State estimation of switched nonlinear systems and systems with bounded inputs: Entropy and bit rates

State estimation is a fundamental problem when monitoring and controlling dynamical systems. Engineering systems interconnect sensing and computing devices over shared bandwidth-limited channels, and therefore, estimation algorithms should strive to use bandwidth optimally. Often, the dynamics of these systems are affected by external factors. In certain cases, these factors would lead the system to switch between different modes. In other cases, they would affect the dynamics of the system continuously in time without leading to explicit mode transitions. In this thesis, we present two notions of entropy for state estimation of nonlinear switched and non-autonomous dynamical systems as lower bounds on the average number of bits needed to be sent from the sensors to the estimators to estimate the states with deterministic (worst case) error bounds. Our approach relies on the notion of topological entropy and uses techniques from control under limited information. Since the computation of these entropies is hard in general, we compute corresponding upper bounds. Additionally, we design a state estimation algorithm for switched systems when their modes cannot be observed. We show that the average bit rate used by the algorithm is optimal in the sense that the efficiency gap is within an additive constant from the gap between the entropy of the considered system and its computed upper-bound. Finally, we apply our theory and algorithms to linear and nonlinear models of systems such as a glycemic index for diabetic patients, a controller of a Harrier jet and a Pendulum

Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations

This work derives a residual-based a posteriori error estimator for reduced models learned with non-intrusive model reduction from data of high-dimensional systems governed by linear parabolic partial differential equations with control inputs. It is shown that quantities that are necessary for the error estimator can be either obtained exactly as the solutions of least-squares problems in a non-intrusive way from data such as initial conditions, control inputs, and high-dimensional solution trajectories or bounded in a probabilistic sense. The computational procedure follows an offline/online decomposition. In the offline (training) phase, the high-dimensional system is judiciously solved in a black-box fashion to generate data and to set up the error estimator. In the online phase, the estimator is used to bound the error of the reduced-model predictions for new initial conditions and new control inputs without recourse to the high-dimensional system. Numerical results demonstrate the workflow of the proposed approach from data to reduced models to certified predictions

Decision-Feedback Detection Strategy for Nonlinear Frequency-Division Multiplexing

By exploiting a causality property of the nonlinear Fourier transform, a novel decision-feedback detection strategy for nonlinear frequency-division multiplexing (NFDM) systems is introduced. The performance of the proposed strategy is investigated both by simulations and by theoretical bounds and approximations, showing that it achieves a considerable performance improvement compared to previously adopted techniques in terms of Q-factor. The obtained improvement demonstrates that, by tailoring the detection strategy to the peculiar properties of the nonlinear Fourier transform, it is possible to boost the performance of NFDM systems and overcome current limitations imposed by the use of more conventional detection techniques suitable for the linear regime

Sum-of-Squares approach to feedback control of laminar wake flows

A novel nonlinear feedback control design methodology for incompressible fluid flows aiming at the optimisation of long-time averages of flow quantities is presented. It applies to reduced-order finite-dimensional models of fluid flows, expressed as a set of first-order nonlinear ordinary differential equations with the right-hand side being a polynomial function in the state variables and in the controls. The key idea, first discussed in Chernyshenko et al. 2014, Philos. T. Roy. Soc. 372(2020), is that the difficulties of treating and optimising long-time averages of a cost are relaxed by using the upper/lower bounds of such averages as the objective function. In this setting, control design reduces to finding a feedback controller that optimises the bound, subject to a polynomial inequality constraint involving the cost function, the nonlinear system, the controller itself and a tunable polynomial function. A numerically tractable approach to the solution of such optimisation problems, based on Sum-of-Squares techniques and semidefinite programming, is proposed. To showcase the methodology, the mitigation of the fluctuation kinetic energy in the unsteady wake behind a circular cylinder in the laminar regime at Re=100, via controlled angular motions of the surface, is numerically investigated. A compact reduced-order model that resolves the long-term behaviour of the fluid flow and the effects of actuation, is derived using Proper Orthogonal Decomposition and Galerkin projection. In a full-information setting, feedback controllers are then designed to reduce the long-time average of the kinetic energy associated with the limit cycle. These controllers are then implemented in direct numerical simulations of the actuated flow. Control performance, energy efficiency, and physical control mechanisms identified are analysed. Key elements, implications and future work are discussed

A New Approach to Linear/Nonlinear Distributed Fusion Estimation Problem

Disturbance noises are always bounded in a practical system, while fusion estimation is to best utilize multiple sensor data containing noises for the purpose of estimating a quantity--a parameter or process. However, few results are focused on the information fusion estimation problem under bounded noises. In this paper, we study the distributed fusion estimation problem for linear time-varying systems and nonlinear systems with bounded noises, where the addressed noises do not provide any statistical information, and are unknown but bounded. When considering linear time-varying fusion systems with bounded noises, a new local Kalman-like estimator is designed such that the square error of the estimator is bounded as time goes to $\infty$. A novel constructive method is proposed to find an upper bound of fusion estimation error, then a convex optimization problem on the design of an optimal weighting fusion criterion is established in terms of linear matrix inequalities, which can be solved by standard software packages. Furthermore, according to the design method of linear time-varying fusion systems, each local nonlinear estimator is derived for nonlinear systems with bounded noises by using Taylor series expansion, and a corresponding distributed fusion criterion is obtained by solving a convex optimization problem. Finally, target tracking system and localization of a mobile robot are given to show the advantages and effectiveness of the proposed methods.Comment: 9 pages, 3 figure