Stability analysis (pole-zero plot) in Signal Processing

Introduction

We use pole-zero plots to check if a system is stable. It helps us see if the system will behave well or go crazy over time.

When designing filters to make sure they don't produce unwanted noise.

When checking if a control system will keep a machine steady.

When analyzing audio effects to ensure they don't cause feedback.

When studying signals to understand their long-term behavior.

Syntax

Signal Processing

import matplotlib.pyplot as plt
from scipy import signal

# Define system with numerator (zeros) and denominator (poles)
sys = signal.TransferFunction(num, den)

# Get poles and zeros
zeros, poles = signal.tf2zpk(num, den)

# Plot poles and zeros
plt.scatter(zeros.real, zeros.imag, marker='o', label='Zeros')
plt.scatter(poles.real, poles.imag, marker='x', label='Poles')
plt.axvline(0, color='gray', lw=1)
plt.axhline(0, color='gray', lw=1)
plt.xlabel('Real')
plt.ylabel('Imaginary')
plt.title('Pole-Zero Plot')
plt.legend()
plt.grid(True)
plt.show()

The poles are values where the system output can become infinite.

For stability, all poles must be on the left side of the plot (negative real part).

Examples

This system is stable because poles are at -1 (left side).

Signal Processing

num = [1]
den = [1, 2, 1]
# Simple stable system with poles at -1 and -1

This system is unstable because one pole has positive real part.

Signal Processing

num = [1]
den = [1, -1, 0.5]
# System with a pole on the right side (unstable)

Sample Program

This code shows the poles and zeros of a simple system and plots them. You can see if the poles are on the left side, meaning the system is stable.

Signal Processing

import matplotlib.pyplot as plt
from scipy import signal

# Define numerator and denominator of transfer function
num = [1]
den = [1, 2, 1]

# Get zeros and poles
zeros, poles = signal.tf2zpk(num, den)

# Print poles and zeros
print('Zeros:', zeros)
print('Poles:', poles)

# Plot pole-zero plot
plt.scatter(zeros.real, zeros.imag, marker='o', label='Zeros')
plt.scatter(poles.real, poles.imag, marker='x', label='Poles')
plt.axvline(0, color='gray', lw=1)
plt.axhline(0, color='gray', lw=1)
plt.xlabel('Real')
plt.ylabel('Imaginary')
plt.title('Pole-Zero Plot')
plt.legend()
plt.grid(True)
plt.show()

Important Notes

Poles on the right side (positive real part) mean the system is unstable.

Poles on the imaginary axis mean the system is marginally stable (may oscillate).

Zeros do not affect stability but shape the system response.

Summary

Poles and zeros help us understand system behavior.

Stable systems have all poles with negative real parts.

Pole-zero plots visually show stability and system characteristics.

Practice

1. What does it mean for a system to be stable in terms of its poles on the pole-zero plot?

easy

A. All poles must lie in the left half of the complex plane (negative real parts).

B. All poles must lie on the imaginary axis.

C. All zeros must lie in the right half of the complex plane.

D. Poles and zeros must be equal in number.

Solution

1. Step 1: Understand pole location and stability

   Stability depends on poles, not zeros. Poles with negative real parts mean the system's response decays over time.

2. Step 2: Identify correct pole location for stability

   Poles in the left half-plane (negative real parts) ensure the system is stable.

3. Final Answer:

   All poles must lie in the left half of the complex plane (negative real parts). -> Option A

4. Quick Check:

   Stable system = poles with negative real parts [OK]

Hint: Stable system means poles left of imaginary axis [OK]

Common Mistakes:

• Confusing zeros with poles for stability
• Thinking poles on imaginary axis are stable
• Assuming zeros affect stability directly

2. Which of the following is the correct Python syntax to plot poles and zeros using the scipy.signal library?

easy

A. signal.pzmap(system, plot=True)

B. signal.pzplot(system)

C. signal.pzplot(system, plot=True)

D. signal.pzmap(system, plot=False)

Solution

1. Step 1: Recall scipy.signal function for pole-zero plot

   The function to plot poles and zeros is pzmap, not pzplot.

2. Step 2: Check correct usage with plotting enabled

   Using pzmap with plot=True plots the pole-zero map correctly.

3. Final Answer:

   signal.pzmap(system, plot=True) -> Option A

4. Quick Check:

   Use pzmap with plot=True for pole-zero plot [OK]

Hint: Use pzmap with plot=True to plot poles and zeros [OK]

Common Mistakes:

• Using non-existent function pzplot
• Forgetting plot=True argument
• Confusing function names

3. Given the poles of a system as poles = [-1+2j, -1-2j, 0.5], what can you say about the system's stability?

medium

A. The system is stable because poles are complex conjugates.

B. The system is stable because all poles have negative real parts.

C. The system is marginally stable because poles are on the imaginary axis.

D. The system is unstable because one pole has a positive real part.

Solution

1. Step 1: Analyze the real parts of each pole

   Poles are -1+2j (real part -1), -1-2j (real part -1), and 0.5 (real part 0.5).

2. Step 2: Determine stability from pole real parts

   Since one pole has positive real part (0.5), the system is unstable.

3. Final Answer:

   The system is unstable because one pole has a positive real part. -> Option D

4. Quick Check:

   Positive real pole means unstable system [OK]

Hint: Check if any pole has positive real part for instability [OK]

Common Mistakes:

• Ignoring positive real part pole
• Assuming complex conjugates guarantee stability
• Confusing marginal stability with instability

4. The following Python code attempts to plot poles and zeros but gives an error:

from scipy import signal
system = signal.TransferFunction([1, 3], [1, 2, 1])
signal.pzplot(system)

What is the error and how to fix it?

medium

A. TransferFunction requires discrete=True for pzplot to work.

B. TransferFunction arguments are reversed; swap numerator and denominator.

C. Function pzplot does not exist; use pzmap(system, plot=True) instead.

D. Import statement is wrong; use import scipy.signal as signal.

Solution

1. Step 1: Identify the function causing error

   The function pzplot does not exist in scipy.signal, causing the error.

2. Step 2: Use correct function to plot poles and zeros

   Use pzmap(system, plot=True) to plot poles and zeros correctly.

3. Final Answer:

   Function pzplot does not exist; use pzmap(system, plot=True) instead. -> Option C

4. Quick Check:

   Use pzmap, not pzplot, for pole-zero plots [OK]

Hint: Use pzmap with plot=True, not pzplot [OK]

Common Mistakes:

• Using non-existent pzplot function
• Swapping numerator and denominator unnecessarily
• Wrong import statement causing confusion

5. You have a system with poles at -0.5, -1+1j, -1-1j and zeros at 0, -2. Which statement best describes the system's stability and response?

hard

A. The system is marginally stable because of zeros at zero.

B. The system is stable with oscillatory decay due to complex poles.

C. The system is unstable because zeros are on the right half-plane.

D. The system is unstable due to poles with positive real parts.

Solution

1. Step 1: Check pole locations for stability

   All poles have negative real parts (-0.5, -1), so system is stable.

2. Step 2: Analyze pole types for response behavior

   Complex conjugate poles (-1±1j) cause oscillations that decay over time.

3. Step 3: Consider zeros' effect on stability

   Zeros do not affect stability directly; zeros at 0 and -2 are not in right half-plane.

4. Final Answer:

   The system is stable with oscillatory decay due to complex poles. -> Option B

5. Quick Check:

   Negative real poles + complex = stable oscillatory decay [OK]

Hint: Stable if poles negative; complex poles cause oscillations [OK]

Common Mistakes:

• Confusing zeros with poles for stability
• Thinking zeros at zero cause instability
• Ignoring oscillatory nature of complex poles