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their performances are compared. Equations that can be used to calculate observer gains are mentioned. Different methods that can be used to implement observers for time-delay systems are illustrated. Various stable and unstable systems are used and H∞ bounds are calculated using these observer designing methods. Delays are assumed to be known constants for all systems. H∞ gains are calculated numerically using disturbance signals and performances of observers are compared.
The primary goal of this thesis is to implement the observer for Time Delay Systems designed using SOS and compare its performance with existing H∞ optimal observers. These observers are more general than other observers for time-delay systems as they make corrections to the delayed state as well along with the present state. The observer dynamics can be represented by an ODE coupled with a PDE. Results shown in this thesis show that this type of observers performs better than other H∞ observers. Sub-optimal observer-based state feedback system is also generated and simulated using the SOS observer. The simulation results show that the closed loop system converges very quickly, and the observer can be used to design full state-feedback closed loop system.
The first class includes linear coupled PDEs with one spatial variable. Parabolic, elliptic or hyperbolic PDEs with Dirichlet, Neumann, Robin or mixed boundary conditions can be reformulated in order to be used by the framework. As an example, the reformulation is presented for systems governed by Schr¨odinger equation, parabolic type, relativistic heat conduction PDE and acoustic wave equation, hyperbolic types. The second form of PDEs of interest are scalar-valued with two spatial variables. An extra spatial variable allows consideration of problems such as local stability of fluid flows in channels and dynamics of population over two dimensional domains.
The approach does not involve discretization and is based on using Sum-of-Squares (SOS) polynomials and positive semi-definite matrices to parameterize operators which are positive on function spaces. Applying the parameterization to construct Lyapunov functionals with negative derivatives allows to express stability conditions as a set of LinearMatrix Inequalities (LMIs). The MATLAB package SOSTOOLS was used to construct the LMIs. The resultant LMIs then can be solved using existent Semi-Definite Programming (SDP) solvers such as SeDuMi or MOSEK. Moreover, the proposed approach allows to calculate bounds on the rate of decay of the solution norm.
The methodology is tested using several numerical examples and compared with the results obtained from simulation using standard methods of numerical discretization and analytic solutions.
The potential threat of Near Earth Objects (NEOs) and the need for effective planetary defense strategies has become increasingly urgent. While a range of mitigation techniques exist, the development of a space elevator could provide significant advantages in planetary defense. The current mitigation strategies require the use of a rocket in order to intercept the NEOs, and therefore launch lighter interceptors at lower velocities. However, the implementation of a space elevator would allow releasing heavier interceptors at much higher velocities. These capabilities combined with faster response times, make space elevators a much more efficient response to planetary defense. By using computational simulations on MATLAB to calculate intercept trajectories and model the new orbit of the NEOs after impact, this paper demonstrates that the use of space elevators can significantly improve the current strategies for planetary defense.
The objective of this report is to discover a skyhook’s ability to change the plane of another spacecraft’s orbit while ensuring that each vehicle’s orbital energy remains constant. Skyhooks are a proposed momentum exchange device in which a tether is attached to a counterweight at one end and at the other, a capturing device intended to intercept rendezvousing spacecraft. Trigonometric velocity vector relations, along with objective comparisons to traditionally proposed uses for skyhooks and gravity-assist maneuvers were responsible for the ultimate parameterization of the proposed energy neutral maneuver. From this methodology, it was determined that a spacecraft’s initial relative velocity vector must be perpendicular to, and rotated about the skyhook’s total velocity vector if it is to benefit from an energy neutral plane change maneuver. A quaternion was used to model the rotation of the incoming spacecraft’s relative velocity vector. The potential post-maneuver spacecraft orbits vary in their inclinations depending on the ratio between the skyhook and spacecraft’s total velocities at the point of rendezvous as defined by the parameter called the alpha criterion. For many cases, the proposed maneuver will serve as a desirable alternative to currently practiced propulsive plane change methods because it does not costly require a substantial amount of propellant. The proposed maneuver is also more accessible than alternative methods that involve gravity-assist and aerodynamic forces. Additionally, by avoiding orbital degradation through the achievement of unchanging total orbital energy, the skyhook will be able to continually and self-sustainably provide plane changes to any spacecraft that belong to orbits that abide by the identified parameters.
We propose parallel algorithms for stability analysis of two classes of systems: 1) Linear systems with a large number of uncertain parameters; 2) Nonlinear systems defined by polynomial vector fields. First, we develop a distributed parallel algorithm which applies Polya's and/or Handelman's theorems to some variants of parameter-dependent Lyapunov inequalities with parameters defined over the standard simplex. The result is a sequence of SDPs which possess a block-diagonal structure. We then develop a parallel SDP solver which exploits this structure in order to map the computation, memory and communication to a distributed parallel environment. Numerical tests on a supercomputer demonstrate the ability of the algorithm to efficiently utilize hundreds and potentially thousands of processors, and analyze systems with 100+ dimensional state-space. Furthermore, we extend our algorithms to analyze robust stability over more complicated geometries such as hypercubes and arbitrary convex polytopes. Our algorithms can be readily extended to address a wide variety of problems in control such as Hinfinity synthesis for systems with parametric uncertainty and computing control Lyapunov functions.
This dissertation presents a control-theoretic analysis of three types of mean-field models proposed in the literature for modelling and control of large-scale multi-agent systems, including robotic swarms. These mean-field models are Kolmogorov forward equations of stochastic processes, and their analysis is motivated by the fact that as the number of agents tends to infinity, the empirical measure associated with the agents converges to the solution of these models. Hence, the problem of transporting a swarm of agents from one distribution to another can be posed as a control problem for the forward equation of the process that determines the time evolution of the swarm density.
First, this thesis considers the case in which the agents' states evolve on a finite state space according to a continuous-time Markov chain (CTMC), and the forward equation is an ordinary differential equation (ODE). Defining the agents' task transition rates as the control parameters, the finite-time controllability, asymptotic controllability, and stabilization of the forward equation are investigated. Second, the controllability and stabilization problem for systems of advection-diffusion-reaction partial differential equations (PDEs) is studied in the case where the control parameters include the agents' velocity as well as transition rates. Third, this thesis considers a controllability and optimal control problem for the forward equation in the more general case where the agent dynamics are given by a nonlinear discrete-time control system. Beyond these theoretical results, this thesis also considers numerical optimal transport for control-affine systems. It is shown that finite-volume approximations of the associated PDEs lead to well-posed transport problems on graphs as long as the control system is controllable everywhere.
This dissertation presents a control-theoretic analysis of mean-field models for which the agent dynamics are governed by either a continuous-time Markov chain on an arbitrary state space, or a discrete-time Markov chain on a continuous state space. Three main problems are investigated. First, the problem of stabilization is addressed, that is, the design of transition probabilities/rates of the Markov process (the agent control parameters) that make a target distribution, satisfying certain conditions, invariant. Such a control approach could be used to achieve desired multi-agent distributions for spatial coverage and task allocation. However, the convergence of the multi-agent distribution to the designed equilibrium does not imply the convergence of the individual agents to fixed states. To prevent the agents from continuing to transition between states once the target distribution is reached, and thus potentially waste energy, the second problem addressed within this dissertation is the construction of feedback control laws that prevent agents from transitioning once the equilibrium distribution is reached. The third problem addressed is the computation of optimized transition probabilities/rates that maximize the speed at which the system converges to the target distribution.