Transfer Function

In this example we demonstrate how to use transfer functions in PathSim using the Pole-Residue-Constant (PRC) form. This representation is particularly convenient for transfer functions with complex poles.

You can also find this example as a single file in the GitHub repository.

PathSim provides multiple transfer function representations:

• TransferFunctionPRC: Pole-Residue-Constant form (poles and residues)

• TransferFunctionZPG: Zero-Pole-Gain form

In this example, we use the PRC form to define a transfer function with complex conjugate poles.

First let’s import the Simulation and Connection classes along with the required blocks:

import matplotlib.pyplot as plt
import numpy as np

# Apply PathSim docs matplotlib style
plt.style.use('../pathsim_docs.mplstyle')

from pathsim import Simulation, Connection
from pathsim.blocks import Source, Scope, TransferFunctionPRC
from pathsim.solvers import RKCK54

Transfer Function Definition

The pole-residue-constant form represents a transfer function as:

\[\mathbf{H}(s) = \mathbf{C} + \sum_{i=1}^{n} \frac{\mathbf{R}_i}{s - p_i}\]

where:

• \(\mathbf{C}\) is the constant term (direct feedthrough)

• \(\mathbf{R}_i\) are the residues

• \(p_i\) are the poles

Complex conjugate poles must come with corresponding complex conjugate residues to ensure a real-valued impulse response.

# Step delay for the input
tau = 5.0

# Simulation timestep
dt = 0.05

# Transfer function parameters
const = 0.0
poles = [-0.3, -0.05+0.4j, -0.05-0.4j, -0.1+2j, -0.1-2j]
residues = [-0.2, -0.2j, 0.2j, 0.3, 0.3]

This transfer function has:

• One real pole at \(s = -0.3\)

• Two pairs of complex conjugate poles at \(s = -0.05 \pm 0.4j\) and \(s = -0.1 \pm 2j\)

The complex poles will produce oscillatory behavior in the step response.

Now let’s create the blocks. We use a Source block to generate a step input and a TransferFunctionPRC block for our system:

# Blocks and connections
Sr = Source(lambda t: int(t >= tau))
TF = TransferFunctionPRC(Poles=poles, Residues=residues, Const=const)
Sc = Scope(labels=["step", "response"])

blocks = [Sr, TF, Sc]

connections = [
    Connection(Sr, TF, Sc),
    Connection(TF, Sc[1])
]

We initialize the simulation with the RKCK54 solver (Runge-Kutta-Cash-Karp 5(4) method) for accurate integration:

# Initialize simulation
Sim = Simulation(blocks, connections, dt=dt, log=True, Solver=RKCK54)

07:10:30 - INFO - LOGGING (log: True)
07:10:30 - INFO - BLOCKS (total: 3, dynamic: 1, static: 2, eventful: 0)
07:10:30 - INFO - GRAPH (nodes: 3, edges: 3, alg. depth: 1, loop depth: 0, runtime: 0.102ms)

Now let’s run the simulation and plot the step response:

# Run simulation
Sim.run(100)

# Plot the results from the scope directly
Sc.plot()
plt.show()

07:10:30 - INFO - STARTING -> TRANSIENT (Duration: 100.00s)
07:10:30 - INFO - --------------------   1% | 0.0s<0.1s | 2561.5 it/s
07:10:30 - INFO - ####----------------  20% | 0.0s<0.1s | 4266.4 it/s
07:10:30 - INFO - ########------------  40% | 0.1s<0.0s | 4171.8 it/s
07:10:30 - INFO - ############--------  60% | 0.1s<0.0s | 4246.0 it/s
07:10:30 - INFO - ################----  80% | 0.1s<0.0s | 4197.3 it/s
07:10:30 - INFO - #################### 100% | 0.1s<--:-- | 4047.1 it/s
07:10:30 - INFO - FINISHED -> TRANSIENT (total steps: 346, successful: 224, runtime: 98.70 ms)