Adaptive Stabilization of Some Multivariable Systems with Nonsquare Gain Matrices of Full Rank
The purpose of this paper is to answer the question of how the pseudoinverse modelbased adaptive approach might be utilized to deal with the uncertain multivariable memoryless system if the number of control inputs is less than the number of outputs. Results. It is shown that the parameter estimates...
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Міжнародний науково-навчальний центр інформаційних технологій і систем НАН України та МОН України
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nasplib_isofts_kiev_ua-123456789-1420912025-02-09T16:30:12Z Adaptive Stabilization of Some Multivariable Systems with Nonsquare Gain Matrices of Full Rank Адаптивна стабілізація деяких багатовимірних систем з прямокутними матрицями коефіцієнтів підсилення повного рангу Адаптивная стабилизация некоторых многомерных систем с прямоугольными матрицами коэффициентов усиления полного ранга Zhiteckii, L.S. Solovchuk, K.Yu. Интеллектуальное управление и системы The purpose of this paper is to answer the question of how the pseudoinverse modelbased adaptive approach might be utilized to deal with the uncertain multivariable memoryless system if the number of control inputs is less than the number of outputs. Results. It is shown that the parameter estimates generated by the standard adaptive projection recursive procedure converge always to some finite values for any initial values of system’s parameters. Based on these ultimate features, it is proved that the adaptive pseudoinverse model-based control law makes it possible to achieve the equilibrium state of the nonsquare system to be controlled. The asymptotical properties of the adaptive feedback control system derived theoretically are substantiated by a simulation experiment. Метою даного дослідження є відповідь на питання про те, чи можна реалізувати адаптивний підхід на основі псевдооберненої моделі для керування невизначеною багатомірною системою без пам'яті, в якій кількість входів керування є менша за кількість вихідних змінних. Результати. Показано, що оцінки параметрів, які формуються стандартною адаптивною рекурентною процедурою проекційного типу, завжди збігаються до деяких скінченних значень за будь-яких початкових оцінок параметрів системи. Доведено, що адаптивний закон керування на основі псевдооберненої моделі дозволяє досягти положення рівноваги системи, яка підлягає керуванню. Асимптотичні властивості системи керування з адаптивним зворотним зв'язком, встановлені теоретично, підтверджуються модельним експериментом. Цель этой статьи — ответить на вопрос, можно ли реализовать адаптивный подход на основе псевдообратной модели для управления неопределенной многомерной системой без памяти, в которой число управляющих входов меньше числа выходных переменных. Результаты. Показано, что оценки параметров, генерируемые стандартной адаптивной рекуррентной процедурой проекционного типа, всегда сходятся к некоторым конечным значениям для любых начальных оценок параметров системы. Доказано, что адаптивный псевдообратный закон управления позволяет достичь положения равновесия управляемой системы. Асимптотические свойства адаптивной системы управления с обратной связью, полученные теоретически, подтверждены модельным экспериментом. 2018 Article Adaptive Stabilization of Some Multivariable Systems with Nonsquare Gain Matrices of Full Rank / L.S. Zhiteckii, K.Yu. Solovchuk // Кибернетика и .вычислительная техника. — 2018. — № 2 (192). — С. 44-60. — Бібліогр.: 31 назв. — англ. 0454-9910 DOI: https://doi.org/10.15407/kvt192.02.044 https://nasplib.isofts.kiev.ua/handle/123456789/142091 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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Интеллектуальное управление и системы Интеллектуальное управление и системы |
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Интеллектуальное управление и системы Интеллектуальное управление и системы Zhiteckii, L.S. Solovchuk, K.Yu. Adaptive Stabilization of Some Multivariable Systems with Nonsquare Gain Matrices of Full Rank Кибернетика и вычислительная техника |
| description |
The purpose of this paper is to answer the question of how the pseudoinverse modelbased adaptive approach might be utilized to deal with the uncertain multivariable memoryless system if the number of control inputs is less than the number of outputs. Results. It is shown that the parameter estimates generated by the standard adaptive projection recursive procedure converge always to some finite values for any initial values of system’s parameters. Based on these ultimate features, it is proved that the adaptive pseudoinverse model-based control law makes it possible to achieve the equilibrium state of the nonsquare system to be controlled. The asymptotical properties of the adaptive feedback control system derived theoretically are substantiated by a simulation experiment. |
| format |
Article |
| author |
Zhiteckii, L.S. Solovchuk, K.Yu. |
| author_facet |
Zhiteckii, L.S. Solovchuk, K.Yu. |
| author_sort |
Zhiteckii, L.S. |
| title |
Adaptive Stabilization of Some Multivariable Systems with Nonsquare Gain Matrices of Full Rank |
| title_short |
Adaptive Stabilization of Some Multivariable Systems with Nonsquare Gain Matrices of Full Rank |
| title_full |
Adaptive Stabilization of Some Multivariable Systems with Nonsquare Gain Matrices of Full Rank |
| title_fullStr |
Adaptive Stabilization of Some Multivariable Systems with Nonsquare Gain Matrices of Full Rank |
| title_full_unstemmed |
Adaptive Stabilization of Some Multivariable Systems with Nonsquare Gain Matrices of Full Rank |
| title_sort |
adaptive stabilization of some multivariable systems with nonsquare gain matrices of full rank |
| publisher |
Міжнародний науково-навчальний центр інформаційних технологій і систем НАН України та МОН України |
| publishDate |
2018 |
| topic_facet |
Интеллектуальное управление и системы |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/142091 |
| citation_txt |
Adaptive Stabilization of Some Multivariable Systems with Nonsquare Gain Matrices of Full Rank / L.S. Zhiteckii, K.Yu. Solovchuk // Кибернетика и .вычислительная техника. — 2018. — № 2 (192). — С. 44-60. — Бібліогр.: 31 назв. — англ. |
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Кибернетика и вычислительная техника |
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ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2018. № 2 (192)
Интеллектуальное
управление и системы
DOI: https://doi.org/10.15407/kvt192.02.044
UDC 681.5
L.S. ZHITECKII, PhD (Engineering),
Acting Head of the Department of Intelligent Automatic Systems
e-mail: leonid_zhiteckii@i.ua
K.Yu. SOLOVCHUK, PhD Student
e-mail: solovchuk_ok@ukr.net
International Research and Training Center for Information Technologies
and Systems of the National Academy of Science of Ukraine
and Ministry of Education and Sciences of Ukraine, Kiev, Ukraine,
Acad. Glushkova av., 40, Kiev, 03187, Ukraine
ADAPTIVE STABILIZATION OF SOME
MULTIVARIABLE SYSTEMS WITH NONSQUARE GAIN
MATRICES OF FULL RANK
Introduction. The paper states and solves a new problem concerning the adaptive stabiliza-
tion of a specific class of linear multivariable discrete-time memoryless systems with non-
square gain matrices at their equilibrium states. This class includes the multivariable systems
in which the number of outputs exceeds the number of control inputs. It is assumed that the
unknown gain matrices have full rank.
The purpose of this paper is to answer the question of how the pseudoinverse model-
based adaptive approach might be utilized to deal with the uncertain multivariable memory-
less system if the number of control inputs is less than the number of outputs.
Results. It is shown that the parameter estimates generated by the standard adaptive
projection recursive procedure converge always to some finite values for any initial values of
system’s parameters. Based on these ultimate features, it is proved that the adaptive pseudo-
inverse model-based control law makes it possible to achieve the equilibrium state of the
nonsquare system to be controlled. The asymptotical properties of the adaptive feedback
control system derived theoretically are substantiated by a simulation experiment.
Conclusion. It is established that the ultimate behavior of the closed-loop control sys-
tem utilizing the adaptive pseudoinverse model-based concept is satisfactory.
Keywords: adaptive control, multivariable system, discrete time, feedback, pseudoinversion,
stability, uncertainty.
L.S. ZHITECKII, K.Yu. SOLOVCHUK, 2018
44
Adaptive Stabilization of Some Interconnected Uncertain System
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2018. № 2 (192) 45
INTRODUCTION
A long-standing problem of the optimal controller design for multivariable systems
has been solved by using different approaches including the l1 optimal control ap-
proach [1, 2]. It remains the important problem in the modern control theory [3–5].
Based on the well-known internal model principle, multivariable control
problem was first approached in the paper [6]. Within the framework of this
principle, the so-called inverse model approach seems to be perspective to deal
with improving MIMO (multi-input multi-output) feedback controls. Since the
pioneering work [7], the problem of inversion of linear time-invariant MIMO
systems has attracted an attention of several researches; see, e.g. [8–10]. Last
time, a significant progress in this scientific area has been achieved in [11–14].
The inverse model approach to ensuring perfect steady-state regulation of linear
discrete-time memoryless multivariable systems was first advanced in [15]. Similar
approach has also been discussed in [10] dealing with the problem of minimal inver-
sion. However, the inverse model approach is quite unacceptable if the MIMO sys-
tems to be controlled are nonsquare.
It turned out that the so-called pseudoinverse (generalized inverse) model
approach first proposed in the paper [9] can be exploited to cope with the non-
inevitability of nonsquare system. Recently, this approach was extended
in [16–19] for controlling a wide class of discrete-time memoryless multivari-
able systems. In particular, in [16] it was first established that pseudoinverse
model-based controller for the steady-state regulation of the MIMO systems
having singular or nonsquare gain matrices is indeed optimal. But such control-
ler can be implemented if system parameters are known a priori. In the case of
the parameter uncertainty, the nonadaptive controller employing a fixed linear
pseudoinverse model can be shown to be acceptable to ensure the robust stability
of multivariable closed-loop systems containing uncertain linear and some
nonlinear memoryless plants [17–19]. Nevertheless, this controller may not be
suitable if parameter uncertainty is great enough.
An adaptation of control law is known as some universal concept to deal
with uncertain systems. Results obtained within the framework of adaptive con-
trols were summarized in many books [20–27], etc. A key question in these con-
trols concerns the stability of resulting systems, i.e. the boundedness of the con-
trol input and output signals [20, 21]. In order to resolve this important question,
two different tools were independently advanced in above two books. Namely,
the so-called Frequency Theorem was exploited in [20, Theorem 4.17.3] to es-
tablish the ultimate boundedness properties of linear adaptive control systems
including multivariable plants with square gain matrices, whereas the so-called
Key Technical Lemma of [21, item 6.2] was used to derive such properties in
MIMO case where the number of outputs does not exceed the number of control
inputs, see [21, subitem 6.3.6]. Unfortunately, these tools seems to be not admis-
sible to an adaptive nonsquare case. To the best of author’s knowledge, there are
no theoretical results concerning adaptive controls of these MIMO systems
while they may appear in practice [21, p.141].
The purpose of this paper is to answer the question of how the pseudoin-
verse model-based adaptive approach might be utilized to deal with the uncertain
multivariable memoryless system in which the number of its output exceeds the
number of control inputs.
L.S. Zhiteckii, K.Yu. Solovchuk
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2018. № 2 (192) 46
PROBLEM STATEMENT
Let
1−= nn Buy (1)
be the difference equation of an static (memoryless) plant that is some MIMO
discrete-time system to be stabilized. In this equation, Tm
nnn yyy ],...,[ )()1(= and
Tr
nnn uuu ],...,[ )()1(= are its m-dimensional output and r-dimensional control input
vectors, respectively, at the nth time instant ),,2,1( …=n and
=
)()1(
)1()11(
mrm
r
bb
bb
B
…
LLL
…
(2)
denotes the time-invariant rm × gain matrix.
Consider a nonsquare system, where
,mr < (3)
i.e., where the number of output variables )1()( miy i
n ≤≤ exceeds the number of
control variables ).1()( rju j
n ≤≤
Suppose that B is some unknown matrix of full rank meaning that
}.,min{rank mrB = Due to (3) we have
.rank rB = (4)
Introducing the vector Tmyyy ],...,[ )(0)1(00 = whose components are the de-
sired output variables (the given set-points for outputs), define the current ith
output error )(i
ne as
.,,1,)()(0)( miyye i
n
ii
n …=−= (5)
Then the output error vector will be given by
.0
nn yye −= (6)
It is assumed that the elements ),,1,,,1()( rjmib ij …… == of B in (2)
are unknown a priori. Moreover, the bounds on these elements are assumed to
be unknown (contrary to [18, 19]) and it is essential.
The problem stated below is as follows. Based on the available observations
of 01 ,,, eee nn …− given by (6), devise an adaptive controller of a general form
),,,,( 01 eeeUu nnnn …−= (7)
such that the closed-loop control system containing the uncertain plant (1) and
the feedback (7) will be stable. More specifically, we require the sequences
Adaptive Stabilization of Some Interconnected Uncertain System
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2018. № 2 (192) 47
…,,:}{ 10 uuun = and …,,:}{ 10 yyyn = to be bounded uniformly
)}{,}({ ∞∞ ∈∈ ll nn yu and for any initial conditions to achieve
,, ee yyuu nnnn → → ∞→∞→ (8)
where the pair ),( ee yu with ee Buy = defines the equilibrium state of the feed-
back control system (1), (6), (7).
Remark 1. Note that it is not required for the errors )()1( ,, m
nn ee … given by
(5) to be asymptotically equal to zero. In fact, m zero errors cannot be achieved
simultaneously except a unique case when ),(0 By ℜ∈ where )(Bℜ denotes the
so-called range of B (the definition of )(⋅ℜ can be found in [28, Exercise
2.8.6]). Thus without less of generality we assume that ).(0 By ℜ∉
ADAPTIVE CONTROLLER DESIGN
Suppose that B is known. Then the pseudoinverse model-based control law of
the form
nnn eBuu +
− += 1 (9)
advanced in [16] can here be chosen. In this equation, +B specifies the so-called
pseudoinverse matrix given by [28, Theorem 3.4]
,)(lim 1
0
T
r
T BIBBB −
→δ
+ δ+= (10)
where qI denotes the identity qq × matrix.
Note that under conditions (3), (4) on ,B instead of (10), a very simple
formula
TT BBBB 1)( −+ = (11)
may be employed to calculate +B for given ;B see [28, Exercise 3.5.3].
Following to the standard identification approach, we will design an adap-
tive control by replacing the unknown matrix B in (9) by its suitable estimate
nB updated at the nth time instant. Then the control law takes the form
,1 nnnn eBuu +
− += (12)
where
.
)()1(
)1()11(
=
mr
n
m
n
r
nn
n
bb
bb
B
…
LLL
…
(13)
L.S. Zhiteckii, K.Yu. Solovchuk
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2018. № 2 (192) 48
To derive the estimation algorithm for updating ,nB we first define the ith
current estimation error )(~ i
ne given as follows:
).,,1(~
1
)(
1
)(
1
)()( miubeee n
Ti
n
i
n
i
n
i
n …=∇+−= −−− (14)
In these expressions, ],,[ )()1()( ir
n
i
n
Ti
n bbb …= is the ith row of nB selected from (13),
and the notation 1: −−=∇ nnn uuu is used.
Remark 2. By virtue of (1) together with (8), (5), it follows that if
,)()( ii
n bb = where ],,[ )()1()( iriTi bbb …= denotes the ith row of B then
0~ )( =ie (15)
will be ensured. Based on (15) define a set nΓ of possible s€ )(ib under which the
estimation errors are equal to zero for given observable .,, 1
)(
1
)(
−− ∇ n
i
n
i
n uee Obvi-
ously, nΓ represents the hyperplane
r
n
Ti
n
i
n
i
n
i
n ubeeb R⊂=∇+−=Γ −−− }0:{ 1
)(
1
)(
1
)()( (16)
belonging to the r-dimensional Euclidean space .rR It is not hard to see that
n
ib Γ∈)( for all .,2,1 …=n □
Now, similar to [21, sect. 3.3], we will choose the adaptation algorithm as
the recursive estimation procedure
),,,1(
||||
~
12
10
)(
)()(
1
)( miu
uc
ebb n
n
i
ni
n
i
n
i
n …=∇
∇+
γ+= −
−
− (17)
where 0c is an arbitrary sufficiently small positive constant )1( 0 <<c needed to
avoid the possibility of division by zero, and )(i
nγ is a scalar possibly time-
varying multiplier (in contrast to Equation (3.3.19) of [21]) satisfying
.20 )()()( <γ≤γ≤γ< ii
n
i (18)
This procedure describes the so-called projection algorithm which is also known
in the literature as the normalized least-mean-squares algorithm [21, p. 52].
There is a simple geometrical interpretation of (17), (18) (in terms of or-
thogonal projection of vector )(
1
i
nb − onto the hyperplane nΓ represented by (16) if
1)( =γ i
n and ).00 =c It is given in Fig. 1, where admissible s)(i
nb for )2,0()( ∈γ i
n
are also shown.
Remark 3. In order to do not deal with the possible division by zero, in-
stead of (17), other estimation algorithm
∇
∇
γ+
=∇
=
−
−
−
−−
otherwise
||||
~
0|||| if
12
1
)(
)()(
1
1
)(
1
)(
n
n
i
ni
n
i
n
n
i
n
i
n u
u
eb
ub
b , (19)
Adaptive Stabilization of Some Interconnected Uncertain System
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2018. № 2 (192) 49
can also be proposed as the adaptive estimation procedure. This algorithm repre-
sents the slightly modified well known Kaczmarz’s algorithm who proposed it in
1937 for solving a set of linear equations (a translation of his original work can
be found in the recent paper [29]). □
Thus, the algorithm (17) (or (19)) together with (16), (18) leads to forming
the estimate matrix nB given by (13). It turns out that it is possible to ensure
…,2,1rank =∀= nrBn (20)
by suitable choice of s)(i
nγ from ],[ )()( ii γγ with arbitrary numbers riii ,,1 …= such
that .1 1 mii r ≤<<≤ … To substantiate this fact, consider the so-called rr × sub-
matrix ],,1|,,[ 1 riiB rn …… of nB consisting of its r rows with the numbers
rii ,,1 … and of all columns (the definition of some submatrix of an arbitrary P and
its symbol notation ]|[ ⋅⋅P have been taken from [30, part 1, item 2.2]).
Following to [20, item 4.2.2] it can be shown that if ],[)( γ ′′γ′∈γ i
n then the
requirement rriiB rn =],,1|,,[rank 1 …… can always be satisfied because
0],,1|,,[det 1 ≠riiB rn ……
may take place at some isolated s.)(i
nγ Thereby, the condition (20) can be met.
This makes it possible to calculate +
nB by
T
nn
T
nn BBBB 1)( −+ = (21)
similarly to (11).
Fig. 1. The algorithm (17), (18) as an orthogonal projection process
for the two-dimensional case (r = 2)
L.S. Zhiteckii, K.Yu. Solovchuk
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2018. № 2 (192) 50
Fig. 2. Configuration of adaptive control system
The choice of )(i
nγ completes the synthesis of the adaptive control algorithm
determined in the expressions (12), (17) together with (6), (13), (14), (18) and
(21) in full detail. To implement this algorithm, the adaptive pseudoinverse
model-based control system is designed as shown in Fig. 2.
As it is seen, the controller of this system contains the discrete integrator
summing the increments
nnn eBu +=∇ (22)
from 0 to n at each nth time instant and giving
∑
=
∇=
n
k
kn uu
0
(in accordance with (12)).
ASYMPTOTIC BEHAVIOR OF ADAPTIVE FEEDBACK CONTROL SYSTEM
To study the ultimate behavior of the adaptive control algorithm (12), (17) to-
gether with (6), (14), (21), the preliminary results formulated in [21, Lemma
3.3.2] are needed. From these results we can derive the following asymptotic
properties:
(i) the scalar variables ||||: )()()( i
n
ii
n bbV −= are the Lyapunov function of the
algorithm (17) meaning
;,,1)(
1
)( miVV i
n
i
n …=∀≤ −
(ii) the sequences }{ )(i
nb satisfy
;,,1as0|||| )(
1
)( minbb i
n
i
n …=∀∞→→− −
Adaptive Stabilization of Some Interconnected Uncertain System
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2018. № 2 (192) 51
(iii) there exist the limits
.,,10
)||||(
~
lim 2/12
10
)(
mi
uc
e
n
i
n
n
…=∀=
∇+ −
∞→
(23)
With the foregoing properties (i) – (iii), the following lemma can be shown
to be valid.
Lemma. For the algorithm (17) together with (14) and subject to (18), it
follows that:
(a) the current estimate nB of unknown B remains always upper bounded
implying
;|||| nBn ∀∞<
(b) the matrix sequence }{ nB satisfies
;0|||| 1 →− ∞→− nnn BB
(c) the zero limit
0
||||
||~||
lim
11
=
∇+ −
∞→ n
n
n uc
e (24)
is achieved, where ne~ represents the estimation error vector defined as
,]~,,~[~ )()1( Tm
nnn eee …= (25)
and .2/1
01 cc =
Proof. Part (a) follows from the property (i) and the definition (13) of .nB
Part (b) holds due to the property (ii).
To prove part (c) we can write
,0
)||||(
||~||
lim 2/12
10
=
∇+ −
∞→
n
n
n uc
e (26)
using (23) and the definition (25) of .~
ne Since 2/1
2
2/1
1
2/1
21 )( hhhh +≤+ for any
numbers ,0, 21 ≥hh the inequality
||||
||~||
)||||(
||~||
11
2/12
10 −− ∇+
≥
∇+ n
n
n
n
uc
e
uc
e
with 2/1
01 cc = is valid. Taking this inequality into account, due to (26) we im-
mediately obtain (24). □
Now, we are able to present some basic result as
Theorem. The adaptation procedure (17), (14) has the following ultimate
property:
.0||~||lim =
∞→ nn
e (27)
L.S. Zhiteckii, K.Yu. Solovchuk
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2018. № 2 (192) 52
Proof. First, recalling that ,],,[ )()1( Tm
nnn eee …= due to (13), (14) we obtain
.~
111 −−− ∇+−= nnnnn uBeee (28)
Substituting (28) into (24) gives
.0
||||
||||
lim
11
111 =
∇+
∇+−
−
−−−
∞→ n
nnnn
n uc
uBee (29)
By virtue of (22), the expression (29) can then be rewritten as follows:
.0
||||
||)(||
lim
11
111 =
∇+
−−
−
−
+
−−
∞→ n
nnnmn
n uc
eBBIe (30)
Consider the equation
11 −− ∇−=− nnn uBee (31)
produced by (1) together with (6). Using this equation, represent (28) and (29) in
the form
,)(~
11 −− ∇−= nnn uBBe (32)
.0
||||
||)(||
lim
11
11 =
∇+
∇−
−
−−
∞→ n
nn
n uc
uBB (33)
It is clear that if |||| 1−∇ nu tends to 0 as n goes to infinity, then (33) is always
satisfied. Assume that .||||suplim 1 ∞=∇ −∞→ nn u To study this case, write
,0
||||/1
||)(||
lim
11
11 =
∇+
∇−
−
−−
∞→ n
nn
n uc
uBB (34)
dividing the numerator and the denominator of (33) by .|||| 1−∇ nu In this expression,
||||/: 111 −−− ∇∇=∇ nnn uuu denotes the unit vector of the same direction as .1−∇ nu
It is not hard to establish that when BBn ≠−1 and ||||sup ),0[ nn u∇∞∈ tends to
∞ then zero limit (34) will be satisfied if and only if
),( 1 BBu nnn −ℵ →∇ −∞→ (35)
where the notation )(Pℵ of the null-space of an arbitrary matrix P taken from
[28, Exercise 2.8.6] has been used.
By definition of ),(⋅ℵ it follows that (35) implies also that
)( 1 BBu nnn −ℵ →∇ −∞→ (36)
becomes the necessary and sufficient condition to achieve the limit (33) for any
}.{ nu∇ Taking (34) in to account, due to (32) which may be rewritten as
,0||)(||lim 11 =∇− −−∞→ nnn
uBB
result (27) follows. □
Adaptive Stabilization of Some Interconnected Uncertain System
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Further, the following proposition is advanced.
Proposition. If the adaptation algorithm (17) together with (14) and with )(i
nγ
chosen as in (18) is applied to the MIMO system (1), then there exist a finite limit
).||(||lim ∞<= ∞∞∞→
BBBnn
(37)
This proposition is based on the observation that each }{ )(i
nb is the so-called
Fejer’s sequence because )(i
nb is pointwise closer than )(
1
i
nb − to the intersection
∩
∞
=ν
+νΓ
n
1 of all the hyperplanes sνΓ defined in (16) (since they contain the point
,)(ib this intersection is non-empty set).
Notice that, in this proposition nothing has been said about the convergence
}{ nB to true ,B and it not necessary, in principle.
By virtue of (27) and (37), from the definition (28) of ,~
ne it follows that
our time-varying control system becomes asymptotical close to a time-invariant
system described by
mnmn eBBIe 0)( 1 =−− −
+
∞∞ (38)
as ,∞→n where T
r
r ]0,,0[:0 321…= denotes the r-dimensional zero vector.
Since (38) produces
∞<= − ||||with 1 nnn eee
for any integer positive n and for any finite ∞<|||| 0e [16, 18], according to
[31], it can be concluded that the adaptive control system given in equations (1),
(6), (12) has the following main ultimate properties:
;0||||lim)1 1 =− −∞→ nnn
ee (39)
there is a finite limit
).||||(lim)2 ∞<= ∞∞∞→
eeenn
(40)
Using the fact that B is the matrix of full rank (see (4)), from (31) we derive
)( 11 −
+
− −−=∇ nnn eeBu
to establish
.|||||||||||| 11 −
+
− −≤∇ nnn eeBu (41)
Due to (39) from (41) it follows that
.0|||| 1 →∇ ∞→− nnu (42)
By (40) and (42) we conclude that }{ ny and }{ nu will go to the equilibrium
state ),( ee yu with ∞−= eyy 0e and ee yBu += as n tends to infinity.
L.S. Zhiteckii, K.Yu. Solovchuk
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2018. № 2 (192) 54
Hence, the problem (8) stated in this paper will be solved.
Comment. It can understand that eu may be specified by solving the vector
equation
0e yBBuB +
∞
+
∞ = (43)
yielding by the condition
rn
e
nn
n
BuyB 0)( 1
0 →− ∞→−
+
43421
under which the equilibrium state should asymptotically be achieved.
A SIMULATION EXAMPLE
To illustrate how the adaptive pseudoinverse model-based control algorithm
performs, a simulation of the closed-loop system consisting of the nonsquare
memoryless MIMO system (1) (the plant) and of the adaptive controller de-
scribed in equation (6), (12), (17) together with (13), (14), (21) was conducted.
The system to be stabilized at an equilibrium state was given by
.
0.51.1
2.40.8
1.40.2
=B
with the matrix B of the full rank ( 2rank =B ).
The desired output vector 0y was taken as Ty ]3,7,2[0 = to ensure
).(0 By ℜ∉ The duration of the simulation experiment was chosen as long as
adaptation of the controller parameters continues.
Table 1 sets out the true system parameters and their initial estimates.
Results of the simulation experiment are presented in Figs. 3 to 5. Fig. 3
shows how the estimate vectors )3,2,1()( =ib i
n move to their final .)(ib∞ We
can see that they differ from .)(ib These final estimates given in Table 1 yield
.
2.172.24
8.074.68
9.635.83
≅∞B
Table 1. System parameters
Parameters b(11) b(12) b(21) b(22) b(31) b(32)
True value
Initial value
Final value
0.2
50
≅ 5.83
1.4
20
≅ 9.63
0.8
30
≅ 4.68
2.4
40
≅ 8.07
1.1
10
≅ 2.24
0.5
10
≅ 2.17
Adaptive Stabilization of Some Interconnected Uncertain System
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2018. № 2 (192) 55
(a) (b)
(c)
Fig. 3. Trajectories of adaptive estimation processes: (a) Vectors ;)1(
nb (b) Vectors ;)2(
nb
(c) Vectors ;)3(
nb
Fig. 4. Norm of estimation error vector
From Fig. 4 it is seen that the norm of the estimation error ne~ converges to 0 as
∞→n as predicted by the theorem above established. Fig. 5 shows the input control
variables )2,1()( =iu i
n and the output variables ).3,2,1()( =iy i
n It is seen from
Fig. 5b that the performance of the adaptive controller is satisfactory because this con-
troller is able to stabilize the output vector ny at some ultimate ∞y given by
Ty ]75.2,88.5,99.2[≅∞
as n tends to infinity. Namely, this vector specifies the equilibrium state of the
adaptive control system: .e
∞= yy
L.S. Zhiteckii, K.Yu. Solovchuk
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2018. № 2 (192) 56
Fig. 5. Systems variables: (a) Control inputs ;)( j
nu (b) Outputs )(i
ny (solid lines)
and desired value )(0 iy (dashed lines)
By solving (43) we numerically determine
Tu ]905.1,634.1[e ≅
that is the same as observed in the simulation example (see Fig.5a).
We also see that although the equilibrium state is asymptotically achieved,
there is a difference between the components )3(0)2(0)1(0 ,, yyy of desired output
vector 0y and their ultimate values ,,, )3()2()1(
∞∞∞ yyy respectively, if ).(0 By ℜ∉
CONCLUSION
This paper shed light on the adaptive pseudoinverse model-based approach to
deal with the stabilization of uncertain nonsquare memoryless MIMO systems in
which the number of the outputs exceeds the number of their control inputs. It
ensure the asymptotical stabilization of such systems at equilibrium states for
any initial estimates of system’s parameters. A simulation experiment has dem-
onstrated a good performance of these systems.
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Adaptive Stabilization of Some Interconnected Uncertain System
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L.S. Zhiteckii, K.Yu. Solovchuk
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2018. № 2 (192) 58
29. Kaczmarz S. Approximate solution of systems of linear equations. Internat. J.
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Elsevier, 1975.
Received 29.03.2018
ЛИТЕРАТУРА
1. Dahleh M.A., Pearson J.B. l1 optimal-feedback controllers for MIMO discrete-time
systems. IEEE Trans. Autom. Contr., 1987, vol. 32, no. 4, pp. 314–322.
2. McDonald J.S., Pearson J.B. l1 optimal control of multivariable systems with output
norm constraints. Automatica, 1991, vol. 27, no. 2, pp. 317–329.
3. Maciejowski J. M. Multivariable Feedback Design. Wokinghan: Addison-Wesley, 1989.
4. Skogestad S., Postlethwaite I. Multivariable Feedback Control . UK, Chichester:
Wiley, 1996.
5. Albertos P., Sala A. Multivariable Control Systems: an Engineering Approach .
London: Springer , 2006.
6. Francis B., Wonham W. The internal model principle of control theory. Automatica,
1976, vol. 12, no. 5, pp. 457–465.
7. Brockett R. W. The invertibility of dynamic systems with application to control. Ph.
D. Dissertation, Case Inst. of Technology, Cleveland, Ohio, 1963.
8. Silverman L. M. Inversion of multivariable linear systems. IEEE Trans. Autom.
Contr., 1969, vol. AC-14, no. 3, pp. 270–276..
9. Lovass-Nagy V., Miller J. R., Powers L. D. On the application of matrix generalized inver-
sion to the construction of inverse systems. Int. J. Control, 1976, vol. 24, no. 5, pp. 733–739.
10. Seraji H. Minimal inversion, command tracking and disturbance decoupling in multi-
variable systems. Int. J. Control, 1989, vol. 49, no. 6, pp. 2093–2191.
11. Marro G., Prattichizzo D., Zattoni E. Convolution profiles for right-inversion of mul-
tivariable non-minimum phase discrete-time systems. Automatica, 2002, vol. 38, no.
10, pp. 1695–1703.
12. Liu C., Peng H. Inverse-dynamics based state and disturbance observers for linear
time-invariant systems. ASME J. Dyn Syst., Meas. and Control , 2002, vol. 124, no. 5,
pp. 376–381.
13. Lyubchyk L. M. Disturbance rejection in linear discrete multivariable systems: inverse
model approach. Prep. 18th IFAC World Congress, Milano, Italy, 2011, pp. 7921–7926.
14. Pushkov S. G. Inversion of linear systems on the basis of state space realization. Journal
of Computer and Systems Sciences International , 2018, vol. 57, vo. 1, pp. 7–17.
15. Пухов Г.Е., Жук К.Д. Синтез многосвязных систем управления по методу об-
ратных операторов, Киев: Наук. думка, 218 с., 1966.
16. Скурихин В.И., Гриценко В.И., Житецкий Л.С., Соловчук К.Ю. Метод обобщенного
обратного оператора в задаче оптимального управления линейными многосвязными
статическими объектами. Доклады НАН Украины, 2014, №8, С. 57–66.
17. Zhiteckii L. S., Azarskov V. N., Solovchuk K. Yu., Sushchenko O. A. Discrete-time
robust steady-state control of nonlinear multivariable systems: a unified approach.
Proc. 19th IFAC World Congress , Cape Town, South Africa, 2014, pp. 8140–8145.
18. Zhitetskii L. S., Skurikhin V. I., Solovchuk K. Yu. Stabilization of a nonlinear multivariable
discrete-time time-invariant plant with uncertainty on a linear pseudoinverse model. Journal
of Computer and Systems Sciences International , 2017, vol. 56, no. 5, pp. 759–773.
19. Zhiteckii L. S., Solovchuk K. Yu. Pseudoinversion in the problems of robust stabiliz-
ing multivariable discrete-time control systems of linear and nonlinear static objects
under bounded disturbances. Journal of Automation and Information Sciences , 2017,
vol. 49, no. 5, pp. 35–48.
20. Фомин В.Н., Фрадков А. Л., Якубович В. А. Адаптивное управление динамиче-
скими объектами. М.: Наук, 448 c., 1981.
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21. Goodwin G.C., Sin K.S. Adaptive Filtering, Prediction and Control . Engewood
Cliffs, NJ.: Prentice-Hall, 1984.
22. Landau I. D., Lozano R., M'Saad M. Adaptive Control. London: Springer, 1997.
23. Кунцевич В. М. Управление в условиях неопределенности: гарантированные ре-
зультаты в задачах управления и идентификации. Киев: Наук. думка, 264c., 2006.
24. Житецкий Л.С., Скурихин В.И. Адаптивные системы управления с параметрически-
ми и непараметрическими неопределенностями. Киев: Наук. думка, 301 с., 2010.
25. Narendra K. S., Annaswamy A. M. Stable Adaptive Systems. NY: Dover Publications, 2012.
26. Ioannou P., Sun J. Robust Adaptive Control. NY: Dover Publications, 2013.
27. Aström K. J., Wittenmark B. Adaptive Control: 2nd Edition. NY: Dover Publications, 2014.
28. Albert A. Regression and the Moore-Penrose Pseudoinverse . New York: Academic
Press, 1972.
29. Kaczmarz S. Approximate solution of systems of linear equations. Internat. J.
Control, 1993. vol. 57, no. 6. pp. 1269–1271.
30. Marcus M., Minc H. A Survey of Matrix Theory and Matrix Inequalities . Boston:
Aliyn and Bacon, 1964.
31. Desoer C.A., Vidyasagar M. Feedback Systems: Input–Output Properties . New York:
Elsevier, 1975.
Получено 29.03.2018
Л.С. Житецький, канд. техн. наук,
в.о. зав. відд. інтелектуальних автоматичних систем
e-mail: leonid_zhiteckii@i.ua
К.Ю. Соловчук, аспірантка
e-mail: solovchuk_ok@ukr.net
Міжнародний науково-навчальний центр інформаційних технологій
та систем НАН України і МОН України, пр. Академіка Глушкова, 40,
м. Київ, 03187, Україна
АДАПТИВНА СТАБІЛІЗАЦІЯ ДЕЯКИХ БАГАТОВИМІРНИХ СИСТЕМ
З ПРЯМОКУТНИМИ МАТРИЦЯМИ КОЕФІЦІЄНТІВ ПІДСИЛЕННЯ
ПОВНОГО РАНГУ
Вступ. У статті поставлено та розв’язано одну нову задачу, яка стосується адаптивної
стабілізації положення рівноваги певного класу лінійних багатовимірних дискретних сис-
тем без пам'яті з прямокутними матрицями коефіцієнтів підсилення. Цей клас включає
багатовимірні системи, у яких кількість виходів перевищує кількість входів керування.
Введено припущення, що невідомі матриці коефіцієнтів підсилення мають повний ранг.
Метою даного дослідження є відповідь на питання про те, чи можна реалізувати
адаптивний підхід на основі псевдооберненої моделі для керування невизначеною
багатомірною системою без пам'яті, в якій кількість входів керування є менша за кіль-
кість вихідних змінних.
Результати. Показано, що оцінки параметрів, які формуються стандартною ада-
птивною рекурентною процедурою проекційного типу, завжди збігаються до деяких
скінченних значень за будь-яких початкових оцінок параметрів системи. Доведено, що
адаптивний закон керування на основі псевдооберненої моделі дозволяє досягти поло-
ження рівноваги системи, яка підлягає керуванню. Асимптотичні властивості системи
керування з адаптивним зворотним зв'язком, встановлені теоретично, підтверджуються
модельним експериментом.
Висновки. Встановлено, що гранична поведінка замкненої системи керування з
використанням адаптивної концепції, основаної на псевдооберненні, є задовільною.
Ключові слова: адаптивне керування, багатовимірна система, дискретний час, зворот-
ний зв'язок, псевдообернення, стійкість, невизначеність.
L.S. Zhiteckii, K.Yu. Solovchuk
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2018. № 2 (192) 60
Л.С. Житецкий, канд. техн. наук,
и.о. зав. отд. интеллектуальных автоматических систем
e-mail: leonid_zhiteckii@i.ua
К.Ю. Соловчук, аспирантка
e-mail: solovchuk_ok@ukr.net
Международный научно-учебный центр информационных технологий
и систем НАН Украины и МОН Украины,
пр. Академика Глушкова, 40, г. Киев, 03187, Украина
АДАПТИВНАЯ СТАБИЛИЗАЦИЯ НЕКОТОРЫХ
МНОГОМЕРНЫХ СИСТЕМ С ПРЯМОУГОЛЬНЫМИ МАТРИЦАМИ
КОЭФФИЦИЕНТОВ УСИЛЕНИЯ ПОЛНОГО РАНГА
Введение. В настоящей статье ставится и решается одна новая задача, касающаяся
адаптивной стабилизации положения равновесия определенного класса линейных
многомерных дискретных систем без памяти с прямоугольными матрицами коэффици-
ентов усиления. Этот класс включает многомерные системы, у которых число выходов
превышает число управляющих входов. Предполагается, что неизвестные матрицы
коэффициентов усиления имеют полный ранг.
Цель этой статьи — ответить на вопрос, можно ли реализовать адаптивный подход на
основе псевдообратной модели для управления неопределенной многомерной системой без
памяти, в которой число управляющих входов меньше числа выходных переменных.
Результаты. Показано, что оценки параметров, генерируемые стандартной адап-
тивной рекуррентной процедурой проекционного типа, всегда сходятся к некоторым
конечным значениям для любых начальных оценок параметров системы. Доказано, что
адаптивный псевдообратный закон управления позволяет достичь положения равновесия
управляемой системы. Асимптотические свойства адаптивной системы управления с
обратной связью, полученные теоретически, подтверждены модельным экспериментом.
Выводы. Установлено, что предельное поведение замкнутой системы управления,
построенной на основе адаптивного псевдообращения, является удовлетворительным.
Ключевые слова: адаптивное управление, многомерная система, дискретное время,
обратная связь, псевдообращение, устойчивость, неопределенность.
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