Arthur E. Bryson

Energy-state approximation in performance optimization of supersonic aircraft

Energy state approximation for supersonic aircraft performance optimization with extension to maximum range problems, noting comparison with complex dynamic models

Attitude Control of a Flexible Spacecraft

An algorithm is presented for designing optimal low order compensators for high order systems and it is applied to the title problem where many vibration modes are excited by the control torque. These low order compensators are compared with the full order optimal compensator and found to be less sensitive to modeling errors and to provide near optimal attitude regulation.

Brief paper: Design of low-order compensators using parameter optimization

An LQG synthesis algorithm is presented that optimizes the parameters of a multi-input, multi-output compensator for a linear time-invariant plant, where the designer specifies the order and the structure of the compensator. An instructive way to use the algorithm is to start with the full-order LQG compensator in minimum-realization modal form, and then omit the more rapidly-damped modes with small residues. The reduced-order compensator is then optimized using the present algorithm. As an example, compensators of order 4, 3, 2 and 1 were designed for a seventh-order model of an airplane with two controls, two outputs, two disturbances and three measurements. Surprisingly little performance is lost by using these lower-order compensators.

Neighboring Optimal Aircraft Guidance in Winds

The technique of Neighboring Optimal Control is extended to handle cases of parameter change in the system dynamic model. This extension is used to develop an algorithm for optimizing horizontal aircraft trajectories in general wind fields using time-varying linear feedback gains. The minimum-time problem for an airplane traveling horizontally between two points in a variable wind field (a type of Zermelo Problem) is used to illustrate how perturbations in system parameters can be accounted for by augmenting the dynamic model with additional bias states. For the special case of a constant wind shear in the cross-track direction, analytical and numerical results are derived for bias perturbations. Numerical simulations are presented to demonstrate the performance of the proposed state-augmentation technique. An additional example is used to demonstrate an algorithm to compute near-optimal trajectories in general wind fields. The algorithm is based on nondimensionalizing the neighboring optimal control solutions and using piecewise linearly varying wind and horizontal wind shear parameters. One proposed application of this technique is to the computation and real-time update of time-optimal trajectories in wind fields by onboard flight management systems and by ground-based air traffic management automation tools.

Optimal landing of a helicopter in autorotation

Gliding descent in autorotation is a maneuver used by helicopter pilots in case of engine failure. The landing of a helicopter in autorotation is formulated as a nonlinear optimal control problem. The OH-58A helicopter was used. Helicopter vertical and horizontal velocities, vertical and horizontal displacement, and the rotor angle speed were modeled. An empirical approximation for the induced veloctiy in the vortex-ring state were provided. The cost function of the optimal control problem is a weighted sum of the squared horizontal and vertical components of the helicopter velocity at touchdown. Optimal trajectories are calculated for entry conditions well within the horizontal-vertical restriction curve, with the helicopter initially in hover or forwared flight. The resultant two-point boundary value problem with path equality constraints was successfully solved using the Sequential Gradient Restoration Technique.

Control Logic for Parameter Insensitivity and Disturbance Attenuation

An approach to the synthesis of control logic that is both insensitive to system parameters and attenuates response to input disturbances (sometimes called robust control logic) is presented and is applied to a simple system with an uncertain vibration frequency. A conclusion drawn from this study is that the lower the order of the compensator dynamics, the lower the sensitivity of the closed-loop system performance to parameter variations. It follows that estimated-state feedback is undesirable when there are large uncertainties in the system parameters. However, quadratic performance indices are minimized with estimated-state feedback, so the designer must trade off parameter insensitivity against disturbance attenuation. This is done here by minimizing the expected value of a sum of quadratic performance indices, each one of which is evaluated with different values of the system parameters; the decision variables in the minimization are parameters in a compensator of specified order that is less than or equal to the order of the system model.

Second-order algorithm for optimal model order reduction

A second-order algorithm is given for optimal model order reduction of continuous single-input/single-output systems. Given a transfer function of order n, it finds the transfer function of a model of order m<n that minimizes the integral-square difference between the impulse responses of the two systems. It is shown that this is exactly equivalent to two other problems: 1) minimizing the integral-square difference between the frequency responses of the two systems and 2) minimizing the mean-square difference between the statistical steady-state responses of the two systems to a white-noise input. The algorithm is based on the known explicit solution of the governing Lyapunov equation, which yields analytical expressions for the first and second derivatives of the performance index with respect to the residues and poles of the reduced-order model. These derivatives are then used in a Newton-Raphson algorithm. The algorithm is applied to find exact solutions for four examples, one of which has an unstable mode. The examples corroborate previous research indicating that Moores truncated balanced realization is not always close to the optimal reduced order model. Matching step response instead of impulse response requires only a slight change in the algorithm.

INVERSE AND OPTIMAL CONTROL FOR DESIRED OUTPUTS

This paper considers the design of control feedforward histories to obtain desired outputs, using inverse and optimal control methods. Some advantages and disadvantages of both methods are presented. The slow roll maneuver of an aircraft is used as an example. A nonlinear inverse solution for this example was given previously; for apparently reasonable specified outputs, namely a roll angle history and a straight flight path, the control inputs and the aircraft sideslip are infeasible and the maneuver is not coordinated (large sideforce). Using optimal control, the mean square control inputs are minimized with only the final roll angle specified. The result is a well-coordinated maneuver with small sideslip, reasonable control amplitudes, and small deviations from the straight flight path.

Optimal Control of Systems with Hard Control Bounds

The numerical solution of the optimal control and trajectory of systems with hard control bounds is considered. A new algorithm, which extends the gradient method [1] to systems with hard control bounds is presented. The algorithm is based on the use of an Adjustable Control-variation Weight (ACW) matrix to enforce the bounds. The application of the algorithm to the minimum-time control of a two-link robot arm is presented as an example.

Methods for Computing Minimum-Time Paths in Strong Winds

This paper describes methods for computing minimum-time flight paths at high altitudes in the presence of strong horizontal wind s. The first method shows how to calculate nonlinear feedback (“dynamic programming”) solutions for minimum-time flight paths. To do this the Zermelo Problem for arbitrary winds is extended from a flat earth model to a spherical earth model as a two-state problem (latit ude and longitude) with one control (heading angle). Many minimum-time paths are calculated backwards from New York JFK Airport and San Francisco International Airport to points in the continental U.S. Then the optimal heading angle and the minimum-time can be found by interpolation given the current latitude and longitude using an interpolant created by Delaunay triangulation. The second method is based on an analytical neighboring optimal control solution that computes neighboring optimal heading commands as a function of the winds along a nominal flight path. As an example, minimum-time paths to New York and San Francisco are determined for the actual jet stream winds on 14 February 2001 at 36 kft.