Nonlinear control with two complementary Lyapunov function

If a Lyapunov function is known, a dynamic system can be stabilized. However, computing or selecting a Lyapunov function is often challenging. This dissertation presents a new approach which eliminates this challenge: a simple control Lyapunov function [CLF] is assumed then the algorithm seeks to reduce the value of the Lyapunov function. If the control effort would have no effect at any iteration, the CLF is switched in an attempt to regain control. There is some flexibility in choosing these two complementary CLF’s but they must satisfy a few characteristics. The method is proven to asymptotically stabilize a wide range of nonlinear systems and was tested on an even broader variety in simulation. It was also tested on an industrial robot to provide compliant behavior. The simulated and hardware demonstrations provide a broad perspective on the algorithm’s usefulness and limitations. In comparison to the ubiquitous PID controller, the algorithm’s advantages include enhanced performance, ease of tuning, and extensions to higher-order and/or coupled systems. Those claimed advantages are validated by a test with four engineering students, which validates the controller as a viable option for nonlinear control (even at the undergraduate level). The algorithm’s drawbacks include the necessity of a dynamic model and, when linearization is required, the reliance on a small simulation time step; however, for the motivating application –interactive industrial robotic systems – both requirements were already met. Finally, the developed software was released to the public as part of the Robot Operating System (ROS) and the details of that release are included in this report.Mechanical Engineerin

Investigations of Model-Free Sliding Mode Control Algorithms including Application to Autonomous Quadrotor Flight

Sliding mode control is a robust nonlinear control algorithm that has been used to implement tracking controllers for unmanned aircraft systems that are robust to modeling uncertainty and exogenous disturbances, thereby providing excellent performance for autonomous operation. A significant advance in the application of sliding mode control for unmanned aircraft systems would be adaptation of a model-free sliding mode control algorithm, since the most complex and time-consuming aspect of implementation of sliding mode control is the derivation of the control law with incorporation of the system model, a process required to be performed for each individual application of sliding mode control. The performance of four different model-free sliding mode control algorithms was compared in simulation using a variety of aerial system models and real-world disturbances (e.g. the effects of discretization and state estimation). The two best performing algorithms were shown to exhibit very similar behavior. These two algorithms were implemented on a quadrotor (both in simulation and using real-world hardware) and the performance was compared to a traditional PID-based controller using the same state estimation algorithm and control setup. Simulation results show the model-free sliding mode control algorithms exhibit similar performance to PID controllers without the tedious tuning process. Comparison between the two model-free sliding mode control algorithms showed very similar performance as measured by the quadratic means of tracking errors. Flight testing showed that while a model-free sliding mode control algorithm is capable of controlling realworld hardware, further characterization and significant improvements are required before it is a viable alternative to conventional control algorithms. Large tracking errors were observed for both the model-free sliding mode control and PID based flight controllers and the performance was characterized as unacceptable for most applications. The poor performance of both controllers suggests tracking errors could be attributed to errors in state estimation, which effectively introduce unknown dynamics into the feedback loop. Further testing with improved state estimation would allow for more conclusions to be drawn about the performance characteristics of the model-free sliding mode control algorithms

Barrier Lyapunov function-based adaptive fuzzy attitude tracking control for rigid satellite with input delay and output constraint

This paper investigates the adaptive attitude tracking problem for the rigid satellite involving output constraint, input saturation, input time delay, and external disturbance by integrating barrier Lyapunov function (BLF) and prescribed performance control (PPC). In contrast to the existing approaches, the input delay is addressed by Pade approximation, and the actual control input concerning saturation is obtained by utilizing an auxiliary variable that simplifies the controller design with respect to mean value methods or Nussbaum function-based strategies. Due to the implementation of the BLF control, together with an interval notion-based PPC strategy, not only the system output but also the transformed error produced by PPC are constrained. An adaptive fuzzy controller is then constructed and the predesigned constraints for system output and the transformed error will not be violated. In addition, a smooth switch term is imported into the controller such that the finite time convergence for all error variables is guaranteed for a certain case while the singularity problem is avoided. Finally, simulations are provided to show the effectiveness and potential of the proposed new design techniques

Nonlinear Burn Condition and Kinetic Profile Control in Tokamak Fusion Reactors

One of the most promising devices for realizing power production through nuclear fusion is the tokamak. In order to maximize performance, it is preferable that tokamaks achieve operating scenarios characterized by good plasma confinement, improved magnetohydrodynamic stability, and a largely non-inductively driven plasma current. Such scenarios could enable steady-state reactor operation with high fusion gain, the ratio of fusion power produced to the external heating power needed to sustain reactions. There are many experimental tokamaks around the world, each exploring different facets of plasma physics and fusion technology. These experiments have reached the point where the power released from fusion is nearly equal to the power input required to heat the plasma. The next experimental step is ITER, which aims to reach a fusion gain exceeding ten for short pulses, and to sustain a gain of five for longer pulses (around 1000 s). In order for ITER to be a success, several challenging control engineering problems must be addressed.Among these challenges is to precisely regulate the plasma density and temperature, or burn condition. Due to the nonlinear and coupled dynamics of the system, modulation of the burn condition (either during ramp-up/shut-down or in response to changing power demands) without a well designed control scheme could result in undesirable transient performance. Feedback control will also be necessary for responding to unexpected changes in plasma confinement, impurity content, or other parameters, which could significantly alter the burn condition during operation. Furthermore, although stable operating points exist for most confinement scalings, certain conditions can lead to thermal instabilities. Such instabilities can either lead to quenching or a thermal excursion in which the system moves to a higher temperature equilibrium point. In any of these situations, disruptive plasma instabilities could be triggered, stopping operation and potentially causing damage to the confinement vessel.In this work, the problem of burn condition control is addressed through the design of a nonlinear control law guaranteeing stability of desired equilibria. Multiple actuation methods, including auxiliary heating, isotopic fueling, and impurity injection, are used to ensure the burn condition is regulated even when actuators saturate. An adaptive control scheme is used to handle model uncertainty, and an online optimization scheme is proposed to ensure that the plasma is driven to an operating point that minimizes an arbitrary cost function. Due to the possible limited availability of diagnostic systems in ITER and future reactors, an output feedback control scheme is also proposed that combines the nonlinear controller with an observer that estimates the states of the burning plasma system based on available measurements. Finally, the control scheme is tested using the integrated modeling code METIS.The control of spatial profiles of parameters, including current, density, and temperature, is also an important challenge in fusion research, due to their effect on MHD stability, non-inductive current drive, and fusion power. In this work, the problem of kinetic profile control in burning plasmas is addressed through a nonlinear boundary feedback control law designed using a technique called backstepping. A novel implementation of the backstepping technique is used that enables the use of both boundary and interior actuation. The backstepping technique is then applied to the problem of current profile control in both low-confinement and high-confinement mode discharges in the DIII-D tokamak based on a first-principles-driven model of the current profile evolution. Both designs are demonstrated in simulations and experimental tests

Robust output stabilization: improving performance via supervisory control

We analyze robust stability, in an input-output sense, of switched stable systems. The primary goal (and contribution) of this paper is to design switching strategies to guarantee that input-output stable systems remain so under switching. We propose two types of {\em supervisors}: dwell-time and hysteresis based. While our results are stated as tools of analysis they serve a clear purpose in design: to improve performance. In that respect, we illustrate the utility of our findings by concisely addressing a problem of observer design for Lur'e-type systems; in particular, we design a hybrid observer that ensures ``fast'' convergence with ``low'' overshoots. As a second application of our main results we use hybrid control in the context of synchronization of chaotic oscillators with the goal of reducing control effort; an originality of the hybrid control in this context with respect to other contributions in the area is that it exploits the structure and chaotic behavior (boundedness of solutions) of Lorenz oscillators.Comment: Short version submitted to IEEE TA

Time-Delay Systems

Time delay is very often encountered in various technical systems, such as electric, pneumatic and hydraulic networks, chemical processes, long transmission lines, robotics, etc. The existence of pure time lag, regardless if it is present in the control or/and the state, may cause undesirable system transient response, or even instability. Consequently, the problem of controllability, observability, robustness, optimization, adaptive control, pole placement and particularly stability and robustness stabilization for this class of systems, has been one of the main interests for many scientists and researchers during the last five decades

Feedback control of sector-bound nonlinear systems with applications to aeroengine control

This dissertation is divided into two parts. In the first part we consider the problem of feedback stabilization of nonlinear systems described by state-space models. This approach is inherited from the methodology of sector bounded or passive nonlinearities, and influenced by the concept of absolute and quadratic stability. It aims not only to regionally stabilize the nonlinear dynamics asymptotically but also to maximize the estimated region of quadratic attraction and to ensure nominal performance at each equilibrium. In close connection to gain scheduling and switching control, a path of equilibria is programmed based on the assumption of centered-epsilon-cover which leads to a sequence of linear controllers that regionally stabilize the desired equilibrium asymptotically. In the second part we tackle the problem of control for fluid flows described by the incompressible Navier-Stokes equation. We are particularly interested in film cooling for gas turbine engines which we model with the jet in cross-flow problem setup. In order to obtain a model amenable to the controller design presented in the first part, the well-known Proper Orthogonal Decomposition (POD)/Galerkin projection is employed to obtain a nonlinear state-space system called the reduced order model (ROM). We are able to stabilize the ROM to an equilibrium point via our design method and we also present direct numerical simulation (DNS) results for the system under state feedback control

Backstepping control with fixed-time prescribed performance for fixed wing UAV under model uncertainties and external disturbances

In this paper, a novel backstepping control scheme with fixed-time prescribed performance is proposed for the longitudinal model of fixed wing UAV subject to model uncertainties and external disturbances. The novel performance function with arbitrarily preassigned fixed-time convergence property is developed, which imposes priori performance envelops on both altitude and airspeed tracking errors. By using error transformed technology, the constrained fixed-time performance envelops are changed into unconstrained equivalent errors. Based on modified error compensation mechanism, a novel backstepping approach is proposed to guarantee altitude tracking equivalent error converges to the specified small neighborhood and presents excellent robustness against model uncertainties and external disturbances, and airspeed controller with fixed-time prescribed performance is designed. The proposed methodology guarantees the transient and steady-state performance of altitude and airspeed tracking errors within constrained fixed-time performance envelops in spite of lumped disturbances. Finally, numerical simulations are used to verify the effectiveness of the proposed control schem