Notation

Plants

$G$ is the plant.

$D_C$ is the controller that feeds input signals to the plant.

Decay is $e^{-\sigma t}$.

flowchart LR
Sigma --> Dc
Dc --> G
G -->|—| Sigma
G --> y(s)
r --> Sigma

Useful Theorems

Final Value Theorem

$$\underset{t\to \infty }{\lim }g(t)=\underset{s\to \infty }{\lim }sG(s)$$

This is only true if the limit exists. The limit can only exist if the Plant is stable.

Initial Value Theorem

$$g(0^{+})=\underset{s\to \infty }{\lim }sG(S)$$

Response

In this class:

• Unit Impulse: $x(t)=\delta (t)$
• Unit Step: $x(t)=1(t)$
• Unit Ramp: $t\cdot 1(t)$
• Sinusoidal: $tbd$

What is the difference between these systems?

$$\begin{array}{r}{G}_1(s)=\frac{2}{s^2+3s+2}=\frac{b(s)}{a(s)},\text{C.E.}a(s)=0\\ G_2(s)=\frac{2}{s^2+2s+2}\\ G_3(s)=\frac{2}{s^2+\sqrt{8s}+2}\end{array}$$

Necessary condition for stability in Higher order ODE: make sure that all coefficients are present and positive.

For second order, to check for stability, we need to get the roots and make sure they are all positive.

$$\begin{array}{r}{G}_1\to s^2+3s+2=0\\ (s+1)(s+2)=0\to s=-1,s=-2\\ G_2\to s^2+2s+2=0\\ (s+1{)}^2+2-1^2\\ (s+1{)}^2+1=0,s=-1\pm j_1\end{array}$$