Nonlinear Systems

Open Mathematics is a challenging notion for theoretical modeling, technical analysis, and numerical simulation in physics and mathematics, as well as in many other fields, as highly correlated nonlinear phenomena, evolving over a large range of time scales and length scales, control the underlying systems and processes in their spatiotemporal evolution. Indeed, available data, be they physical, biological, or financial, and technologically complex systems and stochastic systems, such as mechanical or electronic devices, can be managed from the same conceptual approach, both analytically and through computer simulation, using effective nonlinear dynamics methods. The aim of this Special Issue is to highlight papers that show the dynamics, control, optimization and applications of nonlinear systems. This has recently become an increasingly popular subject, with impressive growth concerning applications in engineering, economics, biology, and medicine, and can be considered a veritable contribution to the literature. Original papers relating to the objective presented above are especially welcome subjects. Potential topics include, but are not limited to: Stability analysis of discrete and continuous dynamical systems; Nonlinear dynamics in biological complex systems; Stability and stabilization of stochastic systems; Mathematical models in statistics and probability; Synchronization of oscillators and chaotic systems; Optimization methods of complex systems; Reliability modeling and system optimization; Computation and control over networked systems

Modeling and Control of mini UAV

International audienc

Analysis and Control of Non-Affine, Non-Standard, Singularly Perturbed Systems

This dissertation addresses the control problem for the general class of control non-affine, non-standard singularly perturbed continuous-time systems. The problem of control for nonlinear multiple time scale systems is addressed here for the first time in a systematic manner. Toward this end, this dissertation develops the theory of feedback passivation for non-affine systems. This is done by generalizing the Kalman-Yakubovich-Popov lemma for non-affine systems. This generalization is used to identify conditions under which non-affine systems can be rendered passive. Asymptotic stabilization for non-affine systems is guaranteed by using these conditions along with well-known passivity-based control methods. Unlike previous non-affine control approaches, the constructive static compensation technique derived here does not make any assumptions regarding the control influence on the nonlinear dynamical model. Along with these control laws, this dissertation presents novel hierarchical control design procedures to address the two major difficulties in control of multiple time scale systems: lack of an explicit small parameter that models the time scale separation and the complexity of constructing the slow manifold. These research issues are addressed by using insights from geometric singular perturbation theory and control laws are designed without making any assumptions regarding the construction of the slow manifold. The control schemes synthesized accomplish asymptotic slow state tracking for multiple time scale systems and simultaneous slow and fast state trajectory tracking for two time scale systems. The control laws are independent of the scalar perturbation parameter and an upper bound for it is determined such that closed-loop system stability is guaranteed. Performance of these methods is validated in simulation for several problems from science and engineering including the continuously stirred tank reactor, magnetic levitation, six degrees-of-freedom F-18/A Hornet model, non-minimum phase helicopter and conventional take-off and landing aircraft models. Results show that the proposed technique applies both to standard and non-standard forms of singularly perturbed systems and provides asymptotic tracking irrespective of the reference trajectory. This dissertation also shows that some benchmark non-minimum phase aerospace control problems can be posed as slow state tracking for multiple time scale systems and techniques developed here provide an alternate method for exact output tracking

Advanced Feedback Linearization Control for Tiltrotor UAVs: Gait Plan, Controller Design, and Stability Analysis

Three challenges, however, can hinder the application of Feedback Linearization: over-intensive control signals, singular decoupling matrix, and saturation. Activating any of these three issues can challenge the stability proof. To solve these three challenges, first, this research proposed the drone gait plan. The gait plan was initially used to figure out the control problems in quadruped (four-legged) robots; applying this approach, accompanied by Feedback Linearization, the quality of the control signals was enhanced. Then, we proposed the concept of unacceptable attitude curves, which are not allowed for the tiltrotor to travel to. The Two Color Map Theorem was subsequently established to enlarge the supported attitude for the tiltrotor. These theories were employed in the tiltrotor tracking problem with different references. Notable improvements in the control signals were witnessed in the tiltrotor simulator. Finally, we explored the control theory, the stability proof of the novel mobile robot (tilt vehicle) stabilized by Feedback Linearization with saturation. Instead of adopting the tiltrotor model, which is over-complicated, we designed a conceptual mobile robot (tilt-car) to analyze the stability proof. The stability proof (stable in the sense of Lyapunov) was found for a mobile robot (tilt vehicle) controlled by Feedback Linearization with saturation for the first time. The success tracking result with the promising control signals in the tiltrotor simulator demonstrates the advances of our control method. Also, the Lyapunov candidate and the tracking result in the mobile robot (tilt-car) simulator confirm our deductions of the stability proof. These results reveal that these three challenges in Feedback Linearization are solved, to some extents.Comment: Doctoral Thesis at The University of Toky

Robust Control Methods for Nonlinear Systems with Uncertain Dynamics and Unknown Control Direction

Robust nonlinear control design strategies using sliding mode control (SMC) and integral SMC (ISMC) are developed, which are capable of achieving reliable and accurate tracking control for systems containing dynamic uncertainty, unmodeled disturbances, and actuator anomalies that result in an unknown and time-varying control direction. In order to ease readability of this dissertation, detailed explanations of the relevant mathematical tools is provided, including stability denitions, Lyapunov-based stability analysis methods, SMC and ISMC fundamentals, and other basic nonlinear control tools. The contributions of the dissertation are three novel control algorithms for three different classes of nonlinear systems: single-input multipleoutput (SIMO) systems, systems with model uncertainty and bounded disturbances, and systems with unknown control direction. Control design for SIMO systems is challenging due to the fact that such systems have fewer actuators than degrees of freedom to control (i.e., they are underactuated systems). While traditional nonlinear control methods can be utilized to design controllers for certain classes of cascaded underactuated systems, more advanced methods are required to develop controllers for parallel systems, which are not in a cascade structure. A novel control technique is proposed in this dissertation, which is shown to achieve asymptotic tracking for dual parallel systems, where a single scalar control input directly affects two subsystems. The result is achieved through an innovative sequential control design algorithm, whereby one of the subsystems is indirectly stabilized via the desired state trajectory that is commanded to the other subsystem. The SIMO system under consideration does not contain uncertainty or disturbances. In dealing with systems containing uncertainty in the dynamic model, a particularly challenging situation occurs when uncertainty exists in the input-multiplicative gain matrix. Moreover, special consideration is required in control design for systems that also include unknown bounded disturbances. To cope with these challenges, a robust continuous controller is developed using an ISMC technique, which achieves asymptotic trajectory tracking for systems with unknown bounded disturbances, while simultaneously compensating for parametric uncertainty in the input gain matrix. The ISMC design is rigorously proven to achieve asymptotic trajectory tracking for a quadrotor system and a synthetic jet actuator (SJA)-based aircraft system. In the ISMC designs, it is assumed that the signs in the uncertain input-multiplicative gain matrix (i.e., the actuator control directions) are known. A much more challenging scenario is encountered in designing controllers for classes of systems, where the uncertainty in the input gain matrix is extreme enough to result in an a priori-unknown control direction. Such a scenario can result when dealing with highly inaccurate dynamic models, unmodeled parameter variations, actuator anomalies, unknown external or internal disturbances, and/or other adversarial operating conditions. To address this challenge, a SMCbased self-recongurable control algorithm is presented, which automatically adjusts for unknown control direction via periodic switching between sliding manifolds that ultimately forces the state to a converging manifold. Rigorous mathematical analyses are presented to prove the theoretical results, and simulation results are provided to demonstrate the effectiveness of the three proposed control algorithms

Four Tilting Rotor Convertible MAV: Modeling and Real-Time Hover Flight Control

International audienceThis paper describes the modeling, control and hardware implementation of an experimental tilt-rotor aircraft. This vehicle combines the high-speed cruise capabilities of a conventional airplane with the hovering capabilities of a helicopter by tilting their four rotors. Changing between cruise and hover flight modes in mid-air is referred to transition. Dynamic model of the vehicle is derived both for vertical and horizontal flight modes using Newtonian approach. Two nonlinear control strategies are presented and evaluated at simulation level to control, the vertical and horizontal flight dynamics of the vehicle in the longitudinal plane. An experimental prototype named Quad-plane was developed to perform the vertical flight. A low-cost DSP-based Embedded Flight Control System (EFCS) was designed and built to achieve autonomous attitude-stabilized flight