Anticausal Systems Research Articles

Revisiting causality using stochastics: 2. Applications

In a companion paper, we develop the theoretical background of a stochastic approach to causality with the objective of formulating necessary conditions that are operationally useful in identifying or falsifying causality claims. Starting from the idea of stochastic causal systems, the approach extends it to the more general concept of hen-or-egg causality, which includes as special cases the classic causal, and the potentially causal and anti-causal systems. The framework developed is applicable to large-scale open systems, which are neither controllable nor repeatable. In this paper, we illustrate and showcase the proposed framework in a number of case studies. Some of them are controlled synthetic examples and are conducted as a proof of applicability of the theoretical concept, to test the methodology witha prioriknown system properties. Others are real-world studies on interesting scientific problems in geophysics, and in particular hydrology and climatology.

Revisiting causality using stochastics: 1. Theory

Causality is a central concept in science, in philosophy and in life. However, reviewing various approaches to it over the entire knowledge tree, from philosophy to science and to scientific and technological applications, we locate several problems, which prevent these approaches from defining sufficient conditions for the existence of causal links. We thus choose to determine necessary conditions that are operationally useful in identifying or falsifying causality claims. Our proposed approach is based on stochastics, in which events are replaced by processes. Starting from the idea of stochastic causal systems, we extend it to the more general concept of hen-or-egg causality, which includes as special cases the classic causal, and the potentially causal and anti-causal systems. Theoretical considerations allow the development of an effective algorithm, applicable to large-scale open systems, which are neither controllable nor repeatable. The derivation and details of the algorithm are described in this paper, while in a companion paper we illustrate and showcase the proposed framework with a number of case studies, some of which are controlled synthetic examples and others real-world ones arising from interesting scientific problems.

Data-driven fault estimation of non-minimum phase LTI systems

Many recently developed data-driven fault estimation methods are restricted to minimum-phase systems so that their practical applications are limited. In this paper, the data-driven fault estimation for non-minimum phase (NMP) systems is studied, for which the main difficulty is that the unstable zeros of an NMP system will result in a growing fault-estimation error. To deal with this problem, the inverse of an NMP system is equivalently formulated as a mixed causal and anti-causal system, and the proposed fault estimator is the sum of a stable causal filter and a stable anti-causal filter. The proposed fault estimator is shown to be asymptotically unbiased and its performance is demonstrated by numerical simulations.

Algebraic structure of the anti-causal system

This paper presents a behavioral framework, as developed by J.C. Willems, of a discrete time anticausal pure predictor system. This study is deduced by identifying that the generalized state space of the system has a natural polynomial module structure over the formal series ring.

Error Analysis of the Anti-Causal Realization of Periodic Inverse Systems

A causal realization of an inverse system can be unstable, and an anti-causal realization is used to deal with this problem to provide a numerically stable procedure to realize the inverse system and compute its input signal. In this paper, we consider the anti-causal realization of the inverse of discrete-time linear periodic systems obtained by an outer-inner factorization approach. It presents an analysis and understanding of the inverse system error caused by different parts of the system components. The analysis shows that the inverse system error due to the mismatching of the system initial state in the anti-causal inverse system is inevitable in practical computations.

A frequency domain subspace algorithm for mixed causal, anti-causal LTI systems

The paper extends the subspacc identification method to estimate state-space models from frequency response function (FRF) samples, proposed by McKelvey et al. (1996) for mixed causal/anti-causal systems, and shows that other frequency domain subspace algorithms can be extended similarly. The method is demonstrated by simulation experiments.

A Subspace Model Identification Solution to the Identification of Mixed Causal, Anti-Causal LTI Systems

This paper describes the modification of the family of MOESP subspace algorithms when identifying mixed causal and anti-causal systems. It is assumed that these class of systems have a regular pencil $zE - A$, where E is possibly singular. The key numerical problem in solving this identification problem is the separation of the extended observability matrix of the causal part from that of the anti-causal part when a mixture of both is determined from the input–output data. For the general mixed causal, anti-causal case, this requires a partial calculation of the Kronecker canonical form of the pencil $zE - A$, where the pair $[ A \, E ]$ has been determined from the recorded input--output data. For the descriptor case, that is, when E is nilpotent, this problem is solved without computing the Kronecker canonical form. All existing members of the MOESP family applicable to causal, linear, time-invariant systems are generalized. This allows a broad scope of identification problems for mixed causal, anti-causal systems to be addressed.

Connections between minimum entropy control and mixed [formula omitted]2/ [formula omitted]∞ control for time-varying systems

In this paper two properties of minimum entropy control for linear time-varying systems are investigated. An entropy formulation for anti-causal systems is first considered. Minimizing the norm of an anticausal system is straightforward because of the fact that the norm of an operator is equal to the norm of its adjoint. This is not true for the entropy measure. For this reason a separate formula for the entropy of anti-causal systems is needed in order to consider dual problems that arise in optimal control theory. Secondly, we investigate the relationship between the entropy measure for time-varying systems and the H 2 and H ∞ norms. For time-invariant systems, minimum entropy control has been shown to be a good mixed H 2/ H ∞ controller. In this paper we show that this is also true in the time-varying case.

Identification of Descriptor Systems and Periodic Linear, Time-Varying Systems

This paper describes the extension of MOESP (standing for multivariable output error state space model identification schemes as introduced in Verhaegen, et.al, 1992) family of algorithms to identify periodically time-varying linear systems and LTI mixed causal and anti-causal systems. The latter class of generalized state space systems include descriptor systems. The identification scheme to identify periodically time-varying systems is applied to multi-rate sampled data systems.

Dynamical structures on pencils, poles, and fundamental subspaces

This paper provides a bridge between singular-system representations, module theory, and the geometric theory of subspaces. By means of a theorem, the “little” pole space associated with a singular system is shown to be isomorphic to the global system pole space, a module-theoretic concept incorporating both finite- and infinite- system-pole information. These results hold true for any regular matrix pencil ( zE − A) and may be interpreted as special cases in terms of causal and anticausal systems. Moreover, fundamental subspaces associated with matrix pencils are isomorphic to k[ z]- and O ∞-modules representing finite and infinite system poles. As a result, dynamical structures on these limit spaces may be determined.

An optimal absorbing boundary condition for finite difference modeling of acoustic and elastic wave propagation

This paper presents an optimal absorbing boundary condition designed to model acoustic and elastic wave propagation in two-dimensional and three-dimensional media using the finite difference method. In this condition, extrapolation on the artificial boundaries of a finite difference domain is expressed as a linear combination of wave fields at previous time steps and/or interior grids. The acoustic and elastic reflection coefficients from the artificial boundaries are derived. They are found to be identical to the transfer functions of two cascaded systems—the inverse of a causal system and an anticausal system. The method makes use of the zeros and poles of reflection coefficients in a complex plane. The optimal absorbing boundary condition described in this paper yields, on the average, reflection coefficients about 10-dB smaller than Higdon’s absorbing boundary condition, and about 20-dB smaller than Reynolds’ absorbing boundary condition.

A new general state-space representation for discrete-time systems

Abstract A new general state-space representation for causal and anticausal discrete-time systems is introduced. Some of its properties related to systems and network theory concepts are developed. The relation to the descriptor form state-space representation is also explained. Applications are given to show the usefulness of the new representation.

A Minimum-Phase LU Factorization Preconditioner for Toeplitz Matrices

A new preconditioner is proposed for the solution of an $N \times N$ Toeplitz system $T_N {\bf x} = {\bf b}$, where $T_N $ can be symmetric indefinite or nonsymmetric, by preconditioned iterative methods. The preconditioner $F_N $ is obtained based on factorizing the generating function $T(z)$ into the product of two terms corresponding, respectively, to minimum-phase causal and anticausal systems and is therefore called the minimum-phase LU (MPLU) factorization preconditioner. Due to the minimum-phase property, $||F_N^{ - 1} ||$ is bounded. For rational Toeplitz matrices $T_N $ with generating function $T(z) = A(z^{ - 1}/Bz^{ - 1} ) + C(z)/D(z)$, where $A(z)$, $B(z)$, $C(z)$, and $D(z)$ are polynomials of orders $p_1 $, $q_1 $, $p_2 $, and $q_2 $, it is shown that the eigenvalues of $F_N^{ - 1} T_N $ are repeated exactly at 1 except at most $\alpha _F $ outliers, where $\alpha _F $ depends on $p_1 $, $q_1 $, $p_2 $, $q_2 $, and the number w of the zeros of $\tilde T(z) = A(z^{ - 1} )D(z) + B(z^{ - 1} )C(z)$ outside the unit circle. A preconditioner $K_N $ in circulant form generalized from the symmetric case is also presented for comparison.

Similarity and Reduction for Time Varying Linear Systems with Well-Posed Boundary Conditions

The classical theory of similarity and reduction of causal and anti-causal systems is generalized and extended to time varying linear systems with well-posed boundary conditions. Two main problems are solved. The first is to determine to what extent a system with boundary conditions can be simplified by similarity and reduction. The second problem is to find the invariants of such a simplification. Special attention is given to the time invariant case.

On Time-Related Properties of Nonlinear Systems

The present article investigates the structure of nonlinear systems with respect to time-related behavior. In a broad sense this study falls within the general context of causality theory. In particular, it is consistent with the recent causality work of Porter [14], [15], and Saeks [16]. To distinguish this paper from the references it is noted that here linear and nonlinear systems as well as causal and noncausal are treated on an equal footing. In this regard a system is modeled as an operator on a group-valued function space and various time-related behavior classes of systems are considered and studied. They consist of the classes of causal, anticausal, and memoryless systems which have already been treated in the technical literature plus an additional class of systems called crosscausal. The main result of the paper can be stated as follows: Every system in a group resolution space is given by the sum of causal, anticausal, memoryless and crosscausal systems.