Open access peer-reviewed chapter

Stochastic Observation Optimization on the Basis of the Generalized Probabilistic Criteria

By Sergey V. Sokolov

Submitted: December 16th 2011Reviewed: January 20th 2012Published: November 28th 2012

DOI: 10.5772/39266

Downloaded: 874

1. Introduction

Till now the synthesis problem of the optimum control of the observation process has been considered and solved satisfactorily basically for the linear stochastic objects and observers by optimization of the quadratic criterion of quality expressed, as a rule, through the a posteriori dispersion matrix [1-4]. At the same time, the statement of the synthesis problem for the optimum observation control in a more general case assumes, first, a nonlinear character of the object and observer, and, second, the application of the non-quadratic criteria of quality, which, basically, can provide the potentially large estimation accuracy[3-6].

In connection with the fact that the solution of the given problem in such a statement generalizing the existing approaches, represents the obvious interest, we formulate it more particularly as follows.

2. Description of the task

Let the Markovian vector process t, described generally by the nonlinear stochastic differential equation in the symmetrized form

ξ˙t=f(ξ,t)+f0(ξ,t)nt,ξ(t0)=ξ0,E1

where f, f0 are known N – dimensional vector and N×M – dimensional matrix nonlinear functions;

nt is the white Gaussian normalized M – dimensional vector - noise; be observed by means of the vector nonlinear observer of form: Z=H(ξ,t)+Wt,

where Z – L N – dimensional vector of the output signals of the meter;

h(,t) – a known nonlinear L- dimension vector - function of observation;

Wt – a white Gaussian L- dimension vector - noise of measurement with the zero average and the matrix of intensityDW.

The function of the a posteriori probability density (APD) of process ρ(ξ,t)=ρ(ξ,t|Zτ,τ[t0,t])is described by the known integro-differential equation in partial derivatives (Stratonovich equation), the right-hand part of which explicitly depends on the observation function h: ρ(ξ,t)t=L{ρ(ξ,t)}+[QQ0]ρ(ξ,t),

where L{ρ(ξ,t)}=div{[f+12f0ξ(f0T)(V)]ρ}+12div{div¯[f0f0Tρ]}– the Focker-Plank- operator,

(А)(V) is the operation for transforming the n×m matrix A into vector (А)(V) formed from its elements as follows: A(V)=|a11a21am1a12a22am2a1na2namn|T,

div¯is the symbol for the operation of divergence of the matrix row, Q=Q(ξ,t)=12[ZH(ξ,t)]TDW1[ZH(ξ,t)],Q0=Q(ξ,t)ρ(ξ,t)dξ.

As the main problem of the a posteriori analysis of the observable process t is the obtaining of the maximum reliable information about it, then the synthesis problem of the optimum observer would be natural to formulate as the definition of the form of the functional dependence h(,t), providing the maximum of the a posteriori probability (MAP) of signal t on the given interval of occurrence of its values ξ*=[ξmin,ξmax]during the required interval of time T = [t0, tk], i.e. in view of the positive definiteness (,t) max{J=Tξ*ρ(ξ,t)dξdt}

Or
min{Tξ*ρ(ξ,t)dξdt}.E2

Generally instead of criterion MAP one can use, for example, the criterion of the minimum of the a posteriori entropy on interval ξ*=[ξmin,ξmax]or the criterion of the minimum of the integrated deviation of the a posteriori density from the density of the given form etc., that results in the need for representing the criterion of optimality J in the more generalized form:J=TξΦ[ρ(ξ,t)]dξdt,

where Ф is the known nonlinear function which takes into account generally the feasible analytical restrictions on the vector t;

T = [t0, tk] is a time interval of optimization;

* is some bounded set of the state parameters t.

In the final forming of structure of the criterion of optimality J it is necessary to take into account the limited opportunities of the practical realization of the function of observation h(,t), as well, that results, in its turn, in the additional restriction on the choice of functional dependence h(,t). The formalization of the given restriction, for example, in the form of the requirement of the minimization of the integrated deviation of function Н from the given form Н0 on interval ξ*during time interval Т allows to write down analytically the form of the minimized criterion J as follows:

J=Tξ*Φ[ρ(ξ,t)]dξdt+Tξ*[H(ξ,t)H0(ξ,t)]T[H(ξ,t)H0(ξ,t)]dξdt=TW*(t)dt.E3

Thus, the final statement of the synthesis problem of the optimum observer in view of the above mentioned reasoning consists in defining function h(,t), giving the minimum to functional (2).

3. Synthesis of observations optimal control

Function APD, included in it, is described explicitly by the integro-differential Stratonovich equation with the right-hand part dependent on h(,t). The analysis of the experience of the instrument realization of the meters shows, that their synthesis consists, in essence, in defining the parameters of some functional series, approximating the output characteristic of the device projected with the given degree of accuracy. As such a series one uses, as a rule, the final expansion of the nonlinear components of vector h(,t) in some given system of the multidimensional functions: power, orthogonal etc.

Having designated vector of the multidimensional functions as|ψ1...ψS|T=ψ, we present the approximation of vector h(,t) as

H(ξ,t)=(EψT)h=ψEh,h=|h11...h1Sh21...h2S...hN1...hNS|T,E4

where hi(ξ,t)=j=1Shij(t)ψj(ξ)is the i-th component of vector h, the factors of which define the concrete technical characteristics of the device,

is the symbol of the Kronecker product.

For the subsequent analytical synthesis of optimum vector - function h(,t) in form of (3) we rewrite the equation of the APD (,t) in the appropriate form

ρt=L[ρ]+hTH1[ρ]hTH2[ρ]h,E5

Where H1[ρ]=[ψETξ*ψET(ξ)ρ(ξ,t)dξ]DW1Zρ(ξ,t),

H2[ρ]=ρ(ξ,t)2[ψETDW1ψEξ*ψET(ξ)DW1ψE(ξ)ρ(ξ,t)dξ].E6

The constructions carried out the problem of search of optimum vector h(,t) is reduced to the synthesis of the optimum in-the- sense -of-(2) control h of the process with the distributed parameters described by Stratonovich equation (in view of representing vector Н0(,t) in the form similar to (3) H0(ξ,t)=ψEh0).

The optimum control of process (,t) will be searched in the class of the limited piecewise-continuous functions with the values from the open area Н*. For its construction we use the method of the dynamic programming, according to which the problem is reduced to the minimization of the known functional [1]

minhH*{dVdt+W*}=0E7

under the final condition V(tk) = 0 with respect to the optimum functional V = V(,t), parametrically dependent on t [t0, tk] and determined on the set of functions satisfying (4).

For the processes, described by the linear equations in partial derivative, and criteria of the form of the above-stated ones, functional V is found in the form of the integrated quadratic form [1], therefore in this case we have: V=ξ*v(ξ,t)ρ2(ξ,t)dξ.

Calculating derivative dVdtdVdt=ξ*(dvdtρ2+2vρdρdt)dξ=ξ*(dvdtρ2+2vρL[ρ]+2vρhTH1[ρ]2vρhTH2[ρ]h)dξ,

the functional equation for v is obtained in the following form: minhHξ*(dvdtρ2+2vρL[ρ]+2vρ(hTH1[ρ]hTH2[ρ]h)+Φ[ρ])dξ++(hh0)Tξ*ψET(ξ)ψE(ξ)dξ(hh0)=0,

whence we have optimum vector hоpt: hоpt={ξ*[ψET(ξ)ψE(ξ)vρ(H2+H2T)]dξ}1ξ*(ψET(ξ)ψE(ξ)h0vρH1)dξ=B(v,ρ)ξ*(ψ1(ξ)h0vρH1)dξ.

Using condition{dVdt+W*}h=hоpt=0, for v(,t) we have the following equation:dvdt=2vρ1L[ρ]ρ2(h0Tψ1ξBTξ*vρH1TdξBT)(2vρH1ψ1h0)++ρ2h0Tψ1(Bψ1ξh0Bξ*vρH1dξ)+ρ2(h0Tψ1ξBTξ*vρH1TdξBT)×

×(2vρH2ψ1)(Bψ1ξh0Bξ*vρH1dξ)ρ2h0Tψ1h0ρ2Φ[ρ],E8

Where

ψ1ξ=ξ*ψ1(ξ)dξ,E9

which is connected with the equation of the APD, having after substitution into it expression hоptthe following form: dρdt=L[ρ]+(h0Tψ1ξBTξ*vρH1TdξBT)H1

(h0Tψ1ξBTξ*vρH1TdξBT)H2(Bψ1ξh0Bξ*vρH1dξ).E10

4. Observations suboptimal control

The solution of the obtained equations (6), (7) exhausts completely the problem stated, allowing to generate the required optimum vector - function h of form (3). On the other hand, the solution problem of system (6), (7) is the point-to-point boundary-value problem for integrating the system of the integro-differential equations in partial derivatives, general methods of the exact analytical solution of which, as it is known, does not exist now. Not considering the numerous approximated methods of the solution of the given problem oriented on the trade-off of accuracy against volume of the computing expenses, then as one of the solution methods for this problem we use the method based on the expansion of function v, p in series by some system of the orthonormal functions of the vector argument : V(ξ,t)=μαμ(t)ϕμ(ξ)=ϕTα,ρ(ξ,t)=μβμ(t)ϕμ(ξ)=ϕTβ,

where is the index running a set of values from (0,...,0) to (М,...,М) [2];

is the vector of the orthonormal functions of argument ;

are vectors of factors of the appropriate expansions.

In this case the solution is reduced to the solution of the point-to-point boundary-value problem for integrating the system of the following equations, already ordinary ones:

β˙=ξ*ϕL[ϕTβ]dξ+ξ*ϕ[h0ψ1ξBT(α,β,ϕ)ξ*ϕTαϕTβH1T(ϕTβ)dξBT(α,β,ϕ)]H1(ϕTβ)dξξ*ϕ[h0Tψ1ξBT(α,β,ϕ)ξ*ϕTαϕTβH1T(ϕTβ)dξBT(α,β,ϕ)]H2(ϕTβ)(B(α,β,ϕ)ψ1ξh0B(α,β,ϕ)ξ*ϕTαϕTβH1(ϕTβ)dξ)dξ,ξ*ϕTαϕTβH1T(ϕTβ)dξBT(α,β,ϕ)](2ϕTαϕTβH1(ϕTβ)ψ1h0)++(ϕTβ)2h0Tψ1(B(α,β,ϕ)ψ1ξh0B(α,β,ϕ)ξ*ϕTαϕTβH1(ϕTβ)dξ)++(ϕTβ)2[h0Tψ1ξBT(α,β,ϕ)ξ*ϕTαϕTβH1T(ϕTβ)dξBT(α,β,ϕ)]××(2ϕTαϕTβH2(ϕTβ)ψ1)(B(α,β,ϕ)ψ1ξh0B(α,β,ϕ)ξ*ϕTαϕTβH1(ϕTβ)dξ)(ϕTβ)2h0Tψ1h0(ϕTβ)2Φ[ϕTβ]}dξE11

under boundary value conditionsα(T)=0,β(t0)=β0, where the values of the components are defined from the expansion of functionρ(ξ,t0)=ρ0.

From the point of view of the practical realization the integration of system (8) under the boundary-value conditions appears to be more simple than integration (6), (7), but from the point of view of organization of the estimation process in the real time it is still hindered: first, the volume of the necessary temporary and computing expenses is great, secondly the feasibility of the adjustment of the vector of factors h in the real time of arrival of the signal of measurement Z - is excluded, the prior simulation of realizations Z appears to be necessary (in this case in the course of the instrument realization, as a rule, one fails to maintain the precisely given values h all the same). Thus, the use of the approximated methods of the problem solution (8) is quite proved in this case, then as one of which we consider the method of the invariant imbedding [3], used above and providing the required approximated solution in the real time.

As the application of the given method assumes the specifying of all the components of the required approximately estimated vector in the differential form, then for the realization of the feasibility of the synthesis of vector h through the given method in the real time we introduce a dummy variable v, allowing to take into account from here on expression hopt as the differential equation v˙=hopt(ϕTα,ϕTβ),

forming with equations (8) a unified system. The application of the method of the invariant imbedding results in this case in the following system of equations: |v˙0β˙0|=|h0ξ*ϕ[ϕTβ0]+h0T(H1[ϕTβ0]H2[ϕTβ]h0)dξ|Dξ*ϕ(ϕTβ0)2Φ[ϕTβ0]dξ,D˙=2ξ*ϕ(β0{L[ϕTβ0]}+h0TH1β0[ϕTβ0]h0TH2β0[ϕTβ0]h0)dξD++2Dξ*ϕ(ϕTβ0)1(ϕTL[ϕTβ0]h0T(ϕTH2[ϕTβ0])h0)dξξ*ϕ(H1T[ϕTβ0]2h0TH2[ϕTβ0])dξμ++2Dξ*ϕ(ϕTβ0)2(2(ϕTβ0)1ϕTΦ[ϕTβ0]β0Φ[ϕTβ0])dξD,μ=2ψ1ξ1ξ*(ϕTβ0)[(ϕTH2)h012H1ϕT]dξ.

By virtue of the fact that matrix D in the method of the invariant imbedding plays the role of the weight matrix at the deviation of the vector of the approximated solution from the optimum one, in this case for variables i0 the appropriate components D characterize the degree of their deviation from the factors of expansion of the true APD (components D0 - are deviations of the parameters at the initial moment). The essential advantage of the approach considered, despite the formation of the approximated solution, is the feasibility of the synthesis of the optimum observation function in the real time, i.e. in the course of arrival of the measuring information.

5. Example

For the illustration of the feasibility of the practical use of the suggested method the numerical simulation of the process of forming vector h=|h1h2|Tof factors of the observer Z=h1ξ+h2ξ2+Wtfor target ξ˙=ξ3+ntwas carried out the normalized Gaussian white noises of the target and meter. As the criterion of optimization the criterion of the maximum of the a posteriori probability of the existence of the observable process on interval ξ*= = [-2.5, 2.5] was chosen that provided the additional restriction in the form of the requirement of the minimal deviation of vector h from the given vector h0=|0.95,0.3|Tthat allows to write down the minimized functional as J=Tξ*ρ(ξ,t)dξdt+T(hh0)TDH(hh0)dt,

Where

DH=ξ*|ξξ2||ξξ2|dξ=|10.40039.1|,T=|0;600|.E12

In this case the equation of the APD has the form ρt=ξ(ξ3ρ)+122ρξ2+hTH1hTH2h,

Where H1=(|ξξ2|ξ*|ξξ2|ρ(ξ,t)dξ)Zρ,

H2=ρ2(|ξ2ξ3ξ3ξ4|ξ*|ξ2ξ3ξ3ξ4|ρ(ξ,t)dξ).E13

The optimum vector h is defined from expression hopt as hоpt=[DHξ*Vρ2(|ξ2ξ3ξ3ξ4|ξ*|ξ2ξ3ξ3ξ4|ρ(ξ,t)dξ)dξ]1××(DHh0Zξ*Vρ2[|ξξ2|ξ*|ξξ2|ρ(ξ,t)dξ]dξ).

Using the Fourier expansion up to the 3-rd order for the approximated representation of functions V, V(ξ,t)=12α0+k=12α1kcoskω0ξ+α2ksinkω0ξ,ρ(ξ,t)=k=12β1kcoskω0ξ+β2ksinkω0ξ,ω0=2π5,(then for v2 the following representation holds truevρ2(ξ,t)=γ0+k=16γ1kcoskω0ξ+γ2ksinkω0ξ,γ0,γik=γ(α,β)are functions linearly dependent on factors αikand quadratically - onβik)

and introducing designationsζC(k,n)=ξ*ξncoskω0ξdξ=2i=1,3ni!Cin(2.5)ni(kω0)i+1sin(kπ+iπ2),n=2;4;ζS(k,m)=ξ*ξmsinkω0ξdξ=2i=0,2mi!Cim(2.5)mi(kω0)i+1cos(kπ+iπ2),m=1;3;vector hopt of the factors of the observer we write down as follows:

hopt=(DH+|γ0(5k=12β1kζC(k,2)10,4)k=16γ1kζC(k,2)5γ0k=12β2kζS(k,3)k=16γ2kζS(k,3)E14
5γ0k=12β2kζS(k,3)k=16γ2kζS(k,3)γ0(5k=12β1kζC(k,4)39,1)k=16γ1kζC(k,4)|)1×E15
×(DHh0Z|k=16γ2kζS(k,1)5γ0k=12β2kζS(k,1)k=16γ1kζC(k,2)5γ0k=12β1kζC(k,2)|)=h(α,β).E16

Then the system of equations for the factors of expansion has the following form:

β˙1(2)i=12K=12β1(2)K(3[ζC(ki,2)±ζC(k+i,2)]E17
kω0[ζS(k+i,3)+ζS(ki,3)])(iω0)22β1(2)i+E18
+ZhT(α,β)|K=12β2(1)K12[ζS(k+i,1)+ζS(ki,1)]β2KζS(k,1)β1(2)iK=12β1(2)K12[ζC(ki,2)±ζC(k+i,2)]β1KζC(k,2)β1(2)i|E19
12hT(α,β)|K=12β1(2)K12[ζC(ki,2)±ζC(k+i,2)]β1KζC(k,2)β1(2)iK=12β2(1)K12[ζS(k+i,3)+ζS(ki,3)]β2KζS(k,3)β1(2)iE20
K=12β2(1)K12[ζS(k+i,3)+ζS(ki,3)]β2KζS(k,3)β1(2)iK=12β1(2)K12[ζC(ki,4)±ζC(k+i,4)]β1KζC(k,4)β1(2)i|h(α,β),E21
β1i(0)=102,β2i(0)=0,i=1,2;E22
α˙0=2(3K=12α1KζC(k,2)+α1K[χC(k,β)+ωC(k,β)]+α2K[χS(k,β)+ωS(k,β)])E23
2ZhT(α,β)|K=12α2KζS(k,1)K=12α1KζC(k,2)|hT(α,β)|K=12α1KζC(k,2)K=12α2KζS(k,3)K=12α2KζS(k,3)K=12α1KζC(k,4)|×E24
×h(α,β)+μ(β)(hh0)Tμ1(β)(hh0),E25
α˙1(2)i=2{32K=12α1(2)K[ζC(ki,2)±ζC(k+i,2)]+E26
+α1K[χ1(2)C(k,i,β)+ω1(2)C(k,i,β)]+α2K[χ1(2)S(k,i,β)+ω1(2)S(k,i,β)]}E27
2ZhT(α,β)(|K=12α2(1)K12[ζS(k+i,1)+ζS(ki,1)]K=12α1(2)K12[ζC(ki,2)±ζC(k+i,2)]|α1(2)i|K=12β2KζS(k,1)K=12β1KζC(k,2)|)E28
hT(α,β)(|K=12α1(2)K12[ζC(ki,2)±ζC(k+i,2)]K=12α2(1)K12[ζS(k+i,3)+ζS(ki,3)]E29
K=12α2(1)K12[ζS(k+i,3)+ζS(ki,3)]K=12α1(2)K12[ζC(ki,4)±ζC(k+i,4)]|E30
α1(2)i|K=12β1KζC(k,2)K=12β2KζS(k,3)K=12β2KζS(k,3)K=12β1KζC(k,4)|)h(α,β)+E31
+μC(S)(i,β)(hh0)Tμ1C(S)(i,β)(hh0),E32
α(tK)=0,E33

where the expressions of factors (determined by the numerical integration in the course of solving) aren’t given as complicated. In the reduced form the system obtained can be given as

β˙=Φβ[β,h(α,β)],E34
α˙=G1(α,h)+G2(β),E35
G2(β)=|μ(β)μC(1,β)μC(2,β)μS(1,β)μS(2,β)|T.E36

The approximated solving of the given boundary-value problem by the method of the invariant imbedding results in the required system of the equations allowing to carry out simultaneously the definition of vector hopt and formation of vector in the real time:

|ν˙0β˙0|=|h0Φβ(β0,h0)|+DG2(β0),E37
D˙=2Φββ(β0,h0)DΦβα(β0,α)|α=0+2DG2β(β0)DDG1α(α,β0,h0)|α=0.E38

The integration of the given system was made by the Runge-Kutta method on interval [0; 600] s. with the step equal to 0,05 s.

For the comparison of efficiency of the approach suggested with that of the existing ones the formation of the optimum- by-the- criterion-of-the-MAP estimation ξ^by two ways was carried out: on the basis of the MAP - filter with the linear observer [4] and by defining the maximum of the function of the APD, approximated by series ϕTβ0(where 0 is the solution of the last system of the estimation equations ), by means of the method of the random draft. The search of the maximum of the APD was carried out on the simulation interval [500; 600] s. for the estimations of vector 0, taken with interval 1 s. The generated test sample of dimension 100 was the normalized Gaussian sequence.

The calculation of the estimation errors was made by comparing the current values of estimations with the target coordinate and subsequent defining of the average values of the errors on interval [500; 600] s. Upon terminating the simulation interval the value of the average error obtained in this way for the estimation equations [4], using the linear observer, has exceeded the average estimation error carried out by the technique suggested, using the information of the optimum observer, by the factor of ~ 1,52.

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Sergey V. Sokolov (November 28th 2012). Stochastic Observation Optimization on the Basis of the Generalized Probabilistic Criteria, Stochastic Modeling and Control, Ivan Ganchev Ivanov, IntechOpen, DOI: 10.5772/39266. Available from:

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