arXiv subjects

Eric C. Kerrigan

Publications and source records attributed to Eric C. Kerrigan.

At least 37 records · Page 2Linked to original sources

arXiv2022

Teaching Predictive Control Using Specification-based Summative Assessments

Including Model Predictive Control (MPC) in the undergraduate/graduate control curriculum is becoming vitally important due to the growing adoption of MPC in many industrial areas. In this paper, we present an overview of the predictive control course taught by the authors at Imperial College London between 2018 and 2021. We discuss how the course evolved from focusing solely on the linear MPC formulation to covering nonlinear MPC and some of its extensions. We also present a novel specification-based summative assessment framework, written in MATLAB, that was developed to assess the knowledge and understanding of the students in the course by tasking them with designing a controller for a real-world problem. The MATLAB assessment framework was designed to provide the students with the freedom to design and implement any MPC controller they wanted. The submitted controllers were then assessed against over 30 variations of the real-world problem to gauge student understanding of design robustness and the MPC topics from the course.

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arXiv2022

Horizon-independent Preconditioner Design for Linear Predictive Control

First-order optimization solvers, such as the Fast Gradient Method, are increasingly being used to solve Model Predictive Control problems in resource-constrained environments. Unfortunately, the convergence rate of these solvers is significantly affected by the conditioning of the problem data, with ill-conditioned problems requiring a large number of iterations. To reduce the number of iterations required, we present a simple method for computing a horizon-independent preconditioning matrix for the Hessian of the condensed problem. The preconditioner is based on the block Toeplitz structure of the Hessian. Horizon-independence allows one to use only the predicted system and cost matrices to compute the preconditioner, instead of the full Hessian. The proposed preconditioner has equivalent performance to an optimal preconditioner in numerical examples, producing speedups between 2x and 9x for the Fast Gradient Method. Additionally, we derive horizon-independent spectral bounds for the Hessian in terms of the transfer function of the predicted system, and show how these can be used to compute a novel horizon-independent bound on the condition number for the preconditioned Hessian.

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arXiv2022

High-level Synthesis using the Julia Language

The growing proliferation of FPGAs and High-level Synthesis (HLS) tools has led to a large interest in designing hardware accelerators for complex operations and algorithms. However, existing HLS toolflows typically require a significant amount of user knowledge or training to be effective in both industrial and research applications. In this paper, we propose using the Julia language as the basis for an HLS tool. The Julia HLS tool aims to decrease the barrier to entry for hardware acceleration by taking advantage of the readability of the Julia language and by allowing the use of the existing large library of standard mathematical functions written in Julia. We present a prototype Julia HLS tool, written in Julia, that transforms Julia code to VHDL. We highlight how features of Julia and its compiler simplified the creation of this tool, and we discuss potential directions for future work.

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arXiv2022

Solving Dynamic Optimization Problems to a Specified Accuracy: An Alternating Approach using Integrated Residuals

We propose a novel direct transcription and solution method for solving nonlinear, continuous-time dynamic optimization problems. Instead of forcing the dynamic constraints to be satisfied only at a selected number of points as in direct collocation, the new approach alternates between minimizing and constraining the squared norm of the dynamic constraint residuals integrated along the whole solution trajectories. As a result, the method can 1) obtain solutions of higher accuracy for the same mesh compared to direct collocation methods, 2) enables a flexible trade-off between solution accuracy and optimality, 3) provides reliable solutions for challenging problems, including those with singular arcs and high-index differential algebraic equations.

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arXiv2021

Towards a Framework for Nonlinear Predictive Control using Derivative-Free Optimization

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.

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arXiv2021

Inaccuracy matters: accounting for solution accuracy in event-triggered nonlinear model predictive control

We consider the effect of using approximate system predictions in event-triggered control schemes. Such approximations may result from using numerical transcription methods for solving continuous-time optimal control problems. Mesh refinement can guarantee upper bounds on the error in the differential equations which model the system dynamics. With the accuracy guarantees of a mesh refinement scheme, we show that the proposed event-triggering scheme -- which compares the measured system with approximate state predictions -- can be used with a guaranteed strictly positive inter-update time. We show that if we have knowledge of the employed transcription scheme or the approximation errors, then we can obtain better online estimates of inter-update times. We additionally detail a method of tightening constraints on the approximate system trajectory used in the nonlinear programming problem to guarantee constraint satisfaction of the continuous-time system. This is the first work to incorporate prediction accuracy in triggering metrics. Using the solution accuracy we can guarantee reliable lower bounds for inter-update times and perform solution dependent constraint tightening.

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arXiv2021

A hybrid green energy-based framework with a multi-objective optimization approach for optimal frost prevention in horticulture

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

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arXiv2021

Dynamic Optimization with Convergence Guarantees

We present a novel direct transcription method to solve optimization problems subject to nonlinear differential and inequality constraints. We prove convergence of our numerical method under reasonably mild assumptions: boundedness and Lipschitz-continuity of the problem-defining functions. We do not require uniqueness, differentiability or constraint qualifications to hold and we avoid the use of Lagrange multipliers. Our approach differs fundamentally from well-known methods based on collocation; we follow a penalty-barrier approach, where we compute integral quadratic penalties on the equality path constraints and point constraints, and integral log-barriers on the inequality path constraints. The resulting penalty-barrier functional can be minimized numerically using finite elements and penalty-barrier interior-point nonlinear programming solvers. Order of convergence results are derived, even if components of the solution are discontinuous. We also present numerical results to compare our method against collocation methods. The numerical results show that for the same degree and mesh, the computational cost is similar, but that the new method can achieve a smaller error and converges in cases where collocation methods fail.

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arXiv2020

Direct Transcription for Dynamic Optimization: A Tutorial with a Case Study on Dual-Patient Ventilation During the COVID-19 Pandemic

A variety of optimal control, estimation, system identification and design problems can be formulated as functional optimization problems with differential equality and inequality constraints. Since these problems are infinite-dimensional and often do not have a known analytical solution, one has to resort to numerical methods to compute an approximate solution. This paper uses a unifying notation to outline some of the techniques used in the transcription step of simultaneous direct methods (which discretize-then-optimize) for solving continuous-time dynamic optimization problems. We focus on collocation, integrated residual and Runge-Kutta schemes. These transcription methods are then applied to a simulation case study to answer a question that arose during the COVID-19 pandemic, namely: If there are not enough ventilators, is it possible to ventilate more than one patient on a single ventilator? The results suggest that it is possible, in principle, to estimate individual patient parameters sufficiently accurately, using a relatively small number of flow rate measurements, without needing to disconnect a patient from the system or needing more than one flow rate sensor. We also show that it is possible to ensure that two different patients can indeed receive their desired tidal volume, by modifying the resistance experienced by the air flow to each patient and controlling the ventilator pressure.

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arXiv2020

An Integral Penalty-Barrier Direct Transcription Method for Optimal Control

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.

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arXiv2020

A Direct Method for Solving Integral Penalty Transcriptions of Optimal Control Problems

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.

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arXiv2020

Information transmission bounds between moving terminals

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.

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arXiv2019

Joint Optimization of Transmission and Propulsion in UAV-Assisted Communication Networks

The communication energy in a wireless network of mobile autonomous agents should be defined to include the propulsion energy as well as the transmission energy used to facilitate information transfer. We therefore develop communicationtheoretic and Newtonian dynamic models of the communication and locomotion expenditures of an unmanned aerial vehicle (UAV). These models are used to formulate a novel nonlinear optimal control problem (OCP) for arbitrary networks of autonomous agents. This is the first work to consider mobility as a decision variable in UAV networks with multiple access channels. Where possible, we compare our results with known analytic solutions for particular single-hop network configurations. The OCP is then applied to a multiple-node UAV network for which previous results cannot be readily extended. Numerical results demonstrate increased network capacity and communication energy savings upwards of 70% when compared to more naive communication policies.

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arXiv2019

External Constraint Handling for Solving Optimal Control Problems with Simultaneous Approaches and Interior Point Methods

Inactive constraints do not contribute to the solution of an optimal control problem, but increase the problem size and burden the numerical computations. We present a novel strategy for handling inactive constraints efficiently by systematically removing the inactive and redundant constraints. The method is designed to be used together with simultaneous approaches under a mesh refinement framework, with mild assumptions that the original problem has feasible solutions, and the initial solve of the problem is successful. The method is tailored for interior point-based solvers, which are known to be very sensitive to the choice of initial points in terms of feasibility. In the example problem shown, the proposed scheme achieves more than a 40% reduction in computation time.

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arXiv2019

Efficient and More Accurate Representation of Solution Trajectories in Numerical Optimal Control

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.

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arXiv2019

Bounding Computational Complexity under Cost Function Scaling in Predictive Control

We present a framework for upper bounding the number of iterations required by first-order optimization algorithms implementing constrained LQR controllers. We derive new bounds for the condition number and extremal eigenvalues of the primal and dual Hessian matrices when the cost function is scaled. These bounds are horizon-independent, allowing for their use with receding, variable and decreasing horizon controllers. We considerably relax prior assumptions on the structure of the weight matrices and assume only that the system is Schur-stable and the primal Hessian of the quadratic program (QP) is positive-definite. Our analysis uses the Toeplitz structure of the QP matrices 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 complexity of trading off the input energy used against the state deviation. An example system shows a three-times increase in algorithm iterations between the two extremes, with the state 2-norm decreased by only 5% despite a greatly increased state deviation penalty.

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arXiv2017

Nonlinear Predictive Control on a Heterogeneous Computing Platform

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 36x 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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arXiv2017

Automatic Software and Computing Hardware Co-design for Predictive Control

Model Predictive Control (MPC) is a computationally demanding control technique that allows dealing with multiple-input and multiple-output systems, while handling constraints in a systematic way. The necessity of solving an optimization problem at every sampling instant often (i) limits the application scope to slow dynamical systems and/or (ii) results in expensive computational hardware implementations. Traditional MPC design is based on manual tuning of software and computational hardware design parameters, which leads to suboptimal implementations. This paper proposes a framework for automating the MPC software and computational hardware co-design, while achieving the optimal trade-off between computational resource usage and controller performance. The proposed approach is based on using a multi-objective optimization algorithm, namely BiMADS. Two test studies are considered: Central Processing Unit (CPU) and Field-Programmable Gate Array (FPGA) implementations of fast gradient-based MPC. Numerical experiments show that optimization-based design outperforms Latin Hypercube Sampling (LHS), a statistical sampling-based design exploration technique.

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