Exponential Stabilisation of Continuous-time Periodic Stochastic Systems by Feedback Control Based on Periodic Discrete-time Observations

Since Mao in 2013 discretised the system observations for stabilisation problem of hybrid SDEs (stochastic differential equations with Markovian switching) by feedback control, the study of this topic using a constant observation frequency has been further developed. However, time-varying observation frequencies have not been considered. Particularly, an observational more efficient way is to consider the time-varying property of the system and observe a periodic SDE system at the periodic time-varying frequencies. This study investigates how to stabilise a periodic hybrid SDE by a periodic feedback control, based on periodic discrete-time observations. This study provides sufficient conditions under which the controlled system can achieve pth moment exponential stability for p > 1 and almost sure exponential stability. Lyapunov's method and inequalities are main tools for derivation and analysis. The existence of observation interval sequences is verified and one way of its calculation is provided. Finally, an example is given for illustration. Their new techniques not only reduce observational cost by reducing observation frequency dramatically but also offer flexibility on system observation settings. This study allows readers to set observation frequencies according to their needs to some extent

Asymptotic stabilization of continuous-time periodic stochastic systems by feedback control based on periodic discrete-time observations

In 2013, Mao initiated the study of stabilization of continuoustime hybrid stochastic differential equations (SDEs) by feedback control based on discrete-time state observations. In recent years, this study has been further developed while using a constant observation interval. However, time-varying observation frequencies have not been discussed for this study. Particularly for non-autonomous periodic systems, it’s more sensible to consider the timevarying property and observe the system at periodic time-varying frequencies, in terms of control efficiency. This paper introduces a periodic observation interval sequence, and investigates how to stabilize a periodic SDE by feedback control based on periodic observations, in the sense that, the controlled system achieve Lp-stability for p > 1, almost sure asymptotic stability and pth moment asymptotic stability for p ≥ 2. This paper uses the Lyapunov method and inequalities to derive the theory. We also verify the existence of the observation interval sequence and explains how to calculate it. Finally, an illustrative example is given after a useful corollary. By considering the time-varying property of the system, we reduce the observation frequency dramatically and hence reduce the observational cost for control

Stability criteria for systems with colored multiplicative noise.

Massachusetts Institute of Technology. Dept. of Electrical Engineering. Thesis. 1974. Ph.D.MICROFICHE COPY ALSO AVAILABLE IN BARKER ENGINEERING LIBRARY.Vita.Bibliography: leaves 165-171.Ph.D

Game-Theoretic Safety Assurance for Human-Centered Robotic Systems

In order for autonomous systems like robots, drones, and self-driving cars to be reliably introduced into our society, they must have the ability to actively account for safety during their operation. While safety analysis has traditionally been conducted offline for controlled environments like cages on factory floors, the much higher complexity of open, human-populated spaces like our homes, cities, and roads makes it unviable to rely on common design-time assumptions, since these may be violated once the system is deployed. Instead, the next generation of robotic technologies will need to reason about safety online, constructing high-confidence assurances informed by ongoing observations of the environment and other agents, in spite of models of them being necessarily fallible.This dissertation aims to lay down the necessary foundations to enable autonomous systems to ensure their own safety in complex, changing, and uncertain environments, by explicitly reasoning about the gap between their models and the real world. It first introduces a suite of novel robust optimal control formulations and algorithmic tools that permit tractable safety analysis in time-varying, multi-agent systems, as well as safe real-time robotic navigation in partially unknown environments; these approaches are demonstrated on large-scale unmanned air traffic simulation and physical quadrotor platforms. After this, it draws on Bayesian machine learning methods to translate model-based guarantees into high-confidence assurances, monitoring the reliability of predictive models in light of changing evidence about the physical system and surrounding agents. This principle is first applied to a general safety framework allowing the use of learning-based control (e.g. reinforcement learning) for safety-critical robotic systems such as drones, and then combined with insights from cognitive science and dynamic game theory to enable safe human-centered navigation and interaction; these techniques are showcased on physical quadrotors—flying in unmodeled wind and among human pedestrians—and simulated highway driving. The dissertation ends with a discussion of challenges and opportunities ahead, including the bridging of safety analysis and reinforcement learning and the need to ``close the loop'' around learning and adaptation in order to deploy increasingly advanced autonomous systems with confidence

Stability of highly nonlinear hybrid stochastic integro-differential delay equations

For the past few decades, the stability criteria for the stochastic differential delay equations (SDDEs) have been studied intensively. Most of these criteria can only be applied to delay equations where their coefficients are either linear or nonlinear but bounded by linear functions. Recently, the stability criterion for highly nonlinear hybrid stochastic differential equations is investigated in Fei et al. (2017). In this paper, we investigate a class of highly nonlinear hybrid stochastic integro-differential delay equations (SIDDEs). First, we establish the stability and boundedness of hybrid stochastic integro-differential delay equations. Then the delay-dependent criteria of the stability and boundedness of solutions to SIDDEs are studied. Finally, an illustrative example is provided

Bayes Linear Strategies for the Approximation of Complex Numerical Calculations Arising in Sequential Design and Physical Modelling Problems.

In a range of different scientific fields, deterministic calculations for which there is no analytic solution must be approximated numerically. The use of numerical approximations is necessary, but introduces a discrepancy between the true solution and the numerical solution that is generated. Bayesian methods are used to account for uncertainties introduced through numerical approximation in a variety of situations.\\ To solve problems in Bayesian sequential experimental design, a sequence of complex integration and optimisation steps must be performed; for most problems, these calculations have no closed-form solution. An approximating framework is developed which tracks numerical uncertainty about the result of each calculation through each step of the design procedure. This framework is illustrated through application to a simple linear model, and to a more complex problem in atmospheric dispersion modelling. The approximating framework is also adapted to allow for the situation where beliefs about a model may change at certain points in the future.\\ Where ordinary or partial differential equation (ODE or PDE) systems are used to represent a real-world system, it is rare that these can be solved directly. A wide variety of different approximation strategies have been developed for such problems; the approximate solution that is generated will differ from the true solution in some unknown way. A Bayesian framework which accounts for the uncertainty induced through numerical approximation is developed, and Bayes linear graphical analysis is used to efficiently update beliefs about model components using observations on the real system. In the ODE case, the framework is illustrated through application to a Lagrangian mechanical model for the interaction between a set of ringing bells and the tower in which they are hung; in the PDE case, the framework is illustrated through application to the heat equation in one spatial dimension

Discontinuous Galerkin based Geostatistical Inversion of Stationary Flow and Transport Processes in Groundwater

The hydraulic conductivity of a confined aquifer is estimated using the quasi-linear geostatistical approach (QLGA), based on measurements of dependent quantities such as the hydraulic head or the arrival time of a tracer. This requires the solution of the steady-state groundwater flow and solute transport equations, which are coupled by Darcy's law. The standard Galerkin finite element method (FEM) for the flow equation combined with the streamline diffusion method (SDFEM) for the transport equation is widely used in the hydrogeologists' community. This work suggests to replace the first by the two-point flux cell-centered finite volume scheme (CCFV) and the latter by the Discontinuous Galerkin (DG) method. The convection-dominant case of solute (contaminant) transport in groundwater has always posed a special challenge to numerical schemes due to non-physical oscillations at steep fronts. The performance of the DG method is experimentally compared to the SDFEM approach with respect to numerical stability, accuracy and efficient solvability of the occurring linear systems. A novel method for the reduction of numerical under- and overshoots is presented as a very efficient alternative to local mesh refinement. The applicability and software-technical integration of the CCFV/DG combination into the large-scale inversion scheme mentioned above is realized. The high-resolution estimation of a synthetic hydraulic conductivity field in a 3-D real-world setting is simulated as a showcase on Linux high performance computing clusters. The setup in this showcase provides examples of realistic flow fields for which the solution of the convection-dominant transport problem by the streamline diffusion method fails

Stability analysis and control of stochastic dynamic systems using polynomial chaos

Recently, there has been a growing interest in analyzing stability and developing controls for stochastic dynamic systems. This interest arises out of a need to develop robust control strategies for systems with uncertain dynamics. While traditional robust control techniques ensure robustness, these techniques can be conservative as they do not utilize the risk associated with the uncertainty variation. To improve controller performance, it is possible to include the probability of each parameter value in the control design. In this manner, risk can be taken for parameter values with low probability and performance can be improved for those of higher probability. To accomplish this, one must solve the resulting stability and control problems for the associated stochastic system. In general, this is accomplished using sampling based methods by creating a grid of parameter values and solving the problem for each associated parameter. This can lead to problems that are difficult to solve and may possess no analytical solution. The novelty of this dissertation is the utilization of non-sampling based methods to solve stochastic stability and optimal control problems. The polynomial chaos expansion is able to approximate the evolution of the uncertainty in state trajectories induced by stochastic system uncertainty with arbitrary accuracy. This approximation is used to transform the stochastic dynamic system into a deterministic system that can be analyzed in an analytical framework. In this dissertation, we describe the generalized polynomial chaos expansion and present a framework for transforming stochastic systems into deterministic systems. We present conditions for analyzing the stability of the resulting systems. In addition, a framework for solving L2 optimal control problems is presented. For linear systems, feedback laws for the infinite-horizon L2 optimal control problem are presented. A framework for solving finite-horizon optimal control problems with time-correlated stochastic forcing is also presented. The stochastic receding horizon control problem is also solved using the new deterministic framework. Results are presented that demonstrate the links between stability of the original stochastic system and the approximate system determined from the polynomial chaos approximation. The solutions of these stochastic stability and control problems are illustrated throughout with examples

Lyapunov exponents for certain stochastic flows

This thesis examines the asymptotic behaviour of solution flows of certain stochastic differential equations utilising the theory of Lyapunov exponents. The approach is taken on two fronts. Initially flows are considered on compact manifolds that arise from embedding the manifold in some Euclidean space - the Gradient Brownian flow. In this case the existence of the Lyapunov exponents is known. Results are obtained for the sum of the exponents - which has the geometrical interpretation as the exponential rate of change of volume under the action of the flow - and for the largest exponent on generalised Clifford Tori and convex hypersurfaces. The situation on non-compact manifolds is then considered - where the existence of the exponents is uncertain due to the fact that the existence of a finite invariant measure is not guaranteed. However, by considering a stochastic mechanical system this problem is overcome and conditions for existence are obtained for both the Lyapunov spectrum and the sum' of the exponents. Some examples are then considered in the noncompact case. Finally in the Appendix a computational method for calculating the largest Lyapunov exponent on a hypersurface is considered