Per Bodin

On the construction of orthonormal basis functions for system identification

Abstract In this paper we present an alternative derivation of orthonormal rational families in H 2 , based on previous work by Roberts and Mullis. The construction is based on balanced state space realizations of all-pass transfer functions, and generalizes recent work of Heuberger et al. The emphasis is put on the simplicity of this approach compared with traditional methods. We also present some historical remarks concerning earlier approaches to this problem, as well as completeness ami uniform boundedness conditions for this family of orthonormal functions.

Thresholding in high order transfer function estimation

A problem in prediction error system identification methods is estimation of pole locations. Typically, iterative numerical optimization methods are used. Reliable initial values are then necessary for good results. The parameterization is often done in the coefficients of transfer function polynomials or some canonical form. In this contribution we discuss a couple of issues related to the above problem. First, we study how all-pass systems can be used to generate suitable model structures. This analysis is based on the relation between balanced realizations of all-pass filters and orthonormal basis transfer functions. Next, we investigate the effects of a priori fixed pole locations, such as in Laguerre and Kautz models. One idea is to use very flexible high-order models. However, the corresponding estimation problem has to be regularized in order to reduce the variance errors due to noise. We will discuss how this can be done by using thresholding of the estimated coefficients.<<ETX>>

Selection of Best Orthonormal Rational Basis

This contribution deals with the problem of structure determination for generalized orthonormal basis models used in system identification. The model structure is parameterized by a prespecified set of poles representing a finite-dimensional subspace of

An algorithm for selection of best orthonormal rational basis

{\cal H}^2

Development, Test and Flight of the SMART-1 Attitude and Orbit Control System

.\ Given this structure and experimental data, a model can be estimated using linear regression techniques. Since the variance of the estimated model increases with the number of estimated parameters, one objective is to find coordinates, or a basis, for the finite-dimensional subspace giving as compact or parsimonious a system representation as possible. In this paper, a best basis algorithm and a coefficient decomposition scheme are derived for the generalized orthonormal rational bases. Combined with linear regression and thresholding this leads to compact transfer function representations. The methods are demonstrated with several examples.

PRISMA: An In -Orbit Test Bed for GNC Experiments

This paper deals with the problem of structure determination for generalized orthonormal basis models used in system identification. The model structure is parametrized by a pre-specified set of poles. Given this structure and experimental data a model can be estimated using linear regression techniques. Since the variance of the estimated model increases with the number of estimated parameters, the objective is to find structures that are as compact/parsimonious as possible. A natural approach would be to estimate the poles, but this leads to nonlinear optimization with possible local minima. In this paper, a best basis algorithm is derived for the generalized orthonormal rational bases. Combined with linear regression and thresholding this leads to compact transfer function representations.

Recursive calculation of expansion coefficients for generalized orthonormal rational bases

*This paper presents results from the development, test and flight of the SMART-1 Attitude and Orbit Control System (AOCS). SMART-1 was captured in lunar orbit on November 16, 2004 after more than one year of travel around the earth since its launch on September 27, 2003. The arrival to the moon marks the completion of the primary mission objective which is to demonstrate the use of Electric Primary Propulsion in a low-thrust transfer trajectory. The arrival to the moon also marked the start of the lunar science observation phase. The Swedish Space Corporation is the prime contractor for the SMART-1 spacecraft. The paper explains how the SMART-1 AOCS was developed and tested. The AOCS flight software was produced using automatic code generation. The use of automatic code generation has proven to be an efficient development approach in which accurate code models have been available throughout the development cycle of the software. The code has been developed starting from early analysis models. These models were integrated into one single AOCS subsystem model which was split up into one simulator part and one flight software part. The simulator part was delivered to the real-time spacecraft system simulator while the flight software part was integrated into the over-all on-board software. All the way through this development, it has been possible to simulate the AOCS subsystem in a MATLAB/Simulink environment using the very parts from which code was generated for the system simulator as well as the on-board software. The availability of these accurate code models has been useful for development of tests on unit and system level involving different degrees of hardware in the loop. The paper explains how the software was developed and how it was used in these different levels of testing. The paper also explains how the accurate code models have been used for analysis of in-flight performance.

A Wavelet Shrinkage Approach for Frequency Response Estimation

The PRISMA formation flying satellite project will demonstrate Guidance, Navigation, and Control strategies for advanced autonomous formation flying. The Swedish Space Corporation (SSC) is the prime contractor for the project which is funded by the Swedish National Space B oard (SNSB). By mid 2008, PRISMA is well into the system integration and test campaign and launch is planned for June, 2009. The mission consists of two spacecraft: MAIN and TARGET. The MAIN satellite has full orbit control capability while TARGET is attitude controlled only. PRISMA will perform a series of GNC related formation flying experiments. SSC is responsible for three main sets of experiments: Autonomous Formation Flying, Proximity Operations and Final Approach/Recede Maneuvers, and Autono mous Rendezvous. This paper focus es on describing these three set s of formation flying experim ents. The system test setup is also described together with real -time system level test results demonstrating the performance of each of the three GNC experiment sets.

Selection of local discriminant generalized orthonormal rational basis

Abstract A recursive scheme is derived for calculation of the expansion coefficients of a known system transfer function into a generalized orthonoimal rational basis. The inherent structure of the basis functions leads to a computationally efficient method, which needs the solution of a low-dimensional Sylvester equation at each step. The method is demonstrated with a brief example. Possible applications are also discussed.

Improvements of Frequency Response Estimation Techniques

Abstract A standard approach to estimate the frequency function of a dynamical system is to use spectral estimation. A problem with these methods is that windowing, which reduces variance, results in loss of resolution. New techniques based on wavelet shrinkage developed by Donoho, seem to avoid the windowing problem. They produce near minimax, optimal smoothers for a large class of functions. In some sense they incorporate frequency adaptation, i. e. the shape and width of the window change with frequency. In this paper it is investigated how these methods can be applied to frequency response estimation. Wavelet shrinkage seems to offer alternative and improved solutions to many system identification problems, and is therefore of general interest. A numerical example is presented showing the improved adaptation property for fine resolution details.