Acta Astronautica, Vol. 2, pp. 897-899. Pergamon Press 1975. Printed in the U.S.A.

Communication

An analytical stochastic controller

V I C T O R L A R S O N t Jet Propulsion Laboratory, California Institute of Technology,

Pasadena, CA 91103, U.S.A.

(Received 3 December 1974; revised 26 June 1975; communicated by Professor George Leitmann,

U.S.A., Corresponding Member, Section 2)

Abstract--This paper addresses the problem of determining an analytical stochastic controller for achieving docking between two vehicles. With the use of simplifying assumptions, analytical RMS docking errors are determined.

The analytical approach presented is considered to be a powerful preliminary design tool in assessing the effects of sensor errors and plant disturbances on docking errors.

Introduction

THIS PAPER provides an analytical solution to a stochastic control problem in which the plant can be adequately modeled as a double integrator. The solution is applied to the problem of determining prel iminary est imates of the RMS docking errors associated with an Apollo-like probe and drogue (Larson, 1971a).

The successful use of optimal stochastic control theory to solve practical problems is becoming increasingly more widespread (Athans, 1971). Larson (1971-1973) provides examples of how the theory can be put into practice to solve practical problems. This paper illustrates the power of the theory very effectively in showing that, in some cases, even analytical results can be obtained.

The main advantages of the analytical approach used herein are (1) the results can be obtained expedit iously and inexpensively, (2) the results can shed light on available design tradeoffs.

For any given problem, however , it is r ecommended that the prel iminary results obtained f rom the analytical solution provided herein be validated by simulating more exact models of the plant and measurement process.

Analytical stochastic controller

The approach taken to arrive at the analytical stochastic controller involves (1) determining the plant and measurement matrices F, G, H, (2) selecting the weighting matrices A, B, $I, (3) determining the variances of the plant and sensor noise, (4) solving algebraically the known equations for the solution to the linear stochastic optimal control problem.

Simpl i f ied mode l s A double integral plant is f requent ly adequate in a prel iminary analysis and

was selected in this s tudy for investigating the docking errors. That is, for the

tMember Technical Staff; Member AIAA.

897

898 V. Larson

position loop, the errors in position and velocity are modeled as

2, = x2;.i , = u + w (I)

where x, is the position error, x2 is the velocity error, u is the control acceleration, and w is a white noise process used to represent disturbance accelerations. Note that for the position channel, x, can refer either to a lateral miss between the probe and drogue, a relative range error, or an attitude misalignment.

The measurement process is modeled as z = x, + v where z is the sensor measurement , x~ is the position error, and v is a white noise process used to represent the sensor uncertainty.

Determinat ion o f Q

The nominal est imate of the RMS translational disturbance accelerat ion is determined in this analysis by considering 10% random variations in the thrust level of an attitude control jet located on the mother vehicle and by considering a 1 deg uncertainty in the direction of the thrust. The resulting values for the one-sigma uncertainties in acceleration are 0.01 ft /sec 2 and 5 x 10-' (rad/sec2), respect ively for translation and rotation. The values of Q were computed f rom the product of the variance of the acceleration uncertainty and the average time be tween thrust reversals. For this work, the nominal values of Q are 1 x 10 4 (ft2/sec 3) and 2.5 x 10-' (rad2/sec3), respect ively for translation and rotation.

Determinat ion o f R

For the measuring processes being considered in this analysis (laser, TV), the measurement process is discrete. Hence , the statistics of the continuous measurement noise R are obtained f rom those for the discrete system R ' according to R = R ' x D T = ~2 x D T where D T is the integration step size and 132 is the variance of v. In general, when a discrete process is modeled as white noise, D T

is the time be tween sensor samples or a measure of the sensor error correlation time. When using a continuous numerical integration scheme to integrate continuous plant equations and discrete measurement equations, D T is the integration stepsize.

Steady-state analytical solution to control problem In this section, the steady state analytical solution to the optimal stochastic

control problem is given. The analytical solution is obtained by: (l) determining the steady-state value of a matrix S and consequent ly the

s teady-state values of the control gains; (2) determining the s teady-state value of the covar iance of the est imator error: (3) determining the steady-state value of the covar iance of the est imate of x; (4) determining the covar iance of x: (5) determining the covar iance of u.

For the problem

~ t 21=x2; .'~z = u + w : J = E ( a , t x ~ 2 + a = x f + b u 2 ) dt (2) o

An analytical stochastic controller 899

the steady-state control and filter gains are

The steady-state variances of the state and control variables are given by

X , 1 = X 1 . + P , , ; X22=P~22+P22; E u 2 = C , 2 f ( , , + 2 C , C 2 f f , 2 + C 2 2 f G 2 (4)

where

P . = X / 2 . R ; P ~ 2 = X / Q R ;

R~, = V Q R

= X/ R ; =

+ a : 2 + _ /_ ( Q \ ,/4 ] Q 1/2

C,C2

Sleady-state RMS results Results were obtained for various values of the Q 's , R ' s , and C's . In this

analysis the nominal values of the parameter ~-ff-' are 0.1 in and 0.1 deg for the laser and 10 in and 10 deg for the TV. The resulting RMS docking errors were 0.01 ft for the lateral miss and 0.3 deg for the angular offset for the laser. For the TV, the RMS lateral miss was 0.1 ft and the angular offset was 2 deg. Hence, for the application at hand in which the tolerable docking errors are ---5 deg angular misalignment and a lateral offset of 1.7 ft, either the laser or TV would be a satisfactory sensor.

References Athans, M. (1971) Editor, Special Issue on Linear-Quadratic Gaussian Problem, IEEE Transactions

on Automatic Control AC-16 (6). Bryson, A. E. and Yu-Chi Ho (1969) Applied Optimal Control, Blaisdell, Waltham, Mass. Larson, V. (1971a) Docking Dynamics Associated with the OOS-C and a Malfunctioned Satellite,

Aerospace Report No. TOR-0059(6531-04)-2. Larson, V. (197 lb) An Estimation Algorithm for Determining the Relative Motion and Relative Attitude

Between the OOS and a Malfunctioned Satellite, Aerospace Report No. TOR-0172(2770-1)-4. Larson, V. (1971c) An Optimal Stochastic Controller for Accurate Position Control (Personal

Transportation Study), Aerospace Corporation Report ATR-72(8124)-I. Larson, V. (1972) A Stochastic Optimal Controller for Achieving Docking of the OOS and a

Malfunctioned Satellite, Aerospace Report TOR-0073(3421-02)-I. Larson, V. (1973) A Suboptimal Stochastic Controller for an N-Body Spacecraft, Paper presented at

the AAS/AIAA Astrodynamics Conference, Vail, Colorado, July 16-18, 1973. Williamson, R. K. (1971) Laser Radar Stationkeeping and Attitude Estimation Algorithm for a

Cooperative Gyrostat Spacecraft, Aerospace Report No. TOR-0172(2770-01)-2,