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SIMULATION OF THREE PARAMETER ESTIMATION ALGORITHMS FOR PATTERN RECOGNITION ARCHITECTURE David Al-Dabass*, Abdalla Zreiba*, David Evans* and Siva Sivayoganathan** *Department of Computing ** Department of Mechanical & Manufacturing Eng'g The Nottingham Trent University Burton Street, Nottingham NG1 4BU, England email: dad@doc.ntu.ac.uk, zab@doc.ntu.ac.uk, dje@doc.ntu.ac.uk, siva.sivayoganatahn@ntu.ac.uk ABSTRACT: To recognise trends embedded in 2. PROBLEM FORMULATION AND patterns, a dynamical model of the system generating the ALGORITHMS patterns can be assumed. A second order model has a wide variety of patterns which can serve well in 2.1 Definitions approximately describing the short-term behaviour of complex physical and biological systems. Apart from Many physical problems are described by the solution of initial conditions, the output pattern of a simple second an initial value problem of the form: order system is completely defined by 3 parameters. Using simulation, three algorithms are investigated in this .x''+2.x'+x=u paper to estimate the parameters of an equivalent second x(0)=x0, x'(0)=x'0 order system from a given trajectory (in time or space) of the pattern. The algorithms combine successive 1st order filters of specified cut-off frequencies, to provide Where is the natural frequency, is the damping ratio, u smoothing and higher order derivative estimation, with is the input and x is the output of the system. The three non-linear static parameter estimators. Accurate parameters may be constants, variables or variable with estimations of and u were made using combinations dynamical behaviour. of time derivatives of the measured output of the system. Results of the simulations are presented which show that 2.2 Algorithm 1: Three-Points in x, x' and x'' the algorithms can cope well with variable parameters. Consider estimating , and u using three sets of x, x' 1. INTRODUCTION and x'': Second order linear equations provide suitable x1'' + 2. x1' + x1 = u (1) mathematical models of important physical processes. The x2'' + 2. x2' + x2 = u (2) motion of a mass on vibrating spring, the torsional x3'' + 2. x3' + x3 = u (3) oscillation of a shaft with a flywheel, the flow of electric current in a simple circuit, can all be described by the Subtracting (2) from (1) and (3) from (1) to give: solution of an initial value problem. A number of approaches have been developed to estimate the .( x1'' - x2'' ) + 2. x1' - x2' ) + ( x1-x2 ) = 0 (4) parameters of dynamical systems. These include least .( x1'' - x3'' ) + 2. x1' - x3' ) + ( x1 - x3 ) = 0 (5) squares [Kailath 1978, Gersch 1974], functional approximation [Man 1995], Fourier transformations Dividing (4) by x1' - x2' ) and (5) by x1' - x3' ): [Cawley 1984], differential equation techniques [Dewolf & Wiberg 1993], Kalman filters [Kalman 1960] and .( x1'' - x2'' ) / x1' - x2' ) + 2. + ( x1 - x2 ) / neural networks and genetic algorithm techniques x1' - x2' ) = 0 (6) [Goodwin 1997, Al-Dabass 1985]. Three hybrid methods of estimating , and u are proposed here and tested for .( x1'' - x3'' ) / x1' - x3' ) + 2. + ( x1 - x3 ) / 3 categories of parameter sets: constants, variables with 1st x1' - x3' ) = 0 (7) order dynamics and variables with 2nd order dynamics. and subtracting gives: These three methods were developed using the following approaches: .[( x1'' - x2'') / x1' - x2') -(x1'' - x3'') / x1' - x3')] + By using three sets of estimated 1st and 2nd time [(x1 - x2) / x1' - x2') - (x1 - x3) / x1' - x3')=0 (8) derivatives of the measured output of the system. By using two sets of estimated 1st, 2nd and 3rd time Using the following notations: derivatives of the measured output of the system. By using a single set of estimated 1st, 2nd, 3rd and 4th (x1- x2), ' x1' - x2'), ''( x1'' - x2'') time derivatives of the measured output of the (x1 - x3), ' x1' - x3'), ''( x1'' - x3'') system. we get expressions for estimated estimated =0 (4) using (4), and estimated u: We get expressions for estimated , estimated using '''''' (2), and estimated u: '' 2 = [x''. x'''' - x'''2] / [x'. x''' - x''2] = [-E.''' = -[E x''' + x'] / [2. Ex''] Eu = E . x1'' + 2...E . x1' + x1 Eu = E. x'' + 2..E . x' + x 2.3 Algorithm 2: Two-Points and One Extra Derivative 3. SIMULATIONS AND PROGRAMS Consider using two sets of x, x', x'' and x'''. Programs implemented in Mathcad are presented here for the 3 categories of parameters: constant and variable with x1'' + 2. x1' + x1 = u (1) 1st and 2nd order dynamics, and are used to illustrate the x2'' + 2. x2' + x2 = u (2) generation of time derivatives. In the next section the 3 algorithms will be applied for each parameter category to test their effectiveness. Subtracting (2) from (1) and dividing by x1' - x2'): .( x1'' - x2'' ) / x1' - x2' ) + 2. + ( x1 - x2 ) / 3.1 Constant Parameters x1' - x2' ) = 0 (3) The output of the system and the cascade of 1st order filters were simulated using the 4th order Runge-Kutta Differentiating (3) with respect to t gives: method in Mathcad. The derivatives vector is shown below. x1' - x2' ). ( x1''' - x2''' )]- ( x1'' - x2'' )2]/ x1' - x2` )2 + 0 +[ x1' - x2' )2 - ( x1 - x2 ).( x1'' - x2')] / x1' - x2' ) = 0 (4) 0 x2 x1 2 x2 u 0 2 2 Using the following notations: 0 G x1 x3 (x1 - x2), ' x1' - x2'), x 0 ''( x1'' - x2'') and '''( x1''' - x2''') 0 D( t x) G G x1 x3 x4 0 G G G x1 x3 x4 x5 We get expressions for estimated , estimated using (3), and estimated u: 0 G G G G x1 x3 x4 x5 x6 G G G G G x1 x3 x4 x5 x6 x7 2 = '). (''' ) - ('' )2]/ [' )2 - .''] Z Rkadapt ( x t 0 t 1 N D ) = [-E.''' - /'] / [2.E ] Eu = E. x1'' + 2...E . x1' + x1 2.4 Algorithm 3: One-Point and Two Extra When the three parameters of the 2nd order system defined Derivatives: in 2.1 were constants, the measured and estimated outputs and their derivatives are shown in Figure 1. Consider using the 1st to 4th time derivatives at a single point. Given the second order system: x'' + 2. x' + x = u (1) x & Ex x` , Ex` & Ex`` 2 20 Differentiate with respect to t: Z Z n 3 n 2 x''' + 2. x'' + x' = 0 Z (2) Z n 5 n 4 1 0 Z n 6 divide by x'': x'''/ x'' + 2. + x'/ x'' = 0 (3) 0 20 0 10 20 0 10 20 n n and differentiate with respect to t again to give: Figure 1: Measured and estimated outputs and their deivatives .[(x''. x'''' - x''' ) / x'' ] + 0 + [(x'' – x'. x''') / x'' ] 2 2 2 2 3.2 Parameters with 1st Order Dynamics x & Ex x` , Ex` & Ex`` 2 100 nd When the three parameters of the 2 order system were Z variables with 1st order dynamics, the derivatives vector is Z n 2 n 3 50 as shown below: Z 1 Z n 11 n 10 0 Z x2 n 12 50 x3 x1 2 x x x x3 x5 2 2 3 4 2 0 t 0 e 0 20 100 0 10 20 0 n t n 20 e 0.01 e t Figure 3: Measured and estimated outputs and their deivatives D( t x) G x1 4 x x6 0 0 G G x1 x6 x7 4. RESULTS AND DISCUSSION 0 G G G x1 x6 x7 x8 0 4.1 Constant Parameters 0 G G G G x1 x6 x7 x8 x9 G G G G G x1 x6 x7 x8 x9 x10 A. Algorithm 1 And the (simulated) measured and estimated outputs and This algorithm uses three points on the time trajectory to their derivatives are shown in Figure 2. provide 3 simultaneous equations in the parameters. In each equation the values of its 1st and 2nd time derivatives x & Ex 10 x` , Ex` & Ex `` are obtained from the filter cascade. D is an n-element 1000 vector-valued function containing the first derivatives of Z Z the system and 3 1st order low pass filters to provide n 2 n 3 0 continuous estimate of x, x' and x''. The half power cut- Z n 4 5 Z n 5 off frequency of these filters is G (which is 1/L in the Z equivalent inductor/resistor low pass 1st order filter) and n 6 1000 was set to 1000 to provide a compromise between good response to systems dynamics and noise filtering. The 0 built-in Runge-Kutta routine was used to find the solution 0 20 n 2000 0 20 of x for the interval t0 and t1, which ranged between 3 and n 6 seconds and resulted in N having the values 30-60. The Figure 2: Measured and estimated outputs and their deivatives separation between the 3 points clearly affects the accuracy of the estimation: the further apart the larger are the differences between the variables and more accurate is 3.3 Parameters with 2nd Order Dynamics the estimation. A separation of 3 to 7 points was found to nd give good accuracy and rapid convergence to the correct When the three parameters of the 2 order system are values. Running program 1 [Zreiba, 1999, Appendix A] variables with 2nd order dynamical behaviour, the produced the estimations of and u shown in figure 4. derivatives vector is as shown below: Ew Ez Eu x2 5.5 0.5 1.5 0 2 2 x3 x5 x2 2 E 5 E Eu 0 x3 x1 x3 x7 n n 0 n 1 10 x4 4.5 u 0 0.5 0.5 o x3 2 o o x4 o uo 2 2 4 0.1 x6 3.5 1 0 0 0 2 4 0 2 4 0 2 4 n n n x 1 2 2 z z x6 2 z x5 z uz 0 Figure 4: Estimated constants , and u using algorithm 1 D( t x) x8 0 u x7 2 u u x8 u uu 2 2 0 Algorithm 2 G x1 0 x9 0 G G x1 x9 x10 This algorithm uses only 2 points from the systems output 0 G G G x1 x9 x10 x11 but requires a further time derivative to provide a continuous estimate of the 3 parameters. The filter G G G G x1 x9 x10 x11 x12 cascade is increased by one to yield the extra derivative G G G G G x1 x9 x10 x11 x12 x13 and the two sets of x, x', x'' and x''' are used in the solution to the 2 simultaneous equations to give the estimated , Z Rkadapt ( x t0 t1 N D ) and u. The separation between the two points can be larger to provide better accuracy; a situation helped by the The measured and estimated outputs as well as the 1st and fact that the parameters are constant. Algorithm 2 (in 2.3) 2nd derivatives of the measured and estimated output are was applied and program 2 [Zreiba, 1999, Appendix A] shown in Figure 3. was run to produce the results shown in figure 5. 6 Ew 0.1 Ez 1.5 Eu ' and u' were similarly arranged to vary from 0.01 to 1, and 1 to 10 respectively, both in 5 sec. E E Eu n 5 n n 1 0 u Algorithm 1 4 0.5 Three points are used in this algorithm, which 3 0 5 0.1 0 5 0 0 5 immediately raises the issue of the effect of the size of the n n n gap between successive points. Parameters do not change Figure 5: Estimated constants , and u using algorithm2 much when the gap is small thus the 3 simultaneous equations are less erroneous in assuming (implying) constant parameters; the larger the gap the less accurate this assumption is and therefore less accurate is the C. Algorithm 3 estimation. Figure 7.a shows the case when the points are separated by a single time-step using program 4 [Zreiba, This algorithm uses a single time point but two further 1999, Appendix A]; from left to right: was changed time derivatives compared to Algorithm-1. The filter from 2 to 20 while setting = 0.1and u = 1; was cascade is increased by one again to provide a continuous changed from 0.1 to 1 while setting and uand estimate of the 4th time derivative x''''. The separation u was changed from 1 to 10 while setting = 10 and problem disappears altogether now to provide a =0.1. G was set to 10,000 throughout. Figure 7.b shows continuous estimate of all parameters at each point on the the case when the points are separated by 4 time-steps, the trajectory. Program 3 [Zreiba, 1999, Appendix A] was errors in the estimation are very clear and indicates the run, and the result of the estimation are given in figure 6; extent by which the assumption for constant parameters which shows fast and accurate convergence. has broken down in the simultaneous equations. There is a counter effect however: a small gap also mean that the Ew Ez Eu variables do not change very much and therefore 10 0.1 1.5 differences will be prone to numerical errors. E E Eu n n n 1 Estimated Omega Est imat ed Zet a a Est imated Zet Est imat ed u 5 0.05 u 0.5 1 15 20 0.5 E E E Eu n n n n 0 10 0 0 0 Z Z Z Z 0 5 0 5 0 5 n 7 10 n 8n 8 0 n 9 n n n 0.5 5 Figure 6: Estimated constants , and u using algorithm3 0 1 1 0 0 20 40 0 0 20 20 40 40 0 20 40 n n n n D. Discussion Figure 7a: Estimated variables (=0.1 u=1), (=10 u=1)and Estimated values for constants parameters were close to u (=10 =0.1) using algorithm 1 separated by 1 time step the desired set values for all three methods. The derived algorithms estimated , and u for a good range of values: from 1 to10, between +/- (0.01 to 1), and u between +/- (0.5-40), and gave accurate estimates. Est imat ed Omega Est imat Zet a Est imateded Zet a Est imated u Estimation errors decreased as increased, particularly 30 10 1 100 for small (less than 0.5): where oscillation provided E wide variation in the variables to decrease errors. The n 20 EEn Eu n n 5 50 differences between the (simulated) system time Z n 7 Z Zn 8 0.5 Z derivatives (x, x' and x'') and their estimates from the filter n 8 n 9 10 cascade depended on G (the cut-off frequency): high G 0 0 provided more accurate estimation of derivatives but 0 made the algorithms prone to noise and vice versa. 0 20 40 0 5 0 20 40 50 n 0 20 40 0 20 40 Another disadvantage of high G from the simulation point n n n of view is that simulation time increased considerably due Figure 7b: Estimated variables (=0.1 u=1), (=10 u=1) to the integration routine adapting to ever smaller steps. and u (=10 =0.1) using algorithm 1 separated by 4 time steps The algorithms provided progressively faster convergence with Algorithm-3 being the fastest to converge. Algorithm 2 st 4.2 Parameters with 1 Order Dynamics Only two points (occupying half the time duration of 3 points) are used here so the constant parameters The parameters were set to vary with time. The derivative assumption is less pronounced. This also allows a wider vector D was modified to include the rate of change of gap to reduce numerical difference errors. Two and u. ' was set to an exponential ramp, typically of simultaneous equations are used with two values of x, x', the form e-t which, when coupled with a suitable initial x'' and x''' instead of three values of x, x' and x''. Program condition, allowed to increases over a good range, say 2 5 [Zreiba, 1999, Appendix A] was run to produce the to 20 rad/sec in a reasonably short duration of about 5 sec. results shown in figure 8: where 8.a and 8.b correspond to Est imat Omega Est imat eded Omega Est imat ed Zet a a Est imat ed Zet Estimated u 30 30 1 0.5 15 gaps of 1 and 4 time-steps respectively. Overall, 1 time- step gap performs better despite large initial errors; the E E n E E n Eu n n 20 n 10 20 0 noticeable reduction in these errors for the larger gap case Z n 9 Z Z Z n 10 0 Z n 11 is probably due to the reduction in numerical errors as the n 9 n 10 10 5 10 0.5 variables have larger values. 0 1 0 0 20 40 10 20 40 0 0 20 40 0 20 n 40 0 20 n 40 ' Est imated Omega Est imat ed Estimated Zeta a Est imat ed Zet Estimated u n n ' n 30 20 1 0.5 15 ' E En n E En Eu Figure 9b: Estimated variables (=0.1 u=1), (=10 u=1) and u 20 n n (=10 =0.1) using algorithm 3 (2.4) G=1000 15 0 10 Z Z 8 Z n 8 Zn 9 0 Z n n 9 n 10 10 10 0.5 5 4.3 Parameters with 2nd Order Dynamics 0 5 1 0 0 20 20 40 40 1 0 20 40 0 0 20 40 0 20 40 In this section the parameters and u are time varying n n n ' n ' n nd ' Figure 8a: Estimated variables (=0.1 u=1), (=10 u=1) and as outputs of 2 order systems to provide a rigorous test of the algorithms. Each parameter subsystem will have its u (=10 =0.1) using algorithm 2(2.3) separated by 1 time step own 3 parameters: for they are o, o and uo; for they are z, z and uz; and for u they are u,u and uu. Est imated Omega Est imat ed Zet Est imat ed Zet aa Est imat ed u 20 12 15 A. Algorithm 1 E n EEn Eu n n 10 0.5 Z n 8 10 Program 7 [Zreiba, 1999, Appendix A] was used to get Z Zn 9 0 Z n 10 n 9 two sets of results corresponding to gaps of 1 and 2 time 0 5 steps between the points needed for the 3 simultaneous 0 equations in this algorithm, figures 10.a and 10.b 0 20 40 2 0 0.5 n 0 0 20 20 40 40 0 20 40 respectively. As expected, severe random errors resulted ' n n n ' ' from the larger gap; using a gap of 2 reduced these errors Figure 8b: Estimated variables (=0.1 u=1), (=10 u=1) considerably but are not shown here for space limitations. and u (=10 =0.1) using algorithm 2(2.3) separated by 4 time steps Est imat ed Omega Est imat ed Estimated Zeta Zet a Est imat ed u 15.83 0.97 11.37 Algorithm 3 E n E En Eu n n Z Z Zn 6 Z n 4 8.91 0.27 n 8 Just one point is used here but with one extra derivative, n 6 0.15 5.69 x''''. As there are no gap effects, the accuracy of the parameters depends entirely on the estimation accuracy of 2 0.81 the time derivatives. These are obtained as by-products of 1 24.5 48 0.68 1 24.5 48 0 1 24.5 48 1 24.5 48 the filter cascade and the accuracy of their instantaneous n ' n n n ' values will depend on the gain. Larger gain, i.e. smaller ' Figure 10a: Estimated 2nd order (=0.1 u=1), (=10 u=1 )and filter time constant and shorter delay, gives more accurate derivatives. However, larger gain also means higher cut u (=10 =0.1) using algorithm 1 separated by 1 time step (o=2, o=0.1, off frequency and therefore less immunity to high z=4, z=0.2, u=2, uz=0.1) frequency noise. Figure 9.a and 9.b show two cases with Estimated Omega Est imat Estimated Zeta ed Zet a Est imat ed u 17.52 2.12 10.41 26.47 gains of 10,000 and 1,000 respectively. E EEn Eu n n n Est imat ed Omega Est imat a Est imat ed Zeted Zet a Z Zn 60.22 Z 20 Est imat ed Zet a 1 0.5 Est imat ed u Z n 4 9.61 n 6 5.17 n 8 6.57 0.5 15 E E E n n n E 0 Eu n n 10 Z Z Z 0.5 n 9 10 Z n 10 n 10 Z 0.06435 13.32 n 10 0 n 11 1.71 1.68 1 21.5 42 1 21.5 42 0.5 1 21.5 42 1 21.5 42 n n ' 5 n n ' ' 0 0 1 Figure 10b: Estimated 2nd order (=0.1 u=1), (=10 u=1) 0 20 n 40 0.5 0 0 20 0 20 n 40 20 40 n 40 0 0 20 40 and u (=10 =0.1) using algorithm 1 separated by 4 time steps (o=2, ' ' n n ' o=0.1, z=4, z=0.2, u=2, uz=0.1) Figure 9a: Estimated variables (=0.1 u=1), (=10 u=1) and u (=10 =0.1) using algorithm 3 (2.4) G=10,000 B. Algorithm 2 Program 8 [Zreiba, 1999, Appendix A] was used to estimate the parameters based on two points. Figures 11.a and 11.b show two sets of results for gaps of 1 and 4 time- steps respectively. The results in 11.a are smoother than 10.a. For the 4 time-steps gap, the large random errors 15.83 Est imat ed Omega Est imated Omega Estimated a Est imat Zet Zeta Est imat ed ed Zet a Estimated u shown in figure 10.b for have reduced considerably 15.63 0.98 11.37 in 11.bfor u, however, the improvement is only marginal. E n E E n E E Eu n n n n There is a marked improvement when the gap is set to 2 Z Z n 4 7.91 n 6 0.32 time-steps, Fig11.c. Z n 4 7.82 Z Z 6 n 6 n 0.35 0.3 Z n 8 5.69 Est imat ed Omega Est imat ed Zet Est imat ed Zet a a Estimated u 0 0.85 15.83 0.29 11.37 1 25.5 50 1 25.5 50 0 0.27 3.7 0 1 n 25.5 50 1 1 n 25.5 25.5 50 50 1 25.5 50 E E n ' n n n n E n Eu '' ' n n Z Z Zn 6 0.15 n 4 8.91 n 6 0.28 Figure 12a: Estimated 2nd order (=0.1 u=1), (=10 u=1)and Z n 8 5.69 u (=10 =0.1) using algorithm 3 G=300 (o=2, o=0.1, z=4, z=0.2 2 1 25 49 0.85 0.69 , u=2,uz=0.1) 11 25 25 49 49 ' 0 n nn 1 25 49 ' n Figure 11: Estimated , and u using algorithm 2 ' Figure 11a: Estimated 2nd order (=0.1 u=1), (=10 13.55 15.84 Estimated Omega Estimated Omega Estimated Zeta Zeta Estimated Estimated Zeta 0.24 8.720.98 11.37 Estimated u u=1)and u (=10 =0.1) using algorithm 2 separated by 1 time step (o=2, o=0.1, z=4, z=0.2, u=2, uz=0.1) E n E n E E E n nn Eu n Z ZZ Z n 4 n 4 6.77 7.92 n Zn 60.3 3.070.47 6 n 6 Z n 8 5.69 Est imat ed Omega Est imat ed Zet a Estimated Zeta Est imat ed u 19.21 2.35 0.93 E EE Eu 0 0.84 n n n n 0 1 25.5 50 2.59 25.5 10.0341 50 0 1 25.5 50 25.5 1 1 25.5 50 50 1 25.5 50 Z Z Z 0.68 n n n 4 10.6 n n 6 0.47 6 Z n 8 15.52 n ' nn '' n ' Figure 12b: Estimated 2nd order (=0.1 u=1), (=10 u=1)and 2 1 u (=10 =0.1) using algorithm 3 G=1000 (o=2, o=0.1, z=4, z=0.2, u=2,uz=0.1) 1 23.5 46 0.011 23.5 46 20.21 n 1 23.5 n 46 ' 1 23.5 46 ' n n ' Figure 11: Estimated , and u using algorithm 2 Figure 11b: Estimated 2 order (=0.1 u=1), (=10 u=1)and u (=10 nd =0.1) using algorithm 2 separated by 4 time steps (o=2, o=0.1, z=4, Est imat ed Omega Est imat Zeta Estimated ed Zet a Estimated u z=0.2, u=2, uz=0.1) 13.5 Estimated Omega 26.22 Estimated Zeta 11.37 15.84 0.24 E E E n Eu Est imat ed Omega Est imat Zet a Est imat eded Zet a Est imat ed u n n n 15.83 0.12 11.37 E E n n Z ZZn 6 11.93 Z n 8 n 4 6.75 n 6 0.48 5.69 E E Z Z n E n n Eu n n 4 7.92 n 6 0.3 Z Z n 4 8.91 Z 6 n 0.35 Z n 6 0.24 n 8 5.69 0 2.35 0.029 0 1 25.5 50 1 1 25.5 25.5 50 50 1 25.5 50 2 0.82 0 0.84 n n 1 24.5 48 0.5 1 24.5 48 0 1 n 25.5 50 1 25.5n 50 ' ' ' n 1 n 24.5 48 ' 1 24.5 48 ' n n n n Figure 11: Estimated , and u using algorithm 2 ' ' Figure 12c: Estimated 2nd order (=0.1 u=1), (=10 u=1)and Figure 11c: Estimated 2 order (=0.1 u=1), (=10 u=1)and u (=10 nd u (=10 =0.1) using algorithm 3 G=2000 (o=2, o=0.1, =0.1) using algorithm 2 separated by 2 time steps (o=2, o=0.1, z=4, z=4, z=0.2, u=2,uz=0.1) z=0.2, u=2, uz=0.1) C. Algorithm 3 5. CONCLUSIONS Program 9 [Zreiba, 1999, Appendix A] was used to obtain results for the single point algorithm. The accuracy of the Three algorithms were developed for the estimation of derivatives is very important and is related to the filter natural frequency (), damping ratio () and input (u) of a gain; three sets of results were obtained to correspond to 3 second order system. These were estimated by using three gain values of 300, 800 and 2000 as shown in figure 12.a, points of x, x' and x'', two points of x, x', x'' and x''', and 12.b and 12.c respectively. one point of x, x', x'', x''' and x''''. The algorithms were tested for 3 categories of parameters: constant, variable with 1st order dynamics and variable with 2nd order dynamics. The algorithms were implemented using Mathcad. For the constant parameter case, they produced accurate estimations of , and u, over a range of set values. Using algorithm 1, the simulations showed better estimations than the other two algorithms because no higher derivatives were used, which normally contribute to numerical errors. For the variable parameter case with 1st order dynamics, Kalman, R, "A New Approach to Linear Filtering and algorithm 3 gave better predictions for and u than Prediction Problems", Tans. Of SAME: Journal of Basic algorithms 1 and 2. For , however, the accuracy of Eng., series D, 82, PP. 35-45 (1960). estimation in algorithm 1 was highest, closely followed by algorithm 2 then algorithm 3. Mathcad 7 Professional Program. Variable parameters with 2nd order dynamics proved to be Zreiba, A, "MathCad Programs for Parameter the most testing as expected. Algorithm 3 gave better Estimation", Research Report, Dept of Computing, The results for and u than algorithms 1 and 2; , however, Nottingham Trent University, Nottingham, 1999. proved to be problematic, with algorithm-1 giving marginally better estimate than algorithm-2 which in turn was better than algorithm-3. This is thought to be due to David Al-Dabass graduated from Imperial College in the increasing use of higher derivatives in the latter 2 1966 with BSc in Electrical Engineering, worked for cases. Different values of gain were tried in algorithm-3 to Redifon Flight Simulation until 1972, completed a PhD in improve the accuracy of the higher derivatives which Parallel Processing at Staffordshire University in 1975 resulted in marginal improvement as the gain was and held post-doctoral and advanced research fellowships increased from 300 to 1000 and finally to 2000. In (76-82) at the Control Systems Centre, UMIST. He joined practice, however, this gain increase will reduce the The Nottingham Trent University in 1983 as a Principal estimator's noise immunity,- there seems to be a clear Lecturer in the Department of Computing. compromise between these two effects. Future work will explore the insights and speed benefits Abdalla Zreiba was born in Tripoli in 1964. He received gained by mapping the algorithms as parallel artificial the BSc. degree in computer engineering from El-Fatah neural net architectures with decision making features to University in 1990. From 1990 until 1996, he worked at select the most suitable algorithm. Estimating is a form the Bironi Remote Sensing Company as computer of information mining using numerical frequency de- engineer. He is currently carrying out research for the modulation. Simultaneously varying more than one MPhil degree at the Dept of Computing, The Nottingham parameter and the effect of measurement noise on the Trent University. His areas of interest include control estimation accuracy need further research. On the systems. theoretical side, the method need to be generalised in terms of matrix formulation to higher order system dimensions. REFERENCES Al-Dabass, D, "Characteristics Estimation of Point Objects using Stereo Vision", IEE Colloquium on Digital Processing, Savoy Place, London, 1985. Goodwin, C, "Real Time Recursive Block Parameter Estimation of Second Order Systems", PhD Thesis, Dept. of Computing, The Nottingham Trent University, Nottingham, 1997. Kailath, T, "Lectures on Linear Least-Squares Estimation", CISM courses and lectures No. 140, Spring- Verlag, New York, 1978. Gersch, W, "Least Squares Estimates of Structural System Parameters using Covariance Function Data", EEE Trans. On auto. Control, 19(6), 1974. Man, Z, "Parameter-Estimation of Continuous Linear Systems using Functional Approximation", Computers and Electrical Eng. Vol. 21, No. 3, pp. 183-187 (1995). Cawley, P, "The reduction of Bias Error in Transfer Function Estimates using FFT-based Analysers", Journal of Vibration, Acoustics, Stress and Reliability in Design, pp.29-35 (1984). Dewolf, D, and D. Wiberg, "An Ordinary Differential- Equation Technique for Continuous Time Parameter Estimation", IEEE Trans. On Auto. Control, Vol. 38, No. 4, PP. 514-528 (1993).