In this note, the concept of wide-band noise is analyzed via a certain integral representation. It is proved that there are infinitely many wide-band noise processes represented in integral form which correspond to the same autocovariance function. Based on this integral representation, a technique of reduction of a wide-band noise driven system to a white noise driven system is presented. This technique is used to modify the separation principle and the Kalman–Bucy filtering to wide-band noise driven systems.

We design a Lyapunov based-boundary feedback controller for achieving mixing in a three-dimensional (3-D) pipe flow governed by Navier–Stokes equations. We show that the control law maximizes a measure related to mixing (that incorporates stretching and folding of material elements), while at the same time minimizing the control effort and the sensing effort. The penalty on sensing results in a static output-feedback control law (rather than full-state feedback).We also derive a lower bound on the gain from the control effort to the mixing measure. Furthermore, we establish input–output-to-state-stability properties for the open-loop system. These results show a form of detectability of mixing in the interior of the pipe from the chosen outputs on the wall. The effectiveness of the optimal control in achieving mixing enhancement is demonstrated in numerical simulations.

We identify a class of feedforward nonlinear systems that are linearizable by a coordinate change. Then we develop explicit expressions for the Lyapunov-based integrator forwarding recursive procedure of Sepulchre, Jankovic, and Kokotovic, which has its roots in a coordinate transformation proposed by Mazenc and Praly. The explicit expressions that we develop allow us to also find closed-form control laws for several classes of systems that are not feedback linearizable, including some that are in the feedforward form and others that are in what we refer to as the “block-feedforward” form. Performance advantages of Lyapunov-based forwarding controllers over nested saturation controllers have been well illustrated in the literature on examples. The analytical expressions for the Lyapunov functions and the control laws allow us to give quantitative performance bounds.

We examine the stability properties of a class of LTV difference equations on an infinite-dimensional state space that arise in backstepping designs for parabolic PDEs. The nominal system matrix of the difference equation has a special structure: all of its powers have entries that are -1, 0, or 1, and all of the eigenvalues of the matrix are on the unit circle. The difference equation is driven by initial conditions, additive forcing, and a system matrix perturbation, all of which depend on problem data (for example, viscosity and reactivity in the case of a reaction-diffusion equation), and all of which go to zero as the discretization step in the backstepping design goes to zero. All of these observations, combined with the fact that the equation evolves only in a number of steps equal to the dimension of its state space, combined with the discrete Gronwall inequality, establish that the difference equation has bounded solutions. This, in turn, guarantees the existence of a state-feedback gain kernel in the backstepping control law. With this approach we greatly expand, relative to our previous results, the class of parabolic PDEs to which backstepping is applicable. (C) 2003 Published by Elsevier B.V.

Extensive effort in controller design of aircraft systems is invested in robustness to ensure safe, stable behavior. Particular attention is placed on anomalous flight-conditions harbored by the atmosphere, especially icing. The thesis presents a regulation trigger-based adaptive controller to cope mathematically with the impact of ice on the aircraft equations of motion and control the aircraft pitch to the commanded angle. Upon an introduction to the problem, a pitch model of an aircraft system is derived, where the impact of icing is modeled. The design of a stabilizing certainty-equivalence controller utilizing backstepping follows, and is succeeded by the introduction of the Batch Least-Squares Identifier (BaLSI). Finally, simulation results of an aircraft experiencing icing demonstrates the effectiveness of the identifier, with the trajectory of the iced system utilizing the proposed identifier closely following that of the nominal system.

This dissertation develops a systematic model-based approach for the boundary control and estimation of freeway traffic congestion problem. Three topics of traffic congestion on a freeway segment are studied and include stop-and-go traffic oscillations, moving traffic shockwave, and downstream traffic bottleneck, which are governed by different partial differential equation (PDE) models and require the advancement and application of three PDE control techniques.

To supress stop-and-go oscillations, we introduce the macroscopic Aw-Rascle-Zhang traffic model, consisting of second-order nonlinear hyperbolic PDEs that govern dynamics of traffic density and velocity. The hetero-directional propagations of information in congested traffic generate the instabilities, motivating us to the stabilization problem for a coupled $2\times 2$ hyperbolic system. Using the backstepping method, a full-state feedback control is designed for ramp metering at outlet to actuate the outgoing traffic flow. We design boundary observer for state estimation and combine it with the full state feedback control to construct an output feedback controller. The observer design is validated with traffic field data. Under model parameter uncertainties, adaptive control design is proposed with on-line parameter estimation. Furthermore, we develop output feedback boundary control for two types of $4\times4$ nonlinear hyperbolic PDEs which arise from two-lane and two-class traffic congestion. Stabilization of two-lane traffic involves regulation of the lane-changing interactions with lane-specific varying speed limits while stabilization of two-class traffic tackles the heterogeneity of vehicles and drivers.

A moving traffic shockwave, caused by changes of local road situations, segregates light traffic upstream and heavy traffic downstream. This density discontinuity travels upstream. As a result, drivers caught in the shockwave experience transitions from free to congested traffic. The interface position is governed by an ordinary differential equation (ODE) dependent on the density of the PDE states, described with Lighthill-Whitham-Richards model. For the coupled PDE-ODE system, the predictor feedback design is applied to compensate the state-dependent input delays. We design bilateral boundary controllers to drive the moving shockwave front to a desirable setpoint position, hindering the upstream propagation of the traffic congestion.

Traffic on a freeway segment with capacity drop at outlet causes a downstream bottleneck. Traffic congestion forms because the traffic at the outlet overflows its capacity. Therefore the incoming flow of the segment needs to be regulated so that the outgoing traffic at the bottleneck area is discharged with its maximum flow rate. Since the traffic dynamics of the bottleneck is hard to model, we apply extremum seeking control, a model free approach for real-time optimization, to obtain the optimal input density at the inlet. The predictor feedback design is combined with the extremum seeking to compensate the delay effect of traffic state of the segment. The maximum flow rate is achieved at the bottleneck by regulating its upstream density at the inlet.

This dissertation presents the study of some advancements in the control theory and application of the systems described by partial differential equations. Three classes of partial differential equations are discussed, mainly concerning three topics: stability analysis, boundary controller design and disturbance rejection.

Stability analysis is a topic of great interest and importance in the control area, for which the Lyapunov method and the operator semigroup theory are used in this dissertation. Stability analysis can serve as a guidance of controller design with the objective of stabilization, and the most frequently employed approach for system controller designs in this work is the backstepping methodology. We design state feedback boundary controllers based on backstepping, and further construct output feedback controllers in assistance with the state observers designed to re- cover the full system states. Moreover, the sliding mode control and the active disturbance rejection control techniques are applied for the purpose of disturbance rejection.

In detail, the following three classes of systems are investigated. The first class consists of the Korteweg-de Vries systems. For a (nonlinear) Korteweg-de Vries equation, the asymptotic stability analysis is conducted. Two methods, one relying on normal forms and the other relying on a Lyapunov approach, are em- ployed based on the center manifold theory. Both prove that the equation is (lo- cally) asymptotically stable around the origin. A class of linear Korteweg-de Vries systems with possible anti-diffusion is also discussed, focusing on the backstepping controller designs. The resulting closed-loop control systems exhibit exponential decay with respect to the corresponding state norms.

The second is a general class of coupled bidirectional hyperbolic systems with spatially varying coefficients, concerning stability analysis, controller design and disturbance rejection problems. This study borrows the controller design idea from an existing result for the systems with constant coefficients, and extends it to deal with time-varying coefficients. As a result, finite-time stability is achieved. With regards to the systems running into control matched uncertainties and disturbances, disturbance rejection and disturbance attenuation are achieved for the closed-loop feedback systems with sliding mode control and active disturbance rejection control, respectively.

Furthermore, the application of estimation techniques to a real-world issue is studied, i.e., the state-of-charge estimation problem in lithium-ion batteries. A thermal-compensated electrochemical model of lithium-ion batteries is proposed. Adding thermal dynamics serves a two-fold purpose: improving the accuracy of state-of-charge estimation and keeping track of the average temperature which is critical for battery safety management. This model can be included by the third class, i.e., the parabolic systems with time-varying coefficients. Note that in these systems, the time dependency of the system coefficients makes the observer design problem nontrivial. This dissertation can be conducive to future related endeavors.

The work in this dissertation summarizes some advancement in the theory of boundary controller design for coupled partial differential equations (PDEs), including a new interpretation of designing bilateral boundary controllers as equivalent coupled PDEs via a method dubbed “folding.” In particular, systems of purely hyperbolic type, purely parabolic type, and mixed (hyperbolic-parabolic) type are all explored in the context of boundary control. The work centers around the method of infinite-dimensional backstepping consisting of an non-trivially invertible spatial transformation mapping to a system with desireable properties, such stability of equilibria and convergence speed. A companion kernel PDE must be solved to properly define the transform. The transformations across the different classes of PDEs are similiar (and in certain cases, identical); however, curious behavior arises from a heterogeneous mixed type PDE, in which the companion kernel PDE becomes non-standard. The thesis studies some preliminary work into mixed type PDEs in an attempt to recover a more general backstepping design for linear PDE.

In the purely hyperbolic work, a special case of a underactuated hyperbolic system is considered. This is in opposition to pre-existing literature, which assumed a fully actuated system. The classical backstepping boundary controller is modified for the underactuated hyperbolic case, admitting a two-tiered trasnformation approach in which the backstepping controller is augmented by a predictor-based controller to achieve a finite-time stability for the trivial solution of the system. In the purely parabolic work, the notion of the folding approach is introduced as an alternative design method to pre-existing bilateral boundary control design work. The folding approach admits additional design parameters for the control designer, allowing the controls to be biased for differing performance indexes. A complimentary state estimator is designed, which allows for collocated point measurement at any arbitrary point in the domain independent of the control design. The two are combined to achieve an output-feedback control result.

Several results are given for mixed type PDE systems of hyperbolic-parabolic type as well. A first result invovles a scalar system with coupling on the boundary and the interior. The interior coupling necessitates more advanced techniques in the analysis of the companion kernel PDE, particularly in showing well-posedness. These ideas are also applied to other higher-order coupled systems of hyperbolic-parabolic type, including delay compensation for systems of parabolic PDEs, and delay compensation for bilateral controller design of parabolic PDEs.

Both the notion of bilateral boundary control and mixed type PDE systems arise in short- wavelength light generation. In the state-of-the-art light generation at the extreme ultraviolet (EUV) wavelengths, instabilities in the light generation process potentially arise due to coupled plasma interactions with the generating process. Phenomena such as ion-acoustic waves, free- electron plasma diffusion, magnetohydrodynamics, and thermofluidics arise due to the interacting plasma, introducing potential modes of instability. This instability neceessitates the introduction of feedback control, which can be introduced via two controllers on either boundary of the process domain.

The dissertation presents results on control and state estimation for a physics-based ``Stefan" model of phase change. Design procedures, theoretical analysis, applications to industrial processes, and experimental validation are addressed. The Stefan model describes a time-evolution of a material's temperature profile during melting/solidification phenomena along with the dynamics of the liquid-solid interface position. The mathematical description comprises a parabolic Partial Differential Equation (PDE), defined on a time-varying spatial domain, whose boundary position dynamics are governed by an Ordinary Differential Equation (ODE) driven by the PDE's state. None of the existing systematic and theoretical control are applicable to this problem due to the system's geometric nonlinearity as well as the infinite dimensionality. We design a boundary heat control to promote the melting so that the liquid-solid interface position is driven to a desired setpoint position. Our design is an extension of the ``PDE backstepping" method to the Stefan system. The closed-loop stability is proven by Lyapunov analysis. The constraints of the temperature state and the heat input are guaranteed by virtue of the maximum principle. Analogous results for the state estimation are also developed to estimate the entire temperature profile from available measurements of the surface temperature and the liquid-solid interface position.

The latter half of the dissertation is devoted to the application of the designed method to several practical problems. First, we introduce a Stefan model of ``sea ice", which has been studied intensively due to the recent rapid melting of sea ice. We verify the desired robust performance of the designed estimator in a numerical simulation, which incorporates further complexity in the model and uncertainties. Second, we focus on ``lithium-ion batteries", which have become ubiquitous in electronic devices, such as laptops and smartphones, and in electric vehicles. The model is described by a Stefan system of the lithium-ion concentration due to a solid-solid phase change in the electrodes during the charging and discharging cycles. Our estimator achieves accurate State-of-Charge estimation in simulation. Third, we apply the designed control method to ``polymer 3D-printing" via screw extrusion for the sake of stabilizing the filament production under a fast printing, by extending the design to deal with the convection and heat loss. Fourth, we focus on ``metal 3D-printing" which has a high impact on products and supply chains in industries. The proposed control method is applied for generating the desired shape of the melt pool, which shows the robust performance with respect to a radiation effect and sensor noise in numerical study. Finally, we conduct experiments of melting paraffin wax as an energy storage material, which shows the successful performance of our PDE-based control algorithm.

The Isotope Production Facility (IPF) receives pulsed H+ beam from the accelerator at the Los Alamos Neutron Science Center (LANSCE) to produce isotopes for various research and medical uses. Work over 2 summers has been conducted to improve diagnostics, reliability, and control for equipment on the IPF beamline, as well as to optimize beam delivered to IPF to increase isotope yield. This second goal is the subject of this document. Specifically, it is desired to maximize beam delivery onto a target without risking damage to the target window or spill too much beam outside a specified aperture. A specified beam profile and repeating beam raster pattern of concentric circles is currently being used to achieve this. The goal addressed here is to determine an improved beam profile and raster pattern. While a specific answer is needed for certain operation conditions, a general method is even better. A constrained optimization problem is formulated, and several approaches for solving it are discussed. It is found that with appropriate design of cost function, a Gradient Descent algorithm is effective at making improvements to a good initial guess. A bounded Extremum Seeking algorithm also shows some effectiveness, while some unbounded versions did not show effectiveness for this problem. The problem formulation, models, methods, results, are presented. They show significant potential to advance IPF’s goals of higher isotope production yield.