Numerical Operability Analysis of Multidimensional Systems with Interval Parameters Using Interval Degeneration Factors
NATALIA DUDARENKO
1, ANATOLY USHAKOV
21
Department of Computer Science and Electrical Engineering Luleå University of Technology
SE-971 87 Luleå SWEDEN
natalia.dudarenko@ltu.se
2
Department of Control Systems and Informatics
The National Research University of Information Technologies, Mechanics and Optics 197101, Saint Petersburg, Kronverkskiy pr., 49
RUSSIA ushakov-avg@yandex.ru
Abstract: - This paper deals with the problem of numerical analysis of multidimensional systems with interval parameters in relation to the operability of MIMO systems. The problem is analyzed for the case when the relative estimation of the interval parameters exceeds 0.5 and based on the assumption that only the state matrix of a MIMO system contains interval parameters. The problem of operability estimation of the MIMO system is solved with interval degeneration factors. Interval degeneration factors are constructed using interval arithmetic and the singular values of the interval criterion matrices of the system. An operability analysis algorithm of multidimensional systems with interval parameters is proposed in the paper. The results are supported with an example.
Key-Words: - Multidimensional systems, Interval parameters, Interval analysis, Degeneration factors.
1 Introduction
System design is one of the main steps before the implementation of a system in real time. A system should meet to a given quality, be reliable, stable, and operable.
Often, changes to system parameters during the implementation phase can lead to a loss of operability. Furthermore, any MIMO system containing human elements can be described using an interval model [1]. Various characteristics of human operators can result in reduced operability [2]. It is, therefore, important to be able to estimate a priori the system operability for varying parameters.
An operability analysis algorithm of multidimensional systems with interval parameters is proposed in the paper. The algorithm is based on a calculation of the interval degeneration factors [3]
that are indexes of system operability. Interval degeneration factors are constructed with singular values of the criterion matrices of the system [3].The problem is considered for the case when the relative estimation of the interval parameters exceeds 0.5.
The interval analysis is used for the calculation of the interval degeneration factors.
The proposed technique allows system operability to be estimated using curves of interval degeneration factors.
2 Degeneration Factors
Any multidimensional system can be described by the linear algebraic equation [3]:
( ) w N w ( , ) ( ) w
η = θ χ , (1)
where N w ( , ) θ is an -dimensional criterion matrix of a system for any ,
m m ×
w θ ; ( ) η w and ( ) χ w are m -dimensional vectors; θ is a p -dimensional parameter that changes the properties of the criterion matrix.
For simplicity, ( , ) N w θ will be written as N .
J denotes degeneration factors of the MIMO
Djsystem (1). The degeneration factors J are
Djcalculated with the singular values α
j( j = 1, m ) of
the criterion matrix N [3] such that { }
( )
1/2
,
: det 0 : 1,
j j
T i
N I N N j m
α
α µ
σ µ µ
⎧ ⎧ = ⎫
⎪ = ⎪ ⎪
⎨ ⎨ ⎬
− = =
⎪ ⎪ ⎩ ⎪ ⎭
⎩
(2)
and satisfy the equation
( ) /
1( ); ,1
Dj j
J = α N α N j m = . (3)
D1
J denotes a global degeneration factor of the
criterion matrix N such that
min max
1
/
J
D= α α , (4)
where α
minand α
maxare minimum and maximum singular values respectively of criterion matrix N . The properties of the degeneration factors are given in [3].
The singular value decomposition (SVD) [4,5] is used for the geometric representation of degeneration factors. Using SVD, the matrix can be represented in the form
N
T
N N N
N U = Σ V , (5) where the matrix Σ =
Ndiag { α
j; j = 1, m } is the
diagonal matrix with singular values on the main diagonal. The diagonal elements of the matrix
are in descending order. The matrix is the matrix of the left singular vector, is the matrix of the right singular vector such that
,
Σ jj
Σ
NU
NV
NT T
N N N N
U U = U U = I V V
N NT= V V
NT N= ; then I and satisfy the equation
U
NV
NNj j Nj
NV = α U , j = m . (6) 1,
Equation (6) corresponds to linear algebraic equation (1), where χ = V
Nj, ( ) η w = α
jU
Nj( j = 1, ) m . In other words, the unit sphere maps to an ellipsoid. The lengths of the semi-axes of the ellipsoid are equal to the singular values of the matrix . The flattening of the ellipsoid depends on the singular values
N
α
j. The ellipsoid transforms into a line for α
1≠ and 0
(
0 ,
j
j m
α = = 2 ; therefore, the MIMO system (1) is )
on the boundary of the global degeneration. For the case α
j= 0 ( j m = ,1 the ellipsoid is a point; hence, )
linear MIMO system (1) is degenerated globally.
In our problem statement the global degeneration of the system corresponds to its full operability loss.
If the global degeneration factor equals zero and other degeneration factors are non-zero, it means that one of the channel of the system does not work or two of them do the same work.
The degeneration factors J allow us to fix the
D1degeneration process of the system as a whole.
2.1 Interval Degeneration Factors
Obviously, multidimensional systems with interval parameters also have interval criterion matrices and interval degeneration factors. So, the degeneration factors of a MIMO system with interval parameters take the form
⎡ ⎣ J
Dj⎤ ⎡ ⎦ ⎣ = α
j( [ ] N ) / ⎤ ⎡ ⎦ ⎣ α
1( [ ] N ) ; ⎤ ⎦ j = m , 1 , (7)
where [ ] N is an -dimensional interval criterion matrix of a multidimensional system. The interval criterion matrix can be represented by the equation [6]
m m ×
[ ] N = N
0+ ∆ [ N ] , (8)
where is the median fixed component of the interval criterion matrix
N
0[ ] N ; [ ] ∆ N is the symmetric interval matrix component of [ ] N ; ⎡ ⎣ α
j( [ ] N ) ⎤ ⎦ and
1
( [ ] N ) α
⎡ ⎤
⎣ ⎦ are interval singular values of the interval criterion matrix [ ] N .
The interval degeneration factors can be decomposed into two components:
0
D j D j D j
J J J
⎡ ⎤ = + ∆ ⎡ ⎤
⎣ ⎦ ⎣ ⎦ , (9)
where J
D j0is the fixed median component of the interval degeneration factors ⎡ ⎣ J
D j⎤ ⎦ ; ⎡∆ ⎣ J
D j⎤⎦ is the symmetric interval component of ⎡ ⎣ J
D j⎤ ⎦ .
Interval values of degeneration factors (7) are calculated with interval arithmetic [6].
The relative estimation of the interval degeneration factors can be calculated in the following form
0 D I D
D j j
j
J
J J
δ =
∆∆ . (10)
The absolute estimation of the interval degeneration factors satisfies the equation
I
J
Dj
∆⎡ J
D⎤
∆ = ⎣ ∆ j ⎦ . (11)
3 Interval Criterion Matrices
Let us consider a multidimensional time-continuous dynamic system with interval parameters in the form x t ( ) = [ ] F x t ( ) + Gg t ( ) ; (0) x ; y t ( ) = Cx t ( ) , (12)
where x t ( ) is an -dimensional state vector, is an -dimensional input vector, is an - dimensional output vector of the plant;
n g t ( )
m y t ( ) m
[ ] F is an
n n × -dimensional state matrix with interval parameters; , are constant matrices of appropriate dimensions.
G C
The interval state matrix can be decomposed into two components
[ ] F = F
0+ ∆ [ F ] , (13)
where F is the fixed median component of the
0matrix [ ] F with the elements F
0ij: i = 1, ; n j = 1, n such that
{ } ( )
0ij ij ij
,
ij0.5
ij ijF = mid ⎡ ⎤ ⎣ ⎦ ⎣ F = ⎡ F F ⎤ ⎦ = F + F . (14)
The second component [ ] ∆ is the symmetric F interval component of the interval state matrix [ ] F
such that
[ ] ∆ F = ∆ ∆ ⎡ ⎣ F , F ⎤⎦ , (15) where
( )
{ }
0 ij ij 0ij
; , 1,
F F F col row F F F i j n
∆ = − = ∆ = − =
( )
{ }
0 ij ij 0ij
; , 1,
F F F col row F F F i j n
∆ = − = ∆ = − =
The relative interval estimation of the interval state matrix [ ] F is calculated by
[ ]
0 I
F F
δ = ∆ F . (16)
The absolute interval estimation of the interval state matrix [ ] F satisfies the equation
[ ]
I
F
∆F
∆ = ∆ . (17)
Notice that, if relative interval estimation of the interval state matrix satisfies the equation
0 ≤ δ
IF ≤ 0.5 , (18) then we should use sensitivity theory [7,8], otherwise we should use the interval analysis that applies to our specific case.
Here we provide examples of interval criterion matrices of MIMO systems with interval parameters for different exogenous disturbances.
For the case of multidimensional harmonic exogenous disturbance the interval criterion matrix
[ ] N satisfies the form [3]
[ ] N = − ( C F [ ]
2+ ω
2jI ) [
−1[ ] F ω
jI ] G , (19) where 0 ≤ ω j ≤ ∞ is the frequencies of the multidimensional harmonic exogenous disturbance.
For the case of stochastic exogenous disturbance
“white noise” w t ( ) with the intensity matrix
; 1,
Q diagQ = jj j = m the interval criterion matrix is the spectral density output matrix ⎡⎣ S
y( ) ω ⎤⎦ with the following representation [3]
⎡ ⎣ S
y( ) ω ⎤ = − ⎦ 2 C F [ ] [ ] ( F
2+ ω
2I )
−1[ ] D
xC
T, (20) where [ ] D is the interval dispersion matrix of state x vector ( ) x t satisfying the matrix Lyapunov equation [9]
. (21)
[ ][ ] [ ][ ] F D
x+ D
xF
T= − GQG
TAssume stochastic exogenous disturbance “colored noise” ( ) ξ t is given by output of a shaping filter that is simulated by “white noise” , where has the intensity matrix Q . In this case:
( )
w t w t ( )
z t
f( ) = Г z t
f f( ) + G
fξ ( ) t ; ( ) ξ t = P z t
f f( ) , (22) where z is an l-dimensional state vector of the
fshaping filter model; Г ,
fG ,
fP are constant
fmatrices of appropriate dimensions. The interval criterion matrix ⎡ ⎣ S
y( ) ω ⎤ ⎦ has form (20), where
[ ] D
x= C D C
x⎡ ⎣
x⎤ ⎦
Tx,
x=
n n×0
n l×x
C I ;
⎡ D ⎤
⎣ ⎦ is the dispersion matrix of the aggregate state vector x = x
Tz
Tf Tand satisfies the Lyapunov equation
, (23)
T T
x x
F D D F GQG
⎡ ⎤ ⎡ ⎤ ⎡ + ⎤ ⎡ ⎤ = −
⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ where
[ ]
0
f f
F GP
F Г
⎡ ⎤ =
⎣ ⎦ , 0
f
G = G .
4 Operability Analysis Algorithm
Using interval degeneration factors with the interval criterion matrices of multidimensional systems with interval parameters allows us to construct the operability analysis algorithm.
1. Define multidimensional system in form (12).
2. Find the value of the relative interval estimation
I
F
δ in form (16).
3. Check if the relative interval estimation satisfies equation (18).
4. Calculate the interval criterion matrix [ ] N of
system (12).
5. Find the median and symmetric interval components of the interval criterion matrix [ ] N .
6. Calculate the degeneration factors of the median and symmetric components of the interval criterion matrix [ ] N .
7. Find the relative estimations of the interval degeneration factors in form (10).
8. Construct the curves of the interval degeneration factors.
9. Find the interval parameters where symmetric components of degeneration factors deviate significantly from median components.
10. Establish the set of critical parameters for further correction.
5 Example
Consider the multidimensional time-continuous
dynamic system in form (12), where
[ ]
1
2
4
0 1 0 0 0
0 0 1 0 0
8 8 4 4.8[ ] 0
0 0 0 0 1
...
0 0 0 0 0
432[ ] 0 0 216 72
0 0 0 0 0
0 0 0 0 0
0 0 0 11664[ ] 0 k
F
k
k
⎡ ⎢
⎢ ⎢ − − − −
⎢ ⎢
= ⎢
⎢ − −
⎢ ⎢
⎢ ⎢
⎢ ⎣
3
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
... 1 0 0 0
12 129[ ] 0 0
0 1 0 0
0 0 1 0
0 5832 648 36
k
⎤ ⎥
⎥ ⎥
⎥ ⎥
⎥ ⎥
− − ⎥
⎥ ⎥
⎥ ⎥
− − − ⎦
3 01
3 02
3 03
0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0
0 0 0 0 0 0 0 0 G
ω
ω
ω
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
= ⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦
;
1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 C
T⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
= ⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦
.
The system consists of three inputs and three outputs.
Every separate channel has the third order.
Connections between the channels are interval parameters [ ] k
1, [ ] k
2, [ k
3] and [ k
4] such that
[ ] k
1= k
10+ ∆ [ ] k
1= 0.5 + − [ 0.5; 0.5 ] ;
[ ] k
2= k
20+ ∆ [ ] k
2= 0.5 + − [ 0.5; 0.5 ] ;
[ ] k
3= k
30+ ∆ [ ] k
3= 0.5 + − [ 0.5; 0.5 ] ;
[ ] k
4= k
40+ ∆ [ ] k
4= 0.5 + − [ 0.5; 0.5 ] ,
where , , , are the median
components of the interval connection parameters;
, , , are the symmetric
components of the interval connection parameters.
Let us consider the system with multidimensional harmonic exogenous disturbance
10
k k
20k
30k
40[ ] ∆ k
1[ ] ∆ k
2[ ] ∆ k
3[ ∆ k
4]
( )
g t such that frequencies ω j of the exogenous disturbance match the bandwidths of the channels.
The values of symmetric elements of the connection parameters are given in Table 1.
Table 1.
k
ij∆ ∆ k
1∆ k
2∆ k
3∆ k
4a) -0.5 -0.5 -0.5 -0.5 b) 0.5 0.5 0.5 0.5 c) -0.5 0.5 -0.5 -0.5 d) 0.5 0.5 -0.5 -0.5 e) -0.5 -0.5 0.5 -0.5 f) 0.5 -0.5 0.5 -0.5 g) -0.5 0.5 0.5 -0.5 h) 0.5 0.5 0.5 -0.5 i) -0.5 -0.5 -0.5 0.5 j) 0.5 -0.5 -0.5 0.5 k) -0.5 0.5 -0.5 0.5 l) 0.5 0.5 -0.5 0.5 m) -0.5 -0.5 0.5 0.5 n) 0.5 -0.5 0.5 0.5 o) -0.5 0.5 0.5 0.5 p) 0.5 0.5 0.5 0.5 The interval criterion matrix [ ] N satisfies equation (19).
The curves of the degeneration factors J
D1∆and of symmetric and median components of the connection parameters respectively are shown in Figure 1.
D
