modred (Control System Toolbox)

Model order reduction

Syntax

rsys = modred(sys,elim)
rsys = modred(sys,elim,'mdc')
rsys = modred(sys,elim,'del')

Description

modred

reduces the order of a continuous or discrete state-space model sys. This function is usually used in conjunction with balreal. Two order reduction techniques are available:

rsys = modred(sys,elim) or rsys = modred(sys,elim,'mdc') produces a reduced-order model rsys with matching DC gain (or equivalently, matching steady state in the step response). The index vector elim specifies the states to be eliminated. The resulting model rsys has length(elim) fewer states. This technique consists of setting the derivative of the eliminated states to zero and solving for the remaining states.
rsys = modred(sys,elim,'del')simply deletes the states specified by elim. While this method does not guarantee matching DC gains, it tends to produce better approximations in the frequency domain (see example below).

If the state-space model sys has been balanced with balreal and the gramians have

small diagonal entries, you can reduce the model order by eliminating the last

states with modred.

Example

Consider the continuous fourth-order model

To reduce its order, first compute a balanced state-space realization with balreal by typing

h = tf([1 11 36 26],[1 14.6 74.96 153.7 99.65])
[hb,g] = balreal(h)
g'

MATLAB returns

ans =
   1.3938e-01   9.5482e-03   6.2712e-04   7.3245e-06

The last three diagonal entries of the balanced gramians are small, so eliminate the last three states with modred using both matched DC gain and direct deletion methods.

hmdc = modred(hb,2:4,'mdc')
hdel = modred(hb,2:4,'del')

Both hmdc and hdel are first-order models. Compare their Bode responses against that of the original model

bode(h,'-',hmdc,'x',hdel,'*')

The reduced-order model hdel is clearly a better frequency-domain approximation of

. Now compare the step responses.

step(h,'-',hmdc,'-.',hdel,'--')

While hdel accurately reflects the transient behavior, only hmdc gives the true steady-state response.

Algorithm

The algorithm for the matched DC gain method is as follows. For continuous-time models

the state vector is partitioned into

, to be kept, and

, to be eliminated.

Next, the derivative of

is set to zero and the resulting equation is solved for

. The reduced-order model is given by

The discrete-time case is treated similarly by setting

Limitations

With the matched DC gain method,

must be invertible in continuous time, and

must be invertible in discrete time.