DECISION SUPPGRT SYSTEM IN THE P3OSLEM OF OBSERVATION PROCESS OPTIMIZATION 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 | http://arc.aiaa.org | DOI: 10.2514/6.1989-3633 Abstract 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, matrices M[Ti f,f] = D l i i Ai,-i , Bi , FL n m , nxm, n x t . 1- The initial state of the system (I) is a s ~ u m e d ~ t h a%,is t Gauss vector, i.e. GEN~~P ?o).~ , The system movement observation (2) is made by carring out the measurements Yi = yiHi13Ci + \i , i= 1,N, (2) Yi, 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, , r Hence, I. Ti = I, in a moment i the measurement is made, if 0 , it isn't made. where w, is a matrix m x n, as well as the restrictions on maxl.mum admissible amount of measurement moments %=i 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 program. 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 control [U~I. 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. 1681 abtuk the initial state of the system (I): ZO= %, 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 and pi=~[(~b-~i,)(&,-&'jT]are poster lor and a priori covariance matrices correspondingly. . 6~= The second problem decision (the control synthesis problem) is defined by a linear law 3' Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1989-3633 %i= - Li %i,-i, (8) where L; is a matrix m x n of feedback coefficients. We note that here the posterior evaluation is a sufficient coordinate vector. Matrix lations Li 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 u i.s defined from the re- (9) Li = xiA;h,A;~ , Fi.=cA&€i+ Bzhi&, Fh - $ sq;are root out of the matrix . f? - ( B Z ~ B ~dh +Ci). We note that for rt 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, A .!?I %aL, = (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 , A = Pw , Matrix $i is a covariance matrix of disturbance &; in the equation for the sufficient coordinate vector evolut ion - 3 ~ ~ = A ~ - i 3 ~ * a + W i B i U i +i-i,N-i &i, According to - (13) [11 Xt P,,%zD$ &,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 ,78lYtic 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 Qi Riccari equation varia les satisfy an identity [ 4 (21) i = i , N , j-I$. ~ s =a! E , f gi, If now in the system (19) we define a boundary,condition at the right end lecJed proceeding from the condition and set a criterion (30) define the corres onding boundary J = ? + ~ \ I~\-")--min .~ (23) ' then as itd' is shown in 4 the I% o ained control problem (19), ( 2 0 ) , (22), ( 23) ,(4) problem: in system (197 it is nec,essry to choose such initial matrices Sd, i N that to transfer the part.i$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. 2=i Y will be equivalent to the initial problem (181, (171, (4). Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | 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= 4- A Numerical Algorithm of Optimization. ?he boundary problem obtained above -be 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 we 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: P F;SA)] 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 a , Qc,. The equation for the conjugate matl$ rix variables is yVrS6 );vh q., ,, a i IT&= A?c-i y.jL+ F, F: hi ( 26 A C - ~ V +~ r i ~D$H.iyii l +W = p-:- \r(] U'o 1,4-rn 3. The trajector;es are formed on the basis of the ystem decito leoft a$ thebounsion (19) from rig?$ Q -XI , dary conditions $4 - 1-1- 1 bi- i=i#, 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). 2r 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 program formulation of a new program respondingly the value (1-w) is rement fraction out of the program (qc) formulation. \ti Thus later on.we can operate only on variables 58 Q $ . rk Proceeding from the Hamiltonian maximization (25) to we find an optimal control structure. moments 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 - . N Exi p0 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 ~ . ~ is lculated on he basis xi (Po) of action 2 at the new program \tii& Then actions 2 t 5 are repeated with a change of iteration number "0" to number 11 I It "I" - "2" etc. 3 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | 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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