Observer design by pole placement

Error dynamics
Our goal is to choose L, so that the state error e(t) → 0 as t → ∞.
The dynamics of e is
σe = σ(x −̂x)
= Ax + Bu − Âx − Bu + L(y − ŷ)
= A(x −̂x)+LC(x −̂x)
= (A+LC) e
} {{ }
A o
i.e., e ∈ B ss (A o ) — an autonomous LTI system.
Therefore, e(t) → 0 as t → ∞ is equivalent to stability of B ss (A o ).
ELEC 3035 (Part I, Lecture 7) Observer design 5 / 12