Nonlinear Signal Processing Vs. Kalman Filtering - Circuits and Systems, 1993., ISCAS '93, 1993 IEEE International Symposium on

Most of the research actlvltles in clrcults and signal processing, whether analog or dlgltal, are conflned to linear signal processing (LSP). The Interest in LSP Is due largely to Its mathematlcal tractabllity and ease of implementation, characteristics that were necessary for the technological climate of the earller decades. However, LSP Is limited In Its usefulness and the technology for Implementing algorithms and analyzlng, modellng and design of complex systems has progressed tremendously. Further, we need to understand and utllize nonllnear signal processlng (NLSP) before we can move to large-scale self-organizing or selflearning systems. In our work we have developed a Unique and powerful paradlgm or methodology for the design of nonlinear dynamical systems and signal processing algorlthms. In thls paper, we wlil discuss some aspects of thls approach and show results of appllcatlon of thls approach In slgnai estlmation. We also compare the obtalned results wlth that of Kalman fllterlng, whlch Is essentially fllterlng uslng a fllter wlth a tlme-varying coefflclent (Kaiman gain). The results Indicate that the new NLSP approach is superior and more robust as compared to Kalman fllterlng. stable systems, and this property holds as long as the individual element values are maintained in their permissible range of values. Thus, to design complex nonlinear systems (a nonlinear signal processor for tracking, for example) and self-organizing systems, one simply has to force the dynamics of those systems to mimic the dynamics of a properly constructed passive nonlinear network, a p!ocess akin to reverse engineering. In our research we have developed the basis for the above approach and applied it with relative ease to a number of problems leading to encouraging results. The fruits of such an approach seems to be endless. For example, the approach can be applied to NLSP, linear and nonlinear controller design (for linear and nonlinear plants), self tuning controllers, model reference adaptive controllers, self-organizing networks, adaptive IIR filter design, adaptive beamforming, two-dimensional systems, fuzzy systems etc. In this paper we provide some details of this approach and show results of application of this approach in signal estimation. We also show results comparing our approach to Kalman filtering, [5], and its superiority. I I . PROPOSED METHOD I . INTRODUCTION Signal processing, which can be considered as a subset of intelligent information processing is stuck primarily at the simple level of linear processing, [l], [2]. However intelligent information processing is by nature nonlinear, and time varying (in terms of memory, application of rules and learning capability), and there are clear indications that the nonlinearhime varying processing account for most of the "intelligent" results outcome, [3], [4]. Thus, the need arises for systematic approaches for the design of nonlinear and time-varying signal processors. This paper is concerned with the first, that of designing stable nonlinear signal processing systems. The analytical approach, that is defining proper models for nonlinear differential equations (NLDE) and analyzing the resulting models is the most commonly used technique in many areas, and NLSP is no exception to this trend. However, this approach has not been very successful in NLSP since it is known that even simple first order NLDE can lead to chaotic signals a situation good to produce nice plots etc, but of not much use in design of stable systems. In our research, which is in its early stages, we have adopted what one may term as an engineering approach or building block approach. Under this approach, we have used the concept of passivity to define a number of elements as building blocks for passive nonlinear electrical circuits. Complex passive nonlinear networks can then be formed by proper interconnection of these various nonlinear elements. When energy storing elements are present in such a network (which themselves can be linear or nonlinear), we can obtain a set of inpuffoutput relationships as nonlinear differential equations. The basic property that the network is lossy (consumes energy) ensures that the nonlinear differential equations obtained from the networks would represent absolutely Our approach for nonlinear signal processor design (which is equally applicable to a number of problem domains in the nonlinear systems arena) is based on an entirely new and interesting paradigm. It may be called an "engineering" or a "building block approach for NLSP design as opposed to the analytical/mathematical point of view adopted by earlier researchers. An engineering or physically motivated approach tries to take into consideration physical properties and constraints based on physical properties at every stage of the design. Passivity formulation (to be defined shortly) is used here to obtain the necessary building blocks for a nonlinear system design. These building blocks can then be interconnected in a proper manner to obtain the general structure for a nonlinear system. To design any system, we can use this general structure and vary the parameters so as to obtain the desired properties. To obtain a digital nonlinear system, we can extract the nonlinear differential equations from the general structure and use a forward difference operator. Passivity is a term commonly used in electrical network theory to indicate consumption of energy. A passive electrical element (linear or nonlinear) is one which always consumes power/energy (lossy) or at most, consumes no powedenergy (lossless). They can be nondynamic (no memorykan't store energy) or dynamic (stores energy and gives it back at some other time). They can be two-terminal (one-port) elements or multi-terminal (multi-port) devices. A passive linearhonlinear network is simply an electrical network formed by proper interconnection of various passive linearhonlinear elements. The interconnections must be such that the basic circuit laws are obeyed. An important property of such networks is that they are stable and remain so as long as the values of individual elements remain in the permissive range 36 0-7803-1254-6/93$03.0