ACTIVE ARAMETRICP IDENTIFICATION OF GAUSSIAN LINEAR ... · summarize results and o er the...

MSC 93E12 DOI: 10.14529/mmp160208

ACTIVE PARAMETRIC IDENTIFICATION OF GAUSSIANLINEAR DISCRETE SYSTEM BASED ON EXPERIMENTDESIGN

V.M. Chubich, Novosibirsk State Technical University, Novosibirsk, RussianFederation, [email protected],O.S. Chernikova, Novosibirsk State Technical University, Novosibirsk, RussianFederation, [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 theapplied problem in various industries, transport, at calculation of systems of automaticcontrol. In this regard development of software for parametrical identication modelsbecomes especially important for fundamental science and practice.

In terms of way of behavior present methods of identication can be divided intopassive and active. In passive identication of dynamic systems only real operating signalsand initial state are used in the modes of not disturbed operation [14].

On the contrary, active identication assumes violation of technological mode and usessome special synthesized designs [59]. So, in a frequency method of parameter estimationof linear stationary models [10, 11] the signal represents the sum of harmonics whichnumber doesn't exceed dimension of space. In this work optimal plan of experiment issearched during the extreme problem solution for some preliminary chosen functional ofinformation (or covariance) vector-matrix of parameters to be estimated. The dicultiesconnected with necessity of violation of a technological mode are provided by increaseof eciency and correctness of research. Methods of active identication give to the

90 Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming& Computer Software (Bulletin SUSU MMCS), 2016, vol. 9, no. 2, pp. 90102

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experimenter considerably greater possibilities in the design of qualitative model incomparison with methods of passive identication. These possibilities are caused by theideology of the active identication based on a combination of traditional parameterestimation techniques with the concept of experiment design.

Works [1215] are devoted to theoretical and applied aspects of the problem of activeparametric identication of stochastic linear discrete systems based on design of inputsignals and initial states.

The authors tried to generalize the results obtained in these papers and constructednew algorithms connected with the simultaneous design inputs and initial states.

The questions of estimating the unknown parameters that we considered, togetherwith the developed methods for designing experiments, enable us to achieve a generalunderstanding of active identication.

1. Problem Denition

Consider the following model of controllable, observable and identiable stochasticlinear 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 themeasurement 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)w

T (ti)]= Qδki,

E [v (tk+1)] = 0, E[v (tk+1) v

T (ti+1)]= Rδki,

E[ν (tk)w

T (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 parametersΘ = (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 ofthe 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.

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91

V.M. Chubich, O.S. Chernikova, E.A. Beriket

2. Procedure of Active Parametric Identication

Procedure of active parametric identication of systems with preliminary chosenmodel structure assumes performance of following stages: calculation of parametersestimates 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 criterionoptimality (stage 2, g. 1) and recalculation of estimates of unknown parameters on themeasuring 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 formodel (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 outaccording to observation data using the criterion of identication χ. Gathering of numericaldata occurs during the identication experiments which are carried out under some designξν .

Assume that experimenter can make ν system starts. He gives signal U1 on the inputof the system with initial state x1(t0) for k1 times, signal U2 with initial state x2(t0) fork2 times etc., at last, signal Uq with initial state xq(t0) for kq times. In this case discrete

normalized experiment design ξν

(q∑

i=1

ki = ν

)is

ξν =

α1, α2, . . . , αq

k1ν, k2

ν, . . . , kq

ν

, αi ∈ Ωα, i = 1, 2, . . . , q, (3)

where the point of design spectrum α has the following structure:

α =[UT ; xT (t0)

]=

[uT (t0) , u

T (t1) , . . . , uT (tN−1) ; x

T (t0)]



and restrictions on the conditions of carrying out experiment are dened by the boundedset

Ωα = ΩU × Ωx(t0) ⊂ RNr+n.

Designate through Y Ti,j

=[(yi,j(t1))

T, (yi,j(t2))

T, ..., (yi,j(tN))

T]the j-th output realization

(j = 1, 2, . . . , ki), corresponding to input signal Ui with initial state xi(t0). Then as aresult of carrying out an identication experiment for design ξν the following set will begenerated

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, asasymptotic unbiasedness, consistency, asymptotic eciency and asymptotic normality.According to this method it is necessary to nd such values of parameters Θ for which

Θ = arg minΘ∈ΩΘ

χ (Θ; Ξ) = arg minΘ∈ΩΘ

[− lnL (Θ; Ξ)] (4)

and (see, for example, [12, 14, 15])

χ(Θ; Ξ)=Nmν

2ln 2π+

1

2

q∑i=1

ki∑j=1

N−1∑k=0

[ε i,j (tk+1)

]TB−1(tk+1)

[ε i,j (tk+1)

]+ν

2

N−1∑k=0

ln detB(tk+1),

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

xi,j(tk+1|tk) = F (tk)

xi,j(tk|tk) + Ψ(tk)u

i (tk) ;

P (tk+1|tk) = F (tk)P (tk|tk)F T (tk) + Γ(tk)QΓT (tk);

yi,j (tk+1|tk) = H(tk+1)xi,j (tk+1|tk) ;

ε i,j (tk+1) = y i,j (tk+1)−

yi,j(tk+1|tk) ;

B (tk+1) = H(tk+1)P (tk+1|tk)HT (tk+1) +R;

K (tk+1) = P (tk+1|tk)HT (tk+1)B−1 (tk+1) ;

xi,j(tk+1|tk+1) =

xi,j(tk+1|tk) +K (tk+1) ε

i,j (tk+1) ;

P (tk+1|tk+1) = [I −K (tk+1)H(tk+1)]P (tk+1|tk) ,

for k = 0, 1, ..., N−1, i = 1, 2, ..., ki, j = 1, 2, ..., q with initial conditions

xi,j(t0|t0) = x (t0),

P (t0|t0) = P (t0).Stage 2. Under continuous normalized design ξ we will understand the collection

ξ =

α1, α2, . . . , αq

p1, p2, . . . , pq

, p i ≥ 0,

q∑i=1

p i = 1, αi ∈ Ωα, i = 1, 2, . . . , q. (5)


93


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

M (ξ) =

q∑i=1

p iM(αi; θ

), (6)

where (Fisher) information matrices for one-point designs M (αi; θ) depend on unknownparameters (this fact allows us to speak further only about locally optimal design) andare calculated in accordance with the algorithm from [16].

The quality of parameter estimation can be improved by construction of experimentdesign which optimizes some convex functional X of information matrix M (ξ), i.e. weshould solve the extreme problem

ξ∗ = argminξ∈Ωξ

X[M(ξ)]. (7)

We will use criterion of D- and À- optimality for which, respectively, X[M(ξ)] =− ln detM (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-optimaldesign we minimize the volume, and for an À-optimal design we minimize the sum of thesquares 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 generalnumerical 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

into account that design spectrum (5) consist of q = s(s+1)2

+1 points. Another approach (itis 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 thanin the direct approach.

Applying of the gradients in the direct or dual procedures leads to the signicantimprove ment of the speed of problem solving. It is impossible without calculation ofderivatives of (Fisher) information matrix by the components of input signal and initialstate. In this case, the gradient of the optimality criterion for points of the spectrum planhas the following structure:

∂X [M (ξ)]

∂αi=

[∂X[M(ξ)]

∂Ui∂X[M(ξ)]∂xi(t0)

)

]=

∂X[M(ξ)]∂ui(t0)

· · ·∂X[M(ξ)]∂ui(tN−1)∂X[M(ξ)]∂xi(t0)

, i = 1, 2, ..., q.

The algorithms for the calculation of gradients ∂X[M(ξ)]∂ui(tk)

and ∂X[M(ξ)]∂xi(t0)

for D optimality

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 fromzero to one. It is easy to notice that in case of given number of possible starts of thesystem ν values k∗i = νp∗i can be noninteger numbers. Carrying out of experiment demands



a rounding o of values k∗i to integers. It is obvious that the design received as a resultof such rounding o will dier from the optimum continuous design. An approach to theoptimum 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 designis given in [19].

Next, we make a discrete design

ξ∗ν =

α1∗, α

2∗, . . . , α

q∗

k∗1ν,k∗2ν, . . . ,

k∗qν

,

conduct the identication experiment and recalculate estimation of unknown parameters.

3. Experimental Results

Consider the following mathematical model of Gaussian linear discrete control systemsof the submarine making the movement with an angular speed ω and longitudinal speedof the course in the vertical plane at an angle of trim µ, taking into account a deviationof fodder horizontal wheels at an angle δ from neutral position (see g. 2) [20]: ˙ω(t)

˙µ(t)˙δ(t)

=

θ1 θ2 θ31 0 00 0 0

ω(t)µ(t)δ(t)

+

001

u(t) +

h0

0.01h

w (t) , (8)

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 byω (tk+1)ψ (tk+1)δ (tk+1)

= F

ω(tk)µ(tk)δ(tk)

+Ψu (tk) + Γw (tk) , (10)

Fig 2. The submarine in the vertical plane

F =

√

θ21+4θ2∗A−θ1∗B

2√

θ21+4θ2− θ2∗B√

θ21+4θ2− θ3∗B√

θ21+4θ2

−B√θ21+4θ2

√θ21+4θ2∗A+θ1∗B

2√

θ21+4θ2

θ3√

θ21+4θ2∗(A−2)+θ1θ3∗B

2√

θ21+4θ2

0 0 1

;


95


Ψ =

θ3

C−D

2θ2√

θ21+4θ2θ3θ1(D−C)

4θ22

√θ21+4θ2

− θ3C+D4θ22

− θ32θ2

0, 5

; Γ =

−h∗B√θ21+4θ2

− θ3h100θ2

+ θ3hA200θ2

+ θ1θ3h∗B200θ2

√θ21+4θ2

h(C−D)

2θ2√

θ21+4θ2− θ3h

200θ2− θ3h(C+D)

400θ22− θ1θ3h(C−D)

400θ22

√θ21+4θ2

h200

;

A =

[e0,25

(θ1−

√θ21+4θ2

)+ e

0,25(θ1+

√θ21+4θ2

)]; B =

[e0,25

(θ1−

√θ21+4θ2

)− e

0,25(θ1+

√θ21+4θ2

)];

C =

(θ1 +

√θ21 + 4θ2

)(e0,25

(θ1−

√θ21+4θ2

)− 1

);

D =

(θ1 −

√θ21 + 4θ2

)(e0,25

(θ1+

√θ21+4θ2

)− 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;

E [v (tk) v (ti)] = 0, 5δki = Rδki; E [x (t0)]=

040

= x (t0) ;

E[x (t0)− x (t0)] [x (t0)− x (t0)]

T=

0, 01 0 00 0, 01 00 0 0, 01

= P (t0) .

Choose the design area Ωα = ΩU × Ωx(t0), where

ΩU = −6 ≤ u (tk) ≤ 6, k = 0, 1, ..., N − 1 ,

Ωx(t0) =

000

≤ x(t0) ≤

144

,

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 receivedestimates of unknown parameters. Realizations of output signal we get by the computermodelling with true values of parameters θ∗1 = −0, 1253, θ∗2 = −0, 004631, θ∗3 = −0, 002198.

The quality of identication in the parameter space and response space will be judgedby values of factors δθ, δ

∗θ and δY , δ

∗Y accordingly. These factors are calculated by following

formulas:

δθ =

∥∥∥θ ∗ − θ∥∥∥

∥θ ∗∥=

√√√√√√√s∑

i=1

(θ∗i − θi

)2

s∑i=1

(θ∗i )2

; δ∗θ =

∥∥∥θ ∗ − θ∗∥∥∥

∥θ ∗∥=

√√√√√√√s∑

i=1

(θ∗i − θ∗i

)2

s∑i=1

(θ∗i )2

;

δY =

√√√√√√√√q∑

i=1

ki∑j=1

N−1∑k=0

(yi,j(tk+1)− yi,j(tk+1 |tk ))2

q∑i=1

ki∑j=1

N−1∑k=0

(yi,j(tk+1))2

; δ∗Y =

√√√√√√√√q∑

i=1

ki∑j=1

N−1∑k=0

(yi,j(tk+1)− yi,j∗ (tk+1 |tk )

)2q∑

i=1

ki∑j=1

N−1∑k=0

(yi,j(tk+1))2

,



where θ ∗ are true values of parameters; θ are estimates of unknown parameters valuesbased on the initial experiment design; θ ∗ are estimates of unknown parameters valuesbased on the synthesized input signal and (or) initial states; y ij (tk+1|tk) and yij∗ (tk+1|tk)are calculated on using the equations of Kallman lter at θ and θ∗ respectively.

Results of the active parametric identication procedure based on D-optimal designof input signals and (or) initial state are presented in Table.

Table

Result of active identication procedures based on design ofinput signals and (or) initial state

Discrete experiment designValues of parameters estimates

and errors of estimation1 2

Initial design

ξν =

α1

k1ν= 1

,

θ1 = −0, 1600

θ2 = −0, 0059

θ3 = −0, 0015α1 =

[UT ; xT (t0)

], input signal UT =

[(6,−6)2,−64, 63,−6, 6,−62,−6, 6,−62, 62,−6, 63,−6, 6,−63, (6,−6)3,−6, 6,−63], initial statex(t0) = (0, 4, 0)T .

δθ = 0, 2770,δY = 0, 1488

Design of initial states

ξν∗ =

α1 α2

k1ν= 2

5k2ν= 3

5

,

θ1 = −0, 1372

θ2 = −0, 0052

θ2 = −0, 0019αi =

[UT ; (xi(t0)

T )], input signal UT =

[(6,−6)2,−64, 63,−6, 6,−62,−6, 6,−62, 62,−6, 63,−6, 6,−63, (6,−6)3,−6, 6,−63], initial statesx1∗(t0) = (0, 4, 0)T , x2∗(t0) = (1, 4, 0)T .

δθ = 0, 0953,δY = 0, 1049

Design of input signals

ξ∗ν =

α1

k∗1ν= 1

5

α2

k∗2ν= 2

5

α3

k∗3ν= 1

5

α4

k∗4ν= 1

5

θ1 = −0, 1301

θ2 = −0, 0049

θ3 = −0, 0019αi =

[(U i

∗)T ; xT∗ (t0)

], input signals (U1

∗ )T = [640]

1 ,(U2

∗ )T = [615,−625], (U3

∗ )T = [640], (U4

∗ )T =

[620,−620], initial state x∗(t0) = (0, 4, 0)T .

δθ = 0, 0380,δY = 0, 0908

Design input signals and initial states

ξ∗ν =

α1

k∗1ν= 2

5

α2

k∗2ν= 1

5

α3

k∗3ν= 1

5

α4

k∗4ν= 1

5

θ1 = −0, 1248

θ2 = −0, 0042

θ3 = −0, 0024αi =

[(U i

∗)T ; (xi∗(t0)

T )], input signals (U1

∗ )T =

[640], (U2∗ )

T = [624,−616], (U3∗ )

T = [612,−628],(U4

∗ )T = [625,−615], initial states x1∗(t0) =

(1, 4, 0)T , x2∗(t0) = x3∗(t0) = x4∗(t0) = (1, 4, 4)T .

δθ = 0, 0055,δY = 0, 0750

1Record [ak, bm, . . .] means that the number a repeats k times, number b−m times.


97


The calculations were performed using the interactive software system of the activeparametric identication (APIS). APIS was developed by the authors at the Departmentof the theoretical and applied informatics NSTU and allows to solve problems of activeparametric identication of nonlinear stochastic discrete and discrete-continuous systemsbased on planning of A- and D- optimal input signals using statistical and temporallinearization. Furthermore, the software system allows optimal estimation of modelparameters for Gaussian linear dynamic systems based on the design of the input signalsand (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 Y = y(t1|t1), y(t2|t2), ..., y(tN |tN),y∗(t2|t2), ..., y∗(tN |tN), Y ∗ = y∗(t1|t1) in accordance with θ = θcp and θ = θ∗cp; average

the results Ycp, Ycp, Y∗

cp, (see g. 3 6).

Fig 3. Graphical representation of Ycp, Ycp(passive identication)

Fig 4. Graphical representation of Ycp, Y∗

cp

(design of the initial states)

Fig 5. Graphical representation of Ycp, Y∗

cp

(design of the input signals)

Fig 6. Graphical representation ofYcp, Y∗

cp

(design of the input signals and initialstates)

The results presented in Table, show that design of the initial states led to betterestimation 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 resultsonly by 23,9% and 5,9%, respectively. The most successful results were obtained for thecase 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 identicationprocedures based on optimal design of input signals and initial states is helpful andadvisable.



Conclusion

For the rst time we considered and solved the problem of active parametricidentication of the Gaussian linear discrete systems in the case when the unknownparameters are contained in the state and control equations as in well as the covariancematrices of the system and measurement noise. The original gradient algorithms for activeidentication 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 signicantimpact on the functional dependence of state variables and model parameters.

The work is executed under the auspices of the Ministry of education and science ofthe Russian Federation ( 2014/138, project 1689).

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identikacija stohasticheskih linejnyh sistem: monograja [Active Parametric Identicationof Stochastic Linear Systems: Monograph]. Novosibirsk, NGTU, 2009. 192 p.

15. Chubich V.M., Chernikova O.S. [Optimal Parameter Estimation for Gaussian Models ofLinear 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 ActiveParametric 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-LinearDiscrete 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 ParametricIdentication 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 ParametricIdentication of Stochastic Dynamical Systems APIS]. XI Mezhdunarodnaja konferencija

aktual'nye problemy jelektronnogo priborostroenija APJeP-2012 [XI International Conferenceon 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 ÈÄÅÍÒÈÔÈÊÀÖÈßÃÀÓÑÑÎÂÑÊÈÕ ËÈÍÅÉÍÛÕ ÄÈÑÊÐÅÒÍÛÕ ÑÈÑÒÅÌÍÀ ÎÑÍÎÂÅ ÏËÀÍÈÐÎÂÀÍÈß ÝÊÑÏÅÐÈÌÅÍÒÀ

Â.Ì. ×óáè÷, Î.Ñ. ×åðíèêîâà, Å.À. Áåðèêåò

Ïðèìåíåíèå ìåòîäîâ òåîðèè ïëàíèðîâàíèÿ ýêñïåðèìåíòà ïðè èäåíòèôèêàöèè äè-

íàìè÷åñêèõ ñèñòåì ïîçâîëÿåò èññëåäîâàòåëþ ïîëó÷èòü áîëåå êà÷åñòâåííóþ ìàòåìàòè-

÷åñêóþ ìîäåëü ïî ñðàâíåíèþ ñ òðàäèöèîííûìè ìåòîäàìè ïàññèâíîé èäåíòèôèêàöèè.



Â ðàáîòå îáîáùàþòñÿ ðàíåå ïîëó÷åííûå àâòîðñêèå ðåçóëüòàòû è ïðåäëàãàþòñÿ àëãî-

ðèòìû 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, óïðàâëå-

íèÿ, âîçìóùåíèÿ, èçìåðåíèÿ, â êîâàðèàöèîííûõ ìàòðèöàõ øóìîâ ñèñòåìû è èçìåðå-

íèé. Ðàçðàáîòàíî ïðîãðàììíî-ìàòåìàòè÷åñêîå îáåñïå÷åíèå, ïîçâîëÿþùåå ðåøàòü çàäà-

÷è àêòèâíîé ïàðàìåòðè÷åñêîé èäåíòèôèêàöèè ñ èñïîëüçîâàíèåì ìåòîäà ìàêñèìàëüíî-

ãî 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 ïðîöå-

äóðû 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 ýêñïåðèìåíòàëüíûìäàííûì / 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: Surveyand 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 äè-íàìè÷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 ñèãíàëàóïðàâëåíèÿ äëÿ îöåíêè ïàðàìåòðîâ ìîäåëåé äèíàìèêè è íàáëþäåíèÿ îáúåêòîâ íåâû-ñîêîãî 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 // Preprintsof 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 ñèñòåìâî âðåìåííîé îáëàñòè / Â.È. Äåíèñîâ, Â.Ì. ×óáè÷, Î.Ñ. ×åðíèêîâà // Ñèáèðñêèé æóð-íàë èíäóñòðèàëüíîé ìàòåìàòèêè. 2003. Ò. 6, 3. Ñ. 7087.

13. Äåíèñîâ, Â.È. Àêòèâíàÿ ïàðàìåòðè÷åñêàÿ èäåíòèôèêàöèÿ ñòîõàñòè÷åñêèõ ëèíåéíûõäèñêðåòíûõ ñèñòåì â ÷àñòîòíîé îáëàñòè / Â.È. Äåíèñîâ, Â.Ì. ×óáè÷, Î.Ñ. ×åðíèêî-âà // Ñèáèðñêèé æóðíàë èíäóñòðèàëüíîé ìàòåìàòèêè. 2007. Ò. 10, 1. Ñ. 7189.


101


14. Àêòèâíàÿ ïàðàìåòðè÷åñêàÿ èäåíòèôèêàöèÿ ñòîõàñòè÷åñêèõ ëèíåéíûõ ñèñòåì: ìîíîãðà-ôèÿ / Â.È. Äåíèñîâ, Â.Ì. ×óáè÷, Î.Ñ. ×åðíèêîâà, Ä.È. Áîáûëåâà. Íîâîñèáèðñê: Èçä-âî ÍÃÒÓ, 2009.

15. ×óáè÷, Â.Ì. Îïòèìàëüíîå îöåíèâàíèå ïàðàìåòðîâ ìîäåëåé ãàóññîâñêèõ ëèíåéíûõ äèñ-êðåòíûõ ñèñòåì íà îñíîâå ïëàíèðîâàíèÿ íà÷àëüíûõ óñëîâèé / Â.Ì. ×óáè÷, Î.Ñ. ×åð-íèêîâà // Íàó÷íûé âåñòíèê ÍÃÒÓ. 2013. 3 (16). Ñ. 1522.

16. ×óáè÷, Â.Ì. Âû÷èñëåíèå èíôîðìàöèîííîé ìàòðèöû Ôèøåðà â çàäà÷å àêòèâíîé ïàðà-ìåòðè÷åñêîé èäåíòèôèêàöèè ñòîõàñòè÷åñêèõ íåëèíåéíûõ äèñêðåòíûõ ñèñòåì / Â.Ì. ×ó-áè÷ // Íàó÷íûé âåñòíèê ÍÃÒÓ. 2009. 1 (34). Ñ. 2340.

17. ×óáè÷, Â.Ì. Èíôîðìàöèîííàÿ òåõíîëîãèÿ àêòèâíîé ïàðàìåòðè÷åñêîé èäåíòèôèêàöèèêâàçèëèíåéíûõ 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÷åñêîéèäåíòèôèêàöèè ãàóññîâñêèõ ëèíåéíûõ äèñêðåòíûõ ñèñòåì / Â.Ì. ×óáè÷ // Íàó÷íûéâåñòíèê 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 ñòîõà-ñòè÷åñêèõ äèíàìè÷åñêèõ ñèñòåì APIS / Â.Ì. ×óáè÷, Î.Ñ. ×åðíèêîâà, Å.Â. Ôèëèïïîâà //XI Ìåæäóíàð. êîíô. ≪Àêòóàëüíûå ïðîáëåìû ýëåêòðîííîãî ïðèáîðîñòðîåíèÿ ÀÏÝÏ-2012≫. 2012. Ò. 6. Ñ. 6673.

Âëàäèìèð Ìèõàéëîâè÷ ×óáè÷, äîêòîð òåõíè÷åñêèõ íàóê, çàâåäóþùèé êàôåä-ðîé, êàôåäðà ≪Òåîðåòè÷åñêàÿ è ïðèêëàäíàÿ èíôîðìàòèêà≫, Íîâîñèáèðñêèé ãîñó-äàðñòâåííûé òåõíè÷åñêèé óíèâåðñèòåò (ã. Íîâîñèáèðñê, Ðîññèéñêàÿ Ôåäåðàöèÿ),[email protected].

Îêñàíà Ñåðãååâíà ×åðíèêîâà, êàíäèäàò òåõíè÷åñêèõ íàóê, äîöåíò, êàôåäðà ≪Òåî-ðåòè÷åñêàÿ è ïðèêëàäíàÿ èíôîðìàòèêà≫, Íîâîñèáèðñêèé ãîñóäàðñòâåííûé òåõíè÷å-ñêèé óíèâåðñèòåò (ã. Íîâîñèáèðñê, Ðîññèéñêàÿ Ôåäåðàöèÿ), [email protected].

Åêàòåðèíà Àëåêñàíäðîâíà Áåðèêåò, ìàãèñòðàíò ôàêóëüòåòà ïðèêëàäíîé ìàòå-ìàòèêè è èíôîðìàòèêè, Íîâîñèáèðñêèé ãîñóäàðñòâåííûé òåõíè÷åñêèé óíèâåðñèòåò(ã. Íîâîñèáèðñê, Ðîññèéñêàÿ Ôåäåðàöèÿ), [email protected].

Ïîñòóïèëà â ðåäàêöèþ 24 àïðåëÿ 2015 ã.


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