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LIBRARY

Michigan State
University

 

This is to certify that the

thesis entitled

COMPUTATION OF OPTIMAL CONTROLS FOR
LINEAR SYSTEMS WITH CONTROL OUTAGE

presented by

Mehdi Kermani

has been accepted towards fulfillment
of the requirements for

1311- D. degree in EE

AZQJZLC). AZZL/

Major professor

Date M

0-7639

 

 

 

 

 

ABSTRACT

COMPUTATION OF OPTIMAL CONTROLS FOR
LINEAR SYSTEMS WITH CONTROL OUTAGE

By

Mehdi Kermani

In this study different aspects of Optimal control problems
are considered for a class Of linear systems for the case in which
the control temporarily fails to function but resumes normal
operation after a certain time period (finite duration of control
outage). Only the class Of linear systems and controls with con-
strained amplitude are treated. The concepts of recoverability
and cost constraint for the general linear Optimal control prob-
lem are also discussed.

Because of the control outage, the control constraint set
is time varying and piecewise continuous. The T-reachable sets
for linear systems with and without control outage are studied.

It is proved that the reachable set in the event of outage is
compact, convex, and varies continuously with time just as it does
in the absence Of outage.

Because of the similarity between the Optimal controls for
some singular systems and systems with control Outage, the con-
ditions for singularity in linear systems are studied, and the

distinctions between the two cases are made explicit.

Mehdi Kermani

The T-reachable sets are derived analytically for time-
invariant systems using the switching time variation technique.
This problem is treated as a series of subcases; e.g., second
order, third order,..., scalar input, vector input, real eigen-
values, and complex eigenvalues. Variation Of the T-reachable
set both with respect to the control outage starting time and
duration of the control outage is studied. Also variation of the
area inscribed by the boundary Of the T-reachable set for second
order systems with respect to the control outage starting time
and duration Of the control outage is treated. The minimum
regulation time for linear time—invariant systems with control
outage and its variation with respect to control outage starting
time, duration of control outage, and initial state Of the system
are investigated.

The convexity Of the T-reachable set for linear systems
with control outage makes possible the use Of known solution
techniques based on the convexity of this set. Gilbert's method
is applied to compute Optimal controls for linear systems with
a finite duration Of control outage. The modification for
calculation Of the contact function for systems with control out-
age is shown. Several examples are solved to demonstrate the
convergence of this algorithm. The convergence rate is slow for
systems of order three and higher. Suggestions are given on

methods for Obtaining faster convergence.

COMPUTATION OF OPTIMAL CONTROLS FOR
LINEAR SYSTEMS WITH CONTROL OUTAGE

By

Mehdi Kermani

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree Of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering and
Systems Science

1972

 

U

Q‘

n..—

TO MY PARENTS

ii

ACKNOWLEDGMENTS

The author wishes to express his gratitude to his major
professor, Dr. Robert O. Barr, for the guidance and inspiration
that he provided throughout the course of this research. In
addition, the author is deeply grateful to the other guidance
committee members, Dr. Richard C. Dubes, Dr. J. Sutherland Frame,
Dr. John B. Kreer, and Dr. Gerald L. Park for their time, interest,
and advice during the writing of this thesis.

Special note should be made of the assistance of
Dr. Frame in developing Section 3.4 of this study.

Gratitude is also expressed to Dr. Shui—Nee Chow for his
time and unselfish interest during the period devoted to this
Work.

Support for the entire Ph.D. program was granted by the
Department of EleCtrical Engineering and Systems Science at
Michigan State University. The grant was in the form of a graduate

teaching assistantship, and is gratefully acknowledged.

 

Chapter

1

TABLE OF CONTENTS

INTRODUCTION

1.1 Review of the Literature

1.2 Organization of the Thesis

1.3 Statement of the Problem

1.4 Optimal Control Problems with Recoverability

Constraint

DEFINITIONS AND GENERAL THEOREMS

2.1
2.2
2.3

2.4
2.5

Definitions

Properties of the T-reachable Regions
Considering Control Outage

Boundary of the Reachable Set and
Extremal Controls

Normality and Singularity
Controllability

COMPUTATION OF THE T-RECOVERABLE REGION AND
OPTIMAL TIME FOR TIME-INVARIANT SYSTEMS

3.7

3.8

Generation Of the T-reachable set

Switching Curves

Calculation Of the Boundary Of the
T-recoverable set

Variation Of the Area Of the T-reachable
Set with Respect to AT and t

Study Of the Minimum Time for Regulation

of the System with Control Outage
Calculation Of the Boundary of the
T-reachable Set for Systems Of Order n
with Distinct Positive Real Eigenvalues and
Scalar Input

Calculation of the Boundary of T-reachable
Set for the System of Order n with
Distinct Positive Real Eigenvalues and
Vector Input

Calculation of the Boundary Of the T-reachable
Set for Systems with Complex Eigenvalues
and Scalar Input

iv

Page

11
19
22
2A
32

33
35

37
49

52

59

67

69

Chapter

4

COMPUTATION OF OPTIMAL CONTROLS FOR THE SYSTEMS
WITH FINITE DURATION OF CONTROL OUTAGE

4.1 Basic Theory
4.2 Computer Evaluation of Contact Function
*
and of to
4.3 Numerical Results for BIP Applied to a
Minimum-Time Example with a Control Outage
4.4 Application of Gilbert's Technique to
Optimal Control Problems with Control Outage

SUMMARY AND CONCLUSIONS

BIBLIOGRAPHY

Page

72
72
76
86
89
92

95

LIST OF TABLES
Table Page

4.2.1 Highlights Of the runs, using (BIP) for the
system given by

'1 O ‘1 0
A = a E : X£O2
O -2 -2 l

and x1 = x302, with 9 = .5, = HEfHZ = .001,

’ X.= “£191 9

('1

TT = TT + I * DELT, 6t = DELT .05 82

*
4.2.2 Highlights of the runs to find t , using (BIP),
for the system given by

0 1 0 4
A = , b_= , x302 =
-1 -1 1 0
* 2
e = .5, e = “E.“ = .001, 6t = .05 sec. 83

4.2.3 (BIP) applied to the system given by

O l O -1
A = , B = , ngz =
O O l O
0 = .5, e = .001, at = .05 sec. 84

4.2.4 (BIP) applied to the system given by

O 1 O O 1
A = O O l , b = O , ngQ = O
O O O 1 0
e = .5, e = .001, St = .05 sec. 85

Figure

1.4.1

3.2.1

3.3.1

3.3.2

3.3.3

3.3.4

3.3.5

3.4.1

3.5.1

3.5.2

3.5.3

LIST OF FIGURES

Variation of the cost functional with respect to
the control-outage starting time t

Switching curves for system defined by equation
(3.1.2) AT = .1 seconds, E = 1 second,

r = *2/X1 = 2

T-recoverable set for the system defined by
equation (3.1.1), with r = xz/xl = 2, T = 1
second, and AT = .18 seconds

T-recoverable set for the system defined by
equation (3.1.1) with r = lex1 = 4, T = 1
second, and AT = .18 seconds

T-recoverable set for the system defined by
equation (3.1.1) with r = lex1 = 4, T = 1
second, and AT = .32 seconds

T-recoverable set for the system defined by
equation (3.1.1) with r = XZ/xl = 10, T = 1
second, and AT = .5 seconds

T-recoverable set for the system defined by
equation (3.1.1) with r = xz/A = 4, T = 4
seconds, and AT = 1.28 seconds

Variation of the area of the T-recoverable set
with respect to t, when r = A /x = 4, T = .5
seconds, for the system defined by equation (3.1.1)

Switching curves and Optimal-time trajectories
for the system given by equation (3.1.2), and
initial state x30} = (-1,0)

Variation Of the minimum.time with respect to
t for the system defined by equation (3.1.2),

xfOZ = (0,1), and r = 2

*
Variation Of t0 with respect to t for the
system defined by equation (3.1.2), when

x30) = (-1,0), and r = 2

vii

Page

36

44

45

46

47

48

53

6O

61

62

Figure

4.2.1

4.2.2

4.3.1

*
Flow chart for finding to , using BIP, and
support function

BIP, for the double integral plant when
T = 1.9 seconds, and t = AT = 0

Theoretical and computed value of minimum time

t0*, for different values of f, when

AT = .2 sec.
plant with

The system is a double integral

x502 = (-1,0)

viii

Page

80

81

88

CHAPTER 1

INTRODUCTION

There exists at present, a considerable amount of
theoretical material, dealing with minimum-time control of linear
systems, with constraint on control amplitude. Bellman,
Glicksberg, Cross [2], Pontryagin [3] and, Neustadt [l] have
shown that the minimal time control is in the familiar "bang-
bang" form.

In some problems it is necessary to constrain the rate
of control, due to inertial or other factors. Luh and Shafran
[4] have studied this class Of problems.

In all preceding studies, it is assumed that control
can be exerted over the process throughout the time interval
of interest. Another problem that can be considered, is that
of loss Of control for some time duration during the process.
That is, what happens to a system if the controller has
normal Operation from an initial time, to some instant E,
fails to function during the interval [t, t + AT], and resumes
normal Operation after t + AT? This loss of control is called
"control outage".

The idea Of control outage forms the basis for a new
class of optimal control problems, in which a "recoverability

constraint" is another term to be considered in the design of

control systems. That is, a question exists as to whether a
system can recover from a loss Of control, and still hit the
target before a specified time threshold is exceeded. Hauer
and Hsu [5], were the first to develop the concept of recover-
ability constraint (control outage), but they only considered

a specific two-dimensional example.

1.1 Review of the Literature

 

To be able to understand minimal-time problems with con-
trol outage, a knowledge Of this class of problems without con-
trol outage is required. In order to apply the results of
Pontryagin's Maximum Principle [3],some methods require a
bounded state-variable process. Berkovitz [6], Gamkrelidze [3],
and Dreyfus [7] have investigated this class of Optimal control
problem . Each author used a different method to obtain
the necessary condition for the Optimal Open—100p control.
Berkovitz used the calculus of variation approach, Gamkrelidze
modified the maximum principle Of Pontryagin, and Dreyfus applied
the method Of dynamic programming. In later work [8, 9], the
equivalence between the variously-derived necessary conditions
was shown.

Many computational techniques have also been develOped
to find the Optimal solution to the general Open-100p bounded
state variable problem, [10, 11, 12, 13]. One technique
involves first replacing the constraint by penalty functions.
Then this new unconstrained problem is solved by using gradient

methods. Another technique involves the direct application Of

the necessary conditions. A certain set Of initial conditions
are guessed. The trajectory is then found by numerical integra-
tion, and the necessary conditions are tested. If they are not
all satisfied, the initial guess is corrected, and the procedure
is repeated.

There are also methods which are based on the convexity
Of the reachable set of the system states. Some Of these methods
require strict convexity of the reachable sets, and exhibit slow
convergence [14].

Gilbert's method [22] does not require strict convexity
of the reachable sets, nor is restricted to bounded state variable
control problems, but exhibits slow convergence especially for
higher-order systems. An extension of Gilbert's method by Barr
[23], guarantees much faster convergence. Using Barr's technique,

a wide variety of Optimal control problems can be solved [24].

1.2 Organization of the Thesis

 

Structurally, there are five chapters in the main body of
this dissertation. In Chapter 1, a general introduction, liter-
ature review, and the problem statement are discussed. Precise
definitions, and mathematical backgrounds for the systems with
and without control outage are presented in the second chapter.
In Chapter 2, it is also proved that the reachable set for
linear systems with control outage is compact, convex, and con-
tinuously varying with time. Chapter 3 is devoted to the
analytical derivation Of the reachable set of the linear time-

invariant systems with control outage. Also time—Optimal

control problems with control outage are studied in this chapter.
In Chapter 4, Gilbert's method is used to find the Optimal con-
trol for systems with a finite duration Of control outage. Al-
though a broad class of optimal control problems could be solved
with this computational technique; in this study most

emphasis is upon the time-Optimal control problems. Finally,
conclusions and suggestions for further study are given in

Chapter 5.

1.3 Statement of the Problem

The class of linear time varying processes to be
investigated are described by the system Of differential equa-

tions

{cm = A(t) x(t) + B(t) u t on t 6 0:0, T] (1.3.1)

where

gig) is an n-dimensional state vector.
ELEL is an r-dimensional control vector.
A(t) is an n X n matrix.

B(t) is an n X r matrix.

x(to) — x0 is an n-dimensional initial state vector.

 

W(t)<: Rn is a nonempty, compact, and continuously moving
target set with respect to the real variable t on tO S t s T.
The matrices A(t) and B(t) are assumed to be piecewise con-
tinuous in t, on t0 3 t s T.

The input vector 2(5) is required to be admissible,

SO that it satisfies the following conditions:

a. ugt) is measurable

b. ugt) E U(t) CZRr, where

0 if tE[E,f+AT]

U(t) =( (1.3.2)
&' otherwise .

 

L

. . . r .
fl' is a unit hypercube in R , and 0, IS the zero vector
in R

t is an arbitrary element (time) in [to, T], and AT

is a finite duration of time such that

tOsEsE+ATsT. (1.3.3)

The set U(t) is a time varying (piecewise continuous),
compact, convex restraint set, which indicates a control outage
starting at t = t, for a duration of AT, during the process.

The unit hypercube &' in Rr could be extended to an arbitrary
convex, compact set, but unit hypercube is used for simplicity.

The general time-Optimal control problem, which is studied
in Chapter 3 Of this thesis, can be described as follows:

Given a system defined by plant equation (1.3.1), with

initial state x(to), and desired terminal state or target func—

 

tion W(t). Find an admissible control 2(3) 6 U(t), where U(t)
is defined by equation (1.3.2), for t E [to, T], which makes
§££)_E W(t) in the smallest possible time, i.e. to*. Such
an admissible control, is called Optimal control uiit), Here

*

to corresponds to the minimum time which it takes the system

7% 7V
to travel from x(to) to the final state at xgto 2 6 W(to ),

 

considering a control outage starting at t, for a duration of

AT, such that

A A *
tost5t+AT$tO .

The time-Optimal control problem is considered a free
terminal time problem. Fixed terminal-time, Optimal control
problems, which can be solved by the method described in Chapter
4, can be stated as follows:

Given a system, defined by plant equation (1.3.1), with

initial state x(to) at to, and given a fixed final time

 

*
T E [to, T]. Find an admissible control u (t), which transfers

the state Of the system from x(to) to W(T), such that:

 

T
J0T(E*) = g(x *(T)) 4”] [x'g02Q(O)X§02 + U*'£Q2R(o)u*§02]do
u t

0

(1.3.4)

T
g min [g(xu(T)) + I [x'go[2(0)éjgl.+ ELLQLR(O)ELQl]dU} °
u§t2€U(t) --——- t

O

JOT(E) is called the cost functional for the systems
with control outage, where the control constraint set is defined
by equation (1.3.2). It should be noted for T E [tO, T],
g(xu(T)) is a given real continuous, convex function from Rn

to R1. Q(t) and R(t) are real n X n continuous symmetric

 

matrices on t0 3 t g T, and Q(t) is assumed to be positive
semidefinite and R(t) is positive definite. Prime indicates
the transpose. The assumptions made here are the same as used

in [19], for Optimal control problems without any control outage.

1.4 Optimal Control Problems with Recoverability Constraint

In fixed or free terminal-time optimal control problems,
a cost functional J0t(g) is given of the form (1.3.4). In time-

Optimal control problems the cost functional is simply

t
J0t(u) = j dt
0

For given initial state x(to), and final state

 

. n . .
X(tfo) E w(tf0) in R , where tf0 18 the t1me when the target

1
is reached, the cost functional JOt (g) E R is a function
fO

Of the control-outage starting time, t, where t E [to, T], and

AT is fixed. Thus for fixed values Of x(t0), x(tfo), and

 

AT; the following equation could be written

JO (2) = F(E) (1.3.5)
t
f0

where in fixed terminal-time problems tf0 is given, but

A

JOt (g) will vary as t changes. In free terminal-time prob-
fO

lems both tf0 and JOt (u) are functions of f.
f0

Given a cost constraint J0CS 6 R1, there may be sub—
intervals 11,12,... in [to, T], such that F(E) > JOCS, if

A

t E Ij, j = 1,2,... . The system is said to be recoverable,

 

if the control-outage starting time, E E Ij, j = 1,2,..., (the

system is not recoverable if E e IJ, J = 1,2,...). In Figure

(1.4.1) the system is not recoverable if f E 11 U I2, and is

recoverable if t E {[to, T] - (I1 U 12)}.

 

 

 

JO
cs

 

 

 

 

Figure (1.4.1): Variation of the Cost Functional with

Respect to the control-outage starting

time t.

W“

CHAPTER 2

DEFINITIONS AND GENERAL THEOREMS

In this chapter precise definitions Of the T-reachable,
and T-recoverable sets are given. The general properties Of
these sets for the cases with and without control outage are
studied. The distinction between the term recoverability-
constraint, which is defined in Chapter 1, and the set of
recoverability is stated. Moreover, the basic theorems which
will be useful for the remainder of the dissertation are pre—
sented. Those theorems which are proved in previous studies
are simply stated, and the reader is referred to the original
papers for proof. The compactness, convexity, and continuity
with respect to time, of the T-reachable set, for the linear
time varying systems with control outage are proved in this

chapter.

2.1 Definitions

 

A11 definitions in this section are given with implicit
reference to the system defined by equation (1.3.1), and control
constraint set given by equation (1.3.2).

DEFINITION 2.1.1. For the given class Of the functions
u$£)_€ U(t), where U(t) is defined by equation (1.3.2), and

initial state x(to) = x0 E R“, and time T E [to’ T], the set

 

of all possible states that the system defined by equation (1.3.1)
9

10

can reach in Rn, at time T E [to, T], by the use Of the controls

from the set U(t), is called the Ifreachablc region, considering

 

control outage. This set will be denoted by R0(T, E, AT). The
T-reachable region in some of the literature is called the set
Of attainability at time T.

DEFINITION 2.1.2. For the given system defined by equa-

tion (1.3.1) and given time T E [to, T] and final state

ng) = W(T); the T-recoverable region with control outage,
relative to xfI), is defined as the set of all initial conditions

at time tO that can be brought tO the final state at time T,

using admissible controls defined by equation (1.3.2). This

set will be denoted by KO(T, f, AT). The relationship between

RO(T, E, AT), and KO(T, t, AT) is shown later in this chapter.
Let n be the control constraint set with no control

outage. That is, consider the admissible control ESSA E Q,

where

O [Q’ for all t E [to, T]] (2.1.1)

where &' is the unit hypercube in Rr

Definitions 2.1.1, and 2.1.2, could be exactly repeated
for the case with no control outage, using 2(5) C n, for
admissible controls. In this study R(T), and R(T) will
represent T-reachable and T-recoverable sets respectively, con-
sidering no control outage.

DEFINITION 2.1.3. A system defined by equation (1.3.1),
is called completely controllable, if given any two states in

n . .
R , there is a bounded control ugt), that Will drive the system

11

from one state to the other in finite time.

It is easy to see that if a system is completely con-
trollable for the case with no control outage, it is also
completely controllable for a finite duration of outage during

the process.

2.2 PrOperties of the T-reachable Regions Considering Control

 

Outage

Given a system defined by equation (1.3.1) and a control
set defined by equation (2.1.1), the reachable set considering

no control outage with respect to the initial state x(to) is

 

t
R(t) = {5; x(t) = ¢(t,to)x(to) +-] ®(t,o)[B(g)u(g)]d0 ,

—--— t

O

ugt) E n and t E [to, T]}

where ¢(t,t0) is a fundamental solution matrix of the homogeneous
differential equation given by equation (1.3.1).

It has been shown [19], that the reachable set with no
control outage R(t) is compact, convex, and continuously moving
with t, where t E [to, T]. Let E E [to, T], be a given control
outage starting time, and AT, a given duration Of control out-

age, such that
to s E s E + AT s T

TO determine the reachable set with control outage for
the system defined by equation (1.3.1), and control set given

by equation (1.3.2), consider the following three cases:

12

 

a)
to s t < E .
Then
t
R0(t,E;AT) = {E} X§t2 = 0(t,tO)X(to) + I ¢(t,o)[B(O)USOZ]dO,
t:O
ugt) e n, t e [to,f)] . (2.2.2)
Therefore
R0(t,E,AT) = R(t) for to s t < E .
b)
E s t < E + AT .
Then
15
XS?) = ¢(E,tO)X(tO) + I 00:,0)[B(0)U(0)]do . (2.2-3)
t
O

 

xgt) = ¢(t,t)x§t)

Thus

RO(t,E,AT) {x} xgt) = ¢(t,E)x§E), if t E [E,E + AT)]

or

RO(t,f,AT) r,,(t;,'E)R(’t‘;) . (2.2.4)

Substituting equation (2.2.3) into equation (2.2.4), yields

A

t
R0(t,t,AT) = {E} x(t) = $(t,to)x(to) + ¢(t,E)£ ¢(t,o)[B(c)g(g)]dg
O

 

ugt) e n, t e [E,E + AT)} . (2.2.5)

13

c)
E + AT 3 t s T
t
x(t) = ¢(t,E + AT)x(t +-AT) + ] $(t,o)[B(o)u(Q)]do (2.2.6)
E+AT
where
xgt +-AT) = ®(E + AT,t)x(E) . (2.2.7)

Substitution Of equation (2.2.6) into (2.2.7) yields

t
xgtz = 0(t,E)§L§1_+ j ¢(t,o)[B(o)U£02]do -
E+AT

Thus

R0(t,E,AT)

(g; x(t) = ¢(t,to)x(to)

 

A

t t
+ ¢(t.’E)f ®<t.o)[B(o>u_e21do +f ¢(t,o)[B(G)9_§Q2_]do,
t
O E+AT

u(t) E O, t e [E + AT,T]} . (2.2.8)

Finally

t
R0(t.E.AT) = ¢(t,E)R(’E) + {1: no =j ¢(t,o)[B(c)ug02]do.
E+AT

U(t) E O, E + AT 3 t s T} . (2.2.9)

Considering equations (2.2.2), (2.2.4), and (2.2.9), the
following theorem can be stated.

THEOREM 2.2.1. For the time-varying linear system given
by equation (1.3.1), the reachable set R(t) with no control
outage and the reachable set RO(t,E,AT), with a control outage

starting at E, and duration of AT, are related as follows:

 

l4

R(t) if to s t < E

¢(t,E)R(E) if E < t < E + AT

RO(t,t,AT) = i t
¢(t,E)R(E) + {x} 11£1_= j ¢(t,o)[B(o)ngL]do,
E+AT

 

U(t) e n, E + AT 5 t s T}. (2.2.10)

For time-invariant linear systems

/

 

R(t) if to s t < E
eA(t_E)R(E) if E s t < E + AT
RO(t,E,AT) = j t
eA(t-E)R(t) + {y} yjtl = J eA(t-O)[B(O)2£gl]do,
E+AT
L g(£)_e n, E + AT s t g T}. (2.2.11)

THEOREM 2.2.2. Consider the linear time-varying control
process in Rn, given by equation (1.3.1). Suppose the control
constraint set, U(t), is given by equation (1.3.2) where U(t) is
a time-varying, piecewise continuous with respect to time,
nonempty, and compact set in Rr. Given E E [tO,T], and AT,
such that tO S f s E + AT 5 T, then the reachable set with control
outage RO(t,t,AT), is compact, convex, and varies continuously
with respect to t, E, and AT, for all t E [tO,T].

Proof: Consider the following three cases:

a)

t s t < E
0
Then by equation (2.2.10)

R0(t,E,AT) = R(t).

15

Since, R(t) is compact, convex, and continuous with
respect to t [19], so is R0(t,E,AT), for t E [tO,E).
b)

E s t < E + AT
By equation (2.2.10)

R0(t,E,AT) = ¢(t.E)R(E).

Since R(t) is compact and convex and ¢(t,t0) is a non—
singular continuous transformation, it follows that RO(t,t,AT),
is compact, convex, and continuous with respect to t, for
t E [E,t + AT).

C)

A

t + AT 5 t S T .

Consider equation (2.2.8)

R0(t,E,AT) = {g; x(t) = ¢(t,t0)x(to)

 

A

t t
+ ¢(t,t>j s(t,o>[B<o)ugs)]ds + f ¢(t,o)[B(O)U(O)]dO,
t x
O t+AT

u(t) e n} . (2.2.12)

I - Compactness

To Show, RO(t,f,AT) is compact in Rn, for

t E [E + AT,T], and t0 3 E S E + AT S T, we have to prove that
every sequence of points in RO(t,E,AT) has a subsequence which

converges to some limit point in R0(t,E,AT).

Let

xr(t) E R0(t,E,AT), for r = 1,2,..

 

16

where

A

t
Xr(t) = ¢(t’to)x(to) + ¢(t,t){ ¢(E,o)[B(o)Ur(o)]do
O

 

 

 

t
+I ¢(t,c)[B(C)Ur(C)]dCa “1.01) E n, r = 132’”
E+AT

 

 

The set Of all controllers ugt) E Q, where O is
defined by equation (2.1.1) is weakly compact [19]. Therefore

there is a subsequence in n which converges to some

figt) E 0. Let R(t) correspond to U(t), thus

A

t
329:) = ¢(t,to)x(to) + Cz,(t,f:)f $(E,o)[B(G)AgQL]do

t
O

 

t
+ I ¢(t,o)[B(o)fi§O)]do
t+AT
Equation (2.2.12), shows that, figt) e R0(t,E,AT). Q.E.D.

II - Convexity

x in R0(t,E,AT). Then there are admissible

Consider x 2

1)

controls u1(t), and u2(t) in n, such that

 

 

 

 

 

 

 

 

E
x1(t> = s<t.to)x<co) + s<t.t>f ¢(E,o)[B(s>u1<o)]ds
to
t
+‘[ ¢(t,o)[B(O)u1(O)]dO .
E+AT
E
x2(t> = s(t,to)x(co> + ¢(t,E){ ¢(E,s>[B(o>u2(s>]ds
O

t
+ I ®(t,o)[B(o)u2(o)]do .
E+AT

 

Now let

17

x3(t) = k x1(t) + (1-x)x2(t) 0 s k S 1

E
¢(t.to)x(to) + ¢(t.E){ ¢(E,O)[B(o)(1 U1(o) +

O

 

 

 

fl

 

 

t
(1'1)u2(o)]do + j ¢(t,o)[B(o)(1 u1(o) +
E+AT

(1-1)U2(o))]do -

 

By virtue Of the convexity of n, u3(t) = X u1(t) +

 

 

(1-x)u2(t) is in n for t E [tO,T] and is admissible.

 

Therefore x3(t) E RO(t,f,AT) and RO(t,t,AT) is convex.

 

III - Continuity with respect to t
Let t1, and t2 in [to, T], be such that

f + AT 3 t1 S t S T. From equation (2.2.12), it follows:

2
t
x(t2) - x(t1) = ¢(t2,to)x(t0) + ¢(t2.E){ ¢(E,o)[B(o)ugo2]do
O

t
2
+ j ¢<t,.s>[B(c>u(cijdo - s<t1.c0>x<to>

 

E+AT
E t1
-¢(t1.E) { $<Ea0)[3(0)slglldo - j @(t1,o)[B(o)U§O)]do .
O - E+AT
tl
Adding, and subtacting the term, I ¢(t2,o)[B(g)ugg)]do
E+AT

to the above equation, yields
t2
\X(t2) - X(t1)l S l{ m(t2,o)[B(o)U(OZ]dO\ +
‘——__- '-_-—_ 1

t

1
\‘Y [$(t230) " $(t130)][B(C)UgOZ[dU\ + ‘[Q§(t2,E) -
t+AT

A

t
¢<t1.t)]j ¢(E,s>[B<o>urci]dc\ + |[¢<t,,to> - s<t1.t0>]x<t0>l
O

18

By the continuity of ¢(t,to), and boundedness of ugt),

it can be shown

lx(t2) ‘ X(t1)\ < Mlltz - t1\ + MQ‘tZ - t1\ +-g3‘t2 - t1\ +

 

 

Maltz ' t1‘

<‘th2 - tll =l5, whenever ‘t - t1| < 6 .

2

IV - Continuity with reSpect to E

Suppose t E [E,E + AT), then, let x1 6 RO(t,E,AT), and

x2 6 RO(t,E + 6E,AT) correspond to control ugt) E n, (assume t,

AT fixed), then

|x2(t,‘t + 512,131") - x1(t,’t,AT)\ = \¢(t,to)x(to) + ¢(t,f: + 51:)

 

 

 

E+5E E

I $(E + OE,C)[B(C)qu2]dC " (D(t’t0)x(t0) ' @(tsf){ $(E90)

t
O

 

O

A

E+5E t
[B(o>2&g23dc\ = \f ¢(t.o)[B(o)glcijdo -‘f ¢(t,o)[B(O)U§O)]dO\

t t

O O

S‘Mjbf‘ = g. whenever ‘OE‘ = [E - E1] < 0 .

2

Suppose t G [E + AT,T], then

 

 

 

lX2(t’E +'5E:AT) ‘ x1(t,E,AT)‘ = |¢(t,t0)x(to) + ¢(t,f + 6E)
E+5E t

l <z><’E + 6%,o>[B<o>stg21do + j" ¢(t,o)[B(O)U(o)]do -
t:o t+5E+AT

A

t:
¢(t.to)x(to) - ¢(t,t)j ¢(E,o)[B(o)ugg)]do -

t
O

t E+6E
j ¢(t.c>[B(c)uggi]dc| s \{ (b(t,o)[B(c)U£o)]do -
O

E+AT

A

t

'[ ¢(t,c>[B<c>2_(si1ds| .

 

19

Thus

\xz - fl] < EM‘6E\ = g. whenever \OE‘ = [E2 - E1| < 0

V - Continuity with respect to AAT
Equations (2.2.2) and (2.2.4) Show that R0(t,E,AT), does
not depend upon AT, if t E [tO,E + AT). To show the continuity

with reSpect to AT, when t E [E + AT,T], consider equation

 

 

 

 

(2.2.12).
t
\x2(t.’E,AT + 0(AT)) - x1(t.t,AT)l = U" o(t,o)£3(o)u(o2]do
E+AT+6(AT)
t Hm
-I ¢<tw>£B<o>e£s11dol = U s(t,c)[B(s)u_(ci]dc\ .
E+AT E+AT+6 (AT)
Let |¢(t,c)[B(o)u(g)]\ $11 .
Thus
E+AT
|x2(t.’E,AT + 0(AT)) - x1(t.’E,AT)\ s my \ = mlsmm = c.
E+AT+5(AT)
Thus whenever \AT2 - AT1\ = 6(AT) < 6, then
|xg - x1\ < §_. Q.E.D.

2.3 Boundary Of the Reachable Set and Extremal Controls

Suppose R0(t,E,AT) is the reachable set, for a system
with control outage, where R0(t,E,AT) is defined by equation

(2.2.10). Let x(to) 6 RH be an initial point in the state

 

space at time to, and assume, that the target can be reached
in finite time using ugt) 6 U(t). Since RO(t,E,AT) is

moving continuously with t, for t E [tO,T], there exists a first

20

time instant,say T E [E + AT,T] at which the trajectory of the
system x31) reaches the target set W(T). It is assumed that
there is a control outage during the process, that the system is
recoverable, and that T > E + AT. It is clear that
x11) E 5R0(T,E,AT), the boundary of the T-reachable set.
DEFINITION 2.3.1. An admissible control §(£)_E U(t),
where U(t) is defined by equation (1.3.2) on t E [tO,T], is
said to be an extremal control if there exists a nontrivial

solution n(t) = [n1(t),...,nn(t)] of the adjoint system

Tut) = -T\(t)A(t) (2.3.1)

such that
R(t)B(t)figt) = max [n(t)B(t)ugt)] for almost all

u(t)EU(t)
t E [tO,E) U (E +AT,T]

= 0 for almost all (2.3.2)
t €[E,E+AT] .
THEOREM 2.3.1. An admissible control figt) E U(t), on
[tO,T] is an extremal control if and only if, there exists a
nontrivial solution of (2.3.1) such that, the trajectory correspond-
ing to E, has the property that, the state ng2 belongs to
the boundary of the T-reachable set, and “(T) is the outward

normal to a support hyperplane Of RO(T,E,AT), at ng).

Proof: Let

n(t) = “(to)¢(to,t) (2.3.3)

21

be the solution for the adjoint differential equation, given by
equation (2.3.1) w1th “(to)= w[n1°o,...,nno]. Let
R(T), be the outward normal to R0(T,E,AT) at R(T). The set

R0(T,E,AT) is convex (Theorem 2.2.2); then the scalar product

R(T)°[§§12 - ng)] 2 O for all x(T) E RO(T,E,AT)~

Considering equations (2.2.12), and (2.3.3), it will yield

t
[I Tl(o)[B(o)§_£_q)_]dO +f mammograms] 2

to E+AT

A

[[0 n(°)[B(O)ELQleO'+ I “(o)fB(O)ugO)]dO, for all
E+AT

U(t) e n] . (2.3.4)

The above equation implies

n(t)[B(t)fi(t)] = max {n(t)[B(t)u(t)]} for
u§t26U(t)

t E [tO,E) U (E + AT,T]

Thus Q(t) is extremal.

Conversely let Q(t) be an extremal which satisfies equa-
tion (2.3.2). Let 3(1)_ and fl(T), be the corresponding
trajectory, and adjoint response which makes Q(t) an extremal.

Then equation (2.3.4) holds, thus

R(T)’[§§II - ng)] 2 0 for all XSTZ E RO(T,E,AT)

By convexity of the R0(T,E,AT), 3(EL_E 5R0(T,E,AT). Q.E.D.
It can be shown, that if an extremal control exists,

then there exists a bang-bang control which is also extremal.

22

To clarify the subject, let 5N} denote the boundary of a unit
hypercube in Rr, and O = 33, the zero vector in Rr, then we
can state the following definition.

DEFINITION 2.3.2. An admissible control ELEL is said to be

a bang-bang control on t E [tO,T], if ugtg E 5U(t), where

o if tE[E,E+AT]
aU(t) = (2.3.5)
by otherwise .
Consider the system defined by equation (1.3.1), and the
constraint control set U(t), given by equation (1.3.2). If
there exists an extremal control Q(t) E U(t) which satisfies
equation (2.3.2), then it is clear there exists a bang-bang con-
trol which is also extremal. This result implies that if there
is a unique extremal control Q(t) which satisfies the maximality

condition (2.3.2), then it is a bang-bang control.

2.4 Normality and Singularity

The question of the uniqueness of the extremal control
is answered by the concept of normality which is stated as follows:
DEFINITION 2.4.1. The system defined by equation (1.3.1)

is normal if given two admissible controls u1(t) and u2(t)

 

in U(t) which drive the system from x(t0) tO the same final

 

state x(T) E 5R0(T,E,AT), then

 

u1(t) = u2(t) a.e. on tO s t S T

This means that, if a system is normal, then the extremal con-

trol which satisfies equation (2.3.2) is almost everywhere unique.

23

It is known that [19], this unique extremal control must be a bang-
bang control in the set defined by the equation (2.3.5). This
bang-bang extremal control is called the Optimal control for
the case with a finite duration of control outage.

For normal systems, the T-reachable set, RO(T,E,AT),
(considering control outage) is strictly convex in R“. The
systems studied in Chapter 3, are assumed to be normal, and the
T-reachable sets are strictly convex. The technique used in
Chapter 4 does not require strict convexity of the reachable
sets; thus the system need not be normal.

There exists a similarity between the forms of the
Optimal controls for systems with control outage and for some

singular systems (bang-bang, including u(t) E 0 for a duration

 

Of time). For this reason it is worth noting some behavior Of
the singular systems.
DEFINITION 2.4.2. Given a constraint control set R,

where
n = {2; ‘ui(t)‘ s 1 for i = 1,...,t, and t e [tO,T]}. (2.4.1)

The system defined by equation (1.3.1) is called a singular system
*
if the Optimal control u (t) E G, which takes the system from

x(to) to xST) E aRO(T,E,AT), and the corresponding adjoint

 

equation have the following prOperty:
There exists at least one interval (t1,t2] E [tO,T],

where

n(t)B(t)u*(t) = 0 for t e (t1,t2] . (2.4.2)

24

Considering equations (2.3.2) and (2.4.2), the similarity
between singular systems and the systems with control outage
could be observed. Equation (2.4.2) is identically zero for
t E (t1,t2] not because the control vector g:(£) is zero in
this interval, but because the nature of B(t) and the adjoint
vector “(t), which depends upon A(t), yields the equation
(2.4.2). For systems with control outage, equation (2.3.2) is

Oin

III

identically zero, for t 6 [E,t + AT], because u(t)
this interval. Since normality is not required, the singular
systems with control outage can be studied using the technique
which is described in Chapter 4. For these systems,

n(t)B(t)§(E), can be zero for two different intervals, one for

the singularity condition and one for the control outage period.

2.5 Controllability

In this section the controllability of the systems with
control outage is investigated.

DEFINITION 2.5.1. The domain of null controllability
C, consists of those initial states in Rn, that can be brought
to the target in finite time using admissible controls.

THEOREM 2.5.1. Given the time-invariant linear system

in R
igc) = A xgt) + B u(t) (2.5.1)

with control constraint set 0 = Rr. The system is completely
controllable if and only if the n X (n X r) controllability

matrix C

25
G = [B,AB,AZB,...,A“‘lB] (2.5.2)

has rank n [19].

THEOREM 2.5.2. Consider the time-invariant system in Rn

xgt) = A x(t) + B u(t) (2.5.3)
where u(t) E U(t), and

ui(t) = 0 for t E [Eft + AT]
U(t) = u- 1 = 1,2,...,r M254)

—. \ui(t)\ S 1 otherwise

 

Assume the G matrix for this system has rank n, and
the system is stable, i.e. Re )1 < 0, where *1 are the eigen-
values of A. Then the domain of null controllability is
C = Rn, or in other words the system is completely controllable.

Proof: For the case with no control outage the theorem
is proved by [19]. For the case with control outage, let
xj§)_€ R“, be the state of the system at t = t, where the con-
trol outage starts. Consider a finite duration of control out-
age AT. The state of the system at t = E + AT, i.e.

x t +' T is in Rn. Thus, xgt + AT) 6 Rn, could be considered
as a new initial state for the system with control constraint

satisfying
U = {2: |ui(t)‘ s 1 for t e [ti-AT,T], 1 =1,...,r}. (2.5.5)

The system is considered to be completely controllable, which_:hn-
plies thenew initial state xgE + AT) E R“, can be driven to the

target in a finite time, using an admissible control satisfying

26

equation (2.5.5). Thus 5391 E Rn is controllable, using a
control vector which satisfies equation (2.5.4). Q.E.D.
This proof illustrates that if a system is completely
controllable for the case with no control outage, it is also
completely controllable for finite durations of control outages
occurring during the process.
THEOREM 2.5.3 [20]. Consider the time-invariant linear

system in Rn

igt) =A fi)_+ [bllb21...\br] E . (2.5.6)

The system is normal if the following holds
2 n-l .
Rank (Gj) = Rank [bj’Abj’A bj,...,A bj] = n, for J = 1,...,r

THEOREM 2.5.4. Given a time-invariant linear system in

_xgt) = A xgt) + B u(t) (2.5.7)

with control constraint set

ui(t) = O t E [t,t + AT]

U(t) = i = 1,...,r . (2.5.8)

— ‘ui(t)‘ s 1 otherwise
Suppose the rank of the controllability matrix C is

less than n. Then there exists a unique linear subspace C C R“,
such that, only inside this region C, the system is completely

controllable (i.e., no points outside C can be brought inside

27

C, or vice versa).

Proof: This theorem is proved for the case with no con-
trol outage by [19]. It will be shown here that C CiRn, the
region of the controllability is the same for both cases, con-
sidering control outage, or no control outage. Let w E Rn be
a target outside the region of the controllability C, where C
is the region of the controllability considering no control out-
age. By the definition w, cannot be reached using any admissible
control, including uL£)_= 0. Thus, if a target outside C (the
region of controllability), cannot be reached in finite time,
with no control outage, it also cannot be reached if a control
outage exists during the process. The same argument could be

used, if the initial condition x(t0) is outside C, (the

 

controllability region), and the target moves inside the con—
trollability region.
THEOREM 2.5.5. Consider the linear time-varying system

in R
igc) = A(t)x(c) + B(t)u(t) (2.5.9)

with control constraint set defined by (2.5.8). The system is

controllable if and only if the n X n matrix W(tO,T)

E
W(tO,T) = I ¢(to,o)B(O)B'(G)¢'(to,o)do
tO
T
+f @(to,o)B(o)B'(o)¢'(t0.o)do (25-10)
t+AT

is nonsingular for all t0 3 E S E + AT g T in R1. Here

¢(t,to) is the fundamental solution of the homogeneous part of

28

equation (2.5.9), with ¢(to,t0) = 1.
Proof: For the case with no control outage, the correspond-
ing W matrix is [19]

T
W(tO,T) = I ¢(CO,G)B(G)B'(G)¢'(to,o)do (2.5-11)
t

0

where admissible controls u(t) E u and n is defined by
equation (2.1.1).
Suppose W(tO,T) is nonsingular. To prove the con-

trollability of the system, let x0 and xf be given initial

and end points. Assume

u(t) = B(t)'¢(to,t)§_ for t E [to,t) U (E + AT,T]

.315) = 0 otherwise
where
g=w(t t)‘1[(t t)x -x]
o’f ®o’f_§ _g
Thus
f£,= ¢(tf’t0)fg.+ ¢(tf.to)w(to.tf)§
or

e
xf — ¢(tf,to):9 + wanton”! ¢(to.o)B(o)B'(o)¢'(t0,o)do

O

t
f
+ I ¢(to’o)B(O)B'(o)¢'(to’0)d0}{B'(t)¢(tO,t)'}-1u(t)
t+AT

t
xf = ¢(tf’to)f9_+ ¢(tf.t0)fi ®(t0,o)B(o)U(o)do
O

t
f
+ f @(to.o)B(o)ugq2do} .
t+AT

29

Therefore the system is controllable.
Assume that the system is controllable. Suppose
W(to,tf) is singular. Then there exists a nonzero constant

vector 3 such that

n'w(to,tf)j\_= 0 .

Thus

E t f

i H'¢(to,o)B(o)B'(c)¢'(to,o)fl§o + I n'¢(to,o)B(O)B'(0)¢'(to,o)fl§o = 0-
o t+AT

(2.5.12)

The system is controllable, therefore the terminal point

¢(tf,to)3 can be reached from the origin such that

E
¢(tf,to)3 = ¢(tf,to)f ¢(to,o)B(C)U§o2do
t

0
cf
+¢(tf,to)j ¢(to,o)3(c)gQLdo
t+AT
or
E tf
I] =I ¢(to,o)B(c)gQ)_do +‘J‘ rf)(tO:C)B(C)USOZdO
to t+AT
But
E tf
“'3 = f “'¢(to,o)B(o)g$gldc + f n'¢(to,o)B(o)g(g)do > 0. (2.5.13)
to t+AT
Results (2.5.12) and (2.5.13) are a contradiction. Thus
W(to,T) is nonsingular. Q.E.D.

THEOREM 2.5.6. Given time-varying linear system in Rn

52m = A(t)X(t) + B(t)u(t) (2.5.14)

30

where u(t) E U(t), for to S t S T, and U(t) is defined by
equation (2.5.8). Let vgt) E U(t) be such that vgt) = ugT-t)
on 0 s t s T - to. Then the T-reachable set for the system

defined by equation (2.5.14) relative to initial state x(to)

 

is the same as the (T - to) recoverable set relative to

x(to) for the same system with time reversed [l8].

 

THEOREM 2.5.7 [18]. Given the time-invariant linear

. n
system in R

21¢) = A ng2 + B ugT) .

Assume *1’A2"°°’An are real, distinct and positive
eigenvalues of A. If the system is normal, then the canonical

form of the system equation is

29;) = P xgt) +Q u(t)

 

where
t = 11¢ and Q = PK,
such that
r— fl
1 0 . O
0 *2/11 ...
P = .
O
>‘n/kl
L 4
l 4k12 ... 'k1r
K = O
L l kn2 ... knr L

 

 

31

THEOREM.2.5.8 [21]. Consider the time-varying linear

. n
system in R

x(t) = A(t)x(t) + B(t)u(t) (2.5.16)

where u(t) E U(t) is given by equation (1.3.2). Assume the
equation for the T-reachable set in Rn+1 is given by $(xfim).
Then the T-reachable set satisfies the following Hamilton-Jacobi
partial differential equation

n am (531')

ggwyrn + 3:;

x,(t) = O . (2.5.17)
1 l i 1

CHAPTER 3

COMPUTATION OF THE T-RECOVERABLE REGION
AND OPTIMAL TIME FOR TIME-INVARIANT SYSTEMS

Sections 1, 2, and 3 Of this chapter are concerned with
the computation of the T-reachable set Of the linear time-
invariant second-order systems, with real eigenvalues and scalar
input, for the cases with and without control outage. By
Theorem (2.5.6), the T-recoverable set for the forward-time system
is the same as the T-reachable set for the reverse-time system.
The computation of the T-reachable set is more tractable, thus
the T-reachable set is studied in these sections.

Section 4 Of this chapter considers the variation of the
area inscribed by the boundary of the T-reachable set, with
respect to the control outage starting time E, and with respect
tO the duration Of the control outage AT.

Section 5 of this chapter studies the Optimal time for
the second order system considering a control outage. The
relationships between the Optimal time and the control outage
starting time t, and between the optimal time and duration of
the control outage AT, are also studied in this section.

In Sections 6 and 7 Of this chapter, the T-reachable set
for nth order systems with distinct real eigenvalues, with scalar
and vector input respectively are computed. Finally Section 8
considers the computation Of the T-reachable set for nth-order

32

33

systems with complex eigenvalues.

3.1 Generation of the T—reachable Set

 

According to Theorem (2.5.7), the normalized equation
for the second order system, with positive, real, and distinct

eigenvalues for the forward time is

x ft} = xgtz + u(t) (3.1.1)
x2/11 A2/11

where X1 and A2 are eigenvalues, u(t) E n, which is defined
by equation (2.1.1). It is easy to show that (3.1.1) is completely
controllable and normal (Theorems 2.5.1 and 2.5.3).
The problem is:
Given x11) as a final state, what is the set of all
initial states for which there is a control in n, that drives
the system to the given final state at given time T? This set
Of initial states, which is called the T-recoverable region
related to £311, is the same as the T-reachable set if the
system with time reversed is considered. To compute, the T—re-

coverable set for (3.1.1) we consider the reverse time system given by

31:) = P xgt) + q u(t) (3.1.2)
where
-l 0 —l
P = , q = (3.1.3)
0

'1 '1
2/)\1 2h,1

34

If there is no control outage at most one switching is
required tO reach any boundary point Of the T-reachable set with
no control outage R(T) [19]. Let the time Of switching be t1.
As t1 ranges from O to T, the locus Of the end points of
the corresponding trajectories generates the boundary 5R(T)

Of the T-reachable set.

If the constraint set is U(t), which is defined by equa-
tion (1.3.2), then given E E [0,T], where the control outage
starts, and AT, duration of the control outage, the boundary
Of the T-reachable set with control outage aRO(T,E,AT) can be
generated considering the following three cases for the switching
time t1:
8)

OSE§E+ATStlsT

TO find aRO(T,t,AT), let

u(t) = k0 for t E [0,E) U [E + AT,t1)
u(t) E O for t E [E,E + AT)
u(t) = -k0 otherwise
where k0 = i.l.
b)
' OSEStlsE+ATsT.

In this case let

u(t) = k0 for t E [0,E)
u(t) E O for t E [E,E + AT)
u(t) = -kO otherwise

35

C)
0 s t1 s E s E + AT s T .
u(t) = ko for t E [0,t1)
u(t) = -ko for t e [t1,E) u [E + AT,T]
u(t) = 0 for t E [E,E + AT) .

3.2 Switching Curves

- . . +
For the systems With no controloutage the sw1tch1ng curve y ,

corresponding tO u = +1 can be generated by starting at

x30) = g, and letting the system run for T seconds with u = +1.

x +
Similarly, given t, and AT, the switching curve yo for the

systems with control outage can be generated by starting at the

origin,

and letting the system run for T-seconds with u(t) E O,

for t E [E,E + AT], and u(t) = +1 otherwise. The switching

curve

+
yO , for the systems with control outage consists of three

segments +, +, and + (Figure 3.2.1), such that
1 2 3

where

0+
Y 3

+ + + +
y0 = yol U yo2 u y03 (3.2.1)

t
[5: xgt) = IeP(t—O)q U(C)d0 = [ePt-I]P_1g
O

for O S t S 3} (3.2.2)

{xz xft) = eP(t_E)x(f) = [ePt-eP(t_E)]P_1q

A

for t s t s E + AT} (3.2.3)

Pt_eP(t-E) + eP(t-E-AT)

{2: 1&2). = [e -131)”,

for t + AT s t S T} . (3.2.4)

without control outage

with control Outage

36

m)

f—l

 

'YO
+
v0 ,1
’9””‘| + I
.,”’ ‘\. Y0 2 I,
‘1
1
Figure (3.2.1):

AT —

Switching curves for system defined by
equation (3.1.2) for

.1 seconds,
t = 1 second, and r = xz/xl = 2.

37

The switching curve yO-, corresponds to u(t) = O, for
t E [t,t + AT] and u(t) = -1, otherwise. This yo’ is
symmetric to yO+ with respect to the origin (Figure (3.2.1))
3.

Given 2, and AT, the equation of the switching curve

and can be described as a union Of in, yo;, and y0

for the system defined by equation (3.1.1) is:

 

 

 

 

 

 

 

 

 

 

 

r \
-t 1
+ e - l
—'= x: xgtz = i. _ for t E [0,E) > (3.2.5)
1 XZ/Xl
- 1
Le _ J
K P A —
_ -... )
+ _ . =
YO? —<2<_. X t i - -x (c-t) >
>‘z/xl 2/11
K Le ‘8 .1 2
for t E [E,E + AT) (3.2.6)
I " ,. W
e-t _ e-t+E + e-t+t+AT _ 1
+
Y0§ fie: .le t =i . . >
-x2/A1t ”AZ/x1(t’t) “AZ/x1(t't'AT)
K [e - e + e - 1])
(3.2.7)

Switching curves for the system given by equation
(3.1.1) are shown in Figure (3.2.1) for the cases with no con-

trol outage and with control outage.

3.3 Calculation Of the Boundary Of the T-Recoverable Set

The T-reachable set for the reverse time system defined
by equation (3.1.3) is calculated in this section. TO compute

aRO(T,f,AT), one can begin at the origin in the reverse time

38

system and search for states that can be reached by admissible
controls in time T, along the minimum time trajectories. That
is, search for the states that can be reached from the origin
in precisely time T, but cannot be reached in time less than
T. This set Of states forms the boundary Of the T-reachable
set for the reverse time system. Following the procedure given
in Section (3.1), it follows:

If

OsEsE+ATStlsT

where t1 is the switching time, then

E . A
an = i [I eP(t‘°)ado1 = : 1e“ — I1P'121
O

 

x E + T = eP<t+AT—t)x t = :ePAT[ePt — I]P_lq
P(tl-'t—AT) t1 P(t1-o)
x(t1) = e xgt + AT) i I e gdo
E+AT
Pt P(t -E) P(t —'t-AT) _
=iEel-e 1 +e 1 -I]Plg_
P(T‘t ) _ "r _
XST) = e 1 x(t1) + I eP<T C>gdo
—--— t

l
where the upper sign corresponds to u = +1 initially and the
lower sign to an initial choice Of u = -1. Therefore the first
segment Of the boundary of the T—reachable set with control out-
age, 5R01(T,E,AT), can be expressed as

PT P(T-C1)

8R01(T,E,AT) = {y xm = -_I-_ [I + e - 2e + eP(T-E—AT)

- eP(T-t)]P-lq for O s t + AT 5 t1} . (3.3.1)

39

If
0 s E s t1 5 E + AT 5 T
then
591;). = i [ePE - I]P-121
x t + T = i epflfizp’E - I]P-19L

A _ T _
= eP(T-t-AT) xft + :T) +16 eP('r <5)ng

t+AT

E

Thus the second part Of the boundary Of the T-reachable

set can be expressed as

6R02(T,E,AT) = {21: xm = i [I + ePT - eP(T'E‘AT) - eP(T‘t)]p‘lg_
for E 3 t1 3 E + AT} . (3.3.2)

 

 

If
OStlsEsE+ATsT,
then
I:
1P(t1-o) Pt1 1
x(t1) = f e qdo = + [e - I]P q
O
P(E-t) __E
29.21 = e 1 mp +j e1“E 0) do
t
1
A P(E"t) _
= [ePt - 2e 1 + I]P 1q_
P(E-t )
x t + T — i ePAT[ Pt - 2 1 + I]P g_

40

or

PT “T‘t 1)
aRO3(T,E,AT) = {x} ngz = :_[I + e — 2e

- eP(T-t-AT) + eP(T-t)]P-1q_ for O s t1 3 E}. (3.3.3)

Thus the boundary Of the T-reachable set with control

outage is
aR0(T.E.AT) = 5R01(T,E.AT) u 5R02(T.E.AT) u aRO3(T.E.AT)

For every value Of t1 6 [0,T], there corresponds a state

x_€ 5R0(T,E,AT) in R“. As t1 ranges from O to E, x_ will
move to generate the boundary of the T-reachable set
aRO(T,E,AT). Suppose ng,E) is a state corresponding to

t = E. Then ng,E) will be the same reSponse for all

t1 6 [E,E + AT]. Thus the following lemma can be stated.

LEMMA 3.3.1. The transformation from R: (positive
real's) to Rn (n-dimensional vector space for X(T,t1)) is not
one to one for t1 6 [E,E + AT].

Proof: Equation (3.3.2) does not depend on t1. There-
fore for fixed values of T, E, and AT, one single state can
be reached in the state Space for all t1 E [E,E + AT]. This
implies that the mapping from t1 to Rn is many to one.

Considering equations (3.3.1), (3.3.2), and (3.3.3),
one can deduce that the boundary Of the T-reachable set with
control outage is symmetric with reSpect tO the origin.

THEOREM 3.3.1. Let x(T,t1) represent a point on the

boundary, aRO(T,E,AT), Of the T-reachable set with control out-

age, generated by switching at t1, where O 3 t1 3 T. Then

41

x(T,t1) moves continuously in Rn as t1 ranges from O to T.
Proof: Suppose x'(T,ti) is an arbitrary state in

aROl(T,E,AT), which is defined by equation (3.3.1). Let

x"(T,t¥) be an arbitrary point in 5R03(T,E,AT), given by

equation (3.3.3). TO show the continuity, suppose e > O is

given; then there exists a 6 > 0, such that

\X'(T,ti) _ X"(T’t$)1 < 6 whenever \ti - t3] < 5 - AT

for all ti 6 [E + AT,T], and t; G [0,E].

Considering equations (3.3.1) and (3.3.3), it follows

P(T-t') A
‘[I + ePT _ 2e 1 + eP(T'E‘AT) _ eP('r-t) _ I _ ePT
P(T-t") A
+ 2e 1 + eP(T'E-AT) .. eP(T‘t)]p'lg‘
_A_ _ _ 'P(T-t" 'P(T-t') _
S 2‘E8P(T t AT) _. eP(T E)]P lg" + 2‘[e 1 _ e 1 JP 11‘.
(3.3.4)

Introducing a vector norm e.g., 2 131‘, and applying
i
the Taylor expansion to the right hand side of the inequality
(3.3.4) for P and 9 given by equation (3.1.3) yields

\X'(T,ti) - x"('r,t"1)‘ < 2(1+r)(AT + “1 - 1'“) = e

where

Thus, let

and

42

\ _.__§._\
522(1+r)/0 Q.E.D.

DEFINITION 3.3.1. The reachable set for infinite travel
time RO(m,E,AT) is called the region Of inescapability, which
simply is

R0(m,E,AT) = R0(T.E.AT) .

T—m

THEOREM 3.3.2. Given a system with positive, real, distinct
eigenvalues, and finite E and AT. Suppose R(m) denotes
the region of inescapability for the case with nO control outage
and let RO(m,E,AT) be the region Of inescapability for the

case with control outage. Then

R(m) = R0 (OD,E 2AT)

Proof: The equation for the boundary of the T-reachable

set with no control outage is:

P(T—t ) _
aR(T)={§=§=iU+ePT-Ze 1]? 121
for O S t1 S T} . (3.3.5)
Let r = T - t1, and considering lim ePT = 0, it follows
T—voo
Pr -1
5R(w) = [x: x = i [I - 2e ]P q for O S r S m} . (3.3.6)

The equation for the boundary of the region of the

inescapability with control outage is

PT _ zePr + ePfi-E-u)

lim aRO(T,E,AT) = {at >1 = i [I + ‘3

”Has

1 eP(T-t)]P-lq for O S r S T} . (3.3.7)

43
For finite E and AT
aRo(m,E,AT) = {5: g = i [I - 2ePr]P_lg_ for o _<. r s ...}. (3.3.8)

Comparing equations (3.3.6) and (3.3.8) proves the theorem.
If the normalized equation for the reverse time system
is given by equation (3.1.2), then the equations for the boundary

of the T-reachable set with control outage are

-(T-t ) A A
1 - - +t+ T - +t
1-2e +e T+e T A -e T

3R01(T,E,AT) = g: E = :
-r(T-t1)

1_2e +e-rT+e-r(T-E-AT)_e-r(T-t)

for E s E + AT s t1 , (3 3.9)
-(T-t ) _
1_2e 1 +e-T_e T+E+AT+e-¢+E

5R02(T,E,AT) = x : x =:

-r(T-t )
1-2e l +e rT_e r(T E AT)+e r(T E)
for O S t1 S E , (3.3.10)
*2
where —- = r > 1. For t1 6 [E,E + AT], equation (3.3.9) with
t1 = E + AT or equation (3.3.10) with t1 = E, can be used.
Thus

aRO(T,E,AT) = 3R01(T,E,AT) u 5R02(T,E,AT)

Digital computer solutions for the 3R0(T,E,AT) are

A

shown in the following figures for different values of t,

12
AT, T, and r = —- .
k1

44

[x

 

 

Figure (3.3.1): T-recoverable
set for the system defined by
equation (3.1.1), with
r = XZ/xl = 2, T = 1 second,

and AT = .18 seconds.

 

x”

1—‘

 

45

 

   
  

Figure (3.3.2): T—recoverable
set for the system defined by
equation (3.1.1) with

r = 12/11 = 4, T = 1 second,

and AT = .18 seconds.

 

x"

46

 

 

Figure (3.3.3): T-recoverable
set for the system defined by

equation (3.1.1) with

r = 12/11 = 4, T = 1 second,

and AT =

 

.32 seconds.

fl)

...:

47

7x

 

T-recoverable

Figure (3.3.4):

set for the system defined by

equation (3.1.1) with

1 second,

12/11 = 10, T

and AT

r

.5 seconds.

 

1x2

sec
0

_
_
_
_

 

 

48

 

 

Figure (3.3.5): T-recoverable
set for the system defined by
equation (3.1.1) with

r = 12/11 = 4, T = 4 seconds,

 

and AT = 1.28 seconds.

49

3.4 Variation of the Area of the T-Reachable Set with Respect
to AT and E

For the system given by equation (3.1.2), the boundary

of the T-reachable set is given by equations (3.3.9) and

(3.3.10):
'(T-t ) _ _ _

1 1_2e 1 +e T-e T+E+AT+e T+E
x
x ------r- -r(T-t )

2 l—2e l +e-rT_e-r(T-E-AT)+e-r(T-E)

for t1 6 [0,E] (3.4.1)

and

‘(T't ) _ _ _
1 +6 T+e T+E+AT_e T+E

2 1-26
x
2
2 = x (m1) =:
x —r(T-t ) A A
2 -2e 1 +e rT+e-r(T-t-AT)_e—r(T-t)
for t1 6 [E + AT,T] (3.4.2)
)2
where r = -— > 1.
K1

Considering the symmetry of the T—reachable set with

respect to the origin, the area Of the T-reachable set is

E T
_ l l 2 2
A - -2 I x2d(x1) + I x2d(x1) . (3.4.3)
0 E+AT
Substituting equations (3.4.1) and (3.4.2) into equation

(3.4.3) yields

A = 4Q[e'(T‘E) - e‘T] + 4s[1 — e'(T‘E'AT)] - 1%? P (3.4.4)

where

50

rT _ e-r(T-t-AT) + e-r(T-E)

Q = [1 + e" 1

s = [1 + e-rT + e-r(T-E-AT) _ e-r(T-E)

1

P = [1 + e'(1+r)(T-E) _ e_(1+r)T _ e‘(1+T)(T-E-AT)

] .

Suppose T and E are fixed. Then

5A _ 4[re'Te-r(T_E'AT)[eT _ et+AT _ eE + 1]

5(AT) —
_ e-T+E+AT[1 _ e—r(T-E—AT) + e-rT _ e-r(T-E)]} (3.4.5)
or
3%353 = Ae-T.e-r(T-E-AT).eE+AT{r[eT-E-AT _ 1 _ e—AT + e—E-AT]
- [er(T'E‘AT) - 1 + e'r(E+AT) - e-rAT]} . (3.4.6)
Let
z1 = eJE-AT s 22 — e"AT s 1 S 23 = eT-E-AT
Q(z) = 2r - rz + r-l (3.4.7)
then
3%353 = 423(1+T)[-¢(zl) + Q(zz) + ¢(1) - Q(z3)] (3.4.8)

Since m(z) = r(zr-1-l), it follows that for r > 1, m(z)
decreases if 0 < z < l and increases if 1 < z, (see the follow-
ing figure).

m(Z)

r—l

 

 

51

Hence

m(zl) > m(22) and m(1) < m(23)

Thus

BA
B(AT) < 0’ for all Values 0f E: AT, and T, satisfying

0 s E s E + AT s T (3.4.9)
THEOREM 3.4.1. The area Of the T-reachable set with con-
trol outage is monotonically decreasing with respect to the dura-
tion Of the outage (AT) if E and T are assumed to be con-
stants.
To Observe the variation Of the area of the T—reachable
region with respect to the control Outage starting time E, it
is assumed that T and AT are constants. Applying the follow-
ing formulas:
B(X)
F(X) = f f(X.y)dy
u(X)

and

B(X)
F'(X) = B'(X)f(X.B(X)) - a'(X)f(x,a(X)) + I f1(x,y)dy

U(X)

tO equation (3.4.3) yields

AA = 4e‘T . eE+AT . e-r(T-E){r(e-AT _ e(r-1)AT))(1_e-E
5E

_e'T'E + eAT) + (e—AT _ 1) (er(T-t) _ 1 _ erAT + e-rt)}. (3.4.10)

For AT : 0, it follows that 3% z 0. This implies that
the area Of the T-reachable set will not change significantly,
no matter where the control Outage occurs during the process if

the duration of the control outage AT is small With respect

to T.

52

Depending upon E, AT, T, and r, the area of the T—reach-
able set could be increasing to reach a maximum for some
0 S t S T. Finding such an extremum analytically is cumbersome.
In fact Figure (3.4.1), shows that the area of the T-reachable
set is not always monotonically decreasing, with respect to E.
This contrasts with Theorem (3.4.1), which showed that the area
of the T—reachable set is always monotonically decreasing with

respect to AT.

3.5 Study Of the Minimum Time for Regulation of the System with
Control Outage

Consider a system defined by equation (3.1.2), and an

X1(0) *
initial state x3 2 = , at time t = 0. Assume t
x2(0) o

is the minimum time which it takes the system to travel from
x30) to the origin with no control outage. As is well known
the switching curves with no control outage divide the state
. . . + - .
space into two regions, i.e., Q and Q , where u = +1 15
+ , —
used when x E Q and u = -l is used when x E Q (see

Figure (3.5.1)). Suppose XfO) E Q-, and let t be the

1
switching time. Thus
Ptl Ptl _1
x(tl) = e x! 2 - [e - I]P g
and
* *
P(t -t1) P(t -t1) _1
g = e X(t1) + [e - I]P q .
Thus
* P(t* t ) *
g = ePt xgo) + [2e 1 - ePt — 111519 . (3.5.1)

 

53

94%?

 

..l ‘\\, # 74_
\. ‘\\\. E
\ \.\
\. ~\
\ '\
' \\
\. -\
\ \\
\.
\\ \'
. \~
\_ \
\ \
. AT = 3 \
- 1 \ \AT = 1
\
The derivative Of
Area (A) ___—_~——____ A with respect __'__”__“-m_.
A be
t O t i (a E )
Figure (3.4.1): Variation of the area of the T—recoverable set

with respect to E, when r = 12/11 = 4,

T = .5 seconds, for the system defined by

 

equation (3.1.1).

 

54

X
For P and g_ given by equation (3.1.3), and T; = 2,
1

there will result

t* = Loge[xi(0) + 2x2(0) - 2x1(0) - 1] ~ Loge[x1(0) - 1 +

 

—\\/;xi(0) - 4x1(0) + 2x2(0) 3 . (3.5.2)

+
If xSO) E Q , the same calculation yields

71‘
* * P(t "t ) .-
O = ePt x30) +[ePt - 2e 1 + I]P 1q . (3.5.3)

In the special case when r = 2

5':

c = Loge[xi(0) + 2x1(0) - 2x2(0) - 1] — Loge[-x1(0) - 1 +

 

2xi(0) + 4x1(0) - 2x2(0) ] . (3.5.4)

To calculate the minimum time when there exists a control
outage, the following three cases are considered (three symmetrical
+
cases occur for x30) E,Q ).

Case a:

xSO) 6Q ,OSEStl, and XSE+AT2 EQ-

 

where t1 is the switching time if there is no control outage.
Thus
A PE Pt ‘1
x(t) = e §(_1 - [e - I]P 9_ (3.5.5)
PAT A
xgE+AT2 =e xgt) . (3.5.6)

Substitution of equation (3.5.5) into equation (3.5.6)

gives

55

_ P(E+AT) P(E+AT) PAT
—e e -e

x E + T ]P-lg_ . (3.5.7)

w-[

The minimum time for the system to move from the new initial

state, i.e. xgE + AT), to the origin must satisfy equation (3.5.2).

Let us denote this minimum time by t). Thus t; satisfies
the following equation
Pt) P(tE-tl) Pt) _1

Q_= e xgE + AT) + [2e - e - I]P 9. (3.5.8)

where ti is the new switching time. It is Obvious that the
*
total minimum time with control outage to is:
* , A
to = tf + t + AT . (3.5.9)

Given ng), E, and AT, one can calculate xgE + AT)

from equation (3.5.7). With a calculated value Of xgE + AT)

and equation (3.5.8), tf

can be found. Finally equation
i
(3.5.9) gives toc, which is the minimum time that it takes the

*
system to travel from x30) to xgto ) = Q.

A
In Special case when r = T; = 2,
l
r E+ 7
e’( AT)eq(0).-1)-+e’5r
x E + T =
e-2(E+AT)(X2(O) _ 1) + e-2AT

 

 

and

2
' _ A A A -
tf — Loge[x1(t + AT) + 2x2(t + AT) - 2x1(t + AT) 1]

- Loge[x1(E + AT) - l +

 

_“\J/2xi(E + AT) - 4x1(E + AT) +-2x2(E + AT) ]. (3.5.10)

56

The computer solution shows that, depending upon the
initial state and AT, the variation of the to* with reSpect
to E is either monotonically decreasing, or reaches a
maximum for some E E [0,t1]. These results are depicted in
Figure (3.5.2)

Case b:

Let wEQ,OSESt1, and xSE-l-AT) EQ+. The
state Of the system at E + AT, xgE + AT), can be calculated from
equation (3.5.7). Motion from xgE + AT) tO the origin is des-
cribed by equation (3.5.3). Let the time which it takes the
system to travel from xgE + AT) to the origin be t%. Then

Pt) PtE P(té-ti) _1
Q.= e xgE + AT) + [e - 2e + I]P g_. (3.5.11)

7':
Equation (3.5.9) gives the minimum time tO . In the

*2

special case when r = I-'= 2, then
1

2 A A A
t) = Loge[x1(t + AT) + 2x1(t + AT) - 2x2(t + AT) - 1]

 

- loge[-x1(E + AT) - 1 +\/2xi(E+AT) + 4x1(E+AT) - 2x2(E+AT) 3.

(3.5.12)

The computer solution indicates that the variation Of

7':
to with respect to E is monotonically increasing for

E E [0,t1], and xgE + AT) 6 Q+ (see Figure (3.5.2)).

Case c:

A * A +
If t1 S t S t and xgt + AT) EQQ , it follows that

Pt1 Pt 1

x(t1) = e x( ) - [e 1 - 1]?” g . (3.5.13)

 

57

+
Now x(t1) E y}, where y satisfies the following equation

 

+ : -PT - I]P-1q_ for O S T S T, where T=tx-t}. (3.5.14)

Therefore t can be calculated by substituting equation

(3.5.13) into equation (3.5.14). Thus

 

 

P(E-tl) P(E-t ) _1

xgE) = e x(t1) + [e - I]P q. (3.5.15)

and
P(E-t +AT)
x E + T = ePAT xgE) = e 1 x(t1) +
P(E-t +AT)
[e 1 - ePAT]P 13 . (3.5.16)
A +

If xgt + AT) E,Q , then it satisfies equation (3.5.3),

such that
Pté Pt) P(tE-tl) _1

Q_= e xgE + AT) + [e - 2e + I]P 9_ (3.5.17)
where té is the minimum time which takes the system to travel
from xgE + AT) to the origin. Here t' is the new switching

l
7%

time. Equation (3.5.9) gives the total minimum time to for
the system to go from x50) to the origin. In the special case,

when r ='—- = 2

2
' — A A - -
tf — Loge[x1(t + AT) + 2x1(t + AT) 2x2(E + AT) 1]

 

- Loge[-x1(E + AT) - 1 ¥\/2xi(E+AT) + 4x1(E+AT) - 2x2(E+AT) 1.
' (3.5.18)

*
The computer solution indicates that in this case to

is monotonically decreasing for E E [t1,T] (see Figure (3.5.2)).

58

Consider now the analysis Of these cases.

Suppose the initial state ij)_ and duration of control
outage AT are given. For different values Of E E [0,T], the
minimum times for the system to travel from. x19)_ to the origin
are different. Let us denote the maximum value Of these different
minimum times by Max(to*). This maximum value of to* occurs
when E, the control outage starting time, is E = t1 + 8. Here
t1 is the switching if there is no control outage and e > O
is very small.

For given ng), and AT, the maximum value Of the

*
minimum times, Max(tO ), can be calculated. Equations (3.5.13)

 

and (3.5.14) give t1. Thus
Ptl Ptl _1
x(t1) : x(E) :‘6 xg ) - [e - I]P g_
and
P(t +AT) P(t +AT)
._ l 1 PAT -]_
XSE + AT) — e ng) - [e - e ]P g.. (3.5.19)

A

From equation (3.5.19) x t + T is calculated. Equa-
tion (3.5.17) gives té, if xgE + AT) is considered to be a new

initial point. Finally

*
Max(tO ) = té + E + AT (3.5.20)

it
where Max(to ) is the maximum value Of minimum times, for

A A P'C
different t, as t changes from O tO t . Thus

Given: x(0) the initial condition, AT the duration of

 

control outage, and tCS > 0, called the recoverability constraint.

 

*
The system is said to be recoverable if Max(to ) S tCS for

 

59

almost all E §W[0,T]. Equation (3.5.2Q) gives a means Of testing

whether or not the system is recoverable.

 

3.6 Calculation Of the Boundary of the TeReachable Set for Systems
of Order n with Distinct Positive Real Eigenvalues and

Scalar Input

Given any system with positive real and distinct eigenvalues,
there exists a real similarity transformation that reduces the

system equation to the following form (Theorem 2.5.7):

 

 

xgt) = P xgt) +-q u(t) (3.6.1)
where
l
O
P = 12/11 (3.6.2)
0 .
L in/xl
and
_ 1 _
12/11
q = . . (3.6.3)
1 xn/XIJ

 

 

First third order systems are considered. For these
systems to reach any point in the state space, at most two

switching times, namely t1 and t2, are required, where

OStlstz ST . (3.6.4)

6O

 

 

ng)
X(ti) _/

 

 

 

 

 

1 Figure (3.5.1): Switching
curves and Optimal-time
trajectories for the system
given by equation (3.1.2),

and initial state XS ) ==[61]

Case c

61

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63

TO calculate the boundary of the T-reachable set with
control outage aRO(T,E,AT),athe sign Of the control component

should change at t1 and t2. For fixed values of E and AT,

as t1 and t2 are ranging from O‘tO T, the six possible cases

and control sequences are as follows (considering u = +1 as'the

initial value of the control component):

a)

E + AT S t S t S T

OStS 1 2

and u(t) is +1, 0, +1, -1, +1 .

b)
O S E S t1 S E + AT S t2 S T
+1, 0, -1, +1 .

c)
0 S t1 S E S E + AT S t2 S T
+1, -1, O, -1, +1 .

d)
O S t1 S E S t2 S E + AT S T
+1, -1.(h +1 .

e)
0 S t1 S t2 S E S E + AT S T
+1, ~1, +1, 0, +1 .

f)

0 S E S t S t S E + AT S T

+1, 0, +1 .

 

64

For case a):

E P(E- ) PE -1
XSE) = I e U qdc = [e - I]P g” u(t) = +1 for t 6 [0,E)
O

eP(E+AT) _ P

x t +' T = e xgE) = [ e AT]P-lq, u(t) E O

for t E [E,E + AT)

 

 

 

P(t -E¥AT) A P(t -E-AT)
_ l P(t+AT) PAT -l -l
X(t1) - e [e - e ]P q_+ [e - I]P q
or
Pt1 P(tl-E) P(tl-E-AT) _
x021) = [e - e + e - I]P g, u(t) = +1
for t E [E + AT,t )
and considering u(t) = -l for t E [t1,t2) yields
Pt P(t -E) P(t -E-AT) P(t -t ) _
x(t2) = [e 2 — e 2 + e 2 - 2e 2 1 + I]P 19
and
A P(T-t ) P(T-t )
P P - - _
XZT: = [e T _ e (T E) + eP(T t AT) _ 2e 1 + 2e 2
- I]P-lq_.
Thus
_« _«_ P(T-t )
aROl(T,E,AT) = {x} §_= :[ePT - eP(T t) + eP<T t AT) - 2e 1
P(T-tz) _1
+ 2e - I]P q, for 0 S E S E + AT S t1 S t2 S T} . (3.6.5)
For case b):
_A P(T-t ) _ _ __
X:T: = [ePT _ eP(T t) +'2e 2 _ eP(T E AT) _ I]P lg,

65

PT P(T-E) P(T-E-AT)
- e - e

5R02(T,E,AT) = {x: a = i [e
P(T-tz) _1
+ 2e - I]P q for 0 S E S t1 S E + AT S t2 S T} . (3.6.6)
For case c):
A _A_ P(T-t )
5R03(T,E,AT) = {5: g = :[ePT + eP(T t) — eP(T t AT) — 2e 1
P(T'tz) _1
+ 2e — I]P q, 0 S t1 S E S E + AT S t2 S T} . (3.6.7)

For ease d):

PT eP(T-t) + eP(T—t-AT)

BR04(T2E:AT) ={ Z )1 = i [e +

|><

P(T-t

)
- 2e 1

_ I]P—13 for 0 S t1 S E S t2 S E + AT S T} . (3.6.8)

For case e):
aROS(T.E,AT) = 5R01(T,E,AT) . (3.6.9)

For ease f

aRO6(T,E,AT) = {5: a = i EePT _ eP(T-t) + eP(T-t-AT) _ I]P—11

for 0 S E S t1 S t2 S E + AT S T} . (3.6.10)

Equation (3.6.10) is a special case of equation (3.6.6)
if t2 = E + AT. Thus the boundary of the T-reachable set for

the third order system is

BRO (T as 2AT) =

"CU"
'—l

aRO.(T,E,AT)
. J

J
For the system Of order n to reach any state in the

state space;at most (n-l) switching times, i.e., t .,t

1,t2,.. n-1

 

66

are required, where

To calculate the boundary of the T-reachable set with

control outage aRO(T,E,AT), the sign of the control component

E + AT S t,

should change at the switching times. Let tj S t s J+1,

then

BRO{(T,E,AT) = {.)£: 2(- = i [ePT + (-1)nI + (-l)j+1eP(T-E)

. n-l . P(T-t.) _
+ (-1)JeP(T’E’AT) + 2 z (-1)le 1 1P 19' when
i=1
tj St St+AT Stj+1 for j =0,1,...,n-1}. (3.6.11)
If
tStht+ATStj+1
then
aRO%(T,E,AT) = {a} §.= :[eP'r + (-1)“1 + (-1)JeP(T’t)
. . n-l , P(T-t,) , P(T-t,)
P -t- T -
+ <-1>Je (T A ) + 2(< z <-1>1e 1) - (-1>Je J )1? 11
i=1
when t s tj S E + AT S tj+l’ for j = 1,...,n-l} . (3.6.12)
Therefore the boundary of the T-reachable set is
n-l j n-l .
BR0(T,E,AT) = ( U aR01(T,E,AT)) U ( U 5R0%(7,E,AT)) - (3.6.13)

3:0 J=1

67

3.7 Calculation of the Boundary of the T-Reachable Set for the

 

System of Order n with Distinct Positive Real Eigenvalues

and Vector Input
Given
igt) = A xgt) + B u(t) (3.7.1)

where u(t) E U'(t), such that

/

III
0

‘ui(t)‘ for t E [E,E + AT]

U'(t) =< u'

. i =1,2,...,r?. (3.7.2)
\u.(t)‘ S k. otherwise
1 1

 

 

K

There exists a real similarity transformation that
diagonalizes A, (Theorem 2.5.7). The resulting system equation

is:

 

 

igc) = P xgt) +-Q u(t) (3.7.3)
where
F1 0 7
xz/xl
P = '. (3.7.4)
0 ° /
>‘n K1
_ J
and
Q = PK
such that:
Fl k k 7
12 °°° 1r
K = 1 k22 ... k2r . (3.7.5)
Ll kn2 ... knr__

 

 

68

In equation (3.7.3) u(t) E U(t), where

Ill

\ui(t)\

U(t) = u(t): i = 1,2,...,r . (3.7.6)
‘ui(t)\ s 1 otherwise

0 for t e [E,E + AT]

For the case of multiple inputs there still exist at
most (n-l) switching times for each component of the control
vector to reach any state in the state space. Assuming the same
switching times for all components of the input vector, and
following the same procedure applied in Section (3.6) the follow-

ing can be obtained:

aR0i<w.E.AT> = {5: xi?) = iiePT + <-1>“1 + <-1>j+1eP(T‘E)

. A n-l . P(T't.)
P -t- T -
+(-1)Je('r A ) + 2 z (-1)1e 1 1? 1Q
i=1
for tj S t S E + AT S tj+1, and j = 0,1,...,n-1} (3.7.7)
and

BROg(T,E,AT) = {5: a = i [ePT + (‘1)nI + (_1)jeP(T-E)

. _ _ n-l . P(T—t.) . P(T‘t.) _
+ <-1>JeP(T E AT) + 2((_z (-1>1e 1 > - (—1>Je J >1P 1Q

1 1

for t s tj s t + AT S t = 1,...,n-1} . (3.7.8)

j+1’ j
The boundary of the T-reachable set for this case is
n-l . n—l j
aR0(T,E,AT) = < u aRoi<w.E,AT>) u < u BR02(T,E,AT)) . (3.7.9)
j=0 j=l

For the case in which different components of u(t) have

different switching times deriving an analytical expression for

 

69

the boundary of the T-reachable set following the procedure
mentioned above is possible, but very cumbersome. Many different

cases must be considered to include all possibilities.

3.8 Calculation of the Boundary of the T-Reachable Set for

 

Systems with Complex Eigenvalues and Scalar Input

Let T* be the time of the nth.switching of the control
component. For all T S T* equation (3.6.13) can be used to
determine the boundary of the T-reachable region. For values of
T, T* < T S 2T*, the following theorem is used:

THEOREM 3.8.1: Given T E [T*,2T*] the boundary of the
T-reachable set and T*—reachable set are related as follows:

a)

* *
if 0 S f S f + AT S T S T S 2T

then

7':
eP(T'T )

* *
5RO(T,E,AT) = aRO(T ,t,AT) + 5R(T-T ) . (3.8.1)

b)
iv A 7'c
O S T S E S t + AT S T S 2T
then
P ( 7': '
_ 7C
5R0(T,E,AT) = e T T )5R(¢ ) + aRO((T-¢*),E,AT). (3.8.2)
C)
if 0 S t S T S t + AT S T S 27
then

P(T-E)

aRO(T,E,AT) = e 5R(t) + 5R(T-t-AT) (3.8.3)

7O

* *
where R(T-T ) indicates the reachable set from origin in (T-T )
seconds considering no control outage.

Proof: For case a) the variation of parameters formula

results in

P(T-o)

* T
ng) = eP(T-T )ng?) +’j e ‘3 u(g)do (3.8.4)
*

T
*
where ng ) can be obtained from equations (3.6.11) or (3.6.12)
A A *
depending upon the location of t, and t + AT in [0,T ]. If
*
T is considered as the initial time and the origin as the

:‘c
initial state, then the boundary of the (T-T ) reachable set

with no control outage is

T
aR(T-T*) = {x} x_= i.[f eP(T-G)q_u(g)do . (3.8.5)
9:
T

Equations (3.8.4) and (3.8.5) prove case a of this theorem.
For case b, since the control vector is present for

A *
t E [0,t], where E > T , then

E .
x_(§)_ = J“ eP(t"’)g U(o)do
0
E A
x§E+AT2 = ePAT xgE) = ePAT I eP(t-G)q_u(o)do
0
A 'T
ng) = e?(T-t-AT) xgE+AT2 + f eP(T_O)q_u(o)do
‘ E+AT
x E A T
= eP(T't) I eP(t-O)g_ U(O)do +J~ eP(T-O)g_ U(O)dc
0 E+AT
7': A
P(T-T") T Pod-0) P(T-t) t P(t-o) '
=e J‘e g_u(o)do'+e [e g_U(O)dO'
0 0
T
+f eP(T-C)g_u(o’)do . (3.8.6)

E+AT

71

*
Since no control outage is assumed for t E [0,T ], then
* *
* T P _
aR(T)={£=a=Ie(T °)
0

g u(o)dc. g e n} . (3.8.7)

* *
Assuming T as initial time,the (T-T ) reachable

set is calculated as follows

P(E-O)

E
*
xgE-I ) = I e g u(o)do
*
T

e g U(o)do

E x
__ PAT j- P(t-0')
T

T
e g_ u(o)do +j‘ eP(T'O)
E+AT

9c
m= g u(o)do. (3.8.8)

P(T-E-AT) PAT ‘ P(t-o)
e e If

*

T

Comparing equations (3.8.6), (3.8.7), and (3.8.8) proves
part b

For case c):

b

X

E x
E = £ eP(t—o)q u(o)do

eP(E-o)g u(o)do

P( eP(T-o)

(D

A ‘T
t 099. U(O’)d0’+lf
E+AT

g U(o)do .

O'—1r‘?>o‘—an>

eP(T'E)

aRO(T,E,AT) = 5R(E) + 3R(T-t-AT) . Q.E.D.

By repeated use of the above argument, Theorem (3.8.1)
k k
can be extended for T E [kT,,(k+1)T>], where k > 1 is a

positive integer.

CHAPTER 4

COMPUTATION OF OPTIMAL CONTROLS FOR THE
SYSTEMS WITH FINITE DURATION OF CONTROL OUTAGE

In Chapter 2 the convexity of the T-reachable set for
linear systems with a finite duration of control outage was
proved. In Chapter 3, the analytical expressions for the T-
reachable set for linear time-invariant systems were calculated.
The convexity of these sets, even when there is a control outage,
leads us to the use of methods which are based upon this char-
acteristic.

Gilbert [22] derived an iterative procedure which
minimizes a quadratic form on a convex set. Barr [23] modified
Gilbert's method to achieve much faster convergence. These
methods require the convexity of the reachable set, which is also
true for linear systems with control outage. In this chapter
Gilbert's method is used to compute optimal controls for linear

systems with a finite duration of control outage.

4.1 Basic Theory

 

In this section the general ideas of the iterative pro-
cedure are introduced [23].

BASIC PROBLEM (BP): Given RO(T,E,AT), the convex reach-
able set for linear systems with control outage in R“, find a

*
point x_ E RO(T,E,AT), such that

72

73

ufuz = min M2 (4.1.1)
i 6 R0(T:E,AT)

*
To find x , the following basic iterative procedure (BIP)

 

is used [23]:

 

Xk+1=x_1£+0’k(s('><_k) '33) k=1.2.--- (4.1.2)
where
ak = 6(33) if 0 S B(ik) S 1
(4.1.3)
(ilk = 1 if 8(3) > 1

Here s(-xk) is called a contact function evaluation for

 

-xk, which is defined as follows. The mapping s(y) from Rn

to the set RO(T,E,AT) is such that y-sgy) = max y-x
g e Ro(w,E,AT).
The expression for B(xk) is:

B(x_k) = Hx—k - “3131125, - (i - “—3” if xk — soils) 34 0

 

 

m
A
N
W
v
I!
C

if xk - s(-xk) = 0

 

(4.1.4)

where xk - (xk - s(—xk)) is the scalar product of the two
vectors. Another function y(xk) is used to measure the

*
"closeness" of xk to x such that:

y(x_k) = MESH-2:13 ' s(-x_k) if H3“ > 0 and Eli-30$) > 0

 

 

Y(EK) = 0 if kaH = 0 or Hi3” > O, and

xk-s(-xk) S 0. (4.1.5)

 

 

74

ABSTRACT PROBLEM (AP), [23]: Given t E [to,T] in R1,

and RO(T,E,AT), a compact, convex set in R“, which moves
continuously with t; find
*
t0 = Min t
t€[to,T], geR0(t,E,AT) . (4.1.6)
*
Note that to need not exist in general. It does exist and is
unique if x§0) lies in the set of null controllability. Hence-
forth, in this chapter x30) is assumed to have this property.

CONTACT FUNCTION: Consider the system defined by

g(£)_= A(t) xgt) + B(t) u(t) (4.1.7)

where u(t) E U(t). The time—varying control constraint set,
is defined by equation (1.3.2). In Section (2.2), the reachable

set for linear systems with control outage was derived as:

R(t) if to S t S E
RO(t,E,AT) = ¢(t,E)R(E) if E < c < E + AT

t
¢(t.E)R(E) + {at Z§t2 = j ¢(t,o)[B(c)ugcz]do
E+AT

if E + AT S t S T (4.1.8)

where

t
R(t) = {5: xgt) = 8(t.co)x(co) + WJJJ" ¢(to,o)FB(o)U(o)]do}- (4.1.9)
— t
0

It is shown in Section 2.2 that RO(t,E,AT) is compact,
convex, and continuous with respect to t, E, and AT; for

almost all t e [tO,T].

75

Given T E [to,T], E E [to,T], AT, the duration of the
control outage, and a vector y E R“, a contact function correspond—
ing to the vector y can be found by considering the following
three cases [24]:

a)

toSTSE

n . .
For given y E R , a contact function 1s:

 

T
sun) = ¢<T,to)x<to) + (magi rmomma«0144,91ch (4.1.10)
0

where
u(t,y) = sgn{B'(t)-n'(t,y)} (4.1.11)

and “(t,y) is the solution of the adjoint differential equation

1’] (t) = —'n(t)A(t), with u(w) = 1' . (4.1.12)
b)
E < T < E + AT
E
8(y_,T) = rb(T.to)X(tO) +f ¢(T,o)[B(o)u(o,x)]do . (4-1-13)
__ to _
c)

E+ATSTST

E
8(X9T) = ¢(T,t0)x(t0) +J‘ @(T,G)[B(U)U(C:D]dc

t
0

 

T
+J" Q)(T.0)[B(o)ll(o,x)]do . (4.1.14)
E+AT

For time-invariant linear systems, a contact function

for a given y E Rn is:

76

eAT{x(O) + e-AOB u(o,y)dc} for O S T S E

—A

S(X.T) = eAT{x§0) + e 0B u(o,1)do} for E < T < E + AT

oc—wrfi o Lwr'fio C—w-‘c

T
eAT{x(0) + e-AOB u(o,y)dg + f e—AOB u(o,y)do}

t+AT

for E + AT S T S T. (4.1.15)

*
4.2 Computer Evaluation of Contact Function and of t0

In equation (4.1.15), the expression for the control
u(t,y) is u(t,y) = sgn{B'(t)-n'(t,y)}. Moreover, for time-
invariant linear systems, the solution of the adjoint differential

equation with final value fl'gT) = y = -xk, is
-A' _
n'(c,1) = e (t “1 for t 6 [0,1’] . (4.2.1)

In hybrid computation the solutions of the state and
adjoint equations can be implemented on an analog computer. If
only a digital computer is used, the solution of the adjoint
differential equation for time—invariant linear system is:

r -l
_ 1 A (T-t)
“'(tal) =[Zil+Z{2(T-t) +...+Zirl—%:)_1zje 1 +

r -1
2 -t
[z' +z' ("r-t) + +z' T‘t ] eizh )+
21 22 2r2 (r2-1)! X

' r -1
° m A (T‘C)
. . _ . 11L_ m
[Zml + Zm2(T t) +. . .+ zmrm (rail)! 11 e (4.2.2)

where Zij are constituent matrices of the matrix A, and

77

11,k2,...,xm are the distinct eigenvalues of A. The scalar
coefficients of the characteristic equation for the square matrix
A can be determined by the Frame method [25, page 212].

The constituent matrices can be calculated from the

following equation:

 

 

 

 

11 U
A
Z12
- = (VT)‘1 A2 (4.2.3)
Z21 :
z A An_1
mr
L... m... _
where
(VTYI = (G'1)VW . (4.2.4)

Entries of both matrices VW and G can be calculated
by using the synthetic division algorithm for the characteristic

polynomial [26]. Reduction of [G, VW] matrix to the form
[U, (G—1)VW] by elementary row operations produces (VT)-1.
After calculation of Zij

gives the forward-time solution of the adjoint differential

, k19'009km, equation (4.2.2)

equation with final value fl'gwg = y. The control which generates

s(y) is

 

u(t,y) = sgn[B'°fi'(t,y)] . (4.2.5)

Substitution of equation (4.2.5) into equation (4.1.15)

yields a contact function evaluation. The ability to make this

78

evaluation is fundamental to the computing procedure for solving
(BP) and (AP).

In Chapter 3, the minimum time to reach the origin for
some linear time-invariant systems with control outage was
calculated. Derivation of an analytical expression for to*
for an arbitrary linear system is impossible. The iterative
procedure described in this chapter is capable of solving any time-
optimal regulator problem with a convex reachable set. The
normality condition is not necessary; thus RO(t,E,AT) is not
required to be strictly convex.

ITERATIVE PROCEDURE FOR (AP). Suppose a linear system
defined by equation (4.1.7) and an initial condition xLQ) is
given. To find the optimal tnne t0*, first consider a given
value of time, i.e., T E [to,T] where T is chosen to be less
than t0*. For example T can be chosen to be 6t, where 6t
is a small preassigned time increment. The basic iterative

procedure (BIP) is used to find an approximation to the point

.1
t

X. e Roofimtr).

Given xk, the use of equation (4.1.2) in (BIP) to

compute xk+1 requires the value of s(-xk), of a contact

function corresponding to -xk. This value is calculated from

equation (4.1.15). For initialization of xk’ let

-x1 = y = -x(0), where x50) is the given initial state. Using

equations (4.1.2) and (4.1.5) the values of xk+1 and v(xk)
are determined. If ka+1H S e (e a preassigned positive

*
value), the program is terminated and to is viewed as equal

to 'T.

79

If ka+lu > 6, then Y<EK) is checked. If Y(:E) < e,
(0 < e < 1 is a preassigned number) with new values of
x_lg=xk+1

For values of y(xk) 2 e, the time T, is increased by an

, and y = -xk+1 another iteration of (BP) is repeated.

. . ' 2 ' =
1ncrement of 6t The support function h [ 3] With :5 xk+1

is constructed:
h(-X_k,'r) = -X_k-S(-X_k,T) -

For fixed value of x the value of the support function

R,
h(-xk,T), is calculated for increasing T in increments of 6t.
This is continued until for some value of T, the support func-

tion h(-xk,T) 2 0 . For this value of T, the basic iterative

procedure (BIP) is used again to determine an approximation to

*

_ E RO(T,E,AT), considering xk = s(—xk,T), and y = -s(-xk,T).
The summary of the iterative procedure is shown in

Figure (4.2.1). Several second—order, time-invariant systems
without control outage were solved successfully with the above
method. Tables (4.2.1), (4.2.2), (4.2.3), and (4.2.4) show the
highlight of the runs for the specified problems.

The system defined by x... = u(t) was tried with initial
condition [2,0,0], on the Control Data 6500 (Table (4.2.4)).
The number of iterations for (BP) is 214, and for finding the zero
crossing for h, the support function, is 17. The total time is
200.6 seconds and the total computing cost is $10.38.

To improve the convergence rate, more than one contact

point at each iteration can be used. Such an improved iterative

procedure (IIP) is discussed by Barr [23].

80

 

 

Given A, B matrices, x50), AT, 9, e,
E |

6t and

 

 
  
    
 

  
     

  

Cal sub conta

Find s(-x],TT)

    

Call subroutine x
compute x i , y(x

 

 

 

 

Terminate the
program
Let t0 = TT

 

 

 

 

   

   
     

Call Sub conta
Find s(-x ,TT)

 

 

*
Figure (4.2.1): Flow Chart for Finding tO

, using BIP and
Support Function.

81

 

 

 

 

Figure (4.2.2): BIP, for the
double integral plant when

T = 1.9 second, and E = AT = 0.

82

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

BASIC ITERATIVE PROCEDURE (BIP), TT = .1 sec.
k xk S(-_X_1_<_, -1) Vol) MIR? x1”
1 0 , 1 -.0951, .6374 .637 l -.095,.637
1
I E: TT H=-x*.s
1 -.95, .637 .15 -.32
4 -.95, .637 .3 -.0866
5 -.95, .637 .35 -.0088
BASIC ITERATIVE PROCEDURE (BIP), TT = .35 sec.
k S('x , -35) Y(X ) U(X ) X
51:. l J; .1; ____k'*'1
2 -.95, .637 -.296, -.0088 054 .857 -.267,.083
4 -.23, .14 -.296, -.008 .915 .238 -.248,.104
*
I _)_<_ TT H
15 -.037, .028 .85 —.00063
7':
BASIC ITERATIVE PROCEDURE (BIP), TT = 85 = t sec
k :E s(-xk, .85) Y<i§) Q(fh) xk+1
27 -.o37, .028 -.226, -.283 .284 .012 -.o4o,.024
28 -.O40, .024 -.0038, .0199 .289 1 -.0038,.Ol9

 

 

 

Table (4.2.1):

-1 0
A = 0 _2 , b

with e = .5, 6

Highlights of the runs, using (BIP) for the
system given by

4:21.44:

J, y = -x§O), and x1

= HEEHZ = .001, TT = TT + 1 * DELT, 6t

m

DELT

.05

 

83

 

BASIC ITERATIVE PROCEDURE (BIP), TT = 1.8 sec.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

k xk S(-EE’ 1'8) Y(Xk) O’(xk) Xk+1
1 4 , 0 .204, -2.339 .051 .763 1.101,-.786
2 1.101, -.786 .783, -1.307 .726 1 .783,—1.307
7':
I g TT H
1 .783, -1.307 1.85 -2.166
10 .783, -1.307 2.3 -.765
18 .783, -1.307 2.7 -1.498
21 .783, -1.307 2.85 .070

 

 

 

 

 

 

 

7':
BASIC ITERATIVE PROCEDURE (BIP), TT = 2.85 sec. = t

 

 

 

 

 

 

k >3 S(-X_k. 2.85) Y(X_k) oz(x—k) xk+1

3 .783, -1.307 -1.491, -.840 o .443 -.225, -1.1
4 —.225, -1.100 -.0246, .082 o .9355 —.o37,.0064
5 —.037, .0064 .734, - 537 0 .0364 .009,-.01

 

 

 

 

 

 

 

Table (4.2.2):

A:

(D
|

Highlights of the runs to find t
for the system given by

[91.1212] 441%]

- .5, e = \\§*H2 = .001, at

k
, using (BIP),

= .05 sec.

84

BASIC ITERATIVE PROCEDURE (BIP), TT = 1 sec.

8( .-1) y ) X)

BASIC ITERATIVE PROCEDURE (BIP), TT = 1.7 sec.
l 80:2. 17) y(X_k)

BASIC ITERATIVE PROCEDURE (BIP) TT = 1.95 sec.
s( , 1.95) ) ( )

BASIC ITERATIVE PROCEDURE (BIP), TT = 1.999 sec =

)

 

7
Table (4.2.3): (BIP) applied to the system given by

A = A3 I], B = [2] , éigl = [E0] ’ 0 = '5’ e = '001: 6t = .05 sec.

 

85

BASIC ITERATIVE PROCEDURE (BIP TT = .85 sec.
( , .85) Y(:E) Q(EE)

.0028 -.36 -

71'
X

O88 -.O26

BASIC ITERATIVE PROCEDURE (BIP), TT = .9 sec.
ES S(-:E’ -9) Y(:E) a(:£)
.088 -.026 . -.022 -.4 -.9 .302

*
BASIC ITERATIVE PROCEDURE (BIP), TT = t
s(-xk, 1.4) Y ) d(

_ ... _

97 . . -.05 —.

 

Table (4.2.4): BIP applied to the system given by

010 o 1
A=001 ,_b_=0 ,x(0)= 0 ,e=-5,
000 1 o

e = .001, At = .05

 

86

4.3 Numerical Results for BIP Applied to a Minimum-Time Example
with a Control Outage

The system to be considered is the "double-integral” plant.

The system equation is:

0
x1 0 1 x1 0

0 = + u(t) (4.3.1)
x2 0 0 x2 1

where u(t) E U(t), which U(t) is defined by equation (1.3.2).
The initial, and final values of the system are
—l O

*
x(0) = , x to = . (4.3.2)
0 o

The solution of this problem is known. Hauer and Hsu [5]

have shown that

* L
tO (E,AT) =AT+2[1 -EAT]2 if OSE S1,Q S0 (4.3.3)

1
t0*(E,AT) = 2E + AT + 2[E2 + EAT — 132 if 0 S E S 1, Q > 0 (4.3.4)

 

t0*(“t.AT) = 2 + AT + qua-E)? if 1 < E s 2, Q > 0 (4.3.5)
where
Q = Q(x(E),AT) = -1 if x2(E) s 0 (4.3.6)
2
x2( ) . .
Q = 2 + X1(t) + x2(t)'AT if X2(E) > 0 (4.3.7)
and
,2
2— ~ 1

xgt) = if E 6 [0,1) (4.3.8)

('7‘)

 

87

_ 2-E 2
2
x(E) if E 6 [1,2] . (4.3.9)
2-E

It is easy to show that in equations (4.3.3) and (4.3.5),
to* is monotonically decreasing with respect to E, but in
equation (4.3.4) to* is monotonicallyincreasing with respect
to E. Figure (4.3.1) represents the variation of t0*
(minimum-time with control outage) with respect to the control
outage starting time E, Such that 0 S E S t*. Here t* is
the minimum time for this system, with the same initial, and
final state, if there is no control outage during the process.

In Figure (4.3.1), the dotted line indicates the
theoretical value of tO* with respect to changes in E, which
follows from equations (4.3.3), (4.3.4), and (4.3.5). The solid
line represents the computed value of t0* using Gilbert's
method as described in Sections (4.1) and (4.2). The control
outage duration AT is assumed to be .2 seconds. The minimum—
time tO* is computed for steps of .1 seconds in E. Figure
(4.3.1) also indicates that the computed value is very close to
theoretical one. The speed of convergence for BIP is slow and
could be improved considerably by using additional contact points

as suggested by Barr [24].

 

88

t0

_._._._._. Theoret ical value

_____________Computed value 1‘
using (BIP) / ’\m\

 

 

 

- ‘4: t
.5 1 1.5
Figure (4.3.1): Theoretical and Computed value of Minimum time
to , for different values of E, when AT = .2 sec.
The system is a double integral plant with
m = g .

 

89

4.4 Application of Gilbert's Technigue to Optimal Control

Problems with Control Outage

The iterative procedures for time—optimal control which
are described in Sections 4.1, 4.2, and 4.3 can be applied to
many other optimal control problems in which there exists a finite
duration of control outage during the process. Consider the system

defined by

x(t) = A(t) x(t) + B(t) u(t) on t E [to,T] (4.4.1)

where pip) E U(t). Here U(t) is a time-varying compact con-
trol set which is defined by equation (1.3.2). The matrix func-
tions A(t) and B(t) are continuous with respect to time. The
continuity of these matrices insure that equation (4.4.1) has a
unique solution x(t,p) for each admissible p(£) and initial

condition x(t ) = x , where t S t S T.
o _p o

 

The family of closed sets W(t) CZEn which are defined
for every t E [tO,T] are called the target sets. It is assumed
that W(t) is convex and continuous in t. In many applications
W(t) either consists of a single point w(t) for each t or
is a k—dimensional linear subspace with 1 S k S n-l.

A cost functional of the following form is considered:

T
JOT(p) = g1(xu(T)) +j [a'(Q) x(o) + g2(g_(g),o)]do (4.4.2)
— t

O

 

where g1(xu(T)) is a given continuous, convex function from

n

R to RI for almost all T E [to,T], a'(o) is a continuous row

vector on [tO,T] and g2(p,o) is a continuous function from

m 1
R x [tO,T] to R .

90

An admissible control Q(gl defined for t E [to, T],
where T E (t°,T], is said to transfer the system state from
x(to) = x0 to W(T) in time T if x(T,§) E W(T). An optimal
control problem will now be described for the two following cases:
Case a):
Let T > to be a fixed point in (to,T]. The problem
is to find an admissible control p:(£), (it is assumed that there

exists one), which transfers the system state from x(to) to

 

W(T) and
JoT(gf) S min (JO (2)).
u(t)€U(t) T

The above problem is called the fixed terminal time optimal
control problem.

Case b):

In the free terminal time optimal control problem, the
objective is to find an admissible control pigp) and an optimal
time t0* 6 [to,T] such that p:(£) transfers the system state

* *
from x(to) to W(tO ) in time to and

 

JO *(23) S min (JO (3)), t e [t ,T].
to u(t)EU(t) t O

Barr and Gilbert [23] show that by extending the abstract
problem AP an entire class of optimal control problems including
minimum fuel, minimum effort, and minimum error rendezvous can be
solved by sequentially applying BIP. The minimum time problem with
moving target set can also be treated. The essential features
required are convexity the ability to compute a contact function.

Theorem (2.2.2) shows that the reachable set for the linear

91

system with control outage is compact, convex, and continuous in
t, E, and AT. A contact function can be evaluated using equa—
tion (2.2.10). Thus optimal computations may be performed for

these broad classes of problems when there is control outage.

CHAPTER V

SUMMARY AND CONCLUSIONS

The problem of computing optimal controls for a class
of linear systems with amplitude bounded inputs and finite
duration of control outage has been considered in this work. It
is shown that in the event of control outage the control constraint
set is time-varying and piecewise continuous with respect to time.
The structure of this thesis is based upon Theorem (2.2.2) which
guarantees the compactness, convexity, and continuity of the
reachable set for linear systems in the case of outage.

It is not possible to calculate a general expression for
the boundary of the T-reachable set for linear systems with con-
trol outage. Analytical expressions for the boundary of these
sets are obtained for linear time-invariant systems with positive
distinct real eigenvalues or complex eigenvalues. These expressions
are restricted to the case where the optimal control for the cases
with and without outage have the same number of switchings (the
change in control from i l to 0 or from 0 to i 1 is not con-
sidered a switching).

The minimum regulation times for linear time-invariant
systems with a finite duration of control outage are investigated.
Specifically, an expression for the minimum regulation time for

a second-order, time-invariant system with negative distinct real

92

93

eigenvalues is derived. The variations of this minimum time with
respect to initial state, control outage starting time, and dura-
tion of outage are shown. The recoverability of the system is
investigated. Equation (3.5.20) determines for which value of
control outage starting time the system is recoverable.

There are many computational methods for computing optimal
controls for linear optimal control problems based on convexity
of the reachable set of system states. Theorem (2.2.2) makes
possible the use of Gilbert's technique which utilizes the con-
vexity of reachable set to compute optimal controls for linear
systems with control Outage. A modified contact function (for
the case of outage) is given by equation (4.1.15). Several
examples for the case with no control outage are solved successfully
with the above techniques.

The minimum regulation time for a double integral plant
is computed using Gilbert's method and considering different
values of control outage starting time. For higher-order systems
and faster convergence rate, Barr's technique [23] can be used.

There are a number of extensions which can be considered
for linear systems with a finite duration of control outage.

1. Linear time-optimal control problems with control
outage, bounded amplitude control, and rate saturation.

II. Extension of Chapter 4 to other linear optimal con-
trol problems, i.e., minimum-error regulation, minimum—fuel,
minimum effort, etc.

III. Application of other computational techniques based

on the convexity of the reachable set, and comparison of the

 

94

convergence rates, computing times, etc.

IV. Multiple control outages.

V. Consideration of linear systems with a disturbance
in input vector, where this disturbance can be a known input or

an unknown realization of a random process.

BIBLIOGRAPHY

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