Proceedings of the Fourth IASTD International Conference POWR AD RGY SYSTMS (AsiaPS 8 April -4, 8 Langawi, Malaysia ISB CD: 978--88986-73- PARAMTR STIMATIO FOR XPOTIAL SIGALS BY TH QUADRATIC ITRPOLATIO Rong-Ching Wu*, Tai-Yi Yang*, Jong-Ian Tsai**, and Ting-Chia Ou*** *Department of lectrical ngineering, I-Shou Uniersity, Kaohsiung, Taiwan, R.O.C **Department of lectronic ngineering, Kao Yuan Uniersity, Kaohsiung, Taiwan, R.O.C. ***Department of lectrical ngineering, ational Sun Yat-Sen Uniersity, Kaohsiung, Taiwan, R.O.C ABSTRACT This paper offers a complete method to find the exact frequency, damping, amplitude, and phase of the exponential molds. A simulated signal is taen to fit the one. When this simulated signal is equal to the one, the parameters of the simulated signal are identical to the alues. This method includes three major steps, initial alue setting, gradient method, and quadratic interpolation. In initial alue setting, this method analyzes the mold parameter with the two highest amplitudes of each mold, and the precise alues will e found. The difference etween simulated and practical signals could e expressed as a least mean square prolem. The gradient method proides the initial condition for the quadratic interpolation. The minimum error search is accomplished y the quadratic interpolation, which could improe the search efficiency and reduce iteration time. After a few iterations, the method will otain the exact harmonic parameters. KY WORDS xponential mold, Parameters, Quadratic interpolation.. Introduction In physical systems, dynamic ehaior can e expressed as differential equations. The results of linear and time-inariant differential equations are mostly composed of exponential forms. Parameters of exponential forms include frequencies, dampings, amplitudes, and phases. There are two types of exponential forms. If the damping is equal to zero, the mode is periodic; conersely, if the damping is not equal to zero, the mode will e aperiodic, and it will decay to zero with time. For the different types of signals, many analysis methods hae een deelop [,]. The Irahim time domain (ITD method uses impulse response function (IRF data to identify modal parameters [3]. The random decrement method aerages the response segments as a free response. The auto-regression- moing-aerage (ARMA method identifies a system and presents future responses from the information of its past inputs and outputs. The least-squares complex exponential (LSC method lins the relationship etween the IRF and its complex poles and residues through a complex exponential [4]. The eolutionary programming algorithm is also applied to the analysis of transient signal [5]. This paper proposes a method to analyze exponent signal, which can improe the accuracy and conergence of parameter estimation. This method taes a simulated signal and fits it to the one. The difference etween the simulated signal and the one is the topic of least mean square. The optimal solution must e found y the iteration of optimization searching. This paper uses quadratic interpolation method to improe searching efficiency. The quadratic interpolation has rapid conergence in optimization and can otain the solutions of non-linear function in a few iterations. The quadratic interpolation regards the function as a quadratic cure. The method infers the minimum of the function according to three nown conditions. Therefore, the calculation of quadratic interpolation must e ased on three nown conditions. This paper proides these three nown conditions y gradient method. That is, the first three iterations are done y the gradient method and the following iterations are taen y the quadratic interpolation. Thus, the searching efficiency will e improed for the quadratic interpolation, and the initial conditions will e otained for the gradient method. The modification of gradient method is ased on the initial alues and their gradients, and the conergence of the iterant is decided y suitale initial alues. A process to find suitale initial alues is also used in this paper. The frequency will e located within two FFT components of each pea. This method taes these two FFT components to find the initial alues of frequency, damping, amplitude, and phase. The following sections completely illustrate theory, procedures, and ealuation. Section illustrates the oject function of optimization. Section 3 deduces the quadratic interpolation method, which is the tool for searching optimal solutions. Section 4 descries the gradient method, which runs the first three iterations. Section 5 proides a procedure to calculate initial alues, which proides the data to the gradient method. Section 6 ealuates accuracy, and compares different methods. Section 7 is the conclusion. 66-5 93
. Oject Function After sampling, a signal can e expressed as x (. This paper regards a signal consist of seeral molds. Moreoer, a simulated signal will e fitted to the one. The mold can e expressed y frequency, damping, amplitude, and phase, that is K α n/ x( A e cos(π fn/ +, ( n,,..., where A is amplitude, and is phase; oth of α and f are damping and frequency respectiely. The actual alues in Hz are α ' α / T,,,..., K ( f ' f / T,,,..., K (3 where T is the whole measurement sampling time. When the distance of the simulated signal and the one is, the simulated signal and the one are identical, and the simulated signal is the one. For this reason, the oject function can e defined as [6] ase n ( x ( (4 where ase n ( x ( quation (4 is within (,, which standardizes the result of estimation. The oject function is influenced from A,, α, and f, and is the high non-linear function with a minimum alue,. The influence of these three parameters on the oject function is shown in Fig.. The purpose of parameter estimation is to mae the oject function descend to. If there is noise existing in the measured signal, this error will not descend to, ut to a minimum alue. Oject function 3. Quadratic Interpolation The quadratic interpolation is that the nown data are estalished in a quadratic function. The minimum of this function will e found. A quadratic function could e expressed as ay + y + c (5 If the independent ariales [ y, y, y ] and their corresponding dependent ariales [,, ] which is near the optimal solution, are nown. This method regards that the relation of these data is a quadratic function. That is ( y y a ( y y (6 ( y y c then the coefficients of this function are a ( y y ( y y (7 c ( y y According to (5, its extreme alue will happen when its first-order differentiation is equal to zero. amely d + ay + + (8 dy Therefore, the following condition must e formed if the alue of the quadratic function is a minimum y + (9 a where y can e sustituted y A,, α,or.when all new independent ariales are otained, the f + new oject function will e calculated y (4. In addition, the exact parameters can e otained in this process. This method searches the optimal solution y three nown conditions which do not require a complex formula howeer, the quadratic interpolation must e started at three nown conditions. Shown in Fig.. The first three conditions could e found y the gradient method. Thus, the searching efficiency will e improed for the gradient method, and the initial conditions will e otained for the quadratic interpolation. Phase Frequency Figure. The influence of different parameters on the oject function 94
xtreme alue Oject function Quadratic function - - + f + η f η f f + α α η α α where η are accelerating factors. (8 (9 ( y 5. Initial Value Setting Figure. Searching for extreme alue y quadratic interpolation 4. Gradient Method The gradient method is a minimization searching method which can deal with multiariales. quation (4 has 4 K unnown ariales, A,, α, and f. To find the minimum of this function is to satisfy the following equation [7]: ( u ( where is the gradient of, that is, is the first order differential for all ariales. ( u [ / u( / u(4k ] T ( The first order partial differential equations of to each parameter are α n/ ( x( e cos( n/ + A ( ase n ase n α n/ ( x ( e sin( n/ + A πa α n/ ( n( x ( e sin( n/ + 4 f ase n A α ase n α n/ ( n( x ( e cos( n/ + (3 (4 (5 The solution of non-linear equation,, can e expressed as + x ( x ( η ( (6 x ( The ariales are in sustitution for x, and the next states of frequency, amplitude, and phase can e found as + A A η A (7 A This section proides a simple and accurate algorithm to ensure iteration is conerged and efficiency improed. The process is illustrated elow [8]. For a clear description, firstly, symols used are shown in Fig. 3. Fig. 3 is the spectrum of a signal, and the two highest amplitudes are X p and X p+ ε, which are located at the FFT components p and p + ε. Where ε is equal to or, respectiely if X p+ X p or not. These 4 data are the references to set initial alues. To find the initial alues, the two auxiliary equations are estalished firstly. Amplitude 振幅 X p X / ( ρ p+ε X p X p + X p p p + p p Frequency 頻率 Amplitude spectrum Figure 3. Reference data for initial alue ρ z ( ρ exp( jπε / then the frequency and the damping can e found: δ arg( z (3 π f p + δ (4 α ln z (5 Once δ and α are determined, the third auxiliary equation can e estalished. X p 95
exp( α jπδ D (6 exp( ( α jπδ / The complex coefficient A is easily determined from (6 A X p / D (7 arg( X p / D (8 6. Procedure This section rearranges the aoe theory as a complete procedure [9]. Step, signal sampling: The sample period T and sample data are decided. In this step, for distinguishing eery and clearly, the sample period must e suitale; for conforming to sampling theorem and calculating aility, the sample data must e suitale. Step, time-frequency transformation: The signal is transformed into spectrum y FFT. Step 3, selection of reference data: Molds will cause their peas on spectrum. From these ands, the method will get the reference data of scales, p, p', X, and. p X p + ε Step 4, initial alue setting: The alues of frequencies and dampings can e found y (4 and (5. Then, referring to the found frequencies, the alues of amplitudes and phases can e calculated y (7 and (8 indiidually. Because the results of parameters approach the ones, it ensures the following iteration is conerging. Step 5, gradient method: Find the next states of frequency, damping, amplitude, and phase y (7 to (. Then the oject function can e found y (4. Step 6, additionally repeat Step 5 twice. Step 7, quadratic interpolation: Coefficients can e otained y (7, and next states can e deried from (9. The next states of frequency, damping, amplitude, and phase are found in turn in this step. Then the oject function can e found y (4. Step 8, conergence examining: If the results haen t conerged to the accepted range, the procedure goes to step 7. The step could also assign the times of iterations, which preents the alues from conerging. Step 9, calculation accomplished. 7. Aility aluation This section ealuates the aility of the method in two parts. The first discusses accuracy; the last compares the results of the different methods. 9. t x( t 4. 65e cos(π 59. 9t. 6 +. 89 cos (π t. 858 + 8. 9 cos(π.4t. 745 + 6.57 e 7. 8t cos (π 4 t. 6 (9 The sample period is sat T(, sec, and the numer of data is sat48. The analysis results are recorded in Tale.The comparison of the simulated signal and one is shown in Fig.4. Satisfactory results will e calculated y the quadratic interpolation, which is shown in the fifth to eighth columns. The fourth column shows results of initial alue setting. Approximate results can e found in this stage. In this example, the conergence is reached after 5 iterations. Real alues can e otained only after a few iterations. Then satisfactory results will e calculated. 7. Comparison Of Different Methods This section compares the quadratic interpolation with gradient method in conergence degree. When analyzing (9, Fig. 5 is the conergence degrees of quadratic interpolation and gradient method. In Fig. 5, the quadratic interpolation can reduce the oject function to * -6 in 5 iterations. With the same initial alues and accuracy required, the gradient method needs more iteration. In all iterations, the conergence degrees of quadratic interpolation are oiously more excellent than the gradient method. Because this method can conerge rapidly, the iterations needed are quite less. Tale Signal analysis Real This method Component Parameter alue Initial st iteration 5th iteration f 59.9 59.89 59.899 59.899 A 4.65 4.6 4.53 4.637 59.9Hz -.6 -.95 -.3 -.4 α -9. -9.84-9.9-9.9 f... A.89.78.84.86.Hz -.858 -.86 -.863 -.859 α.7..4 f.4.399.34.4 A 8.9 8.53 8.54 8.79.4Hz -.745 -.743 -.744 -.745 α.9.6. f 4 4.3 4.47 4.6 A 6.57 6.5 6.569 6.57 4 Hz -.6 -. -.79 -.6 α -7.8-7.8-7.83-7.8 7. Accuracy For proing the accuracy of this method for parameter estimation, this section taes a signal with 4 molds as an example. 96
Amplitude 4 3 - - -3 3 4 Time (ms Real Initial alue Figure 4. Comparison of the simulated signal and one Oject function (*^-5 8 6 4 8 6 4 3 4 5 6 7 8 9 Iteration times Gradient method Quadratic interpolation Figure 5. Conergence of different methods References [].D. yman, Modeling, Simulation, and Control, St. Paul, West Pulication Company, 988. [] T. Soderstorm, P. Stoica, System Identification, ew-jersey: Prentice Hall, 989. [3] J. He, Z.F. Fu, Modal Analysis, Boston: Butterworth- Heinemann, 3. [4] T. Soderstrom, H. Fan, B. Carlsson, and S. Bigi, Least Squares Parameter stimation of Continuous-Time ARX Models from Discrete-Time Data, I Trans. Automatic Control, ol. 4, no. 5, pp. 659-673, May 997. [5] L.L. Lai and J.T. Ma, Application of olutionary Programming to Transient and Sutransient Parameter stimation, I Trans. nergy Conersion, ol., no. 3, pp. 53-53, Sept. 996. [6] R.C. Wu, S.L. Yan, and C.W. Yang, Parameter stimation of the xponential Signals Using the Conjugate Gradient Method, 3 International Conference ICICS, pp. 78-73, 3. [7] B. Li, A Generalized Conjugate Gradient Model for the Mild Slope quation, Coastal ngineering, ol. 3, 994, pp. 5-5. [8] M. Bertocco, C. Offelli, and D. Petri, Analysis of Damped Sinusoidal Signals ia a Frequency-Domain Interpolation Algorithm, I Transactions on Instrumentation and Measurement, ol. 43, no., pp. 45-5, April 994. [9] R.C. Wu, S.L. Yan, and C.W. Yang, Parameter stimation for the Complex xponential Signals y the Second Order Differentiation, The 4 rd Symposiumon lectrical Power ngineering, 3, pp.59-53. 8. Conclusion This paper offers a complete method to find the exact frequency, damping, amplitude, and phase of molds. This method includes three major processes, initial alue setting, gradient method, and quadratic interpolation. In initial alue setting, the process has otained the approximate alues. The gradient method proides the initial condition to the quadratic interpolation. The quadratic interpolation can find the optimal solution in a few iterations. This method possesses the adantages of accuracy and excellent conergence. These features are:. ( Accuracy: The mold parameters are found y least mean square, which maes oject function decrease to minimum. ( xcellent conergence: Since the approximate alues hae een otained in initial alue setting, results are conerged quicly. 97