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USING DESTABILIZATION CONTROL TO IMPROVE THE FUNCTIONING
OF COMPLEX MULTIDIMENSIONAL TECHNOLOGICAL OBJECTS ON
THE TIME INTERVAL
Valery Nikolaevich Shamkin, Dmitriy Yurievich Muromtsev and Alexey Nikolaevich Gribkov
Tambov State Technical University, Tambov, Sovetskaya Street, Russia E-Mail: [email protected]
ABSTRACT
The article addresses the topical issues related to the control of complex energy- and resource-intensive technological objects operating in variable performance modes of product output over long time intervals. The use of non-conventional technologies allows obtaining additional control reserves and achieving an additional economic effect. Such technologies include destabilization optimization of functioning modes of multidimensional objects and original algorithms of their transfer from one type of the process modes to the others, corresponding to the required performance. We consider the problem of running the process of a complex multidimensional object that has additional possibilities of finding optimal technological modes, which can be applied in the case of expanding the domain of control actions of the object under consideration. This makes it possible to improve the object’s static modes on the time interval so that to meet new performance requirements and to minimize the spending of energy or resources. We study the practical cases of the complex object, for which it is possible to obtain an additional effect compared to the solutions where the traditional approach is used. The problem of destabilization optimization for a complex multidimensional technological object operating on a time interval with variable performance on the product output is formalized. The linear two-level multidimensional problem of destabilization optimization has been solved. On the basis of this, a method for analyzing energy-efficient control of multidimensional technological objects has been developed. It provides for the construction and the study of the domain of existence of the problem solution and determination of the optimal control function types. An algorithm for choosing the optimal control action for various possible solutions is proposed. Further it is planned to formulate and solve problems of linear multi-level destabilization optimization, as well as non-linear optimization problems.
Keywords: control action, destabilization, energy- and resource-saving control, multidimensional technological objects, time interval, objective function, optimization.
1. INTRODUCTION
In modern conditions of market economy, the output of an industrial enterprise is determined by consumer demand, while the company needs to save energy and raw materials, which constitute a significant share of the production costs, and in some cases have a decisive impact on the efficiency of its operation. Therefore, there arises the task of managing technological processes in conditions of variable consumption, when their productivity in the output varies repeatedly, which is particularly important for energy- and resource-intensive industries.
For more than two past decades, scientific and engineering issues of energy and resource efficiency of technological processes, given the possibility of finding objects in different states of functioning, have been explored by the school of Professor Yu.L. Muromtsev (Tambov, Russia). The results of the studies were described in the following publications (Muromtsev et al.,
1992; Muromtsev and Lyapin, 1993; Muromtsev et al., 1993; Muromtsev and Orlova, 2001; Muromtsev et al., 2007; Others, 2007; Muromtsev, 2008). At present this scientific direction is developed by his students, whose research results are given in (Artemova and Chernyshov, 2001; Artemova and Artemov, 2012; Muromtsev, 2005; Belousov et al., 2013; Gribkov, 2014; Tyurin, 2005;
Tyurin , 2005; Tyurin, 2011; Tyurin and Gaponov, 2015; Tochka and Shamkin, 2011).
The study of the object functioning on the time interval [0, T], in principle, creates the prerequisites for improving the optimal technological modes corresponding to new performance if it is possible to expand the range of permissible control actions of this object, i.e. to find additional reserves to run the technological process. If such an extension does not occur, then the optimization problem on [0, T] is equivalent to the set of individual optimization problems that are solved for each of the time intervals where the performance is constant. At the same time for steady-state (stabilized) modes, the corresponding material and energy balances, and the balance of momentum are performed. Such technological modes are called stabilized (stabilization) modes, and the corresponding problem of finding the control ensuring the required performance is called the problem of control stabilization.
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Rozhinsky, 1987). The problem was studied most extensively in (Shamkin, 1997), where the findings obtained at that time and the results of his own research in this direction were summarized.
The set of admissible controls can be extended if it is possible to violate balance relations on the interval [0, T]. In this case, there are additional effects that can be independently varied, and a measure of unbalance is the change in some technological variables (level, concentration, temperature, etc.). If the change of any variable in a certain range does not adversely affect the course of the technological process, then it can be safely recommended for further use, while ensuring that finding at any time on [0, T] is within the permissible range. For example, removing the restriction on the constancy of the liquid level in any apparatus of a technological object is equivalent to the appearance of some capacity in which accumulation or activation (in comparison with the established value) of a liquid is possible.
The introduction of such a capacitance causes the occurrence of an integral type constraint in the optimization task of static modes, corresponding to a change in the liquid level in the tank. At such the integral optimization criterion on [0, T] becomes meaningful, and the object itself operates in a destabilized mode.
The problem of finding a control providing the required performance and delivering an extremum to the integral criterion when the corresponding conditions are fulfilled is called the problem of destabilization optimization.
The change in additional controls (in relation to their values in the steady-state stabilized mode) in a certain way affects the value of the integrand characterizing the technological process at a given performance. For different performances, as a rule, this will be a family of parametric dependencies, where the performance of an object is used as a parameter. The solution of the problem depends to a great extent on the type of these dependencies.
It should be noted that the existing technological facilities allow for the possibility of organizing destabilized modes, and the variation in additional controls is usually limited. It is determined both by design features of the object, and by the nature of the performance change. Therefore, the dependencies of the optimality criterion on the control actions are linear or close to linear. Next, we consider one-dimensional and multidimensional optimization problems with a two-level change in the object performance with respect to the product output in the interval [0, T].
2. METHODS
2.1. Basic concepts and formulation of the problem of destabilization optimization
The traditional task of maintaining (stabilizing) the optimal static mode of a certain technological object is to determine a control action uU (U is the range of
permissible controls), uniquely corresponding to the varying performance – perturbation action f F (F is the range of possible perturbations). It is assumed that f is a piecewise constant function. The assumption is justifiable, because for complex energy-intensive and resource-intensive technological objects, which we are dealing with, long periods of their operation in specific steady technological modes are typical.
Mathematically, the problem is formulated as follows It is necessary for the perturbation f (t) given at
time t to find the control u(t)
corresponding to it at this point, for which
0 )) ( ), (
(
ut f t
. (1)
Equation (1) can physically be, for example, the balance equation of mass, component, energy or momentum.
Because of the single-valued dependence of uon
f , we can write (1) in the form
)) ( ( )
(t f t
u
, (2)
where
(
)
is some function.This problem is called the problem of control stabilization
We assume that it is solvable for the areaFU, which is the direct (Cartesian) product areas (sets) of F
and U, if for any
f
(
t
)
F
the inequality:) ( )) ( ( ) ( ) ( )) (
(f t u t ut u f t u t
ul l h h
. (3)
holds true.
Here ul(f(t)),uh(f(t)) are, respectively, the lower and upper permissible control values for the perturbation f(t) at time t.
Condition (1) rigidly fixes the control action u
for each f. In other words, the set Uf
of admissible controls for each f consists of one element, which makes it impossible to formulate a new optimization problem.
The set Uf
can be extended if we consider a certain time interval [0, T], at which a violation of (1) is possible.
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t
dt t f t u C t
0
)) ( ), ( ( )
( , (4)
where C is a constant value.
Note that physically, (t) is a process parameter that changes due to a violation of the corresponding balance (level in the tank, concentration, temperature, pressure).
It follows from (4) that (0) = C, i.e. constant C
is the value of the parameter (t) at the initial time t = 0. The measure (t) depends on the perturbing and control actions on the interval [0, t], denoted ut,ft, respectively, i.e. it is a functional and is written as
) , ( )
(t ut ft
.
Obviously, if (1) and relation (2) are satisfied,
between u()and f()
for each [0,t], the following equality holds:
0 )) ( ), ( (
0
u f dt
, (5)
and measure (t) = (0), is equal to the value of the balance ratio indicator violation, which has already been at time t=0.
Suppose that further landh the permissible limits of the change in the violation indicator of balance relation (1) for any t[0,T], then for all t the relation
h t t
l t u f
≤ () ( , )≤ . (6)
must be satisfied.
Suppose (0) satisfies the condition (6), i.e.
h
l
≤ (0)≤ . (7)
In the light of what has been said, it is possible to formulate several problems.
The simplest optimization problem on the time interval [0, T]
For the given (0) satisfying (7), find function
) (t
u on [0, T] for which the functional
T
dt t f t u Q f u I
0
, )) ( ), ( ( ) , (
(8)
takes a minimum value, where Q() is an integrand, and at any given time t[0,T] the exponent (t) satisfies the condition
h l
t
() (9)
and the control constraint is satisfied
)) ( ( ) ( )) (
(f t u t u f t
ul h . (10)
Obviously, for l h (0)the problem (8)(10) turns into problem (1). Thus, (1) is a special case of the formulated problem. In this case, the stabilizing
control
T
u on the interval [0, T] is included in the set
U
Tof possible controls
u
T. This means that the solution of problem (8)(10), at least, is not worse than the solution on [0, T] of the stabilization problem with the criterion
T
dt t f t u Q f u I
0
. )) ( ), ( ( ) , (
(11)
It is noteworthy that various parametric functions, both linear and nonlinear, can be used as the integrand function Q(). In particular, in solving problems of energy-saving control, the functional for the minimization of energy consumption is used
T
e u f Qu t f t dt I
0 2
. )) ( ), ( ( ) , (
(11 a)
In principle, the problem (8)(10) can be extended if variation (0) is assumed. In this case, the problem is formulated as follows.
The extended optimization problem on the time interval [0,T]
It is necessary to find the value of (0) and the function u(t)defined on the interval [0, T], for which the functional (8) assumes a minimum value and conditions (9), (10) are satisfied at any time t.
We denote by (t) the deviation of the control
) (t
u from the stabilizing controlu(t)
at the moment t, i.e.
) ( ) ( )
(t ut ut
. (12)
Then the extended problem, taking (12) into account, can be reformulated.
The optimization problem on the time interval [0, T] in deviations
It is necessary to find the value of (0) and the function
(t) denoted on the interval [0, T], for which the functionaldt t f t t u Q f
I
t
)) ( ), ( ) ( ( ) , (
0
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assumes a minimum value, and at any moment of time
t[0, T] the conditions
h l
t
() , (14)
dt t f t t u t
t
)) ( ), ( ) ( ( ) 0 ( ) (
0
, (15))) ( ( ) ( )) (
(f t t h f t
l
, (16)
are satisfied, where
)). ( ( )) ( ( )) ( (
)), ( ( )) ( ( )) ( (
t f u t f u t f
t f u t f u t f
h h
l l
(17)
The problem (13), (17) will also be called the problem of destabilization optimization.
It is characteristic that from conditions (3), (17) we have
. 0 )) ( (
, 0 )) ( (
t f
t f h l
(18)
Obviously, when (t)0 for all t[0,T]
0 )) ( ), ( (
(
u f t f t
this problem becomes a stabilization problem (1), the effectiveness of which is estimated by the criterion (11).
2.2 The formulation of a linear two-level multidimensional problem of destabilization optimization
We now consider a more general case. Let the perturbing action f(t) be given by the M-dimensional vector function f(t)=(f1(t),f2(t),...,fM(t)), and the control
action u(t) is searched among N -dimensional vector functions
u(t)=(u1(t),u2(t),...,uN(t)).
We call a function f(t) two-level on the interval [0,] if it satisfies the relations
. if
), ..., (
; 0 if ), ..., (
) (
1 2 , , 22 , 12
1 1
, , 21 , 11
t t f
f f
t t f
f f t f
M M
(19)
For short, we denote this function by a quadruple
) , , , ( )
(t f1 f2 t1 f
,
where
), ,..., ,
( 11 21 ,1
1 f f fM
f
). ,..., ,
( 12 22 ,2
2 f f fM
f
In this case, the m-th component fm(t) of the vector f(t) has the form:
. , 1 ), , , , ( )
(t f 1 f 2t1 m M
fm m m
If is known, for example, =T, where T is the period of the object's operation time, then by default we will denote the periodic two-level function by a triple
) , , ( )
(t f1 f2 t1 f
, and if the switching moment t1 is also
known we will denote is by a double f(t)(f1,f2). A two-level function on the interval [0,] is the function u(t) of the form:
;
if ), ,..., , (
; 0 if ), ,..., , ( ) (
1 2 , 22 12
1 1
, 21 11
t t u
u u
t t u
u u t u
N N
(20)
and denote it by a quadruple
) , , , ( )
(t u1 u2 t1
u ,
where
), ,..., ,
( 11 21 ,1
1 u u uN
u
). ,..., ,
( 12 22 ,2
2 u u uN
u
Correspondingly, the n-th component of un(t) of the vector u(t) has the form:
N n t u u t
un()(n1,n2,1,), 1, .
If is assumed to be given and equal to T, then the periodic two-level control is denoted by
) , , ( )
(t u1u2 t1
u , and n-th component is respectively N
n t u u t
un( )( n1, n2,1), 1,
. If τ and t1 are known, then
we have:
N n u u t u u u t
u ()(1,2), n()(n1,n2), 1, .
Similarly to (1) and (2), we formulate the problem of stabilizing control.
It is necessary for the perturbation f(t)=
(f1(t),f2(t),...,fM(t))F (F is M-dimensional range of changes in perturbation actions) given at time t, to find the
control
U t u t u t u t
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dimensional range of permissible controls), corresponding to it at this point, wherein the balance relations
. , 1 , 0 )) ( ), (
(un t f t n N
n
(21)
re satisfied.
Control u(t)
is called stabilizing.
Because of the uniqueness of the dependence un
on f , (21) can be represented in a somewhat different form
, , 1 )), ( ( )
(t f t n N
un n
(22)
where n(),n1,Nare some functions.
The problem of control stabilization is considered solvable for the domain FU, which is the Cartesian product of the domains F and U if for each f(t)F the inequalities
, , 1 )), ( ( ) ( )) (
(f t u t u f t n N
unl n nh
(23)
hold true, where uln(f(t)) and u (f(t)) h
n are, respectively,
the lower and upper permissible values of controls un(t)
for the perturbation f(t) at time t.
Functional (t)(ut,ft), characterizing the measure of violation of the balance relations (21) on [0, t] in this case is vector. In this case, its dimension is equal to the dimension of N vector and control actions, i.e.
)) , ( ),..., , ( ), , ( ( ) ,
( 1 1 2 2 n tn t
t t t t t
t
f u f
u f u f
u
, where
the components of the vector have the form:
t
n n n
t t n n
n t u f u t f t dt n N
0
. , 1 , )) ( ), ( ( ) 0 ( ) , ( )
( (24)
We assume that the integrandn(un,f) in (24) at each instant of time t is represented by a parametric polynomial linear with respect to the control un(t), i.e.,
, , 1 ), ( )) ( ( )) ( ( )) ( ), (
(un t f t n f t n f t un t n N
n
(25)
where n(f(t)) andn(f(t))are functions of the perturbation action f at the appropriate time moment.
We assume that the integrand Q(u,f) in the functional
T
o
dt t f t u Q f u
I( , ) ( ( ), ())
can be represented on [0,T] as a family of parametric n -dimensional planes of the form:
), ( )) ( ( ))
( ( )) ( ), ( (
1
t u t f b t
f a t f t u
Q n
N
n n
(26)
where a(f(t)),bn(f(t)) are functions of the perturbation action f at the appropriate time moment t.
We now introduce the notation n(t) – the deviation of the component un(t) of the control vector u(t)
from the corresponding components un(t)
of the
stabilizing control u(t)
at time t, i.e.
N n t u t u
t n n
n() () (), 1,
. (27)
According to (25), (21), un(t)
is defined by the expression
. , 1 , )) ( ( ) ( ( )
(t f t f t n N
un n n
(28)
Substituting the result (27) un(t) n(t) un(t)
in (25) and taking into account (28), we have
N n t t f t
f
t n n
n(( ), ()) ( ()) () 1,
(29)Thus, taking into account all the above, the following conditions hold:
- Perturbation action f(t)=(f1(t),f2(t),..., fM(t)) is a
vector-valued periodic function with period T f(t)=f(t+Т),
) , , , ( )
(t f1 f2 t1 f
.
- Control u(t)=(u1(t),u2(t),...,uN(t)) is also a vector-valued
function.
- Perturbation f(t) and control u(t) actions are piecewise-continuous vector-valued functions.
- Integrand Q(u,t) in the functional (25) can be represented as a family of parametric n-dimensional planes of the form (26), i.e., it is linear with respect to un,n1,N.
- Functional (t)(ut,ft)(1(u1t,ft),2(ut2,ft),..., ))
, ( nt t n u f
is vector; its dimension is equal to the dimension of the control vector u, and the components of the vector are determined by (24).
- Function (t)(1(u1,f ),2(u2,f ),...,n(uN,f )) is represented by a family of parametric polynomials (25) that are linear with respect to controls un,n1,N.
- The stabilization problem (21), (23) is solvable in the entire region F of changes in perturbation actions.
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Linear two-level multidimensional problem of destabilization optimization
For a given perturbation periodic action
) , , ( )
(t f1 f2 t1 f
with period T, it is necessary to find the control (t)(1,2,t1)defined on [0,T] and the initial states (0),n1,N, for which the functional
, ) ( )) ( ( ) ( )) ( ( )) ( ( ) , ( 1 0 1 dt t t f b t u t f b t f a f I N n n n T N n n n
(30)takes the minimal value and ant any time t[0,T] forl all
N
n1, the relations
), ( ) ( )
(t n t hn t
l
n
(31)
), ( ) ( )
(t n t hn t
l
n
(32)
are satisfied and we have equalities
n(0)=n(T), (33)
where )), ( ( )) ( ( )) ( ( )
(t f t uln f t un f t
l n l n
(34)
)), ( ( )) ( ( )) ( ( )
(t f t unh f t un f t
h n h n
(35)
t n n nn t f t t dt
0 ) ( )) ( ( ) 0 ( )
( . (36)
3. RESULTS
Theoretical justification of the solution of the linear two-level problem of destabilization optimization.
We prove a number of statements used in the future when developing algorithms for solving a two-level multidimensional destabilization problem (30)( 36).
Auxiliary theorems.
Theorem 1.
For two-level periodic perturbation action
) , , ( )
(t f1 f2 t1 f
with period T and control n(t), so that
N n t
t n n
n()(1,2,1), 1,
function n(t) for any
N
n1, linear in the sections [0,t1] and [t1,T], is defined
by expressions , if ), ( ) ( ) ( ) 0 ( ; 0 if , ) ( ) 0 ( ) ( 1 1 2 2 1 1 1 1 1 1 T t t t t f t f t t t f t n n n n n n n n
n
(37) increases if 1 1) ( n
n f
>0, if 0 t<t1 (38)
or
2 2)
( n
n f
>0, if t1 t<T, (39)
or decreases if
1 1)
( n
n f
<0, if 0tt1 (40)
or
2 2)
( n
n f
<0, if t1t<T. (41)
Proof.
We consider the expression (36) for n(t) on the time interval [0,t1]. By the conditions of the theorem, f(t)
and n(t)on [0,t1] do not change and are equal to
f
1
and
1 n
, respectively, for each n, we obtain the function , ) ( ) 0 ( )(t n n f1 n1 t
n
linear to t on this section.
Coefficients n(0) and n(f1)n1
characterize respectively the initial position of the straight line and the tangent of the slope of the straight line to the time axis. It is obvious that the function n(t) increases ifn(f1)n1
> 0 and decreases if n(f1)n1
<0.
We consider how the function n(t)) behaves if t
changes on [t1,T]. The expression (36) can be represented
here in the form
T t n n t n n nn t f t t dt f t t dt
1 1 ) ( )) ( ( ) ( )) ( ( ) 0 ( ) ( 0 and
taking into account the fact that on [0,t1] f(t) f1
on [0, t1] and n(t)n1, but on [t1,T] f(t) f2
and
2 )
( n
n t
,it is easy to obtain the function
N n t f t f t f
t n n n n n n n
n()(0) (1)11 ( 2)21 ( 2)2 , 1,
,
that is linear with respect to t on [t1,T]. Coefficients
2 2 1 2 2 1 1
1) ( ) and ( )
( ) 0 (
(n n f n t n f n t n f n
characterize respectively the initial position of the straight line and its slope. It is also obvious that n(t) increases if
2 2)
( n
n f
> 0 and decreases if n(f2)n2
<0. Thus, (37)(41) is valid, i.e., Theorem 1 is proved.
Corollary fact.
If f(t) fˆand(t)ˆare constant on interval [t1,t2], function n(t) linear in the section [t1,t2], is defined
by the expression
) ( ˆ ) ˆ ( ) ( )
(t n t1 n f t t1
n
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or decreases if n(fˆ) ˆ <0.
Proof.
The validity of the corollary follows directly from the above arguments.
We now introduce additional notation. We denote as
21 t t n
the difference of functions n(t) at times t1 and t2
, 2
(2) (1)1 2 1 2 1 2
1 f n t n t
t t t t n t t n t t n . , 1 )), ( ), ( ( )) ( ), (
( n t2 f t2 n n t1 f t1 n N
n
Given that
t n n nn t f t t dt
0 ) ( )) ( ( ) 0 ( ) ( , we obtain
( ()) () , 1, .2
1 2
1 f t t dt n N
t t n n t t
n
(42)We denote as 2 1
t t
I
the functional depending on functions
2
1 , 2 1 , 2 2 1 , 1 2 1 2 1 , 2 1 , 2 2 1 , 1 2
1 , ,..., , , ,...,
t t N t t t t t t t t M t t t t t
t f f f u u u u
f , and
defined by the formula
u f
af t b f t u t b f t t dtI I t t n N n n n N n n t t t t t t t
t
2
1 1 1
2 1 2 1 2 1 2
1 , ( ()) ( ()) () ( ()) ()
(43)
Theorem 2.
For the given(t1)(1(t1),2(t1),...,N(t1)), where n(t1), n1,N, satisfies (32), the invariance of perturbation action f (f1,f2,...,fM) and control
(
t
)
)) ( ),..., ( ), (
(1t 2 t N t on [t1,t2], so that for
N n t n(), 1,
, at any time t[t1, t2] (32) is satisfied, when
substituting
(
t
)
with control n(1,2,...,N), thecomponents of which are determined by the formulas
, , 1 , ) ( 1 2 1 1 2 N n dt t t t t t n
n
(44)the assertions:
a)
,
, 2
, 1, ;1 2 1 2 1 2 1 2
1 f f n N
t t n t t n t t t t n t t
n
b)
2 1 2 1 2 1 2 1 2 1 2
1 , ,
t t t t t t t t t t t
t f I f
I
are valid;
c) for any t[t1,t2] the conditions (32) are satisfied.
Proof.
Since according to (42)
2
1 2
1 ( ()) ()
t t n n t t
n f t t dt,
then given the invariance f(t) on [t1, t2], we obtain
2 1 2 1 2 1 21 , ()
t t n n t t t t n t t
n f t dt, (45)
where n n(f(t)). Similarly, for n we have
, 2
(2 1)1 2 1 2
1 f n n t t
t t t t n t t
n
(46)
Taking (44) into account, expressions (45) and (46) are the same and assertion "a" of the theorem is proved.
From the definition (43) for 2 1 t t
I , given the
invariance of f(t), and hence u(t)
on the interval [t1, t2],
we have
N n t t n n N n n n t t t t tt f A B u t t B t dt
I 1 1 2 1 2 1 2 1 2 1 2
1 , ( ) () ,
(47)
where A=а(f(t)), Bn=b(f(t)), unn(f(t)),n1,N
are constants.
Similarly, for n we have
N
n n n N n n n t t t t t
t f A B u t t B t t
I 1 1 2 1 2 1 2 1 2 1 2
1 , ( ) ( )
(48)
Taking (44) into account, expressions (47) and (48) are the same and assertion "b" of the theorem is proved.
In conclusion, we show the validity of the assertion "c".
According to the definition
, ) ( ), ( ) ( ), ( , , ) ( ), ( ) ( ), ( , 1 1 2 2 2 1 2 1 2 1 1 1 2 2 2 1 2 1 2 1 t f t t f t f t f t t f t f n n n n t t t t n t t n n n n n t t t t n t t n N n1,and since by the hypothesis of the theorem
n(t1), f(t1)
n(t1) const,n
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From this, and from the validity of the assertion "a" of the theorem, it follows that
n(t2),f(t2)
n
n,f(t2)
,n 1,N,n
and since by
the hypothesis of the theorem n
n(t2),f(t2)
alsosatisfies the conditions (32), then n
n,f(t2)
alsosatisfies (32) if t=t2.
Thus, the function n
n,f(t2)
,n1,N satisfies (32) in two points if t=t1 and if t=t2, andaccording to the corollary to Theorem 1, this function is linear on the interval [t1, t2]and, consequently, it satisfies
the condition (32) at any point t[t1,t2].
Theorem 2 is proved.
Theorem 3 (the existence theorem).
If the set of admissible controls defined by (31)(36), is not empty, i.e. the problem of linear multidimensional destabilization optimization (30)(36) has the optimal solution for a two-level periodic perturbation action f(t) (f1,f2,t1)
on period T, then
there exists (t)(1,2t1),where
) ,..., ,
( 11 21 1
1
N
, 2(12,22,...,N2)
, which is the optimal solution of this problem.
Proof.
We assume that (t) is a vector-valued function defined on [0, T], which is the optimal solution of the problem (30)(36).
We denote 1 and 2
as mean values of the function
(t) on the intervals [0,t1] and [t1,T],respectively, i.e.
), ,..., , (
), ,..., , (
2 22 12 2
1 21 11 1
N N
where
and consider a two-level function (t) ( 1, 2,t1).
periodic on [0,T].
First, we show that (t)is a permissible control. Since f(t) on [0,t1] is invariable and equal tof1
, then
control 1
is also invariable, with n1,n1,N defined by
(49), wherein if
(
t
)
for all t[t1,T] conditions (32) aresatisfied, then according to Theorem 2, conditions (13) hold for any t[0,t1]. Similarly, according to Theorem 2,
conditions (32) hold for any t[t1,T] if control n2
is of the form (50).
Thus, condition (32) for control is satisfied for any t on [0,T].
According to Theorem 2 for the components of the vector function
(
t
)
on [0,t1] the equalities
,
, 1, ,, 1 01 1 1
1 0 1
0 f n f n N
t n t
n t
n
(51)
hold true; since the initial values of n(0),n1,N for optimal control
(
t
)
and averaged control are unique, it follows from (51) that at the time t=t1 we have
n Nt t f f
t n n
n
n ( ), , , 1,
1 1 1 1
1
. (52)
When (52) is satisfied, assertion “a” of Theorem 2 holds for the interval [t1,T], i.e.
,
2, 2
, 1, ,1 2
1
1 f n f n N
T nt T
nt T
nt
and, therefore, for the time t=T
n NT t f f
T n n
n
n ( ), 2 2, 2 , 1,
. (53)
is valid.
Since for the optimal function
(
t
)
conditions (33) are satisfied, i.e.
n
nT f
n
T n Nn 0 ( ), 2 , 1,
,
then, given (53) we have
n NT t f n
n 2, 2 n(0), 1,
.
Thus, the periodicity condition (33) for control is also satisfied.
Since (t)is determined by the expressions (49), (50), in which for n(t) at each instant of time t[0,T], the
constraints (31) are satisfied, then they also hold for n.
From the validity of conditions (31)(33), when using the control , it follows that this control is permissible. We show that it is optimal.
Functional I when using functions (t) and can be written in the form
,
,
, ),
( 2
1 1 1 1 0 1
0 f I f
I f
7206
,, ,
) ,
( 01 1 1 1 2 2
f I f I f
I t tT (55)
where
1 0
1 1
1 0 1
0 , ( (), )
t t t
dt f t Q f
I ,
T
t T t T
t f Q t f dt
I
1
2 2
1
1 , ( (), )
,
1
0
1 1 1
1 1
0 , ( , )
t t
dt f Q f
I ,
T
t T
t f Q f dt
I
1
2 2 2
2
1 , ( , )
.
According to Theorem 2, and taking account (52), we have
1 1 1 0 1 1 0 1
0 ,f I ,f
It t t ,
2 2 1 2 1
1 ,f I ,f
ITt Tt tT ,
and, consequently, from (54) and (55) we obtain
) , ( ) ,
( f I f
I ,
i.e. control is optimal for the problem (30)(36). Theorem 3 is proved.
Theorem 4.
For two-level periodic perturbation action
) , , ( )
(t f1 f2 t1 f
and control
(
t
)
(1,2,t1)on the period T the fulfilment of constraints (32) for the times 0,t1, T ensures their fulfillment for any t[0, T].Proof.
According to the conditions of the theorem
for the interval [0,t1]the conditions of Theorem 1
are satisfied, n(t) is a straight line on this section. Because according to (56), (58) this line satisfies the constraint (32) on the ends of the interval [0, t1], then
every point of this line n(t)if t[0,t1] satisfies (32).
Similarly for the interval [t1, T] function n(t),
according to Theorem 1, is a straight line. The ends of the line satisfy (32) in accordance with (57), (58), and
therefore any other point of this line n(t) if t[t1, T] also
satisfies (32).
Theorem 5.
If for two-level periodic perturbation action
) , , ( )
(t f1 f2 t1
f and control (t)(1,2,t1)
on period
T, the constraints (32) are satisfied, then they are satisfied for ˆ(t)(ˆ1,ˆ2,t1)
if the initial and final values of the function n(t)are equal, i.e.
, , 1 ), (
ˆ
) 0 (
ˆ
) ( ) 0
( nT n nT n N
n
(59)
and ˆn1[0,n1],n1,N.
Proof.
If t=0 and t=T forˆ(t) hold according to the conditions of the theorem and equality (59).
If we consider time t1 and take arbitrary
N n n n [0, ], 1, ˆ 1 1
, then for t=t1 we have
. , 1 ,
ˆ
) ( ) 0 ( ) (
ˆn t1 n n f1 n1t1 n N
(60)
If ˆn10,n1,N, then ˆn(t1)n(0) and by
the hypothesis of the theorem the constraints (32) are satisfied. If ˆn1n1,n1,N, then
1 1 1 1) (0) ( ) ˆ
(
ˆn t n f n t
and the constraints (32) are
satisfied by the theorem hypothesis. Since ˆn(t1) of the
from (60) is linear relative to ˆn1,n1,N, then constraints (32) are satisfied for any ˆn1[0,n1],n1,N.
Since the constraints (32) are satisfied at the instants of time 0, t1 иT, in accordance with Theorem 4
they are satisfied for all t[0,T].
Theorem 5 is proved.
Theorem 6.
For two-level periodic perturbation action
) , , ( )
(t f1 f2 t1 f
and controls (t) (1,2,t1),
) , ˆ , ˆ ( ) (
ˆ t 12 t1
on the period T and sign-constancy of functions n(f), n1,N from the fulfillment for
) (
ˆ
and )
(t n t
n
conditions:
) (
ˆ
) ( ) 0 (
ˆ
) 0
( n n T n T
n
, (61)
and inequalities
,
ˆ 1
1 n
n
(62)
7207
,ˆ 2
2 n
n
(63)
Proof.
Using the conditions of the theorem, we represent
expression ( ) (0) ( ()) ( ) , 1, ,
0
N n dt t t f
T n
T
n n
n
for the control
(
t
)
(1,2,t1) in the form. , 1 ), ( ) ( )
( ) 0 ( )
(T n n f1 n1 t1 n f2 n2 T t1 n N
n
(64)
Since according to (61) n(T)n(0) for the
control
(
t
)
(1,2,t1), then from (64) we obtain. , 1 , )
( ) (
1 1 1
2 1
2 n N
t T
t
f f
n n
n
n
(65)
Similarly for the control ˆ(t)(ˆ1,ˆ2,t1)
. , 1 ,
ˆ
) (
) (
ˆ 1
1 1
2 1
2 n N
t T
t
f f
n n
n
n
(66)
Subtracting (65) from (66), we have
N n t
T t
f f
n n n
n n
n ( ˆ ), 1,
) (
) (
ˆ 1 1
1 1
2 1 2
2
and for the sign-consistency of functions n(f) from ,
ˆ 1
1 n
n
follows that n2ˆn2, i.e. (63) is satisfied. Theorem 6 is proved.
Corollary 1.
For two-level peridodic on period T the perturbation action f(t) (f1,f2,t1)
and control
(
t
)
) , , (1 2 t1
and sign-consistency of functions
N n f n( ), 1,
from the fulfilment of condition
N n T n
n(0) ( ), 1,
follows the fulfilment of relations
. , 1 , )
( ) (
1 1 1
2 1
2 n N
t T
t
f f
n n
n
n
(67) or
. , 1 , )
( ) (
2 1
1
1 2
1 n N
t t T
f f
n n
n
n
(68)
Proof.
The validity follows immediately from (65).
Corollary 2.
If the conditions of Corollary 1 of the theorem are satisfied, n1 and n2 have different signs and, if n1>
2 n
, then n1>0, n2<0 and vice versa
Proof.
Indeed, taking into account (68) we have
. , 1 , )
( ) (
1 1
1 2
2
1 n N
t t T
f f
n n
n
n
and for sign-consistency of fuctions n(f) we have
, , 1 , 0 2
1 n N
n
n
from which it follows directly that n1 and n2 have
different signs. Obviously, if n1>n2, then n1>0, n2 <0 and vice versa, i.e., if n1<
n2, then
n1<0,
n2>0.Corollary 3.
If the conditions of Corollary 1 of Theorem 6 are satisfied, functions n(t),n1,N, are linear on [0,t1] and
[t1, T], and if on one of the time intervals n(t) increases,
it decreases on the other one.
Proof.
Indeed, if the conditions of Corollary 1 are satisfied, then the assertions (37)(41) of Theorem 1 about the linearity of functions n(t),n1,N, on [0,t1] and
[t1,T] their increase at
, 2 , 1 , , 1 , 0 )
(
n fi ni n Ni
(69)
and decrease at
. 2 , 1 , , 1 , 0 )
(
n fi ni n Ni
(70)
are valid.
Since functions n(f) are sign-consistent, then
1 n
and n2 have different signs (Corollary 2), and from
(69) and (70) immediately follows that the functions
), (t
n
n1,N increase on one of the intervals [0, t1] and
[t1, T]and decrease on the other.
The fundamental theorem
7208
vector-valued function with the same period defined by (20).
It is assumed that the integrand Q(u,f) in the functional I, which characterizes the control efficiency on the time interval [0,T] is represented by a family of parametric N-dimensional planes of the form (26); i.e. they are linear with respect to the control actions un,n=1,N.
The functional itself has the form (30). In addition, the state of the object is characterized by some N-dimensional vector (t), in which its components n(t), n=1,N
determine the measure of violation of the corresponding balance relations of the form (21) and are calculated in accordance with (36).
We introduce the following abbreviations for functions that depend on a two-level perturbation
) , , ( )
(t f1 f2 t1
f . We denote A1= a(f1)
, A2=a(f2)
,
Bn1=bn
(
f
1)
, Bn2=bn(f2)
, n1n(f1),n2n(f2), ) ( 1 1 f l n l n
,hn1hn(f1), nl2 ln(f2), ) ( 2 2 f h n h n
,n=1,N. With this notation, we formulate the following theorem.
Theorem 7 (main).
For two-level periodic perturbation
) , , ( )
(t f1 f2 t1 f
with period T and sign-consistency of functions n(f),n=1,N there are optimal initial values
n(0), n=1,N and optimal control ( 1, 2,t1)
, which is
a two-level periodic function with period T, defined as follows. If 0 1 2 2
1n n n
n B
B
, (71) then , 0 if , , 0 if , ) 0 ( 1 1 n h n n l n
n (72)
), ; ; min( 2 1 1 1 2 1 1 1 l n n n h n •lm n n t t T
(73)
, 1 1 1 2 1 2 n n n n t T t (74) where 1 1 1 t n l n h n •lm
n
. (75)
If
0 1 2 2
1n n n
n B
B , (76)
then , 0 if , ; 0 if , ) 0 ( 1 1 n l n n h n
n (77)
), ; ; max( 2 1 1 1 2 1 1 1 h n n n l n •lm n n t t T
(78)
, 1 1 1 2 1 2 n n n n t T t (79) where 1 1 1 t n l n h n •lm
n
. (80)
The functions p(f),q(f) in the statement of
the theorem can be of different signs for pq.
Proof.
We will solve the optimization problem (30)(36) in the class of two-level functions
(
t
)
(1,2,t1). This can be done, since by Theorem 3 there exists control), , , ( )
(t 1 2 t1
that is the optimal solution of this problem in the class of piecewise continuous functions.
Proof consists of two parts. First we show the validity of (71)( 75), and then comment on the validity of the fulfillment of (76)( 80).
Part 1. According to Corollary 1 of Theorem 6, when the periodicity conditions (33) are fulfilled, i.e. equalities n(0)=n(T), n=1,N, between the controls
1 n
and n2 the relations
, 2 1 1 1 2 1 n n n n t t T
(81)
, 1 1 1 2 1 2 n n n n t T t N
n1, (82)
are established.
For perturbation f(t)(f1,f2,t1) functional of I
form (30) given that un1un1n1
, un2 un2n2
, N
1, =
n , can be represented on [0,T] in the form:
N n n n N n nn t B T t
B A I 1 1 2 2 1 1 1
1 ( ),
