Output feedback stabilization of linear PDEs with finite dimensional input-output maps and Kelvin-Voigt damping

In this paper, we consider systems of partial differential equations with a finite relative degree between the input and the output. In such systems, an output feedback controller can be constructed to regulate the output with the desired convergence properties. Although the zero dynamics are infinite dimensional, we show that the controller alters the boundary conditions in such a way that it leads to a predictable expansion in the stable operating envelope of the system. Moreover, the expansion of the stable envelope depends only on the boundary conditions and the structure of the PDE, and is independent of the system parameters. The methodology is extended to output tracking and time-varying forcing functions as well. The phenomenon investigated in the paper is quite unique to partial differential equations and without any parallel in systems of ODEs

Output feedback stabilization of linear PDEs with finite dimensional input-output maps and Kelvin-Voigt damping

In this paper, we consider systems of partial differential equations with a finite relative degree between the input and the output. In such systems, an output feedback controller can be constructed to regulate the output with the desired convergence properties. Although the zero dynamics are infinite dimensional, we show that the controller alters the boundary conditions in such a way that it leads to a predictable expansion in the stable operating envelope of the system. Moreover, the expansion of the stable envelope depends only on the boundary conditions and the structure of the PDE, and is independent of the system parameters. The methodology is extended to output tracking and time-varying forcing functions as well. The phenomenon investigated in the paper is quite unique to partial differential equations and without any parallel in systems of ODEs

Backstepping-Based Exponential Stabilization of Timoshenko Beam with Prescribed Decay Rate

This is an open access article under the CC BY-NC-ND license.In this paper, we present a rapid boundary stabilization of a Timoshenko beam with anti-damping and anti-stiffness at the uncontrolled boundary, by using PDE backstepping. We introduce a transformation to map the Timoshenko beam states into a (2+2) × (2+2) hyperbolic PIDE-ODE system. Then backstepping is applied to obtain a control law guaranteeing closed-loop stability of the origin in the H1 sense. Arbitrarily rapid stabilization can be achieved by adjusting control parameters. Finally, a numerical simulation shows that the proposed controller can rapidly stabilize the Timoshenko beam. This result extends a previous work which considered a slender Timoshenko beam with Kelvin-Voigt damping, allowing destabilizing boundary conditions at the uncontrolled boundary and attaining an arbitrarily rapid convergence rate

Optimal Actuator Location for Semi-Linear Systems

Actuator location and design are important choices in controller design for distributed parameter systems. Semi-linear partial differential equations model a wide spectrum of physical systems with distributed parameters. It is shown that under certain conditions on the nonlinearity and the cost function, an optimal control input together with an optimal actuator choice exist. First order necessary optimality conditions are derived. The results are applied to optimal actuator location and controller design in a nonlinear railway track model.NSER

Optimal Actuator Design for Nonlinear Systems

For systems modeled by partial differential equations (PDE's), the location and shape of the actuators can be regarded as a design variable and included as part of the controller synthesis procedure. Optimal actuator location is a special case of optimal design. Appropriate actuator location and design can improve performance and significantly reduce the cost of the control in a distributed parameter system. For linear partial differential equations, the existence of an optimal actuator design for a number of cost functions has been established. However, many dynamics are affected by nonlinearities and linearization of the PDE can neglect some important aspects of the model. The existing literature uses the finite-dimensional approximation of nonlinear PDE's to address the optimal actuator design problem. There are new theoretical results on the optimal actuator design of nonlinear PDE's in Banach spaces. This thesis describes new results on optimal actuator design for abstract nonlinear systems on reflexive Banach spaces. Two classes of PDE's have been studied. In the first class, semi-linear systems, a weak continuity assumption on nonlinearities is imposed to establish optimality results. Two examples are provided for this class including nonlinear railway track model and nonlinear wave equation in two space dimensions. The second class is nonlinear parabolic PDE's. For this class, the weak continuity assumption is relaxed at the cost of imposing assumptions on the linear part of the system. The examples provided for this class are Kuramoto-Sivashinsky equation and nonlinear diffusion in two space dimensions. Furthermore, a thorough study of optimal actuator location for nonlinear railway track model was conducted. The study begins with investigating the well-posedness and stability of solutions to this model. It is shown that under certain conditions on inputs, solutions to the railway track model are stable. Further on, using optimization algorithms and numerical schemes, an optimal input and actuator location are computed. Several simulations are run for various physical parameters. The simulations show that the optimal actuator location is not at the center of the track, contrary to a common belief. They also show that an optimally-located actuator significantly improves the performance of the control system

Mathematical Modeling, Motion Planning and Control of Elastic Structures with Piezoelectric Elements

The objective of this work is the development of a motion planning and tracking control approach for elastic structures. Motivated by the morphing wing concept of the field of aerospace engineering a so-called “smart wingsail” defines the center of the presented research. The motion planning and tracking control approach has to achieve different rest-to-rest motions of the wingsail’s transversal displacement. The design of the mechanical structure as well as the control concept of the wingsail relies on the results of proof of concept studies. For this purpose, different systems of interconnected bending beams are considered which emulates parts of the wingsail. The development of the model based control approaches requires an accurate system description. The modeling itself is done by an analytic energy based approach for the beams’ systems, where for the wingsail the finite elements method is used due to the risen complexity of the curved structure. To achieve a precise description of the governing dynamics different parameter identification concepts are discussed and applied. This leads to a precise but rather complex system description which covers the measured behavior of the experimental setups. Considering the objective of a real time capable control approach the complexity has to be reduced without a significant loss of accuracy. For this purpose different model order reduction techniques are discussed and applied. The resulting systems models are the bases of the control designs. Two different control concepts are presented and evaluated. At first the two-degrees-of-freedom control approach is introduced which combines a flatness-based feedforward control approach with a feedback controller. On the other hand, the so-called model predictive control approach is presented which is based on the solution of an optimization problem. Both concepts are evaluated by numeric analyses and by experiments

Optimization based control design techniques for distributed parameter systems

The study presents optimization based control design techniques for the systems that are governed by partial differential equations. A control technique is developed for systems that are actuated at the boundary. The principles of dynamic inversion and constrained optimization theory are used to formulate a feedback controller. This control technique is demonstrated for heat equations and thermal convection loops. This technique is extended to address a practical issue of parameter uncertainty in a class of systems. An estimator is defined for unknown parameters in the system. The Lyapunov stability theory is used to derive an update law of these parameters. The estimator is used to design an adaptive controller for the system. A second control technique is presented for a class of second order systems that are actuated in-domain. The technique of proper orthogonal decomposition is used first to develop an approximate model. This model is then used to design optimal feedback controller. Approximate dynamic programming based neural network architecture is used to synthesize a sub-optimal controller. This control technique is demonstrated to stabilize the heave dynamics of a flexible aircraft wings. The third technique is focused on the optimal control of stationary thermally convected fluid flows from the numerical point of view. To overcome the computational requirement, optimization is carried out using reduced order model. The technique of proper orthogonal decomposition is used to develop reduced order model. An example of chemical vapor deposition reactor is considered to examine this control technique --Abstract, page iii

Nonlinear Adaptive Control of Drilling Processes

This work deals with the modeling and control of automated drilling operations. Advances in drilling automation are of substantial importance because improvements in drilling control algorithms will result in more efficient drilling, which is beneficial from both economic and environmental points of view. While the primary application of the results is extraction of natural resources, potentially there exists a wide range of applications, including offshore exploration, archaeological research, and automated extraterrestrial mining, where implementation of new methods and control algorithms for drilling processes can bring substantial benefits. The main contribution of the thesis is development of new methods and algorithms for control of drilling processes in industrial drilling systems, ensuring stability and high performance characteristics. The problems of regulation of vertical penetration rate and drilling power in rotary drilling systems are solved; as a result, stability and vibration mitigation is ensured. A number of challenges is addressed, such as complexity and nonlinearity of the drilling model, lack of information about environment and parameters of the drilling system itself, and poor communication between downhole sensors and ground-level equipment. Several cases are considered, depending on the amount of information that is available in advance or in real time. Two mathematical models of the drilling system are investigated: one is finite-dimensional, and another is a distributed parameter model. Several solutions are proposed for both of them, using methods of adaptive, robust, and sliding mode control, and comparisons are made. Feasibility and efficiency of the proposed control algorithms are confirmed by simulations in MATLAB/Simulink

Spatio-Temporal Optimization for Control of Infinite Dimensional Systems in Robotics, Fluid Mechanics, and Quantum Mechanics

The majority of systems in nature have a spatio-temporal dependence and can be described by Partial Differential Equations (PDEs). They are ubiquitous in science and engineering, and are of rising interest among the control, robotics, and machine learning communities. Related methods usually treat these infinite dimensional problems in finite dimensions with reduced order models. This leads to committing to specific approximation schemes and the subsequent control laws cannot generalize outside of the approximation schemes. Additionally, related work does not consider spatio-temporal descriptions of noise that realistically represent the stochastic nature of physical systems. This thesis develops a variety of approaches for control optimization and co-design optimization for PDE and stochastic PDE (SPDE) systems from a unified perspective that can be applied to macroscopic systems in robotics and fluid dynamics, as well as microscopic systems in quantum mechanics. These approaches are each developed completely in the infinite dimensional Hilbert spaces where the systems are mathematically described, enabling the frameworks to be agnostic to the discretization scheme used to implement them. The first three developed approaches are applied in simulation to classical systems in fluid dynamics such as the Heat and Burgers equation. The fourth approach is developed for second-order SPDEs that arise in robotic systems, and is applied in simulation to systems in soft-robotics such as the Euler-Bernoulli equation and a biological model of a soft-robotic limb. Finally, several approaches are developed in the context of quantum feedback control of open quantum systems with non-demolition measurement, and one such approach is applied in simulation to perform explicit feedback control of the two qubit open quantum system.Ph.D