Bradley T. Burchett

Model predictive lateral pulse jet control of an atmospheric rocket

Uncontrolled direct fire atmospheric rockets exhibit high impact point dispersion, even at relatively short range, and, as such, have been employed as area weapons on the battlefield. To reduce the dispersion of a direct fire rocket, the use of a small number of short-duration lateral pulses acting as a control mechanism is investigated. A unique control law is reported that combines model predictive control and linear projectile theory for lateral pulse jet control of an atmospheric rocket. The impact point in the target plane is directly controlled. Through simulation, this model predictive flight control law is shown to efficiently reduce direct fire rocket dispersion. A parametric trade study on an example rocket configuration is reported that details the effect of the number and amplitude of individual pulse jets, as well as the effect of the flight control system computation cycle time.

Prediction of swerving motion of a dual-spin projectile with lateral pulse jets in atmospheric flight

Using the linear theory for a dual-spin projectile in atmospheric flight, closed form expressions are obtained for swerving motion under the action of lateral pulse jets. Trajectory results generated by the linear theory equations and a fully nonlinear seven degree-of-freedom dual spin projectile model agree favorably. The analytic solution provides a relatively straightforward and computationally efficient means of trajectory estimation which is useful within smart weapon flight control systems. In order to accurately predict the impact point using the analytic solution, the dual-spin projectile linear model must be updated periodically. Terminal impact point prediction degrades rapidly as the linear model update interval is increased beyond a critical value. Control authority, as defined by the change in impact location due to a pulse jet firing, steadily decreases as a function of projectile down range position.

Fuzzy logic trajectory design and guidance for terminal area energy management

The second-generation reusable launch vehicle will incorporate sophisticated new guidance and control methods to increase the system’s flexibility, in particular making it a practical vehicle for unmanned cargo missions. This project focuses on trajectory design and guidance for the terminal area energy management phase of flight, where traditional methods have produced limited results. Fuzzy logic methods are used to make onboard autonomous gliding return trajectory design robust to the possibility of control surface failures, thus, increasing the flexibility of unmanned gliding recovery and landing. Fuzzy inference systems are used to design the trajectory and provide guidance commands for the vehicle bank angle and negative z axis acceleration. Fuzzy logic provides a practical means of incorporating conflicting objectives and constraints in choosing trajectory parameters. The method has the flexibility to choose these parameters from a continuum of possible values. The design is validated using a high-fidelity six-degree-of-freedom simulation of the X-33 technology demonstrator.

A Comparison of Different Guidance Schemes for a Direct Fire Rocket With a Pulse Jet Control Mechanism

Abstract : Compared to gun launch ammunition, uncontrolled direct fire atmospheric rockets are terribly inaccurate, to the point where they are used most effectively on the battlefield as area weapons. Dispersion characteristics can be dramatically improved by outfitting the rocket with a suitable control mechanism and sensor suite. In the work reported here, a lateral pulse jet control mechanism is considered. The lateral pulse jet mechanism consists of a finite number of small thrusters spaced equally around the circumference of the rocket. Using a simulation model that includes projectile, flight control system, and inertial measurement unit dynamics, three different control laws are contrasted, namely, proportional navigation guidance, parabolic and proportional navigation guidance, and trajectory tracking control laws. When the number of individual pulse jets is small, a trajectory tracking control law provides superior dispersion reduction. However, as the number of pulse jets is increased, the relative performance of the parabolic and proportional navigation guidance control law is slightly better than the trajectory tracking control law. When the number of pulse jets is small, the proportional navigation guidance, as well as the parabolic and proportional navigation guidance control laws, exhibits large mean miss distance. All control laws appear to be equally susceptible to accelerometer and gyroscope errors that corrupt inertial measurement unit rocket state feedback.

QR-Based Algorithm for Eigenvalue Derivatives

Many engineering optimization problems require the calculation of eigenvalue sensitivities. Several straightforward methods have been developed for calculating partial derivatives of distinct complex eigenvalues. Recent work has expanded these methods to include systems with repeated real eigenvalues. A new method is presented that embeds eigenvalue derivative computation into an established numerical eigenvalue algorithm, namely, QR decomposition. The technique is shown to have computational advantages for high-order systems.

QZ-Based Algorithm for System Pole, Transmission Zero, and Residue Derivatives

The QZ algorithm gives a robust way of computing solutions to the generalized eigenvalue problem. The generalized eigenvalue problem is used in linear control theory to find solutions to Ricatti equations, as well as to determine system transmission zeros. In state-space linear system analysis, the system poles and transmission zeros are particularly important for determining system time and frequency response. Here, we embed calculation of the eigenvalue derivatives in the QZ algorithm such that the derivatives of system poles and transmission zeros are computed simultaneously with the poles and zeros themselves. The resulting method is further exercised in finding generalized eigenvalues and their sensitivities required for finding the derivatives of system residues. This technique should open the door to solutions of problems of interest by unconstrained gradient-based methods. Typical numerical results are presented.

Optimal output feedback using dyadic decomposition

The literature is replete with methods to numerically compute the gain matrix for optimal output feedback controllers. Inherent to the vast majority of these methods is the solution of necessary conditions for optimality. This work presents a fundamentally different numerical scheme for computing H/sub 2/ optimal output feedback controllers. The method is based on a direct and analytic solution to the cost function using dyadic decomposition. Analytic solutions for the first and second derivatives of the cost function with respect to gain matrix elements are written in compact form allowing efficient solution using robust second order Newton-Raphson optimization. Using a flight control example, the method is exercised and compared to other optimal output feedback methods.

Aerodynamic Parameter Identification for Symmetric Projectiles: Comparing Gradient Based and Evolutionary Algorithms

Two methods for finding aerodynamic coefficients from free flight range data are compared— a gradient based method and an evolutionary algorithm each using a linear theory solution. The inputs are limited to swerve and yaw measurements such as would be available from spark range or yaw card tests. Each method is tested with minimal inputs. Methods are compared for computational effort and accuracy. Numerical results for two typical finned projectiles are presented.

Feedback linearization guidance for approach and landing of reusable launch vehicles

A feedback linearization guidance system is developed for approach and landing of a reusable launch vehicle. The three-input three-output system is derived through a state transformation using Lie algebra, resulting in a five state linear system. The linearization is shown to be valid for headings within ninety degrees of runway heading and flight path angles within ninety degrees of horizontal. A pseudo control is designed based on a linear quadratic tracker with integrators. Lie algebra then gives the transformation from pseudo control to physical control. An approximate aerodynamic model gives a further transformation from lift, drag, and bank angle, to negative z axis acceleration, bank angle, and speed brake position. The controller is simulated in a Matlab / Simulink environment, and certain controller parameters are optimized using a genetic algorithm.

Euler-Lagrange Optimal Control for Symmetric Projectiles

The linear theory model of a symmetric projectile is well suited to optimal control methods, especially the finite horizon linear optimal regulator. Using a nine–state linear model with gravity treated as an uncontrollable mode, necessary conditions for optimality are derived. These conditions are solved closed–form using a matrix exponential of the Hamiltonian matrix multiplied by distance to go in calibers. Control is thus found without a reference trajectory. A second method allowing system parameters to vary with time is developed and compared. The time–varying Riccati equation is solved recursively backward in time and control at the current state is found without a reference trajectory. Performance is demonstrated on linear and non–linear plant models using forward mounted canards.