James Fisher

Linear quadratic regulation of systems with stochastic parameter uncertainties

In this paper, we develop a theoretical framework for linear quadratic regulator design for linear systems with probabilistic uncertainty in the parameters. The framework is built on the generalized polynomial chaos theory. In this framework, the stochastic dynamics is transformed into deterministic dynamics in higher dimensional state space, and the controller is designed in the expanded state space. The proposed design framework results in a family of controllers, parameterized by the associated random variables. The theoretical results are applied to a controller design problem based on stochastic linear, longitudinal F16 model. The performance of the stochastic design shows excellent consistency, in a statistical sense, with the results obtained from Monte-Carlo based designs.

Polynomial Chaos-Based Analysis of Probabilistic Uncertainty in Hypersonic Flight Dynamics

In this paper, we present a novel computational framework for analyzing the evolution of the uncertainty in state trajectories of a hypersonic air vehicle due to the uncertainty in initial conditions and other system parameters. The framework is built on the so-called generalized polynomial chaos expansions. In this framework, stochastic dynamical systems are transformed into equivalent deterministic dynamical systems in higher dimensional space. Here, the evolution of uncertainty due to initial condition, ballistic coefficient, lift over drag ratio, and atmospheric density is analyzed. The problem studied here is related to the Mars entry, descent, and landing problems. We demonstrate that the polynomial chaos framework is able to predict evolution of uncertainty, in hypersonic flight, with the same order of accuracy as the Monte-Carlo methods but with more computational efficiency.

Stability analysis of stochastic systems using polynomial chaos

A novel framework for stability analysis of linear and polynomial stochastic systems is presented. The framework is built on generalized polynomial chaos theory, which enables analysis of dynamical systems with probabilistic uncertainty on system parameters with various distributions. The theory allows for the transformation of stochastic problems into a higher dimensional deterministic problem, that is able to accurately approximate the evolution of uncertainty in the state trajectories due to stochastic system parameters. The developed theory is applied to analyze a linear flight control design for an F-16 aircraft model. The problem of generating stability certificates for stochastic polynomial systems is also considered.

Optimal Trajectory Generation With Probabilistic System Uncertainty Using Polynomial Chaos

In this paper, we develop a framework for solving optimal trajectory generation problems with probabilistic uncertainty in system parameters. The framework is based on the generalized polynomial chaos theory. We consider both linear and nonlinear dynamics in this paper and demonstrate transformation of stochastic dynamics to equivalent deterministic dynamics in higher dimensional state space. Minimum expectation and variance cost function are shown to be equivalent to standard quadratic cost functions of the expanded state vector. Results are shown on a stochastic Van der Pol oscillator.

On stochastic LQR design and polynomial chaos

In this paper we develop a novel theoretical framework for linear quadratic regulator design for linear systems with probabilistic uncertainty in the parameters. The framework is built on the generalized polynomial chaos theory, which can handle Gaussian, uniform, beta and gamma distributions. In this framework, the stochastic dynamics is transformed into deterministic dynamics in higher dimensional state space, and the controller is designed in the expanded state space. The proposed design framework results in a family of controllers, parameterized by the associated random variables. The theoretical results are applied to a controller design problem based on stochastic linear, longitudinal F16 model. The performance of the stochastic design shows excellent consistency with the results obtained from Monte-Carlo based designs, in a statistical sense.

Spacecraft Momentum Management and Attitude Control using a Receding Horizon Approach

This paper presents the application of the Receding Horizon approach to the spacecraft momentum management problem. Attitude control of a satellite in a circular orbit under the in∞uence of constant disturbances is considered. Control designs are demonstrated for a simplifled planar case as well as for the full dynamics of the spacecraft. The response of the receding horizon controller is compared to a controller derived from Lyapunov conditions. The receding horizon approach demonstrates its ability to align the spacecraft to a torque equilibrium attitude and regulate the momentum of the control even when torque and rate constraints are included in the problem. I. Introduction Spacecraft attitude control has been an important area of research and has received a large amount of attention. Spacecraft dynamics provide a rich area of research because their behavior is nonlinear and they are in many cases to operate over a large range of operating conditions. A common choice of actuators to control the satellite is the momentum exchange device. Momentum exchange devices such as control moment gyros (CMG’s) or reaction wheels maintain control of the satellite by transferring momentum from the spacecraft so that it follows a designed trajectory. If a constant disturbance such as an aerodynamic torque acts on the spacecraft, the momentum in the control can build up as result of the constant control torque required to reject the disturbance. These momentum exchange devices can be desaturated by the alignment of the spacecraft with a torque equilibrium attitude (TEA). When such an attitude is obtained, the gravity gradient torque and gyroscopic torques balance the disturbance torque to create an equilibrium attitude. No control input is required to maintain this attitude, allowing the momentum to desaturate. For a spacecraft in a

Brief paper: Analysis of partial stability problems using sum of squares techniques

A methodology for algorithmic construction of Lyapunov functions for problems concerning the stability of an equilibrium with respect to part of the system variables is proposed. This methodology utilizes the previously developed sum of squares technique to determine Lyapunov certificates. Conditions for stability with respect to part of the variables are developed that allow for Lyapunov functions to be determined in terms of a sum of squares. Asymptotic stability conditions in terms of sum of squares polynomials are developed for autonomous and non-autonomous systems. An example is presented which demonstrates the methodology and gives insight into the new stability conditions.

Construction of Lyapunov Certificates for Partial Stability Problems Using Sum of Squares Techniques

A methodology is proposed that allows for algorithmic construction of Lyapunov functions for problems concerning the stability of an equilibrium with respect to part of the variables. This methodology utilizes the previously developed sum of squares technique to determine Lyapunov certificates. Conditions for stability with respect to part of the variables are developed that allow for Lyapunov functions to be determined in terms of a sum of squares. Some examples are presented that highlight these new conditions.

On generating random systems: a gramian approach

A new procedure for computer generation of random stable, linear, time invariant systems in state space form is described. Controllability and observability gramians are randomly generated according to a desired distribution. Then, a system which has the generated gramians is found. This new random system generation method is compared to several other methods using empirically observed distributions on Hankel singular values, eigenvalue magnitudes, and controllability/observability gramian condition numbers.

Characterization of Dynamical Instability Using Instantaneous Frequency

Dynamical instability induced by the initiation and advancement of mechanical faults in rotary elements is detrimental to the reliability and operation safety of the entire system. The inherent nonlinearity associated with bifurcation presents itself as difficulties in identifying and isolating features indicative of the presence and progression of faults that could lead to eventual mechanical deterioration. The perturbed and deteriorated states of a bearing-shaft system subjected to the actions of two types of commonly seen mechanical faults, namely, rotor speed and imbalance, are investigated using the basic notion of instantaneous frequency. The presented approach realizes temporal events of both short and long time scales as instantaneous frequencies in the joint time-frequency domain and thus effectively uncouples the harmonic components resulted from the coupling of multitude faults. Examples are given to demonstrate the feasibility of applying the approach to the characterization of various deteriorating bearing states and the identification of parameters associated with various modes of instability and chaotic response.Copyright