Control System Toolbox Function Reference

minreal

Minimal realization or pole-zero cancellation

Syntax

sysr = minreal(sys)
sysr = minreal(sys,tol)
[sysr,u] = minreal(sys,tol)

Description

sysr = minreal(sys) eliminates uncontrollable or unobservable state in state-space models, or cancels pole-zero pairs in transfer functions or zero-pole-gain models. The output sysr has minimal order and the same response characteristics as the original model sys.

sysr = minreal(sys,tol) specifies the tolerance used for state elimination or pole-zero cancellation. The default value is tol = sqrt(eps) and increasing this tolerance forces additional cancellations.

[sysr,u] = minreal(sys,tol) returns, for state-space model sys, an orthogonal matrix U such that (U*A*U',U*B,C*U') is a Kalman decomposition of (A,B,C)

Example

The commands

g = zpk([],1,1)
h = tf([2 1],[1 0])
cloop = inv(1+g*h) * g

produce the nonminimal zero-pole-gain model by typing cloop.

Zero/pole/gain:
      s (s-1)
-------------------
(s-1) (s^2 + s + 1)

To cancel the pole-zero pair at , type

cloop = minreal(cloop)

and MATLAB returns

Zero/pole/gain:
      s
-------------
(s^2 + s + 1)

Algorithm

Pole-zero cancellation is a straightforward search through the poles and zeros looking for matches that are within tolerance. Transfer functions are first converted to zero-pole-gain form.

See Also
balreal     Grammian-based input/output balancing

modred      Model order reduction

sminreal    Structured model reduction

margin

modred