Analytical guidance law development for aerocapture at Mars

During the first part of this reporting period research has concentrated on performing a detailed evaluation, to zero order, of the guidance algorithm developed in the first period taking the numerical approach developed in the third period. A zero order matched asymptotic expansion (MAE) solution that closely satisfies a set of 6 implicit equations in 6 unknowns to an accuracy of 10(exp -10), was evaluated. Guidance law implementation entails treating the current state as a new initial state and repetitively solving the MAE problem to obtain the feedback controls. A zero order guided solution was evaluated and compared with optimal solution that was obtained by numerical methods. Numerical experience shows that the zero order guided solution is close to optimal solution, and that the zero order MAE outer solution plays a critical role in accounting for the variations in Loh's term near the exit phase of the maneuver. However, the deficiency that remains in several of the critical variables indicates the need for a first order correction. During the second part of this period, methods for computing a first order correction were explored

Guidance and flight control law development for hypersonic vehicles

During the third reporting period our efforts were focused on a reformulation of the optimal control problem involving active state-variable inequality constraints. In the reformulated problem the optimization is carried out not with respect to all controllers, but only with respect to asymptotic controllers leading to the state constraint boundary. Intimately connected with the traditional formulation is the fact that when the reduced solution for such problems lies on a state constraint boundary, the corresponding boundary layer transitions are of finite time in the stretched time scale. Thus, it has been impossible so far to apply the classical asymptotic boundary layer theory to such problems. Moreover, the traditional formulation leads to optimal controllers that are one-sided, that is, they break down when a disturbance throws the system on the prohibited side of the state constraint boundary

Singular perturbation techniques for real time aircraft trajectory optimization and control

The usefulness of singular perturbation methods for developing real time computer algorithms to control and optimize aircraft flight trajectories is examined. A minimum time intercept problem using F-8 aerodynamic and propulsion data is used as a baseline. This provides a framework within which issues relating to problem formulation, solution methodology and real time implementation are examined. Theoretical questions relating to separability of dynamics are addressed. With respect to implementation, situations leading to numerical singularities are identified, and procedures for dealing with them are outlined. Also, particular attention is given to identifying quantities that can be precomputed and stored, thus greatly reducing the on-board computational load. Numerical results are given to illustrate the minimum time algorithm, and the resulting flight paths. An estimate is given for execution time and storage requirements

Two time scale output feedback regulation for ill-conditioned systems

Issues pertaining to the well-posedness of a two time scale approach to the output feedback regulator design problem are examined. An approximate quadratic performance index which reflects a two time scale decomposition of the system dynamics is developed. It is shown that, under mild assumptions, minimization of this cost leads to feedback gains providing a second-order approximation of optimal full system performance. A simplified approach to two time scale feedback design is also developed, in which gains are separately calculated to stabilize the slow and fast subsystem models. By exploiting the notion of combined control and observation spillover suppression, conditions are derived assuring that these gains will stabilize the full-order system. A sequential numerical algorithm is described which obtains output feedback gains minimizing a broad class of performance indices, including the standard LQ case. It is shown that the algorithm converges to a local minimum under nonrestrictive assumptions. This procedure is adapted to and demonstrated for the two time scale design formulations

Research on output feedback control

In designing fixed order compensators, an output feedback formulation has been adopted by suitably augmenting the system description to include the compensator states. However, the minimization of the performance index over the range of possible compensator descriptions was impeded due to the nonuniqueness of the compensator transfer function. A controller canonical form of the compensator was chosen to reduce the number of free parameters to its minimal number in the optimization. In the MIMO case, the controller form requires a prespecified set of ascending controllability indices. This constraint on the compensator structure is rather innocuous in relation to the increase in convergence rate of the optimization. Moreover, the controller form is easily relatable to a unique controller transfer function description. This structure of the compensator does not require penalizing the compensator states for a nonzero or coupled solution, a problem that occurs when following a standard output feedback synthesis formulation

Research on output feedback control of systems with ill-conditioned dynamics

The accuracy of a two time scale approach to the output feedback regulator design problem is examined. An approximate quadratic performance index which reflects a two time scale decomposition of the system dynamics is developed. A sequential numerical algorithm is defined which obtains output feedback gains minimizing a broad class of performance indices, including the standard LQ case. A procedure for optimally zeroing selected gain elements in an output feedback gain matrix is developed and demonstrated. A summary of conference and journal publications from this research is also provided

Trajectory optimization and guidance law development for national aerospace plane applications

The work completed to date is comprised of the following: a simple vehicle model representative of the aerospace plane concept in the hypersonic flight regime, fuel-optimal climb profiles for the unconstrained and dynamic pressure constrained cases generated using a reduced order dynamic model, an analytic switching condition for transition to rocket powered flight as orbital velocity is approached, simple feedback guidance laws for both the unconstrained and dynamic pressure constrained cases derived via singular perturbation theory and a nonlinear transformation technique, and numerical simulation results for ascent to orbit in the dynamic pressure constrained case

Rapid near-optimal aerospace plane trajectory generation and guidance

Effort was directed toward the problems of the real time trajectory optimization and guidance law development for the National Aerospace Plane (NASP) applications. In particular, singular perturbation methods were used to develop guidance algorithms suitable for onboard, real time implementation. The progress made in this research effort is reported

Rapid near-optimal trajectory generation and guidance law development for single-stage-to-orbit airbreathing vehicles

General problems associated with on-board trajectory optimization, propulsion system cycle selection, and with the synthesis of guidance laws were addressed for an ascent to low-earth-orbit of an air-breathing single-stage-to-orbit vehicle. The NASA Generic Hypersonic Aerodynamic Model Example and the Langley Accelerator aerodynamic sets were acquired and implemented. Work related to the development of purely analytic aerodynamic models was also performed at a low level. A generic model of a multi-mode propulsion system was developed that includes turbojet, ramjet, scramjet, and rocket engine cycles. Provisions were made in the dynamic model for a component of thrust normal to the flight path. Computational results, which characterize the nonlinear sensitivity of scramjet performance to changes in vehicle angle of attack, were obtained and incorporated into the engine model. Additional trajectory constraints were introduced: maximum dynamic pressure; maximum aerodynamic heating rate per unit area; angle of attack and lift limits; and limits on acceleration both along and normal to the flight path. The remainder of the effort focused on required modifications to a previously derived algorithm when the model complexity cited above was added. In particular, analytic switching conditions were derived which, under appropriate assumptions, govern optimal transition from one propulsion mode to another for two cases: the case in which engine cycle operations can overlap, and the case in which engine cycle operations are mutually exclusive. The resulting guidance algorithm was implemented in software and exercised extensively. It was found that the approximations associated with the assumed time scale separation employed in this work are reasonable except over the Mach range from roughly 5 to 8. This phenomenon is due to the very large thrust capability of scramjets in this Mach regime when sized to meet the requirement for ascent to orbit. By accounting for flight path angle and flight path angle rate in construction of the flight path over this Mach range, the resulting algorithm provides the means for rapid near-optimal trajectory generation and propulsion cycle selection over the entire Mach range from take-off to orbit

Research in robust control for hypersonic aircraft

The research during the third reporting period focused on fixed order robust control design for hypersonic vehicles. A new technique was developed to synthesize fixed order H(sub infinity) controllers. A controller canonical form is imposed on the compensator structure and a homotopy algorithm is employed to perform the controller design. Various reduced order controllers are designed for a simplified version of the hypersonic vehicle model used in our previous studies to demonstrate the capabilities of the code. However, further work is needed to investigate the issue of numerical ill-conditioning for large order systems and to make the numerical approach more reliable