School of Astronautics Engineering Harbin Institute of Technology,Harbin,150001, China

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1 Applied Mechanics and Materials Online: ISSN: , Vol. 344, pp -9 doi:.48/ 3 rans ech Publications, Switzerland Research on wo Kinds of Parameter Identification Methods for Short-term ime-varying System JingBo Gao, a, XiaoDan Wang,b, WeiYao Zhang,c and Cong Wang,d School of Astronautics Engineering Harbin Institute of echnology,harbin,, China a gaojb@hit.edu.cn, b wxd6499@63.com, c zhangweiyaotongxue@63.com, d alanwang@hit.edu.cn Key words: forgetting factor; broken line; time varying system; shell experiment Abstract: In this paper, two kinds of time varying parameter identification methods which are called identification method of and identification algorithm with auto-regulation forgetting factor are studied. he two methods are used for short-term time-varying system parameter identification in simulations. According to the results of the simulations, the applicable conditions of the two kinds of identification methods are analyzed. he results of the simulations indicate the effectiveness of the two methods. Furthermore whether the two kinds of identification methods are sensitive to noise or not is studied by setting different noise levels in the simulations. Finally, experiments of a variable mass cylindrical shell are adopted to demonstrate the efficiency of the two kinds of time varying parameter identification methods. Introduction In many practical systems, the parameters often change over time. For instance, in the field of aviation, the aerodynamic parameters of aircraft will change when the height or velocity changes; In the water-exit process of submarine-launched missile, the missile s modal parameters will be changed with the height of water. herefore, time-varying system parameter identification is becoming a more and more popular issue. Currently, there are many researches of the identification methods of time-varying system parameter. Among them, the least square method [] is one of the most widely used methods. In this paper, the identification method of using broken line to approximate time varying parameter [] and the identification algorithm with auto-regulation forgetting factor for fast time varying system [3] are proposed on the basis of the principle of least squares. hese two identification methods are simple in calculation and have high accuracy of the parameters. Principles of Identification Methods. Identification method of he system differential equation is y( k ) a ( k) y( k) a ( k) y( k n ) b ( k) u( k) b ( k) u( k m ) () n n Where k is an integer representing the moment tk corresponding moment of the broken-line s endpoint. Let k, is the sampling period. i is the All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of rans ech Publications, (ID: , Pennsylvania State University, University Park, USA-3//6,3::6)

2 6 Advanced Research on Applied Mechanics, Mechatronics and Intelligent System a j( k) a j( i ) a j( k i ), bj( k) bj( i ) bj( k i ), j,,, n ( k) [ a( k) an( k) b( k) bn( k)] ( i ) [ a( i ) an( i ) b( i ) bn ( i )] [ a a an b b bn ] ( k ) y( k) y( k n) u( k) u( k n) () According to formulas above and least square method, using ( i ) and ( k) instead of ( ) i and,we can obtain ( k ) ( k) K( k ){[ y( k ) ( k ) ( i)]/ ( k i) ( k ) ( k)} K( k ) P( k) ( k )[ ( k ) P( k) ( k )] P( k) ( I K( k) ( k)) P( k ), ( k) ( i) ( k)( k i) (3) he method of resetting covariance is used in case of the covariance matrix Pk ( ).hen let P ( i ) c I, i,,, in which c is a big enough constant. he initial values of ( ) and ( ) can be roughly estimated.. Identification algorithm with auto-regulation forgetting factor In the system differential equation (),let ai ( k) ai ( k) kai ( k), i,,, n; bi ( k) bi ( k) kb i( k), i,,, m ( k) [ a an b bm], ( k) [ a an b bm] ( k) [ ( k) ( k)], ( k) [ ( k) k ( k)] ( k) y( k) y( k n ) u( k) u( k n ) (4) hen the formulas of the identification algorithm with auto-regulation forgetting factor for fast time varying system can be expressed as: e( k) y( k) ( k ) ( k ) () fk e ( k) / {[ ( k l ) P( k ) ( k l )] R} K( k) P( k ) ( k) / [ fk ( k) P( k ) ( k)] (6) ( k) ( k ) K( k) e( k), P( k) [ I K( k) ( k)] P( k ) / f ( k) in which l is the forgotten step. R is either an constant or an variable number. But, it must satisfy the relationship of f( k). At first, let k k, P c I, c is a sufficiently large constant. I is a unit matrix of ( m n) ( m n), is the covariance resetting cycle. If

3 b b b b b b a a a a a a Applied Mechanics and Materials Vol k k ( k ),let k k k ;If k k,let k, I I ( k ) k I ( ), P( k ) I I,in which k is the previous moment; k,,, ; I and are unit I matrix of ( m n) ( m n) and zero matrix. 3 Simulations 3. Simulation he system differential equation is y( k) a( k ) y( k ) a( k ) y( k ) b ( k ) u( k ) b ( k ) u( k ) t. s, a( t) sin( t), a( t). 4, b ( t) t, b ( t). s t s, a( t) sin( t), a( t). 4, b ( t) t, b( t) 7 A white noise is given to generate output signals. he two kinds of identification methods are used, and following results can be obtained: Fig. Estimated values of a Fig. Estimated values of a Fig.3 Estimated values of b Fig.4 Estimated values of b he following conclusions can be obtained in terms of the simulations: )For the parameters of time-invariant and time-varying system, both the two kinds of identification methods can have accuracy estimations. )he two identification methods are satisfactory for identifying the parameters which change in the form of some typical functions (sine, step or slide); For parameters which change in the form of sinusoidal, the method of using broken line is better than the algorithm of auto-regulation forgetting factor; For parameters which change in the form of step, the method of using broken line to approximate time varying parameter has shorter transition period than the algorithm of auto-regulation forgetting factor.

4 b b b b b b b b a a a a a a a a 8 Advanced Research on Applied Mechanics, Mechatronics and Intelligent System 3)It can be seen except the moments near the covariance reseting cycles, we can obtain smaller changes using the algorithm of auto-regulation forgetting factor. And there is less fluctuation in the estimated parameters that adopt the method of using broken line throughout the whole process. 3. Simulation Noise signals are added to the system in simulation, the signal-to-noise ratios (SNR) are 3dB, 4dB, and 6dB. Adopt the two kinds of identification methods to identify the parameters, we can obtain the following results:,snr=3db ,SNR=4dB ,SNR=6dB ,SNR=8dB Fig. Estimated values of,snr=3db ,SNR=4dB ,SNR=6dB ,SNR=8dB a a Fig.6 Estimated values of,snr=3db,snr=3db ,snr=4db ,snr=6db ,snr=8db ,snr=4db ,snr=6db ,snr=8db Fig.7 Estimated values of b Fig.8 Estimated values of b he following conclusions can be obtained in terms of the simulations: )When adding noise signals, results of the algorithm with auto-regulation forgetting factor are more accurate than those of the method of using broken line to approximate time varying parameter. )he greater the SNR, the more accurate results can be obtained by using broken line to approximate time varying parameter. But parameters which use the algorithm with auto-regulation forgetting factor can hardly affected by noise. 4 Experiment Verification Experiments of the variable mass cylindrical shell which is. meters long and.9 meters in diameter, are carried out and the frequencies can be obtained in the different height of water. Adopt the two kinds of identification methods to identify the parameters, the following results can be obtained:

5 frequency(hz) Applied Mechanics and Materials Vol experimental values height of water(cm) Fig.9 he results of the two methods According to the results, the two kinds of methods can identify the frequencies accurately. For this system, the results of the algorithm with auto regulation forgetting factor are more accurate than those. Conclusions According to the simulations and experiments, conclusions can be given: ) he two kinds of identification methods can identify parameters which change in the forms of some typical functions (sine, step or slide); ) Parameters which use broken line to approximate time varying parameter are faster in reaching the real values than those use the algorithm with auto regulation forgetting factor. 3) he identification method of is more sensitive to noise; but the algorithm with auto-regulation forgetting factor can hardly affected by noise Hence, the appropriate parameter identification methods for short-term time-varying system must be selected according to actual situations. References [] Chen Xian-zhi,Chen Chong.Parameter identification of time varying systems based on the least-squares modified algorithm [J].Journal of Fuzhou University,,33(): [] Chen Xin-hai,Yan Xiao-ming,Li Yan-jun.An identitication method of fast time varying parameters adapted to aircraft control systems [J].Acta Aeronautica Et Astronautica Sinica, 99, (9): [3] Yan Xiao-ming,Li Yan-jun, Chen Xin-hai. An identification algrothm with auto-regulation forgetting factor for fast time varying system [J]. Automatica, 99,7(3):

6 Advanced Research on Applied Mechanics, Mechatronics and Intelligent System.48/ Research on wo Kinds of Parameter Identification Methods for Short-erm ime-varying System.48/