Exponential Stabilization of Driftless Nonlinear Control Systems

This dissertation lays the foundation for practical exponential stabilization of driftless control systems. Driftless systems have the form, xdot = X1(x)u1 + .... + Xm(x)um, x ∈ ℝ^n Such systems arise when modeling mechanical systems with nonholonomic constraints. In engineering applications it is often required to maintain the mechanical system around a desired configuration. This task is treated as a stabilization problem where the desired configuration is made an asymptotically stable equilibrium point. The control design is carried out on an approximate system. The approximation process yields a nilpotent set of input vector fields which, in a special coordinate system, are homogeneous with respect to a non-standard dilation. Even though the approximation can be given a coordinate-free interpretation, the homogeneous structure is useful to exploit: the feedbacks are required to be homogeneous functions and thus preserve the homogeneous structure in the closed-loop system. The stability achieved is called p-exponential stability. The closed-loop system is stable and the equilibrium point is exponentially attractive. This extended notion of exponential stability is required since the feedback, and hence the closed-loop system, is not Lipschitz. However, it is shown that the convergence rate of a Lipschitz closed-loop driftless system cannot be bounded by an exponential envelope. The synthesis methods generate feedbacks which are smooth on ℝ^n \ {0}. The solutions of the closed-loop system are proven to be unique in this case. In addition, the control inputs for many driftless systems are velocities. For this class of systems it is more appropriate for the control law to specify actuator forces instead of velocities. We have extended the kinematic velocity controllers to controllers which command forces and still p-exponentially stabilize the system. Perhaps the ultimate justification of the methods proposed in this thesis are the experimental results. The experiments demonstrate the superior convergence performance of the p-exponential stabilizers versus traditional smooth feedbacks. The experiments also highlight the importance of transformation conditioning in the feedbacks. Other design issues, such as scaling the measured states to eliminate hunting, are discussed. The methods in this thesis bring the practical control of strongly nonlinear systems one step closer

Convergence Rates for Nonholonomic Systems in Power Form

This paper investigates the convergence rates of several controllers for low dimenional nonholonomic systems in power form. The method of multiple scales is found to be effective in determining the asymptotic form of the solutions. The general form of the perturbation solutions indicates how parameters in the control laws may be chosen to achieve a desired convergence rate. A detailed analysis of controllers exhibiting exponential convergence is included

The actively controlled jet in crossflow

This study quantifies the dynamics of actuation for the temporally forced, round gas jet injected transversely into a crossflow, and incorporates these dynamics in developing a methodology for open loop jet control. A linear model for the dynamics of the forced jet actuation is used to develop a dynamic compensator for the actuator. When the compensator is applied, it allows the jet to be forced in a manner which results in a more precisely prescribed, temporally varying exit velocity, the RMS amplitude of perturbation of which can be made independent of the forcing frequency. Use of the compensator allows straightforward comparisons among different conditions for jet excitation. Clear identification can be made of specific excitation frequencies and characteristic temporal pulse widths which optimize transverse jet penetration and spread through the formation of distinct, deeply penetrating vortex structures

Integrated low power digital gyro control electronics

Embodiments of the invention generally encompass a digital, application specific integrated circuit (ASIC) has been designed to perform excitation of a selected mode within a vibratory rate gyroscope, damping, or force-rebalance, of other modes within the sensor, and signal demodulation of the in-phase and quadrature components of the signal containing the angular rate information. The ASIC filters dedicated to each channel may be individually programmed to accommodate different rate sensor designs/technology or variations within the same class of sensors. The ASIC architecture employs a low-power design, making the ASIC, particularly suitable for use in power-sensitive applications

Exponential Stabilization of Nonlinear Driftless Systems with Robustness to Unmodeled Dynamics

Exponential stabilization of nonlinear driftless affine control systems is addressed with the concern of achieving robustness with respect to imperfect knowledge of the system's control vector fields. In order to satisfy this robustness requirement, and inspired by \cite{bennani95} where the same issue was first addressed, we consider a control strategy which consists of applying periodically updated open-loop controls that are continuous with respect to state initial conditions. These controllers are more precisely described as continuous time-periodic feedbacks associated with a specific dynamic extension of the original system. Sufficient conditions which, if they are satisfied by the control law, ensure that the control is a robust exponential stabilizer for the extended system are given. Explicit and simple control expressions which satisfy these conditions in the case of $n$-dimensional chained systems are proposed. A constructive algorithm for the design of such control laws, which applies to any (sufficiently regular) driftless control system, is described

Supplementation of Male Pheromone on Rock Substrates Attracts Female Rock Lizards to the Territories of Males: A Field Experiment

Background: Many animals produce elaborated sexual signals to attract mates, among them are common chemical sexual signals (pheromones) with an attracting function. Lizards produce chemical secretions for scent marking that may have a role in sexual selection. In the laboratory, female rock lizards (Iberolacerta cyreni) prefer the scent of males with more ergosterol in their femoral secretions. However, it is not known whether the scent-marks of male rock lizards may actually attract females to male territories in the field. Methodology/Principal Findings: In the field, we added ergosterol to rocks inside the territories of male lizards, and found that this manipulation resulted in increased relative densities of females in these territories. Furthermore, a higher number of females were observed associated to males in manipulated plots, which probably increased mating opportunities for males in these areas. Conclusions/Significance: These and previous laboratory results suggest that female rock lizards may select to settle in home ranges based on the characteristics of scent-marks from conspecific males. Therefore, male rock lizards might attract more females and obtain more matings by increasing the proportion of ergosterol when scent-marking their territories. However, previous studies suggest that the allocation of ergosterol to secretions may be costly and only high quality male