Stages and Main Tasks of the Century-Long Control Theory and System Identification Development. Part IV. Methods and Problems of Designing Robust Control Systems
The article provides a review of the most important methods and problems in the design of robust discrete control systems. The main focus is on the problems of suppressing limited external disturbances, the information about which is presented in the form of constraints on their maximum value. The u...
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| description | The article provides a review of the most important methods and problems in the design of robust discrete control systems. The main focus is on the problems of suppressing limited external disturbances, the information about which is presented in the form of constraints on their maximum value. The use of invariant ellipsoids is considered as the first mathematical apparatus for describing the characteristics of the influence of external disturbances on the trajectory of motion of dynamic systems.
У статті зроблено огляд найважливіших методів і задач проєктування робастних дискретних систем керування. При цьому основну увагу привернуто до задач приглушення обмежених зовнішніх збурень, інформація про які надана у формі обмеження на їх максимальну величину. Для опису характеристики впливу зовнішніх збурень на траєкторію руху динамічних систем як перший математичний апарат розглянуто застосування інваріантних еліпсоїдів.
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© V. ROMANENKO, V. GUBAREV, 2024
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 2 5
АДАПТИВНЕ КЕРУВАННЯ ТА МЕТОДИ ІДЕНТИФІКАЦІЇ
UDC 62-50
V. Romanenko, V. Gubarev
STAGES AND MAIN TASKS
OF THE CENTURY-LONG CONTROL THEORY
AND SYSTEM IDENTIFICATION DEVELOPMENT.
Part IV. METHODS AND PROBLEMS
OF DESIGNING ROBUST CONTROL SYSTEMS
Viktor Romanenko
National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute»,
Institute for Applied Systems Analysis, Kyiv,
ipsa@kpi.ua; romanenko.viktorroman@gmail.com
Vyacheslav Gubarev
Space Research Institute of NAS of Ukraine and SSA of Ukraine, Kyiv,
orcid: 0000-0001-6284-1866
v.f.gubarev@gmail.com
The article provides a review of the most important methods and problems in the
design of robust discrete control systems. In this case, the main attention was fo-
cused on the problems of suppressing limited external disturbances, information
about which is presented only in the form of a limitation on their maximum val-
ue. The use of invariant ellipsoids x is considered as the first mathematical ap-
paratus for describing the characteristics of the influence of external disturb-
ances on the trajectory of motion of dynamic systems. Theorems on the repre-
sentation of invariant ellipsoids x in the form of linear matrix inequalities
(LMI) are formulated, which are further used to synthesize discrete state control-
lers that suppress external disturbances. The solution to a more general problem
of robust suppression of limited disturbances based on the use of LMI in the
presence of system uncertainties in the parameters of the mathematical model of
the control object is considered. The use of H-control theory is considered as
a second mathematical apparatus for suppressing external 2l -limited external
disturbances. In this case, the optimality criterion consists in minimizing the
maximum ratio of the 2l -norm of the vector of output stabilized coordinates to
the 2l -norm of the vector of input disturbances. The problem is solved by reduc-
ing it to the problem of robust control of a discrete dynamic system in space
H based on the Two-Ricatti approach. The standard H-optimization prob-
lem is also considered.
Keywords: state controller, invariant ellipsoid, linear matrix inequalities, robust
control, H-theory, external disturbances.
mailto:ipsa@kpi.ua
mailto:romanenko.viktorroman@gmail.com
mailto:v.f.gubarev@gmail.com
6 ISSN 2786-6491
Introduction
The methods for designing control systems based on state-space object models
and input-output models discussed in previous articles [1, 2] were used to meet the
requirement that these models accurately describe the dynamics of control objects.
To do this, in the third part [3] of this series of articles, methods for identifying the
parameters of mathematical models were considered. However, in modern and clas-
sical control theories, to describe the dynamics of controlled objects operating in a
stochastic environment, it is assumed that white noise, the mathematical expectation
of which is zero, is used as disturbances acting on the plant. In control systems, it is
very rare that real disturbances over limited periods of time can be represented as
white noise. For example, a gust of wind during take off or landing of an airplane in
no way fits into the probabilistic characteristics of white noise. Thus, the uncertain-
ty of the characteristics of external changing disturbances acting on the control ob-
ject is the main problem in the design of controllers that guarantee stability of the
control system in a stochastic environment. Also, in almost every engineering prob-
lem of designing a control system, there is uncertainty, which lies in the fact that
the mathematical model of the object, obtained as a result of identifying its parame-
ters, will be adequate to the true dynamics of the object only for a limited period of
time, after which it will change in a certain area according to an unknown law. As a
result, the problem of controller synthesis under conditions of uncertainty of the
dynamic characteristics of the object and external disturbances required the deve-
lopment of new approaches to the design of control systems in comparison with the
methods that were described in review articles [1–3].
At the end of the 70s of the last century, in the works [4–6], the problem of optimal
suppression of arbitrary limited external disturbances acting on the control object was
formulated. This problem is called 1l -optimization when describing control systems in
discrete time. Later, in [7–9], the linear matrix inequalities (LMI) technique began to be
used to suppress external disturbances with arbitrary characteristics.
The problem of optimal suppression of external disturbances is one of the main
ones in control theory and besides remain one of the difficult problems of linear control
theory [10]. To solve this problem, methods have currently been developed and applied
that belong to two theoretical directions:
1) methods of invariant ellipsoids;
2) methods of H-theory.
The development of these directions over several decades was mainly carried out
for the original mathematical models of control objects in continuous time. In this arti-
cle we will analyze this stage of development of the theory of control systems based on
discrete-time models aimed at the practical implementation of synthesized systems in
microprocessor execution.
In real problems of functioning control systems, there is inevitably system un-
certainty and the control being formed must be operational under the most extreme
conditions. This type of control is called robust. This control must ensure robust
stability of the designed control system when changing external disturbances in giv-
en intervals and for a given family of state matrices of the mathematical model of
the object
0 ,F F= +
where 0F is the nominal matrix of the family, and is the uncertainty. Coefficient
0 determines the range of uncertainty.
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 2 7
1. The problem of suppressing limited external disturbances
using the method of invariant ellipsoids
In [11, 12], invariant ellipsoids are considered as a characteristic of the influence of
external disturbances on the trajectory of a dynamic system, which is represented by a
mathematical model in state space in discrete time
( 1) ( ) ( ),
( ) ( ),
x k Fx k k
y k Cx k
+ = +
=
(1)
where ( ) Rnx k are the phase variables of the system; ( ) Rly k are output coordi-
nates of the system; ( ) Rmk are external disturbances, which are limited as follows:
( ) 1 0,1, 2, ,k , k = (2)
where is a vector norm in Euclidean space. In this way, l -limited external distur-
bances are considered. No other restrictions are imposed on the disturbances, that is, it
is assumed that the they are arbitrary and only bounded. It is also assumed that sys-
tem (1) is stable and the pair ( , )F will be controllable. The matrix C has maximum
row rank.
The family of invariant ellipsoids is defined as follows: an ellipsoid centered at
the origin
T 1{ ( ) R : ( ) ( ) 1}, 0n
x x k x k P x k P− = (3)
is called state ( )x k invariant for a discrete dynamic system (1), (2), if the condition
(0) xx follows that the condition ( ) xx k is satisfied for all discrete moments of
time 0,1, 2,k = . The matrix P is called the ellipsoid matrix.
The following important theorem was formulated and proven in [12].
Theorem 1. Ellipsoid x (3) will be invariant for dynamical system (1) with
l -bounded external disturbance if and only if the matrix P satisfies linear matrix
inequalities (LMI)
T T1 1
0, (0)
(1 )
FPF P P P− +
−
(4)
at some coefficient 0 1.
The objective function is used as an optimality criterion
T( ) tr[ ],f P CPC= (5)
which corresponds to the sum of the squares of the semiaxes of the invariant ellipsoid
T T 1{ ( ) R : ( )( ) ( ) 1}.m
y y k y k CPC y k− =
Function (5) specifies the size of the invariant ellipsoid .y
To synthesize a control system, a controlled discrete model of an object in state
space is considered
1
2
( 1) ( ) ( ) ( ),
( ) ( ) ( ),
x k Fx k G u k k
y k Cx k G u k
+ = + +
= +
(6)
0(0) ,x R .pu At the same time 2 0,TG C = the pair 1( , )F G is controllable.
8 ISSN 2786-6491
To suppress disturbances, a state controller is implemented
( ) ( ),pu k K x k= (7)
which ensures minimization of criterion (5), that is, the minimum size of the invariant
ellipsoid with respect to the output. Based on (6), (7), the model of the closed-loop con-
trol system will have the form
1
2
( 1) ( ) ( ) ( ),
( ) ( ) ( ).
p
p
x k F G K x k k
y k C G K x k
+ = + +
= +
(8)
In [12] a theorem for the synthesis of state controller (7) was formulated and proven.
Theorem 2. If for a discrete system (6) external disturbances are l -limited and
the pair 1( , )F G is controllable, then the problem of synthesizing an optimal state con-
troller (7), which suppresses external disturbances, will be equivalent to the problem of
minimizing a linear function
T
2 2tr[ ] minTCPC G ZG+ → (9)
under restrictions
T
T T T
1 1 1 1
1
[ ] , 1,
(1 )
T TFPF G YF FY G G ZG P
+ + + − +
−
(10)
0, (0),
T
Z Y
P P
Y P
(11)
where ,pY K P= and minimization is performed over variables ,R
T ,n nP P = R
p nY R and T .p pZ Z = R
Minimization of criterion (5) is performed using the variables , ,P ,Y ,Z using
the semidefinite programming method, using SeDuMi Toolbox [13] based on Matlab.
Let ˆ , ˆ,P ˆ,Y Ẑ provide the minimum of criterion (5) under restrictions (10), (11).
Then the matrix ˆ
pK of the optimal controller (7) will be determined as follows:
1ˆ ˆˆ .pK YP−= (12)
The requirement 2 0TG C = is not restrictive. If it is not satisfied, then all the results
of Theorem 2 will be correct, only the objective function (9) will be presented in a dif-
ferent form
T T T
2 2 2 2tr[ ] min.T TCPC G YC CY G G ZG+ + + → (13)
2. Robust suppression of disturbances
In the presence of system uncertainties, the mathematical model of the object (1)
will have the form
( 1) ( ) ( ) ( ) ( ),
( ) ( ),
x k F F x k k
y k Cx k
+ = + + +
=
(14)
where ,n nF R ,n mR l nC R and ( ) nx k R is the phase state with the ini-
tial condition (0)x and output ( ) .ly k R External disturbances ( ) mk R satisfy
constraints (2).
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 2 9
System uncertainties F and have the following structure:
,
,
F F FF A H
A H
=
=
. (15)
where ,Fn P
FA
R ,
n P
A
R ,Fq n
FH
R
q m
H
R are constant matrices,
and the matrix uncertainties F FP q
F
R and
P q
R satisfy the following re-
strictions:
1,F 1. (16)
It is assumed that system (14) is stable, the pair ( , )F is controllable, and the matrix C
has maximum row rank.
Definition. Ellipsoid centred at the origin
T 1{ ( ) R : ( ) ( ) 1}, 0n
x x k x k P x k P− = (17)
is called invariant for system (14), (15) if the condition (0) xx implies the fulfilment
( ) xx k for all discrete moments of time 0,1,k = for all admissible disturban-
ces ( )k and all admissible uncertainties F and .
The following theorem was formulated and proven in [14].
Theorem 3. Ellipsoid (17) is invariant for system (14) for (0) 0,x = if its mat-
rix P satisfies linear matrix inequalities
T
1 2
1
2
0 0
* ( 0 0
0,* * (1 ) 0
* * * 0
* * * *
0
T
F
T T
F F
T
P PF PH
P A A A A
I H
I
I
P
−
− + +
− −
−
−
(18)
at some coefficients 1 2, , . R
This theorem is a robust analogue of Theorem 1.
The problem of robust suppression of external arbitrary disturbances has been
solved in the presence of system uncertainties in the controlled model of the object
1 1
2
( 1) ( ) ( ) ( ) ( ) ( ) ( ),
( ) ( ) ( ),
x k F F x k G G u k k
y k Cx k G u k
+ = + + + + +
= +
(19)
where control ( ) ,pu k R and external disturbance satisfies constraint (2).
System uncertainties of matrices F and have structure (15), and
1 1 11 .G G GG A H = (20)
In this case, the matrix uncertainties 1 2 ,
F Fq q
F
R 1 2
1 ,
G Gq q
G
R 1 2q q
R
satisfy the constraint
1,F 1 1,G 1, (21)
10 ISSN 2786-6491
the pair 1( , )F G is controllable and the pair ( , )F С is observable. Constant matrices
1 ,
Fn q
FA
R 1
1
,
Gn q
GA
R 1 ,
n q
A
R 2 ,
Fq n
FH
R 2
1
,
Gq p
GH
R 2q m
H
R
are given.
The task is to design a controller (7) in the form of a static linear state feed-
back ( )x k that will stabilize a closed-loop control system and suppress the influ-
ence of external arbitrary limited disturbances ( )k in the presence of system un-
certainties (15), (20) based on minimizing the trace of the limiting ellipsoid for the
output ( ).y k
To solve this problem, the following theorem was formulated and proven in [14].
Theorem 4. Let ˆ,P ˆ,Y Ẑ be the solution to the problem for the optimality criteri-
on (13) T T T
2 2 2 2tr[ ] minT TCPC CY G G YC G ZG+ + + → under restrictions
1
T T T
1
1
2
3
0 0
* 0 0 0
* * (1 ) 0 0 0,
* * * 0 0
* * * * 0
* * * * *
T T T
F G
T
P PF Y G PH Y H
I H
I
I
I
− +
− −
−
−
−
(22)
0,
T
Z Y
Y P
, (23)
where ,pY K P=
1 2
1 2 3
T T T
F p G G
P A A A A A A = − + + + with respect to matrix vari-
ables
T ,n nP P = R ,p nY R T p pZ Z = R and scalar variables 1, 2 , 3
and scalar parameter 0 1.
Finding the optimal values of ˆ,P ˆ,Y Ẑ to minimize criterion (13) is per-
formed using the semidefinite programming method in the Matlab system [13].
Then the matrix pK of the robust state controller (7) is determined according
to (12), that is
1ˆ ˆˆ ,pK YP−=
which stabilizes the system (14), and the matrix
T T T
2 2 2 2
ˆ ˆ ˆ ˆT TCPC CY G G YC G ZG+ + +
determines the size of the limiting ellipsoid for the vector of output variables for a
closed-loop control system at (0) 0.x =
3. Synthesis of discrete controllers using the control H∞-theory method
3.1. Standard H∞-optimization problem [15, 16]. When setting up a specific
problem in control H-theory, it is advisable to reduce it to the so-called standard
problem. We consider the general diagram of a closed-loop control system with a dis-
crete controller (Figure), where ( )y z is the vector of output measured coordinates of a
multidimensional controlled object; 1( )y z is vector of output adjustable coordinates;
( )z is vector of external disturbances; ( )u z is vector of control actions. Signals
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 2 11
1( ),v z 2( )v z are introduced to define the concept of stability of a closed-loop system.
For example, a vector 1( )v z arises due to the inaccuracy of actuators, and 2( )v z may
be a vector of measurement noise.
The matrix discrete transfer function (MDTF) ( )W z consists of limited rational
MDFTs of the generalized controlled object, that is ( ) ,W z HR ( )pK Z — matrix
discrete transfer function of the controller.
The matrix ( )W z can be represented as
11 12
21 22
( ) ( )
( )
( ) ( )
W z W z
W z
W z W z
=
, (24)
on the basis of which the dynamics of the controlled object can be represented in the
form of an input-output type model with 1 2 0v v= =
1 11 12( ) ( ) ( ) ( ) ( ),y z W z z W z u z= + (25)
21 22( ) ( ) ( ) ( ) ( ).y z W z z W z u z= + (26)
The dynamics of the controlled object can also be represented in the form of a state
space model
( 1) ( ) ( ) ( ),x k Fx k Gu k k+ = + + (27)
1( ) ( ) ( ).y k C x k D k= + (28)
The equation of stabilized coordinates is introduced separately in the form
1 2( ) ( ).y k C x k= (29)
The correspondence between models (25), (26) and (27)–(29) is carried out as fol-
lows. Let us present the dynamics of the equation of state (27) and measurement (28) in
the form of an input-output model
1 1( ) ( ) ( ) ( ) ( ),x z zI F Gu z zI F z− −= − + − (30)
which we substitute into expressions (28) and (29)
1 1
1 1( ) ( ) ( ) [( ) ] ( ),y z C zI F Gu z C zI F D z− −= − + − + (31)
1 1
1 2 2( ) ( ) ( ) ( ) ( ).y z C zI F Gu z C zI F z− −= − + − (32)
Comparing expressions (32), (31), respectively, with models (25), (26), we obtain the
MDPF values
1( )v z
ξ( )z
( )W z
( )u z
1( )y z
( )pK z
( )y z
2( )v z
12 ISSN 2786-6491
1
11 2
1
12 2
1
21 1
1
22 1
( ) ( ) ,
( ) ( ) ,
( ) ( ) ,
( ) ( ) .
W z C zI F
W z C zI F G
W z C zI F D
W z C zI F G
−
−
−
−
= −
= −
= − +
= −
(33)
Let us represent the regulator control law in the form
( ) ( ) ( ).pu z K z y z= (34)
Using expressions (31), (33), (34), we obtain the equation of the closed-loop sys-
tem for the channel ( ) ( )z y z −
1
22 21( ) [ ( ) ( )] ( ) ( ).py z I W z K z W z z−= − (35)
Based on expression (25), control law (34) and equation (35), we find the equation
of the closed-loop system along the channel ( ) ( )z y z −
1
1 11 12 22 21( ) { ( ) ( ) ( )[ ( ) ( )] ( )} ( ).p py z W z W z K z I W z K z W z z−= + − (36)
Thus, the matrix discrete transfer function of the closed-loop system will be
1
11 12 22 21[ ( ), ( )] ( ) ( ) ( )[ ( ) ( )] ( ).cls p p pW W z K z W z W z K z I W z K z W z−= + − (37)
For the closed-loop system equation (37), the standard H-optimization prob-
lem is formulated as follows: it is necessary to design a controller ( )pK z that will
minimize the H-norm of the matrix discrete function along the channel
1( ) ( )z y z − according to
mininf [ ( ), ( )] .
p
cls p
K
W W z K z
= (38)
In this case, the influence of external disturbances ( )z on the vector of output
regulative coordinates 1( )y z is maximally suppressed.
The dynamics of a closed-loop control system (Figure) can be described by a sys-
tem of equations
1 11 12
21 22 2
1
( ) ( ) ( ) ( ) ( ),
( ) ( ) ( ) ( ) ( ) ( ),
( ) ( ) ( ) ( ),p
y z W z z W z u z
y z W z z W z u z v z
u z K z y z v z
= +
= + +
= +
which can be represented in the form
1 12 11
1
22 21 2
( ) ( ) ( ) ( ) ( ),
( ) ( ) ( ) ( ),
( ) ( ) ( ) ( ) ( ) ( ).
p
y z W z u z W z z
u z K z y z v z
y z W z u z W z z v z
− =
− =
− = +
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 2 13
This system can be written in a generalized vector-matrix form:
112 11
1
22 21 2
( )( ) 0 ( ) 0 0 ( )
0 ( ) ( ) 0 0 ( )
0 ( ) ( ) ( ) 0 ( )
p
y zI W z W z z
I K z u z I v z
W z I y z W z I v z
−
− =
−
. (39)
This system will be well conditioned if the matrix
12
22
( ) 0
0 ( )
0 ( )
p
I W z
I K z
W z I
−
−
−
will be reversible [16]. This system will be internally stable if and only if the nine dis-
crete transfer functions from vectors ( ),z 1( ),v z 2( )v z to vectors 1( ),y z ( ),u z ( )y z
are asymptotically stable. In this case, the synthesized regulator ( ),pK z according
to (38), belongs to the set of internally stabilizing regulators, that is, the regula-
tor ( )pK z will stabilize the object ( ).W z
3.2. The problem of suppressing limited external disturbances based on
H∞-theory. In the early 90s, a number of works appeared devoted to the application of
the H-theory in motion control problems. One of these types of problems is the prob-
lem of forming a control that minimizes the effect of an external disturbance on the con-
trolled object. For example, a typical disturbance is a sudden gust of wind of high inten-
sity, which is especially dangerous during take-off and landing of an aircraft and its
flight at low altitudes.
Currently, a number of H-optimization methods have been developed [17–21],
based on which the so-called Two-Ricatti approach began to be used as the main tool
for solving problems of synthesis of H-optimal controllers. In this case, the synthe-
sized system is presented in Figure. The standard object is specified as
1 11 12
2 21
( )
0
F G
W z C D D
C D
=
, (40)
and the system (Figure) is described by the following system of equations in the state space
1 1 11 12
2 21
( 1) ( ) ( ) ( ),
( ) ( ) ( ) ( ),
( ) ( ) ( ),
x k Fx k Gu k k
y k C x k D u k D k
y k C x k D k
+ = + +
= + +
= +
(41)
where x is the state vector, u is the control vector, y is the vector of output measured
variables; 1y is vector of controlled variables; is vector of external disturbances.
In this case, a generalized sequence of disturbance vectors ( (0), (1),v v v=
, ( ), )v k in the space 2l is considered, the norm of which is determined according
to [16, 17]
1/2
T
0
( ) ( ) .
k
v v k v k
=
=
(42)
14 ISSN 2786-6491
By definition, a sequence 2( )v k l if this series converges, that is, .v
We consider the problem of minimizing the H-norm of a matrix discrete transfer
function of a closed-loop system with a controller
( ) ( )pu k K y k= (43)
to suppress disturbances along the 1( ) ( )k y k → channel, where ( )k is the vector of
all input signals that contain external disturbances.
The solution to this problem will guarantee the robustness of the closed-loop con-
trol system, which consists in the fact that for all possible values of the sequence of dis-
turbances 2( ) ,k l the maximum 2l -norm of the controlled output signal 1( )y k will
be minimized.
The optimality criterion is formulated as follows:
2
1
( )
sup min,
k l
y
J
= →
(44)
where 1y is a sequence of vectors 1{ ( ), 0,1, },y k k = is a sequence { ( ),k
0,1, }k = that belong to the space 2.l Minimization of the criterion J is performed
by forming an optimal sequence of control vectors { ( ), 0,1, }u u k k= = that influence
the vector 1.y In this way, the minimax problem (43) is solved under restrictions
on the norm 2.l
In works [17–21] it is shown that the value J of criterion (44) corresponds to the
norm ( ) ,clsW z
where ( )clsW z is the MDFT of the closed-loop system along the
« 1( ) ( )z y z → » channel, that is 1( ) ( ) ( ),clsy z W z z= and the norm ( )clsW z
is cal-
culated in space H (Hardy space of complex matrix functions, analytic in the unit
disk 1z and limited to circle 1z = to which all MDFTs of stable discrete systems
belong) and is equal to the singular number ( ( ))clsW z for all 1.z Indeed, accord-
ing to Parseval’s theorem, the norm 2l (in the time domain) is equal to the norm 2L (in
the frequency domain), that is
2
1/21/2 2
* T
0 0
1
( ) ( ) ( ) ( ) ( ) ,
2
j j j
L
k
v v k v k v e v e d v e
=
= = =
where ( )jv e is the discrete Fourier transform of an arbitrary discrete time se-
quence ,v
0 0
( ) ( ) ( ) ( ),j j k k
k k
v e v k e v k z v z
= =
= = = ,jz e =
* T( ) ( )j jv e v e − =
is complex conjugate vector. Taking account that 1( ) ( ) ( )clsy z W z z= or 1( )jy e =
( ) ( ),j j
clsW e e = we will have
2
1/2
2
*
1 1 1 1
0
1
( ) ( ) ( )
2
j j j
L
y y e y e y e d
= = =
1/2
2
* *
0
1
( ) ( ) ( ) ( )
2
j j j j
cls clse W e W e e d
=
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 2 15
2
1/2
2
*
0
1/2
2
*
0
1
( ( )) ( ) ( )
2
1
sup ( ( )) ( ) ( )
2
( ) ( ) ( ) .
j j j
cls
j j j
cls
j j
cls clsL
W e e e d
W e e e d
W e e W z
=
= =
Then
2
2 2
2
11
( ) 1( )
( )
sup sup ( ) sup ( ( )) sup ( ( )).
( )j
j
L j
cls cls clsj
k l ze L L
y ey
J W z W e W z
e
= = = =
In [17] it is shown that it is not even inequality that holds, but equality.
Thus, criterion (44) is equivalent to the following optimality criterion
1
( ) sup ( ( )) min.cls cls
uz
J W z W z
= = → (45)
Since ( )clsW z there is an MDPF of a closed-loop control system, it depends on the
control law (43), which forms the control .u In this case, the vector u should not de-
pend on the state vector ,x which may be unmeasurable, but on the vector of measured
variables .y That is, the state vector x must be implicitly evaluated at the same time.
The following theorem was formulated in [21].
Theorem 5. Let system (41) be given and the following conditions be satisfied
a) the pair ( , )F G must be stabilized;
b) the pair 2( , )C F must be detectable;
c) matrices 12D and 21D must have full rank, then the controller, which ensures
the stability condition of the closed-loop system, and the optimality criterion
( ) 1clsJ W z
= is designed in the following form:
contr contr
contr contr
( ) ( 1) ( ),
( 1) ( ) ( ),
dim dim
u k C r k D y k
r k F r k B y k
r x
= + +
+ = +
=
(46)
if and only if there exist non-negative definite symmetric matrices 0,P 0R
such that:
— first Ricatti equation
T
T T
11 1 11 1T 1
1 1
12 1 12 1
( )
T T
T
T T
PF D C PF D C
P F PF C C T P
GPF D C GPF D C
−
= + −
,
T
11 11 11 12
T
12 11 12 12
( )
T T
T T
D D D D
T P P G
D D D D I G
= +
−
;
— closed-loop control system matrix
T
11 11
T
12 1
( ) ( )
T
cp
T
PF D C
W F G T P
G PF D C
−
= −
— asymptotically stable;
16 ISSN 2786-6491
— matrix T
11 11 0,TV P D D= +
T T 1 T
12 12 12 11 11 12( ) ( ) 0.T T TM I D D G PG G P D D V PG D D−= − − + + +
If there is a matrix P that satisfies the specified conditions, then we define the fol-
lowing auxiliary matrices:
T T 1 T
12 1 12 11 11 1( ) ( ),T T TL G PF D C G P D D V PF D C−= + − + +
1 ,pA F GM L−= + 1/2,pE GM−=
1/2 T 1/2 T 1
1 11 1 11 12( ) ( ) ,T T
pC V PF D C V PG D D M L− − −= + + +
1
2 2 21 ,pC C D M L−= + 1/2
21 21 ,pD D M−= 1/2
12 ,pD V=
1/2 T 1/2
11 11 12( ) .T
pD V PG D D M− −= +
Then the matrix R must satisfy the following conditions:
— second Ricatti equation
T
2 21 2 211
1 11 1 11
( )
T T T T
p p p p p p p pT T
p p p p
T T T T
p p p p p p p p
C RA D E C RA D E
R A RA E E H R
C RA D E C RA D E
−
+ +
= + −
+ +
,
where
221 21 21 11
2 1
111 21 11 11
( ) [ ];
T T
pp p p p T T
p pT T
pp p p p
CD D D D
H R R C C
CD D D D
= +
— matrix of a closed system based on observations
2 21 21
11 11
( )
T T
p p p p p
cy p
T T
pp p p p
C RA D E C
W A H R
CC RA D E
−
+
= −
+
must be asymptotically stable;
— matrix 21 21 2 2 0;T T
p p p pQ D D C RC= +
11 11 1 1
1
1 2 11 21 2 1 21 11( ) ( ) 0.
T T
p p p p
T T T T
p p p p p p p p
S I D D C RC
C RC D D Q C RC D D−
= − − +
+ + +
If such matrices ,P R exist, then the designed controller in the form (46) will be
defined as follows:
1 1
contr 12 1 2 11 21
1
contr 12 1 contr 2
1
contr contr 2 21
contr contr 2
( ) ,
( ),
( ) ,
.
T T
p p p p p
p p p
T T
p p p p
cp p
D D C RC D D Q
C D C D C
B D A RC E D Q
A W B C
− −
−
−
= − +
= − +
= − +
= −
(47)
For system (41), the conditions of theorem (b), (c) are satisfied automatically.
Thus, to implement the Two-Ricatti algorithm, one requirement remains — the stabili-
zation of the pair ( , ).F G
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 2 17
To solve problem (41), (44), (46), (47), you can use the hinfsyn function from the
Robust Control Toolbox package (Matlab R2017a) [22]. It is based on some modifica-
tion of the result formulated in the theorem. In this modification, another requirement
( )clsJ W z
= is required instead ( ) 1,clsJ W z
= and the parameter is se-
lected iteratively [18–20]. This procedure is called -iterations. In this case, decreas-
es as long as a solution to the problem exists. In this way, the minimum possible value
of criterion (44) is achieved.
Conclusion
The article provides an overview of the main approaches to the design of control-
lers at the next stage of control theory development — robust control systems.
The first universal approach to solving the problem of suppressing arbitrary limited
external disturbances acting on the controlled object is to design state controllers based
on the method of invariant ellipsoids, which reduces the problem of synthesizing an op-
timal controller to searching for the smallest invariant ellipsoid of a closed-loop control
system. The concept of using invariant ellipsoids makes it possible to represent the
problem of minimizing the size of an ellipsoid in terms of linear matrix inequalities,
which allow the synthesis of a state controller through the use of semidefinite program-
ming tools implemented by numerical methods. This approach was extended to solve
the more general problem of robust suppression of external limited disturbances in the
presence of system uncertainties in the parameters of the model of the controlled object.
This approach to the synthesis of robust control ensures the stability of a closed-loop
system with one controller in the presence of system uncertainty of the object mathe-
matical model and the influence of arbitrary limited disturbances.
The second approach to solving the problem of minimizing the effect of arbitrary
external disturbances on the controlled object is to apply H-theory to solving the
problem of minimizing the maximum ratio of the 2l -norm of the vector of output stabi-
lized coordinates to the 2l -norm of the vector of external disturbances. The H-op-
timization problem was first solved in the form of a standard problem of suppressing
external limited disturbances when describing the dynamics of a controlled object in the
form of a matrix discrete transfer function. This H-controllers synthesis problem was
later implemented for state-space plant models in the Two-Ricatti approach that is now
accepted as a standard.
In practice, the problem of forming a control that minimizes the influence of exter-
nal disturbances on the system was solved by synthesizing a robust aircraft control sys-
tem, which ensures stability of motion in the event of a sudden gust of wind of high in-
tensity during take off and landing of the aircraft and its flight at low altitudes. At the
same time, the mathematical model of the aircraft, as a controlled object, was character-
ized by system uncertainty, which was caused by changes in the mass of the aircraft un-
der different loads.
Robust along with their undeniable advantages, also have certain disadvantages.
This is explained by the fact that a robust control system must remain operational under
the maximum possible disturbances, without having information when this disturbance
will occur. Therefore, robust controllers are configured for the worst case. As a result,
the quality of operation of a robust control system with a small normally distributed dis-
turbance is inferior to the quality of operation of a control system with linear-quadratic
Gaussian controllers.
18 ISSN 2786-6491
В.Д. Романенко, В.Ф. Губарев
ЕТАПИ ТА ОСНОВНІ ЗАДАЧІ
СТОЛІТНЬОГО РОЗВИТКУ ТЕОРІЇ СИСТЕМ
КЕРУВАННЯ ТА ІДЕНТИФІКАЦІЇ.
Частина 4. МЕТОДИ І ЗАДАЧІ ПРОЄКТУВАННЯ
РОБАСТНИХ СИСТЕМ КЕРУВАННЯ
Романенко Віктор Демидович
Національний технічний університет України «Київський політехнічний інститут
імені Ігоря Сікорського», Навчально-науковий інститут прикладного системного
аналізу, м. Київ,
ipsa@kpi.ua, romanenko.viktorroman@gmail.com
Губарев Вячеслав Федорович
Інститут космічних досліджень НАН України та ДКА України, м. Київ,
v.f.gubarev@gmail.com
У статті зроблено огляд найважливіших методів і задач проєктування ро-
бастних дискретних систем керування. При цьому основну увагу привер-
нуто до задач приглушення обмежених зовнішніх збурень, інформація про
які надана у формі обмеження на їх максимальну величину. Для опису ха-
рактеристики впливу зовнішніх збурень на траєкторію руху динамічних
систем як перший математичний апарат розглянуто застосування інваріант-
них еліпсоїдів. Розглянуто теореми про переформулювання проблеми інва-
ріантності еліпсоїдів у терміни лінійних матричних нерівностей, які в по-
дальшому використовуються для синтезу дискретних регуляторів стану
для приглушення зовнішніх збурень. Розглянуто розв’язок більш загальної
задачі робастного приглушення обмежених збурень на основі застосування
лінійних матричних нерівностей за наявності системних невизначеностей
параметрів математичної моделі обʼєкта керування у просторі стану. Як
другий математичний апарат для приглушення зовнішніх 2l -обмежених
зовнішніх обурень розглянуто застосування H-теорії керування. При
цьому критерій оптимальності полягає в мінімізації максимального відношен-
ня 2l -норми вектора вихідних стабілізованих координат об’єкта до 2l -нор-
ми вектора вхідних збурень. Задачу розв’язано за допомогою приведення її
до задачі робастного керування дискретною динамічною системою у просто-
рі H на основі застосування підходу Два-Рікатті. Розглянуто також стан-
дартну задачу H-оптимізації.
Ключові слова: регулятор стану, інваріантний еліпсоїд, лінійні матричні
нерівності, робастне керування, теорія H, зовнішні збурення.
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Submitted 23.02.2024
https://doi.org/10.34229/1028-0979-2024-1-1
http://sedumi.ie.lehigh.edu/
|
| id | nasplib_isofts_kiev_ua-123456789-211145 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0572-2691 |
| language | Ukrainian |
| last_indexed | 2026-03-14T16:30:23Z |
| publishDate | 2024 |
| publisher | Інститут кібернетики ім. В.М. Глушкова НАН України |
| record_format | dspace |
| spelling | Romanenko, V. Gubarev, V. 2025-12-24T21:12:11Z 2024 Stages and Main Tasks of the Century-Long Control Theory and System Identification Development. Part IV. Methods and Problems of Designing Robust Control Systems / V. Romanenko, V. Gubarev // Проблеми керування та інформатики. — 2024. — № 2. — С. 5–18. — Бібліогр.: 22 назв. — англ. 0572-2691 https://nasplib.isofts.kiev.ua/handle/123456789/211145 62-50 10.34229/1028-0979-2024-2-1 The article provides a review of the most important methods and problems in the design of robust discrete control systems. The main focus is on the problems of suppressing limited external disturbances, the information about which is presented in the form of constraints on their maximum value. The use of invariant ellipsoids is considered as the first mathematical apparatus for describing the characteristics of the influence of external disturbances on the trajectory of motion of dynamic systems. У статті зроблено огляд найважливіших методів і задач проєктування робастних дискретних систем керування. При цьому основну увагу привернуто до задач приглушення обмежених зовнішніх збурень, інформація про які надана у формі обмеження на їх максимальну величину. Для опису характеристики впливу зовнішніх збурень на траєкторію руху динамічних систем як перший математичний апарат розглянуто застосування інваріантних еліпсоїдів. uk Інститут кібернетики ім. В.М. Глушкова НАН України Проблеми керування та інформатики Адаптивне керування та методи ідентифікації Stages and Main Tasks of the Century-Long Control Theory and System Identification Development. Part IV. Methods and Problems of Designing Robust Control Systems Етапи та основні задачі столітнього розвитку теорії систем керування та ідентифікації. Частина 4. Методи і задачі проєктування робастних систем керування Article published earlier |
| spellingShingle | Stages and Main Tasks of the Century-Long Control Theory and System Identification Development. Part IV. Methods and Problems of Designing Robust Control Systems Romanenko, V. Gubarev, V. Адаптивне керування та методи ідентифікації |
| title | Stages and Main Tasks of the Century-Long Control Theory and System Identification Development. Part IV. Methods and Problems of Designing Robust Control Systems |
| title_alt | Етапи та основні задачі столітнього розвитку теорії систем керування та ідентифікації. Частина 4. Методи і задачі проєктування робастних систем керування |
| title_full | Stages and Main Tasks of the Century-Long Control Theory and System Identification Development. Part IV. Methods and Problems of Designing Robust Control Systems |
| title_fullStr | Stages and Main Tasks of the Century-Long Control Theory and System Identification Development. Part IV. Methods and Problems of Designing Robust Control Systems |
| title_full_unstemmed | Stages and Main Tasks of the Century-Long Control Theory and System Identification Development. Part IV. Methods and Problems of Designing Robust Control Systems |
| title_short | Stages and Main Tasks of the Century-Long Control Theory and System Identification Development. Part IV. Methods and Problems of Designing Robust Control Systems |
| title_sort | stages and main tasks of the century-long control theory and system identification development. part iv. methods and problems of designing robust control systems |
| topic | Адаптивне керування та методи ідентифікації |
| topic_facet | Адаптивне керування та методи ідентифікації |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211145 |
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