Nonlinear control theory

novel approach to extend the decision-making capabilities of unmanned surface vehicles
(USVs) is presented in this work. A multi-objective framework is described where separate
controllers command different behaviors according to a desired trajectory. Three behaviors
are examined – transiting, station-keeping and reversing. Given the desired trajectory, the
vehicle is able to autonomously recognize which behavior best suits a portion of the
trajectory. The USV uses a combination of a supervisory switching control structure and a
reinforcement learning algorithm to create a hybrid deliberative and reactive approach to
switch between controllers and actions. Reinforcement learning provides a deliberative
method to create a controller switching policy, while supervisory switching control acts
reactively to instantaneous changes in the environment. Each action is restricted to one
controller. Due to the nonlinear effects in these behaviors, two underactuated backstepping
controllers and a fully-actuated backstepping controller are proposed for each transiting, reversing and station-keeping behavior, respectively, restricted to three degrees of freedom.
Field experiments are presented to validate this system on the water with a physical USV
platform under Sea State 1 conditions. Main outcomes of this work are that the proposed
system provides better performance than a comparable gain-scheduled nonlinear controller
in terms of an Integral of Absolute Error metric. Additionally, the deliberative component
allows the system to identify dynamically infeasible trajectories and properly
accommodate them.

There have been much technological advances and research in Unmanned Surface
Vehicles (USV) as a support and delivery platform for Autonomous/Unmanned
Underwater Vehicles (AUV/UUV). Advantages include extending underwater search and
survey operations time and reach, improving underwater positioning and mission
awareness, in addition to minimizing the costs and risks associated with similar manned
vessel operations. The objective of this thesis is to present the design and development a
high-level fuzzy logic guidance controller for a WAM-V 14 USV in order to
autonomously launch and recover a REMUS 100 AUV. The approach to meeting this objective is to develop ability for the USV to intercept and rendezvous with an AUV that is in transit in order to maximize the probability of a final mobile docking maneuver. Specifically, a fuzzy logic Rendezvous Docking controller has been developed that generates Waypoint-Heading goals for the USV to minimize the cross-track errors between the USV and AUV. A subsequent fuzzy
logic Waypoint-Heading controller has been developed to provide the desired heading
and speed commands to the low-level controller given the Waypoint-Heading goals.
High-level mission control has been extensively simulated using Matlab and partially
characterized in real-time during testing. Detailed simulation, experimental results and
findings will be reported in this paper.

Autonomous Underwater Vehicle (AUV) depth control methods typically use a
pressure sensor to measure the depth, which results in the AUV following the trajectory
of the surface waves. Through simulations, a controller is designed for the Ocean
Explorer AUV with the objective of the AUV holding a constant depth below the still
water line while operating in waves. This objective is accomplished by modeling sensors
and using filtering techniques to provide the AUV with the depth below the still water
line. A wave prediction model is simulated to provide the controller with knowledge of
the wave disturbance before it is encountered. The controller allows for depth keeping
below the still water line with a standard deviation of 0.04 and 0.65 meters for wave
amplitudes of 0.1-0.25 and 0.5-2 meters respectively and wave frequencies of 0.35-1.0
𝑟𝑎𝑑⁄𝑠𝑒𝑐, and the wave prediction improves the depth control on the order of 0.03 meters.

The aim of this thesis is to develop a theory for non stationary propulsor flow noise. The model which is proposed is based on Amiet's paper "Acoustic Radiation from an Airfoil in a Turbulent Stream" [1], which describes broad band noise when a simple model of airfoil interacts with a turbulent flow, under the assumption of stationarity. The Karhunen-Loeve method provides a set of modes which describe the turbulent flow without the assumption of stationarity. A method is described to obtain broad band noise calculations when the mean turbulent flow varies with time and produces non stationary turbulence. A comparison of the numerical results obtained with the results from the paper of reference [1] shows the characteristics of time varying sound radiation. The various mathematical formulae will give a starting point to the analysis of real time varying flows, which are not considered in this thesis.

The tasks Autonomous Underwater Vehicles (AUVs) are expected to perform are becoming more and more challenging. Thus, to be able to address such tasks, we implemented a high maneuverability propulsion system: a vectored thruster. The design of a vehicle equipped with such a propulsion system will be presented, from a mechanical, electronic and software point of view. The motion control of the resulting system is fairly complex, and no suitable controller is available in the literature. Accordingly, we will present the derivation of a novel tracking controller, whose adaptive properties will compensate for the lack of knowledge of the system's parameters. Computer simulations are provided and show the performance and robustness of the proposed control algorithm to external perturbations, unmodelled dynamics and dynamics variation. We finally illustrate the advantage of using an adaptive controller by comparing the presented controller to a Proportional Integral Derivative controller.

Underactuated mechanical systems are those possessing fewer actuators than degrees of freedom, making the class a rich one from a control standpoint. The double inverted pendulum is a particular underactuated system and a well-known benchmark case for which many solutions have been offered in the literature. The control objective is to bring the system to its unstable top equilibrium point. The underactuated horizontal double pendulum is a two-link planar robot with only one actuator either at the shoulder or the elbow. Almost no work was done on the underactuated horizontal pendulum, mainly due to the lack of controllability of such a system. The fundamental difference between a double inverted pendulum and an underactuated horizontal double pendulum is that in the latter gravity effects do not exist. Gravity is important to the controllability of the system. Thus, in search for a "gravity substitute," we added springs in the underactuated horizontal double pendulum in order to create a source of potential energy. Two different types of such systems are analyzed: spring coupled underactuated horizontal double pendulums and underactuated horizontal double pendulums with spring-loaded sliding bar constraint. The main contribution of the thesis is in proving that the zero state of the spring coupled systems is globally asymptotically stabilizable. Explicit control laws were developed.

The problem at hand is developing a controller design methodology that is generally applicable to autonomous systems with fairly accurate models. The controller design process has two parts: synthesis and analysis. Over the years, many synthesis and analysis methods have been proposed. An optimal method for all applications has not yet been found. Recent advances in computer technology have made computational methods more attractive and practical. The proposed method is an iterative computational method that automatically generates non-linear controllers with specified global performance. This dissertation describes this method which consists of using an analysis tool, continued propagation cell mapping (CPCM), as feedback to the synthesis tool, best estimate directed search (BEDS). Optimality in the design can be achieved with respect to time, energy, and/or robustness depending on the performance measure used. BEDS is based on a novel search concept: globally directing a random search. BEDS has the best of two approaches: gradient (or directed) search and random search. It possesses the convergence speed of a gradient search and the convergence robustness of a random search. The coefficients of the best controller at the time direct the search process until either a better controller is found or the search is terminated. CPCM is a modification of simple cell mapping (SCM). CPCM maintains the simplicity of SCM but provides accuracy near that of a point map (PM). CPCM evaluates the controller's complete and global performance efficiently and with easily tunable accuracy. This CPCM evaluation guarantees monotonic progress in the synthesis process. The method is successfully applied to the design of a TSK-type fuzzy logic (FL) controller and a Sliding Mode-type controller for the uncertain non-linear system of an inverted pendulum on a cart for large pole angles (+/-86 degrees). The resulting controller's performance compares favorably to other established methods designed with dynamic programing (DP) and genetic algorithms (GA). When CPCM is used as feedback to BEDS, the resulting design method quickly and automatically generates non-linear controllers with good global performance and without much a priori information about the desired control actions.

The objectives of this research as deliberated in this dissertation are two-folded: (i) To study the nonlinear activity in the neural complex (real and artificial) and (ii) to analyze the learning processe(s) pertinent to an artificial neural network in the information-theoretic plane using cross-entropy error-metrics. The research efforts envisaged enclave the following specific tasks: (i) Obtaining a general solution for the Bernoulli-Riccati equation to represent a single parameter family of S-shaped (sigmoidal) curves depicting the nonlinear activity in the neural network. (ii) Analysis of the logistic growth of output versus input values in the neural complex (real and artificial) under the consideration that the boundaries of the sets constituting the input and output entities are crisp and/or fuzzy. (iii) Construction of a set of cross-entropy error-metrics (known as Csiszar's measures) deduced in terms of the parameters pertinent to a perceptron topology and elucidation of their relative effectiveness in training the network optimally towards convergence. (iv) Presenting the methods of symmetrizing and balancing the aforesaid error-entropy measures (in the information-theoretic plane) so as to make them usable as error-metrics in the test domain. (v) Description and analysis of the dynamics of neural learning process in the information-theoretic plane for both crisp and fuzzy attributes of input values. Relevant to these topics portraying the studies on nonlinear activity and cross-entropy considerations vis-a-vis neural networks, newer and/or exploratory inferences are made, logical conclusions are enumerated and relative discussions are presented along with the scope for future research to be pursued.

It is desirable to have robust high performance nonlinear control with a model-free design approach for the real time automatic control of practical industrial processes. The field has seen the application of Sliding Mode Controllers (SMCs). SMCs are nonlinear robust controllers, however most design approaches related to SMCs are model-based approaches. PID controllers and some Fuzzy Logic Controllers (FLCs) are model-free controllers, however their robustness is not integrated into their design parameters directly. This dissertation presents two new types of robust high performance nonlinear controllers with model-free design approaches. One introduces fuzzy logic to a model-free SMC which is a simple saturation function incorporating three design parameters. Due to the interpolative nature of fuzzy control, a TSK type FLC with the model-free SMCs as its rule's consequents will produce a controller with a nonlinear sliding curve and a nonlinear boundary layer. We call this controller a Fuzzy Sliding Controller (FSC). The other uses a new type of Variable Structure Controller (VSC), which intentionally switches from one controller to another controller during a step response. In conventional approaches to VSC, the control surface does not change its shape during a step response. The new type of VSC intentionally changes the shape of the control surface during the step response. This technique is analogous to that technique employed in image processing called "morphing" where a given image gradually changes over time to the image of a different entity. In order to avoid confusion with the conventional approach to a VSC, we use the term "Morphological" Controller (MC) for the VSC of the new type. The performance and robustness with respect to parameter variations, disturbances and slow sample rates of the proposed controllers are studied in detail with a DC motor and an Inverted Pendulum System. As a means to verify the proposed controllers in practical cases, we design the model-free SMC, the FSC and the MC for the highly nonlinear and uncertain dynamics of an Autonomous Underwater Vehicle, Ocean Voyager II. It is shown that the proposed controllers are high performance and high robustness controllers.

Design and Tuning a fuzzy logic controller (FLCs) are usually done in two stages. In the first stage, the structure of a FLC is determined based on physical characteristics of the system. In the second stage, the parameters of the FLC are selected to optimize the performance of the system. The task of tuning FLCs can be performed by a number of methods such as adjusting control gains, changing membership functions, modifying control rules and varying control surfaces. A method for the design and tuning of FLCs through modifying their control surfaces is presented in this dissertation. The method can be summarized as follows. First, fuzzy control surfaces are modeled with Bezier functions. Shapes of the control surface are then adjusted through varying Bezier parameters. A Genetic Algorithm (GA) is used to search for the optimal set of parameters based on the control performance criteria. Then, tuned control surfaces are sampled to create rule-based FLCs. To further improve the system performance, continuity constraints of the curves are imposed. Under the continuity constraints with the same number of tunable parameters, one can obtain more flexible curves that have the potential to improve the overall system performance. An important issue is to develop a new method to self-tune a fuzzy PD controller. The method is based on two building blocks: (I) Bezier functions used to model the control surfaces of the fuzzy PD controller; and, shapes of control surfaces are then adjusted by varying Bezier parameters. (II) The next step involves using a gradient-based optimization algorithm with which the input scaling factors and Bezier parameters are on-line tuned until the controller drives the output of the process as close as possible to the reference position. To protect vendors and consumers from being victimized, various trust models have been used in e-commerce practices. However, a strict verification and authentication process may pose unnecessary heavy cost to the vendor. As an application of the control strategy proposed, this dissertation presents a solution to the reduction of costs of a vendor. With two fuzzy variables (price, credit-history), a trust-surface can be tuned to achieve an optimal solution in terms of profit margin of the vendor. With this new approach, more realistic trust decisions can be reached.