Lyapunov based optimal control of a class of nonlinear systems

Optimal control of nonlinear systems is in fact difficult since it requires the solution to the Hamilton-Jacobi-Bellman (HJB) equation which has no closed-form solution. In contrast to offline and/or online iterative schemes for optimal control, this dissertation in the form of five papers focuses on the design of iteration free, online optimal adaptive controllers for nonlinear discrete and continuous-time systems whose dynamics are completely or partially unknown even when the states not measurable. Thus, in Paper I, motivated by homogeneous charge compression ignition (HCCI) engine dynamics, a neural network-based infinite horizon robust optimal controller is introduced for uncertain nonaffine nonlinear discrete-time systems. First, the nonaffine system is transformed into an affine-like representation while the resulting higher order terms are mitigated by using a robust term. The optimal adaptive controller for the affinelike system solves HJB equation and identifies the system dynamics provided a target set point is given. Since it is difficult to define the set point a priori in Paper II, an extremum seeking control loop is designed while maximizing an uncertain output function. On the other hand, Paper III focuses on the infinite horizon online optimal tracking control of known nonlinear continuous-time systems in strict feedback form by using state and output feedback by relaxing the initial admissible controller requirement. Paper IV applies the optimal controller from Paper III to an underactuated helicopter attitude and position tracking problem. In Paper V, the optimal control of nonlinear continuous-time systems in strict feedback form from Paper III is revisited by using state and output feedback when the internal dynamics are unknown. Closed-loop stability is demonstrated for all the controller designs developed in this dissertation by using Lyapunov analysis --Abstract, page iv

Finite-horizon optimal control of linear and a class of nonlinear systems

Traditionally, optimal control of dynamical systems with known system dynamics is obtained in a backward-in-time and offline manner either by using Riccati or Hamilton-Jacobi-Bellman (HJB) equation. In contrast, in this dissertation, finite-horizon optimal regulation has been investigated for both linear and nonlinear systems in a forward-in-time manner when system dynamics are uncertain. Value and policy iterations are not used while the value function (or Q-function for linear systems) and control input are updated once a sampling interval consistent with standard adaptive control. First, the optimal adaptive control of linear discrete-time systems with unknown system dynamics is presented in Paper I by using Q-learning and Bellman equation while satisfying the terminal constraint. A novel update law that uses history information of the cost to go is derived. Paper II considers the design of the linear quadratic regulator in the presence of state and input quantization. Quantization errors are eliminated via a dynamic quantizer design and the parameter update law is redesigned from Paper I. Furthermore, an optimal adaptive state feedback controller is developed in Paper III for the general nonlinear discrete-time systems in affine form without the knowledge of system dynamics. In Paper IV, a NN-based observer is proposed to reconstruct the state vector and identify the dynamics so that the control scheme from Paper III is extended to output feedback. Finally, the optimal regulation of quantized nonlinear systems with input constraint is considered in Paper V by introducing a non-quadratic cost functional. Closed-loop stability is demonstrated for all the controller designs developed in this dissertation by using Lyapunov analysis while all the proposed schemes function in an online and forward-in-time manner so that they are practically viable --Abstract, page iv

Decentralized adaptive neural network control of interconnected nonlinear dynamical systems with application to power system

Traditional nonlinear techniques cannot be directly applicable to control large scale interconnected nonlinear dynamic systems due their sheer size and unavailability of system dynamics. Therefore, in this dissertation, the decentralized adaptive neural network (NN) control of a class of nonlinear interconnected dynamic systems is introduced and its application to power systems is presented in the form of six papers. In the first paper, a new nonlinear dynamical representation in the form of a large scale interconnected system for a power network free of algebraic equations with multiple UPFCs as nonlinear controllers is presented. Then, oscillation damping for UPFCs using adaptive NN control is discussed by assuming that the system dynamics are known. Subsequently, the dynamic surface control (DSC) framework is proposed in continuous-time not only to overcome the need for the subsystem dynamics and interconnection terms, but also to relax the explosion of complexity problem normally observed in traditional backstepping. The application of DSC-based decentralized control of power system with excitation control is shown in the third paper. On the other hand, a novel adaptive NN-based decentralized controller for a class of interconnected discrete-time systems with unknown subsystem and interconnection dynamics is introduced since discrete-time is preferred for implementation. The application of the decentralized controller is shown on a power network. Next, a near optimal decentralized discrete-time controller is introduced in the fifth paper for such systems in affine form whereas the sixth paper proposes a method for obtaining the L2-gain near optimal control while keeping a tradeoff between accuracy and computational complexity. Lyapunov theory is employed to assess the stability of the controllers --Abstract, page iv

Optimal Adaptive Tracking Control Of Partially Uncertain Nonlinear Discrete-Time Systems Using Lifelong Hybrid Learning

This article addresses a multilayer neural network (MNN)-based optimal adaptive tracking of partially uncertain nonlinear discrete-time (DT) systems in affine form. By employing an actor–critic neural network (NN) to approximate the value function and optimal control policy, the critic NN is updated via a novel hybrid learning scheme, where its weights are adjusted once at a sampling instant and also in a finite iterative manner within the instants to enhance the convergence rate. Moreover, to deal with the persistency of excitation (PE) condition, a replay buffer is incorporated into the critic update law through concurrent learning. To address the vanishing gradient issue, the actor and critic MNN weights are tuned using control input and temporal difference errors (TDEs), respectively. In addition, a weight consolidation scheme is incorporated into the critic MNN update law to attain lifelong learning and overcome catastrophic forgetting, thus lowering the cumulative cost. The tracking error, and the actor and critic weight estimation errors are shown to be bounded using the Lyapunov analysis. Simulation results using the proposed approach on a two-link robot manipulator show a significant reduction in tracking error by $44\%$ and cumulative cost by $31\%$ in a multitask environment

Online Optimal Adaptive Control of Partially Uncertain Nonlinear Discrete-Time Systems using Multilayer Neural Networks

This article intends to address an online optimal adaptive regulation of nonlinear discrete-time systems in affine form and with partially uncertain dynamics using a multilayer neural network (MNN). The actor-critic framework estimates both the optimal control input and value function. Instantaneous control input error and temporal difference are used to tune the weights of the critic and actor networks, respectively. The selection of the basis functions and their derivatives are not required in the proposed approach. The state vector, critic, and actor NN weights are proven to be bounded using the Lyapunov method. Our approach can be extended to neural networks with an arbitrary number of hidden layers. We have demonstrated our approach via a simulation example

Optimal control of nonlinear partially-unknown systems with unsymmetrical input constraints and its applications to the optimal UAV circumnavigation problem

Aimed at solving the optimal control problem for nonlinear systems with unsymmetrical input constraints, we present an online adaptive approach for partially unknown control systems/dynamics. The designed algorithm converges online to the optimal control solution without the knowledge of the internal system dynamics. The optimality of the obtained control policy and the stability for the closed-loop dynamic optimality are proved theoretically. The proposed method greatly relaxes the assumption on the form of the internal dynamics and input constraints in previous works. Besides, the control design framework proposed in this paper offers a new approach to solve the optimal circumnavigation problem involving a moving target for a fixed-wing unmanned aerial vehicle (UAV). The control performance of our method is compared with that of the existing circumnavigation control law in a numerical simulation and the simulation results validate the effectiveness of our algorithm