Qualities of transient response

In this chapter, we explore the qualities of transient response—the response of the system in the interval during which initial conditions dominate.

We focus on characterizing first- and second-order linear systems; not because they’re easiest (they are), but because nonlinear systems can be linearized about an operating point and because higher-order linear system responses are just sums of first- and second-order responses, making “everything look first- and second-order.” Well, many things, at least.

In this chapter, we primarily consider systems represented by single-input, single-output (SISO) ordinary differential equations (also called io ODEs)—with time \(t\), output \(y\), input \(u\), forcing function \(f\), constant coefficients \(a_i,b_j\), order \(n\), and \(m \le n\) for \(n\in\mathbb{N}_0\)—of the form $$\begin{alignat}{8} \label{eq:ioode1} \dfrac{d^n y} {d t^n} & {}+{} & a_{n-1} \dfrac{d^{n-1} y}{d t^{n-1}} & {}+{} & \cdots & {}+{} & a_1 \dfrac{d y} {d t} & {}+{} && a_0 y = f \text{, where} \\ f \equiv b_m \dfrac{d^m u} {d t^m} & {}+{} & b_{m-1} \dfrac{d^{m-1} u}{d t^{m-1}} & {}+{} & \cdots & {}+{} & b_1 \dfrac{d u} {d t} & {}+{} && b_0 u. \label{eq:ioode2} \end{alignat}$$ Note that the forcing function \(f\) is related to but distinct from the input \(u\). This terminology proves rather important.