Continuous

ZeroPole Block

Transfer function via pole-zero form: H(s) = K·prod(s-zeros)/prod(s-poles)

# ZeroPole Block Documentation

Overview

The ZeroPole block represents a transfer function using its poles and zeros:

H(s) = K · (s - z₁)(s - z₂)...(s - zₘ) / (s - p₁)(s - p₂)...(s - pₙ)

This form is intuitive for control system design, stability analysis, and frequency response visualization.

Mathematical Model

H(s) = K · (s - z₁)(s - z₂)...(s - zₘ) / (s - p₁)(s - p₂)...(s - pₙ)

where:

z₁, z₂, ..., zₘ: zeros (roots of numerator)
p₁, p₂, ..., pₙ: poles (roots of denominator)
K: static gain

Inputs

Input: A numeric signal

Outputs

Output: The filtered/processed signal

Parameters

Zeros

Description: Comma-separated list of system zeros
Format Options:
- Real zeros: "1, 2, 3"
- Complex zeros: "1+2j, 1-2j, -3"
- Mixed: "-1, 2+3j"
Default: "-1"
Notes: Complex zeros must appear in conjugate pairs for real systems

Poles

Description: Comma-separated list of system poles
Format: Same as zeros
Default: "-1"
Stability: All poles must have negative real parts for stability

Gain (K)

Description: Static gain multiplier
Type: Real number
Default: 1
Range: Any real number (positive or negative)

Initial Condition

Description: Initial output value
Default: 0
Type: Real number

Pole-Zero Format Examples

First-Order System

Zeros: (empty or "-inf") Poles: "-1" Gain: 1 Represents: 1/(s+1)

Second-Order with Complex Poles

Zeros: (none) Poles: "-1+2j, -1-2j" Gain: 1 Represents: 1/(s² + 2s + 5)

Lead-Lag Compensator

Zeros: "-1" Poles: "-10" Gain: 10 Represents: 10(s+1)/(s+10)

Examples

Example 1: Simple Low-Pass Filter

Zeros: (empty)
Poles: "-1"
Gain: 1

Represents the transfer function H(s) = 1/(s+1), a first-order low-pass filter with cutoff frequency 1 rad/s.

Example 2: Lead Compensator (Phase Boost)

Zeros: "-1"
Poles: "-5"
Gain: 5

Represents H(s) = 5(s+1)/(s+5), useful for phase margin improvement in classical control design.

Example 3: Complex Conjugate Poles (Underdamped System)

Zeros: (none)
Poles: "-1+1.732j, -1-1.732j"
Gain: 1

This represents an underdamped second-order system with damping ratio ζ ≈ 0.5 and natural frequency ωₙ ≈ 2 rad/s.

Common Use Cases

1. Classical control design: Pole-zero form is standard in Bode/Nyquist/Root Locus plots 2. Filter design: Low-pass, high-pass, band-pass filters 3. Compensator design: Lead, lag, lead-lag compensators 4. Stability analysis: Visualize pole locations relative to imaginary axis 5. Frequency response: Zeros/poles directly determine magnitude and phase response

Stability Criteria

Continuous-time: System is stable if all poles have negative real parts (left-half plane)
Marginal stability: Poles on the imaginary axis (jω axis)
Unstable: Any poles in the right-half plane

Frequency Response Interpretation

At a frequency s = jω:

Magnitude: |H(jω)| = K · |∏(jω - zᵢ)| / |∏(jω - pⱼ)|
- Each zero contributes a numerator magnitude
- Each pole contributes a denominator magnitude
- Minima occur near poles, maxima near zeros

Phase: ∠H(jω) = ∠K + Σ∠(jω - zᵢ) - Σ∠(jω - pⱼ)
- Each zero contributes positive phase
- Each pole contributes negative phase

Comparison with Other Blocks

TransferFunction: Use when you have numerator/denominator coefficients
ZeroPole: Use when you have explicit zeros and poles
StateSpace: Use for MIMO systems or when you have state-space realization

Conjugate Pair Assumption

Complex roots in physical (real-coefficient) systems always appear in conjugate pairs. The parser treats each entry independently — if you specify -1+2j, you should also specify -1-2j. Specifying only one complex root will result in complex coefficients in the polynomial, which may produce incorrect or unstable results.

Numerical Integration

Uses forward Euler integration (same as TransferFunction block). The same stability constraints apply: Δt ≤ 1/(5 × |largest pole magnitude|).

Remarks

Conjugate pairs: Complex poles and zeros must be specified in conjugate pairs for real-coefficient systems
Stability: All poles must have negative real parts for a stable system; poles on the imaginary axis are marginally stable
Gain K: Adjusts overall magnitude without affecting phase or pole/zero locations
Empty zeros: Omitted zeros are treated as no zeros (all-pole system); an empty numerator defaults to gain K only
High-order systems: Above ~20 poles, numerical issues may arise; consider model reduction or state-space form