Lyapunov invariant set for affine systems Given a linear system $\dot{x}=Ax$ such that the real part of every eigenvalue of $A$ is less than $0$, Lyapunov's equation $A^T P + P A = -Q$ with $Q$ being any suitably sized positive definite matrix gives us an invariant ellipsoid $x^T P x \leq 1$, i.e. for any initial state $x_0$ such that $x_0^T P x_0 \leq 1$ we know that the states $x$ or rather $x(t)$ (making dependency to $t$ explicit) reachable from $x_0$ remain inside the invariant ellipsoid, i.e. $x(t)^T P x(t) \leq 1 ~\forall t \geq t_0$.
How can this be generalized to affine systems $\dot{x}=Ax + b$ where the real part of every eigenvalue of $A$ is less than $0$? Clearly, transforming the affine system into a linear system by extending state vector by $b$ with $\dot{b}=0$ does not help since we will have eigenvalue(s) $0$ which violates our assumption.

As Evgeny suggested, it is enough to translate the coordinate system, i.e. for $x=y-A^{-1} b$ we have $\dot{y} = \dot{x} = A x + b = A (y - A^{-1} b) + b = A y - b + b = A y$ and thus the invariant ellipsoid is $y^T P y = (x+A^{-1}b)^T P (x+A^{-1}b) \leq 1$. Note that $A^{-1}$ exists since we demanded that the real part of every eigenvalue of $A$ is less than $0$ and it holds that a matrix is invertible if and only if it has no eigenvalue which is $0$.
