Using Modal Algorithms

Rigid Body Dynamics

In many cases, a model's jω-axis poles are important to keep after model reduction, e.g., rigid body dynamics of a flexible structure plant or integrators of a controller. A unique routine, modreal, serves the purpose nicely.

modreal puts a system into its modal form, with eigenvalues appearing on the diagonal of its A-matrix. Real eigenvalues appear in 1-by-1 blocks, and complex eigenvalues appear in 2-by-2 real blocks. All the blocks are ordered in ascending order, based on their eigenvalue magnitudes, by default, or descending order, based on their real parts. Therefore, specifying the number of jω-axis poles splits the model into two systems with one containing only jω-axis dynamics, the other containing the non-jω axis dynamics.

rand('state',5678); randn('state',1234); G = rss(30,1,1);
[Gjw,G2] = modreal(G,1); % only one rigid body dynamics
G2.d = Gjw.d; Gjw.d = 0; % put DC gain of G into G2
subplot(211);sigma(Gjw);ylabel('Rigid Body')
subplot(212);sigma(G2);ylabel('Nonrigid Body')

Further model reduction can be done on G2 without any numerical difficulty. After G2 is further reduced to Gred, the final approximation of the model is simply Gjw+Gred.

This process of splitting jω-axis poles has been built in and automated in all the model reduction routines (balancmr, schurmr, hankelmr, bstmr, hankelsv) so that users need not worry about splitting the model.

The following single command creates a size 8 reduced-order model from its original 30-state model:

rand('state',5678); randn('state',1234); G = rss(30,1,1);
[gr,info] = reduce(G); % choose a size of 8 at prompt
bode(G,'b-',gr,'r--')

Without specifying the size of the reduced-order model, a Hankel singular value plot is shown below.

The default algorithm balancmr of reduce has done a great job of approximating a 30-state model with just eight states. Again, the rigid body dynamics are preserved for further controller design.

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Approximating Plant Models — Multiplicative Error Method

Reducing Large-Scale Models

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