James Anderson

Antonis Papachristodoulou, James Anderson, Giorgio Valmorbida et al.

The release of SOSTOOLS v4.00 comes as we approach the 20th anniversary of the original release of SOSTOOLS v1.00 back in April, 2002. SOSTOOLS was originally envisioned as a flexible tool for parsing and solving polynomial optimization problems, using the SOS tightening of polynomial positivity constraints, and capable of adapting to the ever-evolving fauna of applications of SOS. There are now a variety of SOS programming parsers beyond SOSTOOLS, including YALMIP, Gloptipoly, SumOfSquares, and others. We hope SOSTOOLS remains the most intuitive, robust and adaptable toolbox for SOS programming. Recent progress in Semidefinite programming has opened up new possibilities for solving large Sum of Squares programming problems, and we hope the next decade will be one where SOS methods will find wide application in different areas. In SOSTOOLS v4.00, we implement a parsing approach that reduces the computational and memory requirements of the parser below that of the SDP solver itself. We have re-developed the internal structure of our polynomial decision variables. Specifically, polynomial and SOS variable declarations made using sossosvar, sospolyvar, sosmatrixvar, etc now return a new polynomial structure, dpvar. This new polynomial structure, is documented in the enclosed dpvar guide, and isolates the scalar SDP decision variables in the SOS program from the independent variables used to construct the SOS program. As a result, the complexity of the parser scales almost linearly in the number of decision variables. As a result of these changes, almost all users will notice a significant increase in speed, with large-scaleproblems experiencing the most dramatic speedups. Parsing time is now always less than 10% of time spent in the SDP solver. Finally, SOSTOOLS now provides support for the MOSEK solver interface as well as the SeDuMi, SDPT3, CSDP, SDPNAL, SDPNAL+, and SDPA solvers.

Yuxiao Chen, James Anderson, Karan Kalsi et al.

This paper presents an assume-guarantee reasoning approach to the computation of robust invariant sets for network systems. Parameterized signal temporal logic (pSTL) is used to formally describe the behaviors of the subsystems, which we use as the template for the contract. We show that set invariance can be proved with a valid assume-guarantee contract by reasoning about individual subsystems. If a valid assume-guarantee contract with monotonic pSTL template is known, it can be further refined by value iteration. When such a contract is not known, an epigraph method is proposed to solve for a contract that is valid, ---an approach that has linear complexity for a sparse network. A microgrid example is used to demonstrate the proposed method. The simulation result shows that together with control barrier functions, the states of all the subsystems can be bounded inside the individual robust invariant sets.

Han Wang, Aritra Mitra, Hamed Hassani et al.

We initiate the study of federated reinforcement learning under environmental heterogeneity by considering a policy evaluation problem. Our setup involves $N$ agents interacting with environments that share the same state and action space but differ in their reward functions and state transition kernels. Assuming agents can communicate via a central server, we ask: Does exchanging information expedite the process of evaluating a common policy? To answer this question, we provide the first comprehensive finite-time analysis of a federated temporal difference (TD) learning algorithm with linear function approximation, while accounting for Markovian sampling, heterogeneity in the agents' environments, and multiple local updates to save communication. Our analysis crucially relies on several novel ingredients: (i) deriving perturbation bounds on TD fixed points as a function of the heterogeneity in the agents' underlying Markov decision processes (MDPs); (ii) introducing a virtual MDP to closely approximate the dynamics of the federated TD algorithm; and (iii) using the virtual MDP to make explicit connections to federated optimization. Putting these pieces together, we rigorously prove that in a low-heterogeneity regime, exchanging model estimates leads to linear convergence speedups in the number of agents.

Giorgio Valmorbida, James Anderson

This work addresses the problem of estimating the region of attraction (RA) of equilibrium points of nonlinear dynamical systems. The estimates we provide are given by positively invariant sets which are not necessarily defined by level sets of a Lyapunov function. Moreover, we present conditions for the existence of Lyapunov functions linked to the positively invariant set formulation we propose. Connections to fundamental results on estimates of the RA are presented and support the search of Lyapunov functions of a rational nature. We then restrict our attention to systems governed by polynomial vector fields and provide an algorithm that is guaranteed to enlarge the estimate of the RA at each iteration.

Thomas T. C. K. Zhang, Leonardo F. Toso, James Anderson et al.

A powerful concept behind much of the recent progress in machine learning is the extraction of common features across data from heterogeneous sources or tasks. Intuitively, using all of one's data to learn a common representation function benefits both computational effort and statistical generalization by leaving a smaller number of parameters to fine-tune on a given task. Toward theoretically grounding these merits, we propose a general setting of recovering linear operators $M$ from noisy vector measurements $y = Mx + w$, where the covariates $x$ may be both non-i.i.d. and non-isotropic. We demonstrate that existing isotropy-agnostic representation learning approaches incur biases on the representation update, which causes the scaling of the noise terms to lose favorable dependence on the number of source tasks. This in turn can cause the sample complexity of representation learning to be bottlenecked by the single-task data size. We introduce an adaptation, $\texttt{De-bias & Feature-Whiten}$ ($\texttt{DFW}$), of the popular alternating minimization-descent scheme proposed independently in Collins et al., (2021) and Nayer and Vaswani (2022), and establish linear convergence to the optimal representation with noise level scaling down with the $\textit{total}$ source data size. This leads to generalization bounds on the same order as an oracle empirical risk minimizer. We verify the vital importance of $\texttt{DFW}$ on various numerical simulations. In particular, we show that vanilla alternating-minimization descent fails catastrophically even for iid, but mildly non-isotropic data. Our analysis unifies and generalizes prior work, and provides a flexible framework for a wider range of applications, such as in controls and dynamical systems.

Yuxiao Chen, James Anderson

This paper addresses the problem of designing distributed controllers with state and input constraints in the System Level Synthesis (SLS) framework. Using robust optimization, we show how state and actuation constraints can be incorporated into the SLS structure. Moreover, we show that the dual variable associated with the constraint has the same sparsity pattern as the SLS parametrization, and therefore the computation distributes using a simple primal-dual algorithm. We provide a stability analysis for SLS design with input saturation under the Internal Model Control (IMC) framework. We show that the closed-loop system with saturation is stable if the controller has a gain less than one. In addition, a saturation compensation scheme that incorporates the saturation information is proposed which improves the naive SLS design under saturation.

Aviar Sootla, James Anderson

In this paper, we derive sufficient conditions on drift matrices under which block-diagonal solutions to Lyapunov inequalities exist. The motivation for the problem comes from a recently proposed basis pursuit algorithm. In particular, this algorithm can provide approximate solutions to optimisation programmes with constraints involving Lyapunov inequalities using linear or second order cone programming. This algorithm requires an initial feasible point, which we aim to provide in this paper. Our existence conditions are based on the so-called $\mathcal{H}$ matrices. We also establish a link between $\mathcal{H}$ matrices and an application of a small gain theorem to the drift matrix. We finally show how to construct these solutions in some cases without solving the full Lyapunov inequality.

James Anderson, Nikolai Matni, Yuxiao Chen

Typically when designing distributed controllers it is assumed that the state-space model of the plant consists of sparse matrices. However, in the discrete-time setting, if one begins with a continuous-time model, the discretization process annihilates any sparsity in the model. In this work we propose a discretization procedure that maintains the sparsity of the continuous-time model. We show that this discretization out-performs a simple truncation method in terms of its ability to approximate the "ground truth" model. Leveraging results from numerical analysis we are also able to upper-bound the error between the dense discretization and our method. Furthermore, we show that in a robust control setting we can design a distributed controller on the approximate (sparse) model that stabilizes the dense ground truth model.

Guangchen Lan, Han Wang, James Anderson et al.

Federated reinforcement learning (FedRL) enables agents to collaboratively train a global policy without sharing their individual data. However, high communication overhead remains a critical bottleneck, particularly for natural policy gradient (NPG) methods, which are second-order. To address this issue, we propose the FedNPG-ADMM framework, which leverages the alternating direction method of multipliers (ADMM) to approximate global NPG directions efficiently. We theoretically demonstrate that using ADMM-based gradient updates reduces communication complexity from ${O}({d^{2}})$ to ${O}({d})$ at each iteration, where $d$ is the number of model parameters. Furthermore, we show that achieving an $ε$-error stationary convergence requires ${O}(\frac{1}{(1-γ)^{2}ε})$ iterations for discount factor $γ$, demonstrating that FedNPG-ADMM maintains the same convergence rate as the standard FedNPG. Through evaluation of the proposed algorithms in MuJoCo environments, we demonstrate that FedNPG-ADMM maintains the reward performance of standard FedNPG, and that its convergence rate improves when the number of federated agents increases.

Han Wang, Siddartha Marella, James Anderson

Federated learning is a framework for distributed optimization that places emphasis on communication efficiency. In particular, it follows a client-server broadcast model and is particularly appealing because of its ability to accommodate heterogeneity in client compute and storage resources, non-i.i.d. data assumptions, and data privacy. Our contribution is to offer a new federated learning algorithm, FedADMM, for solving non-convex composite optimization problems with non-smooth regularizers. We prove converges of FedADMM for the case when not all clients are able to participate in a given communication round under a very general sampling model.

Han Wang, Leonardo F. Toso, James Anderson

We study the problem of learning a linear system model from the observations of $M$ clients. The catch: Each client is observing data from a different dynamical system. This work addresses the question of how multiple clients collaboratively learn dynamical models in the presence of heterogeneity. We pose this problem as a federated learning problem and characterize the tension between achievable performance and system heterogeneity. Furthermore, our federated sample complexity result provides a constant factor improvement over the single agent setting. Finally, we describe a meta federated learning algorithm, FedSysID, that leverages existing federated algorithms at the client level.

Leonardo F. Toso, Han Wang, James Anderson

We address the problem of learning linear system models from observing multiple trajectories from different system dynamics. This framework encompasses a collaborative scenario where several systems seeking to estimate their dynamics are partitioned into clusters according to their system similarity. Thus, the systems within the same cluster can benefit from the observations made by the others. Considering this framework, we present an algorithm where each system alternately estimates its cluster identity and performs an estimation of its dynamics. This is then aggregated to update the model of each cluster. We show that under mild assumptions, our algorithm correctly estimates the cluster identities and achieves an approximate sample complexity that scales inversely with the number of systems in the cluster, thus facilitating a more efficient and personalized system identification process.

Bruce D. Lee, Leonardo F. Toso, Thomas T. Zhang et al.

Representation learning is a powerful tool that enables learning over large multitudes of agents or domains by enforcing that all agents operate on a shared set of learned features. However, many robotics or controls applications that would benefit from collaboration operate in settings with changing environments and goals, whereas most guarantees for representation learning are stated for static settings. Toward rigorously establishing the benefit of representation learning in dynamic settings, we analyze the regret of multi-task representation learning for linear-quadratic control. This setting introduces unique challenges. Firstly, we must account for and balance the $\textit{misspecification}$ introduced by an approximate representation. Secondly, we cannot rely on the parameter update schemes of single-task online LQR, for which least-squares often suffices, and must devise a novel scheme to ensure sufficient improvement. We demonstrate that for settings where exploration is "benign", the regret of any agent after $T$ timesteps scales as $\tilde O(\sqrt{T/H})$, where $H$ is the number of agents. In settings with "difficult" exploration, the regret scales as $\tilde O(\sqrt{d_u d_θ} \sqrt{T} + T^{3/4}/H^{1/5})$, where $d_x$ is the state-space dimension, $d_u$ is the input dimension, and $d_θ$ is the task-specific parameter count. In both cases, by comparing to the minimax single-task regret $O(\sqrt{d_x d_u^2}\sqrt{T})$, we see a benefit of a large number of agents. Notably, in the difficult exploration case, by sharing a representation across tasks, the effective task-specific parameter count can often be small $d_θ< d_x d_u$. Lastly, we provide numerical validation of the trends we predict.

Maxfield Parson-Scherban, Kasra Fallah, Navid Rahbariasr et al.

Controllers for motor drives typically require a current reference which will satisfy the requested torque subject to system constraints. This work generalizes existing current reference theory to the case of the Wound Rotor Synchronous Machine (WRSM). By incorporating the additional rotor-current degree-of-freedom, along with magnetic saturation, cross-coupling, and speed-dependent core losses, the problem of finding an optimal current reference is formulated within affine flux regions as a quadratically constrained quadratic program using a piecewise-affine approximation derived from finite-element data. The solution is characterized according to the active constraint regime, yielding closed-form or low-dimensional polynomial solutions in several cases, and a small semidefinite program in the voltage constrained regime. The proposed framework extends unified optimal current reference theory beyond the permanent-magnet setting to three degree-of-freedom WRSMs while remaining computationally tractable. Results on a physical WRSM prototype illustrate the effectiveness of the approach across the torque-speed operating envelope.

Leonardo F. Toso, Han Wang, James Anderson

We investigate the problem of learning an $ε$-approximate solution for the discrete-time Linear Quadratic Regulator (LQR) problem via a Stochastic Variance-Reduced Policy Gradient (SVRPG) approach. Whilst policy gradient methods have proven to converge linearly to the optimal solution of the model-free LQR problem, the substantial requirement for two-point cost queries in gradient estimations may be intractable, particularly in applications where obtaining cost function evaluations at two distinct control input configurations is exceptionally costly. To this end, we propose an oracle-efficient approach. Our method combines both one-point and two-point estimations in a dual-loop variance-reduced algorithm. It achieves an approximate optimal solution with only $O\left(\log\left(1/ε\right)^β\right)$ two-point cost information for $β\in (0,1)$.

Ishaan Mahajan, Jon Arrizabalaga, Andrea Grillo et al.

Semidefinite programming (SDP) provides a principled framework for convex relaxations of nonconvex geometric constraints in motion planning, yet existing solvers are too computationally expensive for real-time control, particularly on resource-constrained embedded systems. To address this gap, we introduce TinySDP, the first semidefinite programming solver designed for embedded systems, enabling real-time model-predictive control (MPC) on microcontrollers for problems with nonconvex obstacle constraints. Our approach integrates positive-semidefinite cone projections into a cached-Riccati-based ADMM solver, leveraging computational structure for embedded tractability. We pair this solver with an a posteriori rank-1 certificate that converts relaxed solutions into explicit geometric guarantees at each timestep. On challenging benchmarks, e.g., cul-de-sac and dynamic obstacle avoidance scenarios that induce failures in local methods, TinySDP achieves collision-free navigation with up to 73% shorter paths than state-of-the-art baselines. We validate our approach on a Crazyflie quadrotor, demonstrating that semidefinite constraints can be enforced at real-time rates for agile embedded robotics.

Kasra Fallah, Leonardo F. Toso, James Anderson

We study adversarially robust multitask adaptive linear quadratic control; a setting where multiple systems collaboratively learn control policies under model uncertainty and adversarial corruption. We propose a clustered multitask approach that integrates clustering and system identification with resilient aggregation to mitigate corrupted model updates. Our analysis characterizes how clustering accuracy, intra-cluster heterogeneity, and adversarial behavior affect the expected regret of certainty-equivalent (CE) control across LQR tasks. We establish non-asymptotic bounds demonstrating that the regret decreases inversely with the number of honest systems per cluster and that this reduction is preserved under a bounded fraction of adversarial systems within each cluster.

Chenyu Zhang, Han Wang, Aritra Mitra et al. · mit

Federated reinforcement learning (FRL) has emerged as a promising paradigm for reducing the sample complexity of reinforcement learning tasks by exploiting information from different agents. However, when each agent interacts with a potentially different environment, little to nothing is known theoretically about the non-asymptotic performance of FRL algorithms. The lack of such results can be attributed to various technical challenges and their intricate interplay: Markovian sampling, linear function approximation, multiple local updates to save communication, heterogeneity in the reward functions and transition kernels of the agents' MDPs, and continuous state-action spaces. Moreover, in the on-policy setting, the behavior policies vary with time, further complicating the analysis. In response, we introduce FedSARSA, a novel federated on-policy reinforcement learning scheme, equipped with linear function approximation, to address these challenges and provide a comprehensive finite-time error analysis. Notably, we establish that FedSARSA converges to a policy that is near-optimal for all agents, with the extent of near-optimality proportional to the level of heterogeneity. Furthermore, we prove that FedSARSA leverages agent collaboration to enable linear speedups as the number of agents increases, which holds for both fixed and adaptive step-size configurations.

Donglin Zhan, Leonardo F. Toso, James Anderson

We study task selection to enhance sample efficiency in model-agnostic meta-reinforcement learning (MAML-RL). Traditional meta-RL typically assumes that all available tasks are equally important, which can lead to task redundancy when they share significant similarities. To address this, we propose a coreset-based task selection approach that selects a weighted subset of tasks based on how diverse they are in gradient space, prioritizing the most informative and diverse tasks. Such task selection reduces the number of samples needed to find an $ε$-close stationary solution by a factor of O(1/$ε$). Consequently, it guarantees a faster adaptation to unseen tasks while focusing training on the most relevant tasks. As a case study, we incorporate task selection to MAML-LQR (Toso et al., 2024b), and prove a sample complexity reduction proportional to O(log(1/$ε$)) when the task specific cost also satisfy gradient dominance. Our theoretical guarantees underscore task selection as a key component for scalable and sample-efficient meta-RL. We numerically validate this trend across multiple RL benchmark problems, illustrating the benefits of task selection beyond the LQR baseline.

Donglin Zhan, James Anderson

Meta-learning methods typically learn tasks under the assumption that all tasks are equally important. However, this assumption is often not valid. In real-world applications, tasks can vary both in their importance during different training stages and in whether they contain noisy labeled data or not, making a uniform approach suboptimal. To address these issues, we propose the Data-Efficient and Robust Task Selection (DERTS) algorithm, which can be incorporated into both gradient and metric-based meta-learning algorithms. DERTS selects weighted subsets of tasks from task pools by minimizing the approximation error of the full gradient of task pools in the meta-training stage. The selected tasks are efficient for rapid training and robust towards noisy label scenarios. Unlike existing algorithms, DERTS does not require any architecture modification for training and can handle noisy label data in both the support and query sets. Analysis of DERTS shows that the algorithm follows similar training dynamics as learning on the full task pools. Experiments show that DERTS outperforms existing sampling strategies for meta-learning on both gradient-based and metric-based meta-learning algorithms in limited data budget and noisy task settings.

Anna Scampicchio, Leonardo F. Toso, Rahel Rickenbach et al.

A major challenge in physics-informed machine learning is to understand how the incorporation of prior domain knowledge affects learning rates when data are dependent. Focusing on empirical risk minimization with physics-informed regularization, we derive complexity-dependent bounds on the excess risk in probability and in expectation. We prove that, when the physical prior information is aligned, the learning rate improves from the (slow) Sobolev minimax rate to the (fast) optimal i.i.d. one without any sample-size deflation due to data dependence.

Leonardo F. Toso, Lintao Ye, James Anderson

We study the problem of learning to stabilize (LTS) a linear time-invariant (LTI) system. Policy gradient (PG) methods for control assume access to an initial stabilizing policy. However, designing such a policy for an unknown system is one of the most fundamental problems in control, and it may be as hard as learning the optimal policy itself. Existing work on the LTS problem requires large data as it scales quadratically with the ambient dimension. We propose a two-phase approach that first learns the left unstable subspace of the system and then solves a series of discounted linear quadratic regulator (LQR) problems on the learned unstable subspace, targeting to stabilize only the system's unstable dynamics and reduce the effective dimension of the control space. We provide non-asymptotic guarantees for both phases and demonstrate that operating on the unstable subspace reduces sample complexity. In particular, when the number of unstable modes is much smaller than the state dimension, our analysis reveals that LTS on the unstable subspace substantially speeds up the stabilization process. Numerical experiments are provided to support this sample complexity reduction achieved by our approach.

Donglin Zhan, Haoting Zhang, Rhonda Righter et al.

Motivated by the growing need for black-box optimization and data privacy, we introduce a collaborative Bayesian optimization (BO) framework that addresses both of these challenges. In this framework agents work collaboratively to optimize a function they only have oracle access to. In order to mitigate against communication and privacy constraints, agents are not allowed to share their data but can share their Gaussian process (GP) surrogate models. To enable collaboration under these constraints, we construct a central model to approximate the objective function by leveraging the concept of Wasserstein barycenters of GPs. This central model integrates the shared models without accessing the underlying data. A key aspect of our approach is a collaborative acquisition function that balances exploration and exploitation, allowing for the optimization of decision variables collaboratively in each iteration. We prove that our proposed algorithm is asymptotically consistent and that its implementation via Monte Carlo methods is numerically accurate. Through numerical experiments, we demonstrate that our approach outperforms other baseline collaborative frameworks and is competitive with centralized approaches that do not consider data privacy.

Kasra Fallah, Leonardo F. Toso, James Anderson

We consider solutions to the linear quadratic Gaussian (LQG) regulator problem via policy gradient (PG) methods. Although PG methods have demonstrated strong theoretical guarantees in solving the linear quadratic regulator (LQR) problem, despite its nonconvex landscape, their theoretical understanding in the LQG setting remains limited. Notably, the LQG problem lacks gradient dominance in the classical parameterization, i.e., with a dynamic controller, which hinders global convergence guarantees. In this work, we study PG for the LQG problem by adopting an alternative parameterization of the set of stabilizing controllers and employing a lifting argument. We refer to this parameterization as a history representation of the control input as it is parameterized by past input and output data from the previous p time-steps. This representation enables us to establish gradient dominance and approximate smoothness for the LQG cost. We prove global convergence and per-iteration stability guarantees for policy gradient LQG in model-based and model-free settings. Numerical experiments on an open-loop unstable system are provided to support the global convergence guarantees and to illustrate convergence under different history lengths of the history representation.

Leonardo F. Toso, Donglin Zhan, James Anderson et al.

We investigate the problem of learning linear quadratic regulators (LQR) in a multi-task, heterogeneous, and model-free setting. We characterize the stability and personalization guarantees of a policy gradient-based (PG) model-agnostic meta-learning (MAML) (Finn et al., 2017) approach for the LQR problem under different task-heterogeneity settings. We show that our MAML-LQR algorithm produces a stabilizing controller close to each task-specific optimal controller up to a task-heterogeneity bias in both model-based and model-free learning scenarios. Moreover, in the model-based setting, we show that such a controller is achieved with a linear convergence rate, which improves upon sub-linear rates from existing work. Our theoretical guarantees demonstrate that the learned controller can efficiently adapt to unseen LQR tasks.

Han Wang, James Anderson

This work considers the problem of learning the Markov parameters of a linear system from observed data. Recent non-asymptotic system identification results have characterized the sample complexity of this problem in the single and multi-rollout setting. In both instances, the number of samples required in order to obtain acceptable estimates can produce optimization problems with an intractably large number of decision variables for a second-order algorithm. We show that a randomized and distributed Newton algorithm based on Hessian-sketching can produce $ε$-optimal solutions and converges geometrically. Moreover, the algorithm is trivially parallelizable. Our results hold for a variety of sketching matrices and we illustrate the theory with numerical examples.

Han Wang, James Anderson

Learning a dynamical system from input/output data is a fundamental task in the control design pipeline. In the partially observed setting there are two components to identification: parameter estimation to learn the Markov parameters, and system realization to obtain a state space model. In both sub-problems it is implicitly assumed that standard numerical algorithms such as the singular value decomposition (SVD) can be easily and reliably computed. When trying to fit a high-dimensional model to data, for example in the cyber-physical system setting, even computing an SVD is intractable. In this work we show that an approximate matrix factorization obtained using randomized methods can replace the standard SVD in the realization algorithm while maintaining the non-asymptotic (in data-set size) performance and robustness guarantees of classical methods. Numerical examples illustrate that for large system models, this is the only method capable of producing a model.

Fengyu Zhou, James Anderson, Steven H. Low

We consider the problem of privately releasing aggregated network statistics obtained from solving a DC optimal power flow (OPF) problem. It is shown that the mechanism that determines the noise distribution parameters are linked to the topology of the power system and the monotonicity of the network. We derive a measure of "almost" monotonicity and show how it can be used in conjunction with a linear program in order to release aggregated OPF data using the differential privacy framework.

James Anderson, John C. Doyle, Steven Low et al.

This article surveys the System Level Synthesis framework, which presents a novel perspective on constrained robust and optimal controller synthesis for linear systems. We show how SLS shifts the controller synthesis task from the design of a controller to the design of the entire closed loop system, and highlight the benefits of this approach in terms of scalability and transparency. We emphasize two particular applications of SLS, namely large-scale distributed optimal control and robust control. In the case of distributed control, we show how SLS allows for localized controllers to be computed, extending robust and optimal control methods to large-scale systems under practical and realistic assumptions. In the case of robust control, we show how SLS allows for novel design methodologies that, for the first time, quantify the degradation in performance of a robust controller due to model uncertainty -- such transparency is key in allowing robust control methods to interact, in a principled way, with modern techniques from machine learning and statistical inference. Throughout, we emphasize practical and efficient computational solutions, and demonstrate our methods on easy to understand case studies.

James Anderson

This paper considers the non-convex problem of finding the nearest Metzler matrix to a given possibly unstable matrix. Linear systems whose state vector evolves according to a Metzler matrix have many desirable properties in analysis and control with regard to scalability. This motivates the question, how close (in the Frobenius norm of coefficients) to the nearest Metzler matrix are we? Dropping the Metzler constraint, this problem has recently been studied using the theory of dissipative Hamiltonian (DH) systems, which provide a helpful characterization of the feasible set of stable matrices. This work uses the DH theory to provide a block coordinate descent algorithm consisting of a quadratic program with favourable structural properties and a semidefinite program for which recent diagonal dominance results can be used to improve tractability.