dkpy 0.1.9

D-K iteration in Python

dkpy

dkpy is a D-K iteration library written in Python, aiming to build upon python-control.

The package is currently a work-in-progress, and no API stability guarantees will be made until version 1.0.0.

D-K iteration

The standard robust control problem has the form:

          ┌─────────┐
          │         │
  w2 ┌────┤    Δ    │◄───┐ z2
     │    │         │    │
     │    └─────────┘    │
     │    ┌─────────┐    │
     └───►│         ├────┘
w1 ──────►│    P    ├──────► z1
     ┌───►│         ├────┐
     │    └─────────┘    │
     │    ┌─────────┐    │
     │    │         │    │
   u └────┤    K    │◄───┘ y
          │         │
          └─────────┘

where P is the generalized plant, K is the controller, and Δ is an uncertain LTI system whose H-infinity norm is less than or equal to 1.

Synthesizing a controller that makes the transfer matrix from w1 to z1 have an H-infinity norm less than 1 guarantees robust stability by the small gain theorem.

When Δ has structure (e.g., Δ = diag(Δ1, Δ2)), this approach is too conservative. Robust stability can instead be achieved by synthesizing a controller whose structured singular value, µ, is less than 1. Robust performance problems can also be viewed as robust stability problems with structured uncertainty.

Minimizing µ is much more challenging than minimizing the H-infinity norm. D-K iteration is one method to do so. It relies on the fact that an upper bound for µ is:

µ(M) ≤ min σ̅(DMD⁻¹)
        D

where D is a complex matrix whose structure commutes with Δ. More specifically, for each full block in Δ, the corresponding entry of D is d I, where d is a scalar and I is the identity matrix. If Δ has any entries of the form δ I, then D has a full block in the corresponding entry.

D-K iteration has the following steps, where D is initially identity.

1. Augment P with D and D⁻¹, then synthesize an H-infinity controller.

2. Compute µ and D for the closed-loop system without the D-scalings over a range of discrete frequencies.

3. Fit a transfer matrix to D and repeat. Stop when µ < 1.

The D-K iteration process is represented by dkpy.DkIteration. The steps of the process are represented by

1. dkpy.ControllerSynthesis,

2. dkpy.StructuredSingularValue, and

3. dkpy.DScaleFit.

Example

import dkpy
import numpy as np

# Load an example
eg = dkpy.example_skogestad2006_p325()

# Set up the D-K iteration method
dk_iter = dkpy.DkIterListOrder(
    controller_synthesis=dkpy.HinfSynSlicot(),
    structured_singular_value=dkpy.SsvLmiBisection(),
    d_scale_fit=dkpy.DScaleFitSlicot(),
    fit_orders=[4, 4, 4],
)

# Synthesize a controller
omega = np.logspace(-3, 3, 61)
block_structure = [
    dkpy.ComplexFullBlock(1, 1),
    dkpy.ComplexFullBlock(1, 1),
    dkpy.ComplexFullBlock(2, 2),
]
K, N, mu, d_scale_fit_info, info = dk_iter.synthesize(
    eg["P"],
    eg["n_y"],
    eg["n_u"],
    omega,
    block_structure,
)

Contributing

To install the pre-commit hook, run

$ pip install -r requirements.txt
$ pre-commit install

in the repository root.

Citation

If you use this software in your research, please cite it as below or see CITATION.cff.

@software{dahdah_dkpy_2025,
    title={{decargroup/dkpy}},
    doi={10.5281/zenodo.14511244},
    url={https://github.com/decargroup/dkpy},
    publisher={Zenodo},
    author={Steven Dahdah and Timothy Everett Adams and James Richard Forbes},
    version = {{v0.1.9}},
    year={2025},
}