Standard Dynamic Elements, Poles & Second-Order Response

Factoring a transfer function into standard elements turns coefficients into mechanisms. A pole contributes a mode; a zero reshapes how modes appear in a particular input/output channel; gain sets scale; delay contributes phase without a finite-dimensional pole-zero factorization. The same factor can be read in time, frequency, and complex-plane views.

Left-half-plane poles decay, right-half-plane poles grow, and imaginary parts create oscillation. Distance from the imaginary axis sets the exponential envelope.

The elementary factors #

An ideal differentiator $Ks$ is improper: its gain grows without bound and it amplifies arbitrary high-frequency noise. Real derivative action includes at least one pole. A pure delay is causal but infinite-dimensional; rational Padé approximations are analysis devices whose added poles and zeros must not be mistaken for plant modes.

First-order response #

For

$$G(s)=\frac{K}{\tau s+1},$$

the zero-state response to unit step is

$$y(t)=K\left(1-e^{-t/\tau}\right),\qquad t\ge0.$$

At $t=\tau$, the response has completed $1-e^{-1}\approx63.2\%$ of its final change. At $3\tau$ it is about $95.0\%$, and at $4\tau$ about $98.2\%$. These are properties of this single-pole monotone response, not universal definitions of settling time.

The pole is $p=-1/\tau$. Moving it left speeds the exponential decay. Changing $K$ changes final scale but not the pole for an open-loop element.

The canonical second-order mode #

Consider

$$G(s)=\frac{\omega_n^2}{s^2+2\zeta\omega_ns+\omega_n^2}.$$

Its poles are

$$s_{1,2}=-\zeta\omega_n\pm\omega_n\sqrt{\zeta^2-1}.$$

• $\zeta>1$: two real negative poles, overdamped.
• $\zeta=1$: repeated real pole, critically damped.
• $0<\zeta<1$: complex pair with decay rate $\sigma=\zeta\omega_n$ and damped frequency $\omega_d=\omega_n\sqrt{1-\zeta^2}$.
• $\zeta=0$: undamped oscillation in the ideal model.
• $\zeta<0$: growing oscillation or divergence.

For $0<\zeta<1$, unit-step percentage overshoot is

$$M_p=100\exp\!\left(-\frac{\pi\zeta}{\sqrt{1-\zeta^2}}\right),$$

and peak time is

$$t_p=\frac{\pi}{\omega_d}.$$

The familiar approximation $t_s\approx4/(\zeta\omega_n)$ corresponds to a roughly 2% exponential envelope for a clean dominant second-order response. It is not reliable with important zeros, extra poles, delay, nonzero initial conditions, or a different settling definition.

Zeros shape what a mode looks like #

Poles determine the natural modes of a minimal transfer function; zeros determine how input and output combine those modes. A left-half-plane zero can speed an initial response and add phase lead. A right-half-plane zero creates nonminimum-phase behavior: a step response may initially move opposite its final direction, and the zero imposes bandwidth/performance limits.

For example,

$$G(s)=\frac{1-s/z}{(s+p_1)(s+p_2)},\qquad z>0,$$

has a zero at $+z$. Increasing loop bandwidth toward or beyond $z$ demands an increasingly aggressive controller and worsens undershoot, sensitivity, or control effort. Cancelling the zero with an unstable controller pole is not a valid internal-stability solution.

Pole-zero cancellation is a structural claim #

If

$$G(s)=\frac{s+a}{(s+a)(s+b)},$$

the input/output expression simplifies to $1/(s+b)$. That cancellation is exact only if the pole and zero truly coincide and the cancelled internal mode is stable and irrelevant to all internal signals. Parameter error leaves a slow residual pair; cancelling an unstable pole can hide an internally divergent state while the selected output appears bounded.

Treat near cancellation as a robustness problem. Inspect a state-space realization, perturb the parameters, and check internal signals.

Relative degree and causality #

For a rational transfer function, relative degree is denominator degree minus numerator degree. A causal finite-dimensional physical plant is normally proper (relative degree at least zero), and many energy-storage plants are strictly proper. An improper inverse or controller requires future information or unbounded bandwidth unless regularized by added dynamics.

High-frequency asymptotes therefore carry physical meaning. Each excess pole eventually contributes about $-20$ dB/decade and $-90^\circ$ phase; each left-half-plane zero contributes the opposite trends. Those asymptotes become the building blocks of Bode analysis.

A reading sequence for any transfer function #

1. Check units, sign, properness, delay, and the zero-frequency gain if finite.
2. Factor stable/unstable poles, minimum/nonminimum-phase zeros, and integrators.
3. Identify dominant modes relative to the required bandwidth—not merely the poles closest to the origin.
4. Predict initial direction, final value, decay envelope, oscillation, and high-frequency slope.
5. Compare those predictions with computed time and frequency responses.
6. Recheck hidden state and uncertainty before accepting cancellations or reductions.

Standard elements are a vocabulary for reasoning, not permission to replace a plant by whichever low-order curve is convenient.