Chapter 6 State-Space Design
Xmath Control Design Module 6-30 ni.com
or the following for the discrete case:
(6-16)
These results derive from the Lyapunov method of stability analysis for
linear systems. Steady state means that at some point the states no longer
change. The derivative term approaches zero in the large for continuous
systems, and xk+1 =xk for discrete-time ones. The state value vector x for
which this is true is defined as the equilibrium state. As stated in [Oga70],
a unique equilibrium state exists for systems with a nonsingular A matrix,
whereas infinitely many equilibrium states exist if A is singular. A system
is described as asymptotically stable if the state values approach the
equilibrium state over time, no matter what value of x one started with.
Such systems will always satisfy the following: for any positive-definite
matrix Q, a positive definite matrix X can be found satisfying in the
Xequation (Equation 6-15) for the continuous case and Yequation
(Equation 6-16) for the discrete case.
Lyapunov equations also can be used to compute system controllability
and observability grammians, which play an important role in internal
balancing and model reduction. This application will be discussed further
in the Balancing a Linear System section.
lyapunov( )X = lyapunov(A,B,{C, discrete})
The lyapunov( ) function provides a solution to both the discrete and
continuous-time Lyapunov equations. When called with three inputs
(A,B,C), it solves the general continuous Lyapunov equation
(Equation 6-10); when called with two inputs (A,C), it solves the special
Lyapunov equation (Equation 6-11). When called with two inputs (A,B)
and the {discrete} keyword, it solves the discrete Lyapunov equation
(refer to Equation 6-12). For examples of discrete, continuous, and special
Lyapunov equation solutions, refer to Example 6-10.
Algorithm
The algorithm for lyapunov( ) uses the Schur decomposition to convert
A and B to upper triangular form, then finds the Lyapunov equation
solution a column at a time by solving. lyapunov( ) warns the user if the
eigenvalues of (A + eye(A)) are close to –1, in which case singularity may
occur and cause the function to terminate. Furthermore, if any combination
ykCxkDuk
+=
YCXC'DQD'+=
x
·