# Riccati & Lyapunov equations

We know that

Lyapunov equation: $A^TP + PA + Q = 0$
Algebraic Riccati equation: $A^TP + PA + Q + PBR^{-1}B^TP= 0$

It seems that the difference between the two lies in $B = 0$ (zero input) in Lyapunov Eq and both are infinite horizon in the case above.

Is there any other engineering-sense difference (not mathematics) between the two Eqs?

Thanks!

• Riccati equation is nonlinear (if $B\ne 0$), Lyapunov equation is linear. – daw Jul 7 '14 at 9:02

Lyapunov equation is used for the stability analysis of a relaxed system, I.e. no input signal. There exists a unique positive definite $P$ for any given positive definite $Q$ if and only if the system $\dot{x}=Ax$ is globally asymptotically stable. This means we can make $\dot{V}$ arbitrarily small where $V$ is the quadratic Lyapunov function $V(x)=x^T Px$.
Replace $A$ by $A-BK$ with $K=R^{-1}B^\top P$ in the Lyapunov equation yields the algebraic Riccati equation. Hope this helps.