In recent years there has been much interest in robustness issues in general and in robust signal processing schemes in particular. Robust schemes are useful in situations where imprecise a priori knowledge of input characteristics makes the sensitivity of performance to deviations from assumed conditions an important factor in the design of good signal processing schemes. In this survey we discuss the minimax approach for the design of robust methods for signal processing. This has proven to be a very useful approach because it leads to constructive procedures for designing robust schemes. Our emphasis is on the contributions which have been made in robust signal processing, although key results of other robust statistical procedures are also considered. Most of the results we survey have been obtained in the past fifteen years, although some interesting earlier ideas for minimax signal processing are also mentioned. This survey is organized into five main parts, which deal separately with robust linear filters for signal estimation, robust linear filters for signal detection and related applications, nonlinear methods for robust signal detection, nonlinear methods for robust estimation, and robust data quantization. The interrelationships among many of these results are also discussed in the survey.

Robust techniques for signal processing: A survey

Necessary and sufficient conditions are derived for the existence of a solution to the continuous-time and discrete-time reduced-order H/sub /spl infin// and L/sub 2/-L/sub /spl infin// filtering problems. These conditions are expressed in terms of linear matrix inequalities (LMIs) and a coupling nonconvex matrix rank constraint. Convex LMI problems are obtained for the full-order and the zeroth-order filtering. An explicit parametrization of all reduced-order filters that correspond to a feasible solution is derived in terms of a contractive matrix, and iterative algorithms are proposed to solve the reduced-order filtering problems using alternating projections.

Reduced-order H/sub /spl infin// and L/sub 2/-L/sub /spl infin// filtering via linear matrix inequalities

The minimax approach to the design of systems that are robust with respect to modeling uncertainties is studied using a game theoretic formulation in which the performance functional and the sets of modeling uncertainties and admissible design policies are arbitrary. The existence and characterization of minimax robust solutions that form saddle points are discussed through various methods that take into account several common features of the games encountered in applications. In particular, it is shown that if the performance functional and the uncertainty set are convex then a certain type of regularity condition on the functional is sufficient to ensure that the optimal strategy for a least favorable element of the uncertainty set is minimax robust. The efficacy of the methods proposed for a general game is tested in the problems of matched filtering, Wiener filtering, quadratic detection, and output energy filtering, in which uncertainties in their respective signal and noise models are assumed to exist. These problems are analyzed in a common Hilbert space framework and they serve to point out the advantages and limitations of the proposed techniques.

On minimax robustness: A general approach and applications

Two phases of substorm-associated magnetospheric dynamics are discussed in terms of particles and fields at synchronous orbit. The first phase corresponds to the 'decreases' of energetic particle flux. The second phase is the conventional expansion phase that begins with the 'onset', characterized in the study by (1) a sudden decrease in the tail current and a return of the inflated magnetosphere to a dipolelike configuration, (2) a sudden shift of trapped high-energy particles toward the tail again following contours of constant B, and at the same time (3) a surge of tail plasma toward the earth as the induced electric field now increases the total convection field.

Dynamics of plasma, energetic particles, and fields near synchronous orbit in the nighttime sector during magnetospheric substorms

Error equations for inertial navigation systems are derived using a perturbation (or true frame) approach and a psi angle (or computer frame) approach in a manner which shows the underlying as sumptions and allows direct comparison of the two methods. The comparison is general since the analysis is not associated with any particular mechanization. Different definitions of velocity errors and misalignment angles result from the two methods of error analysis, and, consequently, have significance in testing and analysis of pure-inertial systems, Doppler-inertial systems, and inertially aided weapon delivery systems. Examples and numerical results are presented for a local-level north-pointing mechanization.

A Comparison of Two Approaches to Pure-Inertial and Doppler-Inertial Error Analysis

The evolution of the application of the Kalman filter in the aerospace arena is traced. The major programs that were the driving forces for the filter's acceptance are noted, as are the specific threads of activity for refining and enhancing the initial contribution. These efforts brought the fundamental ideas presented by Kalman to the point where actual application was possible. Clearly the concepts of the Kalman filter are now "mature." This is also noted and substantiated.

The Kalman Filter Applied to Aerospace and Electronic Systems

A minimax approach to the design of low sensitivity state estimators

A comparison of the error propagation in a local-level reference frame is derived for two inertial navigation systems; one has a local-level configuration, and the other has a space-stable configuration. The error propagation is shown to be equivalent for the two cases considered. This equivalence is demonstrated by starting with the error propagation equations for the space-stable system and transforming them to a local-level reference frame. The transformed equations are then compared with the classical local-level error equations, and the equivalence is noted. The specific implementation used in each case considers velocity damping but not altitude damping.

Comparison of Error Propagation in Local-Level and Space-Stable Inertial Systems

This paper presents an algorithm for a class of suitably constrained reduced-order filters which minimize the variance of the estimated variables. The algorithm generates both the filter gain history and the true estimation error covariance. The algorithm provides a quantitative criterion which can be used to measure the performance of any reduced-order estimator. Both continuous and discrete estimators are considered. Several examples are treated including an application of the technique to a hybrid navigation system of high order.

Applications of Mininum Variance Reduced-State Estimators

In order to stabilize the altitude calculation in an inertial navigation system, an altimeter is commonly used. In a conventional local-level mechanization, this is generally accomplished by correcting the vertical channel integrators with the difference between the inertial system and altimeter indication of vertical position. However, in a space-stable system the procedure is not as clear since a vertical channel is not physically present. Three altitude damping mechanizations for a space-stable inertial navigation system are proposed. The equivalent local-level mechanizations are then found by comparing error propagation equations in a common coordinate frame.

Charles E. Hutchinson

Papers

The Kalman Filter Applied to Aerospace and Electronic Systems

A minimax approach to the design of low sensitivity state estimators

Comparison of Error Propagation in Local-Level and Space-Stable Inertial Systems

Applications of Mininum Variance Reduced-State Estimators

Altitude Damping of Space-Stable Inertial Navigation Systems