Deep neural network approximations for the stable manifolds of the Hamilton-Jacobi equations

As the Riccati equation for control of linear systems, the Hamilton-Jacobi-Bellman (HJB) equations play a fundamental role for optimal control of nonlinear systems. For infinite-horizon optimal control, the stabilizing solution of HJB equation can be represented by the stable manifold of the associated Hamiltonian system. In this paper, we study the neural network (NN) semiglobal approximation of the stable manifold. The main contribution includes two aspects: firstly, from the mathematical point of view, we rigorously prove that if an approximation is sufficiently close to the exact stable manifold of the HJB equation, then the corresponding control derived from this approximation is near optimal. Secondly, we propose a deep learning method to approximate the stable manifolds, and then numerically compute optimal feedback controls. The algorithm is devised from geometric features of the stable manifold, and relies on adaptive data generation by finding trajectories randomly in the stable manifold. The trajectories are found by solving two-point boundary value problems (BVP) locally near the equilibrium and extending the local solution by initial value problems (IVP) for the associated Hamiltonian system. A number of samples are chosen on each trajectory. Some adaptive samples are selected near the points with large errors after the previous round of training. Our algorithm is causality-free basically, hence it has a potential to apply to various high-dimensional nonlinear systems. We illustrate the effectiveness of our method by stabilizing the Reaction Wheel Pendulums.Comment: The algorithm is modified. The main point is that the trajectories on stable manifold are found by a combination of two-point BVP near the equilibrium and initial value problem far away from the equilibrium. The algorithm becomes more effectiv

An experimental study for stabilization of inverted pendulum

Stabilization of Inverted Pendulum is defined as a very basic classical control problem in Control System. The Dynamics of Cart Inverted Pendulum is related to many real life applications like missile launching system, balancing systems like human walking, Aircraft landing pad in sea etc. This is a highly Unstable and non-linear system. This system is a under actuated system and also a non-minimum phase system so design a Controller for Inverted Pendulum System is very complex. This thesis includes system and hardware description of Inverted Pendulum System, Dynamics of the system, State space model, Derivation of Transfer Functions. In Past a lot of research work has already been done in Inverted Pendulum for development of Control Strategy. Here in this thesis we have done a very small work to design Control Strategy and also validate them with real-time experiments. In this thesis two-loop PID, PID+PI & LQR control have been implemented for Inverted Pendulum System and this control strategies gives satisfactory respons

Об управлении неточными быстро-медленными системами Такаги – Сугено

Предложена нечеткая модель некоторого процесса в виде быстро-медленной системы дифференциальных уравнений с параметрическими неточностями и управлением, для описания которой использован набор нечетких предикатных правил. Используя метод функций Ляпунова, предложен вид управления, который обеспечивает асимптотическую устойчивость нулевого состояния равновесия исходной системы и определена область в пространстве параметров, для всех значений параметров из которой такая устойчивость сохраняется, а также предложен вид управления, который обеспечивает требуемый вид устойчивости.Для неточних швидко-повільних систем типу Такагі – Сугено із нелінійними підсистемами побудовано керування, що забезпечує їх асимптотичну стійкість. Оцінено множину значень параметрів, для яких вказана властивість системи зберігається.For the fuzzy uncertain slow-fast systems of the Takagi – Sugeno type with nonlinear subsystems, the control is constructed which provides their asymptotic stability. The set of values of parameters is estimated for which such feature of these systems is preserved

On Stabilization of Cart-Inverted Pendulum System: An Experimental Study

The Cart-Inverted Pendulum System (CIPS) is a classical benchmark control problem. Its dynamics resembles with that of many real world systems of interest like missile launchers, pendubots, human walking and segways and many more. The control of this system is challenging as it is highly unstable, highly non-linear, non-minimum phase system and underactuated. Further, the physical constraints on the track position control voltage etc. also pose complexity in its control design. The thesis begins with the description of the CIPS together with hardware setup used for research, its dynamics in state space and transfer function models. In the past, a lot of research work has been directed to develop control strategies for CIPS. But, very little work has been done to validate the developed design through experiments. Also robustness margins of the developed methods have not been analysed. Thus, there lies an ample opportunity to develop controllers and study the cart-inverted pendulum controlled system in real-time. The objective of this present work is to stabilize the unstable CIPS within the different physical constraints such as in track length and control voltage. Also, simultaneously ensure good robustness. A systematic iterative method for the state feedback design by choosing weighting matrices key to the Linear Quadratic Regulator (LQR) design is presented. But, this yields oscillations in cart position. The Two-Loop-PID controller yields good robustness, and superior cart responses. A sub-optimal LQR based state feedback subjected to H∞ constraints through Linear Matrix Inequalities (LMIs) is solved and it is observed from the obtained results that a good stabilization result is achieved. Non-linear cart friction is identified using an exponential cart friction and is modeled as a plant matrix uncertainty. It has been observed that modeling the cart friction as above has led to improved cart response. Subsequently an integral sliding mode controller has been designed for the CIPS. From the obtained simulation and experiments it is seen that the ISM yields good robustness towards the output channel gain perturbations. The efficacies of the developed techniques are tested both in simulation and experimentation. It has been also observed that the Two-Loop PID Controller yields overall satisfactory response in terms of superior cart position and robustness. In the event of sensor fault the ISM yields best performance out of all the techniques

Visual Feedback Stabilisation of a Cart Inverted Pendulum A

Vision-based object stabilisation is an exciting and challenging area of research, and is one that promises great technical advancements in the field of computer vision. As humans, we are capable of a tremendous array of skilful interactions, particularly when balancing unstable objects that have complex, non-linear dynamics. These complex dynamics impose a difficult control problem, since the object must be stabilised through collaboration between applied forces and vision-based feedback. To coordinate our actions and facilitate delivery of precise amounts of muscle torque, we primarily use our eyes to provide feedback in a closed-loop control scheme. This ability to control an inherently unstable object by vision-only feedback demonstrates an exceptionally high degree of voluntary motor skill. Despite the pervasiveness of vision-based stabilisation in humans and animals, relatively little is known about the neural strategies used to achieve this task. In the last few decades, with advancements in technology, we have tried to impart the skill of vision-based object stabilisation to machines, with varying degrees of success. Within the context of this research, we continue this pursuit by employing the classic Cart Inverted Pendulum; an inherently unstable, non-linear system to investigate dynamic object balancing by vision-only feedback. The Inverted Pendulum is considered to be one of the most fundamental benchmark systems in control theory; as a platform, it provides us with a strong, well established test bed for this research. We seek to discover what strategies are used to stabilise the Cart Inverted Pendulum, and to determine if these strategies can be deployed in Real-Time, using cost-effective solutions. The thesis confronts, and overcomes the problems imposed by low-bandwidth USB cameras; such as poor colour-balance, image noise and low frame rates etc., to successfully achieve vision-based stabilisation. The thesis presents a comprehensive vision-based control system that is capable of balancing an inverted pendulum with a resting oscillation of approximately ±1º. We employ a novel, segment-based location and tracking algorithm, which was found to have excellent noise immunity and enhanced robustness. We successfully demonstrate the resilience of the tracking and pose estimation algorithm against visual disturbances in Real-Time, and with minimal recovery delay. The algorithm was evaluated against peer reviewed research; in terms of processing time, amplitude of oscillation, measurement accuracy and resting oscillation. For each key performance indicator, our system was found to be superior in many cases to that found in the literature. The thesis also delivers a complete test software environment, where vision-based algorithms can be evaluated. This environment includes a flexible tracking model generator to allow customisation of visual markers used by the system. We conclude by successfully performing off-line optimization of our method by means of Artificial Neural Networks, to achieve a significant improvement in angle measurement accuracy.Goodrich Engine Control Systems and Balfour Beatty Rail Technologie

Global Stabilization of an Inverted Pendulum – Control Strategy and Experimental Verification

The problem of swinging up an inverted pendulum on a cart and controlling it around the upright position has traditionally been treated as two separate problems. This paper proposes a control strategy that is globally asymptotically stable under actuator saturation and, in addition, locally exponentially stable. The proposed methodology, which performs swing up and control simultaneously, uses elements from input-output linearization, energy control, and singular perturbation theory. Experimental results on a laboratory-scale setup are presented to illustrate the approach and its implementation

Flachheitsbasierte Methode zum stoßfreien Umschalten von Reglerstrukturen

Die vorliegende Arbeit beschäftigt sich mit der Frage, wie stoßfreie Rekonfigurationen von Systemen zur Laufzeit realisiert werden können. Es werden Anforderungen an die Rekonfiguration definiert und eine neue Methode zur stoßfreien Rekonfiguration vorgestellt, die sowohl bei einfachen Betriebspunktwechseln als auch beim Wechsel der Reglerparameter oder der Reglerstruktur angewendet werden kann. Die Methodik basiert auf der Zwei-Freiheitsgrade-Reglerstruktur und der (differenziellen) Flachheit, einer grundlegenden Eigenschaft des Systems selbst. Die Methodik wird für lineare und nichtlineare Ein- und Mehrgrößensysteme vorgestellt, wobei die Rekonfigurationen immer mittels in Echtzeit berechneter Vorsteuerungs- und Führungsgrößentrajektorien realisiert werden. Anhand von akademischen und praktischen Beispielen wird die neue Methode mit bestehenden Verfahren zur stoßfreien Reglerumschaltung verglichen und die Anwendbarkeit demonstriert.The present thesis deals with a new approach to bumpless transfer for system reconfiguration at runtime. During a system reconfiguration an operating point change, a change of controller parameters or even a change of the control structure can occur. After the definition of requirements which has to be fulfilled during the reconfiguration, a new flatness-based method for bumpless transfer is presented. The flatness-based method draws on the two-degrees-of-freedom control structure and on the (differential) flatness which is a fundamental feature of the controlled system. Bumpless switching is realised by means of feedforward and reference trajectories computed in real time which are applicable with linear and non-linear SISO and MIMO systems. The new method of bumpless switching is compared to existing bumpless-switching procedures and its advantages are evidenced by practical examples.Tag der Verteidigung: 11.12.2014Paderborn, Univ., Diss., 201