“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

SET-Valued Analysis

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Semi Active Suspension Control Design For Vehicles

This paper describes some important classes of two degrees of freedom of underactuated mechanical system and also surveys review of the recent state-of-the-art concerning the mathematical modeling of these systems, their classification, and all the control strategies (linear, nonlinear, and intelligent) that have been made so far (i.e., from the year 2000 to date) to control these systems. Future research and challenges concerning the improvement, the effectiveness, and robustness of the proposed controllers for underactuated mechanical systems are presented.

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A Review on the Control of Second Order Underactuated Mechanical Systems

LMI-based synthesis of a robust saturated controller for an underactuated mechanical system subject to motion constraints

The position control of the ball and beam system using state-disturbance observe-based adaptive fuzzy sliding mode control in presence of matched and mismatched uncertainties

The pendulum is the most known mechanism of everyone. This is due to its importance historical and also to the number of the invention based on its operating principle. The inverted pendulum is an unstable system perfectly described by a nonlinear model obtained by applying the laws of physics. The design goal of the control law is to stabilize the pendulum in its upright unstable equilibrium position. In this article, a review of the different control law that have been applied to the system of the inverted pendulum with a cart is presented. And a comparison of each controller are discussed.

Review: Linear, nonlinear and intelligent controllers for the inverted pendulum problem

This paper presents a Backstepping controller for five degrees of freedom Spherical Inverted Pendulum. Since the system is nonlinear, unstable, underactuated and MIMO and has a nonsquare form, the classic control design cannot be applied to control it. In order to remedy this problem, we propose in this paper a new method based on hierarchical steps of the Backstepping controller taking into a count the nonlinearities that cannot be neglected. Furthermore, a Linear Quadratic Regulator controller and LQR + PID based on the linearized system model are also designed for performance comparison. Finally, a simulation study is carried out to prove the effectiveness of proposed control scheme and is validated using the virtual reality environment that proves the performance of the Backstepping controller over the linear ones where it brings the pendulum from any initial condition in the upper hemisphere while the base is brought to the origin of the coordinates.

Backstepping Controller Design for a 5 DOF Spherical Inverted Pendulum

The Modelling and control design of Two Wheel Vehicle represents an open and a challenging problem in terms of the complexity in these kind of vehicles. This article aims to represent a comparative analysis of two strategies of control which are modern controller LQR and Conventional Controller PID for the two wheeled vehicle. The main goal is to compare their performances in terms of the time specification and to determine the best control strategy. We begin our development with the implementation of the dynamic model of the two wheeled vehicle using Lagrange modeling with holonomic constraints. Further, the article deals with analyzing the eigenvalues of the linearized dynamic system at which the two wheeled vehicle lean and steer are stable. This research targets the development of the two controllers: PID and LQR. Those controllers are used to control both steer angle, and lean rate angle of two wheeled vehicle. The study includes as well a comparative assessment of those control strategies in terms of performance.

A Comparative Assessment between LQR and PID Strategies in Control of Two Wheeled Vehicle

The aim of this paper is to control a spherical inverted pendulum on a quadrotor as a control actuator. The system is modeled using Euler Lagrange formulation considering both the dynamics. And due to the absence of sensors, that can measure states at the ball joints, we propose to use a vision system that will be placed over the Quadrotor in order to determinate the position of the pendulum. For the nonlinear control design, a three-level cascade strategy is proposed. Each level of the cascade system executes a control law designed through the method of backstepping. Simulation results are carried out to corroborate the proposed control strategy using Matlab/Simulink and the Virtual Reality Environment.

Abdelmjid Saka

Papers

A Review on the Control of Second Order Underactuated Mechanical Systems

Review: Linear, nonlinear and intelligent controllers for the inverted pendulum problem

Backstepping Controller Design for a 5 DOF Spherical Inverted Pendulum

A Comparative Assessment between LQR and PID Strategies in Control of Two Wheeled Vehicle

Vision-based control of a flying spherical inverted pendulum