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Identification of dynamic object parameters
Nuclear Instruments and Methods in Physics Research A 502 (2003) 535–536
Identification of dynamic object parameters Z.V. Ilyichenkova, S.M. Ivanova* Moscow State Institute of Electronics and Mathematics, Bolshoy Trehsviatitelsky Lane 3/12, 109028 Moscow, Russia
Abstract Adaptive algorithms for optimal identification of a restoration system are described, which permit correcting the coordinates of a linear dynamic system from a restricted set of observable coordinates in the presence of noise. The algorithms are based on the neural network’s method. The estimations obtained converge to the optimal ones in the sense of the minimum mean square deviation from the exact value. r 2003 Elsevier Science B.V. All rights reserved.
1. Introduction The problem of restoring identification of continued linear dynamic systems can be reduced to the classical neuronet problem. The identification means definition of unknown parameters. This problem is important for different control systems, whose parameters may vary over a wide range. Such systems are used in robotics, nuclear engineering, chemical industry, etc. Conventionally, u is an input signal and x is an output signal in a dynamic object. However, defining the dynamic system parameters is either too complicated or impossible because of measurement noises. So, an adequate model of the object should be constructed. The first requirement of the model is that it would permit identification of its parameters. The structure and the equation describing the object are assumed to be known: n X di xM ai i ¼ u dt i¼0
*Corresponding author. E-mail address: [email protected] (S.M. Ivanova).
where u is the input signal of both the object and the model, xM is the output signal of the object, ai are unknown coefficients of the model (Fig. 1). The loss function as criterion J shows the adequacy of the model to the object: J ¼ MfQ½z; xM ðaÞg; z ¼ x þ v; minai J ¼ J ; where n is the wide-band noise, Q is the loss function. So, such value a of the parameter that is also the minimum of the functional J has to be found. However, there does not exist any strict constraint on the noise characteristics. Besides, the output signal on the presence of wide-band noise is observed. Both the neural network method and the sensitivity theory [1] are proposed to use.
2. The correction of parameters The model of the object according to the equation is as follows: the signal u is the input signal of both the identification object and the object model (OM), x0 is the output signal of the object. The sensitivity model (SM) is proposed to add to a correction block (CB) (Fig. 1). The loss function is usually a square function. The
0168-9002/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0168-9002(03)00493-5
536
Z.V. Ilyichenkova, S.M. Ivanova / Nuclear Instruments and Methods in Physics Research A 502 (2003) 535–536
Fig. 2. Test result of delta reactivity.
and has to be found using the following equation [2]: X X dci dn r b r ¼ n n þ S; li c i þ S ¼ n þ dt l l l dt i i Fig. 1. Block diagram of the identification system.
optimum condition can be obtained by equating the gradient of the functional to zero: rJðaÞ ¼ MfrQ½z; xM ðaÞg ¼ MfrQg ¼ 0: The elements of the vector qQ are defined as qQ qxM qQ rQi ¼ ¼ Si ; qxM qai qxM where Si is the sensitivity function (SF) of the model output xM as regards the parameters of the model ai : The mean of the sensitivity function is calculated in the SM using the output of OM. The SM and OM parameters are treated simultaneously. The adaptive algorithms appear accordingly. If the structures of the object and the model are equivalent, the algorithm ensures convergence of the parameters ai to the parameters of the object. Otherwise, the mean of the parameters ai gives the minimum of the loss function. For the square loss function we have the following equation: dai qxM qxM ¼ gðtÞðz xM Þ ¼ gðtÞe dt q ln ai q ln ai
dci b ¼ li ci þ i n; dt l
i ¼ 1; y; 6;
b¼
X
bi
i
where n; ci ; li ; bi ; l; S are the system parameters. Some preferences can be obtained using the neural algorithms. The main one is the ability of a reactor to work without man’s assistance. Supposing a minor reactivity change and using this algorithm, the delta-reactivity value is corrected online (Fig. 2).
4. Conclusions
where i ¼ 0; 1; y; n; and the coefficient gðtÞ has to satisfy the Robbins-Monro conditions.
Unification of the sensitivity theory method with the neural algorithm makes it possible to effectively solve the identification problem of continued linear dynamic systems. This algorithm can be used successfully to calculate the unknown parameter of a nuclear reactor that is called reactivity. The appropriate tests have demonstrated the efficiency and high noise immunity of the proposed neural algorithms. This method makes it possible online to raise the identification accuracy and the operation effectiveness of the linear dynamic systems.
3. Applicability of the algorithm
References
The algorithm can be used to advantage in nuclear engineering. For example, it may be used to calculate unknown parameters of a nuclear reactor. One of the most important parameters is reactivity. The reactivity r cannot be measured
[1] Ya.Z. Tsypkin, The Informational Identification Theory, in: Proceedings of the State Publishing House of Sciences, Moscow, 1995, in Russian. [2] M.A. Schulz, Control of Nuclear Reactors and Power Plants, 2nd edition, Mc-Craw-Hill Book Company, New York, 1961.
Identification of dynamic object parameters Z.V. Ilyichenkova, S.M. Ivanova* Moscow State Institute of Electronics and Mathematics, Bolshoy Trehsviatitelsky Lane 3/12, 109028 Moscow, Russia
Abstract Adaptive algorithms for optimal identification of a restoration system are described, which permit correcting the coordinates of a linear dynamic system from a restricted set of observable coordinates in the presence of noise. The algorithms are based on the neural network’s method. The estimations obtained converge to the optimal ones in the sense of the minimum mean square deviation from the exact value. r 2003 Elsevier Science B.V. All rights reserved.
1. Introduction The problem of restoring identification of continued linear dynamic systems can be reduced to the classical neuronet problem. The identification means definition of unknown parameters. This problem is important for different control systems, whose parameters may vary over a wide range. Such systems are used in robotics, nuclear engineering, chemical industry, etc. Conventionally, u is an input signal and x is an output signal in a dynamic object. However, defining the dynamic system parameters is either too complicated or impossible because of measurement noises. So, an adequate model of the object should be constructed. The first requirement of the model is that it would permit identification of its parameters. The structure and the equation describing the object are assumed to be known: n X di xM ai i ¼ u dt i¼0
*Corresponding author. E-mail address: [email protected] (S.M. Ivanova).
where u is the input signal of both the object and the model, xM is the output signal of the object, ai are unknown coefficients of the model (Fig. 1). The loss function as criterion J shows the adequacy of the model to the object: J ¼ MfQ½z; xM ðaÞg; z ¼ x þ v; minai J ¼ J ; where n is the wide-band noise, Q is the loss function. So, such value a of the parameter that is also the minimum of the functional J has to be found. However, there does not exist any strict constraint on the noise characteristics. Besides, the output signal on the presence of wide-band noise is observed. Both the neural network method and the sensitivity theory [1] are proposed to use.
2. The correction of parameters The model of the object according to the equation is as follows: the signal u is the input signal of both the identification object and the object model (OM), x0 is the output signal of the object. The sensitivity model (SM) is proposed to add to a correction block (CB) (Fig. 1). The loss function is usually a square function. The
0168-9002/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0168-9002(03)00493-5
536
Z.V. Ilyichenkova, S.M. Ivanova / Nuclear Instruments and Methods in Physics Research A 502 (2003) 535–536
Fig. 2. Test result of delta reactivity.
and has to be found using the following equation [2]: X X dci dn r b r ¼ n n þ S; li c i þ S ¼ n þ dt l l l dt i i Fig. 1. Block diagram of the identification system.
optimum condition can be obtained by equating the gradient of the functional to zero: rJðaÞ ¼ MfrQ½z; xM ðaÞg ¼ MfrQg ¼ 0: The elements of the vector qQ are defined as qQ qxM qQ rQi ¼ ¼ Si ; qxM qai qxM where Si is the sensitivity function (SF) of the model output xM as regards the parameters of the model ai : The mean of the sensitivity function is calculated in the SM using the output of OM. The SM and OM parameters are treated simultaneously. The adaptive algorithms appear accordingly. If the structures of the object and the model are equivalent, the algorithm ensures convergence of the parameters ai to the parameters of the object. Otherwise, the mean of the parameters ai gives the minimum of the loss function. For the square loss function we have the following equation: dai qxM qxM ¼ gðtÞðz xM Þ ¼ gðtÞe dt q ln ai q ln ai
dci b ¼ li ci þ i n; dt l
i ¼ 1; y; 6;
b¼
X
bi
i
where n; ci ; li ; bi ; l; S are the system parameters. Some preferences can be obtained using the neural algorithms. The main one is the ability of a reactor to work without man’s assistance. Supposing a minor reactivity change and using this algorithm, the delta-reactivity value is corrected online (Fig. 2).
4. Conclusions
where i ¼ 0; 1; y; n; and the coefficient gðtÞ has to satisfy the Robbins-Monro conditions.
Unification of the sensitivity theory method with the neural algorithm makes it possible to effectively solve the identification problem of continued linear dynamic systems. This algorithm can be used successfully to calculate the unknown parameter of a nuclear reactor that is called reactivity. The appropriate tests have demonstrated the efficiency and high noise immunity of the proposed neural algorithms. This method makes it possible online to raise the identification accuracy and the operation effectiveness of the linear dynamic systems.
3. Applicability of the algorithm
References
The algorithm can be used to advantage in nuclear engineering. For example, it may be used to calculate unknown parameters of a nuclear reactor. One of the most important parameters is reactivity. The reactivity r cannot be measured
[1] Ya.Z. Tsypkin, The Informational Identification Theory, in: Proceedings of the State Publishing House of Sciences, Moscow, 1995, in Russian. [2] M.A. Schulz, Control of Nuclear Reactors and Power Plants, 2nd edition, Mc-Craw-Hill Book Company, New York, 1961.