Anthony J. Calise

Extended Energy Management Methods for Flight Performance Optimization

This paper develops a singular perturbation approach to extend existing energy managment (EM} methods. A procedure is outlined for modeling altitude and flight path angle dynamics which are ignored in EM solutions. It is shown that nonlinear feedback solutions can be obtained, even for EM problem formulations which currently result in a two-point boundary-value problem. A nonlinear controller for two-dimensional, minimum time aircraft climbs is derived and numerical results for a fighter aircraft are given. The procedure outlined in this paper is general and applicable to solving a wide class of optimal control problems. It avoids the problem of picking the unknown boundary conditions at the initial and terminal times to suppress the unstable modes in the boundary layer.

Singular perturbation methods for variational problems in aircraft flight

The solution of variational problems by singular perturbation methods is discussed. In addition to the benefits of order reduction, these methods also can serve as practical devices for treating the singularities arising in problems where the control appears linearly and/or in state-constrained control problems. Furthermore, approximate feedback solutions can be derived for problem formulations that currently result in a nonlinear two-point boundary value problem. To illustrate an application, a feedback solution for aircraft, three-dimensional minimum lime turns is derived and discussed. Numerical results are presented for an F-106 and an F-4E aircraft.

A new boundary layer matching procedure for singularly perturbed system

This short paper proposes a procedure for applying singular perturbation methods to separately analyze state dynamics even when they vary on the same time scale. The intent is to expand the family of problems to which these methods can be successfully applied. In particular, several problems in flight mechanics are identified. Numerical results for the minimum time to climb problem are given where the procedure is used to separate altitude and flight-path angle dynamics to produce a closed-form solution for lift.

Paper: Optimal output feedback design of systems with ill-conditioned dynamics

Singular perturbation concepts are exploited to develop a procedure for designing a constant gain, output feedback control system. It is assumed that the original system is ill-conditioned in the sense that the plant contains widely separated dynamics, and that an accurate description for the high frequency behavior may not be available. The design procedure attempts to stabilize the system by minimizing a quadratic cost function made up of the control and states associated with a reduced order (low frequency) model for the plant, and a measure of stability for the neglected fast dynamics. The resulting design procedure does not require knowledge of the fast dynamics.

A singular perturbation analysis of optimal thrust control with proportional navigation guidance

This paper derives a nonlinear optimal thrust control law for a missile using proportional navigation guidance to intercept a maneuvering target. It is shown that using singular perturbation techniques combined with a multiple time scaling approach leads to a control solution that has an algebraic feedback form. A state transformation (similar to the energy state transformation used in aircraft analysis) that can be used to extend the analysis is also derived. Numerical results are given for a short range air-launched missile, and comparisons are made to proportional navigation guidance with boost-coast propulsion. The results show that optimal TMC greatly improves performance for missile launches at high aspect angles relative to the target velocity vector, and when launching from a lag condition relative to the line-of-sight.

A servo compensator design approach for variable structure systems

This paper presents an approach for designing a variable structure servo-mechanism for the case of constant disturbance inputs. The design makes use of servo compensators to achieve asymptotic regulation. The effect that disturbances within and outside the range space of the control action have on the design structure are separately discussed. The main result relies on a stability condition which guarantees reaching and existence of the sliding mode.

Near-optimal output feedback regulation of ill-conditioned linear systems

Issues pertaining to the well-posedness of a two time scale approach to the optimal output feedback regulator problem are examined. An approximate quadratic performance index is derived, reflecting a two time scale decomposition of the system dynamics. It is shown that, under mild assumptions, minimization of this cost leads to feedback gains providing a second-order approximation of the full system optimal performance. This is verified with a numerical example. A convergent sequential numerical algorithm for calculating these gains is described.

Output feedback design for aircraft with ill-conditioned dynamics

Singular perturbation concepts are exploited to develop a procedure for designing a dynamic output feedback controller. It is assumed that the original system is ill-conditioned in the sense that the plant contains widely separated dynamics requiring stabilization, and that the measurements are composed of a linear combination of slow and fast variables. A two time scale design approach results which depends on two-way control spillover suppression. An application to active control of aircraft wing flutter modes is used to illustrate the design method.

Hyperstability in variable structure systems

Reaching and existence conditions for variable structure systems are considered using hyperstability theory. A new sufficient condition for linear, time invariant systems is derived, and an example is given to illustrate how the condition is more relaxed than the sufficient condition presently used for such systems.

An analysis of a four state model for pursuit-evasion games

An analysis is given for a four state model useful in medium range air-combat analysis. The problem is formulated as a differential game of degree, with capture time as the performance index. The model is representative of a three-dimensional pursuit-evasion game between two aircraft at medium to large ranges. A complete and exact characterization of the optimal paths and controls is given for arbitrary aerodynamic and propulsion models.