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Internal Daynamics —— Linear sys VS. Nolinear sys

17/04/2011

While reviewing nonlinear control, I relearned many concepts I learned before. Here I want to highlight something.

1. Linear Systems

a) Relative degree

What is the relative degree of a linear system? I learned it long time ago, but didn’t understand it well until today. The subtraction of pole number and zero number is call relative degree. Why relative degree is an important concept? See below.

b) Internal dynamics and zeros

Why relative degree is important? Poles determine the stability of a linear system. Zeros determine the stability of the internal dynamics. We can use feedback control to change the poles of the closed-loop system, but can not change the zeros of the system. (Am I right?) So if there are unstable zeros (with positive real parts), the internal dynamics is unstable no matter what control input we use. This is the famous non-minimum phase problem.

2. Nonlinear Systems

a) Relative degree

What is the relative degree of a non-linear system? Differentiate the output y until the control input u appears in its derivative. Say y^{(r)}=\phi (u). Then r is the relative degree. The name ‘relative degree’ may be more meaningful for linear systems.

b) Internal dynamics

As I mentioned in a previous post, if the relative degree is r, there are still left n-r DOFs undetermined. These undetermined states form the internal dynamics. The stability of the internal dynamics is relevant to the input control. That means different control inputs may result in stable or unstable internal dynamics. It is different from linear cases.

c) Zero dynamics

In order to investigate the stability of internal dynamics, zero dynamics is introduced. One definition of zero dynamics is: if control input makes output be zero, at this time the internal dynamics is called zero dynamics. Another definition is: if the control input of the internal dynamics is zero, the internal dynamics is called zero dynamics. Anyway, zero dynamics is a special case of internal dynamics. People define zero dynamics in order to check the internal stability. If the zero dynamics is globally stable, then the internal dynamics can be locally stable. Nonlinear systems with asymptotically (exponentially) stable zero dynamics is called asymptotically (exponentially) minimum phase systems.

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