Chapter 6 State-Space Design
© National Instruments Corporation 6-37 Xmath Control Design Module
and compare the condition numbers of the balanced system’s grammians:
WcB=lyapunov(Ab,Bb*Bb');
WoB=lyapunov(Ab',Cb'*Cb);
condition(WcB)
ans (a scalar) = 12.7394
condition(WoB)
ans (a scalar) = 12.7394
The condition numbers are now much smaller, and they are equal,
indicating that the system is now equally well conditioned in terms
of its controllability and observability.
Modal Form of a System
The modes of a state-space system are defined as corresponding to the
eigenvalues of the system’s A matrix. The modes of a system are distinct
from the states of a system; because a given system can be arbitrarily
transformed, the states can be arbitrarily assigned. The modes, on the other
hand, do not change from realization to realization of a given system.
The modal decomposition of a system can be obtained mathematically
through a Laplace transform, partial fraction decomposition, and eigen
decomposition as shown in [Kai80]. The key advantage of a modal
decomposition is that it provides a means by which large systems can
be represented as a parallel combination of first-order systems. In addition,
the modal decomposition of a given system representation is often better
conditioned numerically.
The modal form is particularly useful with structured dynamic systems
whose poles primarily occur as complex pairs. When a system model has
been converted to modal form, it can be reduced to focus attention on the
particular modes whose dynamics are of interest.

modal( )

[SysMod, T] = modal(Sys)
The modal( ) function uses eigenvalue decomposition to find the Jordan
form of the system matrix A (all eigenvalues on the diagonal). This
approach is appropriate for models without repeated eigenvalues; modal
decomposition of a system with repeated eigenvalues is numerically
unreliable. If a system with repeated or very closely spaced eigenvalues is
passed to modal( ), a warning appears noting that the results may not be