Interaction matrix uncertainty in active (and adaptive) optics

Uncertainty in the interaction matrix between sensors and actuators can lead to performance degradation or instability in control of segmented mirrors (typically the telescope primary). The interaction matrix is ill conditioned, and thus the position estimate required for control can be highly sensitive to small errors in knowledge of the matrix, due to uncertainty or temporal variations. The robustness to different types of uncertainty is bounded here using the small gain theorem and structured singular values. The control is quite robust to moderate uncertainty in actuator gain, sensor gain, or the ratio of sensor dihedral and height sensitivity. However, the control is extremely sensitive to small errors in geometry, with the maximum error that can be tolerated scaling inversely with the number of segments. The same tools can be applied to adaptive optics; however, the interaction matrix here is better conditioned and so uncertainty is less of an issue, with the tolerable error scaling inversely with the square root of the number of actuators

The sloppy model universality class and the Vandermonde matrix

In a variety of contexts, physicists study complex, nonlinear models with many unknown or tunable parameters to explain experimental data. We explain why such systems so often are sloppy; the system behavior depends only on a few `stiff' combinations of the parameters and is unchanged as other `sloppy' parameter combinations vary by orders of magnitude. We contrast examples of sloppy models (from systems biology, variational quantum Monte Carlo, and common data fitting) with systems which are not sloppy (multidimensional linear regression, random matrix ensembles). We observe that the eigenvalue spectra for the sensitivity of sloppy models have a striking, characteristic form, with a density of logarithms of eigenvalues which is roughly constant over a large range. We suggest that the common features of sloppy models indicate that they may belong to a common universality class. In particular, we motivate focusing on a Vandermonde ensemble of multiparameter nonlinear models and show in one limit that they exhibit the universal features of sloppy models.Comment: New content adde

Dynamic operability assessment : a mathematical programming approach based on Q-parametrization

Bibliography: pages 197-208.The ability of a process plant to guarantee high product quality, in terms of low variability, is emerging as a defining feature when distinguishing between alternative suppliers. The extent to which this can be achieved is termed a plant's dynamic operability and is a function of both the plant design and the control system design. In the limit, however, the closedloop performance is determined by the properties inherent in the plant. This realization of the interrelationship between a plant design and its achievable closed-loop performance has motivated research toward systematic techniques for screening inherently inferior designs. Pioneering research in the early 1980's identified right-half-plane transmission zeros, time delays, input constraints and model uncertainty as factors that limit the achievable closedloop performance of a process. Quantifying the performance-limiting effect of combinations of these factors has proven to be a challenging problem, as reflected in the literature. It is the aim of this thesis to develop a systematic procedure for dynamic operability assessment in the presence of combinations of performance-limiting factors. The approach adopted in this thesis is based on the Q-parametrization of stabilizing linear feedback controllers and involves posing dynamic operability assessment as a mathematical programming problet? In the proposed formulation, a convex objective function, reflecting a measure of closed-loop performance, is optimized over all stable Q, subject. to a set of constraints on the closed-loop behavior, which for many specifications of interest is convex. A discrete-time formulation is chosen so as to allow for the convenient hand.ling of time delays and time-domain constraints. An important feature of the approach is that, due to the convexity, global optimality is guaranteed. Furthermore, the fact that Q parametrizes all stabilizing linear feedback controllers implies that the performance at the optimum represents the best possible performance for any such controller. The results are thus not biased by controller type or tuning, apart from the requirement that the controller be linear

Spacecraft nonlinear attitude control with bounded control input

The research in this thesis deals with nonlinear control of spacecraft attitude stabilization and tracking manoeuvres and addresses the issue of control toque saturation on a priori basis. The cascaded structure of spacecraft attitude kinematics and dynamics makes the method of integrator backstepping preferred scheme for the spacecraft nonlinear attitude control. However, the conventional backstepping control design method may result in excessive control torque beyond the saturation bound of the actuators. While remaining within the framework of conventional backstepping control design, the present work proposes the formulation of analytical bounds for the control torque components as functions of the initial attitude and angular velocity errors and the gains involved in the control design procedure. The said analytical bounds have been shown to be useful for tuning the gains in a way that the guaranteed maximum torque upper bound lies within the capability of the actuator and, hence, addressing the issue of control input saturation. Conditions have also been developed as well as the generalization of the said analytical bounds which allow for the tuning of the control gains to guarantee prescribed stability with the additional aim that the control action avoids reaching saturation while anticipating the presence of bounded external disturbance torque and uncertainties in the spacecraft moments of inertia. Moreover, the work has also been extended blending it with the artificial potential function method for achieving autonomous capability of avoiding pointing constraints for the case of spacecraft large angle slew manoeuvres. The idea of undergoing such manoeuvres using control moment gyros to track commanded angular momentum rather than a torque command has also been studied. In this context, a gimbal position command generation algorithm has been proposed for a pyramid-type cluster of four single gimbal control moment gyros. The proposed algorithm not only avoids the saturation of the angular momentum input from the control moment gyro cluster but also exploits its maximum value deliverable by the cluster along the direction of the commanded angular momentum for the major part of the manoeuvre. In this way, it results in rapid spacecraft slew manoeuvres. The ideas proposed in the thesis have also been validated using numerical simulations and compared with results already existing in the literature