# Dragon Notes

UNDER CONSTRUCTION
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# Linear Systems:Key Relations

$$\ds\boxed{x(\tau)=\bn{\Phi}(\tau,t)x(t)+\int_{t}^{\tau}\bn{\Phi}(\tau,\lambda)\bn{B}(\lambda)u(\lambda)d\lambda}$$
[1] State-space solution

- Works for both time-invariant and time-variant linear systems
- $$\bn{\Phi}(\tau,\lambda) =$$ State-transition Matrix, w/ properties:
$$\Phi(t,t) = \bn{I}$$
$$\bn{\Phi}(t_1,t_3) = \bn{\Phi}(t_1,t_3)\bn{\Phi}(t_2,t_3),\ \forall t_1,t_2,t_3$$
$$\bn{\Phi}(t,\tau)=e^{\bn{A}(t-\tau)}$$
asm
LTI sys.
$$\bn{\Phi}(s)=\mathcal{L}[\bn{\Phi}(t)]=(s\bn{I}-\bn{A})^{-1}$$

$$\ds\boxed{\bkt{f(t),g(t)}=\int_a^b f(t)g(t)dt = 0 \ra f\perp g, t \in [a,b]}$$
asm

$$(f,g)\in \mathbb{R}$$
$$\ds\boxed{\bkt{f(t),g(t)}=\int_a^b f(t)g^{*}(t)dt = 0 \ra f\perp g, t \in [a,b]}$$
asm

$$(f,g)\in \mathbb{C}$$
[2] Orthogonal Functions