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SIMULATION OF PARAMETER ESTIMATION ALGORITHMS

VIEWS: 36 PAGES: 7

									        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. Ex'']
                                    
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 uand
                  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                                EEn                                                       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
                            En
                               n
                                                                             E
                                                                             En                                                         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                                                     EEn                                                                    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 En                                                          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                                                  EEn                                                                     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.bfor 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                                                                  EE                                                   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).

								
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