arXiv subjects

Richard Pates

Publications and source records attributed to Richard Pates.

At least 19 recordsLinked to original sources

arXiv2026

Decentralized Stability Certificates in IBR-Dominated Grids: The Role of the Network State

Small-signal instabilities, such as unforced sub-synchronous oscillations (SSOs), are increasingly observed in inverter-based resource (IBR) dominated grids. While decentralized stability certificates offer a scalable means to avoid instability onset, they are typically derived under restrictive network-state assumptions--such as small angle differences or negligible voltage drops--that cannot capture how departures from these conditions affect system stability. In this paper, we develop a network model and a decentralized analysis framework that explicitly characterizes how reactive power mismatches, line loading, and inverter control parameters jointly determine small-signal stability. We show that increased steady-state reactive power mismatches and line loading lead to more stringent conditions on admissible inverter droop gains. These results make decentralized stability certificates explicitly network-state dependent, showing how network stress shrinks the set of stabilizing local controller parameters.

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arXiv2026

Computing Scaled Relative Graphs of Discrete-time LTI Systems from Data

Graphical methods for system analysis have played a central role in control theory. The Scaled Relative Graph (SRG) has recently emerged as a useful tool for stability analysis of feedback interconnections. In this paper, we further extend its applicability by showing how the SRG of a discrete-time linear-time-invariant (LTI) system can be computed exactly from its state-space representation using linear matrix inequalities. We additionally propose a fully data-driven approach where we demonstrate how to compute the SRG exclusively from input-output data. Furthermore, we introduce a robust version of the SRG, which can be computed from noisy data trajectories and contains the SRG of the actual system.

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arXiv2026

Cholesky factorisation, and intrinsically sparse linear quadratic regulation

We classify a family of matrices of shift operators that can be factorised in a computationally tractable manner with the Cholesky algorithm. Such matrices arise in the linear quadratic regulator problem, and related areas. We use the factorisation to uncover intrinsic sparsity properties in the control laws for transportation problems with an underlying tree structure. This reveals that the optimal control can be applied in a distributed manner that is obscured by standard solution methods.

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arXiv2025

Computable Characterisations of Scaled Relative Graphs of Closed Operators

The Scaled Relative Graph (SRG) is a promising tool for stability and robustness analysis of multi-input multi-output systems. In this paper, we provide tools for exact and computable constructions of the SRG for closed linear operators, based on maximum and minimum gain computations. The results are suitable for bounded and unbounded operators, and we specify how they can be used to draw SRGs for the typical operators that are used to model linear-time-invariant dynamical systems. For the special case of state-space models, we show how the Bounded Real Lemma can be used to construct the SRG. Furthermore, we give an example of how stability analysis of dynamical systems can be done with SRGs.

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arXiv2025

On properties of hydraulic equilibria in district heating networks

District heating networks are an integral part of the energy system in many countries. In future smart energy systems, they are expected to enhance energy flexibility and support the integration of renewable and waste energy sources. An important aspect of these networks is the control of flow rates, which dictates the heat delivered to consumers. This paper concerns the properties of flow rates in tree-structured district heating networks. We show that under mild assumptions of monotonicity in the hydraulic network components, statements regarding the stationary flow rate distribution can be made. In particular, when all consumers in a network incrementally open their valves, an increase in total flow rate throughput is guaranteed, while if one consumer does not open their valve when others do, they will receive a reduced flow rate. These properties are illustrated numerically on a small 2-consumer network as well as on a larger 22-consumer network. Previous works have shown that these properties allow the design and use of efficient control strategies for optimal heat distribution.

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arXiv2025

Performance analysis for cone-preserving switched systems with constrained switching

This paper studies cone-preserving linear discrete-time switched systems whose switching is governed by an automaton. For this general system class, we present performance analysis conditions for a broadly usable performance measure. In doing so, we generalize several known results for performance and stability analysis for switched and positive switched systems, providing a unifying perspective. We also arrive at novel $\ell_1$-performance analysis conditions for positive switched systems with constrained switching, for which we present an application-motivated numerical example. Further, the cone-preserving perspective provides insights into appropriate Lyapunov function selection.

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arXiv2025

On Exact Solutions to the Linear Bellman Equation

This paper presents sufficient conditions for optimal control of systems with dynamics given by a linear operator, in order to obtain an explicit solution to the Bellman equation that can be calculated in a distributed fashion. Further, the class of Linearly Solvable MDP is reformulated as a continuous-state optimal control problem. It is shown that this class naturally satisfies the conditions for explicit solution of the Bellman equation, motivating the extension of previous results to semilinear dynamics to account for input nonlinearities. The applicability of the given conditions is illustrated in scenarios with linear and quadratic cost, corresponding to the Stochastic Shortest Path and Linear-Quadratic Regulator problems.

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arXiv2024

Optimal Control on Positive Cones

An optimal control problem on finite-dimensional positive cones is stated. Under a critical assumption on the cone, the corresponding Bellman equation is satisfied by a linear function, which can be computed by convex optimization. A separate theorem relates the assumption on the cone to the existence of minimal elements in certain subsets of the dual cone. Three special cases are derived as examples. The first one, where the positive cone is the set of positive semi-definite matrices, reduces to standard linear quadratic control. The second one, where the positive cone is a polyhedron, reduces to a recent result on optimal control of positive systems. The third special case corresponds to linear quadratic control with additional structure, such as spatial invariance.

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arXiv2024

Exploiting Heterogeneity in the Decentralised Control of Platoons

This paper investigates the use of decentralised control architectures with heterogeneous dynamics for improving performance in large-scale systems. Our focus is on two well-known decentralised approaches; the 'predecessor following' and 'bidirectional architectures' for vehicle platooning. The former, utilising homogeneous control dynamics, is known to face exponential growth in disturbance amplification throughout the platoon, resulting in poor scalability properties. We demonstrate that by incorporating heterogeneous control system dynamics, this limitation disappears entirely, even under bandwidth constraints. Furthermore, we reveal that introducing heterogeneity in the bidirectional architecture allows the platoon's behaviour to be rendered independent of its length, allowing for highly scalable performance.

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arXiv2024

Lyapunov equations: a (fixed) point of view

The Lyapunov equation is the gateway drug of nonlinear control theory. In these notes we revisit an elegant statement connecting the concepts of asymptotic stability and observability, to the solvability of Lyapunov equations, and discuss how this statement can be proved using the Brouwer fixed-point theorem.

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arXiv2023

A Generalisation of the Secant Criterion

The cyclic feedback interconnection of $n$ subsystems is the basic building block of control theory. Many robust stability tools have been developed for this interconnection. Two notable examples are the small gain theorem and the Secant Criterion. Both of these conditions guarantee stability if an inequality involving the geometric mean of a set of subsystem indices is satisfied. The indices in each case are designed to capture different core properties; gain in the case of the small gain theorem, and the degree of output-strict-passivity in the Secant Criterion. In this paper we identify entire families of other suitable indices based on mappings of the unit disk. This unifies the small gain theorem and the Secant Criterion, as well as a range of other stability criteria, into a single condition.

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arXiv2023

A Frequency Domain Analysis of Slow Coherency in Networked Systems

Network coherence generally refers to the emergence of simple aggregated dynamical behaviours, despite heterogeneity in the dynamics of the subsystems that constitute the network. In this paper, we develop a general frequency domain framework to analyze and quantify the level of network coherence that a system exhibits by relating coherence with a low-rank property of the system's input-output response. More precisely, for a networked system with linear dynamics and coupling, we show that, as the network's \emph{effective algebraic connectivity} grows, the system transfer matrix converges to a rank-one transfer matrix representing the coherent behavior. Interestingly, the non-zero eigenvalue of such a rank-one matrix is given by the harmonic mean of individual nodal dynamics, and we refer to it as the coherent dynamics. Our analysis unveils the frequency-dependent nature of coherence and a non-trivial interplay between dynamics and network topology. We further show that many networked systems can exhibit similar coherent behavior by establishing a concentration result in a setting with randomly chosen individual nodal dynamics.

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arXiv2022

Circuit Analysis using Monotone+Skew Splitting

It is shown that the behavior of an $m$-port circuit of maximal monotone elements can be expressed as a zero of the sum of a maximal monotone operator containing the circuit elements, and a structured skew-symmetric linear operator representing the interconnection structure, together with a linear output transformation. The Condat-V\~u algorithm solves inclusion problems of this form, and may be used to solve for the periodic steady-state behavior, given a periodic excitation at each port, using an iteration in the space of periodic trajectories.

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arXiv2022

Hydraulic Parameter Estimation in District Heating Networks

Using hydraulic models in control design in district heating networks can increase pumping efficiency and reduce sensitivity to hydraulic bottlenecks. These models are usually white-box, as they are obtained based on full knowledge of the district heating network and its parameters. This type of model is time-consuming to obtain, and might differ from the actual behavior of the system. In this paper, a method is proposed to obtain a grey-box hydraulic model for tree-shaped district heating systems: hydraulic parameters are estimated based on pressure measurements in only two locations. While previous works only estimate parameters related to pressure losses in pipes, this work also includes customers valves in the grey-box model structure, an important inclusion for control-oriented applications. Finally, a numerical example illustrates the proposed method on a small district heating network, showing its ability to obtain an accurate model on the basis of noisy measurements.

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arXiv2022

Limitations of time-delayed case isolation in heterogeneous SIR models

Case isolation, that is, detection and isolation of infected individuals in order to prevent spread, is a strategy to curb infectious disease epidemics. Here, we study the efficiency of a case isolation strategy subject to time delays in terms of its ability to stabilize the epidemic spread in heterogeneous contact networks. For an SIR epidemic model, we characterize the stability boundary analytically and show how it depends on the time delay between infection and isolation as well as the heterogeneity of the inter-individual contact network, quantified by the variance in contact rates. We show that network heterogeneity accounts for a restricting correction factor to previously derived stability results for homogeneous SIR models (with uniform contact rates), which are therefore too optimistic on the relevant time scales. We illustrate the results and the underlying mechanisms through insightful numerical examples.

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arXiv2022

Fundamental limitations on the control of lossless systems

In this paper we derive fundamental limitations on the levels of $H_2$ and $H_\infty{}$ performance that can be achieved when controlling lossless systems. The results are applied to the swing equation power system model, where it is shown that the fundamental limit on the $H_2$ norm scales with the inverse of the harmonic mean of the inertias in the system. This indicates that power systems may see a degradation in performance as more renewables are integrated, further motivating the need for new control solutions to aid the energy transition.

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arXiv2022

A Structured Optimal Controller for Irrigation Networks

In this paper, we apply an optimal LQ controller, which has an inherent structure that allows for a distributed implementation, to an irrigation network. The network consists of a water reservoir and connected water canals. The goal is to keep the levels close to the set-points when farmers take out water. The LQ controller is designed using a first-order approximation of the canal dynamics, while the simulation model used for evaluation uses third-order canal dynamics. The performance is compared to a P controller and an LQ controller designed using the third-order canal dynamics. The structured controller outperforms the P controller and is close to the theoretical optimum given by the third-order LQ controller for disturbance rejection.

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arXiv2022

Passive and reciprocal networks: From simple models to simple optimal controllers

Networks constructed out of resistors, inductors, capacitors and transformers form a compelling subclass of simple models. Models constructed out of these basic elements are frequently used to explain phenomena in large-scale applications, from inter-area oscillations in power systems, to the transient behaviour of optimisation algorithms. Furthermore they capture the dynamics of the most commonly applied controllers, including the PID controller. In this paper we show that the inherent structure in these networks can be used to simplify, or even solve analytically, a range of simple optimal control problems. We illustrate these results by designing and synthesising simple, scalable, and globally optimal control laws for solving constrained least squares problems, regulating electrical power systems with stochastic renewable sources, studying the robustness properties of consensus algorithms, and analysing heating networks.

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