A number of emerging applications in the field of optimal control theory require the computation of an open-loop control for a dynamical system with uncertain parameters. In this dissertation we examine a class of uncertain optimal control problems, in which the goal is to minimize the expectation of a predetermined cost functional subject to such an uncertain system. We provide a computational framework for this class of problems based on a discretization of the parameter space. In this approach, a set of nodes from the parameter space and corresponding weights are selected, and the expectation of the cost functional is approximated by a finite sum. This process results in a sequence of standard optimal control problems which can be solved using existing techniques. However, it is well-known that an inappropriately designed discretization scheme for an optimal control problem may fail to converge to the optimal solution, therefore further analysis must be performed to examine the convergence properties of the scheme. We provide this analysis for a scheme based on quadrature methods for the approximation of the expectation in the cost functional. This analysis demonstrates that an accumulation point of a sequence of optimal solutions to the approximate problem is an optimal solution of the original problem. Furthermore, we examine the convergence of the adjoint states for the approximation based on the quadrature scheme, which leads to a Pontryagin-like necessary condition which must be satisfied by these accumulation points. To address the exponential growth of computational cost with respect to the dimension of the parameters, we introduce a numerical algorithm based on sample average approximations, in which an independently drawn random sample is taken from the parameter space, and the expectation in the objective functional is approximated by the sample mean. Using a generalization of the strong law of large numbers, we analyze the convergence properties of this approximation. In addition, we develop an optimality function for the class of uncertain optimal control problems based on the L2-Frechet derivative of the objective functional, which provides a necessary condition for an optimal solution. By demonstrating that an accumulation point of a sequence of stationary points for the approximate problem is a stationary point of the original problem, we demonstrate the approximation scheme based on sample averages is consistent in the sense of Polak. These numerical algorithms for the uncertain optimal control problem are applied to real-world scenarios from the fields of optimal search theory and ensemble control.

Designed for one-semester introductory senior-or graduate-level course, the authors provide the student with an introduction of analysis techniques used in the design of nonlinear and optimal feedback control systems. There is special emphasis on the fundamental topics of stability, controllability, and optimality, and on the corresponding geometry associated with these topics. Each chapter contains several examples and a variety of exercises.

Discrete-Time Inverse Optimal Control for Nonlinear Systems proposes a novel inverse optimal control scheme for stabilization and trajectory tracking of discrete-time nonlinear systems. This avoids the need to solve the associated Hamilton-Jacobi-Bellman equation and minimizes a cost functional, resulting in a more efficient controller. Design More Efficient Controllers for Stabilization and Trajectory Tracking of Discrete-Time Nonlinear Systems The book presents two approaches for controller synthesis: the first based on passivity theory and the second on a control Lyapunov function (CLF). The synthesized discrete-time optimal controller can be directly implemented in real-time systems. The book also proposes the use of recurrent neural networks to model discrete-time nonlinear systems. Combined with the inverse optimal control approach, such models constitute a powerful tool to deal with uncertainties such as unmodeled dynamics and disturbances. Learn from Simulations and an In-Depth Case Study The authors include a variety of simulations to illustrate the effectiveness of the synthesized controllers for stabilization and trajectory tracking of discrete-time nonlinear systems. An in-depth case study applies the control schemes to glycemic control in patients with type 1 diabetes mellitus, to calculate the adequate insulin delivery rate required to prevent hyperglycemia and hypoglycemia levels. The discrete-time optimal and robust control techniques proposed can be used in a range of industrial applications, from aerospace and energy to biomedical and electromechanical systems. Highlighting optimal and efficient control algorithms, this is a valuable resource for researchers, engineers, and students working in nonlinear system control.

This book discusses methods and algorithms for the near-optimal adaptive control of nonlinear systems, including the corresponding theoretical analysis and simulative examples, and presents two innovative methods for the redundancy resolution of redundant manipulators with consideration of parameter uncertainty and periodic disturbances. It also reports on a series of systematic investigations on a near-optimal adaptive control method based on the Taylor expansion, neural networks, estimator design approaches, and the idea of sliding mode control, focusing on the tracking control problem of nonlinear systems under different scenarios. The book culminates with a presentation of two new redundancy resolution methods; one addresses adaptive kinematic control of redundant manipulators, and the other centers on the effect of periodic input disturbance on redundancy resolution. Each self-contained chapter is clearly written, making the book accessible to graduate students as well as academic and industrial researchers in the fields of adaptive and optimal control, robotics, and dynamic neural networks.

A focused presentation of how sparse optimization methods can be used to solve optimal control and estimation problems.

The book consists mainly of two parts: Chapter 1 - Chapter 7 and Chapter 8 - Chapter 14. Chapter 1 and Chapter 2 treat design techniques based on linearization of nonlinear systems. An analysis of nonlinear system over quantum mechanics is discussed in Chapter 3. Chapter 4 to Chapter 7 are estimation methods using Kalman filtering while solving nonlinear control systems using iterative approach. Optimal approaches are discussed in Chapter 8 with retarded control of nonlinear system in singular situation, and Chapter 9 extends optimal theory to H-infinity control for a nonlinear control system.Chapters 10 and 11 present the control of nonlinear dynamic systems, twin-rotor helicopter and 3D crane system, which are both underactuated, cascaded dynamic systems. Chapter 12 applies controls to antisynchronization/synchronization in the chaotic models based on Lyapunov exponent theorem, and Chapter 13 discusses developed stability analytic approaches in terms of Lyapunov stability. The analysis of economic activities, especially the relationship between stock return and economic growth, is presented in Chapter 14.