Specify the height-to-width ratio of the plot:

Label the axes:

Specify the axes origin:

To obtain coordinates, select the graphics and press the period key:

Obtain the coordinates in terms of the damping ratio and natural frequency:

Obtain the coordinates in terms of the damping ratio and natural frequency for a discrete-time system:

The equations to map the coordinates to the damping ratio and natural frequency:

The coordinates tool displays the coordinates in terms of the damping ratio and natural frequency:

For discrete-time systems, Epilog automatically shows a unit circle:

Customize the circle:

Draw a frame around the plot:

Specify labels for the frame:

Label the plot:

Specify the plot range:

Draw polar grid lines:

The damping ratio and natural frequency:

The pole angle:

Draw polar grid lines at the damping ratio and natural frequency of the system:

By default, is used for poles and for zeros:

Specify custom markers:

The SamplingPeriod value determines the domain of a system given as an expression:

A continuous-time system:

Evaluate the response characteristics of a continuous-time system from the dominant pole locations:

A stable system:

A stable system with some overshoot:

A system with undamped oscillations:

An unstable system:

Evaluate the response characteristics of a discrete-time system from the dominant pole locations:

A stable discrete-time system:

A stable discrete-time system with some overshoot:

A discrete-time system with undamped oscillations:

An unstable discrete-time system:

Use PoleZeroPlot along with RegionPlot to see if the poles and zeros are in specific regions:

Analyze the modes of the longitudinal dynamics of an aircraft:

The pole zero plot:

The poles further in the left half-plane are those of the short period mode:

The response converges to the equilibrium:

The poles in the right half-plane and close to the imaginary axis are those of the Phugoid mode:

The response is a divergent oscillation:

Pole-zero plots can be used to analyze the effect of a parameter on a system's transient response. Analyze the effect of the pulse ratio on the response of a DC-DC boost converter:

A PWM-switch model of the circuit:

Construct three state-space models with different pulse ratios:

The poles of the system are closer to the origin as the pulse ratio gets larger:

Increasing the pulse ratio increases the output voltage, and the oscillatory response is because the poles are on the imaginary axis:

Pole-zero plots are useful for pole-placement design. Shape the response of a mass-spring damper by interactively manipulating the pole locations:

A model of the system:

Shape the system's response by manipulating the locations of the poles:

Pole-zero plots are useful in the LQ design to compare the poles of the open- and closed-loop systems. Compare the open- and closed-loop poles of a mixing tank:

A model of the mixing tank:

The open-loop response to nonzero initial conditions is slow:

This is because its poles are close to the unit circle:

Design an LQ controller that improves the tank's response and obtain the closed-loop system:

The closed-loop system is faster:

This is because the closed-loop system's poles are closer to the origin:

The poles correspond to the modes of a system:

The pole-zero plot of the coupled two-mass spring system: »

Its poles and eigenvectors:

The first pair of poles and eigenvectors generates the exponentially decaying oscillatory mode:

In this mode, the motions of the two masses are decaying, oscillatory and out of phase:

The third pole is a purely exponentially decaying mode:

To excite only this mode, the masses are given a set of equal displacements and velocities:

In this mode, the response decays exponentially:

The fourth pole that is at the origin is a purely translational mode:

To excite only this mode, both masses are given equal displacements and no velocities:

This mode is purely translational, in which the two masses move as a single unit:

The general motion of a system can be constructed from its modes:

The modes of the system:

A state response:

The contribution of the modes:

The position of the first mass is the sum total of the individual modes:

So is the position of the second mass:

Poles on the stability boundary result in resonant behavior:

The sinusoid input at the resonant frequency results in a resonant response:

This can also be seen in the Bode plot:

A zero causes blocking:

A system with a zero in the r.h.p at :

The exponential signal is blocked:

The zero does not cause blocking due to cancellation:

A system with a zero and a pole in the r.h.p at :

The exponential signal is not blocked:

The response is same as that of the stable system with unstable pole and zero canceled:

A pair of complex zeros causing blocking:

Its pole-zero plot:

Frequencies close to the zero are partially blocked, while the one matching the zero is completely blocked:

This can also be seen on the Bode plot:

For continuous-time systems, a zero with positive real part indicates an inverse response system:

For discrete-time systems, a zero outside the unit circle indicates an inverse response system:

The open-loop poles and zeros from a RootLocusPlot:

PoleZeroPlot is a special case of RootLocusPlot:

PoleZeroPlot is a special case of ComplexListPlot:

The poles and zeros using TransferFunctionPoles and TransferFunctionZeros:

The pole-zero plot using ComplexListPlot and directly with PoleZeroPlot:

A Butterworth filter has no zeros and has all its poles in the left half-plane and equally spaced on the circle with radius equal to the cutoff frequency:

A Chebyshev type 1 filter has no zeros and has all its poles in the left half-plane and equally spaced on an ellipse:

A Chebyshev type 2 filter has all its zeros as conjugate pairs on the imaginary axis:

The pole-zero plot of an elliptic filter:

A Bessel filter:

A FIR filter is of the form :

All its finite poles are at :

An IIR filter is of the form :

It has no pole at :

The pole count of a causal filter is greater than or equal to its zero count:

Its response to an input sequence:

The response begins with the input:

The pole count of an acausal filter is less than its zero count:

Its response to an input sequence:

The response begins before the input:

A system is stable if its region of convergence (ROC) includes the imaginary axis:

The system's impulse response:

The response is stable:

Its ROC is to the right of the rightmost pole:

The ROC includes the left-hand side of the -plane because it is stable:

A system is unstable if its region of convergence (ROC) does not include the imaginary axis:

The system's impulse response:

The response is unstable:

Its ROC:

The ROC does not include the left-hand side of the -plane because it is not stable:

A discrete-time system is stable if its region of convergence (ROC) includes the unit circle:

The system's impulse response:

The response is stable:

Its ROC:

The ROC includes the unit circle because the system is stable:

A discrete-time system is unstable if its region of convergence (ROC) does not include the unit circle:

Its impulse response:

Its response is unstable:

Its ROC:

Its ROC does not include the unit circle because it is unstable:

A filter's phase is linear if its poles or zeros are symmetric about the unit circle:

A function to generate poles that are symmetric about the unit circle and conjugate:

The pole-zero plot:

The phase is linear:

A filter is invertible if all its zeros are inside the unit circle:

The numerator polynomial:

The invertible filter:

Its inverse:

The inverse filter is stable:

The inverse filter reverses the filter’s zeros and poles:

The inverse filter in series with the filter leaves the input signal unchanged:

A filter is minimum phase if it has the same number of poles and zeros and they are all inside the unit circle:

The numerator polynomial:

The denominator polynomial:

The minimum phase filter:

Its inverse:

The filter and its inverse are stable, as their poles are within the unit circle:

An all-pass filter:

A non-minimum-phase filter:

They have the same magnitude:

But the minimum-phase filter has the lower phase:

An MAProcess[{b₁,…,b_q},…] has q poles at the origin and q zeros:

An ARProcess[{a₁,…,a_p},…] has p zeros at the origin and p poles:

An ARMAProcess[{a₁,…,a_p},{b₁,…,b_q},…] has n poles and n zeros, where n=Max[p,q]:

An ARIMAProcess[{a₁,…,a_p},d,{b₁,…,b_q},…]:

Compared to an ARMAProcess, it additionally has d poles at 1 and d zeros at 0:

A SARMAProcess[{a₁,…,a_p},{b₁,…,b_q},{s,{α₁,…,α_m},{β₁,…,β_r}},…]:

It has n poles and n zeros, where n=Max[p+s m,q+s r]:

The pole-zero plot:

A SARIMAProcess[a,d,b,{s,{α,δ,β}},…]:

Compared to a SARMAProcess, it additionally has d+δ poles at 1 and d+δ zeros at 0:

A time series is not invertible if the zeros of the transfer function lie outside the unit circle:

Check the invertibility of the process:

Since there are no zeros on the unit circle, the time series admits an invertible representation:

The zeros of the transfer function have been reflected over the unit circle:

This can be seen on the pole-zero plot: