Application Of Controlled Random Search Optimization Technique In MMLE With Process Noise

Abstract

Generally in most of the applications of estimation theory using the Method of Maximum Likelihood Estimation (MMLE) to dynamical systems one deals with a situation where only the measurement noise alone is present. However in many present day applications where modeling errors and random state noise input conditions occur it has become necessary for MMLE to handle measurement noise as well as process noise. The numerical algorithms accounting for both measurement and process noise require significantly an order of magnitude higher computer time and memory. Further more, implementation difficulties and convergence problems are often encountered. Here one has to estimate the quantities namely, the initial state error covariance matrix Po, measurement noise covariance matrix R, the process noise covariance matrix Q and the system parameter 0 and the present work deals with the above.

Since the above problem is fairly involved we need to have a good reference solution. For this purpose we utilize the approach and results of Gemson who considered the above problem via the extended Kalman filter (EKF) route to compare the present results from the MMLE route. The EKF uses the unknown parameters as additional states unlike in MMLE which uses only the system states.

Chapter 1 provides a brief historical perspective followed by parameter identification in the presence of process and measurement noises. The earlier formulations such as natural, innovation, combined, and adaptive approaches are discussed.

Chapter 2 deals with the heuristic adaptive tuning of the Kalman filter parameters for the matrices Q and R by Myers and Tapley originally developed for state estimation problems involving satellite orbit estimation. It turns out that for parameter estimation problems apart from the above matrices even the choice of the initial covariance matrix Po is crucial for obtaining proper parameter estimates with a finite amount of data and for this purpose the inverse of the information matrix for Po is used. This is followed by a description of the original Controlled Random Search (CRS) of Price and its variant as implemented and used in the present work to estimate or tune Q, R, and 0 which is the aim of the present work. The above help the reader to appreciate the setting under which the present study has been carried out.

Chapter 3 presents the results and the analysis of the estimation procedure adopted with respect to a specific case study of the lateral dynamics of an aircraft involving 15 unknown parameters. The reference results for the present work are the ones based on the approach of Gemson and Ananthasayanam (1998). The present work proceeds in two phases. In the first case (i) the EKF estimates for Po, Q, and R are used to obtain 0 and in the second case (ii) the estimate of Po and Q together with a reasonable choice of R are utilized to obtain 0 from the CRS algorithm. Thus one is able to assess the capability of the CRS to estimate only the unknown parameters.

The next Chapter 4 presents the results of utilizing the CRS algorithm with R based on a reasonable choice and for Po from the inverse of the information matrix to estimate both Q and 0. This brings out the efficiency of MMLE with CRS algorithm in the estimation of unknown process noise characteristics and unknown parameters. Thus it demonstratesthofcdifficult Q can be estimated using CRS technique without the attendant difficulties of the earlier MMLE formulations in dealing with process noise.

Chapter 5 discusses the - implementation of CRS to estimate the unknown measurement noise covariance matrix R together with the unknown 0 by utilizing the values of Po and Q obtained through EKF route. The effect of variation of R in the parameter estimation procedure is also highlighted in This Chapter. This Chapter explores the importance of Po in the estimation procedure. It establishes the importance of Po though most of the earlier works do not appear to have recognized such a feature. It turned out that the CRS algorithm does not converge when some arbitrary value of Po is chosen. It has to be necessarily obtained from a scouting pass of the EKF. Some sensitivity studies based on variations of Po shows its importance. Further studies shows the sequence of updates, the random nature of process and measurement noise effects, the deterministic nature of the parameter, play a critical role in the convergence of the algorithm.

The last Chapter 6 presents the conclusions from the present work and suggestions for further work.