Adaptive robust multivariable control of noninvertible memoryless systems with bounded disturbances: a generalization
The article is dedicated to robust adaptive control in discrete time for certain classes of undefined memoryless multivariate systems with unmeasurable bounded disturbances, the bounds of which are assumed to be known. The systems considered have a number of control inputs that does not exceed the n...
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| Date: | 2022 |
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Інститут кібернетики ім. В.М. Глушкова НАН України
2022
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| Cite this: | Adaptive robust multivariable control of noninvertible memoryless systems with bounded disturbances: a generalization / S. ZhiteckiiL, K.Yu. Solovchuk // Проблеми керування та інформатики. — 2022. — № 2. — С. 22-38. — Бібліогр.: 29 назв. — англ. |
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| author | Zhiteckii, L.S. Solovchuk, K.Yu. |
| author_facet | Zhiteckii, L.S. Solovchuk, K.Yu. |
| citation_txt | Adaptive robust multivariable control of noninvertible memoryless systems with bounded disturbances: a generalization / S. ZhiteckiiL, K.Yu. Solovchuk // Проблеми керування та інформатики. — 2022. — № 2. — С. 22-38. — Бібліогр.: 29 назв. — англ. |
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| description | The article is dedicated to robust adaptive control in discrete time for certain classes of undefined memoryless multivariate systems with unmeasurable bounded disturbances, the bounds of which are assumed to be known. The systems considered have a number of control inputs that does not exceed the number of outputs.
Стаття присвячена робастному адаптивному керуванню в дискретному часі деякими класами невизначених багатовимірних систем без пам’яті за наявності невимірних обмежених збурень, межі яких припускаються відомими. Розглянуто системи, у яких число керувальних входів не перевищує числа виходів. Основна особливість об’єктів, що підлягають керуванню, полягає у тому, що матриці коефіцієнтів підсилення таких об’єктів не можуть бути обернені. Введено припущення, що елементи цих матриць апріорі невідомі, але є інформація про можливі межі інтервалів, до яких вони належать. Поставлено і вирішено задачу, що полягає в тому, аби у контурі зворотного зв’язку побудувати регулятор, здатний впоратися з матрицями коефіцієнтів підсилення, які не можуть бути обернені, а також з параметричною невизначеністю для пригнічування зовнішніх збурень і забезпечення обмеженості всіх керувань та вихідних сигналів. Для вирішення вищезазначеної задачі використовується робастний адаптивний підхід у поєднанні з так званою концепцією псевдооберненої чи оберненої моделей. Вивчено три різні випадки. У першому випадку побудований робастний адаптивний регулятор, який може застосовуватися до невизначеного об’єкта з виродженою квадратною матрицею коефіцієнтів підсилення. У другому випадку на основі псевдообернених моделей, параметри яких оцінюються за допомогою стандартної рекурентної процедури адаптації, запропоновано робастний метод керування об’єктами з невідомими прямокутними матрицями коефіцієнтів підсилення повного рангу. Запропонований у першому випадку підхід поширюється на третій випадок керування невідомими об’єктами, матриці коефіцієнтів підсилення яких є прямокутними матрицями неповного рангу. Встановлено асимптотичні властивості запропонованих у статті робастно-адаптивних регуляторів. Для підтримки теоретичного дослідження наведено результати числових прикладів.
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| first_indexed | 2026-03-16T12:21:29Z |
| format | Article |
| fulltext |
© L.S. ZHITECKII, K.YU. SOLOVCHUK, 2022
22 ISSN 1028-0979
АДАПТИВНЕ КЕРУВАННЯ
ТА МЕТОДИ ІДЕНТИФІКАЦІЇ
UDC 681.5
L.S. Zhiteckii, K.Yu. Solovchuk
ADAPTIVE ROBUST MULTIVARIABLE
CONTROL OF NONINVERTIBLE
MEMORYLESS SYSTEMS WITH BOUNDED
DISTURBANCES: A GENERALIZATION
Leonid Zhiteckii
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, Kyiv,
leonid_zhiteckii@i.ua
Klavdiia Solovchuk
Poltava Scientific Research Forensic Center of the MIA of Ukraine,
solovchuk_ok@ukr.net
The discrete-time robust adaptive control for some classes of the uncertain multivariable
memoryless (static) systems in the presence of unmeasurable bounded disturbances, whose
bounds are assumed to be known is addressed. The systems where the number of the con-
trol inputs do not exceed the number of their outputs are considered. The main feature of
plants to be controlled is that their gain matrices are noninvertible. The assumption that the
elements of these matrices are unknown a priori but there is information about possible
bounds on these elements. The problem stated and solvesd here is to design a feedback con-
troller to be capable to cope with the noninvertibility of the gain matrices and also with the
parametric uncertainty in order to reject the external disturbances and to ensure the boun-
dedness of all the control and output system signals. To solve the problem above mentioned,
the robust adaptive approach together with the so-called pseudoinverse or inverse model-
based concept is used. Three different cases are studied. In the first case, the robust adaptive
controller applicable to the uncertain plant with the square singular gain matrix is designed.
The robust method employing the pseudoinverse model-based controllers whose parameters
are estimated via a standard recursive adaptation procedure is proposed in the second case to
deal with the unknown nonsquare gain matrices having the full rank. The approach pro-
posed in first case is extended to the third case dealing with the control of the unknown
plants the gain matrices of which represent the nonsquare matrices of not full rank. Asymp-
totic properties of the robustly-adaptive controllers proposed in this paper are established.
Results of numerical examples given to support the theoretic study.
Keywords: discrete time, multivariable memoryless plant, noninvertibility, pseudoinverse
model-based concept, uncertainty, estimation algorithm, robust adaptive control.
Introduction
A long-standing in [1] problem of the optimal controller design for multivariable
system in the presence of unmeasurable disturbances remains an important problem
from both theoretical and practical points of view. Within the framework of this actual
problem, new approaches have been proposed by many researches. The latest results in
this scientific area have been reported in numerous papers including [2] and generalized
in several recent books [3–6] dealing with advanced multivariable control systems.
mailto:leonid_zhiteckii@i.ua
mailto:solovchuk_ok@ukr.net
Міжнародний науково-технічний журнал
«Проблеми керування та інформатики», 2022, № 2 23
Among other methods advanced in the modern control theory, the inverse model-
based method that is an extension of the well-known internal model principle seems to
be perspective in order to cope with arbitrary unmeasurable disturbances and to opti-
mize some classes of multivariable control systems. It turned out that this method first
intuitively devised in [7] makes it possible to optimize the closed-loop control system
containing the multivariable static (memoryless) 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 [8, 9]. However, this approach is
quite unacceptable if the gain matrices are either square but singular or nonsquare be-
cause they are noninvertible.
To optimize the closed-loop control system containing an arbitrary multivariable
static plant, the so-called pseudoinverse (generalized inverse) model-based concept has
been proposed and substantiated in [10] dealing with the possible noninvertibility of
gain matrices whose elements are assumed to be known. This fruitful concept was ex-
tended in [11, 12] to robust control of a noninvertible and uncertain plant with unmeas-
urable bounded disturbances. Unfortunately, the robust control theory may not be em-
ployed if the initial parametric uncertainty is «wide» enough. Meanwhile, the adaptive
approach gives some universal tool to deal with such a type of uncertainty. Foundations
of this approach have been extended and generalized in the books [13–19].
In the recent works [20, 21], the different adaptive control ideas are advanced to
cope with the noninvertible multivariable memoryless system in the presence of para-
metric uncertainties. In particular, the adaptive pseudoinverse model-based control idea
using the standard identification method is exploited in [20, 22] to reject unmeasurable
bounded disturbances acting on the uncertain plants with the nonsquare gain matrices of
full rank. Novel approaches to deal with the multivariable memoryless plants having
unknown square and nonsquare gain matrices of not full rank are reported in [21–23].
The purpose of this paper is to generalize the results achieved in [20–23] and relat-
ed to the case where there are no nonparametric uncertainties.
1. Problem formulation
Let
1n n ny Bu v−= + (1)
be the vector-valued difference equation of a static (memoryless) plant represent-
ing some linear multivariable discrete-time system to be stabilized. In this equa-
tion, ,m r
n ny R u R and
m
nv R are the m dimensional measured output, con-
trol input and unmeasured external disturbance vectors, respectively, at the n th
time instant ( 1, 2, )n = defined by (1) ( ) T (1) ( ) T.[ , , ] ,., .[ , ]m r
n n n n n ny y y u u u= = and
(1) ( ) T...[ , , ] ,m
n n nv v v=
(11) (1 )
( 1) ( )
r
m mr
b b
B
b b
=
(2)
is an arbitrary time-invariant m r gain matrix.
Consider the case when the number of the control inputs
(1) ( ).., ., r
n nu u is not less
than two but does not exceed the number of the outputs
(1) ( ), , m
n ny y meaning that
2 .r m (3)
24 ISSN 1028-0979
Next, suppose that the rank of B satisfies the inequality
rank B r (4)
implying that B may be not a full rank matrix. Note that the rank of 𝐵 satisfying (4)
together with (3) give that B becomes a noninvertible matrix if either r m= but
rank B r or r m irrespective of rank .B
The following basic assumptions with respect to the gain matrix B and the se-
quences
( ) ( )( )
0 1{ } , , ( 1, , )
i ii
nv v v i m= = are made.
A1) all the elements of 𝐵 are all unknown. However, there are some interval esti-
mates defined as
( ) ( ) ( ) , 1, , , 1, , ,ij ij ijb b b i m j r = = (5)
where the upper and lower bounds
( )ijb and ( ) ,ijb respectively, on ( )ijb are assumed to
be known.
A2) ( )s ( =1, ..., )i
nv i m are all the arbitrary scalar sequences bounded in modulus
according to
( ) ( ) 0,1, 2, ...,i i
nv n =
where ( )i s are constant.
A3) The upper bounds, ( )i s are known a priori.
Let 0 0(1) 0( ) T[ , , ]my y y= be a desired output vector 0( ) c( onstiy i =
1, , m= . Suppose that 0(1) 0( ) 0my y++ implying that, at least, one 0( )iy of
0(1) 0( ), ..., my y is nonzero.
Define the output error vector
0 ,n ne y y= − (6)
with the components ( ) 0( ) ( ) ,i i i
n ne y y= − i.e., (1) ( ) T, ..., ] .[ m
n n ne e e=
The problem is to design the feedback controller guaranteeing the ultimate bound-
edness of the sequence 1 2{ } , , ,ne e e= in the form
lim sup n
n
e
→
(7)
provided
lim sup .n
n
u
→
(8)
Remark. The requirement (8) is here introduced additionally since it may not be
satisfied even if (7) takes place.
2. Preliminaries
Assume, for the time being, that B is a known noninvertible matrix. In this case,
the so-called pseudoinverse control
1n n nu u B e+−= + (9)
proposed in several works (see, e.g., [24]) ensures the minimum of the upper bound on
the Euclidean norm 2|| ||ne of the output error vector of the closed-loop control sys-
tem (1), (6), (9) with any bounded sequence 1 2{ } , ,nv v v= Here the notation P+ of
any pseudoinverse matrix introduced in [24, Theorem 3.4] and defined as
Міжнародний науково-технічний журнал
«Проблеми керування та інформатики», 2022, № 2 25
T 1 T
0
,lim ( )rP P P I P+ −
→
= +
where rI denotes the identity r r matrix is used. Recall that if rank ,P r= for some
( ) ,m rP R then the expression of P+ is simplified [25, item 7.46]; we have
T 1 T.( )P P P P+ −= (10)
The closed-loop control system (1), (6), (9) is designed as shown in Fig. 1. In this
control system, the variable 1:n n nu u u −= − produced by the pseudoinverse model
represents the increment of the control action during one step determined as
:n nu B e+=
whereas the signal 𝑢𝑛 is formed as the sum
1
.
n
n k
k
u u
=
=
Fig. 1
3. Robustly-adaptive control of plants with square singular gain matrices
Let 𝐵 be an unknown square singular r r matrix, i.e.,
det 0.B = (11)
Basic idea to deal with a matrix B satisfying the requirement (11) is to replace
adaptive identification of the true plant having the singular gain matrix B to the adaptive
identification of a so-called fictitious plant with the nonsingular gain matrix B of the form
0 ,rBB I= + (12)
where rI denotes the identity r r matrix and 0 is a fixed quantity [26].
Although �̃� as well as 𝐵 remain unknown, the requirement
det 0B (13)
can always be satisfied by the suitable choice of 0 in the expression (12). In fact, each
ith eigenvalue ( )i B of B lies in one of the 𝑟 closed regions of the complex z-plane consist-
ing of all the Gerŝgorin discs [27, p. 146]:
( ) ( )
1
, 1, , .
rii ij
j
j i
z b b i r
=
− = (14)
Since, at least, one of the eigenvalues ( )i B is equal to zero (due to the singularity
of 𝐵), by virtue of (12) there are an integer 𝑖 (1 ≤ 𝑖 ≤ 𝑟) and the numbers
( ) ( ) ( ) ( ) ( )
1
( )
:
1
, :i
j j
j i
r ri i ij i i
j i
ii jb b b b
= =
= − −= (15)
26 ISSN 1028-0979
such that if
( ) ( )i i (16)
then either ( ) 0i but
( ) 0i or ( ) 0i but ( ) 0.i These numbers are defined
as the intersection of the i th Gerŝgorin disc with the real axis of the complex z-plane
as shown in Fig. 2, a and 3, a, respectively. In both cases,
( ) ( ) 0i i if the inequa-
lity (16) is satisfied because ( )i and ( )i cannot have the same sign.
Denoting
(1) ( ) (1) ( ): min{ , ..., }, : max{ ..., },,r r = = (17)
consider the following two cases: (i) | | | |; (ii) | | | | (The case when | | | | =
can be combined with any of two cases.) In order to go to the gain matrix B of the ficti-
tious plant having the form (12) in the case (i), it is sufficient to shift the Gerŝgorin
discs (14) right taking
0 | | (18)
as shown in Fig. 2b. In the case (ii), the discs (14) need to be shifted left according to
0 | |; −
(19)
see Fig. 3, b. In both cases, the nonsingularity of B is guaranteed. Nevertheless, the
conditions (18) and (19) cannot be satisfied, as yet. In fact, the numbers and given
by the expressions (17) depend on
( )i
s and
( )i s defined by (15). But they are un-
known because
( )ijb s are all unknown.
а б
Fig. 2
а б
Fig. 3
Im
Z-plane
Im
Z-plane
Re Re
Im
Z-plane
Im
Z-plane
Re Re
Міжнародний науково-технічний журнал
«Проблеми керування та інформатики», 2022, № 2 27
The following operations are proposed to choose a number 0 satisfying the
reqiurement (13). Introduce
( ) ( ) ( ) ( )
min
1,
( )( ) ( ) ( )
max
1,
: max{| |, | |},
: max{| |, | |}
r
i ii ij ij
j j i
r
iji ii ij
j j i
b b b
b b b
=
=
= −
= +
(20)
minimizing and maximizing in
( )( ) ( )[ , ]
ijij ijb b b the right-hand sides of (15) for
( )i
and ( )i respectively.
Further, the number 0 is found to satisfy the conditions
0 max 0 max maxmin min min
if | | | |, if | | | | − (21)
where maxmin
, represent some quantities defined as follows:
(1) ( )(1) ( )
mix n minmin min min ma, max{ }.: min{ , ..., } : , ...,
rr
= = (22)
It can be clarified that if the conditions (21) together with (20) and (22) are satis-
fied then the condition (13) will without fail be ensured.
After determining the quantity 0 we can proceed to the consideration of the ficti-
tious plant. Since the input variables
(1) ( ), ..., r
n nu u and the disturbances
(1) ( ), ..., r
n nv v of
both true plant and fictitious plant are the same, this feature makes it possible to des-
cribe our fictitious plant by the equation
1 ,n n ny Bu v−= + (23)
similar to the equation (1), where
(1) ( ) T[ , ..., ] ,m
n n ny y y= denotes the output vector of
the fictitious plant.
It is interesting that the components of ny can be measured while the components
of nv in the equation (23) remain unmeasurable. In fact, substituting the expression (12)
into (23), due to (1) we produce
0 1.n n ny y u −= + (24)
It is seen from the equation (24) 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 applicable to
the fictitious plant (23) with the unknown gain matrix B in the presence of arbitrary
bounded disturbances
(1) ( ), ..., .r
n nv v Its solving follows the steps of the section above.
Namely, the adaptive control law is designed in the form
1
1 ,n n n nu u B e−
−= + (25)
28 ISSN 1028-0979
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
0
n ne y y= − (26)
with ny given by the expression (24).
The adaptive identification algorithm used to determine the estimates nB may be
taken as
( ) *( ) 0
1
( )
*( ) *( )
( ) ( )
11 2
1 2
1, ,
if | | ,
sign
otherwise,
|| ||
i i
n in
i
i in
i i n i n
n nn
n
b e
b
e e
b u
u
i r
−
−−
−
=
− + =
(27)
which is similar to that in [22]. In this algorithm,
0
i and i are given by
0 , 1, ,2i i i i r = = (28)
( )T*( ) ( )
11
ii i
n n nne y b u −−= − (29)
represent the i th component of the identification error
*
ne given as
*
1 1,n n n ne y B u− −= − (30)
where
( )( ) ( )
1: ,
ii i
n n ny y y − = − and the notation
( )T ( 1) ( ): [ , , ]...i i ir
n n nb b b= of the i th row of
nB is introduced. The coefficients
( )i
n s are chosen from the intervals
( )0 2i
n (31)
to satisfy
det 0.nB (32)
The feedback adaptive robust control system described in the (1), (25), (27) is
designed as depicted in Fig. 4. In this figure, the notation
*
1 1:n n ny B u− − = is in-
troduced.
The asymptotic properties of the adaptive control system are established in the
theorem below.
Theorem 1. Determine 0 using the formula (21) together with the expres-
sions (20) and (22), and choose an arbitrary initial 0 0 0 rB B I= + with
( )
0 0( )
ij
B b=
whose elements satisfy the conditions
( )( ) ( )
0 .
ijij ijb b b Subject to assumptions
A1–A3, the adaptive controller described in the equations (25) and (27) together
with (24) and (26) when applied to the plant (1) leads to (7) and (8).
Proof. See [26]. □
Міжнародний науково-технічний журнал
«Проблеми керування та інформатики», 2022, № 2 29
Fig. 4
4. Robustly-adaptive control of plants
with nonsquare gain matrices having full rank
Let (1) be the equation of a nonsquare multivariable memoryless system
( 2 r m ) whose gain matrix, B, as full rank:
rank B r= (33)
Suppose that ( )i s are known. For controling this system under condition of the
parametric uncertainty given by (5), the adaptive controller will be designed here as the
adaptive pseudoinverse model-based controller described by
1n n n nu u B e+−= + (34)
with nB+
defined by (10). Noting that the requirement
rank nB r=
(35)
will be satisfied to calculate nB+
by (10), we will update the elements of
( )( ),ij
n nB b=
exploiting the following standard adaptive estimation algorithm proposed in [28, sect. 4.2]:
( )
( ) ( ) 0( )
1
( ) ( ) ( ) ( )
( ) ( )
11 2
1 2
1, ,
if | | ,
sign
{ } otherw )ise (
|| ||
i
i i i
nn
i i i i
n i i n n
n nn
n
b e
b
i
e e
P b u
u
m
−
−−
−
= −
−
=
(36)
30 ISSN 1028-0979
In this algorithm,
( ) ( 1) ( ) T...: [ , , ]i i ir
n n nb b b= is the i th row of ,nB ( ) { }iP w
repre-
sents the projection of w onto the i th set
( 1) ( )( ) ( 1) ( )[ , ] [ , ]
i iri i irb b b b =
and
( ) ( ) 0( ) ( )2 , .i i i i = The coefficients
( )i
n s in (36) are chosen from (31) to
satisfy (35).
The recursive procedure (36) is the on-line identification algorithm need to imple-
ment the adaptive pseudoinverse model-based controller (34). The adaptive closed-loop
control system containing this controller is shown in Fig. 5.
Fig. 5
The convergence and robustness properties of the adaptive pseudoinverse model-
based controller (34), (36) are given below.
Proposition. Provided that Assumptions A1) to A3) are valid, and the conditi-
on (33) is satisfied, the adaptive control algorithm defined in (34), (36) and applied to
the plant (1) gives:
(i) the estimate sequence { }nB converges in the sense of
;n n
B B→
⎯⎯⎯→
(ii) the ultimate boundedness given in the expressions (7), (8) is achieved.
Proof. Due to space limitation, the proof of Proposition is omitted. □
5. Robustly-adaptive control of plants with nonsquare
gain matrices having not full rank
Let B be a nonsquare m r matrix of the form (2) with unknown rank satis-
fying (4). Define the so-called submatrices 1[ [ ], , [ ] |1, , ] r r
rB i k i k r R [43, part I,
subsect. 2.2] whose rows represent the rows of B with the numbers 1[ ], , [ ]ri k i k
( 11 [ ] [ ]ri k i k m ). The quantity of these matrices is equal to .
m
N
r
=
Deno-
ting by [ ]B k the submatrix which corresponds to a k th subset 1{ [ ], , [ ]},ri k i k write
the equations of some k plants as:
Міжнародний науково-технічний журнал
«Проблеми керування та інформатики», 2022, № 2 31
1[ ] [ ] [ ], 1, , ,n n ny k B k u v k k N−= + = (37)
where 1( [ ]) ( [ ])
[ ] [ , , ]ri k i k r
n n ny y Rk y= and 1( [ ]) ( [ ])
[ ] [ , , ] .ri k i k r
n n nv Rk v v=
In accordance with the approach proposed in the previous section, pass from the
equation (37) to the equations of the fictitious plants described by
1[ ] [ ] [ ], 1, , ,n n ny k B k u v k k N−= + = (38)
with the same 1nu − and [ ].nv k In these equations, [ ]ny k denotes the r-dimensional
output vector related to the k th fictitious plant whose gain matrix [ ]B k is defined as
follows:
0[ ] [ ] [ ] ,rB k B k k I= + (39)
where 0[ ]k is a fixed quantity depending on k. This quantity is calculated for each
1, ,k N= using the technique described in the previous section. Namely, with the
constraints (5) in mind, 0[ ]k can always be found to satisfy the conditions
det [ ] 0 1, ,B k k N = (40)
similar to (32).
It follows from the equations (37) to (39) that
0 1[ ] [ ] [ ] .n n ny k y k k u −= +
(41)
This expression shows that although [ ]B k as [ ]B k remain unknown, however, the
components of all N the vectors [ ]ny k can indirectly be «measured» after measuring
the components of ny and 1,nu − and it is essential.
If the conditions (40) are satisfied, then the problem of the adaptive stabilization of
the true plant (1) can be reduced to the problem of simultaneous adaptive stabilization
of all N fictitious plants (38) with unknown but nonsingular r r gain matrices [ ]B k
( 1, ,k N= ) via forming at each n th time instant a set of N different «potentially»
possible controls [1], , [ ]n nu u N and selecting one of them in accordance with cer-
tain choice rule [23] given below.
Following [23], the adaptive control law to be applicable to any fictitious plant is
designed in the form
1
1[ ] [ ] [ ],n n n nu k u B k e k−
−= +
1, , ,k N=
(42)
where
0[ ] [ ] [ ]n ne k y k y k= − with 10( [ ]) 0( [ ])0 T[ ] [ , ..., ]ri k i k
y k y y= defines the output
error vector related to the k th fictitious plant at the n th time instant, and [ ] r r
nB k R
is the current estimate of unknown r r matrix [ ]B k at the same time instant satisfying
det [ ] 0 1, , .nB k k N =
(43)
As the adaptation algorithms, the standard recursive procedures for the adaptive
identification of each k th fictitious plant (37) described by
32 ISSN 1028-0979
( ) *( ) 0
1
*( ) *( )
( )( ) ( )
11 2
1 2
n
1, , , 1, ,
[ ] if | [ ]| ,
[ ] sig [ ]
[ ] [ ] otherwise,
|| ||
i i
n in
i i
ii i n i n
n n nn
n
b k e k
e k e k
b k u
i r N
b k
u
k
−
−−
−
−
= +
= =
(44)
are proposed. In these algorithms,
( )[ ]i
nb k denotes the r-dimensional estimate vector ob-
tained by transposing the i th row of [ ],nB k and
( ) ( )T*( ) ( )
11 1[ ] [ ] [ ] [ ]
i ii i
n n nn ne k y k y k b k u −− −= − −
(45)
represents the scalar variable making sense of the i th component of
*( )[ ]i r
ne k R that is
the identification error vector related to the k th fictitious plant. The coefficients
( )i
n s
are chosen from the ranges (31) to satisfy the requirement (43).
Next, add the adaptation algorithms described in the formulas (44) together with (45)
by an algorithm for estimating unknown B defined as follows:
( ) *( ) 0
1
( ) *( ) *( )
( ) ( )
11 2
1 2
if | | ,
sign
otherwis 1, , ,e,
|| ||
i i
n in
i i i
n i i n i n
n nn
n
b e
b
i
e e
b u m
u
−
−−
−
= −
+
=
(46)
where
( )Ti
nb represents the i th row of the estimate matrix nB and
( ) ( )T*( ) ( )
11 1
i ii i
n n nn ne y y b u −− −= − −
(47)
is the i th component of the identification error vector
*
1 1 1n n n n ne y y B u− − −= − − ( i
and
0
i are given by the conditions (28)).
The estimation procedure defined in the algorithm (46) together with the equation (47)
makes it possible to estimate the m predicted output errors
( )
1[ ]
i
ne k+
( 1, ,i m= ) for
each i th output of true plant (1) at any n using the formula
( ) 0( ) ( )T ( ) .| [ ] | | [ ] | , 1, ,i i i i
n n ne k y b u k i m= − + =
(48)
The synthesis of the adaptive controller is finished by the choice of the control nu
from the set { [1], , [ ]}n nu u N with [ ]nu k given by (42). This choice is implemented
by the rule giving the minimum of the 1-norm of
(1) ( ) T
1 1 1[ ] [ [ ], , [ ]]
m
n n ne k e k e k+ + +=
according to
( )
[ ] 1
arg min | [ ] |,
n
m
i
n n
u k i
u e k
=
= (49)
where
( )
1
[ ]
i
n
e k
+
s are specified by (48).
The asymptotic properties of the adaptive controller described in this section are
given in theorem below.
Theorem 2. Consider the feedback control system containing the plant (1) in
which 𝑟 < 𝑚, and the adaptive controller defined in the equations (44), (49) to-
Міжнародний науково-технічний журнал
«Проблеми керування та інформатики», 2022, № 2 33
gether with (41), (48) and (43). Using the constraints (5), determine 0[1],
0, [ ]N to satisfy the requirement (40). Let assumptions A1)–A3) be valid.
Then, this controller applied to the plant (1) guarantees that the control objecti -
ves (7) and (8) will be achieved.
Proof. See [39]. □
Note that Theorem 2 does not guarantee that the ultimate error lim sup || ||n ne→
will become as in the nonadaptive case when there is no parametric uncertainty and the
pseudoinverse model-based controller proposed in [24] can by applied.
6. Simulation
To demonstrate the behavior of the robust adaptive closed-loop control system de-
signed in Section 3–5, several simulation experiments were conducted.
Simulation experiment 1. In this experiment, the adaptive closed-loop control
system (1), (6), (25) to (32) was simulated. The elements of B were given as: (11) 4,b =
(12) 2,b = (21) 2,b = (22) 1b = (det 0).B = The interval estimates of these elements
were chosen as follows: (11)1 5,b (12)0 2,b (21)0 2,b (22)1 2.b By
the formulas (20) and (22), it was found:
(1)
min
1, = −
(2)
min
1, = −
(1)
max 7, =
(2)
max 4, =
min
1, = − max 7. = It turned out that maxmin
| | | | . Therefore it is required that
0 1 to satisfy the inequalities (21). Namely, 0 1,1 = was put. From the conditions
(11)
0 [1, 5],b
(12)
0 [0, 2],b
(21)
0 [0, 2],b
(22)
0 [1, 2]b the following initial estimates
of nB were taken:
(11)
0 1,b =
(12)
0 1,b =
(21)
0 0,b =
(22)
0 1,9.b = Then
(11)
0 2,1,b =
(12)
0 1,b =
(21)
0 0,b =
(22)
0 3.b =
In this simulation experiment, the sequences
( ){ } ( 1, 2)i
nv i = were generated as
i.i.d. random variables [29, p.40] belonging to [ 1, 1].− It was put:
0 T[1, 3] .y =
The performance of the adaptive estimation algorithm is shown in Fig. 6.
Fig. 6
Fig. 7 shows simulation results illustrating a successful behavior of the robustly-
adaptive control system when this experiment was conducted.
2
1
0
1 2 3 4 5
n =0
n =6
n =13
n =106
n =116 n=314
n =1
n =2 0
1 2
1
2
n =1 n =2
n =9
n =105
n =166
n =534
34 ISSN 1028-0979
Fig. 7
Simulation experiment 2. To verify how the adaptive controller proposed in Sec-
tion 4 performs, we give an illustrative example. In this example, the true B and the ini-
tial 0B were chosen as:
0
0,2 1,4 50 20
0,8 2,4 , 30 40
1,1 0.5 10 10
B B
= =
to ensure 0rank rank 2.B B= = The desired output vector was taken as
0 T[2, 7, 3] .y =
( ){ }i
nv were generated as pseudorandom independently identically dis-
tributed (i.i.d.) sequences satisfying to
(1) [ 0,1; 0,1],nv −
(2) [ 0,2; 0,2]nv − and
(3) [ 0,08; 0,08].nv −
Computer simulations have been carried out to evaluate the behavior of the adap-
tive control system (1), (34), (36). This behavior is presented in Fig. 8. It demonstrates
that the closed-loop control systems containing the adaptive pseudoinverse model-based
controller (34), (36) is successful enough.
Fig. 8
b
a
n
2,8
2,6
2,4
20
3,0
40
b
a
n
60 80
2
1
100
0
20
3
40 60 80 100 0
0
2nu
n
20 40 60 80 100 0
2nu
2ny
10
0
20
0
2
y
4
n
20 40 60 80 100 0
2ny
3,5
3,0
4,0
3,2
2,5
2,0
1,5
Міжнародний науково-технічний журнал
«Проблеми керування та інформатики», 2022, № 2 35
Simulation experiment 3. This simulation experiment was conducted to illustrate
the performance of the adaptive control proposed in Section 5 for the case when 2,r =
3.m = As the gain matrix,
4 2
2 1
3 1,5
B
=
of not full rank ( rank 1B = ) was taken. Since 3,N = it produces the following three
submatrices:
4 2 4 2 2 1
[1] , [2] and [3]
2 1 3 1,5 3 1,5
B B B
= = =
Further, the three vectors
(1) (2) T[1] [ , ] ,n n ny y y=
(1) (3) T[2] [ , ]n n ny y y= and
(2) (3) T[3] [ , ]n n ny y y= were introduced to describe the plants (37) having the gain
matrices [1], [2] and [3]B B B respectively. The quantities 0 0[1] 1,1, [2] 1,2 = = and
0[3] 1,3 = were taken to satisfy the conditions (40) guaranteeing [ ]B k to be nonsingu-
lar were derived from the equation (5). The initial 0 0 0[1], [2] and [3]B B B were chosen
as 0 0 0[ ] [ ] [ ] rB k B k k I= + with the initial elements of 0[ ]B k which were selected from
B inside the corresponding ranges
( ) ( )[ , ]
ij ijb b specified as follows:
(11) [1, 5],b
(12) [0, 2],b
(21) [0, 2],b
(22) [1, 2],b
(31) [1, 4],b
(32) [0, 5].b Namely, we set
(11)
0 1,b =
(12)
0 1,b =
(21)
0 0,b =
(22)
0 1,9,b =
(31)
0 2,b =
(32)
0 2,1.b = The desired output
vector was given as
0 T[1, 3, 7] .y =
The performance of the simulated adaptive closed-loop control system with the
disturbance sequences
( ) ( )( )
0 1{ } , ,
i ii
nv v v= generated as some pseudorandom i.i.d. vari-
ables taken from
(1) (2) (3)0,1 0,1; 0,2 0,2; 0,08 0,08n n nv v v− − − is presented
in Fig. 9.
Fig. 9
a
n
20 40 60 80 100 0
1nB B− 3
4
2
0
1
1
[ ] [ ]nB k B k−
1nB B−
1
[3] [3]nB B−
1
[2] [2]nB B−
1
[1] [1]nB B−
b
n
20 40 60 80 100 0
3
k
2
0
1
36 ISSN 1028-0979
Continuation Fig. 9
Fig. 9, a–d demonstrate that the performance of the proposed adaptive controller
applied to the static multivariable plant having some nonsquare gain matrix with not full
rank is successful enough.
Conclusion
The adaptive control concept together with the pseudoinverse (generalized in-
verse) model-based approach is the suitable tool to deal with discrete-time uncer-
tain multivariable systems whose gain matrices are noninvertible. It has been
shown that a successful behavior of the robustly-adaptive closed-loop control sys-
tem containing the square singular or nonsquare plant can be achieved regardless
of the rank of its gain matrix.
Л.С. Житецький, К.Ю. Соловчук
АДАПТИВНЕ РОБАСТНЕ КЕРУВАННЯ
БАГАТОВИМІРНИМИ СИСТЕМАМИ БЕЗ ПАМ’ЯТІ,
ЩО НЕ МОЖУТЬ БУТИ ОБЕРНЕНИМИ, З
ОБМЕЖЕНИМИ ЗБУРЕННЯМИ: УЗАГАЛЬНЕННЯ
Житецький Леонід Сергійович
Міжнародний науково-навчальний центр інформаційних технологій та систем НАН України і
МОН України, м. Київ,
leonid_zhiteckii@i.ua
Соловчук Клавдія Юріївна
Полтавський науково-дослідний експертно-криміналістичний центр МВС України,
solovchuk_ok@ukr.net
Стаття присвячена робастному адаптивному керуванню в дискретному часі де-
якими класами невизначених багатовимірних систем без пам’яті за наявності не-
вимірних обмежених збурень, межі яких припускаються відомими. Розглянуто
системи, у яких число керувальних входів не перевищує числа виходів. Основна
особливість об’єктів, що підлягають керуванню, полягає у тому, що матриці кое-
фіцієнтів підсилення таких об’єктів не можуть бути обернені. Введено припу-
щення, що елементи цих матриць апріорі невідомі, але є інформація про можливі
межі інтервалів, до яких вони належать. Поставлено і вирішено задачу, що поля-
c
n
20 40 60 80 100 0
1nu
3
4
2
0
1
1ne
d
n
20 40 60 80 100 0
11
10
8
9
12
mailto:leonid_zhiteckii@i.ua
mailto:solovchuk_ok@ukr.net
Міжнародний науково-технічний журнал
«Проблеми керування та інформатики», 2022, № 2 37
гає в тому, аби у контурі зворотного зв’язку побудувати регулятор, здатний впо-
ратися з матрицями коефіцієнтів підсилення, які не можуть бути обернені, а та-
кож з параметричною невизначеністю для пригнічування зовнішніх збурень і за-
безпечення обмеженості всіх керувань та вихідних сигналів. Для вирішення ви-
щезазначеної задачі використовується робастний адаптивний підхід у поєднанні з
так званою концепцією псевдооберненої чи оберненої моделей. Вивчено три різні
випадки. У першому випадку побудований робастний адаптивний регулятор, який
може застосовуватися до невизначеного об’єкта з виродженою квадратною мат-
рицею коефіцієнтів підсилення. У другому випадку на основі псевдообернених
моделей, параметри яких оцінюються за допомогою стандартної рекурентної
процедури адаптації, запропоновано робастний метод керування об’єктами з не-
відомими прямокутними матрицями коефіцієнтів підсилення повного рангу. За-
пропонований у першому випадку підхід поширюється на третій випадок керу-
вання невідомими об’єктами, матриці коефіцієнтів підсилення яких є прямокут-
ними матрицями неповного рангу. Встановлено асимптотичні властивості
запропонованих у статті робастно-адаптивних регуляторів. Для підтримки теоре-
тичного дослідження наведено результати числових прикладів.
Ключові слова: дискретний час, багатовимірний об’єкт без пам’яті, нездатність
до обернення, концепція псевдооберненої моделі, невизначеність, алгоритм оці-
нювання, робастне адаптивне керування.
REFERENCES
1. Davison E. The output control of linear time-invariant multivariable systems with un-
measurable arbitrary disturbances’. IEEE Trans. Autom. Contr. Oct. 1972. AC-17, N 5.
P. 621–631.
2. Gubarev V.F., Mishchenko M.D., Snizhko B.M. Model predictive control for discrete
MIMO linear systems. In: Y.P. Kondratenko, A.A. Chikrii, V.F. Gubarev, J. Kacprzyk
(eds.). Advanced control techniques in complex engineering systems: Theory and applica-
tions. Dedicated to Prof. V.M. Kuntsevich. Studies in systems, decision and control, Spring-
er Nature Switzerland AG, Cham, 2019. 203. P. 63–81. DOI: https://doi.org/10.1007/978-3-
030-21927-7_10.
3. Skogestad S., Postlethwaite I. Multivariable feedback control. Wiley, Chichester, 1996.
4. Albertos P., Sala A. Multivariable control systems: an engineering approach. Springer, London,
2006.
5. Glad T., Ljung L. Control theory: multivariable and nonlinear methods. Taylor & Francis, New
York, 2000.
6. Tan L. A generalized framework of linear multivariable control. Elsevier, Oxford, 2017.
7. Pukhov G.E., Zhuk K.D. Synthesis of interconnected control systems via inverse operator meth-
od. Nauk. dumka, Kiev, 1966 (in Russian).
8. Lyubchyk L.M. Disturbance rejection in linear discrete multivariable systems: inverse model ap-
proach. In: Proc. 18th IFAC World Congress. Milano, Italy, 2011. P. 7921–7926.
9. Lyubchyk L.M. Inverse model approach to disturbance rejection problem, In: Kondratenko Y.P.,
Kuntsevich V.M., Chikrii A.A., Gubarev V.F. (Eds), Advanced control systems: theory and ap-
plications. River Publishers, Gistrup. 2021. P. 129–166.
10. Skurikhin V.I., Zhiteckii L.S., Solovchuk K.Yu. Control of interconnected plants with singular
and ill-conditioned transfer matrices based on pseudo-inverse operator method. Upravlyayush-
chye sistemy i mashiny. 2013. 29, N 3. P. 14−20.
11. Zhiteckii L.S., Solovchuk K.Yu. Pseudoinversion in the problems of robust stabilizing multivari-
able discrete-time control systems of linear and nonlinear static objects under bounded disturb-
ances. Journal of Automation and Information Sciences. 2017. 49, N 5. P. 35–48.
12. Skurikhin V.I., Zhiteckii L.S., Solovchuk K.Yu. Stabilization of a nonlinear multivariable dis-
crete-time time-invariant plant with uncertainty on a linear pseudoinverse model. Journal of
Computer and Systems Sciences International. 2017. 56, N 5. P. 759–773.
38 ISSN 1028-0979
13. Kuntsevich V.M. Control under uncertainty: guaranteed results in control and identification prob-
lems. Nauk. dumka, Kiev, 2006 (in Russian).
14. Fomin V.N., Fradkov A.L., Yakubovich V.A. Adaptive control of dynamic plants. Nauka, Mos-
cow, 1981 (in Russian).
15. Goodwin G.C., Sin K.S. Adaptive filtering, prediction and control. Prentice-Hall, Engewood
Cliffs, 1984.
16. Tao G. Adaptive control design and analysis. John Wiley and Sons, New York, 2003.
17. Narendra K.S., Annaswamy A.M. Stable adaptive systems. Dover Publications, New York,
2012.
18. Ioannou P., Sun J. Robust adaptive control. Dover Publications, New York, 2013.
19. Ǻström K.J. and B. Wittenmark. Adaptive control: 2nd Edition, Dover Publications, New York,
2014.
20. Zhiteckii L.S., Solovchuk K.Yu. Adaptive stabilization of some multivariable systems with
nonsquare gain matrices of full rank. Cybernetics and Computer Engineering. 2018. 192, N 2.
P. 44–60.
21. Zhiteckii L.S., Solovchuk K.Yu. Solving a problem of adaptive stabilization for some static mimo
systems. Cybernetics and Computer Engineering. 2019. 197, N 3. P. 33–50.
22. Zhiteckii L.S., Solovchuk K.Yu. Robust adaptive controls for a class of nonsquare memor-
yless systems. In: Kondratenko, Y.P., Kuntsevich, V.M., Chikrii, A.A., Gubarev, V.F. (Eds),
Advanced Control Systems: Theory and Applications. River Publishers, Gistrup, 2021.
P. 203–226.
23. Zhiteckii L.S., Azarskov V.N., Solovchuk K.Yu. Adaptive robust control of inter-connected static
plants with nonsquare gain matrixes. In: Proc. 13th All-Russian Control Problems Council
(VSPU-2019). IPU, Moscow, 2019. P. 713–718.
24. Albert A. Regression and the Moore-Penrose pseudoinverse. Academic Press, New York,
1972.
25. Voevodin V.V., Kuznetsov Yu.A. Matrices and calculations. Nauka, Moscow, 1984 (in Rus-
sian).
26. Azarskov V.N., Zhiteckii L.S., Solovchuk K.Yu. Parametric identification of the interconnected
static closed-loop system: a special case. In: Proc. 12th All-Russian Control Problems Council
(VSPU-2014). IPU, Moscow, 2014. P. 2764–2776.
27. Marcus M., Minc H. A survey of matrix theory and matrix inequalities. Aliyn and Bacon, Boston,
1964.
28. Zhiteckii L.S., Skurikhin V.I. Adaptive control systems with parametric and nonparametric uncer-
tainties. Nauk. dumka, Kiev, 2010 (in Russian).
29. Walter E., Pronzato L. Identification of parametric models. Springer, Paris/Milan/Barcelona,
1997.
Submitted 01.02.2022
|
| id | nasplib_isofts_kiev_ua-123456789-210873 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0572-2691 |
| language | English |
| last_indexed | 2026-03-16T12:21:29Z |
| publishDate | 2022 |
| publisher | Інститут кібернетики ім. В.М. Глушкова НАН України |
| record_format | dspace |
| spelling | Zhiteckii, L.S. Solovchuk, K.Yu. 2025-12-19T17:14:22Z 2022 Adaptive robust multivariable control of noninvertible memoryless systems with bounded disturbances: a generalization / S. ZhiteckiiL, K.Yu. Solovchuk // Проблеми керування та інформатики. — 2022. — № 2. — С. 22-38. — Бібліогр.: 29 назв. — англ. 0572-2691 https://nasplib.isofts.kiev.ua/handle/123456789/210873 681.5 10.34229/2786-6505-2022-2-2 The article is dedicated to robust adaptive control in discrete time for certain classes of undefined memoryless multivariate systems with unmeasurable bounded disturbances, the bounds of which are assumed to be known. The systems considered have a number of control inputs that does not exceed the number of outputs. Стаття присвячена робастному адаптивному керуванню в дискретному часі деякими класами невизначених багатовимірних систем без пам’яті за наявності невимірних обмежених збурень, межі яких припускаються відомими. Розглянуто системи, у яких число керувальних входів не перевищує числа виходів. Основна особливість об’єктів, що підлягають керуванню, полягає у тому, що матриці коефіцієнтів підсилення таких об’єктів не можуть бути обернені. Введено припущення, що елементи цих матриць апріорі невідомі, але є інформація про можливі межі інтервалів, до яких вони належать. Поставлено і вирішено задачу, що полягає в тому, аби у контурі зворотного зв’язку побудувати регулятор, здатний впоратися з матрицями коефіцієнтів підсилення, які не можуть бути обернені, а також з параметричною невизначеністю для пригнічування зовнішніх збурень і забезпечення обмеженості всіх керувань та вихідних сигналів. Для вирішення вищезазначеної задачі використовується робастний адаптивний підхід у поєднанні з так званою концепцією псевдооберненої чи оберненої моделей. Вивчено три різні випадки. У першому випадку побудований робастний адаптивний регулятор, який може застосовуватися до невизначеного об’єкта з виродженою квадратною матрицею коефіцієнтів підсилення. У другому випадку на основі псевдообернених моделей, параметри яких оцінюються за допомогою стандартної рекурентної процедури адаптації, запропоновано робастний метод керування об’єктами з невідомими прямокутними матрицями коефіцієнтів підсилення повного рангу. Запропонований у першому випадку підхід поширюється на третій випадок керування невідомими об’єктами, матриці коефіцієнтів підсилення яких є прямокутними матрицями неповного рангу. Встановлено асимптотичні властивості запропонованих у статті робастно-адаптивних регуляторів. Для підтримки теоретичного дослідження наведено результати числових прикладів. en Інститут кібернетики ім. В.М. Глушкова НАН України Проблемы управления и информатики Адаптивне керування та методи ідентифікації Adaptive robust multivariable control of noninvertible memoryless systems with bounded disturbances: a generalization Адаптивне робастне керування багатовимірними системами без пам'яті, що не здатні бути оберненими, з обмеженими збуреннями: узагальнення Article published earlier |
| spellingShingle | Adaptive robust multivariable control of noninvertible memoryless systems with bounded disturbances: a generalization Zhiteckii, L.S. Solovchuk, K.Yu. Адаптивне керування та методи ідентифікації |
| title | Adaptive robust multivariable control of noninvertible memoryless systems with bounded disturbances: a generalization |
| title_alt | Адаптивне робастне керування багатовимірними системами без пам'яті, що не здатні бути оберненими, з обмеженими збуреннями: узагальнення |
| title_full | Adaptive robust multivariable control of noninvertible memoryless systems with bounded disturbances: a generalization |
| title_fullStr | Adaptive robust multivariable control of noninvertible memoryless systems with bounded disturbances: a generalization |
| title_full_unstemmed | Adaptive robust multivariable control of noninvertible memoryless systems with bounded disturbances: a generalization |
| title_short | Adaptive robust multivariable control of noninvertible memoryless systems with bounded disturbances: a generalization |
| title_sort | adaptive robust multivariable control of noninvertible memoryless systems with bounded disturbances: a generalization |
| topic | Адаптивне керування та методи ідентифікації |
| topic_facet | Адаптивне керування та методи ідентифікації |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210873 |
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