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Minimal realization or pole-zero cancelation


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


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)

... = minreal(sys,tol,false) and ... = minreal(sys,[],false) disable the verbose output of the function. By default, minreal displays a message indicating the number of states removed from a state-space model sys.


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 cloop.

cloop =
        s (s-1)
  (s-1) (s^2 + s + 1)
Continuous-time zero/pole/gain model.

To cancel the pole-zero pair at s = 1, type

cloopmin = minreal(cloop)

This command produces the following result.

cloopmin =
  (s^2 + s + 1)
Continuous-time zero/pole/gain model.


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.

Introduced before R2006a

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