R . M A S A R OVA , M . J U H A S, B . J U H A S OVA , Z . S U T OVA

FA C U LT Y O F M AT E R I A L S S C I E N C E A N D T E C H N O L O G Y I N T R N AVAS L O VA K U N I V E R S I T Y O F T E C H N O L O G Y I N B R AT I S L AVA

T R N AVAS L O VA K I A

C O P E N H A G E N, J U N E 1 2 T H 2 0 1 5

Inverse Matrix in the Theory of Dynamic Systems

Inverse matrix

• Mathematical model is often used for describing the properties of the dynamic system

• Inverse matrix (using different algorithms) Transfer matrix of a dynamic system

Definition: Let A be a regular matrix (a square matrix with a determinant different from 0). Matrix A-1 is an inverse matrix to matrix A if A . A-1 = A-1 . A = I, where I is an identity matrix.

Masarova, R., Juhasova, B., Juhas, M., Sutova, Z.Inverse Matrix in the Theory of Dynamic Systems

Calculating inverse matrix

Adjusting the matrix (A | I) using either column

or line equivalent operations to get a resulting matrix (I | A).

Using the formula

1. Calculating the characteristic polynomial

2. Auxiliary matrices

3. Inverse matrix

nnnn

n

n

AAA

AAA

AAA

AA

21

22212

12111

1 1

.012

21

1 asasasas nn

nn

nn

IaRAR

IaRAR

IR

nn

n

111

101

0

12

11

01 1

n

nn RsRsRAsI

AsI

Dynamic system

State equations of a continuous linear system with the initial condition x(0) = 0:

Transfer matrix of a system:An inverse dynamic system exists when there

exist an inverse matrix to G(s)-1 , i.e. G(s) is regular (|G(s)|≠0).

tDutCxty

tButAxtx

DBAsICsG 1

Masarova, R., Juhasova, B., Juhas, M., Sutova, Z.Inverse Matrix in the Theory of Dynamic Systems

Existence of inverse matrix

Matrices A, B, C, D are number matrices:

Let us create a matrix . ,

and . As , an inverse dynamic system does

not exist. Both classic and MATLAB calculations confirm the results:

11

22 ,

11

11 ,

11

11 ,

11

10DCBA

1111

2211

1111

1110

DC

BAM 00 M 000

2

2

1

11

MM

021

212

M 0210

ssss

ss

sssG

22

22

2

1212

1

1

0sG

function deter(n, A, B, C, D)syms('s');I=eye(n);M=[A B; C D]invmat=inv(s*I-A);G=C*invmat*B+D;det(G)end

det(G) = 0

Masarova, R., Juhasova, B., Juhas, M., Sutova, Z.Inverse Matrix in the Theory of Dynamic Systems

Masarova, R., Juhasova, B., Juhas, M., Sutova, Z.: Inverse Matrix in the Theory of Dynamic Systems

Conclusion

The problem of finding an inverse matrix in dynamic system theory is much vaster, as the inverse matrix can be found, e.g. by inverting graphs or using dynamic algorithms, etc.

This paper is a part of the VEGA project 1/0463/13.