Bruce K. Walker

Stochastic stability analysis for continuous-time fault tolerant control systems

Active fault tolerant control systems are feedback control systems that reconfigure the control law in real time based on the response from an automatic failure detection and identification (FDI) scheme. The dynamic behaviour of such systems is characterized by stochastic differential equations because of the random nature of the failure events and the FDI decisions. The stability analysis of these systems is addressed in this paper using stochastic Lyapunov functions and supermartingale theorems. Both exponential stability in the mean square and almost-sure asymptotic stability in probability are addressed. In particular, for linear systems where the coefficients of the closed loop system dynamics are functions of two random processes with markovian transition characteristics (one representing the random failures and the other representing the FDI decision behaviour), necessary and sufficient conditions for exponential stability in the mean square are developed.

Aerodynamic Parameter Estimation for High-Performance Aircraft Using Extended Kalman Filtering

The estimation of aerodynamic coefficients in aircraft dynamic models from flight-test data is addressed in this paper. An extended Kalman filter (EKF), implementing the full nonlinear kinematics of the aircraft equations of motion, was used for this purpose. Flight-test data from NASAs X-31 Drop Model and High Angle-of-attack Research Vehicle (HARV) were analyzed. The EKF parameter estimates for the X-31 compared well with wind-tunnel data and flight-data results using other identification techniques. For the HARV, the assumption of pseudonoise in the parameter dynamic model substantially improved the state and parameter estimates. A residual correlation method was used to estimate the process noise intensity matrix for this aircrafts flight data.

Fault Detection Threshold Determination Technique Using Markov Theory

A method for determining time-varying Failure Detection and Identification (FDI) thresholds for singlesample decision functions is described in the context of a triplex system of inertial platforms. A cost function consisting of the probability of vehicle loss due to FDI decision errors is minimized. A discrete Markov model is constructed from which this cost can be determined as a function of the decision thresholds employed to detect and identify the first and second failures. Optimal thresholds are determined through the use of parameter optimization techniques. The application of this approach to threshold determination is illustrated for the Space Shuttles inertial measurement instruments.

Input selection for a second-order mass property estimator

A strategy is developed for selecting online the attitude control jet firings to be commanded during the mass property estimation phase of the operation of any asymmetrical rigid satellite. Using symbolic manipulation, an effort is made to develop a selection stratgegy that accelerates convergence by maximally decreasing the change in the covariance of the estimation error at each time step. These results, which are too complex for online implementation, motivate a suboptimal direct search stragegy. Examples of mass property identification are then demonstrated on a high-fidelity simulation of the Space Shuttle. The results indicate that the suboptimal strategies yield significant reductions in the time to convergence for the mass property estimates and in the cumulative fuel usage relative to a typical predetermined jet firing sequence.

Stochastic Stability Analysis for Continuous Time Fault Tolerant Control Systems

Active fault tolerant control systems are feedback control systems that reconfigure the control law in real time based on the response from an automatic failure detection and identification (FDI) scheme. The dynamic behavior of such systems is characterized by stochastic differential equations because of the random nature of the failure events and the FDI decisions. The stability analysis of these systems is addressed in this paper using stochastic Lyapunov functions and supermartingale theorems. Both exponential stability in the mean square and almost sure asymptotic stability in probability are addressed. In particular, for linear systems where the coefficients of the closed loop system dynamics are functions of two random processes with Markovian transition characteristics (one representing the random failures and the other representing the FDI decision behavior), necessary and sufficient conditions for exponential stability In the mean square are developed.

Stability and Control of Lean Blowout in Chemical Kinetics-Controlled Combustion Systems

This study is motivated by lean-blowout (LBO) detection and control in dry-low-emission (DLE) combustion systems. However, this analysis is confined to chemical kinetics-controlled combustion. Despite its simplicity, some useful insight may still be shed on near-LBO combustion systems, as the chemical reaction rates are rather low near LBO. A third-order linear well-stirred reactor (WSR) model is derived to examine a combustors responses to small deviations from equilibrium points or small external disturbances. Numerical simulation of the normalized, nonlinear, unsteady WSR model is performed to examine a combustors responses to large deviations from equilibrium points or large external disturbances. Eigenvalue analysis shows that, with decreasing equivalence ratio, two real negative eigenvalues will merge and bifurcate into a complex conjugate pair, and will finally cross the imaginary axis and move into the right-half-phase plane. Complex eigenvalues imply the existence of an oscillating mode for which the damping ratio is found to consistently decrease at the approach of LBO. A lower preheat temperature, a higher percentage of incomplete combustion, and more heat loss exacerbate near-LBO combustion stability. The predicted near-extinction oscillating frequency is typically below 25 Hz, and decreases with a larger percentage of incomplete combustion. Comparisons between linear predictions and experiments, where appropriate, are made. Triggered instability is observed (i.e., a WSR may remain stable in the presence of small external disturbances, but will undergo a subcritical bifurcation to complete flame quenching if external disturbances exceed certain thresholds). A slight increase in equivalence ratio, a higher preheat temperature, less heat loss, and a smaller percentage of incomplete combustion are effective in strengthening a WSRs resistance to LBO. This paper numerically demonstrates that zero-mean, small-amplitude fuel modulations based on modern control strategies can be very useful to enhance lean combustion stability and mitigate the danger of LBO.

UAV navigation using predictive vector field control

This paper introduces the predictive vector field controller (PVF). This controller is designed for controlling nonlinear systems with constraints in order to follow artificial vector fields whose integral curves converge to and circulate about a desired path. This is achieved by predicting the state of the system at some future time horizon using a finite set of system inputs by uniformly sampling the configuration space of the system. A cost function relating the vector field and the future state of the system is then evaluated and a continuous mapping between the configuration space and the cost space is computed. The optimal system input with respect to the cost function is determined by minimizing this mapping. To demonstrate the performance of the PVF controller, we compare it to a nonlinear dynamic inversion (NDI) controller for the Dubins vehicle model. We show that, for a given path and vector field, the PVF controller provides accurate path following with significantly less control effort when compared to the NDI controller. Additionally, we demonstrate, that due to the predictive nature of the PVF controller, it does not possess the chattering behavior typically associated with other vector field-based techniques. Finally, we demonstrate the robustness of the PVF algorithm relative to the underlying vector field.

Approximate Evaluation of Semi-Markov Chain Reliability Models,

Abstract A property observed in high reliability fault-tolerant control systems is the relatively rare occurrence of component failures compared to the frequent occurrence of redundancy management decision events. This property leads to a temporal decomposition of the semi-Markov chain reliability model into two time-scales: a slow time-scale for failure events and a fast time-scale for FDI events. Conditions are described under which a semi-Markov chain reliability model of a high reliability fault-tolerant control system can be approximated by an enlarged Markov process, the parameters of which are derived from the parameters of the semi-Markov chain.

Effects of redundancy management on reliability modeling

Two methods are investigated for incorporating the effects of fault detection and isolation (FDI) decision errors and redundancy management (RM) policy into reliability models for a simple single-component-dual-redundant system. These two methods are combinatorial analysis and Markov chain modeling. Reliability analysts have traditionally chosen the classical combinatorial approach. However, the authors show that the existence of time-ordered event sequences resulting from the interaction of FDI decision errors with the RM policy considerably complicates the construction of the combinatorial model. An error analysis illustrates that a simplified combinatorial model, which ignores these time-ordered event sequences, inaccurately predicts the system reliability. The Markov modeling technique is an excellent alternative to the combinatorial approach because it easily and accurately accounts for time-ordered event sequences such as those present in fault-tolerant systems. >

Path Planning of Unmanned Aerial Vehicles in a Dynamic Environment

The main goal of this research effort is to determine the optimal trajectory for an unmanned aerial vehicle (UAV) in a dynamic environment. A Model Predictive Control (MPC) approach is utilized to provide collision avoidance in view of pop-up threats and a random set of moving and stationary obstacles (no fly zones). The UAV path planning needs to adapt in near real-time to the dynamic nature of the operational scenario, and to react rapidly to updates in the situational awareness, given the vehicle’s maneuvering constraints. To achieve this objective, the UAV navigates from a given starting point to a desired target point via selected intermediate waypoints. The possible waypoints are geometrically obtained with additional waypoints placed near the vertices of each polygonshaped obstacle. The MPC optimizer minimizes a cost function at each control cycle using a nonlinear dynamic model of the situation with maneuvering constraints included. The MPC algorithm is selected because it can improve system performance while effectively handling constraints. However, the huge computational effort required for a complete nonlinear realization of the MPC algorithm renders infeasible a comprehensive real-time optimization for this application. To circumvent this problem, discrete-time Laguerre functions are used as basis functions to represent the control inputs instead of using their complete time histories. A representative scenario, consisting of five targets, five static obstacles, nine pop-up threats, and a large, moving no-fly zone, is used to demonstrate this algorithm. Results are presented based on MATLAB® simulation experiments using a UAV model that represents a RQ-7 Shadow 200 to demonstrate the effectiveness of this approach.