Lyapunov equation

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In control theory, the discrete Lyapunov equation is of the form

$A X A^{H} - X + Q = 0$

where $Q$ is a Hermitian matrix and $A^H$ is the conjugate transpose of $A$ . The continuous Lyapunov equation is of form

$AX + XA^H + Q = 0$ .

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.

[edit] Application to stability

In the following theorems $A, P, Q \in \mathbb{R}^{n \times n}$ , and $P$ and $Q$ are symmetric. The notation $P>0$ means that the matrix $P$ is positive definite

Theorem (continuous time version). If there exist $P>0$ and $Q>0$ satisfying $A^T P + P A + Q = 0$ then the linear system $\dot{x}=A x$ is globally asymptotically stable. The quadratic function $V(z)=z^T P z$ is a Lyapunov function that can be used to verify stability.

Theorem (discrete time version). If there exist $P>0$ and $Q>0$ satisfying $A^T P A -P + Q = 0$ then the linear system $x(t+1)=A x(t)$ is globally asymptotically stable. As before, $z^T P z$ is a Lyapunov function.

[edit] Computational aspects of solution

The discrete Lyapunov equations can, by using Schur complements, be written as

$\begin{bmatrix} X^{-1} & A \\ A^H & X-Q \end{bmatrix}=0$

or equivalently as

$\begin{bmatrix} X & XA \\ A^HX & X-Q \end{bmatrix}=0$ .

Specialized software is available for solving Lyapunov equations. For the discrete case, the Schur method of Kitagawa (1977) is often used. For the continuous Lyapunov equation the method of Bartels and Stewart (1972) can be used.

[edit] Analytic Solution

There is an analytic solution to the discrete time equations. Define the $vec(A)$ operator as stacking the columns of a matrix $A$ . Further define $kron(A,B)$ as the kronecker product of $A$ and $B$ . Using the result that $vec(ABC)=kron(C^{T},A)vec(B)$ , one has $(I-kron(A,A))vec(X) = vec(Q)$ where $I$ is a conformable identity matrix^[1] One may then solve for $vec(X)$ by inverting or solving the linear equations. To get $X$ , one must just reshape $vec(X)$ appropriately.