OKID as a general approach to linear and bilinear system identification

This work advances the understanding of the complex world of system identification, i.e. the set of techniques to find mathematical models of dynamical systems from measured input-output data, and exploits well-established approaches for linear systems to address nonlinear system identification problems. We focus on observer/Kalman filter identification (OKID), a method for simultaneous identification of a linear state-space model and the associated Kalman filter from noisy input-output measurements. OKID, developed at NASA, resulted in a very successful algorithm known as OKID/ERA (OKID followed by eigensystem realization algorithm). We show how ERA is not the only method to complete the OKID process, developing novel algorithms based on the preliminary estimation of the Kalman filter output residuals. The new algorithms do not only show potential for better performance, they also cast light on OKID, explicitly establishing the Kalman filter as central to linear system identification in the presence of noise, paralleling its role in signal estimation and filtering. The Kalman filter embedded in the OKID core equation is capable of converting the original problem, affected by random noise, into a purely deterministic problem. The new interpretation leads to the extension of OKID to output-only system identification, providing a new tool for applications in structural health monitoring, and raises OKID to the level of a unified approach for input-output and output-only linear system identification. Any algorithm for linear system identification formulated in the absence of noise can now optimally handle noisy data via a preliminary step consisting in solving the OKID core equation. The OKID framework developed for linear system identification is then extended to bilinear systems, which are of interest because several natural phenomena are inherently bilinear and also because high-order bilinear models are universal approximators for a wide class of nonlinear systems. The formulation of an optimal bilinear observer for bilinear state-space models, similar to the Kalman filter in the linear case, leads to the development of an extension of OKID to bilinear system identification. This is the first application of OKID to nonlinear problems, not only because bilinear systems are themselves nonlinear, but also because one can think of bilinear OKID as a technique to find bilinear approximations of nonlinear systems. Furthermore, the same strategy adopted in this work could be used to extend OKID directly to other classes of nonlinear models

sensorimotor states affect choice in the magnitude judgment of ambiguous durations

66 words) The statistics of the environment seem to exert optimal influence on the organization of functions subserving decision making. In order to make decisions about ambiguous sensory information, predictive coding models suggest that brain generate a template against which to match observed sensory evidence. Here we challenge this notion providing evidence that stochastic choices about the magnitude judgment of visual duration are triggered by bottom-up sensorimotor information. Main Text (1204 words, including acknowledgements and references) The statistics of the environment seem to exert optimal influence on the organization of functions subserving decision making. "Predictive coding" models suggest that whenever a clear outcome is not available, the brain resolves perceptual ambiguity by anticipating the forthcoming sensory environment, generating a template against which to match observed sensory evidence. Accordingly, decisions that we make are often guided by the outcomes of similar decisions made in the past. Nevertheless, everyday life teaches us that higher-level processes, such as voluntary choice, have often proved themselves to be immune to previous experience. Task-irrelevant information may influence behavior, not always orienting decision making toward ecologically optimal deeds. Grounded cognition provided some insight in this direction focusing on the role of the body in cognition, based on widespread findings that bodily states can cause cognitive states and be effects of them. Here we investigate if low-level sensorimotor manipulation affects performance of observers whose attempt to generate magnitude decision about ambiguous durations. To this purpose head turning to the left or to the right space was selectively manipulated in two separated experiments. Lateral head turns are known to reallocate spatial attention in the outside world. Two groups of participants had to judge the duration of a test stimulus as longer or shorter with respect to a reference stimulus, once with their head kept straight (baseline) and once while turning their head. We tested groups' performance in two separate experiments: one during the temporal encoding/storage of the reference stimulus; the other during the retrieval/comparison of the duration of the reference stimulus with that of the test one (Figure 1a, supplementary method). To create an ambiguous vs. unambiguous temporal outcome the duration' stimuli were manipulated by using a loglinear temporal distance from the reference. Data analysis of both baseline sessions specified th

Linear State Representations for Identification of Bilinear Discrete-Time Models by Interaction Matrices

Bilinear systems can be viewed as a bridge between linear and nonlinear systems, providing a promising approach to handle various nonlinear identification and control problems. This paper provides a formal justification for the extension of interaction matrices to bilinear systems and uses them to express the bilinear state as a linear function of input-output data. Multiple representations of this kind are derived, making it possible to develop an intersection subspace algorithm for the identification of discrete-time bilinear models. The technique first recovers the bilinear state by intersecting two vector spaces that are defined solely in terms of input-output data. The new input-output-to-state relationships are also used to extend the Equivalent Linear Model method for bilinear system identification. Among the benefits of the proposed approach, it does not require data from multiple experiments, and it does not impose specific restrictions on the form of input excitation

Bilinear System Identification by Minimal-Order State Observers

Bilinear systems offer a promising approach for nonlinear control because a broad class of nonlinear problems can be reformulated and approximated in bilinear form. System identification is a technique to obtain such a bilinear approximation for a nonlinear system from input-output data. Recent discrete-time bilinear model identification methods rely on Input-Output-to-State Representations (IOSRs) derived via the interaction matrix technique. A new formulation of these methods is given by establishing a correspondence between interaction matrices and the gains of full-order bilinear state observers. The new interpretation of the identification methods highlights the possibility of utilizing minimal-order bilinear state observers to derive new IOSRs. The existence of such observers is discussed and shown to be guaranteed for special classes of bilinear systems. New bilinear system identification algorithms are developed and the corresponding computational advantages are illustrated via numerical examples

Extension of OKID to Output-Only System Identification

Observer/Kalman filter IDentification (OKID) is a successful approach for the estimation, from measured input-output data, of the linear state-space model describing the dynamic behavior of a structure. From such a mathematical model, it is possible to recover the modal parameters, which can be exploited to update a detailed numerical model of the structure, e.g. a Finite Element Model (FEM), to be used to predict the structural response to future excitation and to evaluate damage scenarios. This paper extends OKID to output-only system identification, i.e. to the case where only the response of the structure is measured and the input is unknown. The approach is suitable for structural health monitoring based on modal parameters, in particular for those civil infrastructures whose excitation is random in nature and in the way it is applied to the structure (e.g. wind, traffic) and therefore is difficult to measure. The paper rigorously proves the applicability of the OKID approach to the output-only case, presents the resulting new algorithms and demonstrates them via a numerical example

An All-Interaction Matrix Approach to Linear and Bilinear System Identification

This paper is a brief introduction to the interaction matrices. Originally formulated as a parameter compression mechanism, the interaction matrices offer a unifying framework to treat a wide range of problems in system identification and control. We retrace the origin of the interaction matrices, and describe their applications in selected problems in system identification

Generalized Framework of OKID for Linear State-Space Model Identification

This paper presents a generalization of observer/Kalman filter identification (OKID). OKID is a method for the simultaneous identification of a linear dynamical system and the associated Kalman filter from input-output measurements corrupted by noise. OKID was originally developed at NASA as the OKID/ERA algorithm. Recent work showed that ERA is not the only way to complete the OKID process and paved the way to the generalization of OKID as an approach to linear system identification. As opposed to other approaches, OKID is explicitly formulated via state observers providing an intuitive interpretation from a control theory perspective. The extension of the OKID framework to more complex identification problems, including nonlinear systems, is also discussed

OKID as a Unified Approach to System Identification

This paper presents a unified approach for the identification of linear state-space models from input-output measurements in the presence of noise. It is based on the established Observer/Kalman filter IDentification (OKID) method of which it proposes a new formulation capable of transforming a stochastic identification problem into a (simpler) deterministic problem, where the Kalman filter corresponding to the unknown system and the unknown noise covariances is identified. The system matrices are then recovered from the identified Kalman filter. The Kalman filter can be identified with any deterministic identification method for linear state-space models, giving rise to numerous new algorithms and establishing the Kalman filter as the unifying bridge from stochastic to deterministic problems in system identification

A Linear-Time-Varying Approach for Exact Identification of Bilinear Discrete-Time Systems by Interaction Matrices

Bilinear systems offer a promising approach for nonlinear control because a broad class of nonlinear problems can be reformulated in bilinear form. In this paper system identification is shown to be a technique to obtain such a bilinear approximation of a nonlinear system. Recent discrete-time bilinear model identification methods rely on Input-Output-to-State Representations. These IOSRs are exact only for a certain class of bilinear systems, and they are also limited by high dimensionality and explicit bounds on the input magnitude. This paper offers new IOSRs where the bilinear system is treated as a linear time-varying system through the use of specialized input signals. All the mentioned limitations are overcome by the new approach, leading to more accurate and less computationally demanding identification methods for bilinear discrete-time models, which are also shown via examples to be applicable to the identification of bilinear models approximating more general nonlinear systems

Observers for Bilinear State-Space Models by Interaction Matrices

This paper formulates a bilinear observer for a bilinear state-space model. Relationship between the bilinear observer gains and the interaction matrices are established and used in the design of such observer gains from input-output data. In the absence of noise, the question of whether a deadbeat bilinear observer exists that would cause the state estimation error to converge to zero identically in a finite number of time steps is addressed. In the presence of noise, an optimal bilinear observer that minimizes the state estimation error in the same manner that a Kalman filter does for a linear system is presented. Numerical results illustrate both the theoretical and computational aspects of the proposed algorithms