ACTIVE PARAMETRIC IDENTIFICATION OF ... - Math-Net.Ru

MSC 93E12

DOI: 10.14529/mmp160208

ACTIVE PARAMETRIC IDENTIFICATION OF GAUSSIAN LINEAR DISCRETE SYSTEM BASED ON EXPERIMENT DESIGN V.M. Chubich, Novosibirsk State Technical University, Novosibirsk, Russian Federation, [email protected], O.S. Chernikova, Novosibirsk State Technical University, Novosibirsk, Russian Federation, [email protected], E.A. Beriket, Novosibirsk State Technical University, Novosibirsk, Russian Federation, [email protected]

The application of methods of theory of experiment design for the identication of dynamic systems allows the researcher to gain more qualitative mathematical model compared with the traditional methods of passive identication. In this paper, the authors summarize results and oer the algorithms of active identication of the Gaussian linear discrete systems based on the design inputs and initial states. We consider Gaussian linear discrete systems described by state space models, under the assumption that unknown parameters are included in the matrices of the state, control, disturbance, measurement, covariance matrices of system noise and measurement. The original software for active identication of Gaussian linear discrete systems based on the design inputs and initial states are developed. Parameter estimation is carried out using the maximum likelihood method involving the direct and dual procedures for synthesizing A- and D- optimal experiment design. The example of the model structure for the control system of submarine shows the eectiveness and appropriateness of procedures for active parametric identication. Keywords: parameter estimation; maximum likelihood method; Kalman lter; experiment design; (Fisher) information matrix.

Introduction Mathematical modelling is one of intensively developing scientic directions. Identication is an important element and the most dicult stage of the solution of the applied problem in various industries, transport, at calculation of systems of automatic control. In this regard development of software for parametrical identication models becomes especially important for fundamental science and practice. In terms of way of behavior present methods of identication can be divided into passive and active. In passive identication of dynamic systems only real operating signals and initial state are used in the modes of not disturbed operation [14]. On the contrary, active identication assumes violation of technological mode and uses some special synthesized designs [59]. So, in a frequency method of parameter estimation of linear stationary models [10, 11] the signal represents the sum of harmonics which number doesn't exceed dimension of space. In this work optimal plan of experiment is searched during the extreme problem solution for some preliminary chosen functional of information (or covariance) vector-matrix of parameters to be estimated. The diculties connected with necessity of violation of a technological mode are provided by increase of eciency and correctness of research. Methods of active identication give to the 90

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experimenter considerably greater possibilities in the design of qualitative model in comparison with methods of passive identication. These possibilities are caused by the ideology of the active identication based on a combination of traditional parameter estimation techniques with the concept of experiment design. Works [1215] are devoted to theoretical and applied aspects of the problem of active parametric identication of stochastic linear discrete systems based on design of input signals and initial states. The authors tried to generalize the results obtained in these papers and constructed new algorithms connected with the simultaneous design inputs and initial states. The questions of estimating the unknown parameters that we considered, together with the developed methods for designing experiments, enable us to achieve a general understanding of active identication.

1. Problem Denition Consider the following model of controllable, observable and identiable stochastic linear discrete system in the state space:

x (tk+1 ) = F (tk ) x (tk ) + Ψ (tk ) u (tk ) + Γ (tk ) w (tk ) ,

(1)

y (tk+1 ) = H (tk+1 ) x (tk+1 ) + v (tk+1 ) , k = 0, 1, ..., N − 1.

(2)

Here x(tk ) is the state n-vector; u(tk ) is the deterministic control (input) r-vector; w(tk ) is the process noise p-vector; y(tk+1 ) is the measurement (output) m-vector; v(tk+1 ) is the measurement noise m-vector. Suppose that

• the random vectors w(tk ) and v(tk+1 ) form stationary white Gaussian sequences with [ ] E [w (tk )] = 0, E w (tk ) wT (ti ) = Qδki , [ ] E [v (tk+1 )] = 0, E v (tk+1 ) v T (ti+1 ) = Rδki , [ ] E ν (tk ) wT (ti ) = 0, k, i = 0, 1, ..., N − 1 (E[·] denotes the mathematical expectation, δki is the Kronecker symbol);

• the initial state x(t0 ) is normally distributed with parameters { } E [x (t0 )] = x¯ (t0 ) , E [x (t0 ) − x¯ (t0 )] [x (t0 ) − x¯ (t0 )]T = P (t0 ) and is uncorrelated with w (tk ) and v (tk+1 ) for all values of k ;

• matrices F (tk ), Ψ(tk ), Γ(tk ), H(tk+1 ), Q, R depend on unknown parameters Θ = (θ 1 , θ 2 , . . . , θs ) ∈ ΩΘ . For the mathematical model (1), (2) taking the listed a priori assumptions into account, we need to develop an active parametric identication procedure based on optimal design of the input signals and initial states and study its eciency and reasonability of application. In such mathematical statement this problem is considered and solved for the rst time. Âåñòíèê ÞÓðÃÓ. Ñåðèÿ ≪Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå è ïðîãðàììèðîâàíèå≫ (Âåñòíèê ÞÓðÃÓ ÌÌÏ). 2016. Ò. 9,  2. Ñ. 90102

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2. Procedure of Active Parametric Identication Procedure of active parametric identication of systems with preliminary chosen model structure assumes performance of following stages: calculation of parameters estimates with use of measuring data corresponding to some experiment design (stage 1, g. 1); synthesis of experiment design based on the received estimates and chosen criterion optimality (stage 2, g. 1) and recalculation of estimates of unknown parameters on the measuring data corresponding to the optimal design (stage 3, g. 1).

Fig 1

. Scheme of active parametric identication

We will give more detailed description of stages of active parametric identication for model (1), (2) on the basis of design of the input signals and initial states. Stage 1. Estimation of unknown parameters of mathematical model is carried out according to observation data using the criterion of identication χ. Gathering of numerical data occurs during the identication experiments which are carried out under some design ξν . Assume that experimenter can make ν system starts. He gives signal U1 on the input of the system with initial state x ¯1 (t0 ) for k1 times, signal U2 with initial state x¯2 (t0 ) for q k2 times etc., at last, signal Uq with ( q initial state ) x¯ (t0 ) for kq times. In this case discrete ∑ normalized experiment design ξν ki = ν is i=1

{ ξν =

} α1 , α2 , . . . , αq , αi ∈ Ωα , i = 1, 2, . . . , q, kq k1 k2 , , . . . , ν ν ν

(3)

where the point of design spectrum α has the following structure: [ ] [ ] α = U T ; x¯T (t0 ) = uT (t0 ) , uT (t1 ) , . . . , uT (tN −1 ) ; x¯T (t0 ) 92

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and restrictions on the conditions of carrying out experiment are dened by the bounded set Ωα = ΩU × Ωx¯(t0 ) ⊂ RN r+n . [ ] T T T Designate through Yi,jT = (y i,j (t1 )) , (y i,j (t2 )) , ..., (y i,j (tN )) the j -th output realization

(j = 1, 2, . . . , ki ), corresponding to input signal Ui with initial state x¯i (t0 ). Then as a result of carrying out an identication experiment for design ξν the following set will be generated {( ) } Ξ = αi , Yi,j , j = 1, 2, ..., ki , i = 1, 2, ..., q . The apriori assumptions stated in section II allow to use maximum likelihood (ML) method for parameter estimation. It's known that when performing certain conditions (regularity conditions), ML estimates possess such properties, important for practice, as asymptotic unbiasedness, consistency, asymptotic eciency and asymptotic normality. b for which According to this method it is necessary to nd such values of parameters Θ

b = arg min χ (Θ; Ξ) = arg min [− ln L (Θ; Ξ)] Θ Θ∈ΩΘ

(4)

Θ∈ΩΘ

and (see, for example, [12, 14, 15]) q ki N −1 −1 ∑[ ∑ ]T −1 [ i,j ] νN N mν 1 ∑∑ i,j ε (tk+1 ) B (tk+1 ) ε (tk+1 ) + ln det B(tk+1 ), χ(Θ; Ξ) = ln 2π+ 2 2 i=1 j=1 k=0 2 k=0

where ε i,j (tk+1 ) and B (tk+1 ) are dened according to corresponding recursive equations of discrete Kalman lter: ⌢i,j

⌢i,j

x (tk+1 |tk ) = F (tk )x (tk |tk ) + Ψ(tk )ui (tk ) ;

P (tk+1 |tk ) = F (tk )P (tk |tk ) F T (tk ) + Γ(tk )QΓT (tk ); ybi,j (tk+1 |tk ) = H(tk+1 )b xi,j (tk+1 |tk ) ; ⌢ i,j

ε i,j (tk+1 ) = y i,j (tk+1 ) − y

(tk+1 |tk ) ;

B (tk+1 ) = H(tk+1 )P (tk+1 |tk ) H T (tk+1 ) + R; K (tk+1 ) = P (tk+1 |tk ) H T (tk+1 )B −1 (tk+1 ) ; ⌢i,j

⌢i,j

x (tk+1 |tk+1 ) = x (tk+1 |tk ) + K (tk+1 ) εi,j (tk+1 ) ; P (tk+1 |tk+1 ) = [I − K (tk+1 ) H(tk+1 )] P (tk+1 |tk ) , ⌢ i,j

for k = 0, 1, ..., N −1, i = 1, 2, ..., ki , j = 1, 2, ..., q with initial conditions x (t0 |t0 ) = x ¯ (t0 ), P (t0 |t0 ) = P (t0 ). Stage 2. Under continuous normalized design ξ we will understand the collection

{ ξ=

} q ∑ α1 , α2 , . . . , αq , p i ≥ 0, p i = 1, αi ∈ Ωα , i = 1, 2, . . . , q. p1 , p2 , . . . , pq

(5)

i=1

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For design (5) normalized information matrix M (ξ) is dened by

M (ξ) =

q ∑

( ) p i M αi ; θ ,

(6)

i=1

where (Fisher) information matrices for one-point designs M (αi ; θ) depend on unknown parameters (this fact allows us to speak further only about locally optimal design) and are calculated in accordance with the algorithm from [16]. The quality of parameter estimation can be improved by construction of experiment design which optimizes some convex functional X of information matrix M (ξ), i.e. we should solve the extreme problem

ξ ∗ = arg min X[M (ξ)]. ξ∈Ωξ

(7)

We will use criterion of D- and À- optimality for which, respectively, X[M (ξ)] = − ln det M (ξ) and X[M (ξ)] = −SpM −1 (ξ). Solving the problem of design experiments, we act in a certain way on lower bound in the Rao  Cramer inequality: for a D-optimal design we minimize the volume, and for an À-optimal design we minimize the sum of the squares of axis lengths of the concentration ellipsoid of the estimates for the parameters. We can solve the optimization problem of nding the minimum of (7) using the general numerical methods for nding extrema. The two approaches are possible. The rst (direct) approach is to seek the minimum of the functional X [M (ξ)] taking +1 points. Another approach (it into account that design spectrum (5) consist of q = s(s+1) 2 is called dual) to solving the optimal problem (7) is based on the equivalence theorem [5, 14]. In this case the problem under the study is no longer a convex programming problem, but the size of the parameter vector being varied can turn out substantially smaller than in the direct approach. Applying of the gradients in the direct or dual procedures leads to the signicant improve ment of the speed of problem solving. It is impossible without calculation of derivatives of (Fisher) information matrix by the components of input signal and initial state. In this case, the gradient of the optimality criterion for points of the spectrum plan has the following structure:  ∂X[M (ξ)]  ∂ui (t0 ) [ ] ∂X[M (ξ)]   ∂X [M (ξ)]  ···  ∂Ui = =  , i = 1, 2, ..., q.  ∂X[M (ξ)] ∂X[M (ξ)] ∂αi  ∂ui (tN −1 )  ) ∂x ¯i (t0 ) ∂X[M (ξ)] ∂x ¯i (t0 )

(ξ)] (ξ)] The algorithms for the calculation of gradients ∂X[M and ∂X[M for D  optimality ∂ui (tk ) ∂x ¯i (t0 ) and A  optimality criterions are given in [17, 18]. Stage 3. Recalculation of estimates of unknown parameters on the measuring data corresponding to the optimal design. Practical application of the constructed continuous optimum design is complicated because the weights p∗i are, generally speaking, arbitrary real numbers in the range from zero to one. It is easy to notice that in case of given number of possible starts of the system ν values ki∗ = νp∗i can be noninteger numbers. Carrying out of experiment demands

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a rounding o of values ki∗ to integers. It is obvious that the design received as a result of such rounding o will dier from the optimum continuous design. An approach to the optimum continuous design is better when the number of possible starts is larger. The possible algorithm of "rounding o" of the continuous design to the discrete design is given in [19]. Next, we make a discrete design { 1 2 } α∗ , α∗ , . . . , α∗q ∗ ξν = k1∗ k2∗ , kq∗ , , . . . , ν ν ν conduct the identication experiment and recalculate estimation of unknown parameters.

3. Experimental Results Consider the following mathematical model of Gaussian linear discrete control systems of the submarine making the movement with an angular speed ω and longitudinal speed of the course in the vertical plane at an angle of trim µ, taking into account a deviation of fodder horizontal wheels at an angle δ from neutral position (see g. 2) [20]:          ˙ ω(t) h 0 ω(t) θ1 θ2 θ3  ˙  (8) µ(t) =  1 0 0  µ(t) + 0 u(t) +  0  w (t) , ˙ 0.01h 1 δ(t) 0 0 0 δ(t)

y (tk+1 ) = µ (tk+1 ) + v (tk+1 ) , k = 0, 1, . . . , N − 1.

(9)

The discrete model of a state (8) with discretization step ∆T = 0, 5 is given by     ω(tk ) ω (tk+1 ) ψ (tk+1 ) = F µ(tk ) + Ψu (tk ) + Γw (tk ) , (10) δ(tk ) δ (tk+1 )

Fig 2

√   F = 

. The submarine in the vertical plane

θ12 +4θ2 ∗A−θ1 ∗B

√

2

θ12 +4θ2

√ −B 2

θ1 +4θ2

0

√

− √θ22∗B

θ1 +4θ2

θ12 +4θ2 ∗A+θ1 ∗B

√

2

θ12 +4θ2

0

θ3

√

− √θ32∗B

θ1 +4θ2

   

θ12 +4θ2 ∗(A−2)+θ1 θ3 ∗B  ;

√

2

θ12 +4θ2

1

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

Ψ

C−D √ 2





A= e

√−h∗B − 2



θ1√ θ3 h∗B 2θ2 θ1 +4θ2 θ1 +4θ2 200θ2 θ12 +4θ2  θ θ (D−C)   h(C−D)  θ1 θ3 h(C−D)  θ3 h(C+D) θ3  θ3 h C+D 3 1  =  4θ22 √θ12 +4θ2 − θ3 4θ22 − 2θ2  ; Γ =  2θ2 √θ12 +4θ2 − 200θ2 − 400θ22 − 400θ22 √θ12 +4θ2  ; h 0, 5 200 [ [ ( ) ( )] ( ) ( )] √ √ √ √ 0,25 θ1 − θ12 +4θ2 0,25 θ1 + θ12 +4θ2 0,25 θ1 − θ12 +4θ2 0,25 θ1 + θ12 +4θ2

θ3

θ3 h 100θ2

+

θ3 hA 200θ2

+

+e ; B= e −e ( )( ) ( ) √ √ 0,25 θ1 − θ12 +4θ2 2 C = θ1 + θ1 + 4θ2 e −1 ; ( )( ) ( ) √ √ 0,25 θ1 + θ12 +4θ2 2 D = θ1 − θ1 + 4θ2 e −1 .

;

Assume that −0, 16 ≤ θ1 ≤ −0, 06, −0, 006 ≤ θ2 ≤ −0, 001, −0, 006 ≤ θ3 ≤ −0, 001, N = 40 and all listed a priori assumptions are satised so that

E [w (tk ) w (ti )] = 5, 5δki = Qδki ;   0  E [v (tk ) v (ti )] = 0, 5δki = Rδki ; E [x (t0 )] = 4 = x¯ (t0 ) ; 0   0, 01 0 0 { } 0, 01 0  = P (t0 ) . E [x (t0 ) − x¯ (t0 )] [x (t0 ) − x¯ (t0 )]T =  0 0 0 0, 01 Choose the design area Ωα = ΩU × Ωx¯(t0 ) , where

ΩU = {−6 ≤ u (tk ) ≤ 6, k = 0, 1, ..., N − 1} ,     1   0    0 ≤ x¯(t0 ) ≤ 4 , Ωx¯(t0 ) =   4 0 and D-optimality criterion. In order to weaken the dependence of estimation results on measurement data we will make ve independent starts of system, i.e. (ν = 5) and then we will average the received estimates of unknown parameters. Realizations of output signal we get by the computer modelling with true values of parameters θ1∗ = −0, 1253, θ2∗ = −0, 004631, θ3∗ = −0, 002198. The quality of identication in the parameter space and response space will be judged by values of factors δθ , δθ∗ and δY , δY∗ accordingly. These factors are calculated by following formulas: v v )2 )2 u∑ u∑ s ( s (



u u ∗ ∗ ∗ ˆ ˆ

∗ ˆ ∗ ∗ θ − θi θ − θi u u

θ − θˆ

θ − θ u i=1 i u i=1 i ∗ =u ∑ =u ∑ δθ = ; δθ = ; s s t t ∥θ ∗ ∥ ∥θ ∗ ∥ (θi∗ )2 (θi∗ )2 i=1

i=1

v v u q ki N −1 u q ki N −1 ( )2 ∑ ∑ ∑ u u ∑ ∑ ∑ i,j 2 i,j i,j u u y (tk+1 ) − yˆ∗i,j (tk+1 |tk ) (y (tk+1 ) − yˆ (tk+1 |tk )) u i=1 j=1 k=0 u i=1 j=1 k=0 ∗ u δY =u ; δ = , Y u u q ∑ q ∑ ki N∑ ki N∑ −1 −1 ∑ ∑ t t 2 2 i,j i,j (y (tk+1 )) (y (tk+1 )) i=1 j=1 k=0

96

i=1 j=1 k=0

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where θ ∗ are true values of parameters; θˆ are estimates of unknown parameters values based on the initial experiment design; θˆ ∗ are estimates of unknown parameters values based on the synthesized input signal and (or) initial states; yˆ ij (tk+1 |tk ) and yˆ∗ij (tk+1 |tk ) are calculated on using the equations of Kallman lter at θˆ and θˆ∗ respectively. Results of the active parametric identication procedure based on D-optimal design of input signals and (or) initial state are presented in Table. Table

Result of active identication procedures based on design of input signals and (or) initial state Discrete experiment design 1 Initial { design } α1 , ξν = k1 =1 ν [ T T ] α1 = U ; x¯ (t0 ) , input signal U T = [(6, −6)2 , −64 , 63 , −6, 6, −62 , −6, 6, −62 , 62 , −6, 63 , −6, 6, −63 , (6, −6)3 , −6, 6, −63 ], initial state T x¯(t0 ) = (0, 4, 0) . Design states { of initial } 1 2 α α ∗ ξ ν = k1 , = 25 kν2 = 35 ν [ T ] αi = U ; (¯ xi (t0 )T ) , input signal U T = [(6, −6)2 , −64 , 63 , −6, 6, −62 , −6, 6, −62 , 62 , −6, 63 , −6, 6, −63 , (6, −6)3 , −6, 6, −63 ], initial states x¯1∗ (t0 ) = (0, 4, 0)T , x¯2∗ (t0 ) = (1, 4, 0)T . of input signals } {Design 1 α2 α3 α4 α ∗ ξν = k1∗ k∗ k∗ k∗ = 15 ν2 = 25 ν3 = 51 ν4 = 15 ν [ ] αi = (U∗i )T ; x¯T∗ (t0 ) , input signals (U∗1 )T = [640 ]1 , (U∗2 )T = [615 , −625 ], (U∗3 )T = [640 ], (U∗4 )T = [620 , −620 ], initial state x¯∗ (t0 ) = (0, 4, 0)T . Design input signals and initial states { } 1 α α2 α3 α4 ∗ ∗ ∗ ∗ ∗ ξν = k1 k k k = 25 ν2 = 15 ν3 = 51 ν4 = 15 ν ] [ xi∗ (t0 )T ) , input signals (U∗1 )T = αi = (U∗i )T ; (¯ 2 T [640 ], (U∗ ) = [624 , −616 ], (U∗3 )T = [612 , −628 ], (U∗4 )T = [625 , −615 ], initial states x¯1∗ (t0 ) = (1, 4, 0)T , x¯2∗ (t0 ) = x¯3∗ (t0 ) = x¯4∗ (t0 ) = (1, 4, 4)T .

1 Record

Values of parameters estimates and errors of estimation 2 θˆ1 = −0, 1600 θˆ2 = −0, 0059 θˆ3 = −0, 0015

δθ = 0, 2770, δY = 0, 1488 θˆ1 = −0, 1372 θˆ2 = −0, 0052 θˆ2 = −0, 0019 δθ = 0, 0953, δY = 0, 1049 θˆ1 = −0, 1301 θˆ2 = −0, 0049 θˆ3 = −0, 0019 δθ = 0, 0380, δY = 0, 0908 θˆ1 = −0, 1248 θˆ2 = −0, 0042 θˆ3 = −0, 0024 δθ = 0, 0055, δY = 0, 0750

[ak , bm , . . .] means that the number a repeats k times, number b − m times.

Âåñòíèê ÞÓðÃÓ. Ñåðèÿ ≪Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå è ïðîãðàììèðîâàíèå≫ (Âåñòíèê ÞÓðÃÓ ÌÌÏ). 2016. Ò. 9,  2. Ñ. 90102

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The calculations were performed using the interactive software system of the active parametric identication (APIS). APIS was developed by the authors at the Department of the theoretical and applied informatics NSTU and allows to solve problems of active parametric identication of nonlinear stochastic discrete and discrete-continuous systems based on planning of A- and D- optimal input signals using statistical and temporal linearization. Furthermore, the software system allows optimal estimation of model parameters for Gaussian linear dynamic systems based on the design of the input signals and (or) the initial states [21]. We executed ve starts of system, given a pseudorandom binary signal U . For each start at θ = θ∗ we simulate data measurements using equations (8), (10) Y = {y(t1 ), y(t2 ), ..., y(tN )}, write down Yˆ = {ˆ y (t1 |t1 ), yˆ(t2 |t2 ), ..., yˆ(tN |tN )}, ∗ ∗ ∗ ∗ ∗ ˆ yˆ (t2 |t2 ), ..., yˆ (tN |tN )}, Y = {ˆ y (t1 |t1 )} in accordance with θ = θˆcp and θ = θˆcp ; average ∗ ˆ ˆ the results Ycp , Ycp , Ycp , (see g. 3  6).

. Graphical representation of Ycp , Yˆcp (passive identication)

. Graphical representation of Ycp , Yˆcp∗ (design of the initial states)

Fig 3

Fig 4

. Graphical representation of Ycp , Yˆcp∗ (design of the input signals)

Fig 6

Fig 5

. Graphical representation ofYcp , Yˆcp∗ (design of the input signals and initial states)

The results presented in Table, show that design of the initial states led to better estimation only by 18,2% in the parameter space, and 4,4% in the space of responses, while design of the input signals contributed to the improvement of evaluation results only by 23,9% and 5,9%, respectively. The most successful results were obtained for the case of design of the inputs and initial states (improvement by 27,2% and 7,4%). Thus, the authors consider that applying of the active parametric identication procedures based on optimal design of input signals and initial states is helpful and advisable. 98

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Conclusion For the rst time we considered and solved the problem of active parametric identication of the Gaussian linear discrete systems in the case when the unknown parameters are contained in the state and control equations as in well as the covariance matrices of the system and measurement noise. The original gradient algorithms for active identication based on optimal design of input signals and initial states were developed. It was shown that design of the input signals and the initial states has a signicant impact on the functional dependence of state variables and model parameters. The work is executed under the auspices of the Ministry of education and science of

the Russian Federation ( 2014/138, project  1689).

References 1. Kash'jap R.L. Postroenie dinamicheskih stohasticheskih modelej po jeksperimental'nym dannym [Construction of Dynamic Stochastic Models Based on Experimental Data]. Moscow, Nauka, 1983. 384 p. 2. L'jung L. Identikacija sistem: Teorija pol'zovatelja [System Identication: Theory User]. Moscow, Nauka, 1991. 432 p. 3. Cypkin Ja.Z. Informacionnaja teorija identikacii [Information Theory of Identication]. Moscow, Nauka, 1995. 336 p. 4. Walter E., Pronzato L. Identication of Parametric Models from Experimental Data. Berlin, Springer-Verlag, 1997. 413 p. 5. Mehra R.K. Optimal Input Signals for Parameter Estimation in Dynamic Dystems: Durvey and New Results. IEEE Trans. on Automat. Control, 1974, vol. 19, no. 6, pp. 753768. DOI: 10.1109/TAC.1974.1100701 6. Krug G.K., Sosulin Ju.A., Fatuev V.A. Planirovanie jeksperimenta v zadachah identikacii i jekstrapoljacii [Planning Experiment in Problems of Identication and Extrapolation]. Moscow, Nauka, 1977. 208 p. 7. Ovcharenko V.N. Planning of Identies Input Signals in Linear Dynamic Systems. Automation and Remote Control, 2001, vol. 62, no. 2, pp. 236247. DOI: 10.1023/A:3A1002894223036 8. Jauberthie C., Denis-Vidal L., Coton P., Joly-Blanchard G. An Optimal Input Design Procedure. Automatica, 2006, vol. 42, pp. 881884. DOI: 10.1016/j.automatica.2006.01.003 9. Voevoda A.A., Troshina G.V. [Using the Fisher Information Matrix when Selecting a Control Signal for Estimating the Parameters of Dynamic Models and Observation of Objects of Low Order]. Sbornik nauch. trudov NGTU [Collection of Scientic Works NSTU], 2006, no. 3 (45), pp. 1924. (in Russian) 10. Alexandrov A.G. Finite-Frequency Method of Identication. Preprints of 10th IFAC Symposium on System Identication, 1994, vol. 2, pp. 523527. 11. Aleksandrov A.G., Orlov Ju.F. [Finite-Frequency Identication: Dynamic Algorithm]. Problemy upravlenija [Problems of Control], 2009, no. 4, pp. 28. (in Russian) 12. Denisov V.I., Chubich V.M., Chernikova O.S. [Active Identication of Stochastic Linear Discrete Systems in the Time Domain]. Journal of Applied and Industrial Mathematics, 2003, vol. 6, no. 3, pp. 7087. (in Russian) Âåñòíèê ÞÓðÃÓ. Ñåðèÿ ≪Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå è ïðîãðàììèðîâàíèå≫ (Âåñòíèê ÞÓðÃÓ ÌÌÏ). 2016. Ò. 9,  2. Ñ. 90102

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13. Denisov V.I., Chubich V.M., Chernikova O.S. Active Identication of Stochastic Linear Discrete Systems in the Frequency Domain. Journal of Applied and Industrial Mathematics, 2007, vol. 10, no. 1, pp. 183200. DOI: 10.1134/S1990478909020057 14. Denisov V.I., Chubich V.M., Chernikova O.S., Bobyleva D.I. Aktivnaja parametricheskaja identikacija stohasticheskih linejnyh sistem: monograja [Active Parametric Identication of Stochastic Linear Systems: Monograph]. Novosibirsk, NGTU, 2009. 192 p. 15. Chubich V.M., Chernikova O.S. [Optimal Parameter Estimation for Gaussian Models of Linear Discrete Systems Based on the Planning of the Initial Conditions]. Scientic Bulletin of NSTU, 2013, no. 3 (36), pp. 1522. (in Russian) 16. Chubich V.M. [Computation the Fisher Information Matrix for the Problem of Active Parametric Identication of Stochastic Nonlinear Discrete Systems]. Scientic Bulletin of NSTU, 2009, no. 1 (34), pp. 2340. (in Russian) 17. Chubich V.M. [Information Technology Active Parametric Identication of Quasi-Linear Discrete Systems]. Informatics and Applications, 2011, vol. 5, no. 1, pp. 4657. (in Russian) 18. Chubich V.M. [Planning the Initial Conditions in the Problem of Active Parametric Identication of Gaussian Linear Discrete Systems]. Scientic Bulletin of NSTU, 2011, no. 1 (42), pp. 3946. (in Russian) 19. Ermakov S.M., Zhigljavskij A.A. Matematicheskaja teorija optimal'nogo jeksperimenta [Mathematical theory of optimal experiment]. Moscow, Nauka, 1987. 320 p. 20. Veremej E.I. Linejnye sistemy s obratnoj svjaz'ju [Linear Systems with Feedback]. St. Petersburg, Lan', 2013. 448. p. 21. Chubich V.M., Chernikova O.S., Filippova E.V. [Software System Active Parametric Identication of Stochastic Dynamical Systems APIS]. XI Mezhdunarodnaja konferencija aktual'nye problemy jelektronnogo priborostroenija APJeP-2012 [XI International Conference on Actual Problems of Electronic Instrument Engineering APEIE-2012]. Novosibirsk, 2012, vol. 6, pp. 6673. (in Russian) Received April 24, 2015

ÓÄÊ 618.5.015

DOI: 10.14529/mmp160208

ÀÊÒÈÂÍÀß ÏÀÐÀÌÅÒÐÈ×ÅÑÊÀß ÈÄÅÍÒÈÔÈÊÀÖÈß ÃÀÓÑÑÎÂÑÊÈÕ ËÈÍÅÉÍÛÕ ÄÈÑÊÐÅÒÍÛÕ ÑÈÑÒÅÌ ÍÀ ÎÑÍÎÂÅ ÏËÀÍÈÐÎÂÀÍÈß ÝÊÑÏÅÐÈÌÅÍÒÀ Â.Ì. ×óáè÷, Î.Ñ. ×åðíèêîâà, Å.À. Áåðèêåò

Ïðèìåíåíèå ìåòîäîâ òåîðèè ïëàíèðîâàíèÿ ýêñïåðèìåíòà ïðè èäåíòèôèêàöèè äèíàìè÷åñêèõ ñèñòåì ïîçâîëÿåò èññëåäîâàòåëþ ïîëó÷èòü áîëåå êà÷åñòâåííóþ ìàòåìàòè÷åñêóþ ìîäåëü ïî ñðàâíåíèþ ñ òðàäèöèîííûìè ìåòîäàìè ïàññèâíîé èäåíòèôèêàöèè.

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ÌÀÒÅÌÀÒÈ×ÅÑÊÎÅ ÌÎÄÅËÈÐÎÂÀÍÈÅ

 ðàáîòå îáîáùàþòñÿ ðàíåå ïîëó÷åííûå àâòîðñêèå ðåçóëüòàòû è ïðåäëàãàþòñÿ àëãîðèòìû àêòèâíîé ïàðàìåòðè÷åñêîé èäåíòèôèêàöèè ãàóññîâñêèõ ëèíåéíûõ äèñêðåòíûõ ñèñòåì íà îñíîâå ñîâìåñòíîãî ïëàíèðîâàíèÿ âõîäíûõ ñèãíàëîâ è íà÷àëüíûõ óñëîâèé. Ðàññìàòðèâàþòñÿ ìàòåìàòè÷åñêèå ìîäåëè â ïðîñòðàíñòâå ñîñòîÿíèé, â ïðåäïîëîæåíèè, ÷òî ïîäëåæàùèå îöåíèâàíèþ ïàðàìåòðû ñîäåðæàòñÿ â ìàòðèöàõ ñîñòîÿíèÿ, óïðàâëåíèÿ, âîçìóùåíèÿ, èçìåðåíèÿ, â êîâàðèàöèîííûõ ìàòðèöàõ øóìîâ ñèñòåìû è èçìåðåíèé. Ðàçðàáîòàíî ïðîãðàììíî-ìàòåìàòè÷åñêîå îáåñïå÷åíèå, ïîçâîëÿþùåå ðåøàòü çàäà÷è àêòèâíîé ïàðàìåòðè÷åñêîé èäåíòèôèêàöèè ñ èñïîëüçîâàíèåì ìåòîäà ìàêñèìàëüíîãî ïðàâäîïîäîáèÿ, à òàêæå ïðÿìîé è äâîéñòâåííîé ãðàäèåíòíûõ ïðîöåäóð ïîñòðîåíèÿ À- è D- îïòèìàëüíûõ ïëàíîâ. Íà ïðèìåðå ìîäåëüíîé ñòðóêòóðû ñèñòåìû óïðàâëåíèÿ ïîäâîäíûì àïïàðàòîì ïîêàçàíà ýôôåêòèâíîñòü è öåëåñîîáðàçíîñòü ïðèìåíåíèÿ ïðîöåäóðû àêòèâíîé ïàðàìåòðè÷åñêîé èäåíòèôèêàöèè íà îñíîâå ñîâìåñòíîãî ïëàíèðîâàíèÿ âõîäíûõ ñèãíàëîâ è íà÷àëüíûõ óñëîâèé. Êëþ÷åâûå ñëîâà: îöåíèâàíèå ïàðàìåòðîâ; ìåòîä ìàêñèìàëüíîãî ïðàâäîïîäîáèÿ; ôèëüòð Êàëìàíà; ïëàíèðîâàíèå ýêñïåðèìåíòà; èíôîðìàöèîííàÿ ìàòðèöà.

Ëèòåðàòóðà 1. Êàøüÿï, Ð.Ë. Ïîñòðîåíèå äèíàìè÷åñêèõ ñòîõàñòè÷åñêèõ ìîäåëåé ïî ýêñïåðèìåíòàëüíûì äàííûì / Ð.Ë. Êàøüÿï, À.Ð. Ðàî.  Ì.: Íàóêà, 1983. 2. Ëüþíã, Ë. Èäåíòèôèêàöèÿ ñèñòåì: Òåîðèÿ ïîëüçîâàòåëÿ / Ë. Ëüþíã.  Ì.: Íàóêà, 1991. 3. Öûïêèí, ß.Ç. Èíôîðìàöèîííàÿ òåîðèÿ èäåíòèôèêàöèè / ß.Ç. Öûïêèí.  Ì.: Íàóêà, 1995. 4. Walter, E. Identication of Parametric Models from Experimental Data / E. Walter, L. Pronzato.  Berlin: Springer-Verlag, 1997. 5. Mehra, R.K. Optimal Input Signals for Parameter Estimation in Dynamic Systems: Survey and New Results / R.K. Mehra // IEEE Trans. on Automat. Control.  1974.  V. 19,  6.  P. 753768. 6. Êðóã, Ã.Ê. Ïëàíèðîâàíèå ýêñïåðèìåíòà â çàäà÷àõ èäåíòèôèêàöèè è ýêñòðàïîëÿöèè / Ã.Ê. Êðóã, Þ.À. Ñîñóëèí, Â.À. Ôàòóåâ.  Ì.: Íàóêà, 1977. 7. Îâ÷àðåíêî, Â.Í. Ïëàíèðîâàíèå èäåíòèôèöèðóþùèõ âõîäíûõ ñèãíàëîâ â ëèíåéíûõ äèíàìè÷åñêèõ ñèñòåìàõ / Â.Í. Îâ÷àðåíêî // Àâòîìàòèêà è òåëåìåõàíèêà.  2001.   2.  C. 7587. 8. Jauberthie, C. An Optimal Input Design Procedure / C. Jauberthie, L. Denis-Vidal, P. Coton, G. Joly-Blanchard // Automatica.  2006.  V. 42.  P. 881884. 9. Âîåâîäà, À.À. Èñïîëüçîâàíèå èíôîðìàöèîííîé ìàòðèöû Ôèøåðà ïðè âûáîðå ñèãíàëà óïðàâëåíèÿ äëÿ îöåíêè ïàðàìåòðîâ ìîäåëåé äèíàìèêè è íàáëþäåíèÿ îáúåêòîâ íåâûñîêîãî ïîðÿäêà / À.À. Âîåâîäà, Ã.Â. Òðîøèíà // Ñá. íàó÷. òð. ÍÃÒÓ.  Íîâîñèáèðñê, 2006.   3 (45).  C. 1924. 10. Alexandrov, A.G. Finite-Frequency Method of Identication / A.G. Alexandrov // Preprints of 10th IFAC Symposium on System Identication.  1994.  V. 2  P. 523527. 11. Àëåêñàíäðîâ, À.Ã. Êîíå÷íî-÷àñòîòíàÿ èäåíòèôèêàöèÿ: äèíàìè÷åñêèé àëãîðèòì / À.Ã. Àëåêñàíäðîâ, Þ.Ô. Îðëîâ // Ïðîáëåìû óïðàâëåíèÿ.  2009.   4.  Ñ. 28. 12. Äåíèñîâ, Â.È. Àêòèâíàÿ èäåíòèôèêàöèÿ ñòîõàñòè÷åñêèõ ëèíåéíûõ äèñêðåòíûõ ñèñòåì âî âðåìåííîé îáëàñòè / Â.È. Äåíèñîâ, Â.Ì. ×óáè÷, Î.Ñ. ×åðíèêîâà // Ñèáèðñêèé æóðíàë èíäóñòðèàëüíîé ìàòåìàòèêè.  2003.  Ò. 6,  3.  Ñ. 7087. 13. Äåíèñîâ, Â.È. Àêòèâíàÿ ïàðàìåòðè÷åñêàÿ èäåíòèôèêàöèÿ ñòîõàñòè÷åñêèõ ëèíåéíûõ äèñêðåòíûõ ñèñòåì â ÷àñòîòíîé îáëàñòè / Â.È. Äåíèñîâ, Â.Ì. ×óáè÷, Î.Ñ. ×åðíèêîâà // Ñèáèðñêèé æóðíàë èíäóñòðèàëüíîé ìàòåìàòèêè.  2007.  Ò. 10,  1.  Ñ. 7189. Âåñòíèê ÞÓðÃÓ. Ñåðèÿ ≪Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå è ïðîãðàììèðîâàíèå≫ (Âåñòíèê ÞÓðÃÓ ÌÌÏ). 2016. Ò. 9,  2. Ñ. 90102

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14. Àêòèâíàÿ ïàðàìåòðè÷åñêàÿ èäåíòèôèêàöèÿ ñòîõàñòè÷åñêèõ ëèíåéíûõ ñèñòåì: ìîíîãðàôèÿ / Â.È. Äåíèñîâ, Â.Ì. ×óáè÷, Î.Ñ. ×åðíèêîâà, Ä.È. Áîáûëåâà.  Íîâîñèáèðñê: Èçäâî ÍÃÒÓ, 2009. 15. ×óáè÷, Â.Ì. Îïòèìàëüíîå îöåíèâàíèå ïàðàìåòðîâ ìîäåëåé ãàóññîâñêèõ ëèíåéíûõ äèñêðåòíûõ ñèñòåì íà îñíîâå ïëàíèðîâàíèÿ íà÷àëüíûõ óñëîâèé / Â.Ì. ×óáè÷, Î.Ñ. ×åðíèêîâà // Íàó÷íûé âåñòíèê ÍÃÒÓ.  2013.   3 (16).  Ñ. 1522. 16. ×óáè÷, Â.Ì. Âû÷èñëåíèå èíôîðìàöèîííîé ìàòðèöû Ôèøåðà â çàäà÷å àêòèâíîé ïàðàìåòðè÷åñêîé èäåíòèôèêàöèè ñòîõàñòè÷åñêèõ íåëèíåéíûõ äèñêðåòíûõ ñèñòåì / Â.Ì. ×óáè÷ // Íàó÷íûé âåñòíèê ÍÃÒÓ.  2009.   1 (34).  Ñ. 2340. 17. ×óáè÷, Â.Ì. Èíôîðìàöèîííàÿ òåõíîëîãèÿ àêòèâíîé ïàðàìåòðè÷åñêîé èäåíòèôèêàöèè êâàçèëèíåéíûõ äèñêðåòíûõ ñèñòåì / Â.Ì. ×óáè÷ // Èíôîðìàòèêà è åå ïðèìåíåíèå.  2011.  Ò. 5,  1 (34).  Ñ. 4657. 18. ×óáè÷, Â.Ì. Ïëàíèðîâàíèå íà÷àëüíûõ óñëîâèé â çàäà÷å àêòèâíîé ïàðàìåòðè÷åñêîé èäåíòèôèêàöèè ãàóññîâñêèõ ëèíåéíûõ äèñêðåòíûõ ñèñòåì / Â.Ì. ×óáè÷ // Íàó÷íûé âåñòíèê ÍÃÒÓ.  2011.   1 (42).  Ñ. 3946. 19. Åðìàêîâ, Ñ.Ì. Ìàòåìàòè÷åñêàÿ òåîðèÿ îïòèìàëüíîãî ýêñïåðèìåíòà / Ñ.Ì. Åðìàêîâ, À.À. Æèãëÿâñêèé.  Ì.: Íàóêà, 1987. 20. Âåðåìåé, Å.È. Ëèíåéíûå ñèñòåìû ñ îáðàòíîé ñâÿçüþ / Å.È. Âåðåìåé.  ÑÏá.: Ëàíü, 2013. 21. ×óáè÷, Â.Ì. Ïðîãðàììíàÿ ñèñòåìà àêòèâíîé ïàðàìåòðè÷åñêîé èäåíòèôèêàöèè ñòîõàñòè÷åñêèõ äèíàìè÷åñêèõ ñèñòåì APIS / Â.Ì. ×óáè÷, Î.Ñ. ×åðíèêîâà, Å.Â. Ôèëèïïîâà // XI Ìåæäóíàð. êîíô. ≪Àêòóàëüíûå ïðîáëåìû ýëåêòðîííîãî ïðèáîðîñòðîåíèÿ ÀÏÝÏ2012≫.  2012.  Ò. 6.  Ñ. 6673.

Âëàäèìèð Ìèõàéëîâè÷ ×óáè÷, äîêòîð òåõíè÷åñêèõ íàóê, çàâåäóþùèé êàôåäðîé, êàôåäðà ≪Òåîðåòè÷åñêàÿ è ïðèêëàäíàÿ èíôîðìàòèêà≫, Íîâîñèáèðñêèé ãîñóäàðñòâåííûé òåõíè÷åñêèé óíèâåðñèòåò (ã. Íîâîñèáèðñê, Ðîññèéñêàÿ Ôåäåðàöèÿ), [email protected]. Îêñàíà Ñåðãååâíà ×åðíèêîâà, êàíäèäàò òåõíè÷åñêèõ íàóê, äîöåíò, êàôåäðà ≪Òåîðåòè÷åñêàÿ è ïðèêëàäíàÿ èíôîðìàòèêà≫, Íîâîñèáèðñêèé ãîñóäàðñòâåííûé òåõíè÷åñêèé óíèâåðñèòåò (ã. Íîâîñèáèðñê, Ðîññèéñêàÿ Ôåäåðàöèÿ), [email protected]. Åêàòåðèíà Àëåêñàíäðîâíà Áåðèêåò, ìàãèñòðàíò ôàêóëüòåòà ïðèêëàäíîé ìàòåìàòèêè è èíôîðìàòèêè, Íîâîñèáèðñêèé ãîñóäàðñòâåííûé òåõíè÷åñêèé óíèâåðñèòåò (ã. Íîâîñèáèðñê, Ðîññèéñêàÿ Ôåäåðàöèÿ), [email protected]. Ïîñòóïèëà â ðåäàêöèþ 24 àïðåëÿ 2015 ã.

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