Eric C. Kerrigan

Martin P. Neuenhofen, Eric C. Kerrigan

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.

Mason Thammawichai, Eric C. Kerrigan

We present a number of novel algorithms, based on mathematical optimization formulations, in order to solve a homogeneous multiprocessor scheduling problem, while minimizing the total energy consumption. In particular, for a system with a discrete speed set, we propose solving a tractable linear program. Our formulations are based on a fluid model and a global scheduling scheme, i.e. tasks are allowed to migrate between processors. The new methods are compared with three global energy/feasibility optimal workload allocation formulations. Simulation results illustrate that our methods achieve both feasibility and energy optimality and outperform existing methods for constrained deadline tasksets. Specifically, the results provided by our algorithm can achieve up to an 80% saving compared to an algorithm without a frequency scaling scheme and up to 70% saving compared to a constant frequency scaling scheme for some simulated tasksets. Another benefit is that our algorithms can solve the scheduling problem in one step instead of using a recursive scheme. Moreover, our formulations can solve a more general class of scheduling problems, i.e. any periodic real-time taskset with arbitrary deadline. Lastly, our algorithms can be applied to both online and offline scheduling schemes.

Ian McInerney, Eric C. Kerrigan, George A. Constantinides

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.

Eduardo M. G. Vila, Eric C. Kerrigan, Paul Bruce

This paper presents a pseudo-spectral method for Dynamic Optimization Problems (DOPs) that allows for tight polynomial bounds to be achieved via flexible sub-intervals. The proposed method not only rigorously enforces inequality constraints, but also allows for a lower cost in comparison with non-flexible discretizations. Two examples are provided to demonstrate the feasibility of the proposed method to solve optimal control problems. Solutions to the example problems exhibited up to a tenfold reduction in relative cost.

Luke Rickard, Paola Falugi, Eric C. Kerrigan

Distributed energy systems with prosumers require new methods for coordinating energy exchange among agents. Coalitional control provides a framework in which agents form groups to cooperatively reduce costs; however, existing bottom-up coalition-formation methods typically require full information sharing, raising privacy concerns and imposing significant computational overhead. In this work, we propose a limited information coalition-formation algorithm that requires only limited aggregate information exchange among agents. By constructing an upper bound on the value of candidate coalitions, we eliminate the need to solve optimisation problems for each potential merge, significantly reducing computational complexity while limiting information exchange. We prove that the proposed method guarantees cost no greater than that of decentralised operation. Coalition strategies are optimised using a distributed approach based on the Alternating Direction Method of Multipliers (ADMM), further limiting information sharing within coalitions. We embed the framework within a model predictive control scheme and evaluate it on real-world data, demonstrating improved economic performance over decentralised control with substantially lower computational cost than full-information approaches.

Lorenzo Sabug, Eric Kerrigan

We revisit the problem of physics-informed regression, and propose a method that directly computes the state at the prediction point, simultaneously with the derivative and curvature information of the existing samples. We frame each prediction as a constrained optimisation problem, leveraging multivariate Taylor series expansions and explicitly enforcing physical laws. Each individual query can be processed with low computational cost without any pre- or re-training, in contrast to global function approximator-based solutions such as neural networks. Our comparative benchmarks on a reaction-diffusion system show competitive predictive accuracy relative to a neural network-based solution, while completely eliminating the need for long training loops, and remaining robust to changes in the sampling layout.

Benjamin Biggs, Ian McInerney, Eric C. Kerrigan et al.

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.