Associate Professor Adrian Wills

This article addresses the problem of computing fixed-interval smoothed state estimates of a linear time-varying Gaussian stochastic system. There already exist many algorithms that perform this computation, but all of them impose certain restrictions on system matrices in order for them to be applicable, and the restrictions vary considerably between the various existing algorithms. This article establishes a new sufficient condition for the fixed-interval smoothing density to exist in a Gaussian form that can be completely characterized by associated means and covariances. It then develops an algorithm to compute these means and covariances with no further assumptions required. This results in an algorithm more generally applicable than any one of the multitude of existing algorithms available to date.

This paper considers parameter estimation for nonlinear state-space models, which is an important but challenging problem. We address this challenge by employing a variational inference (VI) approach, which is a principled method that has deep connections to maximum likelihood estimation. This VI approach ultimately provides estimates of the model as solutions to an optimisation problem, which is deterministic, tractable and can be solved using standard optimisation tools. A specialisation of this approach for systems with additive Gaussian noise is also detailed. The proposed method is examined numerically on a range of simulated and real examples focusing on the robustness to parameter initialisation; additionally, favourable comparisons are performed against state-of-the-art alternatives.

Sequential Monte Carlo methods¿also known as particle filters¿offer approximate solutions to filtering problems for nonlinear state-space systems. These filtering problems are notoriously difficult to solve in general due to a lack of closed-form expressions and challenging expectation integrals. The essential idea behind particle filters is to employ Monte Carlo integration techniques in order to ameliorate both of these challenges. This article presents an intuitive introduction to the main particle filter ideas and then unifies three commonly employed particle filtering algorithms. This unified approach relies on a nonstandard presentation of the particle filter, which has the advantage of highlighting precisely where the differences between these algorithms stem from. Some relevant extensions and successful application domains of the particle filter are also presented.

Polycrystals illuminated by high-energy X-rays or neutrons produce diffraction patterns in which the measured diffraction peaks encode the individual single crystal strain states. While state of the art X-ray and neutron diffraction approaches can be used to routinely recover per grain mean strain tensors, less work has been produced on the recovery of higher order statistics of the strain distributions across the individual grains. In the setting of small deformations, we consider the problem of estimating the crystal elastic strain tensor probability distribution from diffraction data. For the special case of multivariate Gaussian strain tensor probability distributions, we show that while the mean of the distribution is well defined from measurements, the covariance of strain has a null-space. We show that there exist exactly 6 orthogonal perturbations to this covariance matrix under which the measured strain signal is invariant. In particular, we provide analytical parametrisations of these perturbations together with the set of possible maximum-likelihood estimates for a multivariate Gaussian fit to data. The parametric description of the null-space provides insights into the strain PDF modes that cannot be accurately estimated from the diffraction data. Understanding these modes prevents erroneous conclusions from being drawn based on the data. Beyond Gaussian strain tensor probability densities, we derive an iterative radial basis regression scheme in which the strain tensor probability density is estimated by a sparse finite basis expansion. This is made possible by showing that the operator mapping the strain tensor probability density onto the measured histograms of directional strain is linear, without approximation. The utility of the proposed algorithm is demonstrated by numerical simulations in the setting of single crystal monochromatic X-ray scattering. The proposed regression methods were found to robustly reject outliers and accurately predict the strain tensor probability distributions in the presence of Gaussian measurement noise.

This letter considers the problem of determining an optimal control action based on observed data. We formulate the problem assuming that the system can be modeled by a nonlinear state-space model, but where the model parameters, state and future disturbances are not known and are treated as random variables. Central to our formulation is that the joint distribution of these unknown objects is conditioned on the observed data. Crucially, as new measurements become available, this joint distribution continues to evolve so that control decisions are made accounting for uncertainty as evidenced in the data. The resulting problem is intractable which we obviate by providing approximations that result in finite dimensional deterministic optimization problems. The proposed approach is demonstrated in simulation on a nonlinear system.

This paper presents a method for calculating the smoothed state distribution for Jump Markov Linear Systems. More specifically, the paper details a novel two-filter smoother that provides closed-form expressions for the smoothed hybrid state distribution. This distribution can be expressed as a Gaussian mixture with a known, but exponentially increasing, number of Gaussian components as the time index increases. This is accompanied by exponential growth in memory and computational requirements, which rapidly becomes intractable. To ameliorate this, we limit the number of allowed mixture terms by employing a Gaussian likelihood mixture reduction strategy, which results in a computationally tractable, but approximate smoothed distribution. The approximation error can be balanced against computational complexity in order to provide an accurate and practical smoothing algorithm that compares favourably to existing state-of-the-art approaches.

Jump Markov linear systems (JMLS) are a useful model class for capturing abrupt changes in system behaviour that are temporally random, such as when a fault occurs. In many situations, accurate knowledge of the model is not readily available and can be difficult to obtain based on first principles. This paper presents a method for learning parameter values of this model class based on available input¿output data using the maximum-likelihood framework. In particular, the expectation¿maximisation method is detailed for this model class with attention given to a deterministic and numerically stable implementation. The presented algorithm is compared to state-of-the-art methods on several simulation examples with favourable results.

We provide a proof-of-concept for a novel state-space modelling approach for predicting monthly deaths due to political violence. Attention is focused on developing the method and demonstrating the utility of this approach, which provides exciting opportunities to engage with domain experts in developing new and improved state-space models for predicting violence. The prediction is made on a grid of cells with spatial resolution of 0.5 × 0.5 degrees, and each cell is modeled to have two mathematically well-defined unobserved/latent/hidden states that evolves over time and encode the "onset risk" and "potential severity", respectively. This offers a certain level of interpretability of the model. By using the model for computing the probability distribution for a death count at a future time conditioned on all data observed up until the current time, a predictive distribution is obtained. The predictive distribution typically places a certain mass at the death count 0 (no violent outbreak) and the remaining mass indicating a likely interval of the fatality count, should a violent outbreak appear. To evaluate the model performance we¿lacking a better alternative¿report the mean of the predictive distribution, but the access to the predictive distribution is in itself an interesting contribution to the application. This work merely serves as a proof-of-concept for the state-space modeling approach for this type of data and several possible directions for further work that could improve the predictive performance are suggested.

The modern world contains an immense number of different and interacting systems, from the evolution of weather systems to variations in the stock market, autonomous vehicles interacting with their environment, and the spread of diseases. For society to function, it is essential to understand the behavior of the world so that informed decisions can be made that are based on likely future outcomes. For instance, consider the spread of a new disease such as COVID-19 coronavirus. It is of great importance to be able to predict the number of people that will be infected at different points in time to ensure that appropriate health-care facilities are available. It is also of interest to be able to make decisions based on accurate information to best attenuate the spread of disease. Moreover, understanding specific attributes of a disease - such as the incubation time and number of unreported cases - and how certain we are about this knowledge are also crucial.

In this paper we present a novel quasi-Newton algorithm for use in stochastic optimisation. Quasi-Newton methods have had an enormous impact on deterministic optimisation problems because they afford rapid convergence and computationally attractive algorithms. In essence, this is achieved by learning the second-order (Hessian) information based on observing first-order gradients. We extend these ideas to the stochastic setting by employing a highly flexible model for the Hessian and infer its value based on observing noisy gradients. In addition, we propose a stochastic counterpart to standard line-search procedures and demonstrate the utility of this combination on maximum likelihood identification for general nonlinear state space models.

In 1959, the integrated circuit (IC) was invented simultaneously by Jack Kilby of Texas Instruments and Robert Noyce of Shockley Semiconductor [Ki lby, 2000]. This development has been considered one of mankind's most significant innovations.

Maximum likelihood (ML) estimation using Newton's method in nonlinear state space models (SSMs) is a challenging problem due to the analytical intractability of the loglikelihood and its gradient and Hessian. We estimate the gradient and Hessian using Fisher's identity in combination with a smoothing algorithm. We explore two approximations of the log-likelihood and of the solution of the smoothing problem. The first is a linearization approximation which is computationally cheap, but the accuracy typically varies between models. The second is a sampling approximation which is asymptotically valid for any SSM but is more computationally costly. We demonstrate our approach for ML parameter estimation on simulated data from two different SSMs with encouraging results.

This paper develops and illustrates a new maximum-likelihood based method for the identification of Hammerstein-Wiener model structures. A central aspect is that a very general situation is considered wherein multivariable data, non-invertible Hammerstein and Wiener nonlinearities, and colored stochastic disturbances both before and after the Wiener nonlinearity are all catered for. The method developed here addresses the blind Wiener estimation problem as a special case. © 2012 Elsevier Ltd. All rights reserved.

This paper addresses the design of gradient based search algorithms for multivariable system estimation. in particular, the work here considers so-called 'full parametrization' approaches, and establishes that the recently developed 'Data Driven Local Coordinate' (DDLC) methods can be seen as a special case within a broader class of techniques that are designed to deal with rank-deficient Jacobians. This informs the design of a new algorithm that, via a strategy of dynamic Jacobian rank determination, is illustrated to offer enhanced performance. Copyright © 2005 IFAC.

This paper describes a MATLAB-based software package for estimation of dynamic systems. A wide range of standard estimation approaches are sup- ported. These include the use of non-parametric, subspace-based and prediction- error algorithms coupled (in the latter case) with either MIMO state space or MISO polynomial model structures. A key feature of the software is the implementation of several new techniques that have been investigated by the authors. These include the estimation of non-linear models, the use of non-standard model parametrizations, and the employment of Expectation Maximisation (EM) methods. Copyright © 2005 IFAC.

This paper considers the system architecture and design issues for implementation of on-line Model Predictive Control (MPC) in Field Programmable Gate Arrays (FPGAs) and Application Specific Integrated Circuits (ASICs). In particular, the computationally itensive tasks of fast matrix QR factorisation, and subsequent sequential quadratic programming, are addressed for control law computation. An important aspect of this work is the study of appropriate data word-lengths for various essential stages of the overall solution strategy.

This article reviews authors' recently developed algorithm for identification of nonlinear state-space models under missing observations and extends it to the case of unknown model structure. In order to estimate the parameters in a state-space model, one needs to know the model structure and have an estimate of states. If the model structure is unknown, an approximation of it is obtained using radial basis functions centered around a maximum a posteriori estimate of the state trajectory. A particle filter approximation of smoothed states is then used in conjunction with expectation maximization algorithm for estimating the parameters. The proposed approach is illustrated through a real application. © 2009 IFAC.

The standard continuous time state space model with stochastic disturbances contains the mathematical abstraction of continuous time white noise. To work with well defined, discrete time observations, it is necessary to sample the model with care. The basic issues are well known, and have been discussed in the literature. However, the consequences have not quite penetrated the practise of estimation and identification. One example is that the standard model of an observation being a snapshot of the current state plus noise independent of the state cannot be reconciled with this picture. Another is that estimation and identification of time continuous models require a more careful treatment of the sampling formulas. We discuss and illustrate these issues in the current contribution. An application of particular practical importance is the estimation of models based on irregularly sampled observations. Copyright © 2007 International Federation of Automatic Control All Rights Reserved.

The Wiener model is a block oriented model having a linear dynamic system followed by a static nonlinearity. The dominating approach to estimate the components of this model has been to minimize the error between the simulated and the measured outputs. We show that this will in general lead to biased estimates if there is other disturbances present than measurement noise. The implications of Bussgang's theorem in this context are also discussed. For the case with general disturbances we derive the Maximum Likelihood method and show how it can be efficiently implemented. Comparisons between this new algorithm and the traditional approach confirm that the new method is unbiased and also has superior accuracy. Copyright © 2007 International Federation of Automatic Control All Rights Reserved.

The nonlinearity that arises in constrained MPC (model predictive control) satisfies an IQC (integral quadratic constraint), provided zero is feasible. It is thus possible to construct a robust stability test against any model uncertainty that also satisfies an IQC. In particular the test may be applied to structured and unstructured infinity-norm bound uncertainty. The test may be applied with predictive controllers of arbitrary horizon. The test is illustrated for several simple MPC schemes and simulation results are shown for a two-input two-output plant with left matrix fraction uncertainty. Copyright © 2006 IFAC.

We show that a sufficient condition for the robust stability of constrained linear model predictive control is for the plant to be open-loop stable, for zero to be a feasible solution of the associated quadratic programme and for the input weighting be sufficiently high. The result can be applied equally to state feedback and output feedback controllers with arbitrary prediction horizon. If integral action is included a further condition on the steady state modelling error is required for stability. Copyright © 2005 IFAC.

In constrained linear model predictive control a quadratic program must be solved on-line at each control step. If zero is feasible the resultant static nonlinearity is sector bound. We show that the nonlinearity is also monotone nondecreasing and slope restricted; furthermore it may be expressed as the gradient of a convex potential function. Hence we show the existence of Zames-Falb multipliers for such a nonlinearity. For completeness, we construct such multipliers both for the general case of multi-input multi-output static nonlinearities and for the particular case where the nonlinearity arises from a quadratic program. We also express the results in terms of integral quadratic constraints. These multipliers may be used in a general and versatile analysis of the robust stability of constrained model predictive control. © 2005 IEEE.

The theory of self-concordant barriers was introduced by Nesterov and Nemirovskii (1994) in the context of interior-point methods for convex optimisation. Their development is general, elegant and enjoys widespread implementation in state-of-the-art. algorithms. In this paper we exploit the theory of self-concordant functions with application to nonlinear MPC. In particular we construct an invariant terminal constraint set via properties of self-concordant functions. We also extend earlier results on recentred barrier function MPC to nonlinear MPC (model predictive control) with state constraints. We show nominal closed-loop stability for a wide class of nonlinear systems under full state feedback. Copyright

We discuss the design of cross-directional controllers which are guaranteed to be robustly stabilizing while incorporating a quadratic program for steady state performance. In particular we propose implementing cross-directional controllers in modal form with a constrained internal model control structure. Nominal optimal steady state performance is guaranteed via a non-linear element that incorporates a quadratic program. The quadratic program can be expressed as a continuous sector bounded nonlinearity together with two linear transformations. Thus the multivariable circle criterion can be used to guarantee closed-loop stability in the presence of disturbances and modeling uncertainties.

© 2003 EUCA. Actuator constraint handling is necessary for many cross-directional controllers. We discuss how optimal steady state performance can be guaranteed by modifying an internal model control structure with a non-linear element. For the simple dynamics associated with most web processes this also gives good closed-loop dynamic behaviour. Thus unconstrained control design techniques may be applied directly to the constrained control problem.

Actuator constraint handling is necessary for many cross-directional controllers. We discuss how optimal steady state performance can be guaranteed by modifying an internal model control structure with a non-linear element. For the simple dynamics associated with most web processes this also gives good closed-loop dynamic behaviour. Thus unconstrained control design techniques may be applied directly to the constrained control problem.