## Publications

165 results found

Rodriguez Bernuz JM, McInerney I, Junyent Ferre A,
et al., 2021, Design of a linear time-varying Model Predictive Control energy regulator for grid-tied VSCs, *IEEE Transactions on Energy Conversion*, Vol: 36, Pages: 1425-1434, ISSN: 0885-8969

This paper presents an energy regulator based on a Model Predictive Control (MPC) algorithm for a Voltage Source Converter (VSC). The MPC is formulated to optimise the converter performance according to the weights defined in an objective function that trades off additional features, such as current harmonic distortion, reactive power tracking and DC bus voltage oscillation. Differently from most approaches found in the research literature, the MPC proposed here considers the coupling dynamics between the AC and DC sides of the VSC. This study is focused on the example case of a single-phase VSC, which presents a nonlinear relationship between its AC and DC sides and a sustained double-line frequency power disturbance in its DC bus. To reduce the burden of the MPC, the controller is formulated to benefit from the slow energy dynamics of the system. Thus, the cascaded structure typically used in the control of VSCs is kept and the MPC is set as an energy regulator at a reduced sampling frequency while the current control relies on a fast inner controller. The computational burden of the algorithm is further reduced by using a linear time-varying approximation. The controller is presented in detail and experimental validation showing the performance of the algorithm is provided.

McInerney I, Nita L, Nie Y, et al., 2021, Towards a framework for nonlinear predictive control using derivative-free optimization, 7th IFAC Conference on Nonlinear Model Predictive Control, Publisher: International Federation of Automatic Control

The use of derivative-based solvers to compute solutions to optimal control problems with non-differentiable cost or dynamics often requires reformulations or relaxations that complicate the implementation or increase computational complexity. We present an initial framework for using the derivative-free Mesh Adaptive Direct Search (MADS) algorithm to solve Nonlinear Model Predictive Control problems with non-differentiable features without the need for reformulation. The MADS algorithm performs a structured search of the input space by simulating selected system trajectories and computing the subsequent cost value. We propose handling the path constraints and the Lagrange cost term by augmenting the system dynamics with additional states to compute the violation and cost value alongside the state trajectories, eliminating the need for reconstructing the state trajectories in a separate phase. We demonstrate the practicality of this framework by solving a robust rocket control problem, where the objective is to reach a target altitude as close as possible, given a system with uncertain parameters. This example uses a non-differentiable cost function and simulates two different system trajectories simultaneously, with each system having its own free final time.

Atam E, Abdelmaguid TF, Keskin ME, et al., 2021, A hybrid green energy-based framework with a multi-objective optimization approach for optimal frost prevention in horticulture, Publisher: arXiv

In this paper, first we propose a novel hybrid renewable energy-basedsolution for frost prevention in horticulture applications involving activeheaters. Then, we develop a multi-objective robust optimization-basedformulation to optimize the distribution of a given number of active heaters ina given large-scale orchard. The objectives are to optimally heat the orchardby the proposed frost prevention system and to minimize the total length of theenergy distribution pipe network (which is directly related to the installationcost and the cost of energy losses during energy transfer). Next, the resultingoptimization problem is approximated using a discretization scheme. A casestudy is provided to give an idea of the potential savings using the proposedoptimization method compared to the result from a heuristic-based design, whichshowed a 24.13% reduction in the total pipe length and a 54.29% increase infrost prevention.

McInerney I, Kerrigan EC, Constantinides GA, 2020, Horizon-independent preconditioner design for linear predictive control, Publisher: arXiv

First-order optimization solvers, such as the Fast Gradient Method, areincreasingly being used to solve Model Predictive Control problems inresource-constrained environments. Unfortunately, the convergence rate of thesesolvers is significantly affected by the conditioning of the problem data, withill-conditioned problems requiring a large number of iterations. To reduce thenumber of iterations required, we present a simple method for computing ahorizon-independent preconditioning matrix for the Hessian of the condensedproblem. The preconditioner is based on the block Toeplitz structure of theHessian. Horizon-independence allows one to use only the predicted system andcost matrices to compute the preconditioner, instead of the full Hessian. Theproposed preconditioner has equivalent performance to an optimalpreconditioner, producing up to a 6x speedup for the Fast Gradient Method inour numerical examples. Additionally, we derive horizon-independent spectralbounds for the Hessian in terms of the transfer function of the predictedsystem, and show how these can be used to compute a novel horizon-independentbound on the condition number for the preconditioned Hessian.

Nie Y, Kerrigan EC, 2020, Solving dynamic optimization problems to a specified accuracy: an alternating approach using integrated residuals, Publisher: arXiv

We propose a novel direct transcription and solution method for solvingnonlinear, continuous-time dynamic optimization problems. Instead of forcingthe dynamic constraints to be satisfied only at a selected number of points asin direct collocation, the new approach alternates between minimizing andconstraining the squared norm of the dynamic constraint residuals integratedalong the whole solution trajectories. As a result, the method can 1) obtainsolutions of higher accuracy for the same mesh compared to direct collocationmethods, 2) enables a flexible trade-off between solution accuracy andoptimality, 3) provides reliable solutions for challenging problems, includingthose with singular arcs and high-index differential algebraic equations.

Solis-Lemus JA, Costar E, Doorly D,
et al., 2020, A simulated single ventilator/dual patient ventilation strategy for acute respiratory distress syndrome during the COVID-19 pandemic, *ROYAL SOCIETY OPEN SCIENCE*, Vol: 7, ISSN: 2054-5703

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Kerrigan E, Nie Y, Faqir O, et al., 2020, Direct transcription for dynamic Optimization: a tutorial with a case study on dual-patient ventilation during the COVID-19 pandemic, 59th IEEE Conference on Decision and Control 2020, Publisher: IEEE

A variety of optimal control, estimation, sys-tem identification and design problems can be formulatedas functional optimization problems with differential equalityand inequality constraints. Since these problems are infinite-dimensional and often do not have a known analytical so-lution, one has to resort to numerical methods to computean approximate solution. This paper uses a unifying notationto outline some of the techniques used in the transcriptionstep of simultaneous direct methods (which discretize-then-optimize) for solving continuous-time dynamic optimizationproblems. We focus on collocation, integrated residual andRunge-Kutta schemes. These transcription methods are thenapplied to a simulation case study to answer a question thatarose during the COVID-19 pandemic, namely: If there arenot enough ventilators, is it possible to ventilate more than onepatient on a single ventilator? The results suggest that it ispossible, in principle, to estimate individual patient parameterssufficiently accurately, using a relatively small number of flowrate measurements, without needing to disconnect a patientfrom the system or needing more than one flow rate sensor. Wealso show that it is possible to ensure that two different patientscan indeed receive their desired tidal volume, by modifyingthe resistance experienced by the air flow to each patient andcontrolling the ventilator pressure.

Neuenhofen MP, Kerrigan E, 2020, An integral penalty-barrier direct transcription method for optimal control, 59th IEEE Conference on Decision and Control 2020, Publisher: IEEE

Some direct transcription methods can fail to converge, e.g. when there are singular arcs. We recently introduced a convergent direct transcription method for optimal control problems, called the penalty-barrier finite element method (PBF). PBF converges under very weak assumptions on the problem instance. PBF avoids the ringing between collocation points, for example, by avoiding collocation entirely. Instead, equality path constraint residuals are forced to zero everywhere by an integral quadratic penalty term.We highlight conceptual differences between collocation- and penalty-type direct transcription methods. Theoretical convergence results for both types of methods are reviewed and compared. Formulas for implementing PBF are presented, with details on the formulation as a nonlinear program (NLP), sparsity and solution. Numerical experiments compare PBF against several collocation methods with regard to robustness, accuracy, sparsity and computational cost. We show that the computational cost, sparsity and construction of the NLP functions are roughly the same as for orthogonal collocation methods of the same degree and mesh. As an advantage, PBF converges in cases where collocation methods fail. PBF also allows one to trade off computational cost, optimality and violation of differential and other equality equations against each other.

Neuenhofen MP, Kerrigan E, 2020, A direct method for solving integral penalty transcriptions of optimal control problems, IEEE Conference on Decision and Control, Publisher: IEEE

We present a numerical method for the minimization of objectives that are augmented with large quadratic penalties of overdetermined inconsistent equality constraints. Such objectives arise from quadratic integral penalty methods for the direct transcription of equality constrained optimal control problems. The Augmented Lagrangian Method (ALM) has a number of advantages over the Quadratic Penalty Method (QPM) for solving this class of problems. However, if the equality constraints of the discretization are inconsistent, then ALM might not converge to a point that minimizes the unconstrained bias of the objective and penalty term. Therefore, in this paper we explore a modification of ALM that fits our purpose. Numerical experiments demonstrate that the modified ALM can minimize certain quadratic penalty-augmented functions faster than QPM, whereas the unmodified ALM converges to a minimizer of a significantly different problem.

Faqir OJ, Kerrigan EC, Gunduz D, 2020, Information transmission bounds between moving terminals, *IEEE Communications Letters*, Vol: 24, Pages: 1410-1413, ISSN: 1089-7798

In networks of mobile autonomous agents, e.g. for data acquisition, we may wish to maximize data transfer or to reliably transfer a minimum amount of data, subject to quality of service or energy constraints. These requirements can be guaranteed through both offline node design/specifications and online trajectory/communications design. Regardless of the distance between them, for a stationary point-to-point transmitter-receiver pair communicating across a single link under average power constraints, the total data transfer is unbounded as time tends to infinity. In contrast, we show that if the transmitter/receiver is moving at any constant speed away from each other, then the maximum transmittable data is bounded. Although general closed-form expressions as a function of communication and mobility profile parameters do not yet exist, we provide closed-form expressions for particular cases, such as ideal free space path loss. Under more general scenarios we instead give lower bounds on the total transmittable information across a single link between mobile nodes.

Atam E, Kerrigan EC, 2020, Optimal partitioning of multi-thermal zone buildings for decentralized control, Publisher: arXiv

In this paper, we develop an optimization-based systematic approach for thechallenging, less studied, and important problem of optimal partitioning ofmulti-thermal zone buildings for the decentralized control. The proposed methodconsists of (i) construction of a graph-based network to quantitativelycharacterize the thermal interaction level between neighbor zones, and (ii) theapplication of two different approaches for optimal clustering of the resultingnetwork graph: stochastic optimization and robust optimization. The proposedmethod was tested on two case studies: a 5-zone building (a small-scaleexample) which allows one to consider all possible partitions to assess thesuccess rate of the developed method; and a 20-zone building (a large-scaleexample) for which the developed method was used to predict the optimalpartitioning of the thermal zones. Compared to the existing literature, ourapproach provides a systematic and potentially optimal solution for theconsidered problem.

Brown J, Su D, Kong H,
et al., 2020, Improved noise covariance estimation in visual servoing using an autocovariance least-squares approach, *Mechatronics*, Vol: 68, ISSN: 0957-4158

Position based visual servoing is a widely adopted tool in robotics and automation. While the extended Kalman filter (EKF) has been proposed as an effective technique for this, it requires accurate noise covariance matrices to render desirable performance. Although numerous techniques for updating or estimating the covariance matrices have been developed in the literature, many of these suffer from computational limits or difficulties in imposing structural constraints such as positive semi-definiteness (PSD). In this paper, a relatively new framework, namely the autocovariance least-squares (ALS) method, is applied to estimate noise covariances using real world visual servoing data. To generate the innovations data required for the ALS method, we utilize standard position based visual servoing methods such as EKF, and also an advanced optimization-based framework, namely moving horizon estimation (MHE). A major advantage of the proposed method is that the PSD and other structural constraints on the noise covariances can be enforced conveniently in the optimization problem, which can be solved efficiently using existing software packages. Our results show that using the ALS estimated covariances in the EKF, instead of hand-tuned covariances, gives more than 20% mean error reduction in visual servoing, while utilising MHE to generate the ALS innovations provides a further 21% accuracy improvement.

McInerney I, Kerrigan E, Constantinides G, 2020, Modeling round-off error in the fast gradient method for predictive control, 2019 IEEE 58th Conference on Decision and Control (CDC), Publisher: IEEE, Pages: 1-6

We present a method for determining the smallest precision required to have algorithmic stability of an implementation of the Fast Gradient Method (FGM) when solving a linear Model Predictive Control (MPC) problem in fixed-point arithmetic. We derive two models for the round-off error present in fixed-point arithmetic. The first is a generic model with no assumptions on the predicted system or weight matrices. The second is a parametric model that exploits the Toeplitz structure of the MPC problem for a Schur-stable system. We also propose a metric for measuring the amount of round-off error the FGM iteration can tolerate before becoming unstable. This metric is combined with the round-off error models to compute the minimum number of fractional bits needed for the fixed-point data type. Using these models, we show that exploiting the MPC problem structure nearly halves the number of fractional bits needed to implement an example problem. We show that this results in significant decreases in resource usage, computational energy and execution time for an implementation on a Field Programmable Gate Array.

Faqir O, Kerrigan E, 2020, Mesh refinement for event-triggered nonlinear model predictive control, 21st IFAC World Congress, Publisher: IFAC Secretariat, ISSN: 2405-8963

We consider the effect of using approximate system predictions in event-triggeredcontrol schemes. Such approximations often result from numerical transcription methods forsolving continuous-time optimal control problems. Mesh refinement schemes guarantee upperbounds on the error in the differential equations used to model system dynamics. In particular,we show that with the accuracy guarantees of a mesh refinement scheme, then event-triggeringschemes based on bounding the difference between predicted and measured state can be usedwith a guaranteed strictly positive inter-update time. We determine a lower bound for this timeand show that additional knowledge of the employed transcription method and evaluation ofthe approximation errors may be used to obtain better online estimates of inter-update times.This is the first work to consider using the solution accuracy of an optimal control problem asa metric for triggering new control updates.

Nie Y, Kerrigan E, 2020, Efficient Implementation of Rate Constraints for Nonlinear Optimal Control, *IEEE Transactions on Automatic Control*, ISSN: 0018-9286

We propose a general approach to directly implement rate constraints on the discretization mesh for all collocation methods, for both state and input variables. Unlike conventional approaches that may lead to singular control arcs, the solution of this on-mesh implementation has better properties. Moreover, computational speedups of more than 30% can be achieved by exploiting the properties of the resulting linear constraint equations.

McInerney I, Kerrigan E, Constantinides G, 2020, Closed-form preconditioner design for linear predictive control, 21st IFAC World Congress, Publisher: IFAC Secretariat, ISSN: 2405-8963

Model Predictive Control (MPC) with linear models and constraints is extensivelybeing utilized in many applications, many of which have low power requirements and limitedcomputational resources. In these resource-constrained environments, many designers chooseto utilize simple iterative first-order optimization solvers, such as the Fast Gradient Method.Unfortunately, the convergence rate of these solvers is affected by the conditioning of the problemdata, with ill-conditioned problems requiring a large number of iterations to solve. In order toreduce the number of solver iterations required, we present a simple closed-form method forcomputing an optimal preconditioning matrix for the Hessian of the condensed primal problem.To accomplish this, we also derive spectral bounds for the Hessian in terms of the transferfunction of the predicted system. This preconditioner is based on the Toeplitz structure of theHessian and has equivalent performance to a state-of-the-art optimal preconditioner, withouthaving to solve a semidefinite program during the design phase.

Nie Y, Kerrigan EC, 2020, External constraint handling for solving optimal control problems with simultaneous approaches and interior point methods, *IEEE Control Systems Letters*, Vol: 4, Pages: 7-12, ISSN: 2475-1456

Inactive constraints do not contribute to the solution but increase theproblem size and burden the numerical computations. We present a novel strategyfor handling inactive constraints efficiently by systematically removing theinactive constraints and redundant constraint sets under a mesh refinementframework. The method is tailored for interior point-based solvers, which areknown to be very sensitive to the choice of initial points in terms offeasibility. In the example problem shown, the proposed scheme achieves morethan 40% reduction in computation time.

Nie Y, Kerrigan E, 2020, Efficient and more accurate representation of solution trajectories in numerical optimal control, *IEEE Control Systems Letters*, Vol: 4, Pages: 61-66, ISSN: 2475-1456

We show via examples that, when solving optimal control problems, representing the optimal state and input trajectory directly using interpolation schemes may not be the best choice. Due to the lack of considerations for solution trajectories in-between collocation points, large errors may occur, posing risks if this solution is to be applied. A novel solution representation method is proposed, capable of yielding a solution of much higher accuracy for the same discretization mesh. This is achieved by minimizing the integral of the residual error for the overall trajectory, instead of forcing the errors to be zero only at collocation points. In this way, the requirement for mesh resolution can be significantly reduced, leaving the problem dimensions relatively small. This particular formulation also avoids some of the drawbacks found in the earlier work of integrated residual minimization, leading to more efficient computations.

Iftikhar S, Faqir O, Kerrigan E, 2019, Nonlinear model predictive control of an overhead laboratory-scale gantry crane with obstacle avoidance, 2019 IEEE Conference on Control Technology and Applications (CCTA), Publisher: IEEE

Gantry cranes are complex nonlinear electrome- chanical systems representing a challenging control problem. We propose an optimization-based controller for guiding the crane through arbitrary obstacles. Solving path planning problems with obstacles typically requires a two-stage approach. First, a path is found that is feasible w.r.t. system dynamics and obstacles. The path is then interpreted as a series of set points by a lower-level controller that guides the system. We instead generate a path, and the associated control input to move along that path, from a single optimization problem using a nonlinear model predictive control framework. In doing so, we generate a trajectory that is locally optimal and feasible w.r.t. system dynamics and obstacles. Multiple obstacle avoidance constraint formulations are proposed as smooth, differentiable functions. Objects are approximated either as the union of a set of smooth shapes or as smooth indicator functions. The formulations presented in this work are applicable to (non-)convex problems in 2-D or 3-D spaces. Numerical methods are used to solve the proposed problems for both 2-D (fixed string length) and 3-D (varying string length) models of the gantry crane, resulting in consistently lower costs than nodal or sampling based algorithms.

Brown J, Su D, Kong H, et al., 2019, Improved noise covariance estimation in visual servoing using an autocovariance least-squares approach, Joint 12th IFAC Conference on Control Applications in Marine Systems, Robotics, and Vehicles 1st IFAC Workshop on Robot Control, Publisher: Elsevier, Pages: 37-42, ISSN: 2405-8963

For pose estimation in visual servoing, by assuming the relative motion over one sample period to be constant, many existing works adopt a linear time invariant (LTI) dynamic model. Since the standard feature point transformation is nonlinear, extended Kalman filtering (EKF) has become popular due to its simplicity. Thus, the problem at hand becomes filtering of an LTI system with a time-varying output matrix. To obtain satisfactory performance, accurate knowledge of the noise covariances is essential. Various methods have been proposed on how to adaptively update their values to improve performance. However, these techniques cannot guarantee the positive semidefiniteness (PSD) of the covariance estimates. In this paper, we propose to apply the autocovariance least-squares (ALS) approach to covariance identification in pose estimation. The ALS approach can provide reliable estimates of the covariance matrices while maintaining their PSD and imposing desired structural constraints. Our tests show that using the covariance estimates from the ALS method in EKF can reduce the average pose estimation error by more than 30% in simulation, and the average position estimation error by about 30% using experimental data, respectively, compared to a hand-tuned EKF.

Zhang L, Kerrigan E, Pal B, 2019, Optimal communication scheduling in the smart grid, *IEEE Transactions on Industrial Informatics*, Vol: 15, Pages: 5257-5265, ISSN: 1551-3203

This paper focuses on obtaining the optimal communication topology in the smart grid architecture, i.e. what is the optimal communication setup of smart meters in a smart building. The fact that smart meters also consume energy, more often than not, gets ignored by researchers and engineers. In this paper, we will show that smart meter networks can consume significantly less energy with optimal scheduling. Numerical results show that the overall energy consumption can be reduced by implementing the optimal communication architecture and transmission rate setup, rather than implementing a straightforward communication architecture with uniform channel bandwidth.

Zhang L, Kerrigan E, Pal B, 2019, Optimal Communication Scheduling in the Smart Grid, *IEEE Transactions on Industrial Informatics*, ISSN: 1551-3203

McInerney I, Kerrigan EC, Constantinides GA, 2019, Bounding computational complexity under cost function scaling in predictive control, Publisher: arXiv

We present a framework for upper bounding the number of iterations requiredby first-order optimization algorithms implementing constrained LQRcontrollers. We derive new bounds for the condition number and extremaleigenvalues of the primal and dual Hessian matrices when the cost function isscaled. These bounds are horizon-independent, allowing for their use withreceding, variable and decreasing horizon controllers. We considerably relaxprior assumptions on the structure of the weight matrices and assume only thatthe system is Schur-stable and the primal Hessian of the quadratic program (QP)is positive-definite. Our analysis uses the Toeplitz structure of the QPmatrices to relate their spectrum to the transfer function of the system,allowing for the use of system-theoretic techniques to compute the bounds.Using these bounds, we can compute the effect on the computational complexityof trading off the input energy used against the state deviation. An examplesystem shows a three-times increase in algorithm iterations between the twoextremes, with the state 2-norm decreased by only 5% despite a greatlyincreased state deviation penalty.

Faqir O, Nie Y, Kerrigan E, et al., 2018, Energy-efficient communication in mobile aerial relay-assisted networks using predictive control, 6th IFAC Conference on Nonlinear Model Predictive Control, Publisher: Elsevier, Pages: 197-202, ISSN: 2405-8963

Energy-efficient communication in wireless networks of mobile autonomous agents mandates joint optimization of both transmission and propulsion energy. In Faqir et al. (2017) we developed communication-theoretic data transmission and Newtonian flight mechanics models to formulate a nonlinear optimal control problem. Here we extend the previous work by generalizing the communication model to include UAV-appropriate slow fading channels and specifically investigate the potential from joint optimization of mobility and communication over a multiple access channel. Numerical results exemplify the potential energy savings available to all nodes through this joint optimization. Finally, using the slow fading channel problem formulation, we generate a chance-constrained nonlinear model predictive control scheme for control of a terrestrial network served by a single UAV relay. Closed-loop simulations are performed subject to uncertainties in both transmission and mobility models.

Nie Y, Kerrigan EC, 2018, How should rate constraints be implemented in nonlinear optimal control solvers?, 6th IFAC Conference on Nonlinear Model Predictive Control, Publisher: IFAC Secretariat, Pages: 362-367, ISSN: 2405-8963

This paper investigates the problem of implementing rate constraints when solvingnonlinear optimal control problems with direct transcription methods. We generalize theapproach of directly implementing rate constraints on the discretization mesh to all typesof collocation methods (h,pandhp), for both state and input variables. This “on mesh”implementation replaces the additional dynamic equations and nonlinear path constraints inclassical implementations with linear equations. Thus, there is no contribution to the Hessianand the contribution to the Jacobian can be precomputed, enabling faster iterations. Throughan example, the benefits of the proposed approach are demonstrated, both in terms of obtainingsingular arc-free solutions, as well as reductions in computation time of more than 20%.

Faqir OJ, Kerrigan EC, Gunduz D, 2018, Information transmission bounds in mobile communication networks, UKACC 12th International Conference on Control (CONTROL), Publisher: IEEE, Pages: 99-99

Faqir OJ, Kerrigan EC, Gunduz D, 2018, Energy-optimal control in mobile aerial relay-assisted networks, UKACC 12th International Conference on Control (CONTROL), Publisher: IEEE, Pages: 100-100

Neuenhofen MP, Kerrigan EC, 2018, Dynamic optimization with convergence guarantees, Publisher: arXiv

We present a novel direct transcription method to solve optimization problemssubject to nonlinear differential and inequality constraints. In order toprovide numerical convergence guarantees, it is sufficient for the functionsthat define the problem to satisfy boundedness and Lipschitz conditions. Ourassumptions are the most general to date; we do not require uniqueness,differentiability or constraint qualifications to hold and we avoid the use ofLagrange multipliers. Our approach differs fundamentally from state-of-the-artmethods based on collocation. We follow a least-squares approach to findingapproximate solutions to the differential equations. The objective is augmentedwith the integral of a quadratic penalty on the differential equation residualand a logarithmic barrier for the inequality constraints, as well as aquadratic penalty on the point constraint residual. The resulting unconstrainedinfinite-dimensional optimization problem is discretized using finite elements,while integrals are replaced by quadrature approximations if they cannot beevaluated analytically. Order of convergence results are derived, even ifcomponents of solutions are discontinuous.

Nie Y, Kerrigan EC, 2018, Efficient Implementation of Rate Constraints for Nonlinear Optimal Control, 2018 UKACC 12th International Conference on Control (CONTROL), Publisher: IEEE

Khusainov B, Kerrigan EC, Suardi A,
et al., 2018, Nonlinear predictive control on a heterogeneous computing platform, *Control Engineering Practice*, Vol: 78, Pages: 105-115, ISSN: 0967-0661

We propose an implementation of an interior-point-based nonlinear predictive controller on a heterogeneous processor. The workload can be split between a general-purpose CPU and a field-programmable gate array to trade off the contradicting design objectives of control performance and computational resource usage. A new way of exploiting the structure of the KKT matrix yields significant memory savings. We report an 18x memory saving, compared to existing approaches, and a 6x speedup over a software implementation with an ARM Cortex-A9 processor. We also introduce a new release of Protoip, which abstracts low-level details of heterogeneous programming and allows processor-in-the-loop verification.

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