dkpy: Robust Control for LTI Systems

• dkpy is an open-source Python package for robust control of LTI systems with structured uncertainties using μ-analysis and DK-iteration.
• It integrates uncertainty characterization from frequency-response data and generalized-plant construction to support iterative robust controller synthesis.
• The package leverages modular abstractions with interfaces like python-control and SLICOT, enabling scalable robust analysis and dynamic D-scale fitting.

Searching arXiv for the specified paper and closely related robust-control references. arxiv_search.query({"^^^^^^^^6search_query6 arxiv_search.query({"^^^^^^^^6search_query6 Robust Control with Structured Uncertainty in Python\"","^^^^^^^^6start6 dkpy is an open-source Python package for the analysis and synthesis of robust controllers for linear time-invariant (LTI) systems subject to structured uncertainty. Its core functionality is built around PRESERVED_PLACEHOLDER_^^^^6search_query6^^^^-analysis (structured singular-value analysis) and PRESERVED_PLACEHOLDER_^^^^6id:(Adams et al., 17 Nov 2025)6^^^^-synthesis (DK-iteration) to assess robust stability and performance, and to design controllers that guarantee stability and performance across all admissible perturbations. It also provides tools to characterize unstructured uncertainty from frequency-response data of perturbed plant models (multi-model uncertainty characterization). The package is presented in "dkpy: Robust Control with Structured Uncertainty in Python" (&&&^^^^6search_query6^^^^&&&).

^^^^6id:(Adams et al., 17 Nov 2025)6^^^^. Scope and problem setting

Models used for control design are, to some degree, uncertain. Model uncertainty must be accounted for to ensure the robustness of the closed-loop system. In this setting, PRESERVED_PLACEHOLDER_^^^^6start6^^^^-analysis and PRESERVED_PLACEHOLDER_^^^^6max_results6^^^^-synthesis methods allow for the analysis and design of controllers subject to structured uncertainties. These tools can also be applied to robust performance problems, because such problems are fundamentally robust control problems with structured uncertainty (&&&^^^^6search_query6^^^^&&&).

The package addresses LTI systems with structured uncertainty and provides two main capabilities. First, it supports robust controller analysis and synthesis through PRESERVED_PLACEHOLDER_^^^^6search_query6^^^^-analysis and DK-iteration. Second, it offers tools for model uncertainty characterization using data from a set of perturbed systems. The open-source project is available at https://github.com/decargroup/dkpy.

A plausible implication is that dkpy is intended to support a workflow in which uncertainty characterization, generalized-plant construction, robust analysis, and robust synthesis are performed within a single Python environment. That implication follows from the coexistence of uncertainty-weight fitting, structured singular-value computation, and DK-iteration in the same package.

^^^^6start6^^^^. Mathematical framework for structured uncertainty

In many multi-input multi-output systems, uncertainties arise in specific subsystems, such as actuator dynamics or sensor gains, and can be modeled by a block-diagonal perturbation PRESERVED_PLACEHOLDER_^^^^6all:\6^^^^ in which each block is either a repeated scalar uncertainty $\delta_i I$ or a full complex block $\Delta_j$ (&&&^^^^6search_query6^^^^&&&). The uncertainty structure is defined as

$$\Delta_\omega = \{ \operatorname{diag}(\{\delta_i I_{r_i}\}, \{\Delta_j\}) \mid \delta_i\in\mathbb{C}, \Delta_j\in\mathbb{C}^{m_j\times m_j} \},$$

and the set of admissible perturbation systems is

$$\Gamma = \{ \Delta(s)\in RH_\infty \mid \|\Delta(s)\|_\infty\le 1 \text{ and } \Delta(j\omega)\in\Delta_\omega \ \forall\ \omega \}.$$

Given a complex matrix PRESERVED_PLACEHOLDER_^^^^6id:(Adams et al., 17 Nov 2025)6search_query6^^^^ and uncertainty structure PRESERVED_PLACEHOLDER_^^^^6id:(Adams et al., 17 Nov 2025)6id:(Adams et al., 17 Nov 2025)6^^^^, the structured singular value is

PRESERVED_PLACEHOLDER_^^^^6id:(Adams et al., 17 Nov 2025)6start6^^^^

for which no closed form exists. In practice, one uses the upper bound

PRESERVED_PLACEHOLDER_^^^^6id:(Adams et al., 17 Nov 2025)6max_results6^^^^

where

PRESERVED_PLACEHOLDER_^^^^6id:(Adams et al., 17 Nov 2025)6search_query6^^^^

and PRESERVED_PLACEHOLDER_^^^^6id:(Adams et al., 17 Nov 2025)6all:\6^^^^ denotes the maximum singular value.

For a feedback interconnection PRESERVED_PLACEHOLDER_^^^^6id:(Adams et al., 17 Nov 2025)66^^^^, robust stability holds if and only if

PRESERVED_PLACEHOLDER_^^^^6id:(Adams et al., 17 Nov 2025)6^^^^7

This framework makes the package specifically aligned with structured uncertainty descriptions based on block-diagonal perturbations rather than generic unstructured robustness margins.

^^^^6max_results6^^^^. PRESERVED_PLACEHOLDER_^^^^6id:(Adams et al., 17 Nov 2025)6^^^^8-analysis, PRESERVED_PLACEHOLDER_^^^^6id:(Adams et al., 17 Nov 2025)6^^^^9-synthesis, and DK-iteration

The package implements PRESERVED_PLACEHOLDER_^^^^6start6search_query6^^^^-analysis through computation of an upper bound on the structured singular value together with the associated scaling. This is the analysis step used to assess robust stability and robust performance over a frequency grid.

For controller design, the relevant optimization problem is the non-convex problem

PRESERVED_PLACEHOLDER_^^^^6start6id:(Adams et al., 17 Nov 2025)6^^^^

dkpy approaches this through DK-iteration, which alternates between controller synthesis and scaling computation. The procedure is described as follows (&&&^^^^6search_query6^^^^&&&):

^^^^6id:(Adams et al., 17 Nov 2025)6^^^^. initialize PRESERVED_PLACEHOLDER_^^^^6start6start6^^^^ ^^^^6start6^^^^. PRESERVED_PLACEHOLDER_^^^^6start6max_results6^^^^ ^^^^6max_results6^^^^. compute PRESERVED_PLACEHOLDER_^^^^6start6search_query6^^^^ and PRESERVED_PLACEHOLDER_^^^^6start6all:\6^^^^ at grid PRESERVED_PLACEHOLDER_^^^^6start66^^^^ ^^^^6search_query6^^^^. fit PRESERVED_PLACEHOLDER_^^^^6start6^^^^7 to PRESERVED_PLACEHOLDER_^^^^6start6^^^^8 ^^^^6all:\6^^^^. repeat until PRESERVED_PLACEHOLDER_^^^^6start6^^^^9 or max iterations reached

The significance of this organization is practical rather than purely formal. The synthesis step uses an PRESERVED_PLACEHOLDER_^^^^6max_results6search_query6^^^^ controller design routine for fixed scaling, while the analysis step updates the frequency-dependent scale and the fitted dynamic PRESERVED_PLACEHOLDER_^^^^6max_results6id:(Adams et al., 17 Nov 2025)6^^^^. This suggests that dkpy is designed for iterative workflows in which analysis and synthesis are tightly coupled but implemented by separable software components.

^^^^6search_query6^^^^. Software architecture and implementation

dkpy is built on python-control and slycot. Its modular architecture is centered on abstract base classes (ABCs) and their implementations. The main abstractions cover controller synthesis, structured singular-value computation, dynamic PRESERVED_PLACEHOLDER_^^^^6max_results6start6^^^^-scale fitting, and DK-iteration composition (&&&^^^^6search_query6^^^^&&&).

ABC	Role	Implementations
ControllerSynthesis	abstract synthesize method	HinfSynSlicot, HinfSynLmi, HinfSynLmiBisection
StructuredSingularValue	abstract compute_ssv method	SsvLmiBisection
DScaleFit	abstract fit method	DScaleFitSlicot
DkIteration	composes synthesis, SSV, and D-fit	DkIterFixedOrder, DkIterListOrder, DkIterAutoOrder, DkIterInteractiveOrder

The concrete synthesis implementations are differentiated by algorithmic backend. HinfSynSlicot uses SLICOT SB^^^^6id:([2511.13927](/papers/2511.13927))6search_query6^^^^AD for PRESERVED_PLACEHOLDER_^^^^6max_results6max_results6^^^^ synthesis; HinfSynLmi is an LMI-based PRESERVED_PLACEHOLDER_^^^^6max_results6search_query6^^^^ synthesis method; and HinfSynLmiBisection uses an LMI formulation with bisection on PRESERVED_PLACEHOLDER_^^^^6max_results6all:\6^^^^. For structured singular-value computation, SsvLmiBisection uses an LMI formulation with bisection to compute PRESERVED_PLACEHOLDER_^^^^6max_results66^^^^ and scaling PRESERVED_PLACEHOLDER_^^^^6max_results6^^^^7. For scale fitting, DScaleFitSlicot uses SLICOT SB^^^^6id:([2511.13927](/papers/2511.13927))6search_query6^^^^YD to fit stable minimum-phase PRESERVED_PLACEHOLDER_^^^^6max_results6^^^^8 to a given PRESERVED_PLACEHOLDER_^^^^6max_results6^^^^9.

The DK-iteration layer is compositional: a DkIteration object contains one ControllerSynthesis, one StructuredSingularValue, and one DScaleFit. Its subclasses vary only in how they choose the order of the PRESERVED_PLACEHOLDER_^^^^6search_query6search_query6^^^^ fit over iterations. This separation of concerns indicates a design in which synthesis, analysis, and rational fitting can be replaced independently within a common iteration structure.

^^^^6all:\6^^^^. Uncertainty characterization and usage patterns

In addition to structured robust analysis and synthesis, dkpy provides high-level functions for multi-model uncertainty characterization from frequency-response data of perturbed plant models. The three functions are:

• compute_uncertainty_residual_response
• compute_uncertainty_weight_response
• fit_uncertainty_weight

The first function can, for example, solve PRESERVED_PLACEHOLDER_^^^^6search_query6id:(Adams et al., 17 Nov 2025)6^^^^. The second uses an LMI to get optimal PRESERVED_PLACEHOLDER_^^^^6search_query6start6^^^^. The third uses a log-Chebyshev method to fit stable, minimum-phase PRESERVED_PLACEHOLDER_^^^^6search_query6max_results6^^^^ (&&&^^^^6search_query6^^^^&&&).

The installation paths are:

$\delta_i I$^^^^6max_results6^^^^

or

$\delta_i I$^^^^6search_query6^^^^

A basic PRESERVED_PLACEHOLDER_^^^^6search_query6search_query6^^^^-analysis example is given in the following form:

$\delta_i I$^^^^6all:\6^^^^

A basic DK-iteration example is given as:

$\delta_i I$6

These examples show that the expected workflow is explicit: define the uncertainty structure, construct either the loop interconnection or the generalized plant, compute PRESERVED_PLACEHOLDER_^^^^6search_query6all:\6^^^^ for analysis or invoke DK-iteration for synthesis, and inspect the peak upper bound.

6. Reported applications, domains, and current limitations

Two example applications are described in the package documentation associated with the paper. The first is multi-model uncertainty characterization for an actuator bundle. The sequence is: load nominal PRESERVED_PLACEHOLDER_^^^^6search_query66^^^^ and off-nominal PRESERVED_PLACEHOLDER_^^^^6search_query6^^^^7 data; compute PRESERVED_PLACEHOLDER_^^^^6search_query6^^^^8 for each PRESERVED_PLACEHOLDER_^^^^6search_query6^^^^9; compute PRESERVED_PLACEHOLDER_^^^^6all:\6search_query6^^^^; and fit PRESERVED_PLACEHOLDER_^^^^6all:\6id:(Adams et al., 17 Nov 2025)6^^^^. This yields multiplicative input weights PRESERVED_PLACEHOLDER_^^^^6all:\6start6^^^^ and PRESERVED_PLACEHOLDER_^^^^6all:\6max_results6^^^^ for the generalized plant (&&&^^^^6search_query6^^^^&&&).

The second example is robust controller synthesis for lateral aircraft dynamics. The steps are: build PRESERVED_PLACEHOLDER_^^^^6all:\6search_query6^^^^ with aerodynamic model, actuator model PRESERVED_PLACEHOLDER_^^^^6all:\6all:\6^^^^, PRESERVED_PLACEHOLDER_^^^^6all:\66^^^^, and performance weights; use DkIterListOrder with ^^^^6max_results6^^^^ iterations and ^^^^6search_query6^^^^th-order PRESERVED_PLACEHOLDER_^^^^6all:\6^^^^7 fits; obtain a result in which PRESERVED_PLACEHOLDER_^^^^6all:\6^^^^8 reduces below PRESERVED_PLACEHOLDER_^^^^6all:\6^^^^9; and simulate step responses across ^^^^6search_query6search_query6^^^^ sampled $\delta_i I$^^^^6search_query6^^^^. The reported outcome is good tracking and limited performance degradation observed.

The domains listed for application are aerospace flight control, automotive chassis control, power electronics, process control, and robotics—anywhere structured uncertainty modeling reduces conservatism. This suggests that the package is intended for settings in which uncertainty has a meaningful subsystem structure and where preserving that structure is preferable to collapsing it into a single unstructured bound.

The paper also states current limitations. In version v^^^^6search_query6^^^^.^^^^6id:([2511.13927](/papers/2511.13927))6^^^^.9, dkpy supports only complex block-diagonal uncertainty and LTI perturbations, unstructured multi-model characterization for additive or multiplicative uncertainty, and uses a fixed set of solver algorithms (SLICOT, MOSEK). Planned enhancements include support for real-parametric uncertainty (real $\delta_i I$^^^^6id:(Adams et al., 17 Nov 2025)6^^^^), mixed complex-real block structures, alternative scaling and $\delta_i I$^^^^6start6^^^^-analysis methods such as DK-pole shifting, integration of time-delay uncertainty, improved auto-scheduling of fit orders and grid refinement, and expanded examples and tutorials in robotics and power systems. With its open architecture, dkpy invites the community to contribute new synthesis routines, structured uncertainty classes, and improved characterizations within a fully Pythonic, open-source ecosystem (&&&^^^^6search_query6^^^^&&&).