Simulation of Real Time Parameter Estimation Algorithms

for Time Varying Systems

MPhil Thesis, March 2000

Abdalla Zreiba

Abstract

To recognise trends embedded in patterns, a dynamical model of the system generating the patterns can be assumed. A second order model has a wide variety of patterns which can serve well in approximately describing the short-term behaviour of complex physical, financial, societal and biological systems. Apart from initial conditions, the output pattern of a simple second order system is completely defined by 3 parameters: natural frequency (w), damping ratio (z) and external input (u).

Three algorithms are proposed and investigated in this study to estimate the parameters of an equivalent second order system from a given trajectory (in time or space) of the pattern. The algorithms combine successive 1^st order filters of specified cut-off frequencies, to provide smoothing and higher order derivative estimation, with non-linear static parameter estimators.

A complete simulation environment is devised enabling the three-parameter estimation algorithms to be tested for 3 categories of parameter sets: constant, variable with 1^st order dynamics and variable with 2^nd order dynamics. When the parameters have 2^nd order dynamics, they themselves may be modelled as having their own unique time varying patterns, i.e. have dynamical behaviour. This leads to a hierarchical parameter estimation process where on-line algorithms are needed to work concurrently with the actual system to provide a continuous estimate of the first level parameters. When these parameters are time varying, then they in turn are submitted as input to another level of parameter estimation algorithm to estimate the parameters of their own dynamics. This process may be repeated, in theory at least, to as many levels as necessary until a set of parameters is found which is constant.

Accurate estimations of w, z and u were made using non-linear combinations of time derivatives of the measured output of the system. Results of the simulations are presented which show that the algorithms can cope well with variable parameters.

The effect of measurement noise on the estimation accuracy is considered when the incoming trajectories are corrupted with random noise. Noise is simulated using a random number generator with zero-mean and added to the simulated system output. Analysis of the simulation results show varying abilities of the algorithms to cope with the noise perturbations. In some instances high prediction robustness were achieved, in others, simulations showed high sensitivity to noise.