Solving a Problem of Adaptive Stabilization for Some Static MIMO Systems
The purpose of the paper is to show that it is possible to stabilize any uncertain multivariable static plant which gain matrix may be either square or nonsquare and may have an arbitrary rank remaining unknown for the designer. Methods. The methods based on recursive point estimation of unknown pla...
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| Cite this: | Solving a Problem of Adaptive Stabilization for Some Static MIMO Systems / L.S. Zhiteckii, V.N. Azarskov, K.Yu. Solovchuk // Cybernetics and computer engineering. — 2019. — № 3 (197). — С. 33-50. — Бібліогр.: 29 назв. — англ. . |
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nasplib_isofts_kiev_ua-123456789-1617552025-02-23T18:35:06Z Solving a Problem of Adaptive Stabilization for Some Static MIMO Systems Розв'язування однієї задачі адаптивної стабілізації деяких статичних МІМО систем Решение одной задачи адаптивной стабилизация некоторых статических МИМО систем Zhiteckii, L.S. Azarskov, V.N. Solovchuk, K.Yu. Intelligent Control and Systems The purpose of the paper is to show that it is possible to stabilize any uncertain multivariable static plant which gain matrix may be either square or nonsquare and may have an arbitrary rank remaining unknown for the designer. Methods. The methods based on recursive point estimation of unknown plant parameters are utilized to design the adaptive inverse model-based controller. Results. The asymptotic properties of the adaptive controllers have been established. Simulation results have been presented to support the theoretic studies. Мета статті — показати, що можна стабілізувати довільний невизначений багатовимірний статичний об'єкт, матриця коефіцієнтів підсилення якого може бути квадратною або прямокутною і мати довільний ранг, залишаючись невідомою конструктору системи. Результати. Встановлено асимптотичні властивості адаптивних регуляторів. Щоб підкріпити теоретичні дослідження, надано результати моделювання. Цель статьи — показать, что можно стабилизировать любой неопределенный многомерный статический объект, матрица коэффициентов усиления которого может быть квадратной или прямоугольной и иметь произвольный ранг, оставаясь неизвестной конструктору системы. Результаты. Установлены асимптотические свойства адаптивных регуляторов. Чтобы подкрепить теоретические исследования, представлены результаты моделирования. 2019 Article Solving a Problem of Adaptive Stabilization for Some Static MIMO Systems / L.S. Zhiteckii, V.N. Azarskov, K.Yu. Solovchuk // Cybernetics and computer engineering. — 2019. — № 3 (197). — С. 33-50. — Бібліогр.: 29 назв. — англ. . 2663-2578 DOI: https://10.15407/kvt197.03.033 https://nasplib.isofts.kiev.ua/handle/123456789/161755 681.5 en Кибернетика и вычислительная техника application/pdf Міжнародний науково-навчальний центр інформаційних технологій і систем НАН України та МОН України |
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Intelligent Control and Systems Intelligent Control and Systems |
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Intelligent Control and Systems Intelligent Control and Systems Zhiteckii, L.S. Azarskov, V.N. Solovchuk, K.Yu. Solving a Problem of Adaptive Stabilization for Some Static MIMO Systems Кибернетика и вычислительная техника |
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The purpose of the paper is to show that it is possible to stabilize any uncertain multivariable static plant which gain matrix may be either square or nonsquare and may have an arbitrary rank remaining unknown for the designer. Methods. The methods based on recursive point estimation of unknown plant parameters are utilized to design the adaptive inverse model-based controller. Results. The asymptotic properties of the adaptive controllers have been established. Simulation results have been presented to support the theoretic studies. |
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Article |
| author |
Zhiteckii, L.S. Azarskov, V.N. Solovchuk, K.Yu. |
| author_facet |
Zhiteckii, L.S. Azarskov, V.N. Solovchuk, K.Yu. |
| author_sort |
Zhiteckii, L.S. |
| title |
Solving a Problem of Adaptive Stabilization for Some Static MIMO Systems |
| title_short |
Solving a Problem of Adaptive Stabilization for Some Static MIMO Systems |
| title_full |
Solving a Problem of Adaptive Stabilization for Some Static MIMO Systems |
| title_fullStr |
Solving a Problem of Adaptive Stabilization for Some Static MIMO Systems |
| title_full_unstemmed |
Solving a Problem of Adaptive Stabilization for Some Static MIMO Systems |
| title_sort |
solving a problem of adaptive stabilization for some static mimo systems |
| publisher |
Міжнародний науково-навчальний центр інформаційних технологій і систем НАН України та МОН України |
| publishDate |
2019 |
| topic_facet |
Intelligent Control and Systems |
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https://nasplib.isofts.kiev.ua/handle/123456789/161755 |
| citation_txt |
Solving a Problem of Adaptive Stabilization for Some Static MIMO Systems / L.S. Zhiteckii, V.N. Azarskov, K.Yu. Solovchuk // Cybernetics and computer engineering. — 2019. — № 3 (197). — С. 33-50. — Бібліогр.: 29 назв. — англ.
. |
| series |
Кибернетика и вычислительная техника |
| work_keys_str_mv |
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| first_indexed |
2025-11-24T11:13:57Z |
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2025-11-24T11:13:57Z |
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| fulltext |
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 3 (197)
DOI: https://10.15407/kvt197.03.033
UDC 681.5
L.S. ZHITECKII1, PhD (Engineering),
Acting Head of the Intelligent Automatic Systems Department
e-mail: leonid_zhiteckii@i.ua
V.N. AZARSKOV2, DSc. (Engineering), Professor,
Chief of the Aerospace Control Systems Department,
e-mail: azarskov@nau.edu.ua
K.Yu. SOLOVCHUK3,
Assistant of the Department of Computer Information Technologies and Systems
e-mail: solovchuk_ok@ukr.net
1 International Research and Training Center for Information Technologies
and Systems of the National Academy of Sciences of Ukraine
and Ministry of Education and Science of Ukraine,
40, Acad. Glushkov av., Kyiv, 03187, Ukraine
2 National Aviation University, Kyiv, Ukraine.
1, Kosm. Komarova av., Kyiv, 03680, Ukraine
3 Poltava National Technical Yuri Kondratyuk University, Poltava, Ukraine.
24, Pershotravneva av., Poltava, 36011, Ukraine
SOLVING A PROBLEM OF ADAPTIVE STABILIZATION
FOR SOME STATIC MIMO SYSTEMS
Introduction. The adaptive stabilization of some classes of uncertain multivariable static
plants with arbitrary unmeasurable bounded disturbances is addressed in this article. The
cases where the number of the control inputs does not exceed the number of the outputs are
studied. It is assumed that the plant parameters defining the elements of its gain matrix are
unknown. Again, the rank of this matrix may be arbitrary. Meanwhile, bounds on external
disturbances are supposed to be known. The problem stated and solved in this work is to
design adaptive controllers to be able to ensure the boundedness of the all input and output
system’s signals in the presence of parameter uncertainties.
The purpose of the paper is to show that it is possible to stabilize any uncertain multi-
variable static plant which gain matrix may be either square or nonsquare and may have an
arbitrary rank remaining unknown for the designer.
Methods. The methods based on recursive point estimation of unknown plant parame-
ters are utilized to design the adaptive inverse model-based controller.
Results. The asymptotic properties of the adaptive controllers have been established.
Simulation results have been presented to support the theoretic studies.
© L.S. ZHITECKII, V.N. AZARSKOV, K.Yu. SOLOVCHUK, 2019
33
Zhiteckii L.S., Azarskov V.N., Solovchuk K.Yu.
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 3 (197) 34
Conclusion. The adaptive control laws proposed in the paper can guarantee the bound-
edness of all the signals generated by the feedback control systems. However, this important
feature will achieve via an “overparameterization” of these systems. Nevertheless, the simu-
lation experiments demonstrate their efficiency.
Keywords: adaptive control, boundedness, discrete time, estimation algorithm, feedback,
multivariable system, uncertainty.
INTRODUCTION
The problem of efficient control of multivariable systems with arbitrary unmeasur-
able external disturbances stated several decades ago remains important both from
theoretical and practical points of view until recently. Novel results in this scientific
area have been reported in numerous papers and generalized in several books includ-
ing [1–3]. This problem attracts an attention of many researchers dealing with the
design of optimal controllers for controlling the so-called multi-inputs multi-outputs
(MIMO) system by using different approaches.
Among other methods advanced in the modern control theory, the inverse
model-based method that is an extension of the well-known internal model prin-
ciple seems to be perspective in order to cope with arbitrary unmeasurable dis-
turbances and to optimize some classes of multivariable control systems. It
turned out that this method first intuitively advanced in [4] makes it possible to
optimize the closed-loop control system containing the MIMO static (memory-
less) plants whose gain matrices are square and nonsingular. Since the beginning
of the 21st century, a significant progress has been achieved utilizing the inverse
model-based approach, e.g., [5] and other works. Nevertheless, it is quite unac-
ceptable if the MIMO plants to be controlled have singular square or else any
nonsquare gain matrices because they are noninvertible.
To optimize the closed-loop control system containing an arbitrary MIMO
static plant, the pseudoinverse model-based approach has been proposed and
substantiated in [6]. Naturally enough that its gain matrix must be known to
implement this approach. In practice, however, the plant parameters defining the
elements of gain matrices may not be known a priori. In this case, the problem
of designing the so-called robust multivariable control system may be stated.
The monographs [7–9] give a fairly full picture concerning the results achieved
in the robust control theory to the beginning of the 2000s. Within the framework of
this theory, the pseudoinverse model-based method has been modified in [10–12] to
stabilize some classes of uncertain interconnected linear and nonlinear systems
whose gain matrices are arbitrary. (Note that the problem of robust control of some
nonlinear one-dimensional static plant has before been solved in the work [13].).
Unfortunately, the pseudoinverse model-based controller having fixed parameters
may not be suitable if the parameter uncertainty is great enough.
An adaptation concept plays a role of some universal tool to deal with the
control of uncertain systems [8, 14–20], et al. This concept has been employed
in the papers [21–23] in which adaptive controllers for controlling fix linear and
nonlinear multivariable static plants have been designed and studied, assuming
that their gain matrices are nonsingular square matrices. The latest results with
respect to robust adaptive control of the linear and some nonlinear static plants
having one output and several control inputs can be found in [8, chap. 3].
Solving a Problem of Adaptive Stabilization for Some Static Mimo Systems
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 3 (197) 35
In [24], the adaptive pseudoinverse model-based control has been proposed
to stabilize a nonsquare MIMO plant having the gain matrix of full rank in the
absence of disturbances. Recently, the problem of the stabilization of single-
input multi-outputs (SIMO) static systems with bounded disturbances has been
solved in [25]. In [26, 27] the adaptive control systems containing the intercon-
nected plants with both square and nonsquare matrices of the nonfull ranks in the
presence of bounded disturbances have been designed and argued.
Difficulties that take place when adaptive control use the point estimation
algorithms are how to guarantee the stability (the boundedness) of the closed-
loop system [28]. See also [14, 15]. To overcome these difficulties in the case of
the singular square system, the so-called fictitious plant to be controlled adap-
tively has been introduced in the closed-loop circuit [26]. The idea of the simul-
taneous adaptive control of the true and of fictitious plants advanced in this work
turned out fruitful to deal with adaptive stabilization of any MIMO static plants
irrespective of the ranks of their gain matrices [27].
The purpose of the paper is to generalize results obtained in [26, 27] and
to show that within the framework of the adaptive approach, it is possible to
stabilize the arbitrary MIMO static uncertain plant without knowledge concern-
ing both the elements and also rank of its gain matrix.
STATEMENT OF THE PROBLEM
Let
11 −− += nnn vBuy (1)
be the equation describing a MIMO plant with measurable m-dimensional output
vector, unmeasured m-dimensional disturbance vector and the r-dimensional
control vectors related to the nth discrete time ),2,1( K=n are
,],...,[ )()1( Tm
nnn yyy = Tm
nnn vvv ],...,[ )()1(= and ,],...,[ )()1( Tr
nnn uuu = respectively.
B represents some time-invariant rm× gain matrix given by
⎟⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎝
⎛
=
)()1(
)1()11(
mrm
r
bb
bb
B
K
LLL
K
. (2)
Consider the class of MIMO plants, where the number r of the control inputs is not
less than two, but does not exceed the number m of the outputs, i.e., .2 mr ≤≤
The following assumptions with respect to the gain matrix B and the se-
quences K,,}{ )(
1
)(
0
)( iii
n vvv = are made.
A1. The elements of the matrix B in (2) are all unknown. However, there
are some interval estimates
rjmibbb
ijijij ,,1,,,1,
)()()(
KK ==≤≤ (3)
with the known upper and lower bounds )(ijb and ,
)(ij
b respectively. This im-
plies that B may be singular, in principle.
Zhiteckii L.S., Azarskov V.N., Solovchuk K.Yu.
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 3 (197) 36
A2. The rank of B remaining unknown, in general, may be arbitrary num-
ber which satisfies
}).,min{(rank1 mrrB =≤≤ (4)
A3. }{ )(i
nv ),,1( mi K= are all the arbitrary scalar sequences bounded in
modulus according to
,|| )( ∞<ε≤ i
i
nv (5)
where siε are constant. For simplicity of exposition, it is assumed that they are
known.
Denote by Tmyyy ],...,[ )(0)1(00 = the desired m-dimensional output vector.
Without loss of generality, suppose 0|||| )(0)1(0 ≠++ myy K implying that
∞<< ||||0 0y const( )(0 ≡iy ).,,1 mi K=∀
Define the output error vector
.0
nn yye −= (6)
of the current errors )()(0)( i
n
ii
n yye −= for each ith output )(i
ny giving
.],...,[ )()1( Tm
nnn eee = Then the control objective is to design an adaptive controller
stabilizing the unknown plant (1). More exactly within the framework of as-
sumptions A1) – A3), it is required to guarantee the ultimate boundedness of the
sequences }{ ne and }{ nu in the form
,||||suplim ∞<
∞→
n
n
e (7)
.||||suplim ∞<
∞→
n
n
u (8)
THE CASE OF SQUARE NONSINGULAR GAIN MATRICES
Suppose that B is a square nonsingular rr × matrix meaning mr = and
.0det ≠B (9)
In this case, the control law may be chosen as in [17, sect. 4.2] setting
,1
1 nnnn eBuu −
− += (10)
where ne is given by (6), and 1−
nB denotes the matrix obtained via the inversion
of the current estimate matrix nВ for unknown B.
According to [17, sect. 4.2], the rows of nВ defining the vectors
),,1(],,[ T)()1()( ribbb ir
n
i
n
i
n KK == are updated by exploiting the recursive adap-
tation algorithm
Solving a Problem of Adaptive Stabilization for Some Static Mimo Systems
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 3 (197) 37
⎪
⎩
⎪
⎨
⎧
∇
∇
ε−
γ−
ε≤
=
−
−
−
−
otherwise.
||||
sign
,|| if
12
21
)()(
)()(
1
0)()(
1
)(
n
n
i
ni
i
ni
n
i
n
i
i
n
i
n
i
n u
u
eeb
eb
b (11)
In this expression, )(i
ne represents the ith component of ne given by (6). 2|||| x
denotes the Euclidean norm of some s-dimensional vector T)()1( ],,[ sxxx K=
determined as .][][|||| 2)(2)1(
2
sxxx ++= K The variable 1: −−=∇ nnn uuu is the
increment of .nu s0
iε are arbitrary fixed numbers satisfying
.,,1,20 riiii K=ε=ε>ε (12)
The coefficients s)(i
nγ are chosen as
20 )()()( <γ≤γ≤γ< ii
n
i (13)
to ensure
.0det ≠nB (14)
The asymptotical behavior of the adaptive control algorithm (10), (11) to-
gether with (12) to (14) is given in the theorem below.
Theorem 1. Consider the closed-loop stabilization system containing the
plant (1) and the feedback adaptive controller described in the expressions
(10)–(14). If the conditions (5) and (9) are satisfied then the control objectives
(7) and (8) are achieved.
Proof. Follows from the results presented in [14, subsect. 4.2.3]. □
THE CASE OF SQUARE SINGULAR GAIN MATRICES
Let B be a square singular rr × matrix, i.e.,
.0det =B (15)
Basic idea to deal with a matrix B satisfying (15) is the transition from the adaptive
identification of the true plant having the singular gain matrix B to the adaptive
identification of a fictitious plant with the nonsingular gain matrix B~ of the form
,~
0 rIBB δ+= (16)
where rI denotes the identity rr × matrix and 0δ is a fixed quantity [26].
Although B~ as well as B remain unknown, the requirement
0~det ≠B (17)
can always be satisfied by the suitable choice of . 0δ in (16). In fact, each ith
eigenvalue )(Biλ of B lies in one of the r closed regions of the complex z-
plane consisting of all the Gerŝgorin discs [29, p. 146]:
Zhiteckii L.S., Azarskov V.N., Solovchuk K.Yu.
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 3 (197) 38
.,,1,||||
1
)()( ribbz
r
ij
j
ijii K=≤− ∑
≠
=
(18)
Since, at least, one of the eigenvalues )(Biλ is equal to zero (due to the sin-
gularity of ),B by virtue of (17) there are the numbers
∑∑
≠
=
≠
=
+=β−=β
r
ij
j
ijiiir
ij
j
ijiii bbbb
1
)()()(
1
)()()( ||:,||: (19)
such that if
0|||| )()1( ≠++ iri bb K , (20)
then either 0)( ≤β i but 0
)(
>β
i
or 0)( <β i but .0
)(
≥β
i
These numbers are de-
fined as the intersection of the ith Gerŝgorin disc with the real axis of the com-
plex z-plane as show in Figs 1 and 2, respectively, left. In both cases, 0
)()( ≤ββ
ii
if (20) is satisfied because )(iβ and
)(i
β cannot have the same sign.
Fig. 1. The Gerŝgorin discs for r=2 in the case ||||
)1()2( β<β
Fig. 2. The Gerŝgorin discs for r=2 in the case |||| )1()2(
β<β
Solving a Problem of Adaptive Stabilization for Some Static Mimo Systems
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 3 (197) 39
Denoting
},,,max{:},,,min{:
)()1()()1( rr ββ=βββ=β KK (21)
consider the following two cases: a) |;||| β<β b) |||| β>β (The case when
|||| β=β can be combined with any two cases.) In order to go to the gain matrix
B~ of the fictitious plant having the form (16) in the case a), it is sufficient to
shift the Gerŝgorin disc (18) right taking
|,|0 β>δ (22)
as shown in Fig. 1, right. In the case b), the discs (9) need to be shifted left ac-
cording to
.||0 β−<δ (23)
See Fig. 2, right. In both cases, the nonsingularity of B~ is guaranteed. Neverthe-
less, the conditions (22) and (23) cannot be satisfied, as yet. In fact, the numbers
β and β given by the expressions (21) depend of )(iβ and s)(iβ defined by
(19). But they are unknown because s)(ijb are all unknown.
The following actions are proposed to choose a number 0δ satisfying (17).
Introduce
|},|,|max{|:
|},|,|max{|:
)(
1
)()()(
max
)(
1
)()()(
min
ijr
ij
j
ijiii
ijr
ij
j
ijiii
bbb
bbb
∑
∑
≠
=
≠
=
+=β
−=β
(24)
minimizing and maximizing in ],[
)()()( ijijij bbb ∈ the right sides of (19) for )(iβ
and ,
)(i
β respectively.
Further, the number 0δ is found to satisfy the conditions
|,|||if
|,|||if
maxminmax0
maxminmin0
β>ββ−<δ
β<ββ−>δ
(25)
where maxmin
, ββ represent some quantities defined as follows:
}.,,max{:
},,,min{:
)(
max
)1(
maxmax
)(
min
)1(
minmin
r
r
ββ=β
ββ=β
K
K
(26)
Zhiteckii L.S., Azarskov V.N., Solovchuk K.Yu.
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 3 (197) 40
It can be clarified that if (25) together with (24) and (26) will be satisfied
then the condition (17) will without fail be ensured.
After determining the quantity 0δ we can proceed to the consideration of
the fictitious plant. Since the input variables )()1( ,, r
nn uu K and the disturbances
)()1( ,, N
nn vv K of both true plant and fictitious plant are the same, this feature
makes it possible to describe our fictitious plant by the equation
,~~
11 −− += nnn vuBy (27)
similar to (1). In this equation, T)()1( ]~,,~[~ r
nnn yyy K= denotes the output vector of
the fictitious plant.
It is interesting that the components of ny~ can be measured while the com-
ponents of nv in (28) remain unmeasurable. In fact, substituting (16) into (27)
due to (1) we produce
.~
10 −δ+= nnn uyy (28)
It is seen from (28) that ny~ can always be found indirectly having nu and ny to
be measured.
Now, our problem reduces to the known problem of adaptive control appli-
cable to the fictitious plant (27) with the unknown gain matrix B~ in the presence
of arbitrary bounded disturbances .,, )()1( r
nn vv K Its solving follows the steps of the
section above. Namely, the adaptive control law is designed in the form
,~~ 1
1 nnnn eBuu −
− += (29)
in which, instead of the current estimate nB of ,B another nB~ is exploited, and
the error vector ne defined in (6) is replaced by
nn yye ~~ 0 −= (30)
with ny~ given by the expression (28).
The adaptive identification algorithm used to determine the estimates nB~
may be taken as
⎪
⎩
⎪
⎨
⎧
=∇
∇
ε−
γ+
ε≤
=
−
−
∗∗
−
∗
−
,,,1otherwise,
||||
~sign ~~
,|~| if ~
~
12
21
)()(
)()(
1
0)()(
1
)(
riu
u
eeb
eb
b
n
n
i
ni
i
ni
n
i
n
i
i
n
i
n
i
n
K
(31)
which is similar to (11). In this algorithm, 0
iε and iε are given by (12).
1
T)(
1
)()( ~~~
−−
∗ ∇−∇= n
i
n
i
n
i
n ubye (32)
represent the ith component of the identification error ∗
ne~ given as
Solving a Problem of Adaptive Stabilization for Some Static Mimo Systems
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,~~~
11 −−
∗ ∇−∇= nnnn uBye (33)
where ,~~:~ )(
1
)()( i
n
i
n
i
n yyy −−=∇ and the notation ]~,,~[:~ )()1(Т)( ir
n
i
n
i
n bbb K= of the ith row
of nB~ is introduced. The coefficients s)(i
nγ are chosen as in (13) to
.0~det ≠nB (34)
The feedback adaptive robust control system described in the equations (1),
(29), (31) is designed as depicted in Fig. 3. In this figure, the notation
11
~:~
−−
∗ ∇=∇ nnn uBy is introduced.
The asymptotic properties of the adaptive control system are established in
the following theorem.
Theorem 2. Determine 0δ using the formula (25) together with (24) and
(26), and choose an arbitrary initial IBB 000
~
δ+= with }{ )(
00
ijbB = whose ele-
ments satisfy the conditions .
)()(
0
)( ijijij bbb ≤≤ Subject to assumptions A1 – A3,
the adaptive controller described in the equations (29), (31) together with (28),
(30) when applied to the plant (1) yields (7), (8).
Proof. See [26].
Fig. 3. Configuration of adaptive stabilization system
Zhiteckii L.S., Azarskov V.N., Solovchuk K.Yu.
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 3 (197) 42
THE CASE OF NONSQUARE GAIN MATRICES
WITH ARBITRARY RANKS
Let B be a nonsquare rm× matrix of the form (2) with unknown rank satisfy-
ing (4). Define the so-called submatrices rr
r rkikiB ×∈R],,1|][,],[[ 1 KK [29,
part I, subsect. 2.2] whose rows represent the rows of B with the numbers
][,],[1 kiki rK ).][][1( 1 mkiki r ≤<<≤ K The quantity of these matrices is equal
to .⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
r
m
N Denoting by ][kB the submatrix which corresponds to a kth subset
]},[,],[{ 1 kiki rK write the equations of some k plants as:
,,,1],[][][ 11 NkkvukBky nnn K=+= −− (35)
where ,.],,[][ T])[(])[( 1 rki
n
ki
nn
ryyky R∈= K and .],,[][ T])[(])[( 1 rki
n
ki
nn
rvvkv R∈= K
In accordance with the approach proposed in the previous section, pass from
(35) to the equations of the fictitious plants described by
NkkvukBky nnn ,,1],[][~][~
11 K=+= −− (36)
with the same 1−nu and ].[1 kvn− In these equations, ][~ kyn denotes the r-
dimensional output vector related to the kth fictitious plant whose gain matrix
][~ kB is defined as follows:
,][][][~
0 rIkkBkB δ+= (37)
where ][0 kδ is a fixed quantity depending on k. This quantity is calculated for
each Nk ,,1 K= using the technique described in the previous section. Namely,
taking into account the constraints (3), ][0 kδ can always be found to ensure
.,,10][~det NkkB K=∀≠ (38)
It follows from (35) to (37) that
.][][][~
10 −δ+= nnn ukkyky (39)
This expression shows that although as ][~ kB as ][kB remain unknown, how-
ever, the components of all N the vectors ][~ kyn can indirectly be “measured”
after measuring the components of ny and ,1−nu and it is essential.
If the conditions (38) are satisfied, then the problem of the adaptive stabili-
zation of the true plant (1) can be reduced to the problem of simultaneous adap-
tive stabilization of all N fictitious plants (36) with unknown but nonsingular
rr × gain matrices ),,1(][~ NkkB K= via forming at each nth time instant a
set of N different “potentially” possible controls ][,],1[ Nuu nn K and selecting
one of them in accordance with certain choice rule [27] given below.
Solving a Problem of Adaptive Stabilization for Some Static Mimo Systems
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Following to [27], the adaptive control law to be applicable to any fictitious
plant is designed in the form
,,,1],[~][~][ 1
1 NkkekBuku nnnn K=+= −
− (40)
where ][~][][~ 0 kykyke nn −= with T])[(0])[(00 ],,[][ 1 kiki ryyky K= defines the out-
put error vector related to the kth fictitious plant at the nth time instant, and
rr
n kB ×∈R][~ is the current estimate of unknown rr × matrix ][~ kB at the same
time instant satisfying
.,,10][~det NkkBn K=∀≠ (41)
As the adaptation algorithms, the standard recursive procedures for the
adaptive identification of each kth fictitious plant (35) described by
⎪
⎪
⎩
⎪
⎪
⎨
⎧
==
∇
∇
ε−
γ+
ε≤
= −
−
∗∗
−
∗
−
Nkri
u
u
kekekb
kekb
kb n
n
i
ni
i
ni
n
i
n
i
i
n
i
n
i
n
,,1,,,1
otherwise,
||||
][~sign ][~
][~
,|][~| if ][~
][~
12
21
)()(
)()(
1
0)()(
1
)(
KK
(42)
are proposed. In these algorithms, ][~ )( kb i
n denotes the r-dimensional estimate
vector obtained by transposing the ith row of ],[~ kBn and
1
T)(
1
)(
1
)()( ][~][~][~][~
−−−
∗ ∇−−= n
i
n
i
n
i
n
i
n ukbkykyke (43)
represents the scalar variable making sense of the ith component of r
n ke R∈∗ ][~
that is the identification error vector related to the kth fictitious plant. The coef-
ficients s)(i
nγ are chosen from the ranges ],[ )()( ii γγ (similarly to that in (13)) to
ensure the requirement (41).
Next, add the adaptation algorithms described in the formulas (42) together
with (43) by an algorithm for estimating unknown B defined as follows:
⎪
⎩
⎪
⎨
⎧
=∇
∇
ε−
γ+
ε≤
=
−
−
∗∗
−
∗
−
,,,1otherwise,
||||
sign
,|| if
12
21
)()(
)()(
1
0)()(
1
)(
miu
u
ee
b
eb
b
n
n
i
ni
i
ni
n
i
n
i
i
n
i
n
i
n
K
(44)
where T)(i
nb represents the ith row of the estimate matrix ,nB and
1
T)(
1
)(
1
)()(
−−−
∗ ∇−−= n
i
n
i
n
i
n
i
n ubyye (45)
Zhiteckii L.S., Azarskov V.N., Solovchuk K.Yu.
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 3 (197) 44
is the ith component of the identification error vector 111 −−−
∗ ∇−−= nnnnn uByye
( iε and 0
iε are given by (12)) .
The estimation procedure defined in (44) together with (45) makes it possi-
ble to estimate the m predicted output errors ),,1(][)(
1 mike i
n K
r
=+ for the each
ith output of true plant (1) at any n using the formula
.,,1,|][||][| )(T)()(0)(
1 mikubyke i
n
i
n
ii
n K
r
=ε+−=+ (46)
The synthesis of the adaptive controller is finished by the choice of the con-
trol nu from the set ]}[,],1[{ Nuu nn K with ][kun given by (40). This choice is
implemented by the rule giving the minimum of the 1-norm of
T)(
1
)1(
11 ]][,],[[][ kekeke m
nnn +++ =
r
K
rr
as
,|][|minarg
1
)(
1][ ∑
=
+=
m
i
i
nkun keu
n
r (47)
where s][)(
1 ke i
n+
r are specified by (46).
Remark. The definition of the 1-norm 1|||| ⋅ can be found in [7, p. 260].
The asymptotic properties of the adaptive controller described in this section
are given in theorem below (the main result).
Theorem 3. Consider the feedback control system containing the plant (1)
in which ,mr < and the adaptive controller defined in (42), (47) together with
(39), (46) and (41). Using the constants (3), determine ][,],1[ 00 Nδδ K to satisfy
(38). Let assumption A1–A3 be valid. Then, this controller applied to plant (1)
guarantees that the control objectives (7) and (8) will be achieved.
Proof. Follows the lines of [14, chap. 4]. (Due to space limitation, de-
tails are omitted.)
Note that Theorem 3 does not guarantee that the ultimate error
||||suplim nn e∞→ will become as in the nonadaptive case when there is no pa-
rameter uncertainty and the pseudoinverse model-based controller proposed in
[6] can by applied.
SIMULATION
A simulation experiment was conducted to illustrate the performance of the pro-
posed adaptive control in the case when .3,2 == mr As the gain matrix,
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
=
5.13
12
24
B
Solving a Problem of Adaptive Stabilization for Some Static Mimo Systems
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 3 (197) 45
with nonfull rank )1rank ( =B was taken. Since ,3=N it produces the follow-
ing three submatrices:
.
5.13
12
]3[and
5.13
24
]2[,
12
24
]1[ ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
= BBB
Further, the three vectors ,],[]1[ T)2()1(
nnn yyy = T)3()1( ],[]2[ nnn yyy = and
T)3()2( ],[]3[ nnn yyy = was introduced to describe the plants (35) having the gain
matrices ],3[and]2[],1[ BBB respectively.
The quantities 3.1]3[and2.1]2[,1.1]1[ 000 =δ=δ=δ guaranteeing ][~ kB
to be nonsingular were derived from (3). The initial ]3[~and]2[~],1[~
000 BBB were
chosen as rIkkBkB ][][][~
000 δ+= with the initial elements of ][0 kB which were
selected from B inside the corresponding ranges ],[
)()( ijij bb specified as fol-
lows: ],2,0[],5,1[ )12()11( ∈∈ bb ],2,0[)21( ∈b ],2,1[)22( ∈b ],4,1[)31( ∈b
].5,0[)32( ∈b Namely, we set ,1)11(
0 =b 1)12(
0 =b ,0)21(
0 =b ,9.1)22(
0 =b ,2)31(
0 =b
.1.2)32(
0 =b The desired output vector was given as .]7,3,1[ T0 =y
The performance of the simulated adaptive control system with the distur-
bance sequences K,,}{ )(
1
)(
0
)( iii
n vvv = generated as some pseudorandom i.i.d.
variables taken from ,1.01.0 )1( ≤≤− nv ,2.02.0 )2( ≤≤− nv 08.008.0 )3( ≤≤− nv is
presented in Figs. 4 and 5.
Fig. 4. Variables describing the adaptive estimation processes
Zhiteckii L.S., Azarskov V.N., Solovchuk K.Yu.
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 3 (197) 46
Fig. 5. The behavior of the control system: a) the current number k
of control ][kun chosen from ]}3[],2[],1[{ nnn uuu at given n; b)
the 1-norm of control vector; c) the 1-norm of output vector in adap-
tive case (solid line) and in nonadaptive optimal case (dashed line)
Figs. 5, a — c demonstrate that the performance of the proposed adaptive
controller applied to the static MIMO plant having some nonsquare gain matrix
with nonfull rank is successful enough.
CONCLUSION
It has been established that the adaptive control laws can guarantee the bound-
edness of all the signals generated by the feedback control systems. However,
this important feature will achieve via an “overparameterization” of these sys-
tems. Nevertheless, the simulation experiments demonstrate their efficiency.
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Received 30.05.2019
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turbances. Journal of Automation and Information Sciences. 2017. vol. 49. no. 5. P. 35–48.
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uncertainty. Autom. Remote Control. 1987. no. 1. P. 76–88.
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Отримано 30.05.2019
Л.С. Житецький1, канд. техн. наук,
в.о. зав. відд. інтелектуальних автоматичних систем
e-mail: leonid_zhiteckii@i.ua
В.М. Азарсков2, д-р техн. наук, професор,
зав. каф. аерокосмічних систем керування
e-mail: azarskov@nau.edu.ua
К.Ю. Соловчук3, аспірантка,
асистентка каф. комп’ютерних інформаційних
технологій та систем
e-mail: solovchuk_ok@ukr.net
1 Міжнародний науково-навчальний центр інформаційних
технологій та систем НАН України і МОН України,
пр. Акад. Глушкова, 40, м. Київ, 03187, Україна
2 Національний авіаційний університет,
пр. Космонавта Комарова, 1, м. Київ, 03680, Україна
3 Полтавський національний технічний університет імені Юрія Кондратюка,
пр. Першотравневий, 24, м. Полтава, 36011, Україна
РОЗВ'ЯЗУВАННЯ ОДНІЄЇ ЗАДАЧІ АДАПТИВНОЇ СТАБІЛІЗАЦІЇ
ДЕЯКИХ СТАТИЧНИХ МІМО СИСТЕМ
Вступ. У статті розглянуто задачу адаптивної стабілізації деяких класів невизначених бага-
товимірних статичних об'єктів з довільними невимірними обмеженими збуреннями. Дослі-
джено випадки, коли кількість входів керування не перевищує кількість виходів. Припуще-
но, що параметри об'єкта, що визначають елементи матриці коефіцієнтів підсилення, неві-
домі. Окрім того, ранг цієї матриці може бути довільним. Водночас, межі зовнішніх збу-
рень повинні бути відомі. Поставлена та вирішена у роботі задача полягає в тому, щоб
побудувати адаптивний регулятор, здатний забезпечити обмеженість всіх вхідних і вихід-
них сигналів системи за наявності параметричних невизначеностей.
Мета статті — показати, що можна стабілізувати довільний невизначений багато-
вимірний статичний об'єкт, матриця коефіцієнтів підсилення якого може бути квадрат-
ною або прямокутною і мати довільний ранг, залишаючись невідомою конструктору
системи.
Методи. Методи, що базуються на рекурентному точковому оцінюванні невідо-
мих параметрів об'єкта, використовуються для побудови адаптивного регулятора на
основі оберненої моделі.
Zhiteckii L.S., Azarskov V.N., Solovchuk K.Yu.
ISSN 2663-2586 (Online), ISSN 2663-2578 (Print). Cyb. and comp. eng. 2019. № 3 (197) 50
Результати. Встановлено асимптотичні властивості адаптивних регуляторів. Щоб
підкріпити теоретичні дослідження, надано результати моделювання.
Висновки. Адаптивні закони керування, що пропонуються в статті, можуть гаранту-
вати обмеженість всіх сигналів, що генеруються системами керування зі зворотним зв'яз-
ком. Однак це важлива властивість буде досягатися за рахунок «зверхпараметризації» цих
систем, і модельні експерименти показують їхню ефективність.
Ключові слова: адаптивне керування, обмеженість, дискретний час, алгоритм оціню-
вання, зворотний зв'язок, багатовимірна система, невизначеністью.
Л.С. Житецкий1, канд. техн. наук,
и.о. зав. отд. интеллектуальных автоматических систем
e-mail: leonid_zhiteckii@i.ua
В.Н. Азарсков2, д-р. техн. наук, профессор,
зав. кафедрой аэрокосмических систем управления
e-mail: azarskov@nau.edu.ua
К.Ю. Соловчук, аспирантка
ассистентка каф. компьютерных информационных технологий и систем
e-mail: solovchuk_ok@ukr.net
1 Международный научно-учебный центр информационных технологий
и систем НАН Украины и МОН Украины,
пр. Аккад. Глушкова, 40, г. Киев, 03187, Украина
2 Национальный авиационный университет,
пр. Космонавта Комарова, 1, г. Киев, 03680, Украина
3 Полтавский национальный технический университет имени Юрия Кондратюка,
пр. Первомайский, 24, г. Полтава, 36011, Украина
РЕШЕНИЕ ОДНОЙ ЗАДАЧИ АДАПТИВНОЙ СТАБИЛИЗАЦИЯ
НЕКОТОРЫХ СТАТИЧЕСКИХ МИМО СИСТЕМ
Введение. В статье рассмотрена задача адаптивной стабилизации некоторых классов
неопределенных многомерных статических объектов с произвольными неизмеряемыми
ограниченными возмущениями. Исследованы случаи, когда количество входов управ-
ления не превышает количество выходов. Предположено, что параметры объекта,
определяющие элементы ее матрицы коэффициентов усиления, неизвестны. Кроме
того ранг этой матрицы может быть произвольным. Между тем, границы внешних
возмущений должны быть известны. Задача, которая была поставлена и решена в рабо-
те, состоит в том, чтобы построить адаптивный регулятор, способный обеспечить ог-
раниченность всех входных и выходных сигналов системы при наличии параметриче-
ских неопределенностей.
Цель статьи — показать, что можно стабилизировать любой неопределенный
многомерный статический объект, матрица коэффициентов усиления которого может
быть квадратной или прямоугольной и иметь произвольный ранг, оставаясь неизвест-
ной конструктору системы.
Методы. Методы, основанные на рекуррентном точечном оценивании неизвест-
ных параметров объекта, используются для построения адаптивного регулятора на
основе обратной модели.
Результаты. Установлены асимптотические свойства адаптивных регуляторов. Что-
бы подкрепить теоретические исследования, представлены результаты моделирования.
Выводы. Адаптивные законы управления, предложенные в статье, могут гарантиро-
вать ограниченность всех сигналов, генерируемых системами управления с обратной свя-
зью. Однако это важное свойство будет достигаться за счет «сверхпараметризации» этих
систем. В тоже время, модельные эксперименты показывают их эффективность.
Ключевые слова: адаптивное управление, ограниченность, дискретное время, алго-
ритм оценивания, обратная связь, многомерная система, неопределенность.
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/SUO <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>
/SVE <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>
/ENU (Use these settings to create Adobe PDF documents for quality printing on desktop printers and proofers. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /NoConversion
/DestinationProfileName ()
/DestinationProfileSelector /NA
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure true
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles true
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /NA
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /LeaveUntagged
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|