Discrete-Time Steady-State Control of Interconnected Systems Based on Pseudoinversion Concept
The purpose of the paper is to generalize the results obtained by the authors in their last works which are related to the asymptotic properties of the pseudoinverse model-based method for designing an efficient steady-state control of interconnected systems with uncertainties and arbitrary bounded...
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irk-123456789-1249892017-10-14T03:03:14Z Discrete-Time Steady-State Control of Interconnected Systems Based on Pseudoinversion Concept Zhiteckii, L.S. Solovchuk, K.Yu. Интеллектуальное управление и системы The purpose of the paper is to generalize the results obtained by the authors in their last works which are related to the asymptotic properties of the pseudoinverse model-based method for designing an efficient steady-state control of interconnected systems with uncertainties and arbitrary bounded disturbances and also to present some new results. Розглянуто концепцію псевдообернення як деяку уніфіковану концепцію керування усталеними станами багатозв'язних систем за наявності невимірюваних обмежених збурень з повною і неповною інформацією про параметри лінійної номінальної моделі, по якій будується зворотний зв'язок. Припускається, що ранг матриці коефіцієнтів підсилення цієї моделі може бути довільним. Встановлено достатні умови граничної обмеженості всіх сигналів у замкнених системах керування, що реалізують запропоновану концепцію. Наведено результати моделювання Рассмотрена концепция псевдообращения как некоторая унифицированная концепция управления установившимися состояниями многосвязных систем при наличии неизмеряемых ограниченных возмущений с полной и неполной информацией о параметрах линейной номинальной модели, по которой строится обратная связь. Предполагается, что ранг матрицы коэффициентов усиления этой модели может быть произвольным Установлены достаточные условия предельной ограниченности всех сигналов в замкнутых системах управления, реализующих предлагаемую концепцию. Приведены результаты моделирования. 2017 Article Discrete-Time Steady-State Control of Interconnected Systems Based on Pseudoinversion Concept / L.S. Zhiteckii, K.Yu. Solovchuk // Кибернетика и вычислительная техника. — 2017. — Вип. 3 (189). — С. 29-43. — Бібліогр.: 23 назв. — англ. 0452-9910 DOI: doi.org/10.15407/kvt188.02.049 http://dspace.nbuv.gov.ua/handle/123456789/124989 681.5 en Кибернетика и вычислительная техника Міжнародний науково-навчальний центр інформаційних технологій і систем НАН України та МОН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Интеллектуальное управление и системы Интеллектуальное управление и системы |
| spellingShingle |
Интеллектуальное управление и системы Интеллектуальное управление и системы Zhiteckii, L.S. Solovchuk, K.Yu. Discrete-Time Steady-State Control of Interconnected Systems Based on Pseudoinversion Concept Кибернетика и вычислительная техника |
| description |
The purpose of the paper is to generalize the results obtained by the authors in their last works which are related to the asymptotic properties of the pseudoinverse model-based method for designing an efficient steady-state control of interconnected systems with uncertainties and arbitrary bounded disturbances and also to present some new results. |
| format |
Article |
| author |
Zhiteckii, L.S. Solovchuk, K.Yu. |
| author_facet |
Zhiteckii, L.S. Solovchuk, K.Yu. |
| author_sort |
Zhiteckii, L.S. |
| title |
Discrete-Time Steady-State Control of Interconnected Systems Based on Pseudoinversion Concept |
| title_short |
Discrete-Time Steady-State Control of Interconnected Systems Based on Pseudoinversion Concept |
| title_full |
Discrete-Time Steady-State Control of Interconnected Systems Based on Pseudoinversion Concept |
| title_fullStr |
Discrete-Time Steady-State Control of Interconnected Systems Based on Pseudoinversion Concept |
| title_full_unstemmed |
Discrete-Time Steady-State Control of Interconnected Systems Based on Pseudoinversion Concept |
| title_sort |
discrete-time steady-state control of interconnected systems based on pseudoinversion concept |
| publisher |
Міжнародний науково-навчальний центр інформаційних технологій і систем НАН України та МОН України |
| publishDate |
2017 |
| topic_facet |
Интеллектуальное управление и системы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/124989 |
| citation_txt |
Discrete-Time Steady-State Control of Interconnected Systems Based on Pseudoinversion Concept / L.S. Zhiteckii, K.Yu. Solovchuk // Кибернетика и вычислительная техника. — 2017. — Вип. 3 (189). — С. 29-43. — Бібліогр.: 23 назв. — англ. |
| series |
Кибернетика и вычислительная техника |
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AT zhiteckiils discretetimesteadystatecontrolofinterconnectedsystemsbasedonpseudoinversionconcept AT solovchukkyu discretetimesteadystatecontrolofinterconnectedsystemsbasedonpseudoinversionconcept |
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2025-07-09T02:21:38Z |
| last_indexed |
2025-07-09T02:21:38Z |
| _version_ |
1837134197932687360 |
| fulltext |
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 3 (189)
Интеллектуальное
управление и системы
DOI: https://doi.org/10.15407/kvt188.02.049
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, Postgraduate Student
e-mail: solovchuk_ok@mail.ru
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
DISCRETE-TIME STEADY-STATE CONTROL OF INTERCONNECTED SYS-
TEMS BASED ON PSEUDOINVERSION CONCEPT
Introduction. The problem of controlling interconnected systems subjected to arbitrary un-
measurable disturbances remains actual up to now. It is important problem from both theo-
retical and practical points of view. During the last decades, the internal model control prin-
ciple becomes popular among other methods dealing with an improvement of the control
system. A perspective modification of the internal model control principle is the so-called
model inverse approach. Unfortunately, the inverse model approach is quite unacceptable if
the systems to be controlled are square but singular or if they are nonsquare. It turned out
that the so-called pseudoinverse (generalized inverse) model approach can be exploited to
cope with the noninevitability of singular square and also nonsquare system.
The purpose of the paper is to generalize the results obtained by the authors in their
last works which are related to the asymptotic properties of the pseudoinverse model-based
method for designing an efficient steady-state control of interconnected systems with uncer-
tainties and arbitrary bounded disturbances and also to present some new results.
Results. In this paper, the main effort is focused on analyzing the asymptotic properties
of the closed-loop systems containing the pseudoinverse model-based controllers. In the
framework of the pseudoinversion concept, new theoretical results related to the asymptotic
behavior of these systems are obtained. Namely, in the case of nonsingular gain matrices
with known elements, the upper bounds on the ultimate norms of output and control input
vectors are found. Next, in the case of nonsquare gain matrices whose elements are also
known, the asymptotic behavior of the feedback control systems designed on the basis of
pseudoinverse approach are studied. Further, the sufficient conditions guaranteeing the
boundedness of the output and control input signals for the linear and certain class of
nonlinear interconnected systems in the presence of uncertainties are derived.
L.S. ZHITECKII, K.Yu. SOLOVCHUK, 2017
29
L.S. Zhiteckii, K.Yu. Solovchuk
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 3 (189) 30
Conclusion. It has been established that the pseudoinverse model-based concept can be
used as a unified concept to deal with the steady-state regulation of the linear interconnected
discrete-time systems and of some classes of nonlinear interconnected systems with possible
uncertainties in the presence of arbitrary unmeasured but bounded disturbances.
Keywords: discrete time, feedback, pseudoinversion, interconnected systems, optimality,
stability, uncertainty.
INTRODUCTION
The problem of controlling interconnected systems subjected to arbitrary un-
measurable disturbances stated several decades ago in the work [1] remains ac-
tual up to now [2, 3]. It is important problem from both theoretical and practical
points of view [4, 5]. During the last decades, the internal model control princi-
ple becomes popular among other methods dealing with an improvement of the
control system. Based on this method, interconnected control problem was first
approached in [6]. A perspective modification of the internal model control prin-
ciple is the so-called model inverse approach. The perfect output control per-
formance is an important interconnected control problem closely related to in-
verse systems. Since the pioneering work [7], the problem of inversion of linear
time-invariant interconnected systems has attracted an attention of several re-
searches. See [8–11]. Recently, a significant progress in this research area has
been achieved in [2, 3, 12]. Most of these works except [3, 12] dealt with con-
tinuous-time interconnected systems.
An inverse model approach to ensuring perfect steady-state regulation in
linear discrete-time interconnected systems was first advanced in [13] and inde-
pendently in [14]. Similar discrete-time counterpart of interconnected process
control systems containing the table inverse model was proposed in [15]. The
steady-state control of linear interconnected system discussed in [11] in the
framework of the problem of minimal inversion, has also been studied in the
paper [16] dealing with nonlinear discrete-time interconnected control systems.
Unfortunately, the inverse model approach is quite unacceptable if the systems
to be controlled are square but singular or if they are nonsquare. Several re-
searches whose works are cited in [17] observed that the inverse model-based
controller may be also not admissible for designing some process control sys-
tems which contain ill-conditioned plants since they may become (almost) non-
invertible in the presence of an uncertainty.
It turned out that the so-called pseudoinverse (generalized inverse) model
approach first proposed in the paper [10] can be exploited to cope with the non-
inevitability of nonsquare system. Recently, this approach was extended in
[18–20] for controlling a wide class of discrete-time interconnected systems.
The purpose of the paper is to generalize the results obtained by the au-
thors in their last works which are related to the asymptotic properties of the
pseudoinverse model-based method for designing an efficient steady-state con-
trol of interconnected systems with uncertainties and arbitrary bounded distur-
bances and also to present some new results.
Discrete-Time Steady-State Control of Interconnected Systems Based on
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 3 (189) 31
THE DESCRIPTION OF CONTROL SYSTEM AND PROBLEM STATEMENT
Basic assumptions. Suppose the plant to be regulated is a nonlinear intercon-
nected time-invariant system whose static characteristic is
y = )(uϕ (1)
where Tmyyy ],...,[ )()1(= denotes the m-dimensional output vector,
Truuu ],...,[ )()1(= denotes the r-dimensional input (control) vector, and
mr RR →⋅ϕ :)( represents some nonlinear vector-valued function given by
.)](),...,([)( )()1( Tm uuu ϕϕ=ϕ (2)
Consider a class of systems in which the number of inputs is not more than
the number of outputs:
.r m≤
The following assumption with respect to the nonlinearity )(uϕ will be re-
quired.
Assumption 1. The components )(),...,( )()1( uu mϕϕ of )(uϕ in (2) are all
the continuously differentiable functions of the variables .,..., )()1( ruu
In order to implement the discrete-time control, the signals
)(),...,( )()1( tyty m given in the continuous time t need to be sampled with a
sampling period 0T to yield the sequences )},({ 0
)( nTy i whereas the control
signals are of zero-order sampled-hold type, i.e.,
)()( 0
)()( nTutu ii = for ,)1( 00 TntnT +<≤ .,,1 ri …=
Assumption 2. As in [14] and [16], suppose that the sampling period 0T is
large enough so that the transient stage caused by stepwise changes of inputs
)(),...,( )()1( tutu r at each (n–1)th time instant 0)1( Tnt −= may practically be
completed during the time interval ).,)1[( 00 nTTn − In view of (1), this narra-
tive description of the discrete-time steady-state control gives that the steady
state of this interconnected system can be mathematically modeled by the first-
order nonlinear difference equation
)( 1−ϕ= nn uy (3)
similar to that in [16], if any disturbances are absent. In this equation, the nota-
tions )(: 0nTyyn = and )(: 0nTuun = are introduced (for the simplicity of expo-
sition).
In practical applications, the outputs )(),...,( )()1( tyty m are usually influ-
enced by certain classes of persistent external disturbances ),(),...,( )()1( tdtd m
respectively. Then, instead of (3), another equation
L.S. Zhiteckii, K.Yu. Solovchuk
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 3 (189) 32
11)( −− +ϕ= nnn duy (4)
with the disturbance vector Tm
nnn ddd ],...,[: )()1(= as a steady-state model of
system will be further considered. Now, the following assumption about }{ nd is
introduced.
Assumption 3. The components of nd are upper bounded in modulus by an
iε for all ,,2,1 …=n i.e.,
∞<ε≤ i
i
nd || )( ).,,1( mi …= (5)
Let Tmyyy ],,[: )*()1*(* …= )const( )*( ≡iy be some vector defining the de-
sired output vector (a given set-point). The following assumption with respect to
this vector is made.
Assumption 4. ∗y is not the m-dimensional zero-vector ,]0,...,0[:0 T
m =
i.e., 0|||| ≠∗y implying that
.0|||| )()1( ≠++ ∗∗ myy … (6)
Regulation strategy using pseudoinverse model-based control ap-
proach. Let
=
)(
0
)1(
0
)1(
0
)11(
0
0
mrm
r
bb
bb
B
…
M
…
be a fixed rm× matrix chosen further by the designer to deal with some linear
model of (1). Define the so-called pseudoinverse (generalized inverse) mr ×
matrix )( )(
00
ijB β=+ specified as
,)(lim 0
12
0000
T
r
T BIBBB −
→δ
+ δ+= (7)
where NI denotes the identity NN × matrix. (Note that the limit (7) exist for
any 0
m rB ×∈R [21].)
According to [19], [20] the control law utilizing the pseudoinverse model-
based control strategy to regulate ny around ∗y is given by
,01 nnn eBuu +
− += (8)
where ne represents the output error vector at nth time instant 0nTt = specified
as
,nn yye −= ∗ (9)
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The equations (8), (9) describe the some linear interconnected controller of
the integral action. Namely, to implement the control law (8), one needs the
discrete integrator whose output is
,
1
∑
=
∆=
n
k
kn uu (10)
where
.0 nn eBu +=∆ (11)
Due to (11) together with (10), this controller plays the role of an I-type in-
terconnected discrete-time controller with a matrix gain +
0B (Fig. 1).
Regulation problems. To formulate the goals of the regulation, we before
need the following definition.
Definition 1 [22]. The closed-loop control system containing the plant de-
scribed by (4) and the feedback (8), (9) is said to be BIBS (bounded-input
bounded-state) stable if there exist some nonnegative numbers Cu, Cy, Cd such
that
,||||sup||||sup||||suplim
00
n
n
dn
n
un
n
dCuCy
≥≥∞→
+≤ (12)
||||sup||||sup||||suplim
00
n
n
dn
n
yn
n
dCyCu
≥≥∞→
+≤ (13)
are satisfied.
Now, introduce the performance index
||||suplim: n
n
eJ
∞→
= (14)
evaluating the asymptotic behavior of the control system (4), (8), (9). Then, one
of the following control objectives may be stated [22].
Fig. 1. Configuration of the regulation system (4), (9), (10), (11)
L.S. Zhiteckii, K.Yu. Solovchuk
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 3 (189) 34
• The optimization: it is required to minimize J defined by (14) in the sense
that
}{
inf||||suplim
nun
n
e =
∞→
(15)
must be achieved.
• Quasi-optimization: it is necessary to minimize an upper bound of J given
in the inequality
.||||suplim Jen
n
≤
∞→
(16)
• Stability (robust stability): the closed-loop system (4), (8), (9) must be
stable (in the sense of Definition 1) by suitable choice of 0B .
LINEAR CASE
Regulation without parameter uncertainty. In the linear case, )(uϕ in (1) is
defined as ,)( Buu =ϕ where )( )(ijbB = represents some numerical rm ×
matrix with the elements )(ijb whose rank satisfies .rank1 rB ≤≤ In this case,
the equation (4) becomes
.11 −− += nnn dBuy (17)
Let mr = and .rank rB = Clearly, it implies that B is non-singular. Then
the inverse matrix 1−B exists and .1−+ = BB Assume that there is no parameter
uncertainty, i.e., B is known a priori. We can derive immediately the inverse-
model based control law
,1
1 nnn eBuu −
− += (18)
followed from (8) after setting .0 BB =
It turns out the control law above guarantees the optimality of the closed-
loop system (17), (18), (9) (in the sense of (15)). This fact is established in the
theorem below.
Theorem 1 [22]. Let the plant to be regulated be described by (17). Sup-
pose B is the known non-singular square matrix ).0(det ≠B Then, the control-
ler (18), (9) when applied to (17) achieves the regulation objective (15). Fur-
thermore, subject to Assumptions 4, it yields
∞<−≤
∞<+≤
−
∞<≤
∞<≤
−−
∞→
||||sup
,||||sup||||||||||||||||suplim
1
0
0
1*1
nn
n
n
n
n
n
ddJ
dByBu
(19)
for any initial .|||| 0 ∞<u
Discrete-Time Steady-State Control of Interconnected Systems Based on
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Corollary. Under the conditions of Theorem 1, in the terms of the Euclid-
ean norm ,|||| 2⋅ the asymptotic properties of the controller (18), (9) are given
by
ε≤
ε+≤
∞→
−−
∞→
2||||suplim
,||||||||||||||||suplim
2
2
1
2
*
2
1
2
n
n
n
n
e
ByBu
(20)
with 2/12)*(2)1*(
2
* ]|||[||||| Myyy ++= … and .][ 2/122
1 mε++ε=ε …
Proof. Immediate from (19) together with (5) and from the definition of *y
taking into account the definitions of the Euclidean vector and matrix norms
[23].□
Let B be a known nonsquare matrix ).( mr < In this case, instead of (18),
nnn eBuu +
− += 1 (21)
is chosen as the control law. The equation (21) together with (9) describes the
pseudoinverse model-based controller.
The following result can be shown to be valid.
Theorem 2 [20]. The controller (21), (9) applied to (17) leads to a stable
closed-loop system (in the sense of Definition 1). Moreover, subject to Assump-
tion 4, it gives that quasi-optimality property of the form (16) is ensured with the
minimal J such that
.2)||(||||||||||suplim
,||||||||||||||||suplim
2
*
22
22022
∞<ε+ε+−≤
∞<ε+−−≤−
+
∞→
++
∞→
yBBIe
BuuBBIuu
mn
n
e
r
e
n
n (22)
Remark 1. Note that if r=m and 0det ≠B yielding ,1−+ = BB then the
inequalities (22) finally leads to (20), respectively.
Regulation in the presence of parameter uncertainty. Consider the
steady-state model of the plant given in the form (17) with an arbitrary nonzero
matrix ).( )(ijbB = Assume that s)(ijb are unknown but the bounds, )(
max
)(
min , ijij bb
of the intervals
),,1;,,1()(
max
)()(
min rjmibbb ijijij …… ==≤≤ (23)
to which they belong are known. Additionally, let
.0 )(
max
)(
min ∞<< ijij bb (24)
Denote by Ξ the set of possible ˆsB whose elements, ( )ˆ ijb satisfy
( ) ( ) ( )
min max
ˆ [ , ].ij ij ijb b b∈ This means that
L.S. Zhiteckii, K.Yu. Solovchuk
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 3 (189) 36
( ) ( ) ( ) ( )
min max
ˆ{( ) : 1, , , 1, , }.ij ij ij ijb b b b i m j rΞ = ≤ ≤ = =… … (25)
Further, choose a matrix B0 from the set Ξ provided 0det 0 =B if this set
contains at least one singular matrix ˆ.B Thus,
),,1;,,1()(
max
)(
0
)(
min rjmibbb ijijij …… ==≤≤
has to be met.
The sufficient condition guaranteeing the boundedness of }{ ny and }{ nu is
established in the following theorem.
Theorem 3. Consider the feedback system (17), (8), (9). Let the require-
ments (23), (24) hold and the requirements on the choice of B0 above mentioned
be met. Assume that the equilibrium state of the feedback system (17), (8), (9)
defined by the pair ),( ee yu which is the solution of the equation
*
00 yBBuB e ++ =
together with ee Buy = exists. Introduce the matrix .0 BB −=∆ If the condi-
tion
1<q (26)
with
||||max 0)(: 0
∆= +
Ξ∈∆−∆
Bq
B
(27)
is satisfied, then the closed-loop control system containing the plant (17) and the
pseudoinverse model-based controller
)( *
01 nnn yyBuu −+= +
− (28)
will be the robust BIBS stable. Moreover, subject to Assumption 3, this control-
ler makes it possible to achieve
.)1(2]2||[||||||||||suplim
,||||)1(||||||||)1(||||suplim
1
202002
20
1
20200
1
2
∞<−ε+ε+−≤
∞<−ε+−−−≤−
−+
∞→
+−+−
∞→
qeBBIe
BquuBBIquu
mn
n
e
r
e
n
n (29)
Proof. Due to space limitation, details are omitted.□
By virtue of (12), (13), the condition (26) together with the expression (27)
guarantee the boundedness of }{ ny and }{ nu as ∞→n (according to (29)).
Note that this condition can simply be verified by setting
10)(: ||||max
0
∆= +
Ξ∈∆−∆ Bq B and by using the linear programming technique
( 1|||| P denotes here the 1-norm of arbitrary matrix P; the definition of 1|||| P
can be found in [23]).
Discrete-Time Steady-State Control of Interconnected Systems Based on
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A numerical example and simulation. To illustrate the robust stability
properties derived from Theorem 3, a numerical example was considered setting
2== mr and ( ) (11) (12) (21){ : 0.4 1.4, 1.2 0.5, 0.8 2.8,ijb b b bΞ = ≤ ≤ − ≤ ≤ − ≤ ≤
(22)2.7 0.7}b− ≤ ≤ − Such set was chosen to ensure the singularity of some ˆsB
belonging to .Ξ In this example,
−
−
=
70.18.1
85.09.0
0B
was put. Such a choice of 0B gives Ξ∈0B and .0det 0 =B Using the formula
(7),
−−
=+
613/136613/68
613/144613/72
0B
was found. By exploiting the linear programming technique, it was established
that .1572,0||||max 10)(: 0
<≈∆= +
Ξ∈∆−∆ Bq B Thus, requirement (26) together
with (27) will be satisfied.
Next, taking ,878.0)11( =b ,864.0)12( −=b ,082.1)21( =b 096.1)22( −=b
under which )( )(ijbB = will satisfy ,Ξ∈B a simulation experiment with the
closed-loop control system described by (17), (8), (9) was conducted. In this
experiment, )2()1( , nn dd were simulated as the pseudo-random variables within
].07.0,07.0[− The components of *y were chosen as follows:
*(1) *(2)0.4 if 0 50, 0.6 if 0 50,
and
0.2 if 50 100 0.8 if 50 100.
n n
y y
n n
≤ ≤ ≤ ≤
= = < ≤ < ≤
Results of the simulation experiment are depicted in Figs. 2 and 3.
Fig. 2. The norm of control input vector
L.S. Zhiteckii, K.Yu. Solovchuk
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 3 (189) 38
Fig. 3. The norms of output vector (solid line) and of set-point vector (dashed line)
We observe that system behavior is successful while B and B0 are different.
REGULATION OF UNKNOWN NONLINEAR SYSTEM
Case 1. Now, consider the nonlinear interconnected system described by (4).
Recalling Assumption 1 and denoting ,/)(:)( )()()( jiij uuub ∂ϕ∂= introduce the
matrix
=
)()(
)()(
)(
)()1(
)1()11(
ubub
ubub
uB
mrm
r
…
M
…
(30)
which represents the rm × Jacobian matrix whose elements )()( ub ij play a role
of some “dynamical” gains from the jth input, )( ju to the ith output, )(iy for
each fixed .ru R∈ Next, the following two additional assumptions regarding
the nonlinearity )(uϕ will be required.
Assumption 5. s)()( ub ij in (30) do not change its sign and remain uni-
formly bounded for all u from rR according to (24) and to
).,,1;,,1(,)( )(
max
)()(
min rjmibubb ijijij …… ==≤≤ (31)
Assumption 6. In case 1 to be studied, Ξ represents the set of matrices
having the full rank: .€rank rB =
Under these assumptions we first choose a Ξ∈0B and design again the
pseudoinverse model-based controller of the form (28). The asymptotic proper-
ties of this controller are formulated in the theorem below.
Theorem 4 [19]. Consider the feedback control system described by (4),
(28). Let the equilibrium state defined by
)(,)( *
00
eee uyyBuB ϕ==ϕ ++
Discrete-Time Steady-State Control of Interconnected Systems Based on
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 3 (189) 39
exist. Suppose that Assumptions 1 and 3 to 6 are valid. Then this system will be
robust BIBS stable for any nonlinearity )(uϕ satisfying (31) together with (24)
if the requirement (26) in which
,maxmax
1 1
)()(
0
],[1 )()()(
∑ ∑
= =δδ∈δ≤≤
δβ=
r
i
m
j
jikj
rk ijijij
q (32)
where )(
0
)(
max
)()(
0
)(
min
)( , ijijijijijij bbbb −=δ−=δ is met. Furthermore,
},,max{||||)1(||||suplim 110
1
mn
n
Bqu εε−≤ +−
∞
∞→
…
will take place, where ∞|||| x denotes the ∞ -norm of a vector x.
As in the linear case before studied, the condition (26) but with q given by
(32) can be verified via the linear programming tool.
Case 2. In this case, instead of Assumptions 1, 5 and 6, another assumption
with respect to )(uϕ is introduced.
Assumption 7. The nonlinearity )(uϕ can be represented as the sum
),()( ugBuu +=ϕ (33)
in which )( )(ijbB = is a numerical rm × matrix and )(ug is a nonlinear vec-
tor-valued function satisfying
∞<≤
∈
Cug
ru
||)(||sup
R
(34)
with some C.
Due to the expression (33) given in Assumption 7, the system equation be-
comes
.)( 111 −−− ++= nnnn dugBuy (35)
As in the linear case with unknown B, it is assumed that ,Ξ∈B where Ξ
is given by (25). Similarity to this case, we choose Ξ∈0B so that 0det 0 =B if
r=m and there is at least a singular matrix .€ Ξ∈B Next, the pseudoinverse
model-based controller of the form (28) is designed to regulate the plant (35).
The following theorem establishes stability results of the closed-loop system
(35), (28).
Theorem 5. Under the conditions of Theorem 3 added by Assumption 7, the
closed-loop system containing the controller (28) and the plant (35) will be ro-
bust BIBS stable.
Proof. Proceeds along the lines of the proof of Theorem 3 after replacing
∞<∞<≤ ||||sup0 nn d by .||||sup0 ∞<+∞<≤ Cdnn □
Remark 2. In contrast with [19], it is not required that )(,),( )()1( uu mϕϕ …
in (2) to be smooth functions of u.
L.S. Zhiteckii, K.Yu. Solovchuk
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 3 (189) 40
Remark 3. Note that )(ug may not be the Lipchitz function, i.e.,
|| ( ) ( ) || || || , (0 )rg u g u L u u u u L′ ′′ ′ ′′ ′ ′′− ≤ − ∀ ∈ < < ∞R
is not necessary. However, due to (34) it has to be bounded as .|||| ∞→u
Comment. Contrary to the case 1, the set Ξ may contain singular ˆsB and it
is essential.
CONCLUSION
In this paper, the main effort has been focused on analyzing the asymptotic
properties of the closed-loop systems containing the pseudoinverse model-based
controllers. We have established that the pseudoinverse model-based concept
can be used as a unified concept to deal with the steady-state regulation of the
linear interconnected discrete-time systems and of some classes of nonlinear
interconnected systems with possible uncertainties in the presence of arbitrary
unmeasured but bounded disturbances. In the framework of this concept, new
theoretical results related to the asymptotic behavior of these systems have been
presented.
REFERENCES
1. Davison E. The output control of linear time-invariant multivariable systems with un-
measurable arbitrary disturbances. IEEE Trans. Autom. Contr., 1972, vol. AC-17, no. 5,
pp. 621–631.
2. 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.
3. Lyubchyk L. M. Disturbance rejection in linear discrete Multivariable systems: inverse
model approach. Prep. 18th IFAC World Congress, Milano, Italy, 2011, pp. 7921–
7926.
4. Skogestad S., Postlethwaite I. Multivariable Feedback Control . UK, Chichester: Wiley,
1996.
5. Freudenberg J. and Middleton R. Properties of single input, two output feedback sys-
tems. Int. J. Control, 1999, vol. 72, no. 16, pp. 1446–1465.
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. Sain M. K., Massey J. L. Invertibility of linear time-invariant dynamical systems. IEEE
Trans. Autom. Contr.,1969, vol. AC-14, no. 2, pp. 141–149, Apr. 1969.
9. Silverman L. M. Inversion of multivariable linear systems. IEEE Trans. Autom. Contr.,
1969, vol. AC-14, no. 3, pp. 270–276..
10. Lovass-Nagy V., Miller J. R., Powers L. D. On the application of matrix generalized
inversion to the construction of inverse systems. Int. J. Control, 1976, vol. 24, no. 5,
pp. 733–739.
11. Seraji H. Minimal inversion, command tracking and disturbance decoupling in multi-
variable systems. Int. J. Control, 1089, vol. 49, no. 6, pp. 2093–2191.
12. Marro G., Prattichizzo D., Zattoni E. Convolution profiles for right-inversion of multi-
variable non-minimum phase discrete-time systems. Automatica, 2002, vol. 38, no. 10,
pp. 1695–1703.
13. Pukhov G. E., Zhuk K. D. Synthesis of Interconnected Control Systems via Inverse Op-
Discrete-Time Steady-State Control of Interconnected Systems Based on
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 3 (189) 41
erator Method. Kiev: Nauk. dumka, 1966 (in Russian).
14. Lee T., Adams G., Gaines W. Computer Process Control: Modeling and Optimizatio n.
New York: Wiley, 1968.
15. Skurikhin V. I., Procenko N. M., Zhiteckii L. S. Multiple-connected systems of techno-
logical processes control with table of objects. Proc. IFAC Third Multivariable Tech.
Systems Symp., Manchester, U.K., 1974, pp. S 35-1 – S 35-4.
16. Katkovnik V. Ya., Pervozvansky A. A. Methods for the search of extremum and the
synthesis problems of multivariable control systems. Adaptivnye Avtomaticheskie Sis-
temy, Moscow: Sov. Radio, pp. 17–42, 1973 (in Russian).
17. Skogestad S., Morari M., Doyle J. Robust control of ill-conditioned plants: high purity
distillation. IEEE Trans. Autom. Contr., 1988, vol. 33, no. 12, pp. 1092–1105.
18. 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.
Upravlyayushchye sistemy i mashiny , 2013, no. 3, pp. 14−20, 29 (in Russian).
19. 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.
20. Skurikhin V. I., Gritsenko V. I., Zhiteckii L. S., Solovchuk K. Yu. Generalized inverse
operator method in the problem of optimal controlling linear interconnected static
plants. Dopovidi NAN Ukrainy, no. 8, pp. 57–66, 2014 (in Russian).
21. Albert A. Regression and the Moore-Penrose Pseudoinverse . New York: Academic
Press, 1972.
22. Zhiteckii L. S., Skurikhin V. I. Adaptive Control Systems with Parametric and Non-
parametric Uncertainties. Kiev: Nauk. dumka, 2010 (in Russian).
23. Lancaster P., Tismenetsky M. The Theory of Matrices: 2nd ed. With Applications .
N.Y.: Academic Press, 1985.
Received 17.02.2017
ЛИТЕРАТУРА
1. Davison E. The output control of linear time-invariant multivariable systems with un-
measurable arbitrary disturbances. IEEE Trans. Autom. Contr., 1972, vol. AC-17, no. 5,
pp. 621–631.
2. 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.
3. Lyubchyk L. M. Disturbance rejection in linear discrete Multivariable systems: inverse
model approach. Prep. 18th IFAC World Congress, Milano, Italy, 2011, pp. 7921–
7926.
4. Skogestad S., Postlethwaite I. Multivariable Feedback Control . UK, Chichester: Wiley,
1996.
5. Freudenberg J. and Middleton R. Properties of single input, two output feedback sys-
tems. Int. J. Control, 1999, vol. 72, no. 16, pp. 1446–1465.
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. Sain M. K., Massey J. L. Invertibility of linear time-invariant dynamical systems. IEEE
Trans. Autom. Contr.,1969, vol. AC-14, no. 2, pp. 141–149, Apr. 1969.
9. Silverman L. M. Inversion of multivariable linear systems. IEEE Trans. Autom. Contr.,
1969, vol. AC-14, no. 3, pp. 270–276.
10. Lovass-Nagy V., Miller J. R., Powers L. D. On the application of matrix generalized
inversion to the construction of inverse systems. Int. J. Control, 1976, vol. 24, no. 5,
pp. 733–739.
11. Seraji H. Minimal inversion, command tracking and disturbance decoupling in multi-
variable systems. Int. J. Control, 1089, vol. 49, no. 6, pp. 2093–2191.
12. Marro G., Prattichizzo D., Zattoni E. Convolution profiles for right-inversion of multi-
L.S. Zhiteckii, K.Yu. Solovchuk
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 3 (189) 42
variable non-minimum phase discrete-time systems. Automatica, 2002, vol. 38, no. 10,
pp. 1695–1703.
13. Pukhov G. E., Zhuk K. D. Synthesis of Interconnected Control Systems via Inverse Op-
erator Method. Kiev: Nauk. dumka, 1966 (in Russian).
14. Lee T., Adams G., Gaines W. Computer Process Control: Modeling and Optimizatio n.
New York: Wiley, 1968.
15. Skurikhin V. I., Procenko N. M., Zhiteckii L. S. Multiple-connected systems of techno-
logical processes control with table of objects. Proc. IFAC Third Multivariable Tech.
Systems Symp., Manchester, U.K., 1974, pp. S 35-1 – S 35-4.
16. Katkovnik V. Ya., Pervozvansky A. A. Methods for the search of extremum and the
synthesis problems of multivariable control systems. Adaptivnye Avtomaticheskie Sis-
temy, Moscow: Sov. Radio, pp. 17–42, 1973 (in Russian).
17. Skogestad S., Morari M., Doyle J. Robust control of ill-conditioned plants: high purity
distillation. IEEE Trans. Autom. Contr., 1988, vol. 33, no. 12, pp. 1092–1105.
18. 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.
Upravlyayushchye sistemy i mashiny , 2013, no. 3, pp. 14−20, 29 (in Russian).
19. 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.
20. Skurikhin V. I., Gritsenko V. I., Zhiteckii L. S., Solovchuk K. Yu. Generalized inverse
operator method in the problem of optimal controlling linear interconnected static
plants. Dopovidi NAN Ukrainy, no. 8, pp. 57–66, 2014 (in Russian).
21. Albert A. Regression and the Moore-Penrose Pseudoinverse . New York: Academic
Press, 1972.
22. Zhiteckii L. S., Skurikhin V. I. Adaptive Control Systems with Parametric and Non-
parametric Uncertainties. Kiev: Nauk. dumka, 2010 (in Russian).
23. Lancaster P., Tismenetsky M. The Theory of Matrices: 2nd ed. With Applications . N.Y.:
Academic Press, 1985.
Получено 17.02.2017
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@mail.ru
Міжнародний научно-навчальний центр інформаційних технологій
та систем НАН України і МОН України, пр. Академіка Глушкова, 40,
м. Київ, 03187, Україна
ДИСКРЕТНЕ КЕРУВАННЯ УСТАЛЕНИМИ СТАНАМИ
БАГАТОЗВ’ЯЗНИХ СИСТЕМ НА ОСНОВІ
КОНЦЕПЦІЇ ПСЕВДООБЕРНЕННЯ
Розглянуто концепцію псевдообернення як деяку уніфіковану концепцію керування
усталеними станами багатозв'язних систем за наявності невимірюваних обмежених
збурень з повною і неповною інформацією про параметри лінійної номінальної моделі,
по якій будується зворотний зв'язок. Припускається, що ранг матриці коефіцієнтів
підсилення цієї моделі може бути довільним. Встановлено достатні умови граничної
обмеженості всіх сигналів у замкнених системах керування, що реалізують запропоно-
вану концепцію. Наведено результати моделювання.
Ключові слова: дискретний час, зворотний зв'язок, псевдообернення, багатозв'язні
системи, оптимальність, стійкість, невизначеність.
Discrete-Time Steady-State Control of Interconnected Systems Based on
ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 3 (189) 43
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@mail.ru
Международный научно-учебный центр информационных технологий
и систем НАН Украины и МОН Украины,
пр. Академика Глушкова, 40, г. Киев, 03187, Украина
ДИСКРЕТНОЕ УПРАВЛЕНИЕ УСТАНОВИВШИМИСЯ
СОСТОЯНИЯМИ МНОГОСВЯЗНЫХ СИСТЕМ НА ОСНОВЕ
КОНЦЕПЦИИ ПСЕВДООБРАЩЕНИЯ
Рассмотрена концепция псевдообращения как некоторая унифицированная концепция
управления установившимися состояниями многосвязных систем при наличии неизме-
ряемых ограниченных возмущений с полной и неполной информацией о параметрах
линейной номинальной модели, по которой строится обратная связь. Предполагается,
что ранг матрицы коэффициентов усиления этой модели может быть произвольным.
Установлены достаточные условия предельной ограниченности всех сигналов в за-
мкнутых системах управления, реализующих предлагаемую концепцию. Приведены
результаты моделирования.
Ключевые слова: дискретное время, обратная связь, псевдообращение, многосвяз-
ные системы, оптимальность, устойчивость, неопределенность.
|