Active parametrical identification of stochastic linear continuous-discrete systems based on the experiment design in th.pdf Development of information technologies for identification of complex dynamic systems of stochastic nature is an important area of research and has attracted considerable interest. Application of the optimal experiment theory methods in parametrical identification improves the quality of the results by taking into account more fully the properties of the dynamic object and data collection procedures [1-7]. Thus, given the structure of the mathematical model, the procedure of active parametrical identification involves the following steps: - Calculation of parameter estimates based on measurement data corresponding to a test input signal. - Synthesis based on the obtained estimates of the optimal input signal (experiment design). - Recalculation of estimates of unknown parameters according to the measured data corresponding to the synthesized signal. Traditionally, the estimation of unknown parameters is carried out based on the classical Kalman filter, which makes it possible to find estimates of the state vector and corresponding covariance matrices under the assumption of the normality of the noise distribution of the system and measurements. When solving practical problems (for example, problems of communication, navigation, control and radar) there are cases when the usual mechanism of formation of observation data is broken and the appearance of anomalous observations that do not contain information about the object under study. In this case the specified filter can lead to biased estimates or even diverge. At the moment, numerous robust modifications of the Kalman filter, resistant to the appearance of outliers, have been developed. In this regard, it is advisable to consider robust estimation methods that provide good quality results. This work is devoted to the development of mathematical and software procedures of active identification of stochastic continuous-discrete systems based on robust parameter estimation. The efficiency of the developed procedure is demonstrated by the example of one model structure. 1. Problem statement Consider the following controlled, observed, identifiable dynamic system model in state space: dx(t) = F(t)x(t) + T(t)u(t) + r(t)w(t), t e[t0,tN], (1) У(tk+i ) = H (tk+i)x(tk+i) + v(tk+i), k = 0,l5-5N -1, (2) where x (t) is the state «-vector; u (t) is deterministic control (input) r-vector; w (t) is the process noise p-vector; y (tk+1) is the measurement (output) «-vector; v( tk+1) is the measurement error «-vector. Let us suppose that - the random vectors w (t) and v( tk+1) form a white Gaussian noise, for which E[w(t)] = 0, E w(t)wT(x) = Q(t)5(t-t), E[v(tk+i )] = 0, E[v(tk+i) vT (tl+i)] = R(tk+i)5fe-, v( tk+1) wT(T) E = 0 (here E[ ] is operator of mathematical expectation, 5(t - t) is delta function, 5fo is the Kronecker symbol); - initial state x (t0) has a normal distribution with parameters E[ x (to)] = x (t0 ) , E{[ x(t0 ) - x (t0 )][ x(t0 )- x (t0 )]T} = P(t0 ) and is uncorrelated with w (t) and v(tk+1) for all values of k; - output data may contain outliers; - unknown parameters are summarized in the 5-vector 0, including the elements of matrices F (t), T(t), Г(t), H(tk+1), Q(t), R(tk+1), P (t0) and vector x (t0) in various combinations. For the mathematical model (1), (2), taking into account the a priori assumptions, it is necessary to develop procedures for the active parametrical identification of stochastic continuous-discrete systems based on robust parameter estimation and conduct a numerical study of the effectiveness of its application. 2. Methods of research Let us consider the main theoretical aspects of the active identification procedure. Parameter estimation. Unknown parameters estimations of the mathematical model (1), (2) are carried out according to observational data S by using some criterion of identification x . The collection of numerical data occurs during identification experiments which are carried out under some discrete design ^v : 5v =1 k1 v Я V Here v is the total number of launches of the system, q is the number of points of the design, kt is the num ber of experiments corresponding to the signal u1 (t), Q u is the set of design (determined by restrictions on the conditions of the experiment). [ y1J (0]т[ y1J(tN )]T Let us denote through Y? realization of the output signal with number j (J = 1,...,k1) corresponding to the input signal u1 (t). Then S = |(u1 (t),Yj), J = 1,2,...,ki,i = 1,2,...,qj, ik =v . Due to the fact that the measurement data contain anomalous observations, we will calculate quasilikelihood estimates [8], solving the following optimization problem: 0 = argmin [x(0; S)] = arg min [-ln L (0; S)] . (3) 0gQq 0gQq 1 q ki N-1 r- .. -T r -i-1 г- -i x(0; S) = Nf^1п2л +Tz ZZ[sjOk+i)] (tk+i)] |>j(tk+i)]+ 2 t=i j=i k=0 L ] L ] L ] Here (4) 1 q N-i + - 2 k Z 1ndet B (tk+i), 2 i=i k=0 where sj (tk+i) and B (tk+i) determined by the recurrent equations of a robust filter. The calculation of the conditional minimum (3) will be carried out by the method of sequential quadratic programming [9, 10], implemented in the optimization Toolbox MATLAB package and assuming the calculation of the gradient. Differentiating the equality (4) by 0a (a = i,...,5) taking into account expression d lndet B(tk j) dB(tk+1) O0„ B~\\tk+i) = Sp O0„ =- B- 4+,) ^ B-i(ti+1) and symmetry of the matrix B(tk+i), we obtain -|T Osj (tk+i) Oy(0; s) q ki n-i M ' = Z Z Z ^ П-i Г B (tk+i) s (tk+i) O0„ O0„ i=i j=i k=0 ^ OB (tk+i> Ai (tk+i Г ~T Г . (tk+i) B (tk+i) (tk+i) ij + s s O0„ 1 qfN-^[n,^ \\]-i OB (tk+i) +tZ k Z SP \\ B (tk+i) -^- 2 i=i k=0 - ] O0„ ) determined by the equations arising from the corresponding rela- Osi] (tk+i) OB (tk+, Derivatives of---- and --- O0a O0a tions of the robust filter. In [11] the authors conducted a comparative analysis of the efficiency of some modern robust filters for non-stationary linear continuous-discrete systems. While best and quite comparable the results showed correntropy filters Izanloo-Fakoorian-Yazdi-Simon [12] and Chen-Liu-Zhao-Principe [13]. From the point of view of the organization of calculations and, as a consequence, the software implementation of the first of these filters is much easier. In this regard, it seems appropriate to use the Izanloo-Fakoorian-Yazdi-Simon filter when estimating the parameters of models of stochastic linear continuous-discrete systems in the presence of anomalous observations. The corresponding recurrence relations for a single system startup are shown below. Izanloo-Fakoorian-Yazdi-Simon filter. Initialization: x(t0 110) = x(t0), P(t0 110) = P(t0); c = c0. To run in a loop on k = 0, N - i: dx (t | tk ) = F (t) x (t | tk ) + ^(t )u (t), t e[tk, tk+i ]; d_ dt P (t|tk ) = F (t) P (t|tk) + P (t|tk) FT (t ) + r(t )Q (t )ГГ (t) ,t e [tk ,tk+i ]; s (tk+i)R (tk+i )s(tk+i) f s(tk+i) = y(tk+i) - H(tk+i)x(tk+i| tk ); L (tk+i) = exP B(tk+i)-H(tk+i)P(tk+i \\tk)1 (tk+i)1(tk+i)+1 (tk+i); ;(tk+i)_P(tk+i 1)1(tk+i)1(tk+i)1 1 (^k+i); x (tk+i I tk+i ) = x (tk+i I fk ) + K (tk+i )s(tk+i); P(tk+i 1 fk+i) = [ " [(tk+i)H(tk+i)]](tk+i 1 tk ] End of loop. Algorithms for calculating the maximum likelihood criterion and its gradient based on robust filtering for linear continuous-discrete models are presented in [14]. Experiment design. Let us consider the features of input signal design for models of continuous-discrete systems (1), (2). Continuous normalized design in this case can be specified as f = - >Pi >fZp, =l,u‘ (t)enu,i=\\,2,...,q . (5) У Pi , Pi , - , Pq J '=i Unlike discrete design fv, weights pt in continuous design f can take any values, including irrational number. Information matrix M (f) for design (5) is determined by the relation M (s) = z pm (u (t), e), in which the information matrices of single-point design depend on the unknown parameters to be estimated and are calculated in accordance with [15]. We find the optimal experiment design for some convex functional X information matrix M (f) by solving the following extremal problem f * = arg min X[M (4)]. (6) The construction of optimal design can be associated with the representation of the components of the input signals in the form of linear combinations of basic functions (as such, you can use orthogonal polynomials Legendre, Chebyshev, Walsh function, etc.) and then search for the coefficients of such linear combinations. Another approach is related to the assumption that the input signals are piecewise-constant functions preserving their values on the interval between adjacent measurements. In [16] have demonstrated the effectiveness and applicability of the piecewise-constant approximation of the input signal, which makes it possible to calculate the derivatives of the information matrix Fisher from the components of the input signal by recurrent analytical formulas and, consequently, to apply gradient procedures for the synthesis of optimal signals. This means that U (t) = [U‘ (t0 ),U (ti ).■■■,U (tN-i J = U, . Based on the method of sequential quadratic programming, we present a combined procedure for constructing D- or A-optimal continuous design for pre-computed estimates of the parameters 0 , which using direct and dual approaches [17-19] for solving the extremal problem (6). 1. Set the initial nondegenerate design lf/° U° U° I | Ui , U 2, ■■■, U q , | 0 0 0 РЬ P2 , ■ ■■, Pq 0 i л о P, =-, i = i, 2, ■■■, q . q U0 60 fo = Calculate the information matrix of a single-point designs M(uf,0) for i = i,_,q and put k = 0 . 2. Counting the design weight pk,■■■,pk are fixed, find the design f+i=^k minu x[m ou)]. Ui ,■■■,Uq 60U Calculate the information matrix of a single-point designs M(uk+i,0), i = 1,■■■,q. 3. Having fixed the points of the design spectrum 4i+1, we find the design 4i+1 = arg min х[м(4+1)], pi > 0, i pi =1 i =1,...,q. Pl ,...,Pq i=1 4. If an inequality holds for a small positive 51 (Pi+1 _ Pi )2 U+1 _ иЧ -51 , q + i i=1 then let's put 40 = +1 (the execution of the direct procedure is finished), i = 0 and go to step 5. Otherwise, take i = i +1 and go to step 2. 5. Calculate the information matrix M (4i). 6. Find the local maximum Uk = argmax v(U, ^). U gQ„ If the condition ^(^, 4i )_Ц ц, let's move on the step 7, otherwise, to seek a new local maximum. 7. Find xi Tk = argmn X [M fa+1)]. 0 3. Simulation we perform identification experiments and recalculate estimates of unknown parameters. We assume that all a priori assumptions made in the problem definition are fulfilled. Following [20], consider a position control system consisting of an antenna and a direct current (DC) motor. Let the first component of the state vector be responsible for the angular position of the antenna, the second - for its angular velocity. The input signal is the voltage at the input of the DC amplifier controlling the motor. The angular position is measured using a potentiometer. Then the models of state and observation can be determined by the relations: 0 d 0 1 0 -01 y(tk+i ) = [1 0]x(tk+i) + v(tk+i), k = 0,-,N - 1 Here 01,02 are unknown parameters and Q0 = {1 < 01 < 10, 0

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