modred

Eliminate states from state-space models

Syntax

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

Description

rsys = modred(sys,elim) 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 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. 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' (default): Enforce matching DC gains. The state-space matrices are recomputed as described in Algorithms.
'Truncate': Simply delete X2.

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.

Examples

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Order Reduction by Matched-DC-Gain and Direct-Deletion Methods

Open Live Script

Consider the following continuous fourth-order model.

$h (s) = \frac{s^{3} + 11 s^{2} + 36 s + 26}{s^{4} + 14.6 s^{3} + 74.96 s^{2} + 153.7 s + 99.65} .$

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

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

Examine the gramians.

g'

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');

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

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

The reduced-order model hdel is clearly a better frequency-domain approximation of h. Now compare the step responses.

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

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

Limitations

With the matched DC gain method, A₂₂ must be invertible in continuous time, and I – A₂₂ must be invertible in discrete time.

Algorithms

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

$\begin{array}{l} \dot{x} = A x + B y \\ y = C x + D u \end{array}$

the state vector is partitioned into x₁, to be kept, and x₂, to be eliminated.

$\begin{array}{l} [\begin{matrix} {\dot{x}}_{1} \\ {\dot{x}}_{2} \end{matrix}] = [\begin{matrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] + [\begin{matrix} B_{1} \\ B_{2} \end{matrix}] u \\ y = [\begin{matrix} C_{1} & C_{2} \end{matrix}] x + D u \end{array}$

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

$\begin{array}{l} {\dot{x}}_{1} = [A_{11} - A_{12} A_{22}^{- 1} A_{21}] x_{1} + [B_{1} - A_{12} A_{22}^{- 1} B_{2}] u \\ y = [C_{1} - C_{2} A_{22}^{- 1} A_{21}] x + [D - C_{2} A_{22}^{- 1} B_{2}] u \end{array}$

The discrete-time case is treated similarly by setting

$x_{2} [n + 1] = x_{2} [n]$

Documentation