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Veniamin V. Malyshev, Michail N. Krasilshchikov
Moscow Aviation Institute
Volokolamskoye shosse, 4 , Moscow, USSR
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | | DOI: 10.2514/6.1989-3633
Now, many results of the optimal
stochastic control theory make a methodological base for the appropriate automated
design and decision support systems. In
particular, the design problem decision of
the optimal filters and controllers is
automated in a linear dynamic system by
a quality quadratic criterion. The further
development of these achievements would be
a realization of the observation process
optimization algorithms (optimal experimental design) at the program level. Here
the main problem is the development of the
constructive optimization methods. Reference [I] gives a general theoretical effect,
the problem of optimization is reduced to
the problem of the Riccati-type equation
determinate control for the covariance
matrix of the estimation errors. But the
non-linearity and the matrix structure of
the equation cause the great difficulties
in the decision of the control problem. In
this connection the paper suggests a new
approach, that was developed in Ref. 2 and
that reduces the given problem of the control to a linear one by using the analytic
properties of the Riccati equation.
solved under conditions of the restrictions on total energy expenditure
Statement of a Problem. Let a movement of a dynamic system be described by
where 'Xi, is a phase vector n x I: U ; ~ Ta
system control m x I; { i)-'?,~.i the succession of noncorrelatel Gauss isturbances with the characteristics M gi,l=o,
M[Ti f,f] = D l i i Ai,-i , Bi , FL
n m , nxm, n x t .
The initial state of the system (I)
is a s ~ u m e d ~ t h a%,is
Gauss vector, i.e.
?o).~ ,
The system movement observation (2)
is made by carring out the measurements
Yi = yiHi13Ci + \i ,
i= 1,N,
is a measurement vector 1 x I ,
Xi - a matrix L x n,
- the successions of noncorrelated auss errors of
the measurements with characteristics
The succession
M1'1;3 =@,MtJi jiY=%i
is an observation prog, ,
Hence, I. Ti = I, in a moment i the
measurement is made, if
0 , it isn't
is a matrix m x n, as well as the restrictions on maxl.mum admissible amount
of measurement moments
As an optimality criterion we choose
some root-mean-square characteristic of
a control terminal accuracy
where K is a matrix n x n.
We note that in the relation (5)
the minimization operation is made according to the measuring information processing algorithm, to the observation program,
to the control law.
The problem stated above will be
solved in two stages, beginning from a
synthesis of the filtration algorithm and
the control law at a fixed program ['fi] ,
and then carring out an additional optimization of the criterion according to this
The first stage realization is made
in accordance with Ref.1. For this purpose
we reduce the stated problem to classic
form by combinin the criterion (5) and
the restriction 73);
where- d is Iiakrange multiplier defined
by equation decision (3) at an optimal
The decision of the problems (I),
( 2 ) , (6) is stated in the so called partition theorem I. The main point of the
theorem is that the initial problem is
divided into two ones: the filtration
problem and the optimal control synthesis
problem to complete data. The first problem is solved by Kalman filter equations
The initial conditions in ( 7 ) are
defined by a priori statistic information
According to observation data (2)
the system control problem (I) will be
Copyright O American Institute of Aeronautics and
Astronautics. lnc., 1989. All [ights reserved.
abtuk the initial state of the system (I):
%, PO*=Po
We note that :
X is
xi afposterior evaluation of the,state
ter measurin in i moment, XX;~
a priori
gvaluat ion,
M[(%i-%i*) (CXi-?$*)ll
pi=~[(~b-~i,)(&,-&'jT]are poster lor and
a priori covariance matrices correspondingly.
The second problem decision (the
control synthesis problem) is defined by
a linear law
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | | DOI: 10.2514/6.1989-3633
%i= - Li %i,-i,
where L; is a matrix m x n of feedback
coefficients. We note that here the posterior evaluation is a sufficient coordinate vector.
of different factors characterizing the
initial control problem on its meaning.
Thus in relation (T6: the first summand
means "contribution" to the criterion.
The "contribution" is limited by the system initial state mean deviation from the
zero value, the second summand is limited
by the a priori indeterminacy of the initial state, the third - by the random disturbance action in the system movement
by the filtration
channel, the forth
process errors.
Hence the observation program choice
is easily seen to manifest itself
.n the forth summand value in (17).
Consequently the observation process optimization is made according to the criterion
i.s defined from the re-
Li = xiA;h,A;~
, Fi.=cA&€i+ Bzhi&,
Fh - $ sq;are root out of the matrix .
f? - ( B Z ~ B ~dh
We note that for
where a matrix parameter h i - ~ x n enters
the expression for the coming loss funct ion
it is necessary to
ma't&ix calculation
realize a recurrent procedure (11) that
descrjbes the evolution of a matrix
and it isa'svet by a recurrent relation
According to what has been said the
observation process opti.miza ion problem
is Riccati
equation control f1,
= (P;' + '$'~H:D$~
with a boundary condition h w = K
The scalar parameter C i in (10)
is also counted in the process of the recurrent relation use from right to left
with boundary conditions
CN =Q ,
Pw ,
Matrix $i is a covariance matrix
of disturbance &; in the equation for
the sufficient coordinate vector evolut ion
3 ~ ~ = A ~ - i 3 ~ * a + W i B i U i +i-i,N-i
According to
Xt P,,%zD$
&,I%: +D$)I$CJt$ 14)
By the coming loss function meaning (10)
the optimal value of the criterion (6) is
recorded as
After simple transformations -the criterion
(15) can be represented in the form
We note that such representation of
a conventional optimality criterion is
suitable for the analysis of the impact
in accordance with the criterion (17)
taking into account the restriction (4).
An Equivalent Problem. The problem
decision(I8), (I1/), ( 4 ) witfi large values
n will entail ereat difficulties. The lat-ter is conditizned by non-linearity of
a controlable system (18) and by its matrix structure. To overcome determined difficulties the problem decision will be made on the basis of its equivalent transformation to linear with the help
properties of Riccati equation
In the consideration we introduce
a N-bundle of Hamiltonian systems for
matrix variables Si-nxm, 0,;-n*m:
We get the initial conditions in
(19) as follows
I n accordance with analytic roperties of
Riccari equation varia les
satisfy an identity [
i = i , N , j-I$.
~ s =a!
E ,
If now in the system (19) we define a
boundary,condition at the right end
lecJed proceeding from the condition
and set a criterion
(30) define the corres onding boundary
J = ? + ~ \ I~\-")--min
then as itd' is shown
in 4 the I%
o ained
control problem (19), ( 2 0 ) , (22), ( 23) ,(4)
problem: in system (197 it is nec,essry to
choose such initial matrices Sd, i N
that to transfer the part.i$ularsysiem
from thy state (59 :
,Sd), i,N in t h e
(where pa.- is free), &i,N
state (l, {
of the coAkrol determl ed
with thd
from conditions (29), (30; under the rgstriction at the control resource Nx=Nz,
c Ti,=Rz.
will be equivalent to the initial problem
(181, (171, (4).
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | | DOI: 10.2514/6.1989-3633
The equivalence is proved primarily
on the basis of a e n t i t y ;21), written in
= ilN)
and on the bamoments d-1
sis of restriction (22):
The relations (Ig), ( 2 0 ) , (22), (29),
A Numerical Algorithm of Optimization. ?he boundary problem obtained above
solved on the basis of KrylovChernousko method. In Ref. 4 the decision
is reduced to the realization of the following succession of actions.
I. It is set an initial (standard) observation programi~ (the upper index is an
iteration number
To find an optimal control \ i
Xiccause a discrete maximum principle. n accor- 2. According to the program
ti equation (18) is solved and matrices
dance with the principle we write down
of the dimension n x m are defined:
multiplier limit'ed
+ fiI+&
latter bein
0.n m e boundary, i.e. & P O
d=i,J(- conjugate variables of the dlmensi0.n n x , m corresponding to the matrices
, Qc,.
The equation for the conjugate matl$
rix variables is yVrS6
,, a i
A?c-i y.jL+ F, F: hi ( 26
A C - ~ V +~ r i ~D$H.iyii
3. The trajector;es
formed on the basis of the ystem decito leoft a$ thebounsion (19) from rig?$
Q -XI ,
dary conditions $4 -
1-1- 1
4. In accordance with formula (30) a program succession {M;] is defined.
5 A n observation program \b\yl is defined from the Hamiltonian maximization condition (29).
Boundar. Sbndit ions in (26) are defined in
moments i = j T
where the param&er @ is'a fraction of
a measurey%nt number corresponding to the
land chosen out of iti
formulation of a new program
respondingly the value (1-w) is
rement fraction out of the program (qc)
Thus later on.we can operate only on variables 58 Q $ .
Proceeding from the Hamiltonian maximization (25) to
we find an optimal
control structure.
the very maximum values. The same number
of measurement moments with minimal vaar removed from the renewal
lues \h0]
program 16:\.
The parameter
the condition
ment s
is chosen out of
M ~ = ~ ~-~a=i
, \ z Q ~ H : D < , H . ~ Q ~ ? ) )'3(
achieves ita Nc largest values. In other
words )$i = I , ~f R%)I$ where & is se-
an equivalent crite ion yalue-defin"ed'according to (23) a s ~ ~ \ ~ ~ where
~ ~ ~ t ~ . ~
lculated on he basis
xi (Po)
of action 2 at the new program
Then actions 2 t 5 are repeated with
a change of iteration number "0" to number
11 I It
"I" - "2" etc.
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | | DOI: 10.2514/6.1989-3633
As it is shown in Ref. 4 the ite&ated
algorithm described above is convergent
in a criterion. In other words in some
number of iterations a practical decision
of observation process optimization problem will be achieved.
A numerical algorithm under consideration is realized in the form of an application package for observation problem
decision in arbitrary linear systems.
A user can set a particular model of a
controlable object in an interactive mode,
as well as the necessary initial data.
The algorithm efficiency was shown in the
process of the specific technical problem
decision. Thus in a computer simulation
(with a speed of 300 000 operations per
second) of the observation optimization
problems in the movin objects control
(dimension n = 6 + 307 the processing
time expenditures made up I + 3 minutes.
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