0.22 I/o and i/s/o relationships in time and frequency

I/o and i/s/o representation of siso linear systems

Definition of state from i/o description

Let $H(s)=D+\frac{\langle p(s)\rangle }{q(s)}$ , $\mathrm{deg}(\langle p\rangle )< \mathrm{deg}(q)$ . Define $w$ so that $\frac{d q}{d t}w(t)=u(t)$ , $(y(t)=\frac{d \langle p\rangle }{d t}w+Du(t))\implies (x^T=\begin{pmatrix}w & w^1 & \dots & w^{(n-1)}\\ \end{pmatrix}\in \mathbb{R}^n)$ , $n$ : degree of $q(s)$ .

Various responses

Zero-input or free response

response due to initial conditions alone.

Zero-state or forced response

response due to input (forcing function) alone (zero initial condition).

Homogeneous solution

general form of free-response (arbitrary initial conditions).

Particular solution

forced response.

Steady-state response

response obtained for large balues of time $T\to$∞ .

Transient response

full response minus steady minus state response.