The Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical systems.
, while the continuous-time Lyapunov equation is In the following theorems
Theorem (continuous time version).
is globally asymptotically stable.
is a Lyapunov function that can be used to verify stability.
Theorem (discrete time version).
is globally asymptotically stable.
The Lyapunov equation is linear; therefore, if
entries, the equation can be solved in
time using standard matrix factorization methods.
However, specialized algorithms are available which can yield solutions much quicker owing to the specific structure of the Lyapunov equation.
For the discrete case, the Schur method of Kitagawa is often used.
[3] For the continuous Lyapunov equation the Bartels–Stewart algorithm can be used.
as stacking the columns of a matrix
, the continuous time and discrete time Lyapunov equations can be expressed as solutions of a matrix equation.
is "stable", the solution can also be expressed as an integral (continuous time case) or as an infinite sum (discrete time case).
by inverting or solving the linear equations.
is stable (in the sense of Schur stability, i.e., having eigenvalues with magnitude less than 1), the solution
can also be written as For comparison, consider the one-dimensional case, where this just says that the solution of
is Using again the Kronecker product notation and the vectorization operator, one has the matrix equation where
denotes the matrix obtained by complex conjugating the entries of
Similar to the discrete-time case, if
is stable (in the sense of Hurwitz stability, i.e., having eigenvalues with negative real parts), the solution
can also be written as which holds because For comparison, consider the one-dimensional case, where this just says that the solution of
is We start with the continuous-time linear dynamics: And then discretize it as follows: Where
indicates a small forward displacement in time.
Substituting the bottom equation into the top and shuffling terms around, we get a discrete-time equation for
Now we can use the discrete time Lyapunov equation for
It stands to reason that we should also recover the continuous-time Lyapunov equations in the limit as well.
which is the continuous-time Lyapunov equation, as desired.