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

Technological Educational Institute of PIRAEUS
Computer Systems Engineering Department

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

International Scientific Conference
eRA-5
2010

Introduction
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.

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

Introduction

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]


State Equations

u (k 1)
x(k 1) Ax ( k ) Bu ( k ), x (0) x0 u (k 2)
x(k ) Ak x(0) [ B  AB ... Ak 1B ]

y (k ) Cx ( k ) u (0)

u (k 1)
k k 1
u (k 2)
where: x(k ) A x(0) [ B  AB ... A B ]

u (0)



 Controllable Definition
For | A | 0 the state equation x(k 1) is controllable if is possible a
control sequence {u(0), u(1),u(q 1)} to be found, that can, in finite
time even q, lead the system from any initial state x(0) to any final
state x(q) , Rn
u (q 1)
u (q 2)
Then: q
x(0) [ B  AB    Aq 1 B]

.
u (0)

It is known from linear algebra, that this equation has solution when:
rank [ B    A q 1 B | q
x(0)] rank [ B    A q 1B]


To apply the above equation, for any arbitrary final
state should : rank [ B    A q 1B] n, q N
From Cayley-Hamilton theorem, conditions A j B, for j n
are linearly dependent on the first n terms ( B, AB, , A n 1B)

The above equality must be satisfied for q=n. This
means: rank [ B    A n 1B] n



Finally the system x( k 1) Ax( k ) Bu ( k ), x(0) x is 0

controllable, when rank S=n , S [ B  AB  An 1B]

Where the nxnm table S called controllability
table



 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.



 Observable definition

Specifically the state space model is observable
C
CA
If rank R n , R=

CAn 1

Where the npxn table R called observability
table


The loss of controllable and observable because of
sampling
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.

Concepts of information and code theory

Group 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 a, b, c G
 2. Existence of identity: There exists e G such that
e*a=a and a*e=a for all a G
 3.Existence of inverses: For each a G there exists a 1 G
such that a * a 1 e and a 1 * a e



Cyclic Groups definition

For each positive integer p, there is a group called the cyclic
group of order p, with set of elements
Z p {0,1, ,( p 1)}
and operation defined by i j i j
If i j p , where (+ )denotes the usual operation
of addition of integers, and i j i j p
If i j p ,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 ( Z p , ) ” , or simply the cyclic group Z p



Ring definition

A ring is a triple ( R, , ) consisting of a set R, and two operations + and , referred to
as addition and multiplication, respectively, which satisfy the following conditions:
1.Associativity of +: a (b c) (a b) c, for all a,b,c R
2.Commutativity of +: a b b a for all a,b R
3.Existence of additive identity: there exists 0 R such that 0 a a and a 0 a for all a R
a ( a) 0
4.Existence of additive inverses: for each a R there exists a R such that and ( a ) a 0
5.Associativity of : a (b c) (a b) c, for all a, b, c R

6. Distributivity of over +: a (b c) (a b)(a c), for all a, b, c R

Evariste Galois
1811-1832


Cyclic rings definition

For every positive integer p, there is a ring ( Z p , , ) , called the cyclic ring of order p,
with set of elements

Zp {0,1, , ( p 1)}

and operations + denoting addition modulo p, and denoting multiplication modulo p

Evariste Galois
1811-1832


Linear codes definition
n
 A binary block code is a subset of B for some n .Elements of the code are called
code words
n
 Linear code : A linear code is a linear subspace of B
 Minimum distance : The minimum distance of a linear code is the minimum of the
weights of the non –zero code words

Evariste Galois
1811-1832


Generator 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 B
4

0 0 0 1
G=
1 0 0 0

To find the code words from the generator matrix , we perform the following multiplications:

0 0 0 1 0 0 0 1
0 0 0 0 0 0 0 1 1 0 0 0
1 0 0 0 1 0 0 0

0 0 0 1 0 0 0 1
1 0 0 0 0 1 1 1 1 0 0 1
1 0 0 0 1 0 0 0



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 B n is in canonical form if it is
of the form G [ I : A] where I is a k k identity matrix and A is arbitrary k (n k )
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.


Combining the theories
For a common noisy digital channel of the following diagram

we propose a state space equations modeling according to the digital control
theory.
x ( k 1) Ax ( k ) Bu ( k )

y(k ) Cx ( k )


x (k 1) Ax (k ) Bu (k )

y (k ) Cx (k )
Where:

U(k) u ( k ) {b0 , b1 ,...bk } B n

A [aij ], ij 1, 2,..., n

Aij Bij Cij B [bij ], ij 1, 2,..., n
C [cij ], ij 1, 2,..., n


Modulo-2 Arithmetic

From the cyclic groups definition we have:
Zp {0,1, , ( p 1)}

The equation becomes for p=2:
Z2 {0,1}

Where Z 2 is a cyclic group in modulo-2
arithmetic.
The operations stands as referred previously.


Generator Matrix
 From the principles of code and
information theory , we use a generator
matrix for the encoding of the information
channel.
G [b 'ij ], ij 1,2,..., nwith 'b Bn
 For example:
1 0 0 1
1 1 1 0
G
0 1 0 1
1 0 0 0


 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


Controllable
We 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.

S [ B  AB  An 1 B]
where Aij , Bij Bn ' with n' means the length of the code word

 The system is controllable if: rank [ B    A n 1 B] n
where n is the dimension of the quadratic matrix S.
 In different case the system isn’t controllable.


Observable
We 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. C
CA
R=

CAn 1

n'
where Aij , Cij B with n' means the length of the code w ord
 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.


Determinant of quadratic matrix in
modulo-2
To calculate the controllable and observable is necessary to
calculate the determinant of the quadratic matrix R, S in
modulo-2
k11 k12  k1n
k21 k22  k2 n
K b Bn
   
kn1 kn 2  knn

D K ki1K i1 ki 2 K i 2 ... kin K in
where: Kij ( 1)i j Dij


Example implementation
x (k 1) Ax (k ) Bu (k )

y (k ) Cx (k )
Suppose we have a system which described by the following
state space tables:
1 1 0 1
A 0 1 1 , B= 1 , C= 1 1 0 with n=3
1 1 1 0
0 1
AB 1 , A2 B 1
CA 1 0 1 , CA 2 [0 0 1]
0 1
1 0 1 C 1 1 0
R [B AB A2 B ] 1 1 1 , R 1 0 R CA 1 0 1 1 0
0 0 1 CA2 0 0 1
The system is controllable The system is observable

Conclusion

As we observe before it’s possible the design of a
digital 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.

Technological Educational Institute of PIRAEUS
Computer Systems Engineering Department

END OF PRESENTATION

Thank you for your attention

Ch. N. Tasiopoulos, A. A. Fotopoulos, D. Voukalis,
P. H.Yannakopoulos

International Scientific Conference
eRA-5
2010 29

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

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