Daniel Choukroun

Robust Nonlinear H∞ Output-Feedback for Spacecraft Attitude Control

We propose a computational scheme for solving the robust output-feedback H∞ control problem, for a class of nonlinear systems with polynomial vector field. By converting the resulting Hamilton-Jacobi inequalities from rational forms to their equivalent polynomial forms, we overcome the non-convex nature and numerical difficulty. Using quadratic Lyapunov functions, both the robust state-feedback and robust output-feedback problems are reformulated as semi-definite optimization conditions, while locally tractable solutions can be obtained through sum of squares (SOS) programming. The methodology is implemented on a spacecraft attitude control problem, where presence of unmodelled dynamics are matched and unstructured.

Aircraft Longitudinal Guidance Based on a Spatial Reference

In this study, instead of using time as the independent variable to describe the guidance dynamics of an aircraft, distance to land, which can be considered today to be available online with acceptable accuracy and availability, is adopted. A new representation of aircraft longitudinal guidance dynamics is developed according to this spatial variable. Then a nonlinear inverse control law based-on this new representation of guidance dynamics is established to make the aircraft follow accurately a vertical profile and a desired airspeed. The desired airspeed can be regulated to make the aircraft overfly different waypoints according to a planned time-table. Simulations results with different wind conditions for a transportation aircraft performing a descent approach for landing under this new guidance scheme are displayed.

Nonlinear Output-Feedback H ∞ Control for Spacecraft Attitude Control

In this paper, a novel computational scheme is proposed in order to solve the output-feedback H ∞ control problem for a class of nonlinear systems with polynomial vector field. By converting the resulting Hamilton-Jacobi inequalities from rational forms to their equivalent polynomial forms, we overcome the non-convex nature and numerical difficulty. Using quadratic Lyapunov functions, both the state feedback and output-feedback problems are reformulated as semi-definite optimization conditions, while locally tractable solutions can be obtained through sum of squares (SOS) programming. A numerical example shows that the proposed computational scheme results in a better disturbance attenuation closed-loop system, as compared to standard methods, by using classical quadratic Lyapunov functions. The novel methodology is applied in order to develop a robust spacecraft attitude regulator.

Spacecraft Attitude Estimation and Gyro Calibration via Stochastic H ∞ Filtering

A filter for estimating spacecraft attitude quaternion and gyro drift from vector measurements in the presence of white noises in the gyro error, in the drift dynamics, and in the line-of-sight measurement error is developed. The variance parameters of the white noises are unknown, and are modeled as non-anticipative second order stochastic processes. The approach taken in this work consists in estimating the attitude quaternion and the gyro drift while attenuating the transmission from the unknown variances to the estimation error. The resulting H ∞ filter involves the solution of a set of (differential) linear matrix inequalities. In the case of gyro white noises, extensive Monte-Carlo simulations were run showing that the proposed filter performs well from the standpoint of attitude estimation per se, in a wide range of gyro noise and line-of-sight noise intensities. The guaranteed disturbance attenuation level seems to be slightly dependent on the noises intensities. The actual level of disturbance attenuation is improving when the noises levels increase and admits as worst scenario the case of(ideal) noise-free sensors, as expected from the analysis. When compared with a matched quaternion Kalman filter, the H ∞ filter produces higher Monte-Carlo standard deviations of the estimation error, but lower Monte-Carlo means. The higher the level of noises are, the less obvious the advantage of the Kalman filter is. When estimating the quaternion only, and as opposed to standard quaternion Kalman filters, the H ∞ filter’s gains can be computed independently from the quaternion estimates, which makes it insensitive to estimation errors. This favorable feature is further emphasized when comparing its performances with those of unmatched Kalman filters. When provided with too high or too low noise covariances, the Kalman filter is outperformed by the H ∞ filter, which delivers essentially identical levels of errors within a wide range of noise intensities.

Fault-Tolerant Spacecraft Magnetic Attitude Control

This work is concerned with the development of a suboptimal control algorithm for Markovian jump-linear systems, and its application to fault-tolerant spacecraft magnetic attitude control. For completeness, the jump-linear quadratic optimal controller with full state and mode information is presented. Relaxing the assumption of perfect mode information, a similar optimal control problem is formulated where the mode is observed via discrete measurements. The elements of the measurement matrix, i.e. the probabilities for correct and wrong mode observations are assumed known. The optimal controller is developed, which requires an exponentially growing computational burden, and a suboptimal controller is proposed that only requires knowledge of the current mode measurement. This controller is finite memory and possess some of the classical linear quadratic regulator features such as the linear state feedback structure and a state quadratic optimal cost-to-go. The performances of the suggested algorithm are illustrated through extensive Monte-Carlo simulations on a simple numerical example. A realistic fault-tolerant spacecraft magnetic attitude controller is developed based on the proposed approach. The attitude controller succeeds in mitigating the destabilizing effect of corrupted mode observations while being computationally efficient.

Investigation of a Symmetric Vibrating Gyroscope Characteristics Using a Simplified Dynamic Model

In this work a dynamical model for MEMS vibrational gyroscopes is developed that generalizes a previous work, allows for simpler but accurate qualitative and quantitative analysis of several sources of angular velocity measurement errors, and opens avenues for future developments in MEMS vibrational gyroscopes designs. The proposed model equations govern the dynamics of the amplitudes rather than the dynamics of the rapid oscillatory processes. The characteristics of this approximate model are significantly slower than the driving frequency. It allows a linear time-invariant analysis of the angular velocity measurement errors. These may be direct like a bias caused by the structural damping, or indirect, due e.g. to the unmatched frequencies between the drive and the sense channels. The approximate model was validated on a particular numerical example by examination and comparison of the frequency responses. A simple proportional feedback control was designed for both the drive and the quadrature loops, showing the potential impact of the feedback gains on the low-frequencies error. Being a linear time-invariant model, this model will easily lend itself to the development of more advanced control strategies.

Spacecraft Attitude Estimation via Stochastic H ∞ Filtering

A filter for estimating spacecraft attitude quaternion from line-of-sight (LOS) measurements in the presence of white noises in the gyro output and in the attitude sensing is developed. The variance parameters of the white noises are assumed unknown and modeled as non-anticipative second order stochastic processes. The approach taken in this work consists in estimating the attitude quaternion while attenuating the transmission gain from the unknown variances to the estimation error. The resulting H1 filter involves the solution of a set of (differential) linear matrix inequalities, which do not depend on the estimated quaternion. Extensive Monte-Carlo simulations were run showing that the proposed filter performs well from the standpoint of attitude estimation per se, in a wide range of gyro and LOS intensities. The guaranteed disturbance attenuation level seems to be slightly dependent on the noises intensities. The actual level of disturbance attenuation is improving when the noises levels increase and admits as worst scenario the case of(ideal) noise-free sensors, as expected from the analysis. When compared with two different matched quaternion Kalman filters (KF), the H1 filter produces higher Monte-Carlo standard deviations of the estimation error, but lower Monte-Carlo means. The higher the level of noises are, the less obvious the advantage to the Kalman filters is. As opposed to standard quaternion Kalman filters, the H 1 filter’s gain process can be computed independently from the quaternion estimate process, which makes it unsensitive to estimation errors. This nice feature is further emphasized when comparing its performances with those of unmatched Kalman filters. When provided with too high or too low noise covariances, the Kalman filter is outperformed by the H1 filter, which delivers essentially identical levels of errors within a wide range of noise intensities.