Ma ix Equa ions in Mul i a iable Con ol
ROMAN PROKOP, JIŘÍ KORBEL
Tomas Ba a Uni e si y in Zlín
Nám. T.G.M. 5555, 760 01 Zlín
CZECH REPUBLIC
[email p o ec ed] h p://www.u b.cz/ ai
Abs ac : - The con ibu ion is ocused on a con ol design and simula ion o mul i inpu ou pu (MIMO) linea
con inuous- ime sys ems. Sui able and e icien ools o desc ip ion and con olle de i a ion a e algeb aic
no ions as ings, polynomial ma ices, and Diophan ine equa ions. The gene alized MIMO PI con olle design
is s udied o s able and uns able sys ems. A uni ied app oach h ough ma ix Diophan ine equa ion can be
applied in bo h cases. All s abilizing eedback con olle s a e ob ained ia solu ions o a ma ix Diophan ine
equa ion. The me hodology allows de ining scala pa ame e s (one o mo e) o uning and in luencing o
con olle pa ame e s. A Ma lab-Simulink p og am implemen a ion was de eloped o simula ion and
e i ica ion o he s udied app oach. Illus a i e examples show he e ec i eness and lexibili y o he p oposed
me hod o some simple MIMO sys ems.
Key-Wo ds: - Polynomial ma ices, Diophan ine equa ions, Mul i a iable sys ems, S abiliza ion.
1 In oduc ion
The s udy o mul i inpu –mul i ou pu (MIMO)
sys ems has a ac ed scien i ic a en ion o
decades. Analysis and con ol me hods and ools
ha e been de eloped in many monog aphs (e.g. [1],
[2], [4], [8], [12], [14]) as well as in jou nal and
con e ence con ibu ions (e.g. [3], [9], [10], [16],
[18]) o in p og am oolboxes, e.g. [17].
Mul i a iable sys ems ep esen an in e es ing
esea ch ield also om ma hema ical poin o iew.
Many no ions, me hods and ools o single inpu –
ou pu (SISO) sys ems canno be simply and
i ially gene alized in o mul i a iable cases. The
main p oblem ela es o he ma ix non-commu a i e
mul iplica ion. Howe e , many algeb aic no ions
and ools can be success ully u ilized also in he
non-commu a i e case. The main ool o
con inuous- ime sys ems is he Laplace ans o m
and b ie ly speaking, mul i a iable linea
con inuous- ime sys ems a e desc ibed and
exp essed by a se o linea di e en ial equa ions.
So, scala polynomials desc ibing single inpu –
ou pu linea sys ems a e eplaced by polynomial
ma ices. Algeb aic no ions and modules emain a
sui able and e ec i e ool o analysis and con ol
design o MIMO sys ems. T ans e unc ions as a
a io o wo polynomials a e in MIMO cases
conside ed as ma ix ac ions and due o non-
commu a i e ma ix mul iplica ion he denomina o
can be in he le o igh side o he ma ix ac ion
([2], [8], [12]) in disc e e and con inuous- ime case.
Also, a scala linea Diophan ine equa ion is
gene alized in o a ma ix one, see e.g. [3], [14],
[15]. The con ibu ion is scheduled as ollows. The
basic no ions a e men ioned in sec ion II, he sys em
desc ip ion o MIMO sys ems is in oduced in
sec ion III. Sec ion IV deals wi h ma ix
Diophan ine equa ions and he nex sec ion ou lines
and summa izes a con ol design. Some i s o de
examples and de i a ions a e p esen ed in sec ion
VI. The p oposed me hodology b ings one o
se e al scala which can une and in luence he
con ol beha io in an easy way. Simula ions a e
p esen ed in Sec ion VII, he las sec ion concludes
he con en s o he con ibu ion.
2 Polynomial Ma ices
Polynomial ma ices a e called l x m ma ices whe e
all elemen s o ma ices a e polynomials in an
inde e mina e s. This inde e mina e can be
conside ed in linea sys ems as he Laplace ope a o
and he se o polynomial ma ices is Rlm(s). I l =
m, hen he se o polynomial ma ices cons i u es a
non-commu a i e ing. A uni in his ing (an
in e se elemen exis s in he ing) is a ma ix wi h
eal nonze o de e minan and all uni s a e called
unimodula . Gene ally, l

m se Rlm(s) is no mo e a
ing. I A = BC hen B is a le di iso o A and A is
a igh mul iple o B, while C is a igh di iso o A
and A is a le mul iple o C. Simila ly, g ea es
common le and igh di iso s a e in oduced. Two
ma ices A, B a e le ( igh equi alen , i A = U1 B
(A = B U2) wi h unimodula U1, U2. When A = U1 B
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U2 hen A, B a e simply called equi alen . Ma ices
wi h he same numbe o columns a e le cop ime i
hei all le di iso s a e unimodula ma ices and
ma ices wi h he same numbe o aw a e igh
cop ime i hei all igh di iso s a e unimodula
ones.
The known ex ended (scala ) Euclidean
algo i hm o can be gene alized in mul i a iable
cases. A g ea es le common di iso G1(s) can be
calcula ed o A, B wi h he same numbe o aw by
1 1 1
11
( ) ( ) ( ) ( ) ( ),
( ) ( ) ( ) ( ) 0,
A s P s B s Q s G s
A s R s B s S s


(1)
Mo eo e , L = AR1 = - BS1 is he leas common
igh mul iple o A, B. A g ea es igh common
di iso G2(s) can be calcula ed o A, B wi h he
same numbe o columns by
2 2 2
22
( ) ( ) ( ) ( ) ( ),
( ) ( ) ( ) ( ) 0,
P s A s Q s B s G s
R s A s S s B s


(2)
Also, L = R2A = - S2B is he le common
mul iple o A, B. Rela ions (1), (2) a e he basic
algeb aic no ions o Diophan ine equa ions, see e.g.
[1], [3], [10].
3 Sys em Desc ip ion
A linea con inuous- ime mul i a iable (MIMO)
sys em is desc ibed by a se o linea di e en ial
equa ions and hen i can be easily exp essed by he
Laplace ans o m echnique in he o m
( ) ( ) ( ) ( ),A s Y s B s U s
(3)
whe e
( ), ( )A s B s
a e polynomial ma ices in he
Laplace ans o m a iable s. Fo con ol design, i
is use ul o cha ac e ize MIMO linea ime-in a ian
sys ems in e ms o hei ans e unc ion ma ices.
The gene aliza ion o single inpu -ou pu linea
sys em o MIMO ones is e y simple in he s a e
space desc ip ion
( ) ( ) ( ),
( ) ( ) ( )
x Fx u
y Hx Lu
  

(4)
The sys em wi h l inpu s ad m ou pu s in (4) has
he s a e ec o x( ) wi h alues in Rn and eal
ma ices F,

, H, L ha e dimensions (nxn), (nxl),
(mxn), (mxm), espec i ely. Any decomposi ion
1
( ) ( )G s H sI F L

  
(5)
de ines a a ional sys em´s ans e unc ion ma ix.
A ealiza ion (5) is minimal when he s a e ec o
dimension n is as small as i can be and his alue is
called he MacMillan deg ee and i ep esen s he
o de o he sys em. So, G(s) in (5) is a a ional
ma ix unc ion, i means ha all en ies a e a ional
unc ions o s. This ma ix unc ion can be hen
exp essed by he le o igh ma ix ac ion
11
( ) ( ) ( ) ( ) ( )
RR
G s A s B s B s A s


(6)
whe e A, B, AR, BR a e polynomial ma ices, mo e
de ails can be ound i.e. in [8]. No e, ha bo h
ma ices A(s), AR(s) a e squa ed bu no necessa ily
o he same dimension. In he case o sys ems wi h l
inpu s and m ou pu s, he le denomina o A(s) has
dimension lxl, while he igh denomina o AR(s) has
he dimension mxm. Howe e , bo h ma ices a e
associa es and he cha ac e is ic polynomial
ollowing om he s a e-space desc ip ion (4) is also
associa es. I means ha all oo s o he men ioned
polynomials a e same. I means
de ( ) de ( ) de ( )
R
A s A s sI F
(7)
whe e F is he squa ed sys em ma ix in (4).
Wi h ela ion (3) he no ion o s abili y is closely
connec ed. A linea sys em is asymp o ic (in e nal
s able), i all de e minan s in (3) a e s able, o
con inuous- ime sys ems i means ha all oo s lie in
he open le hal o he complex plane.
4 Ma ix Diophan ine Equa ions
Diophan ine equa ions de ined in commu a i e ings
a e linea equa ions o he o m
,ax by c
(8)
whe e a, b, c a e known gi en en ies and x, y a e
unknown ones in he ing. I is well known (see e.g.
[1], [2], [8]) ha equa ion (8) has a solu ion i and
only i he g ea es common di iso o a, b di ides
c, b ie ly gcd(a,b) / c. Mo eo e , i x0, y0 is a pai o
pa icula solu ions o (8) , hen all x, y gi en by
00
00
,
,
x x b
y y a


(9)
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whe e is an a bi a y elemen o he ing and
a0=a/gcd(a,b), b0=b/gcd(a,b).Then, wi hou loss o
gene ali y, equa ion (8) can be supposed wi h
cop ime a, b and he solu ion o (8) exis s o any c.
The si ua ion is mo e complex in non-
commu a i e ings, such is a se o polynomial
ma ices. Due o he non-commu a i i y o ma ix
mul iplica ion, equa ion (8) is spli in o h ee kinds
o linea ma ix equa ions o e he ing. A na u al
gene aliza ion o his equa ion is ei he he equa ion
1 1 1,A X BY C
(10)
o he equa ion
2 2 2,XA YB C
(11)
Bo h equa ions a e called unila e al ones. The
las equa ion is called a bila e al one and i has he
o m
3 3 3,A X YB C
(12)
In he case o equa ion (10), ma ices (A1, B1, C1)
ha e he same numbe o ows, while in equa ion
(11) ma ices in he iple (A2, B2, C2) ha e he same
numbe o columns. The sol abili y o equa ions
(10) – (12) is s udied e.g. in [1], [2], [8]. The esul s
can be b ie ly o mula ed in he enginee ing
pa lance as ollows:
a) Equa ion (10) has a solu ion i and only i he
g ea es le common di iso o ma ices A, B is a
le di iso o C.
b) Equa ion (11) has a solu ion i and only i he
g ea es common igh di iso o ma ices A, B is a
igh di iso o C.
c) Equa ion (12) has a solu ion i and only i he
ma ices
0,
00
A A C
BB
   
   
   
(13)
a e equi alen . This case is ou o he in e es o his
con ibu ion and some de ails can be ound in [2].
I a pa icula solu ion o a gi en linea
Diophan ine equa ion exis s, he e exis a se o all
solu ions. In he case o (10), (11) he se s o
solu ions a e gi en
0 1 0 1
,,X X BT Y Y AT   
(14)
whe e X0, Y0 a e pa icula solu ions o (10) and T is
an a bi a y polynomial ma ix o he app op ia e
dimension and
1 1 1 1.AB B A
(15)
Solu ions o (11) a e
0 2 0 2
,,X X TB Y Y TA   
(16)
and again X0, Y0 a e pa icula solu ions o (11), T is
an a bi a y polynomial ma ix o he app op ia e
dimension and
2 2 2 2.B A A B
(17)
Rela ions (15), (17) a e no hing else han he
opposi e ma ix ac ion. Sui able and con enien
ools o he solu ion o linea ma ix equa ions a e
o e ed by a Polynomial oolbox [17] which
con ains a se o use iendly Ma lab unc ions o
a ious con ol sys em pu poses.
As a simple example sol e equa ion (10) o
ma ices
1 1 1
1 2 1.5 1 0
,,
3 2 2 0 1
s
A B C
ss

     
  
     

     
(18)
Ma lab unc ion AXBYC in Polynomial oolbox
gi es he pa icula solu ion
 
0
0
0 0.33 ,
2 0.67 0.67
2 0.67 0.67
Xs
Ys




  
(19)
All solu ion a e gi en in he o m (14) wi h
11 2
1 2 0.94 0.47
,.
3 2 2.4 2.8 0.94
ss
AB
s s s

   

   
  
   
(20)
The ee polynomial ma ix T has he o m
 
12
( ) ( )T s s
(21)
wi h 1(s), 2(s) a bi a y polynomials. Really, he
p oduc
1 1 1 1
AB B A
gi es he same esul in he o m
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2
23
5.7 4.2 1.4
7.5 1.9 4.7 0.94
ss
s s s



  

(22)
which con i ms equa ion (15).
5 Con ol Design
The mos equen scheme o a basic eedback
con ol sys em is depic ed in Fig. 1. All signals in
he MIMO case a e ec o ones. Inpu signals o he
eedback sys em in Fig. 1 is a e e ence (se poin )
signal w = Fw-1(s) Gw(s) and a load dis u bance
signal d = Fd-1(s) Gd(s) de ined by hei ma ix le
ma ix ac ions.
Fig. 1 one deg ee o eedom (1DOF) con ol
sys em
All s abilizing con olle s o he 1DOF eedback
sys em in Fig. 1 a e gi en by any solu ion o ma ix
Diophan ine equa ion
( ) ( ) ( ) ( ) ( ),
RR
A s P s B s Q s M s
(23)
whe e P-1(s)Q(s) = QR(s) PR-1(s) is a le and igh
ma ix ac ion o he con olle C and A-1(s)B(s) =
BR(s)AR-1(s) is a le and igh ma ix ac ion o he
con olled plan G. Mo e de ails ca be ound e.g. in
[1], [5], [12], [15], [16].
Howe e , o asymp o ic acking and dis u bance
ejec ion mus be ul illed u he condi ions. B ie ly
speaking, denomina o o he con olle mus be
di isible by he denomina o s o inpu signals. I is a
eason o a p e-compensa o F in Fig. 2 which
ep esen s he condi ions o di isibili y. In he case
o asymp o ic acking only, i is F=Fw. In he case
o simul aneous asymp o ic acking and dis u bance
ejec ion F=FwFd. The basic s abili y and
asymp o ic acking in he sense o Fig. 2 is hen he
con olle QR(s)PR-1(s) gi en by he solu ion o
ma ix Diophan ine equa ion
( ) ( ) ( ) ( ) ( ) ( ),
RR
A s F s P s B s Q s M s
(24)
whe e M(s) is a s able polynomial ma ix wi h
p esc ibed poles o i s de e minan . Resul ing
ma ices PR, QR ep esen he igh ma ix ac ion
11
( ) ( ) ( ) ( )
RR
P s Q s Q s P s


(25)
Fig. 2 eedback 1DOFcon ol sys em wi h p e-
compensa o
The con ol law is hen go e ned by he equa ion
1
( ) ( ) ( ) ( )( ( ) ( )),P s F s U s Q s W s Y s

(26)
which can be easily ew i en in o di e en ial
equa ions. Now, i is necessa y o p opose he
me hod o solu ion o ma ix equa ion (2). Fo
simple cases, he solu ion can be ound by means o
elemen a y column ope a ion, acco ding o he
scheme
elemen a y column
ope a ions

(27)
Elemen a y column ope a ions (27) may always
be lead in he way ha he polynomial ma ix PR(s)
emains as uni ma ix and he con e sion (25) is
i ial and also a uni one. Then no in e sion in (26)
is necessa y and he ealiza ion o he con ol law is
e y simple. In mo e complex cases, he s anda d
echniques based on Euclidean algo i hms can be
used, see [2], [8], [15]. Mo e complex ma ix
polynomial equa ions can be con enien ly sol ed by
Polynomial oolbox [17] as i is shown in Sec ion
IV.
6 Illus a i e Examples
Illus a i e examples 1 - 3 in his con ibu ion a e
i s o de s able, uns able and in eg a ing ones wo
inpu – wo ou pu (TITO) sys ems a e ep esen ed
by he ma ix equa ion
1 2 1 2
11
3 4 3 4
22
( ) ( )
( ) ( )
s a a b b
Y s U s
a s a b b
Y s U s

   
   

   
   
   
   
(28)
The s abiliza ion ma ix Diophan ine equa ion
(24) akes he o m
1
2
0
R
R
M
PZ
QZ





0
0
AF B
I
I





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1 2 1 2 1 2
3 4 3 4 3 4
2
1 0 5 4 0
2
3 2 7 6 0
0
0
( ) 0
0 ( )
s a a p p b b
s
a s a p p b b
s
q s q q s q sm
q s q q s q sm

     
 
     


     






 


(29)
Example 1: Le a TITO linea con inuous- ime
sys em be exp essed by he Laplace ans o m
echnique in he o m
1 1 2 1 2
2 2 1 1 2
'( ) 2 ( ) 0.8 ( ) 5 ( ) 6 ( )
'( ) 1.5 ( ) 0.6 ( ) 2 ( ) 3 ( )
y y y u u
y y y u u
   
   
(30)
The Laplace ans o m o equa ions (6) gi es
ma ices A, B
2 0.8 5 6
( ) , ( )
0.6 0.6 2 3
s
A s B s
s

   

   

   
(31)
The sys em desc ibed in (31) is e iden ly s able
because de A = s2 + 2.6s + 0.72 is a s able
polynomial. Then he scheme (27) can be applied
and he esul is in he o m o gene alized PI
con olle :
1 1 1 0 1 5 2 4 2
2 3 1 2 1 7 2 6 2
( ) ( )
( ) ( )
u q e q e d q e q e d
u q e q e d q e q e d
   
   
   

   

(32)
whe e con olle pa ame e s we e ob ained by
elemen a y column ope a ions acco ding scheme (5)
in he o m:
10
2
00
30
2
20
2 0.8
41
33
2
3
qm
qm
qm
qm


  

50
2
40
70
2
60
4 2.2
2
10 5.9
33
5
3
qm
qm
qm
qm
  



(33)
In (9) ei a e na u ally acking e o s and m0>0 is
a uning pa ame e in luencing con ol beha iou .
Example 2: Le an uns able TITO linea
con inuous- ime sys em can be exp essed by
di e en ial equa ions
1 1 2 1 2
2 2 1 1 2
'( ) ( ) ( ) ( ) 0.5 ( )
'( ) 0.5 ( ) 2 ( ) 0.8 ( ) 2 ( )
y y y u u
y y y u u
   
   
(34)
and he ma ix exp ession has he o m
11
22
( ) ( )
1 1 1 0.5
( ) ( )
2 0.5 0.8 2
Y s U s
s
Y s U s
s
   
   

   
   

   
   
(35)
Ma ix equa ion (27) gi es he con olle
ma ices PR, QR
1 0 5 4
3 2 7 6
10
01
RR
q s q q s q
P and Q q s q q s q





 
 
(36)
whe e pa ame e s a e
10
2
00
30
2
20
2.5 0.65
1.25
0.75
0.5
qm
qm
qm
qm


  

50
2
40
70
2
60
0.6 1.1
0.3
1.25 0.19
0.625
qm
qm
qm
qm
  



(37)
The o m o he con ol law (32) is again a
gene alized PI con olle .
Example 3: Le an in eg a ing (also uns able)
TITO linea con inuous- ime sys em can be
exp essed by di e en ial equa ions
1 2 1 2
2 1 1 2
'( ) ( ) ( ) 0.5 ( )
'( ) 0.5 ( ) 0.6 ( ) 1.5 ( )
y y u u
y y u u
  
  
(38)
De e minan o A(s) = s2 – 0.5 is e iden ly an
uns able one. The con olle is de i ed in a simila
way bu a he igh hand side o (29) is he s able
ma ix M(s) in he o m
2
112
2
2
( ) 0
( ) , , 0
0 ( )
sm
M s m m
sm






(39)
The choice o di e en mi > 0 gi es he
possibili y o di e en dynamics in bo h con olled
ou pu s. The con ol law is again in he o m o (32)
wi h he ollowing se o pa ame e s qi:
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11
2
01
31
2
21
2.5 0.21
1.25
0.42
0.5
qm
qm
qm
qm


  

52
2
42
72
2
22
0.83 1.25
0.42
0.5 1.67
0.83
qm
qm
qm
qm
  



(40)
The las example ep esen s a simple
asymme ical case, a wo inpu – one ou pu sys em
o he i s o de gi es also a kind o a gene alized
PI con olle .
Example 4: A con olled sys em is a wo inpu –
single ou pu (TISO) sys em desc ibed by he
di e en ial equa ion
1 1 1 2
'( ) 0.5 ( ) 0.5 ( ) 1.2 ( )y y u u   
(41)
Sys em (41) is e iden ly an uns able one. The
ini ial and inal s a e o he scheme o s abilizing
equa ion (24) is
and he con ol law akes he o m o wo equa ions
which can be also conside ed as a gene alized PI
con olle
1 0 1
2
2 0 1
( ) (4 1) ( )
( ) 0.833 ( )
u m e
u m e d



(42)
An impo an ema k is ha he e exis an in ini e
numbe o easible s abilizing con olle s. I depends
how o choose elemen a y column ope a ions in
educ ion (27). Con ol law (42) ep esen s
p opo ional con olle in u1( ) and an in eg a ing
one in he u2( ) con ol loop. Polynomial oolbox
[17] gi es a simila solu ion bu no necessa ily he
same one.
7 Simula ion Resul s
Ma lab and Simulink o e a sui able en i onmen
o modelling and simula ion o dynamic sys ems.
The Simulink scheme o wo inpu – wo ou pu
uns able sys em (7) wi h con olle (10) is depic ed
in Fig. 3.
Con ol esponses o s able TITO sys em
(Example 1) o uning pa ame e m0=1.5 and m0=3
a e shown in Fig. 4. The con ol esponses o he
uns able TITO sys em (example 2) o uning
pa ame e m0=1.5 and m0=3 a e shown in Fig. 5.
Fig. 3 simulink scheme o eedback uns able sys em
Examples 1 and 2 illus a e ha uning pa ame e
m0 > 0 in luences he dynamical beha io o he
con olled a iable in he s able as well as in he
uns able case. The pa ame e m0 > 0 ep esen s a
mul iple pole o he eedback cha ac e is ic
polynomial.
Fig. 4 con ol esponses o m0=1.5 and m0=3
(Example 1)
0 5 10 15 20 25 30 35 40
0
0.5
1
1.5
2
2.5
Time (s)
w, y
w1
y1 (m=1.5)
y2 (m=1.5)
w2
y1 (m=3)
y2 (m=3)
22
200
0
2
0
2
0.5 0.5 1.2 0.5 1.2
1
1 0 0 0 0
(4 1)
0 1 0 1 0
0 0 1 0 1
0.833
s m m
ss
ms
m
















WSEAS TRANSACTIONS on SYSTEMS and CONTROL
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Fig. 5 con ol esponses o m0=1.5 and m0=3
(Example 2)
In some cases, i can be use ul e e y con olled
a iable in luence in di e en dynamics. I is easily
ob ained by a di e en choice o poles in eedback
loops. The si ua ion is shown in Fig. 6 and Fig. 7 o
TITO in eg a ing sys em. While he esponse in Fig.
6 is o m1 = m2 = 1, he esponses in Fig. 7 a e o
he choice m1 = 1.5, m2 = 2.
Fig. 8 and Fig. 9 illus a e con ol esponses o he
wo inpu – single ou pu sys em sol ed in Example
4. Tuning pa ame e m0 > 0 again in luences he
con ol beha iou and dynamics.
Fig. 6 con ol esponses (Example 3) o in eg a ing
sys em o uning pa ame e s m1 = 1, m2 = 1.
Fig. 7 con ol esponses (Example 3) o in eg a ing
sys em o uning pa ame e s m1 = 1.5, m2 = 2.
Fig. 8 con ol esponses (Example 4) o TISO
sys em o uning pa ame e s m0=0.5.
Fig. 9 con ol esponses (Example 4) o TISO
sys em o uning pa ame e s m0=1.
0 5 10 15 20 25 30 35 40
-0.5
0
0.5
1
1.5
2
2.5
Time (s)
w, y
w1
y1 (m=1.5)
y2 (m=1.5)
w2
y1 (m=3)
y2 (m=3)
0 5 10 15 20 25 30 35 40
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
Time (s)
w, y, u
w1
y1
y2
w2
u1
u2
0 5 10 15 20 25 30 35 40
-2
-1
0
1
2
3
4
5
Time (s)
w, y, u
w1
y1
y2
w2
u1
u2
0 5 10 15 20 25 30 35 40
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Time (s)
w, y, u
w
y
u1
u2
0 5 10 15 20 25 30 35 40
-2
-1
0
1
2
3
4
5
Time (s)
w, y, u
w
y
u1
u2
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8 Conclusion
The pape deals wi h mul i a iable con ol o simple
con inuous- ime linea sys ems. The con olle
design is pe o med h ough a solu ion o a ma ix
Diophan ine equa ion. This app oach enables o
de ine one o a couple scala uning pa ame e s o
in luencing o con ol beha iou . The uning
pa ame e s ep esen poles o he cha ac e is ic
eedback equa ion. In he i s o de cases, he
solu ion and a inal con olle can be ob ained in
simple and explici o m pe o ming by elemen a y
column ope a ion o he gi en ma ices. Resul ing
con olle s hen a e o gene alized PI con olle s. All
simula ions and esul s a e clea ly demons a ed in
he Ma lab-Simulink en i onmen .
Acknowledgemen
This wo k was suppo ed by he Eu opean Regional
De elopmen Fund unde he p ojec CEBIA-Tech
No. CZ.11.05./2.1.00/03.0089.
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WSEAS TRANSACTIONS on SYSTEMS and CONTROL
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Volume 10, 2015