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This paper considers the problem of determining an optimal control action based on observed data. We formulate the problem assuming that the system can be modelled 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 optimisation problems. The proposed approach is demonstrated in simulation on a nonlinear system., Comment: in IEEE Control Systems Letters, 2021

This paper considers the problem of computing Bayesian estimates of both states and model parameters for nonlinear state-space models. Generally, this problem does not have a tractable solution and approximations must be utilised. In this work, a variational approach is used to provide an assumed density which approximates the desired, intractable, distribution. The approach is deterministic and results in an optimisation problem of a standard form. Due to the parametrisation of the assumed density selected first- and second-order derivatives are readily available which allows for efficient solutions. The proposed method is compared against state-of-the-art Hamiltonian Monte Carlo in two numerical examples.

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.

This paper considers Bayesian parameter estimation of dynamic systems using a Markov Chain Monte Carlo (MCMC) approach. The Metroplis-Hastings (MH) algorithm is employed, and the main contribution of the paper is to examine and illustrate the efficacy of a particular proposal density based on energy preserving Hamiltonian dynamics, which results in what is known in the statistics literature as ``Hamiltonian Monte--Carlo'' (HMC). The very significant utility of this approach is that, as will be illustrated, it greatly reduces (almost to the point of elimination) the typically very high correlation in the Metropolis--Hastings chain which has been observed by several authors to restrict the application of the MH approach to only very low dimension model structures. The paper illustrates how the HMC approach may be applied to both significant dimension linear and nonlinear model structures, even when the system order is unknown, and using both simulated and real data.

Jump Markov linear systems (JMLS) are a useful class which can be used to model processes which exhibit random changes in behavior during operation. This paper presents a numerically stable method for learning the parameters of jump Markov linear systems using the expectation-maximisation (EM) approach. The solution provided herein is a deterministic algorithm, and is not a Monte Carlo based technique. As a result, simulations show that when compared to alternative approaches, a more likely set of system parameters can be found within a fixed computation time, which better explain the observations of the system.

This paper presents a Bayesian method for identification of jump Markov linear system parameters. A primary motivation is to provide accurate quantification of parameter uncertainty without relying on asymptotic in data-length arguments. To achieve this, the paper details a particle-Gibbs sampling approach that provides samples from the desired posterior distribution. These samples are produced by utilising a modified discrete particle filter and carefully chosen conjugate priors.

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 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.

Pseudo-marginal Metropolis-Hastings (pmMH) is a versatile algorithm for sampling from target distributions which are not easy to evaluate point-wise. However, pmMH requires good proposal distributions to sample efficiently from the target, which can be problematic to construct in practice. This is especially a problem for high-dimensional targets when the standard random-walk proposal is inefficient. We extend pmMH to allow for constructing the proposal based on information from multiple past iterations. As a consequence, quasi-Newton (qN) methods can be employed to form proposals which utilize gradient information to guide the Markov chain to areas of high probability and to construct approximations of the local curvature to scale step sizes. The proposed method is demonstrated on several problems which indicate that qN proposals can perform better than other common Hessian-based proposals., Comment: 45 pages and 11 figures. Submitted to journal. Fixed typos

The computation of Bayesian estimates of system parameters and functions of them on the basis of observed system performance data is a common problem within system identification. This is a previously studied issue where stochastic simulation approaches have been examined using the popular Metropolis--Hastings (MH) algorithm. This prior study has identified a recognised difficulty of tuning the {proposal distribution so that the MH method provides realisations with sufficient mixing to deliver efficient convergence. This paper proposes and empirically examines a method of tuning the proposal using ideas borrowed from the numerical optimisation literature around efficient computation of Hessians so that gradient and curvature information of the target posterior can be incorporated in the proposal., Comment: 16 pages, 2 figures. Accepted for publication in the Proceedings of the 18th IFAC Symposium on System Identification (SYSID)

This paper considers the problem of estimating linear dynamic system models when the observations are corrupted by random disturbances with nonstandard distributions. The paper is particularly motivated by applications where sensor imperfections involve significant contribution of outliers or wrap-around issues resulting in multi-modal distributions such as commonly encountered in robotics applications. As will be illustrated, these nonstandard measurement errors can dramatically compromise the effectiveness of standard estimation methods, while a computational Bayesian approach developed here is demonstrated to be equally effective as standard methods in standard measurement noise scenarios, but dramatically more effective in nonstandard measurement noise distribution scenarios., Comment: 17 pages, 4 figures. Accepted for publication in the Proceedings of the 18th IFAC Symposium on System Identification (SYSID). Typos corrected

A Bayesian filtering algorithm is developed for a class of state-space systems that can be modelled via Gaussian mixtures. In general, the exact solution to this filtering problem involves an exponential growth in the number of mixture terms and this is handled here by utilising a Gaussian mixture reduction step after both the time and measurement updates. In addition, a square-root implementation of the unified algorithm is presented and this algorithm is profiled on several simulated systems. This includes the state estimation for two non-linear systems that are strictly outside the class considered in this paper.

In this paper, we study networks of discrete-time linear time-invariant subsystems. Our focus is on situations where subsystems are connected to each other through a time-invariant topology and where there exists a base-station whose aim is to control the subsystems into any desired destinations. However, the base-station can only communicate with some of the subsystems that we refer to as leaders. There are no direct links between the base-station and the rest of subsystems, known as followers, as they are only able to liaise among themselves and with some of the leaders. The current paper formulates this framework as the well-known reachability problem for linear systems. Then to address this problem, we introduce notions of leader-reachability and base-reachability. We present algebraic conditions under which these notions hold. It turns out that if subsystems are represented by minimal state space representations, then base-reachability always holds. Hence, we focus on leader-reachability and investigate the corresponding conditions in detail. We further demonstrate that when the networked system parameters i.e. subsystems' parameters and interconnection matrices, assume generic values then the whole network is both leader-reachable and base-reachable., Comment: The paper is submitted for possible publication in a journal