Report ESL-R -339 SAMPLED-DATA CONTROL OF HIGH ... - Core

Report ESL-R -339

SAMPLED-DATA CONTROL OF HIGH-SPEED TRAINS by Alexander H. Levis and Michael Athans

This study was conducted at the Electronic Systems Laboratory with support extended by the U. S. Department of Transportation under Contract C-85-65 (Project Transport, DSR Project No. 76105), Clearinghouse No. 177 669. This report represents tax supported research. It is not copyrighted and permission is hereby granted to reproduce all or part with credit to the source.

Electronic Systems Laboratory Department of Electrical Engineering Massachusetts Institute of Technology Cambridge, Massachusetts 02139 January,

1968

ABSTRACT This report deals with the control of the positions and velocities of high-speed vehicles in a single guideway. It is assumed that each and every train measures its position and velocity every T seconds. The appropriate accelerations or decelerations to be applied to each vehicle are constrained to be constant during the sampling interval. Through the use of a control cost functional, which penalizes the system for any deviations from the desired headway and velocity, the required control accelerations and decelerations are obtained by deriving the system equations in discrete-time and, through the use of available results in the theory of discrete optimal control, the optimal linear time-invariant sampled-data feedback control system is determined. The general results, as well as the general purpose digital computer programs, are presented and are used to study the effect of changing the sampling time T upon the control-system performance. Since, in general, the cost of the communication system (in terms of required channel capacity, bandwidth, etc.) decreases with increasing values of the sampling time, the system designer has the capability of conducting trade-off studies involving the deterioration of the control system performance vs. the decrease in the cost of communication as the sampling time is increased.

ACKNOWLEDGEMENT The authors gratefully acknowledge the helpful comments of Messrs. Roberto D. Canales, Keith M. Joseph and William S. Levine. Steven R. Powell's contribution to the computer algorithm is greatly appreciated. The reports and comments of Dr. David L. Kleinman were most helpful toward a better understanding of the regulator problem. The computations were performed at the M. I. T. ing Services Center.

iv 0o

Information Process-

CONTENTS I.

INTRODUCTION

II.

DEFINITION OF THE OPTIMIZATION PROBLEM

3

III.

THE EQUIVALENT DISCItETE-TIME OPTIMIZATION PROBLEM

8

pal

1

IV.

SOLUTION OF THE OPTIMIZATION PROBLEM

11

V.

EXAMPLE:

14

VI.

DISCUSSION OF RESULTS

21

VII.

CONCLUSIONS

26

VIII.

REFERENCES

27

APPENDIX:

28

A STRING OF FIVE TRAINS

THE COMPUTER PROGRAM AND ITS USE

To~~~~~~~~~~~~~~~~~~~

.~~~~~

LIST OF FIGURES 1.

Typical Piecewise Constant Element of the Control Vector

page

5

The Structure of the Optimal Sampled-Data Feedback Control System

13

3.

Initial Conditions of Five Vehicle System

16

4.

The Normalized Optimal Cost' as a Function of the Sampling Period. Cases I A, IB

19

The Normalized Optimal Cost as a Function of the Cases IIA , IIB Sampling Period.

19

Trade-off Curves for Normalized Cost vs. Sampling Interval

23

Flow Chart of MAIN Program

32

2.

5. 6. 7.

vi

I.

INTRODUCTION

This study was motivated by the continuing interest in developing simple but efficient methods for controlling the spacing, velocity and acceleration of each individual vehicle in a tightly packed string of high-speed trains.

In work reported previouslyl1

the equations of

motion of the vehicles were linearized about a set of operating conditions and the optimal control system was determined.

The cri-

terion with respect to which the performance of the trains was optimized, was a quadratic functional that penalized the system for deviations from the desired average velocity, for deviations from the specified separation distance between adjacent trains and, finally, for the use of large corrective forces.

The optimal design was found

to be a linear, time -invariant, state feedback control.

Simulation on

an analog computer showed that the design was quite satisfactory. Nevertheless, actual implementation in large systems became problematic because of the large number of feedback loops required.

This

realization motivated research in two directions--both having the same objective:

the reduction of the size and complexity of the required

communication system, or, alternatively, the reduction of the amount of information that had to be transmitted. The first approach was based on the practical realization,

sub-

stantiated by theoretical results, that a vehicle's behavior is affected mostly by the behavior of adjacent vehicles.

This permitted the de-

velopment of suboptimal control designs that required a drastically reduced number of feedback loops for their implementation.

Further-

more, results from simulation of typical systems on the analog computer compared well with respect to the optimal system. 1 The alternative approach, i.e., the reduction of the amount of information to be transmitted, motivated the introduction of sampleddata feedback.

In this case, the position and velocity of each vehicle

Superscripts refer to numbered items in the References. -1-

in the string are measured once every

T seconds, (sampling). .On

the basis of this information, corrective forces are applied that remain constant until the next sampling of the data. The basic model and the design philosophy used in this report are similar to those used in the continuous feedback case.

In addition,

the theoretical development is parallel, but in the sampled-data .;

case the discrete minimum principle is used.

2

The optimal control was found to be a linear one, i.e., over

:|

a sampling interval each corrective force could be expressed as a weighted linear combination of the values of the position and velocity deviations measured at the beginning of the interval.

Evaluation of

the weighting coefficients required the solution of a general discretetype matrix Riccati equation.

This was implemented successfully on

a high-speed digital computer. In Section II of this report the model of the system is described and the optimization problem is formulated.

In Section III, the

sampled-data model is transformed to a discrete-time one and an equivalent discrete -time optimization p r o bl e m

is

s t a t e d.

Section IV, the analytical solution to the problem is presented.

In As an

illustrative example, numerical results for a string of five trains obtained from the digital computer solution of the problem are presented in Section V.

Se ction VI,

In

the cost of communication

systems necessary for the implementation of the optimal control is Finally, in the Appendix, the digital computer program is

discussed.

listed along with instructions for its use. Although the research reported here is centered on the control of vehicles moving in a single guideway, the results are also applicable to the sampled-data control of high-speed trains for the merging problem.

The reason is that the merging problem can be transformed

into a problem involving the control of vehicles in a single guideway. This approach is explained in more detail in the report by Athans and Levine

and it is not discussed here.

For the purpose of this report,

it suffices to state that the same type feedback system presented here can be used to control the vehicles in the merging problem. .En

suffices

to

state

that

the

same

type

feedback

system

presented

here

II.

DEFINITION OF THE OPTIMIZATION PROBLEM

The investigation is restricted,

as in the past, to the properties

of a string of vehicles moving along a single guideway at high speed. The problem of injection (i.e.,

merging) of vehicles from the guide-

way is treated in Ref.

n

3.

The

vehicles that comprise the string

are taken to be identical and are moving with an average,

"cruising"

v . This implies that the starting and stopping operations 0 near stations are excluded fromi the model. velocity

Since the description of the physical model and the derivation of the governing mathematical relations are presented in detail in Ref.

1, only a brief exposition of the salient features of the resulting

system of equations will be given here. Each vehicle is.modeled as a second-order dynamical system with nonlinear damping.

If,

instead of actual vehicle position, the

deviation from a desired separation distance between adjacent trains is

considered as the position state variable, then the equations of Furthermore, if instead of actual

adjacent vehicles become coupled.

vehicle velocity, the deviation from the prescribed mean velocity v

0

then the equations of

is considered as the velocity state variable,

motion can be linearized. The linearized string equations are d dt 6wk(t) d m dt6 Yk(t) where

6ok(t)

=

6Yk(t) - 6Yk+l(t)

= -a

6

yk(t) + 5fk(t)

of the k-th vehicle,and

6f k

plied to the k-th vehicle.

lations.

6

yk

is the velocity deviation

is the incremental corrective force apSince the vehicles were assumed identical,

is the mass of each vehicle and

force.

(2.2)

is the deviation from the desired separation distance

between the k-th and the (k+1st vehicles,

m

(2.1)

ao6yk(t)

is the linearized drag

The separation distance does not appear explicitly in the reIt should be noted that Eq. 2. 1 is valid for

while Eq. 2.2 is valid for k=l,,..,n.

-3-

k=l,...,

n-l

-4The introduction of the following dimensionless variables simplifies the equations and makes the results applicable to many dynamically similar models. For the k-th vehicle:

mv

dimensionless distance deviation:

Xk(t)

= 6(t)/(

)

(2. 3)

0

dimensionless velocity deviation: dimensionless incremental force:

= 6yk(t)/v

v

k(t)

dimensionless time:

T

=

k(t) (2.4)

fk(t)/aov

(2.5)

= t/(m/ao)

(2.6)

The equations of motion (2.2) and(2. 1) then take the normalized form

d e

dT

@(T)

=

-(T)

+

(2.7)

k(T)

d d

Xk(T)

= /'(T)

- /+1(T)

By interlacing the velocity deviations from the desired position

(2.8)

Ok with the deviations

Xk, the following matrix representation of

the dynamical equations of motion for a string of N vehicles is obtaine d:

-1

0

XI

1

0

-1

X1

0

0

-1

b

0

0

dT

X(T)

1

A

0

.1

1

00

0

0

0

0

o

0

0

1

0

0

0

0

00

x-T)

B

01

0

u (T)

(2.9) Using the indicated vector-matrix notation,Eq. 2.9 can be written compactly as

dX(T) = A AX(T ) +± x-x(-r) Ax(T)

BU(T ) BU(T)

(2.10) (2.10)

-5where: state vector of the system

x

is the (2n-1)

u

is the n control vector of the system

A

is the (2n-1) x (2n-1) 'system matrix

B

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(2. 11)

Tf - T

It is then assumed that the control vector u takes some constant value over each interval. U((T)

=

u(T +kT)

- u

k

;

+kT < T


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