Arthur Bryson

• Doctor of Philosophy
• Professor Emeritus at Stanford University

This paper describes methods for computing minimum-time flight paths at high altitudes in the presence of strong horizontal winds. 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 sph...

Theoretical stability derivatives are determined for a slender missile through use of the concept of inertia coefficients of its cross section. The inertia coefficients are determined by the velocity potentials for two-dimensional motion of the cross section in two mutually perpendicular directions and for rotation about an axis perpendicular to th...

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 far an airplane traveling horizontally betw...

This paper gives an applied view of linear-quadratic worst-case control and relates it to linear-quadratic-Gaussian smoothing. It extends the worst-case control problem formulation to include tracking of desired output histories and nonzero terminal constraints and separates the problem into future,past, and present problems, each of which must sat...

Despite the attractions of linear-quadratic-Gaussian (LQG) controllers, they have had limited acceptance in practice due to lack of knowledge about their robustness to uncertainties in the plant model. A suitable measure of robustness, like the gain margin or the phase margin of classical controllers, was lacking. A new robustness measure, based on...

A method for analytic synthesis of periodic observer gains for linear periodic systems is presented. The eigenvalues of the transition matrix after one period (the Floquet poles) may be assigned; thus the method is similar to synthesis of time-invariant observers by pole placement. The closed-loop transition matrix is al so obtained analytically. T...

Recent research in control theory has centered on worst case design, i. e. determining controller parameters that minimize some cost measure for the ‘worst’ inputs. If the worst inputs are defined as those that maximize the cost, the designer is faced with a minimaz design problem, since the worst inputs depend on the controller design and the best...

The linear-quadratic best controller for the worst bounded disturbances (LQW controller), or so-called H ∞ controller, is derived as a differential game between the controller and disturber using the calculus of variations. This derivation explicitly shows that for the full-order LQW controller the worst measurement disturbances is zero (!) and the...

A parameter identification algorithm for linear systems is presented. It is based on smoothing test data with successively improved sets of system model parameters. The smoothing pass through the data provides all of the information needed to compute the gradients of the smoothing performance measure with respect to the parameters. The parameters a...

A controller design method is presented that gives the best linear-quadratic-Gaussian closed-loop performance over a set of worst plant parameter changes. The design algorithm combines a multiplant optimal design code, SANDY, with a new worst parameter algorithm that uses a quadratic norm on parameter changes. The minimax algorithm is unique in the...

An approximate time-optimal of a solar sail from a geosynchronous orbit to the sun-earth L2 libration point is found using a combined method of local optimization and single shooting. The local optimization strategy is based on maximizing the time rate of change of an energy variable at each time. This strategy overcomes the numerical difficulties...

Minimax methods are proposed for the analysis and design of controllers for the best controls with the worst initial conditions, worst parameter changes with specified quadratic norms, and worst disturbances with specified integral-square norms. The worst initial conditions are the only forcing functions; disturbances are regarded as an added set o...

We present a nonlinear feedback law for safe aircraft penetration through downbursts on landing approach. It is based on following a nominal approach path relative to the ground, subject to a minimum airspeed constraint. The guidance design uses a point-mass aircraft model and considers the flight path to be in the vertical plane. We modify the sla...

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 reason...

A controller design method is presented that gives the best state feedback gains for the worst disturbance functions with integral-square norms, and the worst parameter changes and initial conditions with quadratic norms. The method is applied to a benchmark problem of controlling a system with a rigid-body mode and a vibration mode.

Dynamic optimization and feedback control system design techniques are used to determine proper guidance laws for aircraft in the presence of downbursts, and insensitivity to downburst structures is emphasized. Avoidance is the best policy. If an inadvertent encounter occurs when the aircraft is already close to or even in the downburst, the pilot...

Equality constraints represent a general class of path constraints in trajectory optimization, since many types of constraints can be converted into path equality constraints. Miele has developed an algorithm named Sequential Gradient Restoration Algorithm (SGRA), which can solve general trajectory optimization problems. Goh and Teo proposed a unif...

This paper describes a reliable, general algorithm for modal decomposition in real arithmetic and its use in analyzing and synthesizing control logic for linear dynamic systems. The numerical difficulties are described associated with computing the Jordan canonical form when the system has repeated, or nearly repeated, eigenvalues. A new algorithm...

Guidance schemes are designed to approximate the optimal survival and optimal performance paths through downbursts, which were determined in the previous paper. Specifically, climb-rate command following is used to achieve performance, and altitude command following is used to enhance survivability. Nonlinear simulations are conducted to investigat...

This paper is concerned with the modeling of a flexible toroidal structure for attitude and structural control studies. A two-dimensional elastic system, consisting of a flexible toroid and a pretensioned membrane, is analyzed in terms of the transfer function from a control torque input to the collocated sensor output. The critical tension for out...

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...

A class of optimization problems of interest in the field of robotics is one that seeks to minimize the time required to force a manipulator to travel a specified distance. Robots employ multiple, bounded control inputs. This work describes a new, very fast algorithm for determining minimum-time trajectories for such systems. We have modified the s...

Aircraft takeoff and landing in the presence of downbursts are addressed. Dynamic optimization and feedback control system design techniques are used to determine proper guidance laws for aircraft in the presence of downbursts, and insensitivity to downburst structures is emphasized. Avoidance is the best policy. If an inadvertent encounter occurs...

Quasi-steady analysis and energy-state approximation are employed to study flight paths through windshears and downdrafts. Takeoff flight is mainly considerd while landing flight is briefly discussed. The relation between angle of attack and airspeed is found to be almost independent of horizontal wind and slightly dependent on vertical wind. In sl...

Three dimensional incompressible vortex rings are used to model downburst flow field, and modifications are made to express the induced velocities analytically. As a result, this three dimensional downburst model can be used in trajectory optimization, as well as in flight simulations. Two such vortex rings are superposed to match the JAWS AB Corri...

The control of an aircraft's takeoff path through a downburst is presently formulated as a dynamic optimization problem with minimum-altitude constraint and two different performance measures; a landing path through a downburst is also discussed. Paths are determined which, in addition to maximizing an airspeed/altitude combination immediately afte...

A gradient algorithm is developed that determines optimal trajectories with path equality constraints and terminal constraints. A generalized gradient is formed which improves both the performance index and the path equality constraints simultaneously. The algorithm is extended to treat terminal constraints by using Bryson's impulse response techni...

Generic models of flexible space structures are investigated from the infinite discrete-spectrum viewpoint of distributed parameter systems. The models are simple enough to treat analytically, yet complicated enough to demonstrate the practical usefulness of the transcendental-transfer-function modeling for the purposes of preliminary contro synthe...

Transfer-function realizations (TFR's) of multi-input, multi-output (MIMO) linear dynamic systems can be converted into minimum modal state variable realizations (MSR's) by the use of singular-value decompositions (SVD's) of the residue matrices for each pole. Each residue matrix is decomposed into the sum of outer products of vectors that form col...

A method is described for designing time-invariant controllers to minimize root-mean-square target-tracking errors with root-mean-square control constraints in the presence of random initial conditions, target motions, and sensor errors. The maximum control responses usually occur during the initial transient period before the statistical steady st...

Wind and wind shear components are estimated and used for controlling an externally-blown flap STOL aircraft and a B-727 during landing approach under the JAWS microburst profile. A combination of feedforward and feedback concepts is used. The objective is tracking of airspeed and glide slope. Both an asymptotic disturbance rejection scheme, which...

Feedback logic is given that produces satisfactory penetration of an aircraft through a severe downburst. The logic has two parts: (1) A switch that calls for full throttle and tight climb-rate-hold feedback logic when the magnitude of headwind-rate exceeds a threshold level; this switch stays on until re-set by the pilot after penetration of the d...

Wind and wind shear components are estimated and used for controlling an externally-blown flap STOL aircraft and a B-727 during landing approach under the JAWS microburst profile. A combination of feedforward and feedback concepts is used. The objective is tracking of airspeed and glide slope. Both an asymptotic disturbance rejection scheme, which...

This Note discusses two examples that have been used to demonstrate the superiority of multivariable control analysis/synthesis techniques based on singular values over classical techniques. Classical stability analysis, by breaking loops one at a time, is known to be an unreliable way of testing robustness to simultaneous perturbations of all the...

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 th...

Minimum time paths for a two-link robot arm are determined so that the tip moves a specified distance, starting from rest and ending at rest. Two torquers are used, one at the shoulder and one at the elbow, each having bounded torque. The optimal locations of the two end points in the robot coordinate system are determined as part of the solution....

Minimum time paths for a two-link robot arm are determined so that the tip moves a specified distance, starting from rest and ending at rest. Two torquers are used, one at the shoulder and one at the elbow, each having bounded torque. The optimal locations of the two end points in the robot coordinate system are determined as part of the solution....

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, an...

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 approach to the synthesis of control logic that is both insensitive to system parameters and at tenuates response to input disturbances (sometimes called robust control logic) is presented and is applied to a simplesystem with an uncertain vibration frequency. A conclusion drawn from this study is that the lower the order ofthe compensator dynam...

Large structures are being considered for future missions in space. They will make efficient use of material and hence are likely to be quite flexible. Flexible structures are distributed parameter systems with infinite-dimensional discrete spectra. They are often approximated by using a finite number of modes. When this is done, using finite eleme...

The effects of actuator and sensor locations on transfer function zeros are investigated, using uniform bars and beams as generic models of flexible space structures. It is shown how finite element codes may be used directly to calculate transfer function zeros. The impulse response predicted by finite-dimensional models is compared with the exact...

Not Available

A procedure is presented for synthesizing time-invariant control logic to cause the outputs of a linear plant to track the outputs of an unforced (or randomly forced) linear dynamic system. The control logic uses feedforward of the reference system state variables and feedback of the plant state variables. The feedforward gains are obtained from th...

For the purposes of aircraft control system design and analysis, the wind can be characterized by a mean component which varies with height and by turbulent components which are described by the von Karman correlation model. The aircraft aero-dynamic forces and moments depend linearly on uniform and gradient gust components obtained by averaging ov...

Some three-dimensional minimum-time paths to a specified final line (or point), heading, and energy are presented for an example supersonic aircraft. The optimum maneuvers have been determined using the calculus of variations and the energy-state approximation. These are compared with suboptimal solutions obtained with the additional assumption tha...

A study of three-dimensional, minimum-time turning maneuvers for supersonic fighter aircraft is described. For turns which do not specify the end-point position, the optimum maneuvers comprise a one-parameter family of flight paths. All the optimum turns, and their associated control variables can be presented in simple graphical form for a specifi...

A set of sufficient conditions for a weak minimum is derived for the nonsingular Bolza problem of variational calculus, expressed in control notation. The initial state and time are assumed fixed; the terminal state and time are variable, subject to a set of equality constraints. Specified terminal time is considered as a special case. The sufficie...

The report deals with two subject areas; (a) Navigational System Optimal Integration, and (b) Stabilization for Precision Pointing and Tracking. Under the first subject area a theoretical investigation was conducted on the possibility of using all of the navigational aids (DME, VOR, TACAN, inertial systems, Air data systems, etc) aboard aircraft to...

Some three-dimensional, minimum-time paths to a specified terminal line (or point), heading and energy are presented for an example supersonic aircraft. These paths were calculated using numerical solution techniques implemented in a digital computer program for the CDC-6600 computer. The optimum maneuvers have been determined using the calculus of...

Automatic control is used in connection with goal-oriented, man-made systems. There are several, usually overlapping, reasons for using automatic control: (a) to relieve human operators from tedious, repetitive tasks (e.g., dial telephones, traffic signals, automatic elevators, autopilots), (b) to speed up and/or lower costs of production processes...

Optimal Control theory is used to determine the thrust and bank angle programs for minimum time turns of a supersonic aircraft to a specified point or line when constant altitude is maintained. A stall constraint, a maximum velocity placard limit, a maximum normal load constraint, and maximum and minimum thrust limits are imposed. Numerical results...

The initial phase of a study of techniques for real-time, on-line optimum flight path control is described. A review of the historical development of aircraft performance optimization during the past three decades is presented. This review indicates that climbing maneuvers in the vertical plane (zero bank angle) have received considerable study, wh...

A simplified mathematical model of a VTOL aircraft is developed that treats ground rangeto- go as the independent variable, altitude and crossrange as state variables, and the three commanded velocity components as control variables. By considering perturbations about a nominal path and minimizing a performance index which is a quadratic function o...

Optimal control theory is used to determine thrust, hank-angle, and angle- of-attack programs for minimum time turns of a supersonic airplane at constant altitude for three different terminal conditions: 1) both heading angle and velocity specified, 2) only heading angle specified, 3) only velocity specified. The angle-of-attack is constrained to b...

The energy-state approximation, properly interpreted, is quite adequate for performance optimization of supersonic aircraft. In addition to its use for determining flight profiles for minimum time and minimum fuel to climb to a given altitude and speed, this paper shows that the energy-state approximation may also be used to determine flight profil...

Conjugate gradient methods have recently been applied to some simple optimization problems and have been shown to converge faster than the methods of steepest descent. The present paper considers application of these methods to more complicated problems involving terminal constraints. As an example, minimum time paths for the climb phase of a V/STO...

Maximum trajectory for glider entering earth atmosphere at supercircular velocity, subject to maximum altitude constraint

A new set of necessary conditions are presented for solutions to optimal programming problems with a state variable inequality constraint (SVIC). On a constrained arc, the dimension of the state space is reduced, and the influence functions associated with this reduced state space are shown to be unique and continuous across junctures with unconstr...