State-space model

In small signal analysis, a system model consists of $n$ coupled first-order non-linear differential equations, with $n$ state variables, $m$ excitation input variables, and $k$ output variables. We call the differential equations state space equations.

$$\frac{d{x}_1}{dt}=f_1(x_1(t)\dots x_n(t),u_1(t)\dots u_m(t))$$ $$\frac{d{x}_n}{dt}=f_n(x_1(t)\dots x_n(t),u_1(t)\dots u_m(t))$$

Observe that the first-order time derivatives of the state variables are on the left side. The functions on the right have inputs of the state variables and the excitation inputs.

We also define $k$ output equations that depend on the $n$ state variables and $m$ input signals:

$$y_1=g_1(x_1(t)\dots x_n(t),u_1(t)\dots u_m(t))$$ $$y_k=g_k(x_1(t)\dots x_n(t),u_1(t)\dots u_m(t))$$

And we define system equations as an equation set comprised of state space equations (that describe the dynamics of the system) and output equations.