Peter M. Dower

Deep learning has been a critical part of designing inverse design methods that are computationally efficient and accurate. An example of this is the design of photonic metasurfaces by using their photoluminescent spectrum as the input data to predict their topology. One fundamental challenge of these systems is their ability to represent nonlinear relationships between sets of data that have different dimensionalities. Existing design methods often implement a conditional generative adversarial network in order to solve this problem, but in many cases, the solution is unable to generate structures that provide multiple peaks when validated. It is demonstrated that in response to the target spectrum, the bidirectional adversarial autoencoder is able to generate structures that provide multiple peaks on several occasions. As a result, the proposed model represents an important advance toward the generation of nonlinear photonic metasurfaces that can be used in advanced metasurface design.

A min-of-quadratics representation for the cost function underlying a popular approach to classical coverage control is used to motivate and develop connections to function approximation via semiconcave and semiconvex duality. By considering functions that are both semiconcave and semiconvex, upper and lower bounds are developed via their semiconcave and semiconvex relaxations. Min-of-quadratics and max-of-quadratics representations for these bounds are provided by construction, using quadratic basis functions that are synonymous with agents. These representations induce a pair of Voronoi tessellations, whose constituent polytopes delineate regions of space where agents are individually responsible for the upper and lower bounds. The difference in these bounds, weighted by the usual notion of density function, defines a function approximation error that is differentiable via Leibniz. Using the derivatives obtained, a generalized distributed control architecture is developed for minimizing the function approximation error involved. In the special case of spatially invariant function approximation, it is observed that the well-known classical coverage control architecture is recovered.

This paper develops an algorithm for upper- and lower-bounding the value function for a class of linear time-varying games subject to convex control sets. In particular, a two-player zero-sum differential game is considered where the respective players aim to minimise and maximise a convex terminal state cost. A collection of solutions of a single-player dynamical system subject to a trimmed control set is used to characterise a viscosity supersolution of a Hamilton-Jacobi (HJ) equation, which in turn yields an upper bound for the value function. Analogously, a collection of hyperplanes is used to characterise a viscosity subsolution of the HJ equation, which yields a lower bound. The computational complexity and memory requirement of the proposed algorithm scales with the number of solutions and hyperplanes that characterise the bounds, which is not explicitly tied to the number of system states. Thus, the algorithm is tractable for systems of moderately high dimension whilst preserving rigorous guarantees for optimal control and differential game applications.

In Hamilton-Jacobi reachability analysis, reachable sets are characterised by sublevel sets of the value function of an optimal control problem. Although grid-based approaches provide a general-purpose means of numerically computing this value function, the curse-of-dimensionality often limits their use in all but simple applications. In this paper, inner-approximations of the backwards reachable set are characterised through the design of a feedback control law for feedback linearisable systems subject to input constraints. This structure is exploited to propose a selection of control gains via an optimisation problem. The computational cost and memory requirement of the proposed scheme are tied to the number of system states in a polynomial manner. Thus, the proposed approximation scheme is computationally tractable for feedback linearisable systems of relatively high dimension.

Reachable sets for a dynamical system describe collections of system states that can be reached in finite time, subject to system dynamics. They can be used to guarantee goal satisfaction in controller design or to verify that unsafe regions will be avoided. However, conventional grid-based methods for computing these sets suffer from the curse of dimensionality, which typically prohibits their use for systems with more than a small number of states, even if they are linear. In this paper, we demonstrate that local viscosity supersolutions and subsolutions of a Hamilton-Jacobi-Bellman equation can be used to generate, respectively, under-approximating and over-approximating reachable sets for time-varying nonlinear systems. Based on this observation, we derive dynamics for a union and intersection of ellipsoidal sets that, respectively, under-approximate and over-approximate the reachable set for linear time-varying systems subject to an ellipsoidal input constraint and an ellipsoidal terminal (or initial) set. The dynamics for these ellipsoids can be selected to ensure that their boundaries coincide with the boundary of the exact reachable set along a collection of solutions of the system. The ellipsoids can be generated with polynomial computational complexity in the number of states, making our approximation scheme computationally tractable for continuous-time linear time-varying systems of relatively high dimension.

Multifidelity models integrate data from multiple sources to produce a single approximator for the underlying process. Dense low-fidelity samples are used to reduce interpolation error, while sparse high-fidelity samples are used to compensate for bias or noise in the low-fidelity samples. Deep Gaussian processes (GPs) are attractive for multifidelity modelling as they are non-parametric, robust to overfitting, perform well for small datasets, and, critically, can capture nonlinear and input-dependent relationships between data of different fidelities. Many datasets naturally contain gradient data, especially when they are generated by computational models that are compatible with automatic differentiation or have adjoint solutions. Principally, this work extends deep GPs to incorporate gradient data. We demonstrate this method on an analytical test problem and a realistic partial differential equation problem, where we predict the aerodynamic coefficients of a hypersonic flight vehicle over a range of flight conditions and geometries. In both examples, the gradient-enhanced deep GP outperforms a gradient-enhanced linear GP model and their non-gradient-enhanced counterparts.

The coverage control problem involves spatially disseminating a network of mobile agents using distributed control laws, or coverage controllers, over a desired region to locally minimize an associated cost function. Mobile robots using coverage controllers will be subject to various disturbances, such as uncertainty, errors and delays. Asymptotic stability of Cortés et al.'s coverage controller and its variations have been studied, however rates of convergence are omitted. Recent work has provided convergence rates for a coverage controller under certain assumptions, including regions where the network locally converges exponentially. The main contribution of this work is to show that a variation of Cortés et al.'s coverage controller also features robust stability properties, specifically input-to-output stability, under the same set of assumptions. Conservative bounds are used to provide theoretical guarantees on stability, and simulations are used to verify the results and highlight practical performance of the controller.