A new approach in specifying the inverse quadratic matrix in modulo -2 for controllable and observable information channels

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

TEI of PIRAEUS, Computer Systems Engineering Department International Scientific Conference eRA-5 2

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

TEI of PIRAEUS, Computer Systems Engineering Department International Scientific Conference

eRA-5 3

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


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


1

( 1)

( 2)( ) (0) [ ... ]

(0)

k k

u k

u kx k A x B AB A B

u

State Equations


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:


| | 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


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:


1rank [ A ] , qB B n q N

, for jA B j n1( , AB, , A )nB B

1rank [ A ]nB B n


Controllable DefinitionFinally the system

is controllable, when rank S=n ,

Where the table S called controllability

table


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


1

, R=

n

C

CArank R n

CA

npxn


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.


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


, , 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


{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 +:


( , , )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


( , , )pZ

{0,1, , ( 1)}pZ p


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



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:


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.


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.


( 1) ( ) ( )

( ) ( )

x k Ax k Bu k

y k Cx k

Combining the theories


( 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 examle:


[ ' ], 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


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.


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.


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


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


( 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.


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

A new approach in specifying the inverse quadratic matrix in modulo -2 for controllable and observable information channels

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