Control System Toolbox™

modred - Model order reduction

Syntax

modred rsys = modred(sys,elim,'method')

Description

modred reduces the order of a continuous or discrete state-space model sys by eliminating the states found in the vector elim. The full state vector X is partitioned as X = [X1;X2] where X2 is to be discarded, and the reduced state is set to Xr = X1+T*X2 where T is chosen to enforce matching DC gains (steady-state response) between sys and rsys.

elim can be a vector of indices or a logical vector commensurate with X where true values mark states to be discarded. This function is usually used in conjunction with balreal. Use balreal to first isolate states with negligible contribution to the I/O response. If sys has been balanced with balreal and the vector g of Hankel singular values has M small entries, you can use modred to eliminate the corresponding M states. For example:

[sys,g] = balreal(sys)  % Compute balanced realization
elim = (g<1e-8)         % Small entries of g are negligible states
rsys = modred(sys,elim) % Remove negligible states

rsys = modred(sys,elim,'method') also specifies the state elimination method. Choices for 'method' include

'MatchDC': Enforce matching DC gains (default)
'Truncate': Simply delete X2 and sets Xr = X1.

The 'Truncate' option tends to produces a better approximation in the frequency domain, but the DC gains are not guaranteed to match.

If the state-space model sys has been balanced with balreal and the grammians have m small diagonal entries, you can reduce the model order by eliminating the last m 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'

These commands produce the following result.

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

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

hmdc = modred(hb,2:4,'MatchDC')
hdel = modred(hb,2:4,'Truncate')

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

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

The reduced-order model hdel is clearly a better frequency-domain approximation of h(s). 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 x₁, to be kept, and x₂, to be eliminated.

Next, the derivative of x₂ is set to zero and the resulting equation is solved for x₁. The reduced-order model is given by

The discrete-time case is treated similarly by setting

Limitations