Preview-Based Stable-Inversion for Output Tracking - NTRS - NASA

V

v

Preview-Based

Stable-Inversion Qingze

Zou

for Output

Santosh

*

Devasia

Tracking

t

Abstract Stable

Inversion

However,

techniques

for nonminimum

has to be precomputed for pre-specification to trajectory

can

phase

using

systems,

problems

output

to achieve

the

a pre-specified

of the desired

planning

be used

inverse

high-accuracy is noncausal

desired-output restricts

(for non_minimum

output - hence

trajectory.

This

the use of inversion-based phase

systems).

tracking. the

inverse

requirement approaches

In the present

article,

%.¢

it is shown

that

inversion-based quantified %F

of the structure

preview output

in terms

system).

The

information tracking

of linear

of the tracking

error

methodology

and experimental

*Graduate Student,

of the desired

Mechanical

results

systems. and

is applied

output

can be used

The amount

the internal

of preview-time

dynamics

to the online

to achieve

output

online

needed

of the system tracking

(zeros

of a flexible

are presented.

Engineering

Department,

University

of Utah, SLC, UT, 84112.

Email: [email protected] tAssistant

Professor,

Mechanical

Email: santosh_eng.utah.edu

V

Engineering

Department,

University

is

of Utah, SLC, UT, 84112.

i W

1

Introduction

Inversion

of system

dynamics

can

be used

[10, 22]. For systems

with

sion

to be unbounded

techniques

tracking. phase

tend

Recently, systems,

stable

which

inversion-based

inversion

yield

output

manipulators

nonminimum

[20, 16], aircraft

phase and

therefore,

inputs

has

been

systems

inputs

which

dynamics,

approaches

bounded

tracking

to find

inputs cannot

studied

found

been

[18, 24, 4], and

standard

for practical

developed

g

inveroutput-

for nonminimum

The

nonminimum

high precision

output-tracking

through

output-tracking.

for several

exact

to be used

[5, 11, 1] have for exact

achieve

application systems,

positioning

of such

like flexible

of piezo-probes O

for nano-technology

[2]. A critical

inverse

is noncausal

(for non-minimum

jectory

must

can

be pre-specified.

be a substantial

limits

the

of this

paper

also

quantifies

and

the

proach with

approach, the

location

A major linear

linear

systems Byrnes

[13].

with

The

the regulator

the

the

needed

desired

of the

applications.

process

can

to be applied

in terms

Implementation

approach,

The

inverse

of the

desired

issues

are

it to the

output

tracking

solution

of the

output

output

the tra-

output-trajectory

inversion-based

noncausal

the inversion

by applying

tracking

[9]. These

desired

is easily

non-causal planning

systems,

system.

therefore

is that

main

and

it

contribution

be computed online.

using

This

tracking

discussed

article

outputs

designed approach

is the

and

of a flexible

results are

were assumed

generalized

for the

to be generated

by solving

a manageable

is that

the exosystem

regulation nonlinear

the

apg

structure

set of linear states

problem case

by an exosystem equations.

are often

switched

w

accuracy

dynamics.

in output

by Francis

regulator

however,

verified

phase

result

of the

and

approaches

for pre-specification

to trajectory

allows

inversion-based

systems)

of the

of preview-time

zeros

is experimentally

use

for linear

which

amount

of the

nonminimum

and

that,

these

requirement the

approach

is to show

a preview-based

on

with

phase

This

limitation

inversion-based

difficulty

for

by Isidori and

the

A problem, to describe

the

desired

output;

switching-caused v

[5, 7].

tracking

of a particular

Thus,

dynamics

trajectory

finding

bounded

of time

Stable

This

motivates

using

preview

based

output

Other

track

(but

systems

inverse

the

implies

the

work,

information tracking

and

the

have

which

used

output inversion

exactly

tracks

the standard

this

entire

to output when

based

approach,

a single

specified

approaches

shows

that

enables

the

preview

the

phase

by

[5, 11].

to be known

(for nonminimum

online

output

inverse-inputs

needs

noncausal

the

to unbounded

trajectories

trajectory

exact-

Inversion

lead

of unbounded

input-state

Such

tracking

approaches

problem

output

for nonminimum

also

instants.

approaches

In the

exact-tracking

thereby

technique

approaches

resolve

that

switching

as in the case of the output-regulator).

use of inversion-based

current

that

since

noncausal)

the

inversion-based

to find the input

techniques

after

inversion-based

is required.

is challenging

possibly

restricts

to use

trajectory

errors

by using

a class of outputs

inversion

of the

which

can be avoided

output

phase

[10].

errors

tracking

it is advantageous

than

for nonminimum inputs

to transient

[10, 22] is inverted

(rather

noncausality

leads

transient

tracking

system V

this

phase

inverse

can

implementation

The

ahead

systems).

be found

of the

by

inversion-

systems.

information

of the

desired

output

trajectory

for

v

output by

tracking,

exploiting

is in linear

actuator

time

increases.

v

v

of the of the

other

framework

(see

works,

inputs.

desired

to achieve

(e.g.,

goals [8]).

like

goal

The

resulting

shown

tracking

that

is to trade-off

Trading

based

controller

controllers

controller tracking

approaches accuracy

is also

possible

is noncausal,

phase

information

the performance

the

off the

minimization inverse

to nonminimum

use of previewed output

inversion

trajectory. vibration

due

of the infinite-preview

In contrast,

output

Another

[23]) have

the

problems

optimal

the performance

In these

the

[25].

(lq-based)

coworkers

approach

magnitude control

and

to alleviate

redundancy

quadratic-based

4). Tomizuka controllers

for example,

dynamics

of the

(see

output

[17], Chapter

of finite-time-preview as the amount

requirement aim to achieve output-tracking within which

the can

of preview

to reduce high

the

accuracy

requirement inversion-based also

be imple-

I mented The and 2.

using

the

preview-based

paper

is organized

its dependence The

problem

mentation

issues

are

is applied

to the

output

presented.

Discussions

2

Stable

solution is then

studied

Section,

the

solving

the inversion

solved

tracking are

format.

of the

3.

The

preview

our

problem

output

conclusions

is equivalent

are

approach

to finding

of the

bounded

and

and

inversion

results

in Section

6.

Phase

Systems

is presented.

imple -

approach

experimental

solutions

scheme in Section

output,

preview-based

structure

tracking

is presented

information 4, the

tracking

output

dynamics

for Nonminimum

inversion-based

article.

inversion-based

of a flexible

5 and

current

internal

In section

control

in section

in the

system's using

in Section

Inversion

In this

discussed

in the following

on the

inversion

controller

It is shown

to the systems's

are

that

internal

dynamics.

2.1

Output

tracking

using

inversion

of system

dynamics Q

Consider

which that

a linear

has the

the

system

same

system

described

number

by

it(t)

=

A z(t)

y(t)

=

C x(t)

of inputs

is stabilizable.

Let

as outputs,

Yd(')

be the

+ B u(t) (1)

u(t),y(t) desired

C _P,

output

and

trajectory

x(t)

E _n

U

We assume

to be tracked.

Then W

in the that

inversion-based satisfies

the

approach

system

equations

we, first,

find a nominal

(1) and

2_e:(t)

=

A xr_:(t)

yd(t)

=

C x_l(t

yields

the

+ B u::(t)

input-state

desired

_

trajectory,

output

exactly,

Vt E

xre:(')]

i.e.,

(2)

J

)

[u:/(-),

3

1!

and,

second,

back

(see Figure

achieved.

v

1).

While

standard

inverse

phase

trajectory

Thus

x(t)

of the

input-state This

preview

yielding

---* xrel(t)

[14] like state

dynamics. using

the exact-output

stabilization

techniques

to find the mum

we stabilize

and

trajectory paper

y(t)

reference

feedback

information

of the

of the

the

trajectory,

---, yd(t)

state

lull(-),

addresses

state

trajectory form

K[x(t)

can

on-line

computation

main

for systems

is

through

challenge

with

inverse

feed-

tracking

achieved

the

of the

state

output

be easily

- xrel(t)],

- especially

trajectory,

by using

as t --_ oc and

xr_:(.)]

desired

xr_/(.),

is

nonmini-

input-state

Ya.

Feed forward

u(t)

Input uff(t)

y(t)

Plant Previewed

On line

Desired

Inversion

Trajectory

of Plant

Reference

Dynamics

State Xref(t)

Trajectory Tracker

%J

Figure

1. The

output

Actual

tracking

control

State x(t)

scheme.

%,2

2.2

Stable

In this

subsection,

finding

bounded

a well given

defined

inversion

scheme

it is shown solutions

vector

relative

that

to the

finding

system's

degree,

the

inverse

internal

r := [rl,

r_,

input-state

dynamics. ...,

rp].

Let Then

trajectory the the

linear output's

is equivalent system

to

(1) have

derivatives

are

as:

V

d"_Y---_k= CkA"_x dt_k

+ CkA"k-lBu

V

4

V

(3)

i

g

where

Ck is the

k th row

of C, and

1 < k < p.

In vector

notation

let equation

(3) be rewritten

as

y(r)(t)

=

Axx(t)

+ B_u(t)

(4)

where y(r)

:=

[ drlyl, dt_l

I

ddt_2 _2y2

C2A _2 x

B_ is invertible

tivates

the

choice

because

By

of the

of the

control

for all t E (-c_, is maintained,

c_).

law of the

Substituting

J

] T

B

:-_-

CpA _p- I B

well-defined

uil(t

drPyp dtrp

C2Ar_-IB

:=

CpA_p and

Q

CI A_I-I

C1 Ar_

n

I

relative

degree

assumption.

Equation

(4) mo-

form

) = B_ 1 [yd(_)(t)-

this control

Axx(t)]

law in equation

(5)

(4), it is seen

that

exact

tracking

i.e.,

= To study

the

effect

of this

control

law consider

_(t)

where

_¢(t) consists

_(t):=

of the

output

[Yl,Yl,...,

and

a change

of coordinates

T such

that

=Tx(t)

(6)

its time-derivatives

dn - lyl dt_,_l 'Y2'_]2"'"

_._r2- 1..y2 ' dt_2_ 1 ,...,yp,

. yp,...,

t !

d_- 1yp dtrp-1 _]' "

The

system

equation

(1) can

then

be re-written

in the

new-coordinates

as

_(t) = AI_ + A2n + B,_

(7)

9(t) = A3¢ + A_n + _2_

(8)

where ,4

In the

new

written

as

:=

TAT

coordinates,

-I

the

:=

;

control

u1i(t

law

and

for maintaining

) = B_ -1 [yd(*)(t)-

=

/3 :=

exact

A_d(t)-

tracking

TB

(Equation

(5))

can

be

(9)

A,_7(t)]

where

Note

that

time

derivatives

chosen

the

such

Equations

desired are

that

_(.)

specified.

exact

(7) and

is known This

tracking

when

the

desired

:= A.T

-1.

desired

output

((.)

is maintained,

is defined y(r)(t)

trajectory as _a(').

= y(r)(t)

Ya(') Since

we also

and

the

have

the

control _(t)

output's law

= _u(t),

was and

(8) become

_(t)

= _.(t)

n(t)

=

(10)

A3G(t) + A4_(t) + .B2B_'

[ya(*)(t)-

A¢¢a(t)-

A,_r/(t)].

(11)

:= AT n(t) + B, Y.(t) %2

where

:= v

V

A4-

[32B;1A,

;

B,

:=

[[32B;

1

Az-

132B;'A¢]

;

and

Ya

:=

y_(t) (_(t) 1

(12)

V

=

t

This

is the inverse

If a bounded feedforward

system,

solution, input

r/d('),

can

and in particular, to the internal

be found

through

ufl(t

and

the

reference

state

Equation

dynamics

(11) represents

(11)

equation

(9) as

) = B_ 1 [yd(_)(t)-

A_a(t)-

trajectory

can be found

can be found

the internal then

the

dynamics.

exact-tracking

(13)

A,Tr/,_(t)]

t

as

(14) t Thus the

a bounded output

2.3

solution

tracking

scheme

Bounded

We restrict the zeros

to the internal shown

solutions

the following of the system

dynamics

in Figure

to the

discussion (Equation

(11) is required

to find the inverse

for applying

1.

internal

to systems

t

dynamics

with

hyperbolic

1) lies on the imaginary

internal

dynamics,

axis of the complex

plane

i.e., none

of

(a technique t

to address there

the

exists

stable

nonhyperbolic

a transformation

subsystem

(as)

and

case

can be found

U such an unstable

that

the

in

[4]). If the internal

internal

subsystem

dynamics

(11)

dynamics can

is hyperbolic,

be decoupled

into

a

(a_): B

as(t) = A,¢,(t) + Ssv_(t)

(15)

d_,(t)

(16)

= ._,,o'u(t)

+ /3_Ya(t)

where

_(t) := Bounded

solution

to the

internal

dynamics

= ur/(t) in the

transformed

(17) coordinates

can

then

be found

S

as

7 W

i:l!li

a_(t)

=

t

e2"(t-*)[_Ya(T)dT

(18)

e']_(_-t)[_uya(T)dr

(19)

OO

tru(t ) = j(t

In the

new

coordinates,

the

ulI(t

where

U -1

This

:=

using

(18)

in Fig. the .

°

v

(17).

solution

output

must

which

restricts (19)

(20)i Which

Note

that

to the unstable specified

Based we discuss

preview-information

(including

future

to trajectory

by using

(where

state

Equation

This

and

input

scheme

is needed

Equation

_Ta, by

to compute

(18).

However,

(19),

is the

shown

the

main

desired problem,

it may be acceptable

to solve

that

the

of the are

Inversion online

desired

specified

on-line.

computation

output

and

tracking

and

trajectory,

We begin

error. 8

V

by using

through

dynamics,

in the control

information).

planning

found

reference

information

dynamics (a_)

internal

the

used

(17).

solutions

to the

by finding

past

as:

(20)

of cr in Equation

the bounded

are then

subsystem

V

preview-time

partition

only the

of the internal

can be written

- A,U_a_(t)]

solution

completed

(13)

off-line).

section,

trajectories

a bounded

is then

(14) and

use of inversion

to the

To summerize:

to find

subsystem

be completely

A,U_a_(t)

according

tracking.

solution

the

used

inversion

output

Preview this

The

to the stable

a bounded

In

are

law in Equation

A_(t)-

technique.

Equations

1 to obtain

to find

3

(19)

by using

Equation

v

and

control

B; 1 [y_(t)-

the inversion

Equation

trajectories

=

[U_ _r_] is partitioned

completes

Equations

)

feedforward

°c9

by

implementation which

quantifying

enables the

of the the

tracking

relationship

inverse

using

of output between

the

v

Let

the

seconds,

desired

i.e., at time

preview

Yd (and

t, ]fd(7)

information

to approximate The

output,

is used

the

(defined

time-derivatives)

in Equation

to approximate

bounded

approximated

its

solution

solution,

(12))

given

is known

the solution

to the

#_ is found

be

for

for all t _< 7-
j_---_

_

Y_

g

from

user

Figure

3. On-line

4.3

Online

The

preview-based

gral

Equation

formula

__

generation

of the

implementation solution (21),

was

desired

trajectory

and

of preview-Based

to the

unstable

computed

subsystem

online

its time

derivatives

[12].

inversion

of the

by discretization

internal

by using

dynamics, the

i.e.,

fourth-order

the

inte-

Simpson

[15]

5_,(t)

=

- ft+Tp

e-a"('-t)[3,_Yn(r)d

"r

Jt

h

_

N

g _M(i)

(38)

_'a(t, i)

i=O

where

the

sampling

time

is Ts such

that

N := Tp/Ts

is an integer,

Yu(t,

i)

=

Ya(t

+ i • Ts)

and: if/

_(i)

-2

* e-a"(i*r°)

[3.

-4

* e-A"(i*T°)J_u

= {

=

0

if i > 0 and

i is even

if i is odd (39)

15

_:1t i

Note

that

tions.

the

This

matrices

.h4(i)

can

computation-scheme

discretization,

be precomputed is shown

can be reduced

by choosing

and

stored

in Figure

4.

The

a small

enough

to reduce

the

online

computation-error,

sampling

computa-

due

to time-

time(Ts). ! ! I

,' _

Dynamics Internal

'_d(t+Tp)

7_,ero-OrderSample I*

_u(n) _[ Zero-Order Hold

Discrete Time Calculation

Figure

4. The

schematic

of on-line

calculation

of the

unstable

Continuous Time Signal

part

of internal

dynamics.

V

4.4

Experimental

Experimental

Results

results

T_ = 50 seconds,

for output-tracking

are

shown

in Figure

with

two-different

5, which

preview-time,

illustrates

the

Tp = 20 seconds

improvement

and

in output-tracking

v

as preview-time to account and

imbalance

orientation).

preview

output

in Figure time,

and

experiments

dynamics

of the

It is noted

desired

As shown

The

for unmodeled

static

these

increases.

discs,

that

trajectories

in the which

trajectories

specification

of the

two

on-line improves output

16

v

(like

friction

a tendency

for the

inversion

feedback-stabilization,

system

created

were generated

5, preview-based online

include

cases

in the

in the are

discs

different

which experimental to settle

trajectory

performance is possible.

in a specific 5 because

preview-time with

added system

in Figure

for the two different tracking

was

cases.

increasing

W

g

/I

°°V

,/

\

-20

I 0

8O

20

40

60

I

!

!

I

80 1O0 (a) time (sec)

i

!

i

i

I

120

140

160

!

I

I

6O

_4o

0 I

I

20

I

40

60

I

I

I

I

I

80

100

120

140

160

(b)

Figure represents

5. Experimental the output

results.

trajectory.

The Plot

solid

desired

preview

trajectory,

preview

the

time,

and

dashed plot

line

(b) is for 50

time.

Discussion

For nonminimum high-accuracy scribed

line is the

(a) is for 20 seconds

seconds

5

time (sec)

for the

phase output off-line

systems, tracking. computation

recent

stable

However, of the

inversion-based

the

entire

inverse 17

desired - this

approaches

can be used

output-trajectory is a significant

has limitation

to achieve to be presince

the

desired

trajectory

preview-based tracking

cannot approach

controllers.

to systems

like

change

desired

the

preview

of the

drawback

The

is that

the

preview

time.

The

that

imaginary

preview

right-half

plane

In summary,

the

time zeros

allowed

plane(see

amount

can

3) can

be made

are far away approach

of the

arbitrarily

current

time

initial

by choosing imaginary

in the

phase

systems

inversion

design

of the

implementation

ap-

sufficiently of the

a

tracking

large

location

of

of a system.

In

design-parameters axis

if a

is not

output

with

in terms

to

technique

information

preview-based small

aid in the

the

will help

by the

approach

it is necessary

for nonminimum

smaller

from

where

current output-

inversion-based

for high-accuracy

of preview

section

by the

inversion-based

for preview

result

be made

alleviated

systems

this is necessary

can

of the

needed

of the

requirement

An important

error

preview-based

application

are

been of the

on output-tracking

[21].

(if any)

the

This

Rather,

quantification

has

implementation

online-changes

limitations

available

limitation

servo-positioning

is possible.

output-tracking

in the the

and

controller.

is not

proach

allows

Such

performance

if output-preview

particular,

methodology

trajectory

This

the online

manipulators

current

are

online.

allows

trajectory.

desired

there

system-zeros

which

flexible

of the

because

be changed

such

complex

plane.

of inversion-based

V

control

laws

6

for high-accuracy,

output-tracking

developed

and

implemented

for non-minimum

phase

required

in output

accuracy

discussed

experimental

and flexible

the

systems.

preview-based The

tracking,

technique

preview and

was

verified

needed

to the by

structure.

output

time

related

18

V

of nonminimum

phase

systems.

Conclusion

We have

were

on-line

tracking was

quantified

system-zeros.

applying

using

it to the

on-line

inversion

in terms

of the

Implementation

issues

output

of an

tracking

Q

7

Acknowledgment

Financial

support

9813080

are

from

gratefully

NASA

Ames

Research

Center

Grant

NAG

2-1042

and

NSF

Grant

CMS

acknowledged.

References

W

[1] E. Bayo. robot.

A finite-element Journal

approach

of Robotic

to control

System.s,

the

4(1):63-75,

end-point

motion

of a single-link

flexible

1987. I

[2] D.

Croft,

S. Stilson,

Nanotechnology,

[3] C.A.

1998.

Desoer

Press,

and

S. Devasia.

Accepted

Avenue,N.Y.

Output

tracking

Optimal

tracking

of piezo-based

nano-positioners.

for Publication.

M. Vidyasagar.

111 Fifth

[4] S. Devasia.

and

Feedback 10003,

Systems:

1975.

Input-Output

(Chapter

with nonhyperbolic

Properties.

Academic

g

2).

and near nonhyperbolic

internal

dynamics: B

Helicopter

hover

control.

Journal

of Guidance,

Control,

and

Dynamics,

20(3):573-580,

1997.

[5] S. Devasia,

D.

Transactions

[6] S. Devasia

IEEE

[7] S. Devasia, rameter

and

on Automatic

and

systems.

Chen,

B. Paden. Transactions

B. Paden,

jumps.

and

International

B. Paden.

Nonlinear

Control,

41(7):930-943,

Stable

inversion

Journal

Control,

output

tracking.

IEEE

1992.

for nonlinear

on Automatic

C. Rossi.

inversion-based

nonminimum-phase

43(2):283-288,

Exact-output

tracking

of Control,

67(1):117-131,

Feb.

theory

li

1999.

for systems

1997.

19

time-varying

with

pa-

W

v

[8] J.S.

Dewey,

K. Leang, redesign

for flexible

ment,

Control,

120:456-461,

tion,

and

The

15:486-505,

on Automatic

G. Meyer,

on Automatic

Control,

G. Robert

Irvine.

Englewood

Cliffs,

and

edition,

[15]

and

and

trajectories.

results

of Dynamic

in output-

Systems,

Measure-

problem.

Journal

of Control

Optimiza-

NJ

07632,

system.

IEEE

Transactions

for linear

systems.

IEEE

Transactions

1996.

Characteristics

York,

Output

1981.

and

(Chapter

regulation

35(2):131-140,

Systems.

control

inverses

Amplifier New

nonlinear

1979.

41(4):608-611,

W. Cheney.

Applications.

Prentice-Hall.,

9).

of nonlinear

systems.

IEEE

Transactions

1990.

Prentice

Numerical

Hall,

Upper

Saddle

River,

NJ

07458,

second

Analysis.

Brooks/Cole,

Pacific

Grove,

CA,

93950,

7).

W. J. Book.

An inverse

In Proceedings

F. L. Lewis edition,

regulator

and R. Su. Noncausal

Nonlinear

(Chapter

[16] D. Kwon

[17]

Journal

theoretical

1996.

D. Kincaid 1991.

and

98.

24(6):855-865,

Operational

Control,

[14] H. K. Khalil.

Dec.

ASME

of multivariable

C. I. Byrnes.

on Automatic

structures.

multi-variable

Invertibility Control,

[11] L. R. Hunt,

A. Isidori

linear

Experimental

1977.

[10] R. M. Hirschorn.

[13]

S. Devasia.

trajectory

[9] B. A. Francis.

[12]

and

and

V. L. Syrmos.

dynamic

of the American

Optimal

method Control

Control.

1995.

2O

John

yielding Conference,

Wiley

and

flexible

manipulator

state

1990.

Sons,

New

York,

second

+.

[18]

G. Meyer, zero

L. R. Hunt,

dynamics.

and

NASA

R. Su. Nonlinear

Technical

system

Memorandun,

guidance

(4661),

in the presense

Jan.

of transmission

1995. ,i

[19]

R. B. Miller

and

of Guidance,

[20]

M. Pachter.

Control,

B. Paden,

D. Chen,

multi-joint

flexible

Control,

[21] L. Qiu

Systems,

Yin

of flexible and

E. Bayo.

ASME

Journal

limitations

Automatica,

29:337-349,

of multivariable 1969.

On

of digital

the

design and

and Controls

S. N. Singh.

macro-micro

Control,

and

14(3):370-376,

model.

and

control

20(4):729-734,

Performance

Inversion

J. Lygeros,

aircraft

flight

with

actuator

constraints.

Journal

1997.

Exponentially

stable

of Dynamic

Systems,

tracking

control

Measurement,

for and

1993.

Measurement,

[24] C. Tomlin,

[25] W.

L.

Control,

M. Tomizuka.

ctoi

manipulators.

problem.

[22] M. Silverman,

[23]

R. Ledesma,

E. J. Davison.

servomechanism

matic

and Dynamics,

115(I):53-59,

and

Maneuvering

Control,

S. Sastry.

manipulators.

119(3):412-420,

Sept.

March

linear

tracking

Output

systems.

June

tracking Conference,

inverse

and

ASME

Journal

1997.

21

phase

systems

in the

1993.

controllers.

115:412-418,

and Decision

Nonlinear

of non-minimum

IEEE

Transactions

ASME

Journal

on Auto-

of Dynamic

1993.

for a nonminimum pages

predictive

phase

1867-1872,

end

of Dynamic

point

dynamic

1995,

trajectory

Systems,

control

Measurement,