Poles and Zeros: Location and Significance

For a rational transfer function written in factored form,

$$G(s)=\frac{N(s)}{D(s)} =K\frac{(s-z_1)(s-z_2)\cdots(s-z_m)} {(s-p_1)(s-p_2)\cdots(s-p_n)},$$

the roots $z_i$ and $p_i$ expose much of the input-output dynamics without solving the differential equation explicitly.

The zeros are roots of the numerator:

$$N(z_i)=0.$$

For an uncancelled zero, $G(z_i)=0$. Zeros weight and reshape the forced response, but zeros alone do not determine stability.

The poles are roots of the denominator:

$$D(p_i)=0.$$

An uncancelled pole is a singularity of $G(s)$ and contributes a natural mode of the form $e^{p_i t}$; a repeated pole contributes polynomial factors such as $t e^{p_i t}$.

Second-Order Pole Geometry

The characteristic polynomial of the standard second-order system is

$$s^2+2\zeta\omega_n s+\omega_n^2=0.$$

For $0<\zeta<1$, its poles are

$$p_{1,2}=-\zeta\omega_n\pm j\omega_d, \qquad \omega_d=\omega_n\sqrt{1-\zeta^2}.$$

Thus the radial distance from the origin is $\omega_n$, the horizontal distance to the imaginary axis is $\zeta\omega_n$, and the angle measured from the negative real axis satisfies $\cos\theta=\zeta$.

Geometry of an underdamped second-order pole pair in the $s$-plane.

Pole Location and Stability

For a finite-dimensional, proper, continuous-time rational transfer function, BIBO stability requires every uncancelled pole to lie strictly in the open left half-plane. Pole location determines both the mode shape and its decay or growth.

Pole location	Natural-response term	Conclusion
Negative real axis	Decaying exponential $e^{-at}$, $a>0$	Asymptotically decaying
Complex pair in LHP	Decaying sinusoid $e^{-\sigma t}\sin\omega t$	Asymptotically decaying
Positive real axis	Growing exponential $e^{at}$	Unstable
Complex pair in RHP	Growing sinusoid $e^{\sigma t}\sin\omega t$	Unstable
Simple pole at origin	Constant natural mode; step input to an integrator gives a ramp	Not BIBO stable
Simple pair on $j\omega$ axis	Sustained sinusoidal natural mode	Not BIBO stable; marginal only in the internal-mode sense
Repeated pole on $j\omega$ axis	Polynomially growing oscillation	Unstable

Significance of continuous-time pole locations

Note

Key Point — BIBO stability versus marginal modes

A simple, nonrepeated imaginary-axis eigenvalue can be called marginally stable for the zero-input state response when all remaining eigenvalues are in the LHP. Its impulse response is not absolutely integrable, however, so the corresponding transfer function is not BIBO stable. A pole at the origin likewise represents integration, not BIBO stability.

Pole regions, dominant-pole distance, representative zeros, and their response signatures.

Dominant Poles

In a stable system, poles nearest the imaginary axis have the smallest decay rates and usually dominate the long transient. Therefore:

• poles far to the left decay rapidly;

• poles near the imaginary axis decay slowly;

• a dominant complex-conjugate pair largely sets ringing, overshoot, and settling time.

For the pair $-\zeta\omega_n\pm j\omega_d$, the envelope decays as $e^{-\zeta\omega_n t}$, so the distance $\zeta\omega_n$ from the imaginary axis sets the decay rate.

Tip

Exam Focus — When a dominant-pole approximation is valid

The nearest pole is not automatically dominant. Its residue must be appreciable, it must not be nearly cancelled by a zero, and neglected poles should normally decay several times faster. A common engineering check is that nondominant real parts are at least about five times farther left.

How Zeros Shape the Response

Zeros alter modal weights and frequency response rather than creating unforced modes. Depending on location and relative strength, a zero can speed the initial response, increase overshoot, or partially suppress a pole’s visible contribution. A right-half-plane zero is a non-minimum-phase zero; it commonly produces an initial motion in the wrong direction (undershoot or inverse response) and limits achievable feedback bandwidth.

Note

Key Point — Minimum-phase qualification

A stable rational system is minimum phase only when its zeros are also in the open LHP (with delay and relative-degree qualifications). Moving an RHP zero to the LHP preserves magnitude on the $j\omega$ axis but changes phase and transient direction.

Pole-Zero Cancellation

Consider

$$G(s)=\frac{s+2}{s^2+5s+6} =\frac{s+2}{(s+2)(s+3)}.$$

The displayed numerator zero is $-2$ and the denominator poles are $-2$ and $-3$. Exact algebraic cancellation gives the reduced input-output model

$$G_{\mathrm{red}}(s)=\frac{1}{s+3}.$$

Note

Law — Cancellation does not remove physical dynamics

Exact cancellation is a property of an ideal model. Parameter uncertainty leaves a nearby pole-zero pair, and an internal mode can remain hidden if it is uncontrollable or unobservable at the chosen terminals. In particular, never rely on a controller zero to cancel an unstable plant pole: the external transfer expression may look stable while internal signals remain unstable.

[!NOTE]

Example — Poles, zeros, and the stability qualification

Find the poles and zeros of

$$G(s)=\frac{5(s+4)}{s(s+2)(s^2+2s+5)}$$

and comment on stability.

The zero is $s=-4$. The poles are

$$s=0,\qquad s=-2,\qquad s^2+2s+5=0 \Rightarrow s=-1\pm j2.$$

The LHP poles decay, but the pole at the origin makes this open-loop transfer function an integrator and hence not BIBO stable. Calling the origin mode “marginal” is valid only for a simple internal zero-input mode, not as a BIBO claim. These open-loop poles also do not by themselves decide closed-loop stability; the closed-loop characteristic equation must be formed and tested. For example, under negative unity feedback it would be

$$1+G(s)=0 \quad\Longleftrightarrow\quad s^4+4s^3+9s^2+15s+20=0,$$

whose Routh first column changes sign, so that particular closed loop is unstable.

[!TIP]

Exam Focus — Fast pole-zero reading

Factor the numerator and denominator, mark zeros with $\circ$ and poles with $\times$, check the RHP first, then the imaginary axis, and only then identify dominant LHP poles. State explicitly whether the poles belong to an open-loop or a closed-loop transfer function.