Eliminate states from state-space models
rsys = modred(sys,elim)
rsys = modred(sys,elim,'method')
reduces the order of a continuous or discrete state-space model
rsys = modred(
by eliminating the states found in the vector
elim. The full state
vector X is partitioned as X =
[X1;X2] where X1 is the
reduced state vector and X2 is discarded.
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 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
'MatchDC'(default): Enforce matching DC gains. The state-space matrices are recomputed as described in Algorithms.
'Truncate': Simply delete X2.
'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
Order Reduction by Matched-DC-Gain and Direct-Deletion Methods
Consider the following continuous fourth-order model.
To reduce its order, first compute a balanced state-space realization with
h = tf([1 11 36 26],[1 14.6 74.96 153.7 99.65]); [hb,g] = balreal(h);
Examine the gramians.
ans = 1×4 0.1394 0.0095 0.0006 0.0000
The last three diagonal entries of the balanced gramians are relatively small. Eliminate these three least-contributing states with
modred, using both matched-DC-gain and direct-deletion methods.
hmdc = modred(hb,2:4,'MatchDC'); hdel = modred(hb,2:4,'Truncate');
hdel are first-order models. Compare their Bode responses against that of the original model.
The reduced-order model
hdel is clearly a better frequency-domain approximation of
h. Now compare the step responses.
hdel accurately reflects the transient behavior, only
hmdc gives the true steady-state response.
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