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CH. N. TASIOPOULOS, A. A. FOTOPOULOS, D. VOUKALIS,
P. H. YANNAKOPOULOS
A new approach in specifying the inverse quadratic matrix in modulo-2
for controllable and observable information channels
Technological Educational Institute of PIRAEUSComputer Systems Engineering Department
2010
International Scientific ConferenceeRA-5
Introduction
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Noisy Communication System
[Figure 5.1, page 200, “Fundamentals of Information Theory and Coding Design”, R. Togneri, Ch. deSilva]
In the above diagram we can see the use of channel & source coders in modern digital communication systems provide efficient and reliable transmission of information in the presence of noise.
Introduction
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A common noisy communication system is described by the channel encoder, the source encoder, the digital channel, the channel decoder and the source decoder.We can model such a system, according to the principles of digital control theory, as a digital communication channel between transmitter and receiver. The function of the new modeled system can be described from the state space equations that will be analyzed bellow. For the encoding of information we will use, from the principles of code and information theory, a generator matrix . In this presentation we will use the state space matrixes as components of the generator matrix for the channel encoding. Finally we will investigate the controllable and observable theorems for the above suggested encoding using modulo-2 arithmetic in Galois field.
Concepts of Digital Control Theory
The controllable of a system refers to the possibility for it to be transferred from a given initial state in any final, in finite time, and the observable constitutes the dualism of the controllable. [Kalman 1960]
4TEI of PIRAEUS, Computer Systems Engineering Department International Scientific Conference
eRA-5
Concepts of Digital Control Theory
0
1
( 1) ( ) ( ), (0)
( ) ( )
where:
( 1)
( 2)( ) (0) [ ... ]
(0)
k k
x k Ax k Bu k x x
y k Cx k
u k
u kx k A x B AB A B
u
TEI of PIRAEUS, Computer Systems Engineering Department International Scientific Conference eRA-5 5
1
( 1)
( 2)( ) (0) [ ... ]
(0)
k k
u k
u kx k A x B AB A B
u
State Equations
Concepts of Digital Control Theory
Controllable Definition For the state equation is controllable
if is possible a
control sequence to be found, that can, in finite
time even q, lead the system from any initial state to any final
state
Then: .
It is known from linear algebra, that this equation has solution when:
TEI of PIRAEUS, Computer Systems Engineering Department International Scientific Conference eRA-5 6
| | 0A ( 1)x k
{ (0), (1), ( 1)}u u u q (0)x
( ) , nx q R
1
( 1)
( 2)(0) [ ]
(0)
q q
u q
u qx B AB A B
u
1 1rank [ A | (0)] [ A ]q q qB B x rank B B
Concepts of Digital Control Theory
Controllable DefinitionTo apply the above equation, for any
arbitrary final state should :
From Cayley-Hamilton theorem,
conditions are linearly dependent on the first n
termsThe above equality must be satisfied for
q=n. This means:
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1rank [ A ] , qB B n q N
, for jA B j n1( , AB, , A )nB B
1rank [ A ]nB B n
Concepts of Digital Control Theory
Controllable DefinitionFinally the system
is controllable, when rank S=n ,
Where the table S called controllability
table
TEI of PIRAEUS, Computer Systems Engineering Department International Scientific Conference eRA-5 8
0( 1) ( ) ( ), (0)x k Ax k Bu k x x
1[ ]nS B AB A B
nxnm
Concepts of Digital Control Theory Observable definition
The state space equations model is observable
If there exists finite q, such that knowledge of
inputs {u(0), u(1),…, u(q-1)} and outputs
{y(0),y(1), …, y(q-1)}, can uniquely determine
The initial state x(0) of the system.TEI of PIRAEUS, Computer Systems Engineering
Department International Scientific Conference eRA-5 9
Concepts of Digital Control Theory Observable definition
Specifically the state space model is observable
If
Where the table R called observability
table
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1
, R=
n
C
CArank R n
CA
npxn
Concepts of Digital Control Theory
It is known that when sampling a continuous time system, we have a discrete time system with tables that depend on the sampling period T. A discrete time system is controllable if the continuous time system is also controllable. This is because the control signals in a sampled system are a subset of control signals of the continuous time system. Nevertheless it is possible to lose controllability for some values of the sampling period. While the original continuous system is
controllable, the equivalent discrete system may not be controllable. Similar problems occur with observable.
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The loss of controllable and observable because of sampling
Concepts of information and code theoryGroup definition
A group (G,*) is a pair consisting of a set G and an operation
* on that set , that is a function from the Cartesian product
GxG to G , with the result of operating on a and b denoted
by a*b , which satisfies 1. associativity : a*(b*c)= (a*b)*c for all 2. Existence of identity: There exists such that
e*a=a and a*e=a for all 3.Existence of inverses: For each there exists
such that and
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, , Ga b ce G
a Ga G
1a G 1*a a e 1 *a a e
Concepts of information and code theoryCyclic Groups definition
For each positive integer p, there is a group called the cyclic
group of order p, with set of elements
and operation defined by
If , where (+ )denotes the usual operation
of addition of integers, and
If ,where (-) denotes the usual operation of subtraction of integers.
The operation in the cyclic group is addition modulo p. We shall use the sign +
instead of to denote this operation in what follows and refer to “the cyclic
Group ” , or simply the cyclic group
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{0,1, , ( 1)}pZ p
i j i j i j p
i j i j p i j p
( , )pZ pZ
Concepts of information and code theoryRing definition
A ring is a triple consisting of a set R, and two operations + and , referred to
as addition and multiplication, respectively, which satisfy the following conditions:
1.Associativity of +:
2.Commutativity of +:
3.Existence of additive identity: there exists
4.Existence of additive inverses: for each there exists such that
5.Associativity of :
6. Distributivity of over +:
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( , , )R
( ) ( ) , for all a,b,c a b c a b c R
for all a,b a b b a R 0 such that 0 and 0 for all R a a a a a R
a R a R ( ) 0
and ( ) 0
a a
a a
( ) ( ) , for all , ,a b c a b c a b c R
( ) ( )( ), for all , ,a b c a b a c a b c R
Evariste Galois1811-1832
Concepts of information and code theoryCyclic rings definition
For every positive integer p, there is a ring , called the cyclic ring of order p,
with set of elements
and operations + denoting addition modulo p, and denoting multiplication modulo p
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( , , )pZ
{0,1, , ( 1)}pZ p
Evariste Galois1811-1832
Concepts of information and code theoryLinear codes definition
A binary block code is a subset of for some n .Elements of the code are called code words
Linear code : A linear code is a linear subspace of
Minimum distance : The minimum distance of a linear code is the minimum of the
weights of the non –zero code words
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Evariste Galois1811-1832
n
B
n
B
Concepts of information and code theoryGenerator Matrix
A generator matrix for a linear code is a binary matrix whose rows are the code words
belonging to some basis for the code
Example : The code { 0000, 0001, 1000, 1001} is a two- dimensional linear code in
{0001, 1000} is a basis for this code , which give us the generator matrix
To find the code words from the generator matrix , we perform the following multiplications:
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4
B
0 0 0 1G=
1 0 0 0
0 0 0 10 0 0 0 0 0
1 0 0 0
0 0 0 10 1 1 0 0 0
1 0 0 0
0 0 0 11 0 0 0 0 1
1 0 0 0
0 0 0 11 1 1 0 0 1
1 0 0 0
Concepts of information and code theory
Elementary Row Operation & Canonical Form
An elementary row operation on a binary matrix consists of replacing a row of the matrix
with the sum of that row and any other row.
The generator matrix G of a k-dimensional linear code in is in canonical form if it is of the form where I is a identity matrix and A is arbitrary binary matrix
If the generator matrix G is in canonical form, and w is any k-bit word, the code word s=wG is in systematic form and the first k bits of s are the same as the bits of w.
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n
B[ : A]G I k k ( )k n k
Combining the theoriesFor a common noisy digital channel of the
following diagram
we propose a state space equations modeling according to the digital control
theory.
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( 1) ( ) ( )
( ) ( )
x k Ax k Bu k
y k Cx k
Combining the theories
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( 1) ( ) ( )
( ) ( )
x k Ax k Bu k
y k Cx k
Where:
U(k)0 1( ) { , ,... } n
ku k b b b B
ijA ijB ijC[ ], 1, 2,...,
[ ], 1, 2,...,
[ ], 1, 2,...,
ij
ij
ij
A a ij n
B b ij n
C c ij n
Modulo-2 Arithmetic
From the cyclic groups definition we have:
The equation becomes for p=2:
Where is a cyclic group in modulo-2
arithmetic.The operations stands as referred
previously. TEI of PIRAEUS, Computer Systems Engineering
Department International Scientific Conference eRA-5 21
{0,1, , ( 1)}pZ p
2 {0,1}Z
2Z
Generator Matrix From the principles of code and
information theory , we use a generator matrix for the encoding of the information channel.
For example:
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[ ' ], 1,2,..., with ' nijG b ij n b B
1 0 0 1
1 1 1 0
0 1 0 1
1 0 0 0
G
Combining the theories It is known that a generator matrix of the above form G can be divided into submatrixes. In the proposed model we use 3 submatrixes, that come from the state space tables A,B,C. The suggested generator matrix G consists itself an encoding channel protocol, with the tables A,B,C accruing from different encoding protocols
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ControllableWe will investigate whether the matrixes A,B,C of the
proposed generator matrix model of a specific protocol in
modulo-2 arithmetic of Galois field can lead the system in
controllable Form.
where
The system is controllable if: where n is the dimension of the quadratic matrix S. In different case the system isn’t controllable.
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1[ ]nS B AB A B ', with n' means the length of the code wordn
ij ijA B B
1rank [ A ]nB B n
ObservableWe will investigate whether the matrixes A,B,C of the proposed generator matrix model of a specific protocol in modulo-2 arithmetic of Galois field can lead the system in observable form.
where The system is observable if rank R=n where n is the dimension of the
quadratic matrix R. In different case the system isn’t observable.
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1
R=
n
C
CA
CA
', with n' means the length of the code wordnij ijA C B
Determinant of quadratic matrix in modulo-2To calculate the controllable and observable is
necessary to
calculate the determinant of the quadratic matrix R, S in
modulo-2
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11 12 1
21 22 2
1 2
n
n n
n n nn
k k k
k k kK b B
k k k
1 1 2 2 ...
where: ( 1)
i i i i in in
i jij ij
D K k K k K k K
K D
Example implementation
Suppose we have a system which described by the following
state space tables:
The system is controllable The system is observable
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( 1) ( ) ( )
( ) ( )
x k Ax k Bu k
y k Cx k
1 1 0 1
0 1 1 , B= 1 , C= 1 1 0 with n=3
1 1 1 0
A
2
2
0 1
1 , 1
0 1
1 0 1
[ ] 1 1 1 , 1 0
0 0 1
AB A B
R B AB A B R
2
2
1 0 1 , CA [0 0 1]
1 1 0
1 0 1 1 0
0 0 1
CA
C
R CA
CA
Conclusion
As we observe before it’s possible the design of adigital system in controllable and observable form encoded in modulo-2 arithmetic.Key advantage of the proposed model is the study of controllability and observability in a binary(modulo-2) information channel.The direct connection between machine language and digital controllers can create many opportunities and application in design level.
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END OF PRESENTATION
Thank you for your attention
29
Technological Educational Institute of PIRAEUSComputer Systems Engineering Department
Ch. N. Tasiopoulos, A. A. Fotopoulos, D. Voukalis,
P. H. Yannakopoulos
International Scientific ConferenceeRA-5
2010