32,091 research outputs found
Adaptive Discrete Second Order Sliding Mode Control with Application to Nonlinear Automotive Systems
Sliding mode control (SMC) is a robust and computationally efficient
model-based controller design technique for highly nonlinear systems, in the
presence of model and external uncertainties. However, the implementation of
the conventional continuous-time SMC on digital computers is limited, due to
the imprecisions caused by data sampling and quantization, and the chattering
phenomena, which results in high frequency oscillations. One effective solution
to minimize the effects of data sampling and quantization imprecisions is the
use of higher order sliding modes. To this end, in this paper, a new
formulation of an adaptive second order discrete sliding mode control (DSMC) is
presented for a general class of multi-input multi-output (MIMO) uncertain
nonlinear systems. Based on a Lyapunov stability argument and by invoking the
new Invariance Principle, not only the asymptotic stability of the controller
is guaranteed, but also the adaptation law is derived to remove the
uncertainties within the nonlinear plant dynamics. The proposed adaptive
tracking controller is designed and tested in real-time for a highly nonlinear
control problem in spark ignition combustion engine during transient operating
conditions. The simulation and real-time processor-in-the-loop (PIL) test
results show that the second order single-input single-output (SISO) DSMC can
improve the tracking performances up to 90%, compared to a first order SISO
DSMC under sampling and quantization imprecisions, in the presence of modeling
uncertainties. Moreover, it is observed that by converting the engine SISO
controllers to a MIMO structure, the overall controller performance can be
enhanced by 25%, compared to the SISO second order DSMC, because of the
dynamics coupling consideration within the MIMO DSMC formulation.Comment: 12 pages, 7 figures, 1 tabl
Qualitative Studies of Nonlinear Hybrid Systems
A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior. Hybrid systems arise in a wide variety of important applications in diverse areas, ranging from biology to computer science to air traffic dynamics. The interaction of continuous- and discrete-time dynamics in a hybrid system often leads to very rich dynamical behavior and phenomena that are not encountered in purely continuous- or discrete-time systems. Investigating the dynamical behavior of hybrid systems is of great theoretical and practical importance.
The objectives of this thesis are to develop the qualitative theory of nonlinear hybrid systems with impulses, time-delay, switching modes, and stochastic disturbances, to develop algorithms and perform analysis for hybrid systems with an emphasis on stability and control, and to apply the theory and methods to real-world application problems.
Switched nonlinear systems are formulated as a family of nonlinear differential equations, called subsystems, together with a switching signal that selects the continuous dynamics among the subsystems. Uniform stability is studied emphasizing the situation where both stable and unstable subsystems are present. Uniformity of stability refers to both the initial time and a family of switching signals. Stabilization of nonlinear systems via state-dependent switching signal is investigated. Based on assumptions on a convex linear combination of the nonlinear vector fields, a generalized minimal rule is proposed to generate stabilizing switching signals that are well-defined and do not exhibit chattering or Zeno behavior.
Impulsive switched systems are hybrid systems exhibiting both impulse and switching effects, and are mathematically formulated as a switched nonlinear system coupled with a sequence of nonlinear difference equations that act on the switched system at discrete times. Impulsive switching signals integrate both impulsive and switching laws that specify when and how impulses and switching occur. Invariance principles can be used to investigate asymptotic stability in the absence of a strict Lyapunov function. An invariance principle is established for impulsive switched systems under weak dwell-time signals. Applications of this invariance principle provide several asymptotic stability criteria. Input-to-state stability notions are formulated in terms of two different measures, which not only unify various stability notions under the stability theory in two measures, but also bridge this theory with the existent input/output theories for nonlinear systems. Input-to-state stability results are obtained for impulsive switched systems under generalized dwell-time signals. Hybrid time-delay systems are hybrid systems with dependence on the past states of the systems. Switched delay systems and impulsive switched systems are special classes of hybrid time-delay systems. Both invariance property and input-to-state stability are extended to cover hybrid time-delay systems.
Stochastic hybrid systems are hybrid systems subject to random disturbances, and are formulated using stochastic differential equations. Focused on stochastic hybrid systems with time-delay, a fundamental theory regarding existence and uniqueness of solutions is established. Stabilization schemes for stochastic delay systems using state-dependent switching and stabilizing impulses are proposed, both emphasizing the situation where all the subsystems are unstable. Concerning general stochastic hybrid systems with time-delay, the Razumikhin technique and multiple Lyapunov functions are combined to obtain several Razumikhin-type theorems on both moment and almost sure stability of stochastic hybrid systems with time-delay.
Consensus problems in networked multi-agent systems and global convergence of artificial neural networks are
related to qualitative studies of hybrid systems in the sense that dynamic switching, impulsive effects, communication time-delays, and random disturbances are ubiquitous in networked systems. Consensus protocols are proposed for reaching consensus among networked agents despite switching network topologies, communication time-delays, and measurement noises. Focused on neural networks with discontinuous neuron activation functions and mixed time-delays, sufficient conditions for existence and uniqueness of equilibrium and global convergence and stability are derived using both linear matrix inequalities and M-matrix type conditions.
Numerical examples and simulations are presented throughout this thesis to illustrate the theoretical results
Interpreting Quantum Mechanics in Terms of Random Discontinuous Motion of Particles
This thesis is an attempt to reconstruct the conceptual foundations of quantum mechanics. First, we argue that the wave function in quantum mechanics is a description of random discontinuous motion of particles, and the modulus square of the wave function gives the probability density of the particles being in certain locations in space. Next, we show that the linear non-relativistic evolution of the wave function of an isolated system obeys the free Schrƶdinger equation due to the requirements of spacetime translation invariance and relativistic invariance. Thirdly, we argue that the random discontinuous motion of particles may lead to a stochastic, nonlinear collapse evolution of the wave function. A discrete model of energy-conserved wavefunction collapse is proposed and shown to be consistent with existing experiments and our macroscopic experience. In addition, we also give a critical analysis of the de Broglie-Bohm theory, the many-worlds interpretation and dynamical collapse theories, and briefly analyze the problem of the incompatibility between quantum mechanics and special relativity
The Time Invariance Principle, Ecological (Non)Chaos, and A Fundamental Pitfall of Discrete Modeling
This paper is to show that most discrete models used for population dynamics
in ecology are inherently pathological that their predications cannot be
independently verified by experiments because they violate a fundamental
principle of physics. The result is used to tackle an on-going controversy
regarding ecological chaos. Another implication of the result is that all
continuous dynamical systems must be modeled by differential equations. As a
result it suggests that researches based on discrete modeling must be closely
scrutinized and the teaching of calculus and differential equations must be
emphasized for students of biology
Superselection from canonical constraints
The evolution of both quantum and classical ensembles may be described via
the probability density P on configuration space, its canonical conjugate S,
and an_ensemble_ Hamiltonian H[P,S]. For quantum ensembles this evolution is,
of course, equivalent to the Schroedinger equation for the wavefunction, which
is linear. However, quite simple constraints on the canonical fields P and S
correspond to_nonlinear_ constraints on the wavefunction. Such constraints act
to prevent certain superpositions of wavefunctions from being realised, leading
to superselection-type rules. Examples leading to superselection for energy,
spin-direction and `classicality' are given. The canonical formulation of the
equations of motion, in terms of a probability density and its conjugate,
provides a universal language for describing classical and quantum ensembles on
both continuous and discrete configuration spaces, and is briefly reviewed in
an appendix.Comment: MiKTex 2.3, no figures, minor clarifications, to appear in J. Phys.
Variational Integrators for Nonvariational Partial Differential Equations
Variational integrators for Lagrangian dynamical systems provide a systematic
way to derive geometric numerical methods. These methods preserve a discrete
multisymplectic form as well as momenta associated to symmetries of the
Lagrangian via Noether's theorem. An inevitable prerequisite for the derivation
of variational integrators is the existence of a variational formulation for
the considered problem. Even though for a large class of systems this
requirement is fulfilled, there are many interesting examples which do not
belong to this class, e.g., equations of advection-diffusion type frequently
encountered in fluid dynamics or plasma physics. On the other hand, it is
always possible to embed an arbitrary dynamical system into a larger Lagrangian
system using the method of formal (or adjoint) Lagrangians. We investigate the
application of the variational integrator method to formal Lagrangians, and
thereby extend the application domain of variational integrators to include
potentially all dynamical systems. The theory is supported by physically
relevant examples, such as the advection equation and the vorticity equation,
and numerically verified. Remarkably, the integrator for the vorticity equation
combines Arakawa's discretisation of the Poisson brackets with a symplectic
time stepping scheme in a fully covariant way such that the discrete energy is
exactly preserved. In the presentation of the results, we try to make the
geometric framework of variational integrators accessible to non specialists.Comment: 49 page
Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms
This paper studies variational principles for mechanical systems with symmetry and their applications to integration algorithms. We recall some general features of how to reduce variational principles in the presence of a symmetry group along with general features of integration algorithms for mechanical systems. Then we describe some integration algorithms based directly on variational principles using a
discretization technique of Veselov. The general idea for these variational integrators is to directly discretize Hamiltonās principle rather than the equations of motion in a way that preserves the original systems invariants, notably the symplectic form and, via a discrete version of Noetherās theorem, the momentum map. The resulting mechanical integrators are second-order accurate, implicit, symplectic-momentum algorithms. We apply these integrators to the rigid body and the double spherical pendulum to show that the techniques are competitive with existing integrators
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