Στην παρούσα διπλωματική εργασία παρουσιάζεται η μεθοδολογία εξαγωγής διαφορικών εξισώσεων κατά Hamilton. Οι προκύπτουσες διαφορικές εξισώσεις είναι n το πλήθος (όπου n είναι το πλήθος των βαθμών ελυθερίας του συστήματος) διαφορικές εξισώσεις πρώτου βαθμού και περιγράφουν την δυναμική συμπεριφορά ενός μηχανικού συστήματος. Παρουσιάζεται το εκτενές θεωρητικό πλαίσιο που απορρέει απο τις εξισώσεις του Ηamilton και αξιοποιούνται οι θεωρητικές αρχές κατα την περιγραφή και τον έλεγχο του συστήματος σφαίρας-ράβδου. / In the diploma thesis we present the Hamiltonian formalism in order to derive first order differential equations which represent the dynamical behaviour of a mechanical system. The derived differential equations are n first order equations (where n is the number of system's degrees of freedom). We present the extensive theoretical framework that is derived from the Hamilton's equations which we use so that to describe and control the ball and beam system.

The ball and beam system is a very simple and powerful control system problem. The easy construction of this system combined with its challenging control design requirement makes it one of the most favorable example models for control engineers. The model contains a horizontal beam which can pivot about its center; a DC Motor whose shaft is connected to the center of the beam; and a ball that can freely roll on top of the beam. The basic idea is to accurately tilt the beam about its center, using the motor, to indirectly control the position of the ball that freely rolls on the beam. In this thesis, the L1 adaptive control technique is considered for precise positioning of the rolling ball on the beam. Two different architectures of L1 adaptive control namely, the L1 adaptive state feedback control and the L1 adaptive output feedback control are designed and verified in simulation. L1 adaptive control guarantees transient performance and robustness in presence of fast adaptation without introducing or enforcing persistence of excitation.

A ”Haptic servo system” is here understood as a servo system whereforces from a controlled system are fed back to an operator. This thesis workis a design work where the work among other things comprises the choice ofsuitable motors, one for operating the beam and another one for operatingthe steering wheel. Data for the beam and ball are assumed to be known.Data for the feed back torque to the steering wheel is assumed to be specifiedin advance. Two models to represent the human response are suggested. Asimulation study is carried out to show that the system works according tosome specification. The ball and beam process is simulated with hardwarein the loop. The hardware in the loop is a Maxon motor. The motor is usedas the steering wheel and the motor will also propagate the torque feedbackto the operator.The task of the thesis work could then be formulated as: Can a human, withtorque feedback, manually control the ball on the beam without looking atthe ball and the beam?

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Research into observer design for non-linear discrete-time systems has produced many design methods. There is no general design method however and that provides the motivation to search for a new simple and realizable design method. In this thesis, an observer for non-linear discrete-time systems is designed using the sliding mode technique. The equation of the observer error is split into two parts; the first part being a linearized model of the system and the second part an uncertain vector. The sliding mode technique is introduced to eliminate the uncertainty caused by the uncertain vector in the observer error equation. By choosing the sliding surface and the boundary layer, the observer error is attracted to the sliding surface and stays within the sliding manifold. Therefore, the observer error converges to zero. The proposed technique is applied to two cases of observers for nonlinear discrete-time systems. The second case is chosen to be a particular practical system, namely the non-linear discrete-time ball and beam system. The simulations show that the sliding mode technique guarantees the convergence of the observer error for both systems.

Master of Science / Department of Mechanical and Nuclear Engineering / Warren N. White / This thesis describes the process of testing and implementing ultrasonic transducers for ball position feedback on a ball-and-beam apparatus. Also included are specifications for equipment to allow feedback and command signals to be wireless, not hardwired to the control computer. The author presents various ball-and-beam configurations as well as details about the specific configuration used for this work. These details include choices in sensors, materials, hardware, construction, and controller. After the apparatus has been described, the author provides information to support claims about system performance. The conclusions presented specify the necessary hardware to make the system wireless and indicate that acoustic sensors can complete a successful ball-on-beam balancing system.

In this thesis different control methods are applied to the beam and ball system. Test setup for the previous thesis is handled, circuit assemblies and hardware redesigned. As the system is controlled by the control law partitioning method by a computer, discrete time system model is created. The controllability and the observability of the system are analyzed and a nonlinear controller by using control law partitioning in other words computed torque is designed. State feedback control algorithm previously designed is repeated. In case of calculating the non measurable state variables two different reduced order observers are designed for these two different controllers, one for control law partitioning controller and the other for state-feedback controller. Two controller methods designed for the thesis study are tested in the computer environment using modeling and simulation tools (Also a different controller by using sliding mode controller is designed and tested in the computer environment using simulation tools). A controller software program is written for the designed controller algorithms and this software is tested on the test setup. It is observed that the system is stable when we apply either of the control algorithms.

This thesis dissertation mainly compares and investigates laboratory study of different implementation methodologies of applied control systems and how they can be adopted in industrial, as well as commercial, automation applications. Namely the research paper aims to assess or evaluate eventual feedback control loops' performance and robustness over multiple conventional or state-of-the-art technologies in the field of applied industrial automation and instrumentation by implementing a laboratory case study setup: the ball on beam system. Hence, the paper tries to close the gap between industry and academia by: first, conducting a historical study and background information of main evolutional and technological eras in the field of industrial process control automation and instrumentation. Then, some related basic theoretical as well as practical concepts are reviewed in Chapter 2 of the report before displaying the detailed design. After that, the next Chapter, analyses the ball on beam control system problem as the case studied in the context of this research through reviewing previous literature, modeling and simulation. The following Chapter details the proposed design and implementation of the ball on beam case study as if it is under the introduced distributed industrial automation architecture. Finally, Chapter 5 concludes this work by listing several points leaned, remarks, and observations, and stating possible development and the future vision of this research.