CONTROLLABILITY AND REACHABILITY OF LINEAR DIDCRETE-CONTINUOUS SYSTEMS

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7
-
85

CONTROLLABILITY AND
REACHABILITY

OF

LINEAR
DIDCRETE
-
CONTINUOUS

SYSTEMS


Prof. Grigory Agranovich


Ariel University Center of
Samaria,
Dept. of Electrical and Electronic Engineering
,
Ariel, Israel


ABSTRACT.

The linear discrete
-
continuous system
s,

are studi
ed in this work,
contain

the two coupled subsystems: subsystem with continuous
-
time dynamics and
subsystem with discrete
-
time dynamics. Continuous dynamics
is

described by
ordinary linear differential equations, and a discrete one is described by differenc
e
equations for system's state jumps in prescribed time moments. For this class of
models with time
-
varying, periodical and constant parameters
the reachability and the
controllability properties are investigated. As well controls are found

for such system
s
state’s transition.
Those solutions were applied for dynamical systems control design.


INTRODUCTION



Discrete
-
continuous
systems
(DCS) are those that combine both discrete and
continuous dynamics
.

Many examples of DCS can be found in manufacturin
g
systems, intelligent vehicle systems
,

robots etc. All computer
-
controlled systems are
included in this wide class of

dynamics
.

Modern complex engineering and other
automatic control systems have hierarchical architecture. Common practice is to use
digita
l stabilizing local controllers also at the lower levels. Therefore, the controlled
system for an upper level control has discrete
-
continuous nature.


At the

same

time
hybrid systems constitute a wider class in which a controller has essentially nonlinear
control

logic
.

Because of great practical importance of hybrid, and specifically
discrete
-
continuous, systems they have been the subject of intensive research for
many years. An important characteristic of these systems from a control point of view
is that

they are simultaneously driven at each time instant by a continuous
-
time
control signal plus by a discrete
-
time control signal at each preceding sampling time
instants. The continuous
-
time control and the discrete
-
time control have not only a
different ph
ysical nature, but they have also a different dynamical meaning.

Indeed a
continuous
-
time control excites the system’s state time variations by means of
velocity change, however, a discrete
-
time control results in an instantaneous system’s
state changes (“
jumps”). In this way, there are two different control signals acting on
the plant at the same time and the two kinds of a subsystem dynamics. The
mathematical models of such systems are well known in literature [
1
], [2], [5], [8],
[10], [11] etc.



Great effort was concentrated on optimal control and filtering problems' solution,
such as LQG [1], [2], [3], [10]
2
H
and

H

[2], [3], [10]. Various stabilization

techniques were developed for hybrid systems [9]
, [10], [11] etc
.



At the same time, problems of reachability and controllability analysis of
linear discrete
-
continuous systems (LDSC), which are well
-
known for continuous
-
time and for discrete
-
time models and are effective design tools of a pract
ical
engineer, still remain unsolved in full.

This paper is devoted to comparative analysis
of reachability and controllability properties of time
-
variable LDSC, time
-
periodical
LDSC, and time
-
constant LDSC. The reachability property of the system means an

existence of a control signal, which transposes the system from zero initial state to

7
-
86

any designed final state. The controllability of the system means an existence of a
control signal, which transposes the system from any initial state to final zero stat
e.
For continuous
-
time nonsingular linear systems these properties are coinciding. But
for

both discrete
-
time and

discrete
-
continuous linear systems, which transition matrix
is singular as a rule, this is not the case. In this work for such systems the Kal
man
-
like
reachability and controllability criteria are developed.


STATE MODEL AND TRANSITION MATRIX OF LDCS



Consider the set


of prescribed discrete time instants







0
h
t
t
:
,
t
,
t
,
t
k
1
k
3
2
1








,


(1)


and time functions


t
),
t
(
k
, defined on this set as









)
t
(
k
k
k
t
t
t
,
t
t
,
Q
t
:
k
max
)
t
(
k






,



(2)


which specify the operating time of a discrete
-
time subsystem.


I
f in equation (1) the interval
h

between successive time instants is constant, then the
equation (1)
and functions (2)
can be
represented

as:











h
h
t
t
t
,
h
t
)
t
(
k
,
k
:
kh
t
k










.

(3)


Equations (2) ar
e useful for describing an interaction between continuous
-
time and
discrete
-
time subsystems of discrete
-
continuous system. Consider linear model of
DCS in the form of a system with jumps ([1], [2], [3], [5], [6], [8], [10]).



DEFINITION 1. The
)
(

C

denotes a space of real
-
valued bounded functions,
continuous on time segments
)
t
,
t
[
1
k
k

, and right
-
continuous in


k
t
.



Equations of a state vector evolution of the linear discrete
-
continuous system
(LD
CS) are




,
t
),
t
(
u
B
)
o
t
(
X
A
)
t
(
X
:
t
t
t
),
t
(
u
B
)
t
(
X
A
)
t
(
X
k
k
d
d
k
d
k
1
k
k
c
c
c













(4)


where


-

)
(
)
(


n
C
t
X

is the state vector;

-

)
(
)
(


C
m
c
C
t
u
, and
d
m
k
d
R
t
u

)
(

are continuous
-
time and discrete
-
time control
excitations;


-

,
A
,
B
,
A
d
c
c

and
d
B

are the coefficient matrices of compatible dimensions,
elements of continuous
-
time subsystem matrices
c
c
B
,
A

belong to
)
(

C
.


We shall consider LDCS of the fo
llowing three degrees of generality, depending
on the structure of the discrete
-
time instants set


and time
-
dependence of model (4)

coefficient matrices:


DEFINITION 2.
(Time
-
variable, time
-
periodical and time
-
invariant LDCS)


7
-
87

2.1.
A

system (4)

is said to be

time
-
varying

L
DCS

(LTVDCS or in brief LDCS) if



at least on of the assertions holds:

(i)

its discrete
-
time instants set

is defined by (1),

(ii)

at least one element of its continuous
-
time part coefficient matric
es
c
c
B
,
A

is
a time
-
varying function of class
)
(

C
, and the rest are constants,

(iii)

at least one element of its discrete
-
time part coefficient matrices
d
d
B
,
A

is a
time
-
varying function, and the rest are con
stants.

2.2. A system is said to be

h
-
periodical LDCS

(LTPDCS) if the following
assertions



hold

(i)


its discrete
-
time instants set


is defined by (3),

(ii)


at least one element of its continuous
-
time part coefficient matrices
c
c
B
,
A

is
h
-
periodical function of class
)
(

C
, and the rest are constants,

(iii)


all elements of its discrete
-
time part coefficient matrices
d
d
B
,
A

are
constants .


2.3. A system
is said to b
e

time
-
invariant
LDCS

(LTIDCS) if the following



assertions are hold

(i)


its discrete
-
time instants set


is defined by (3),

(ii)

all elements of its coefficient matrices are constants.



The state vector of LDCS (4) can be represented as follow
s ([1], [2],
[5]
,
[
6]
,
[
10]
):













t
t
t
k
t
k
q
q
d
q
d
q
c
c
t
u
t
B
t
t
d
u
B
t
X
t
t
t
X
0
0
)
(
1
)
(
0
0
)
(
)
(
)
,
(
)
(
)
(
)
,
(
)
,
(
)
(





(
5
)


where




t
t
),
,
(

is the transition matrix of LDCS, which satisfies the following
equations


















t
t
t
t
t
A
t
t
t
k
c
t
k
k
q
q
q
c
q
d
t
k
c
),
,
(
)
,
(
)
(
)
,
(
)
,
(
)
(
)
(
1
)
(
1
)
(
.

(
6
)


)
,
(

t
c


in (
6
) is a
transition matrix

of the continuous
-
time part of LDCS (4). The
direct corollary of (
6
) is the semigroup property



f
0
0
f
0
f
t
t
t
),
t
,
t
(
)
t
,
t
(
)
t
,
t
(










(
7
)

of the LDCS.



The following expression is an equivalent representation of the expression (
6
)






















t
t
t
A
t
t
t
t
t
k
c
q
d
t
k
k
q
q
q
c
t
k
c
),
,
(
)
(
)
,
(
)
,
(
)
,
(
1
)
(
)
(
1
)
(
1
1
)
(
.

(
8
)



It should be remind that for continuous
-
time system
)
t
,
(
)
,
t
(
1
c
c






, but this
identity is not valid for LDC
S with singular discrete
-
time part coefficient
matrix
)
(
k
d
t
A
.


7
-
88


For
h
-
periodic LTPDCS a transition matrix representations (
6
), (
8
) take the
following two equivalent forms

















t
),
t
,
(
)
h
(
)
t
,
t
(
)
,
t
(
)
(
k
1
c
)
(
k
)
t
(
k
dc
)
t
(
k
c

)
0
,
(
)
(
h
A
h
c
d
dc



,


(
9
)

and

















t
),
,
t
(
)
h
(
)
t
,
t
(
)
,
t
(
1
)
(
k
c
)
(
k
)
t
(
k
cd
1
)
t
(
k
1
c
,
d
c
cd
A
h
h
)
0
,
(
)
(



.

(1
0
)


For LTIDCS a transition matrix of the continuous
-
time part can be expressed by the
matrix exponent
t
A
c
c
e
t


)
(
, and therefore


















t
e
h
e
t
c
c
A
k
t
k
dc
t
A
,
)
(
)
,
(
}
{
)
(
)
(
}
{
,
h
A
d
dc
c
e
A
h


)
(
,

(1
1
)

and
















t
e
h
e
t
h
A
k
t
k
cd
h
t
A
c
c
,
)
(
)
,
(
})
{
(
)
(
)
(
)
}
({
,
d
h
A
cd
A
e
h
c


)
(
.

(1
2
)


It is essential that the transition matrix of LTIDCS, equations (1
1
) and (1
2
), be a
function of two arguments,
t

and

, as long as the system (4) with constant
coefficients remains to be periodic by its nature (i.e. time
-
variant).




The main part,
)
(
k
)
t
(
k
dc




(or
)
(
)
(

k
t
k
cd


) of the transition matrices representation
s
(
9
)


(1
2
) of
LTPDCS

and LTIDCS is the function of the difference

)
(
k
)
t
(
k


, the
rest multipliers are the
h
-
periodic nonsingular matrices. This is crucial property of the
LTPDCS

(LTIDCS) transition matrix, which, for example, yields immedia
tely the
following necessary and sufficient condition of system stability.


CRITERION 1.
(Stability of a LTPDCS)
A
LTPDCS

)
4) is exponentially stable if
and only if the modules of all eigenvalues of its
dc

component (or, equivalently,
cd

), are less than one.



It makes sense to define eigenvalues of
dc


as spectrum of the operator
described by equation (4). It should also be recalled that matrix
cd


(1
1
) is similar
to
matrix
dc


(1
0
) and therefore has the same eigenvalues. The singularity (or non
-
singularity) of a linear system's transition matrix is of primary importance for its
analysis. It follows from (
9
), (1
0
) that the transition matrix of LTPD
CS is singular if
and only if discrete
-
time subsystem dynamical matrix
d
A

is singular. It should be
noted that, as a rule, the transition matrix of a LTPDCS controlled by a built
-
in
computer is singular. This is due to a data
-
processin
g time delay that occurs as a
time shift between the input and output signals of the discrete
-
time subsystem. This
time delay coincides with zero eigenvalue of matrix
d
A
, which implies singularity of
dc


.



REACHA
BILITY AND CONTROLLABILITY OF LDCS



The majority of definitions of the notions reachability (and controllability) can be
d
ivided on the two following types. The first type determines
reachability

as property
of a system in preset instant of time

f
t
, the second
-

determines
reachability

as
property of a system on preset time interval
]
t
,
t
[
f
0
. The
reachability

of a system in
time instant is important condition for solution of the control, the optimization

7
-
89

problems.
The second type of definitions is more convenient for derivation of
reachability and controllability

criteria. In the case of time
-
invariant continuous
-
time
systems the technique for those two approaches are coincide. Since LDCS, even with
constant coeffic
ients, do not keep properties of time
-
invariant continuous
-
time
system, we will consider the both types of the definitions.



D
EFINITION

3
.
(Reachability and Controllability on a time interval)


3.1.
A LDCS (4) is said to be
reachable on the time inter
val
]
t
,
t
[
f
0

if for any
state
f
f
X
)
t
(
X


exists admissible control
d
c
u
,
u

on time interval
]
t
,
t
[
f
0
which
converts zero initial state
0
)
t
(
X
0


to the state
f
X
.


3.2.

A LDCS is said to be
controllable on the time interval
]
t
,
t
[
f
0

if for any
state
0
0
X
)
t
(
X


exists admissible control
d
c
u
,
u

on time interval
]
t
,
t
[
f
0
which
converts zero initial state
0
X

to the zero final state
0
)
t
(
X
f

.



D
EFINITION

4
.
(Reachability and Controllability at a time moment)


4.1. A LDCS (4) is said to be
reachable at time
f
t

if for any state
f
f
X
)
t
(
X


exi
sts time instant
)
t
t
(
t
f
0
0




and admissible control
d
c
u
,
u

on time interval
]
t
,
t
[
f
0

which converts zero initial state
0
)
t
(
X
0


to the state
f
X
.

4.2. A LDCS is said to be
con
trollable at time
0
t

if for any initial state
0
0
X
)
t
(
X


exists time instant
)
t
t
(
t
f
0
f




and admissible control
d
c
u
,
u

on time interval
]
t
,
t
[
f
0

which converts the initial state
0
X

to the zero final state
0
)
t
(
X
f

.


D
EFINITION

5
.
(Complete Reachability and Controllability)

5.1.


A
L
DCS (4) is said to be
completely reachable

if it is
reach
able
at

any time,
including limiting time instants



k
k
t
,
0
t
.

5.2.


A
L
DCS is said to be
completely
controll
able

if it is
reach
able
at

any time,
including limiting time instants



k
k
t
,
0
t
.



From (3) it follows, that the space of states of the LDCS, reachable on
]
t
,
t
[
f
0

is
represented by











f
0
f
0
t
t
)
t
(
k
1
)
t
(
k
q
q
d
q
d
q
c
c
f
0
)
t
(
u
)
t
(
B
)
t
,
t
(
d
)
(
u
)
(
B
)
,
t
(
)
t
,
t
(







.


(13)


where
d
c
u
,
u

-

admissible continuous and discrete controls.


Then for reachability of LDCS on
]
t
,
t
[
f
0

it is necessary and sufficient tha
t

n
f
0
R
)
t
,
t
(


. It can be proven that



)
t
,
t
(
dim
)
t
,
t
(
dim
f
0
f
0





for all
0
0
t
t



(14)

and


)
t
,
t
(
dim
)
t
,
t
(
dim
f
0
f
0





for
h
kh
t
t
kh
f
f






. (15)


By analogy with continuous
-
time systems [13] it is shown in [7], [11] that
reachability space
)
t
,
t
(
f
0


and range space of the symmetric positively semidefinite
reachability matrix


7
-
90








)
t
(
k
1
)
t
(
k
q
q
T
q
T
d
q
d
q
t
t
T
c
T
c
c
f
0
f
0
f
0
)
t
,
t
(
)
t
(
B
)
t
(
B
)
t
,
t
(
d
)
,
t
(
)
(
B
B
)
(
B
)
,
t
(
)
t
,
t
(
R










(16)

are coincide. Therefore LTIIS is reachable on the segment
]
t
,
t
[
f
0

if and only if
reachability matrix (
16
) is nonsingular.

It follows from state vector LDCS
representation (5) and (13) that system’s re
achability on
]
t
,
t
[
f
0

imply its
controllability on this time segment. B
ut

the inverse proposition is no always the true.
It is true for nonsingular transition matrix
)
t
,
t
(
0
f

.


REACHABILITY
CRITERIA OF LDCS


CRITERION 2.

(Rea
chability on a time interval)
.


A LTDCS (4) is
reach
able on the
time
interval

]
t
,
t
[
f
0

if and only if the
r
eachability matrix
)
t
,
t
(
R
f
0
(16)
is
nonsingular.


The following corollaries are useful for verification of system’s

reachability in a
prescribed

time moment.


COROLLARY 2.1. If LDCS is

reach
able

on the time interval
]
t
,
t
[
f
0
,

then it is
reach
able

on any time interval
]
t
,
t
[
f
0

, where
0
0
t
t


.

COROLLARY 2.2. If LDCS is rea
chable in the time moment
f
t

and for
f
f
t
t



the transition matrix
)
t
,
t
(
f
f



is nonsingular, then the system remains
reachable in the time moment
f
t

. If
f
t

belongs to continuity interval
1
k
f
k
t
t
t




, then the system is reachable in any time moment
f
t

, which belong to
the time segment
1
k
f
f
t
t
t




.



For h
-
periodical LTPDCS, which ha
ve the following additional properties



)
h
t
,
h
t
(
R
)
t
,
t
(
R
),
h
,
h
t
(
)
,
t
(
0
f
0
f











(17)


the standard time interval for reachability test is defined by the next criterion.


CRITERION 3. (Reachability of h
-
periodical LDCS in a time
moment). A LTPDCS
is reachable in time moment
f
t

if and only if it is reachable on the time interval
]
t
,
nh
t
[
f
f

.




For h
-
periodical and, as a special case, for LTIDCS the following assertion is
valid.


ASSERTI
ON 1. A LTPDCS (LTIDCS) is completely reachable, if and only if it is
reachable in arbitrary

time moment


k
t
. If in addition, the matrix
d
A

is
nonsingular, then for complete reachability it is necessary and suffici
ent the system
reachability in arbitrary time moment
t
.


Using this assertion, the following Kalman
-

like criterion is obtained.


CRITERION 4.

(
C
omplete reachability of LTIDCS) A LTIDCS is completely

7
-
91

reachable completely reacha
ble if and only if
one of the two following equivalent
assertions is fulfilled:

1)





1
n
0
l
l
T
dc
T
dc
dc
l
dc
dc
))
h
(
(
B
B
))
h
(
(
R



(18)


where




d
c
1
n
c
d
c
2
c
d
c
c
d
c
d
dc
B
B
A
A
B
A
A
B
A
A
B
A
B





(19)



is nonsingular;

2) Rank of the matrix




dc
1
n
dc
dc
2
dc
dc
dc
dc
dc
B
)
h
(
B
)
h
(
B
)
h
(
B
R







(20)



is equal to
n
.


CONTROLLABILITY CRITERIA OF LDCS



By analogy to reachability, controllabilit
y properties of LDCS can be investigate
of base of linear subspace
)
t
,
t
(
f
0
C

of of system’s states
)
t
(
X
0
, controllable on a
time interval
]
t
,
t
[
f
0
. This subspace
)
t
,
t
(
f
0
C


of a LDCS state space
n
R

is defined
as the solution of the linear set
-
theoretical equation



)
t
,
t
(
range
)
t
,
t
(
)
t
,
t
(
)
t
,
t
(
0
f
f
0
f
0
C
0
f






. (21)



Using (21), the following criterion is proved.


CRITERION 5.

(Controllability o
n a time interval).

A LTDCS (4) is controllable on
the time interval
]
t
,
t
[
f
0

if and only if
one of these equivalent statements is valid



)
t
,
t
(
R
rank
)]
t
,
t
(
R
)
t
,
t
(
[
rank
f
0
f
0
0
f



(22)



)
t
,
t
(
R
rank
)]
t
,
t
(
R
)
t
,
t
(
)
t
,
t
(
[
rank
f
0
f
0
0
f
T
0
f




(23)



The following corollaries are useful for verification of LDCS controllability in
the time moment and its complete controllability.


COROLLARY 5.1. If
a
LDCS is controllable on the time in
terval
]
t
,
t
[
f
0
, then it is
controllable on any including time interval
],
t
,
t
[
f
0


where
f
f
0
0
t
t
,
t
t




.


COROLLARY 5.
2
. If
a
LDCS is controllable
in

the time
moment
0
t
, then it is
controllable
i
n an
y
time moment
0
0
t
t


.



A
h
-
periodical LDCS has the following additional properties.


COROLLARY 5.
3
.

A LTPDCS

is
controll
able

in time moment
t

if and only if it is
controll
able
o
n time

interval
]
h
n
t
,
t
[

.

COROLLARY 5
.
4
.

A LTPDCS

is completely
controll
able, if and only if it is
controll
able in arbitrary

time moment
t
.



By analogy to reachability criterion 4 it
was

obtain
ed the

following the Kalman
-

like algebraic criterion of complet
e controllability of LTIDCS.



CRITERION 6.
(Complete controllability of LTIDCS) A LTIDCS (4) is completely
controllable if and only if if
one of the two following equivalent assertions is fulfilled:


7
-
92

1)



dc
dc
n
dc
R
rank
]
R
)
h
(
[
rank


,

(
24
)


2)

dc
dc
n
dc
R
rank
]
R
)
h
(
[
rank



. (25)



CONCLUSIONS


This paper has been devoted to the derivation of Kalman
-

like reachability and
controllabi
lity criteria for LDCS time
-
varying, time
-
periodical and time
-
invariant
dynamical systems. The analogous problems of observability and constructibility are
solved by means of the duality principle. From these results known Kalman
-

like
criteria for LTI co
ntinuous
-
time and discrete
-
time systems
follow as a special cases
[4], [12], [13], [14].
These solutions give mathematical foundation for applications of
optimal control and other methods for discrete
-
continuous systems design.


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1.

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-
discrete Dynamic System's
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-

24.


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-
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utomatic
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3.

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st

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-
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-
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-
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nd

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