Functions represent physical systems, i.e., a circuit
A system—with inputs, internal state, and outputs.
Full credits to Joe for the tedious LaTeX!
quiz 4 small signal modelling (2021)
Functions represent physical systems, i.e., a circuit
A system—with inputs, internal state, and outputs.
System equations that describe a system are comprised of state space equations and output equations:
State space equations are differential equations describing change in internal state:
$$ \begin{align*} \frac{dx_1}{dt} &= f_1(x_1(t), \dots, x_n(t), u_1(t), \dots, u_m(t)) \\ \vdots \\ \frac{dx_n}{dt} &= f_n(x_1(t), \dots, x_n(t), u_1(t), \dots, u_m(t)) \end{align*} $$
Output equations are normal equations describing the output of a system
$$
\begin{align*} y_1 &= g_1(x_1(t), \dots, x_n(t), u_1(t), \dots, u_m(t)) \\ \vdots \\ y_k &= g_n(x_1(t), \dots, x_n(t), u_1(t), \dots, u_m(t)) \end{align*} $$
Both the state space equations and output equations are functions of the input and the current state
Inputs are assumed to be small perturbations/variations $(\hat u_1(t), \dots, \hat u_m(t))$around equilibrium/centre points $(U_1, \dots, U_m)$
$$
\begin{align*} u_i(t) &= U_i + \hat u_i(t) \\ x_i(t) &= X_i + \hat x_i(t) \\ y_i(t) &= Y_i + \hat y_i(t) \end{align*} $$
Although the system equations may be non linear, they can be linearly approximated if input perturbations are small.
Using the same idea as linear approximations from the previous section, we have
$$ \begin{bmatrix}\frac{d\hat x_1}{dt} \\ \vdots \\ \frac{d\hat x_n}{dt}\end{bmatrix} = J_f\begin{bmatrix}\hat{x_1} \\ \vdots \\ \hat{x_n}\end{bmatrix} + B_f \begin{bmatrix}\hat{u_1}(t) \\ \vdots \\ \hat{u_m}(t)\end{bmatrix}
$$
…where $J_f$ and $B_f$ are Jacobians:
…giving us the output variables: $\begin{bmatrix}\hat y_1 \\ \vdots \\ \hat y_k\end{bmatrix} = J_g\begin{bmatrix}\hat{x_1} \\ \vdots \\ \hat{x_n}\end{bmatrix} + B_g \begin{bmatrix}\hat{u_1}(t) \\ \vdots \\ \hat{u_m}(t)\end{bmatrix}$
Assigned HW: 1,9,13,29,36,43,47,51,52,63