Forced Linear Systems and State-Space Realizations

Corresponds to lectures: Continuous-time linear systems; Discrete-time systems and standard forms; Realizations and interconnections. Extends the autonomous system $\dot x = Ax$ to the full state-space model $\dot x = Ax + Bu$, $y = Cx + Du$, where an external input $u$ drives the state and an output $y$ is read off from it — and gives the parallel discrete-time model $x_{k+1} = Ax_k + Bu_k$. A realization is the reverse direction: given only an input-output behavior (e.g. a transfer function), find state-space matrices $(A,B,C,D)$ that produce it — not unique, which is exactly why the canonical forms in the next note matter. Covers how realizations interconnect (series, parallel, feedback) to build complex systems from simple ones.

Outline (TODO — flesh out each)

• Continuous-time linear systems — the state-space model
• Discrete-time systems and standard forms
• Realizations — from input-output behavior to state-space
• Interconnections of state-space systems