What is the format for the state-space form?
$$ \dot{x} = \mathbf{A} x + \mathbf{B} u $$
What is the name of $ \mathbf{A} $ in the state-space form?
The Jacobian Matrix
For non-linear systems, what format must the state-space form Jacobian matrix follow for a 2 x variable system?
$$ A = \begin{bmatrix} \frac{\partial \dot{x_1}}{\partial x_1} & \frac{\partial \dot{x_1}}{\partial x_2} \\ \frac{\partial \dot{x_2}}{\partial x_1} & \frac{\partial \dot{x_2}}{\partial x_2} \end{bmatrix} $$
In the state-space form, what is matrix $ \mathbf{B} $ often referred to as?
The control influence matrix, or external inputs to a system.
The Laplace Transform is a change between which two domains?
The time and frequency domains.
How do you find the poles in a Laplace Transform?
By setting the denominator equal to zero and solving for $ s $.
What does a pole in a Laplace Transform represent?
It represents where the transform goes to infinity.
What is the Laplace Transform for an impulse function, $ \delta (t) $?
$$ \mathcal{L} \lbrace \delta (t) \rbrace = A $$
What is the Laplace Transform for a step function, $ u(t) $?
$$ \mathcal{L} \lbrace u(t) \rbrace = \frac{1}{s} $$
What is the Laplace Transform for an exponential function, $ A u(t) e^{-at}, a>0 $?
$$ \mathcal{L} \lbrace A u(t) e^{-at} \rbrace = \frac{A}{s+a} $$
What is the Laplace Transform for a ramp function, $ A u(t) t $?
$$ \mathcal{L} \lbrace A u(t) t \rbrace = \frac{A}{s^2} $$
What is the Laplace Transform for a monomial function, $ A u(t) t^k $?
$$ \mathcal{L} \lbrace A u(t) t^k \rbrace = A \frac{k!}{s^{k+1}} $$
What is the Laplace Transform for a sinusoidal function, $ A u(t) \sin(\omega t) $?
$$ \mathcal{L} \lbrace A u(t) \sin(\omega t) \rbrace = A \frac{\omega}{s^2 + \omega^2} $$