Lyapunov equation

Lyapunov equation
In control theory, the discrete Lyapunov equation (also known as Stein equation) is of the form
where is a Hermitian matrix and is the conjugate transpose of .
The continuous Lyapunov equation is of the form
.
The Lyapunov equation occurs in many branches of control theory, such as stability analysis and optimal control. This and related equations are named after the Russian mathematician Aleksandr Lyapunov.
Application to stability
In the following theorems , and and are symmetric. The notation means that the matrix is positive definite.
Theorem (continuous time version). Given any , there exists a unique satisfying if and only if the linear system is globally asymptotically stable. The quadratic function is a Lyapunov function that can be used to verify stability.
Theorem (discrete time version). Given any , there exists a unique satisfying if and only if the linear system is globally asymptotically stable. As before, is a Lyapunov function.
Computational aspects of solution
The Lyapunov equation is linear, and so if contains entries 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. For the continuous Lyapunov equation the Bartels–Stewart algorithm can be used.
Analytic solution
Defining the vectorization operator as stacking the columns of a matrix and as the Kronecker product of and , the continuous time and discrete time Lyapunov equations can be expressed as solutions of a matrix equation. Furthermore, if the matrix is "stable", the solution can also be expressed as an integral (continuous time case) or as an infinite sum (discrete time case).
Discrete time
Using the result that , one has
where is a conformable identity matrix and is the element-wise complex conjugate of . One may then solve for by inverting or solving the linear equations. To get , one must just reshape appropriately.
Moreover, if 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
.
Continuous time
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
.
Relationship Between Discrete and Continuous Lyapunov Equations
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 .
Where we've defined . Now we can use the discrete time Lyapunov equation for :
Plugging in our definition for , we get:
Expanding this expression out yields:
Recall that is a small displacement in time. Letting go to zero brings us closer and closer to having continuous dynamics—and in the limit we achieve them. It stands to reason that we should also recover the continuous-time Lyapunov equations in the limit as well. Dividing through by on both sides, and then letting we find that:
which is the continuous-time Lyapunov equation, as desired.
See also
Sylvester equation, which generalizes the Lyapunov equation
Algebraic Riccati equation
Kalman filter
References
Control theory