Adaptive Suboptimal Control of Some Discrete-Time Plants with Nonstochastic Bounded Disturbances
The paper addresses adaptive suboptimal control of linear, discrete-time, stationary, minimum-phase, scalar plants in the presence of nonstochastic bounded unmeasurable disturbances with asymmetric upper and lower bounds assumed to be unknown a priori. An additional assumption is that the order of t...
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Інститут кібернетики ім. В.М. Глушкова НАН України
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| Cite this: | Adaptive Suboptimal Control of Some Discrete-Time Plants with Nonstochastic Bounded Disturbances / V.M. Azarskov, K.Y. Solovchuk, O.Y. Volkov, L.S. Zhiteckii // Проблеми керування та інформатики. — 2024. — № 2. — С. 20–32. — Бібліогр.: 23 назви. — англ. |
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| author | Azarskov, V.M. Solovchuk, K.Y. Volkov, O.Y. Zhiteckii, L.S. |
| author_facet | Azarskov, V.M. Solovchuk, K.Y. Volkov, O.Y. Zhiteckii, L.S. |
| citation_txt | Adaptive Suboptimal Control of Some Discrete-Time Plants with Nonstochastic Bounded Disturbances / V.M. Azarskov, K.Y. Solovchuk, O.Y. Volkov, L.S. Zhiteckii // Проблеми керування та інформатики. — 2024. — № 2. — С. 20–32. — Бібліогр.: 23 назви. — англ. |
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| description | The paper addresses adaptive suboptimal control of linear, discrete-time, stationary, minimum-phase, scalar plants in the presence of nonstochastic bounded unmeasurable disturbances with asymmetric upper and lower bounds assumed to be unknown a priori. An additional assumption is that the order of the difference equation describing the plant is known beforehand. The unique feature of the stated problem is that neither the bounds on unmeasured disturbances nor the admissible region to which the unknown parameters of the plant belong are known a priori. Adaptive point and membership set estimation procedures are used to solve this problem
Розглядається адаптивне субоптимальне керування лінійними, дискретними, стаціонарними, мінімально-фазовими, скалярними об’єктами за наявності нестохастичних обмежених невимірюваних збурень, верхня та нижня межі яких можуть бути асиметричними та вважаються апріорі невідомими. Додаткове припущення полягає в тому, що порядок різницевого рівняння, яке описує об’єкт, апріорі відомий. Відмінною особливістю поставленої в статті задачі є те, що межі ані невимірюваних збурень, ані допустимої області, до якої належать невідомі параметри об’єкта, не вважаються апріорі відомими. Для вирішення цієї задачі використовуються адаптивні процедури точкового і множинного оцінювання.
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© V. AZARSKOV, K. SOLOVCHUK, O. VOLKOV, L. ZHITECKII, 2024
20 ISSN 2786-6491
UDC 681.5
V. Azarskov, K. Solovchuk, O. Volkov, L. Zhiteckii
ADAPTIVE SUBOPTIMAL CONTROL
OF SOME DISCRETE-TIME PLANTS
WITH NONSTOCHASTIC BOUNDED DISTURBANCES
Valerii Azarskov
National Aviation University, Kyiv,
orcid: 0009-0004-9206-2475
azarskov@nau.edu.ua
Klavdiia Solovchuk
International Research and Training Center for Information Technologies and Systems
of the National Academy of Sciences of Ukraine and the Ministry of Education and Scie-
nce of Ukraine, Kyiv,
orcid: 0009-0000-5191-939X
solovchuk.ok@gmail.com
Oleksandr Volkov
International Research and Training Center for Information Technologies and Systems
of the National Academy of Sciences of Ukraine and the Ministry of Education and Scie-
nce of Ukraine, Kyiv,
orcid: 0000-0002-5418-6723
director@irtc.org.ua
Leonid Zhiteckii
International Research and Training Center for Information Technologies and Systems
of the National Academy of Sciences of Ukraine and the Ministry of Education and Scie-
nce of Ukraine, Kyiv,
orcid: 0000-0002-4560-5113
leonid_zhiteckii@i.ua
This paper deals with the adaptive suboptimal control of linear, discrete-time,
time-invariant, minimum phase, scalar plants in the presence of nonstochastic
bounded unmeasurable disturbances whose upper and lower bounds, which may
be asymmetric, are assumed to be unknown a priori. Additional assumption is that
an order of the difference equation describing the plant is known a priori. The dis-
tinguishing feature of the problem stated in this paper is that neither bounds on the
unmeasured disturbances, nor bounds on an allowable region to which the un-
known plant parameters belong are assumed to be known a priori. To solve this
problem, adaptation procedures for the point and membership set estimation are
utilized. The standard recursive procedure with adjustable dead zone is employed
in order to derive the point estimates of unknown plant parameters together with
the point estimate of time-invariant disturbance component. The size of this dead
zone depends on the previous point estimate of the bounds on the time-varying
disturbance component and also on a fixed suboptimality index chosen by the de-
signer. The estimates generated by the point estimation procedure are directly ex-
ploited to derive the adaptive control law. The main idea advanced in this paper is
that, instead of unknown a priori membership set of these parameters, their peculi-
ar hypothetical a posteriori membership sets are designed via the use of the meas-
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 2 21
ured system’s signals together with the current point estimate of bounds on the
time-varying disturbance component. Contrary to the usual membership set esti-
mation approach, this set is updated if only it is discovered that the unknown pa-
rameter vector does not belong in reality to this set. To this end, a remarkable
property of the point estimation procedure is utilized. Such an approach makes it
possible to reconstruct this set and to update the previous estimate of the bounds on
the time-varying disturbance component. The finite convergence of the adaptation
procedures and also the ultimate boundedness of system’s signals are established.
To demonstrate an efficiency of the adaptive controller and support the theoretical
study, simulation results are presented.
Keywords: adaptive control, point estimation, membership set estimation, dis-
crete-time, nonstochastic disturbance.
Introduction
The problem of a perfect performance of modern control systems with parametric
and nonparametric uncertainties remains an actual problem from both theoretical and
practical points of view. Within the framework of this scientific problem, novel ideas
have been advanced in recent work [1] dealing with the model predictive control ex-
tended last time in the modern control theory [2]. An essential progress has been
achieved by many researches using adaptive control approaches presented in [3–5]. Im-
portant theoretical results have been reported in numerous papers and generalized in
several books including, in particular, [6–9]. Some practical application of these adap-
tive control approaches for a manufacturing systems with uncertainties has been pre-
sented in [10].
There are exist stochastic and nonstochastic approaches to the adaptive control
design in the presence of unmeasurable external disturbances. The nonstochastic
approach has been advanced by V.M. Kuntsevich who studied methods for the so-
called membership set estimation of unknown plant parameters in his book [11] and
also by V.A. Yakubovich dealing with the traditional point estimation of these pa-
rameters in the book [12, chap. 4]. Their works have been extended for over last
two decades in [13, 14]. Recent results achieved in this scientific field can be found
in the papers [4, 15].
A distinguishing feature of the disturbances considered in the works above men-
tioned is that they have no probabilistic nature [12, p. 137]. Namely, their sequences do
not have any stochastic characteristics such as the expected value, variance and others.
Nevertheless, it is known [12, item 4.2.1] that, in contrast to stochastic case, the exact
identification of a plant subjected to unmeasured nonstochastic disturbances is impossi-
ble, in principle, even when the asymptotic behavior of a plant to be controlled is con-
sidered. One of the areas which attracts an attention of the adaptive control community
up-to-date remains the rejection of arbitrary bounded nonstochastic disturbances.
Meanwhile, traditional techniques for rejecting random disturbances based on stochastic
approaches become here unacceptable.
The adaptive optimal control of the linear discrete-time systems without a pure delay
in the presence of arbitrary bounded disturbances with known bounds has originally been
proposed in the paper [16]. To establish the asymptotic properties of the adaptive closed-
loop control system, the Key Technical Lemma reported in [17, p. 181] is utilized. The
adaptive suboptimal controller for rejecting nonstochastic bounded disturbance has been
proposed before by V.A. Yakubovich together with his disciple V.A. Bondarko utilizing
their Frequency Theorem [12, sec. 4.2]. It turns out that adaptive optimal and suboptimal
controls based on the point estimation without the membership set estimation are possible
if only the time delay is absent even when bounds on the disturbances are known.
22 ISSN 2786-6491
The method for adaptive robust control of the systems with bounded disturbances
whose bounds are unknown has been developed by G. Feng in [18] who has estab-
lished the ultimate boundedness of all signals in the closed-loop control system, as-
suming that a lower bound on the high frequency gain of the open-loop system and al-
so its sign are known a priori. However, no suboptimality of his adaptive controller
has been achieved. To cope with the absence of knowledge concerning the bounds on
these disturbances, V.A. Bondarko devised the adaptive suboptimal control algorithm
using a remarkable property of the finitely convergent estimation procedure examined
in [12, chap. 2]; see [19]. In order to implement this algorithm, certain knowledge of
a priori bounded membership set of unknown plant parameters was needed. The
adaptive suboptimal feedback controller for the first-order nonlinear plant in the pres-
ence of bounded disturbances with unknown bounds is designed in [13]. Fruitful ideas
of this paper have been extended in [14], where the problem of adaptive suboptimal
control of the first-order plant with two unknown parameters and unmodelled dynamics
is studied. One of the basic assumptions introduced in [13, 14] is that some constraints
on the allowable values of unknown parameters are known a priori. A disadvantage of
the works [13, 14, 18, 19] is that the bounds on the disturbance are assumed to be sym-
metric. This disadvantage has been overcome in [20]. Unfortunately, some a priori
knowledge of the membership set is also required to implement this algorithm.
A common feature of the works above mentioned is that they require certain
a priori information about a bounded membership set of unknown plant parameters.
Difficulties that may arise in practice are how to get suitable estimates of the
bounds on admissible parameters which will not be conservative and simultaneous-
ly to guarantee that each unknown parameter will lie within corresponding bounds.
To the best of our knowledge, the existing works do not provide an answer to this
question.
This paper proposes a new adaptive suboptimal control method to deal with ar-
bitrary bounded disturbances having unknown bounds in the absence of any infor-
mation about a priori membership set of unknown parameters. Its original version
was presented in [21] at 22nd IFAC World Congress which was held in Yokohama
(Japan) in July 2023.
1. Problem statement
Let the system to be controlled be a linear discrete-time, time-invariant single-input
single-output system with no pure delay described by
1 1 1 1 1( 0),t t n t n t n t n ty a y a y b u b u v b− − − −+ + + = + + (1)
where ,ty tu and tv denote the scalar output, input (control) and unmeasurable dis-
turbance, respectively, at time instant t. Using the backward shift operator
1q−
defined
by
1
1,t tq x x−
−= rewrite (1) in the form
1 1( ) ( ) ,t t tA q y B q u v− −= + (2)
where
1( )A q−
and
1( )B q−
are the polynomials of the order n given as
1 1
1( ) 1 ,n
nA q a q a q− − −= + + +
1 1
1( ) .n
nB q b q b q− − −= + +
The following assumptions about the system (2) are made:
A1) the integer n is known a priori;
A2) the system (1) is the minimum phase plant;
A3) the coefficients of
1( )A q−
and
1( )B q−
are unknown;
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 2 23
A4) the disturbance tv represents the so-called nonstochastic but bounded varia-
ble [11, 12] defined by
min maxtv t− (3)
with possibly asymmetric lower and upper bounds, i.e., min max ;−
A5) min and max are unknown a priori.
An illustrative example of nonstochastic bounded disturbance tv with asymmetric
bounds is depicted in Fig. 1 below.
Fig. 1
Remark 1. Note that the open-loop minimum phase system (2) may be unstable, in
principle. Besides,
1( )A q−
and
1( )B q−
may not be relatively prime, i.e., the pole-zero
cancellation is valid. See [12, item. 4.2.1].
Denoting by
0y a desired output of the system (1), the regulation problem when
0 consty will be considered. To this end, introduce the ultimate performance index
0( , ) lim sup | |t
t
J u v y y
→
= −
depending on the control sequence 1 2: { } , ,tu u u u= = generated by some controller
and on the disturbance sequence 1 2: { } , ,tv v v v= = . It is clear that
0( ) inf ( , ) inf lim sup | |t
u u t
J v J u v e
→
= =
where is the standard notation of the space of any bounded sequence, characterizes
the optimal asymptotic behavior of the output error
0
t te y y= − (4)
for a given disturbance sequence v.
Definition [19, 20]. The closed-loop control system is said to be suboptimal with a
suboptimality index 0 if the requirement
min max
0
: [ , ]
lim sup sup ( )
t
t
v vt
e J v
→
+ (5)
is satisfied for any positive .
Remark 2. In contrast with [12], the expression (5) defines the suboptimality index
in the additive but not in multiplicative form, and it is here essential.
With assumptions A1) to A5) in mind, the problem is to design the adaptive subop-
timal controller guaranteeing the achievement of the control goal (5) for any positive
chosen by the designer.
tv
max
min
v
t
24 ISSN 2786-6491
2. Optimal nonadaptive control (ideal case)
Suppose, for the time being, that the coefficients of
1( )A q−
and
1( )B q−
are
known. To derive the optimal control satisfying the requirement
min max
0
: [ , ]
lim sup | | sup ( ),
t
t
v vt
e J v
→
(6)
write the variable tv in the following form:
,t tv v v= + (7)
where v and tv are the time-invariant and the time-varying components of ,tv respec-
tively. Due to (3) they are given as
min max( ) / 2.v = +
, 0,1, 2,tv t = (8)
with
max min( ) / 2. = − (9)
Next, substituting (7) into (1) and shifting the indices of ty and ,tu one obtains
1 1 1 1 1 1.t t n t n t n t n ty a y a y b u b u v v+ − + − + ++ + + = + + + (10)
Following the standard steps in deriving the optimal control of the discrete-time
systems in the presence of bounded disturbances taken, e.g., from [12], one puts
0
1 1 1 1t n t n t n t na y a y b u b u v y− + − +− − − + + + =
to get the control law
1 0
1 1 1 2 1 1[ ].t t n t n t n t nu b y a y a y b u b u v−
− + − − += + + + − − − (11)
By virtue of (10) and the constraints (8), this control law yields
0
ty y t−
implying that the optimality requirement (6) rewritten as
0
: [ , ]
lim sup sup ( ),
t
t
v vt
e J v
− →
will be achieved with 0
: | |sup ( ) ,
tv v J v where (4) has been utilized.
3. Main result
Basic ideas. Return to the true situation when the components of the 2n-dimen-
sional vector
T
1 1[ , , , , , ]n na a b b = and also the (2 1)n + -dimensional extended
vector
T T[ , ]v = are unknown. According to the so-called certainty equivalence
adaptive control principle [17], replace unknown by some updated estimate
T
1[ ( ), ,t a t = 1( ), ( ), , ( ), ( )]n na t b t b t v t which will be found later. Then from (11),
the adaptive control law in the form
1 0
1 1 1 2 1 1( )[ ( ) ( ) ( ) ( ) ( )]t t n t n t n t nu b t y a t y a t y b t u b t u v t−
− + − − += + + + − − − − (12)
can directly be written.
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 2 25
Further, recalling that the upper bound, on { }tv defined in (9) remains unknown
(since min and max are unknown), a current estimate, t of will be exploited to
devise the adaptive algorithm for updating both the parameter estimate vector
2 1R n
t
+
and scalar .t
Define some strip tS in the (2 1)thn+ Euclidean space of extended vectors
T
1 1[ , , , , , , ]n na a b b v = as
T 2 1
1: { : | | } R ,n
t t ttS y +
−= − (13)
where
TT
1 1 1 1,[ , 1] [ , , , , ,t t t t n ty y u− − − − − = = − − ,1]t nu − denotes the extended re-
gression vector. It follows from (10) together with (8) that tS will contain the unknown
vector
T T[ , ]v = if [0, )sup | | .t t tv
It turns out that two different cases shown in Fig. 2 are possible. In the case depict-
ed in Fig. 2 left, a current estimate t of unknown upper bound on the absolute value
of the disturbance v is not less than its true value . Then the unknown extended pa-
rameter vector will belong to the strip defined by the expression (13). However, in
the other case depicted in this figure right when ,t the unknown expended parame-
ter vector may not belong to this strip.
Fig. 2
Now, consider some intersection written in the recursive form
1t t tS− = (14)
as depicted in Fig. 3, a, b.
a b
Fig. 3
tS
tS
1t− 1t−
t
0 0
tS
{ }
{ }
tS
26 ISSN 2786-6491
It can be understood that if 1t− and also tS then t (see Fig. 3, a)
whereas if 1t− but tS then t becomes empty (see Fig. 3, b) meaning that
the set of the inequalities
T
1 0 0
ˆ
( , 1, , )k k ky k k k t−− = +
with respect to unknown will be incompatible. With this fact in mind, put
1 / 2t t− = + (15)
if it is discovered that ,t = and
1t t− = (16)
otherwise. It is clear that the increase of the estimates st of the upper bound on | |tv
cannot be infinite, since t becomes nonempty if [0, )sup | | .t t tv
To avoid the case ,t = fix a 0,kt put 1 2k k kt t t n+ + = = = ( 0,1, )k =
and consider the intersection [ ]k of the 2 1n + strips tS defined in (13) at the time
instants , 1, , 2k k kt t t t n= + + as
1 2[ ] .
k k kt t t nk S S S+ + = (17)
Obviously, the set [ ]k defined in (17) represents a convex bounded polytope iff
the vectors 2, ,
k kt t n+ are linearly independent as illustrated in Fig. 4. Its Cheby-
shev center
c
and radius rad are specified as follows:
c
2
arg min max ,
= −
c ( )
2
1, ,
rad max ,i
i N
h
=
= −
where ( )ih is the notation of ith vertex of and N is the number of these vertices, and
rad denotes the radius of ,
2
is the Euclidean vector norm.
Fig. 4
c
2kt nS +
kt
S
[ ]k
(1)h
(2)h
( )h
( )Nh
2kt n+
kt
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 2 27
One of the basic ideas advanced in this paper is to utilize the set [ ]k given
by (17) as some hypothetical a posteriori membership set of unknown since [ ]k
may not contain in reality if
kt
is sufficiently small. On the other hand, the condi-
tion [ ]k will be guaranteed via a sequential increase of
kt
from a time kt to some
time 1kt + for 0,1,k = . (Such an approach will be described in the subsection below.)
Adaptive law. The adaptation algorithm is implemented via the following subse-
quent steps.
Step 1. Starting from 0,k = fix some initial 0 ,t t= put
0
/ 2,t = and find the
vertices
(1) ( ), , Nh h of [0] using the measured vectors t at 0 0, , 2 .t t t n= +
These vectors must be linearly independent to ensure the boundedness of [ ].k To do
it, choose arbitrarily the initial control sequence
0 0 01 2, , ,t t t nu u u+ + so that the
Gram’s determinants t satisfy
0t for 0 0[ , 2 ],t t t n + (18)
where
0 00 0
0
0 00 0
T T
T T
det , 1, , 2 .
t t jt t
t j
t j t jt t j
j n
+
+
+ ++
= =
(Note that the requirement (18) defines the necessary and sufficient condition for the
linear independence of these vectors [22, chap. 9, item 3].)
Step 2. Compute the Chebyshev center
c
2kt n+ using, e.g., [23] and the radius
of [ ].k
Step 3. Choose the initial estimate 2kt n+ moving it to the Chebyshev center
c
2 ,
kt n+ i.e., set
c
2 2 .
k kt n t n+ + =
Step 4. Set 0t = at 2 ,kt t n= + where t denotes an auxiliary variable deter-
mined below.
Step 5. Update t s using the adaptive estimation procedure with the adjustable
dead zone as
0
1 1
1 12
1 2
( , , )
,t t t
t t t t
t
f e − −
− −
−
= −
(19)
at 2 1,kt t n + + where
0 / 2,t t = + (20)
0
0 0 0
0
if ,
( , , ) 0 if (0 ),
if
e e
f e e
e e
−
=
+ −
denotes the dead-zone function caused by the identification algorithm of [12, chap. 2]
and t is the coefficient chosen free from
28 ISSN 2786-6491
0 2t (21)
to ensure 1( ) 0.b t
Step 6. If kt t then calculate t as follows:
0
1 1 1
2 2 2
1 1 21
if ( , , ) 0,
(2 )(| | ) || || otherwise.
t t t t
t
t t t t t t
f e
e
− − −
−
− − −
=
=
+ − −
(22)
Step 7. Verify the condition
2(rad [ ]) .t k (23)
If (23) is violated then put 0t = and increase k by 1 and instead of 1t− the updated
t specified in (15) exploit, reconstruct [ ]k using the same initial vectors t given in
Step 1 with updated t and return to Step 2.
Remark 3. It can be proved that the violation of (23) indicates that [ ]k for
given .
kt
This remarkable fact makes it possible to discover indirectly that the inter-
section (17) is indeed empty without performing complex computations.
Convergence and ultimate boundedness analysis. A remarkable property of the
procedure (19)–(21) leading to (23) is given in the following lemma:
Lemma. If [0, )sup | |t t tv and
c
0 [ 1]k = − then
1
22 2 0 2
1 1 1 2
(2 ) ( , , ) (rad [ 1]) .
k
t t t t t tt t
f e k
−
−
− − −
− −
Proof. Follows from the features of adaptive estimation procedure established
in [12, chap. 2]. □
With this lemma, the ultimate properties of the adaptive controller proposed here is
summarized in next theorem.
Theorem. Let assumptions A1) to A5) be satisfied. Then the resulting closed-loop
control system including the plant (1), the control law (12) and the adaptation algo-
rithm described in Steps 1 to 7 has the following properties:
i) the adaptive estimation algorithm determined by (15), (16), (19)–(23) together
with (13) and (17) converges at a finite time;
ii) the system signals are ultimately bounded;
iii) the suboptimal control objective (5) is achieved.
Proof. Due to space limitation, details are omitted. □
4. A numerical example and simulations
To demonstrate the features of the proposed adaptive controller, the simplest first-
order system described by
1 12,4t t t ty y u v− −− = +
was considered. (Although the instability is not inherent for many industrial plants to be
controlled, firstly, for open process control systems,
1( )A q−
was purposely chosen to
be unstable (as in [14]), in order to show that even in such a «bad» case, the adaptive
suboptimal performance can be achieved.) In this example, the suboptimality index was
chosen as 0,1. =
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 2 29
Before going to simulation experiments, the parameters of sets [ ]k of vectors
T
1 1
ˆˆ ˆ[ , ],a b = 0,1, 2, 3k = for t = 0,05; 0,1; 0,15; 0,2 were calculated and summa-
rized in the Table. These sets are depicted in Fig. 5, showing that [3] which contains
unknown is an appropriate a posteriori membership set of this vector while
[0], [1] and [2] are not.
Table
Ordinal number k Chebyshev center Radius of [ ]k
0 [– 2,28, 1,26] 0,092
1 the same 0,126
2 the same 0,181
3 the same 0,320
Results of the simulation experiment conducted for [ 0,2, 0,2]tv − and
0 5y =
are given in Fig. 6. They show the behavior of the adaptive control system in this exam-
ple. It can be seen from Fig. 6, a that if the variable t exceeds the square of the radius
of [ ]k designed at the time instant kt t= ( 0,1, 2, 3),k = then by virtue of the algo-
rithm determined by (15), (16), (19)–(23) together with (13) and (17) the variable t is
reset to zero, whereas the current estimate of the bound on disturbance increases with-
out exceeding a certain threshold / 2 0, 25 + = (Fig. 6, b). One observes that the
components of
T
1 1[ ( ), ( )]t a t b t = jump moving toward the Chebyshev center c at the
same time instants kt t= as shown in Fig. 6, c and d.
Fig. 5
For comparison, the behavior of nonadaptive control system with the same plant
and fixed parameters of the controller specified by c (instead of unknown ) is pre-
sented in Fig. 7. It turned out that this control system was also stable and its output error
remained bounded. However, one can observe that the performance of the adaptive con-
trol system is essentially better than in the nonadaptive case. Moreover, it is seen that
the suboptimality property of the proposed adaptive controller is ensured (Fig. 6, e).
1a
1,0
1,2
1,4
1,6
1b̂
1b
c
Chebyshev center
True parameter vector
– 2,8 – 2,4 – 2,0 1̂a
30 ISSN 2786-6491
Fig. 6
Fig. 7
– 1,2
0 500 1000 1500
e(t)
– 0,8
t
– 0,6
– 1,0
1500
e
0,00
0 500 1000 1500
t
a
t
Zeroing
t
at t = t0, t1, t2
Zeroing
t
at t = t3
0,01
0,02
0,10
0 500 1000 1500
t
b
0,03
t
Change in estimating
at t = t0, t1, t2
Change in estimating
at t = t3
0,14
– 2,38
0 500 1000 1500
c
a
1
(t)
Initialization of adaptation
process at t = t0, t1, t2
Initialization of adaptation
process at t = t3
First component of Chebyshev center at t = t0,…, t3
– 2,34
– 2,30
0,98
0 500 1000 1500
b
1
(t)
Initialization of adaptation
process at t = t0, t1, t2
Initialization of adaptation
process at t = t3
Second component of Chebyshev center at t = t0,…, t3
1,06
d
t
t
– 1,5
0 500 1000
e(t)
– 0,5
1,26
t
0,5
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 2 31
Conclusion
The adaptive suboptimal discrete-time closed-loop system containing the linear
uncertain scalar minimum phase plant subjected to nonstochastic bounded disturbances
with unknown bounds in the absence of any a priori information about the membership
set of its parameters is addressed in this paper. The asymptotical features of the adaptive
controller above proposed are established. They shed some light on the possibility of
achieving any suboptimality index utilizing the adaptive controller. In future, this cont-
roller can be improved via the use of additional ellipsoid estimation procedure in order
to accelerate the adaptation process. The MIMO case will also be studied.
В.М. Азарсков, К.Ю. Соловчук, О.Є. Волков, Л.С. Житецький
АДАПТИВНЕ СУБООПТИМАЛЬНЕ
КЕРУВАННЯ ДЕЯКИМИ ДИСКРЕТНИМИ
ОБ’ЄКТАМИ З НЕСТОХАСТИЧНИМИ
ОБМЕЖЕНИМИ ЗБУРЕННЯМИ
Азарсков Валерій Миколайович
Національний авіаційний університет, м. Київ,
azarskov@nau.edu.ua
Соловчук Клавдія Юріївна
Міжнародний науково-навчальний центр інформаційних технологій та систем
НАН України та МОН України, м. Київ,
solovchuk.ok@gmail.com
Волков Олександр Євгенович
Міжнародний науково-навчальний центр інформаційних технологій та систем
НАН України та МОН України, м. Київ,
director@irtc.org.ua
Житецький Леонід Сергійович
Міжнародний науково-навчальний центр інформаційних технологій та систем
НАН України та МОН України, м. Київ,
leonid_zhiteckii@i.ua
Розглядається адаптивне субоптимальне керування лінійними, дискретними,
стаціонарними, мінімально-фазовими, скалярними об’єктами за наявності не-
стохастичних обмежених невимірюваних збурень, верхня та нижня межі яких
можуть бути асиметричними та вважаються апріорі невідомими. Додаткове
припущення полягає в тому, що порядок різницевого рівняння, яке описує
об’єкт, апріорі відомий. Відмінною особливістю поставленої в статті задачі є те,
що межі ані невимірюваних збурень, ані допустимої області, до якої належать
невідомі параметри об’єкта, не вважаються апріорі відомими. Для вирішення
цієї задачі використовуються адаптивні процедури точкового і множинного
оцінювання. Стандартна рекурентна процедура з регульованою зоною нечутли-
вості застосовується для отримання точкових оцінок невідомих параметрів
об’єкта разом з точковою оцінкою постійної компоненти збурення. Розмір цієї
зони нечутливості залежить від попередньої точкової оцінки меж змінної у часі
компоненти збурення, а також від фіксованого показника субоптимальності,
вибраного конструктором системи. Оцінки, отримані за допомогою процедури
точкового оцінювання, безпосередньо використовуються для отримання закону
адаптивного керування. Головна ідея, висунута в статті, полягає в тому, що за-
мість невідомої апріорної множини належності цих параметрів використову-
ються їхні своєрідні гіпотетичні апостеріорні множини належності, розроблені
mailto:leonid_zhiteckii@i.ua
32 ISSN 2786-6491
за допомогою вимірюваних сигналів системи разом з поточною точковою оцін-
кою меж змінної у часі компоненти збурення. На відміну від звичайного підхо-
ду до оцінки множини належності цей набір оновлюється лише тоді, коли вияв-
ляється, що невідомий вектор параметрів насправді не належить цій множині.
Для цього застосовується певна властивість процедури точкового оцінювання.
Такий підхід дає змогу реконструювати цю множину і оновити попередню оці-
нку меж компоненти збурення, змінної в часі. Встановлено скінченну збіжність
процедур адаптації, а також граничну обмеженість сигналів системи. Для де-
монстрації ефективності адаптивного регулятора та підтримки теоретичного до-
слідження репрезентовані результати моделювання.
Ключові слова: адаптивне керування, точкове оцінювання, множинне оці-
нювання, дискретний час, нестохастичне збурення.
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Submitted 02.04.2024
https://jais.net.ua/index.php/files/issue/view/4
https://jais.net.ua/index.php/files/issue/view/4
https://doi.org/10.34229/2786-6505-2022-3-1
https://jais.net.ua/index.php/files/issue/view/4
https://jais.net.ua/index.php/files/issue/view/4
https://doi.org/10.34229/2786-6505-2022-2-2
https://doi.org/10.34229/2786-6505-2022-2-2
|
| id | nasplib_isofts_kiev_ua-123456789-211146 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0572-2691 |
| language | English |
| last_indexed | 2026-03-17T00:18:18Z |
| publishDate | 2024 |
| publisher | Інститут кібернетики ім. В.М. Глушкова НАН України |
| record_format | dspace |
| spelling | Azarskov, V.M. Solovchuk, K.Y. Volkov, O.Y. Zhiteckii, L.S. 2025-12-24T21:17:43Z 2024 Adaptive Suboptimal Control of Some Discrete-Time Plants with Nonstochastic Bounded Disturbances / V.M. Azarskov, K.Y. Solovchuk, O.Y. Volkov, L.S. Zhiteckii // Проблеми керування та інформатики. — 2024. — № 2. — С. 20–32. — Бібліогр.: 23 назви. — англ. 0572-2691 https://nasplib.isofts.kiev.ua/handle/123456789/211146 681.5 10.34229/1028-0979-2024-2-2 The paper addresses adaptive suboptimal control of linear, discrete-time, stationary, minimum-phase, scalar plants in the presence of nonstochastic bounded unmeasurable disturbances with asymmetric upper and lower bounds assumed to be unknown a priori. An additional assumption is that the order of the difference equation describing the plant is known beforehand. The unique feature of the stated problem is that neither the bounds on unmeasured disturbances nor the admissible region to which the unknown parameters of the plant belong are known a priori. Adaptive point and membership set estimation procedures are used to solve this problem Розглядається адаптивне субоптимальне керування лінійними, дискретними, стаціонарними, мінімально-фазовими, скалярними об’єктами за наявності нестохастичних обмежених невимірюваних збурень, верхня та нижня межі яких можуть бути асиметричними та вважаються апріорі невідомими. Додаткове припущення полягає в тому, що порядок різницевого рівняння, яке описує об’єкт, апріорі відомий. Відмінною особливістю поставленої в статті задачі є те, що межі ані невимірюваних збурень, ані допустимої області, до якої належать невідомі параметри об’єкта, не вважаються апріорі відомими. Для вирішення цієї задачі використовуються адаптивні процедури точкового і множинного оцінювання. en Інститут кібернетики ім. В.М. Глушкова НАН України Проблеми керування та інформатики Адаптивне керування та методи ідентифікації Adaptive Suboptimal Control of Some Discrete-Time Plants with Nonstochastic Bounded Disturbances Адаптивне субоптимальне керування деякими дискретними об’єктами з нестохастичними обмеженими збуреннями Article published earlier |
| spellingShingle | Adaptive Suboptimal Control of Some Discrete-Time Plants with Nonstochastic Bounded Disturbances Azarskov, V.M. Solovchuk, K.Y. Volkov, O.Y. Zhiteckii, L.S. Адаптивне керування та методи ідентифікації |
| title | Adaptive Suboptimal Control of Some Discrete-Time Plants with Nonstochastic Bounded Disturbances |
| title_alt | Адаптивне субоптимальне керування деякими дискретними об’єктами з нестохастичними обмеженими збуреннями |
| title_full | Adaptive Suboptimal Control of Some Discrete-Time Plants with Nonstochastic Bounded Disturbances |
| title_fullStr | Adaptive Suboptimal Control of Some Discrete-Time Plants with Nonstochastic Bounded Disturbances |
| title_full_unstemmed | Adaptive Suboptimal Control of Some Discrete-Time Plants with Nonstochastic Bounded Disturbances |
| title_short | Adaptive Suboptimal Control of Some Discrete-Time Plants with Nonstochastic Bounded Disturbances |
| title_sort | adaptive suboptimal control of some discrete-time plants with nonstochastic bounded disturbances |
| topic | Адаптивне керування та методи ідентифікації |
| topic_facet | Адаптивне керування та методи ідентифікації |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211146 |
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