition?
I am dealing with a nonlinear system in which having a positive value on the inputs (cable tension) is one of the necessary condition for the system to be active. I would like to know how can the zero dynamics of the system be calculated in a non-zero initial condition.
We discuss zero dynamics of a multivariable system (zero dynamics of multiple input, multiple output "MIMO" system):
x˙ = Ax + Bu y = Cx,
where (A, B, C) is minimal and A is n × n. From now we always assume that both B and C have full rank. For the time being we assume that the number of inputs and the number of the outputs are the same (a square system), namely B is n × m and C is m × n. Correspondingly we also have the frequency domain representation as: G(s) = C(sI − A) −1B.
Unfortunately we do not have a straightforward way to extend the concept of transmission zero for a single-input, single-output (SISO) system to the MIMO case.
When a system has non-minimum phase zeros, high-gain feedback (e.g. F matrix with large norm) cannot be used to stabilize the system, since it can be shown that some of the poles of the feedback system tend to its zeros as the matrix F increases. This is a similar phenomenon as that what occurs with root-locus diagrams. Consequently, the selection of feedback matrices in the non-minimum phase case is very delicate, since feedback should be high enough to have some effect but simultaneously low enough to avoid instabilities. Finally, it can be shown that the sensitivity of a non-minimum phase system to disturbances acting at the input of the plant is severely limited both when the open-loop plant has unstable poles and non-minimum phase zeros.
I hope it is helpful to you.
All the best.
if you have a quadratic MIMO system it is the same as SISO system. just have a look on isidori nonlinear book
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