1,327 research outputs found
Stability Analysis and Decentralized Control of Coupled Oscillators with Feedback Delays
Most dynamic systems do not react instantaneously to actuation signals. The temporal evolution of some others is based on retarded communications or depends on information from the past. In such cases, the mathematical models used to describe these systems must include information about the past dynamics of the states. These models are often referred to as delay or retarded systems. Delays could channel energy in and out of a system at incorrect time intervals producing instabilities and rendering controllers\u27 performance ineffective. The purpose of this research is two folds. The first investigates the effect of inherent system delays on the stability of coupled oscillators subjected to decentralized control and the second studies the prospectus of augmenting the delay into a larger delay period that could actually stabilize the coupled system and enhance its damping characteristics. Towards these ends, a system of two linearly-coupled oscillators with decentralized delayed-proportional feedback is considered. A comprehensive linear stability analysis is utilized to generate maps that divide the controllers\u27 gain and delay domain into regions of stability for different coupling values. These maps are then used to draw definite conclusions about the effect of coupling on the stability of the closed-loop in the presence of delay. Once the stability maps are generated, the Lambert-W function approach is utilized to find the stability exponents of the coupled system which, in turn, is used to generate damping contours within the pockets of stability. These contours are used to choose gain-delay combinations that could augment the inherent feedback delays into a larger delay period which can enhance the damping characteristics and reduce the system settling time significantly. An experimental plant comprised of two mass-spring-damper trios coupled with a spring is installed to validate the theoretical results and the proposed control hypothesis. Different scenarios consisting of different gains and delays are considered and compared with theoretical findings demonstrating very good agreement. Furthermore, the proposed delayed-proportional feedback decentralized controller is tested and its ability to dampen external oscillations is verified through different experiments. Such a research endeavor could prove very beneficial to many vital areas in our life. A good example is that of the coupled system of the natural and artificial cardiac pacemakers where the natural pacemaker represents a rhythmic oscillating system and the coupled artificial pacemaker provides a stabilizing signal through a feedback mechanism that senses the loss in rhythm. In this system, even the minute amount of delay in the sensing-actuating could prove very detrimental. The result of this research contributes to the solution of this and similar problems
Pole Placement and Reduced-Order Modelling for Time-Delayed Systems Using Galerkin Approximations
The dynamics of time-delayed systems (TDS) are governed by delay differential equa-
tions (DDEs), which are infinite dimensional and pose computational challenges. The
Galerkin approximation method is one of several techniques to obtain the spectrum of DDEs
for stability and stabilization studies. In the literature, Galerkin approximations for DDEs
have primarily dealt with second-order TDS (second-order Galerkin method), and the for-
mulations have resulted in spurious roots, i.e., roots that are not among the characteristic
roots of the DDE. Although these spurious roots do not affect stability studies, they never-
theless add to the complexity and computation time for control and reduced-order modelling
studies of DDEs. A refined mathematical model, called the first-order Galerkin method, is
proposed to avoid spurious roots, and the subtle differences between the two formulations
(second-order and first-order Galerkin methods) are highlighted with examples.
For embedding the boundary conditions in the first-order Galerkin method, a new
pseudoinverse-based technique is developed. This method not only gives the exact location
of the rightmost root but also, on average, has a higher number of converged roots when
compared to the existing pseudospectral differencing method. The proposed method is
combined with an optimization framework to develop a pole-placement technique for DDEs
to design closed-loop feedback gains that stabilize TDS. A rotary inverted pendulum system
apparatus with inherent sensing delays as well as deliberately introduced time delays is used
to experimentally validate the Galerkin approximation-based optimization framework for the
pole placement of DDEs.
Optimization-based techniques cannot always place the rightmost root at the desired
location; also, one has no control over the placement of the next set of rightmost roots.
However, one has the precise location of the rightmost root. To overcome this, a pole-
placement technique for second-order TDS is proposed, which combines the strengths of the
method of receptances and an optimization-based strategy. When the method of receptances
provides an unsatisfactory solution, particle swarm optimization is used to improve the
location of the rightmost pole. The proposed approach is demonstrated with numerical
studies and is validated experimentally using a 3D hovercraft apparatus.
The Galerkin approximation method contains both converged and unconverged roots
of the DDE. By using only the information about the converged roots and applying the
eigenvalue decomposition, one obtains an r-dimensional reduced-order model (ROM) of the
DDE. To analyze the dynamics of DDEs, we first choose an appropriate value for r; we
then select the minimum value of the order of the Galerkin approximation method system
at which at least r roots converge. By judiciously selecting r, solutions of the ROM and the
original DDE are found to match closely. Finally, an r-dimensional ROM of a 3D hovercraft
apparatus in the presence of delay is validated experimentally
Reflected Brownian motions in the KPZ universality class
This book presents a detailed study of a system of interacting Brownian
motions in one dimension. The interaction is point-like such that the -th
Brownian motion is reflected from the Brownian motion with label . This
model belongs to the Kardar-Parisi-Zhang (KPZ) universality class. In fact,
because of the singular interaction, many universal properties can be
established with rigor. They depend on the choice of initial conditions.
Discussion addresses packed and periodic initial conditions, stationary initial
conditions, and mixtures thereof. The suitably scaled spatial process will be
proven to converge to an Airy process in the long time limit. A chapter on
determinantal random fields and another one on Airy processes are added to have
the notes self-contained. This book serves as an introduction to the KPZ
universality class, illustrating the main concepts by means of a single model
only. It will be of interest to readers from interacting diffusion processes
and non-equilibrium statistical mechanics.Comment: arXiv admin note: text overlap with arXiv:1502.0146
Quasilinear Control of Systems with Time-Delays and Nonlinear Actuators and Sensors
This thesis investigates Quasilinear Control (QLC) of time-delay systems with nonlinear actuators and sensors and analyzes the accuracy of stochastic linearization for these systems. QLC leverages the method of stochastic linearization to replace each nonlinearity with an equivalent gain, which is obtained by solving a transcendental equation. The idea of QLC is to stochastically linearize the system in order to analyze and design controllers using classical linear control theory. In this thesis, the existence of the equivalent gain for a closed-loop time-delay system is discussed. To compute the equivalent gain, two methods are explored. The first method uses an explicit but complex algorithm based on delay Lyapunov equation to study the time-delay, while the second method uses Pade approximant. It is shown that, under a suitable criterion, Pade approximant can be effectively applied for QLC of time-delay systems. Furthermore, the method of Saturated-Root Locus (S-RL) is extended to nonlinear time-delay systems. It turns out that, in a time-delay system, S-RL always terminates prematurely as opposed to a delay-free system, which may or may not terminate prematurely. Statistical experiments are performed to investigate the accuracy of stochastic linearization compared to a system without time-delay. The impact of increasing the time-delay in the approach of stochastic linearization is also investigated. Results show that stochastic linearization effectively linearizes a nonlinear time-delay system, even though delays generally degrade accuracy. Overall, the accuracy remains relatively high over the selected parameters. Finally, this approach is applied to pitch control in a wind turbine system as a practical example of a nonlinear time-delay system, and its performance is analyzed to demonstrate the efficacy of the approach
Synchronization of Different Fractional Order Time-Delay Chaotic Systems Using Active Control
Chaos synchronization of different fractional order time-delay chaotic systems is considered. Based on the Laplace transform theory, the conditions for achieving synchronization of different fractional order time-delay chaotic systems are analyzed by use of active control technique. Then numerical simulations are provided to verify the effectiveness and feasibility of the developed method. At last, effects of the fraction order and the time delay on synchronization are further researched
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