An Iterative Algebraic Geometric Approach for Identification of Switched ARX Models with Noise

The algebraic geometric (AG) approach has been used to identify switched auto regressive exogenous (SARX) models in hybrid systems, and it has several advantages over other SARX identification methods. This paper is focused on improving the estimation accuracy of the AG approach for systems corrupted with indispensable noises. A stochastic hybrid decoupling polynomial (SHDP) is constructed by reformulating the hybrid decoupling polynomial (HDP) used in the original AG method. An iterative scheme is developed to estimate parameters of the SHDP, which are used to calculate the SARX model parameters. This estimation involves linear regression with multiplicative noises, therefore a novel approach is proposed to solve this regression problem. Then, the parameters are recovered from the SHDP. Finally, all these steps for SARX model identification are summarized in an algorithm called the iterative algebraic geometric (IAG) approach. Simulations and experimental validation results are shown to demonstrate the effectiveness of and the improvement made by the proposed IAG method.     

Equivalence and identifiability analysis of uncontrolled nonlinear dynamical systems

The problem of parameter identifiability has been considered from different points of view in the case of nonlinear dynamical systems. For analytic systems the standard approach for uncontrolled systems is the Taylor series approach (Pohjanpalo, Math. Biosciences 41 (1978) 21), or the approaches based on differential algebra for polynomial and rational systems. The similarity transformation approach, based on the local state isomorphism theorem, gives a sufficient and necessary condition for global identifiability of nonlinear controlled systems. But it leads only to a necessary condition for identifiability in the case of some uncontrolled systems. Our contribution consists in using the equivalence of systems, based on the straightening out theorem, to analyse the identifiability of uncontrolled systems. From this theory, we state the necessary or sufficient identifiability conditions, some of them depending on the state variable dimension.     

How non-zero initial conditions affect the minimality of linear discrete-time systems

From the state-space approach to linear systems, promoted by Kalman, we learned that minimality is equivalent with reachability together with observability. Our past research on optimal reduced-order LQG controller synthesis revealed that if the initial conditions are non-zero, minimality is no longer equivalent with reachability together with observability. In the behavioural approach to linear systems promoted by Willems, that consider systems as exclusion laws, minimality is equivalent with observability. This article describes and explains in detail these apparently fundamental differences. Out of the discussion, the system properties weak reachability or excitability, and the dual property weak observability emerge. Weak reachability is weaker than reachability and becomes identical only if the initial conditions are empty or zero. Weak reachability together with observability is equivalent with minimality. Taking the behavioural systems point of view, minimality becomes equivalent with observability when the linear system is time invariant. This article also reveals the precise influence of a possibly stochastic initial state on the dimension of a minimal realisation. The issues raised in this article become especially apparent if linear time-varying systems (controllers) with time-varying dimensions are considered. Systems with time-varying dimensions play a major role in the realisation theory of computer algorithms. Moreover, they provide minimal realisations with smaller dimensions. Therefore, the results of this article are of practical importance for the minimal realisation of discrete-time (digital) controllers and computer algorithms with non-zero initial conditions. Theoretically, the results of this article generalise the minimality property to linear systems with time-varying dimensions and non-zero initial conditions.     

A tutorial on the positive realization problem

This paper is a tutorial on the positive realization problem, that is the problem of finding a positive state-space representation of a given transfer function and characterizing existence and minimality of such representation. This problem goes back to the 1950s and was first related to the identifiability problem for hidden Markov models, then to the determination of internal structures for compartmental systems and later embedded in the more general framework of positive systems theory. Within this framework, developing some ideas sprang in the 1960s, during the 1980s, the positive realization problem was reformulated in terms of a geometric condition which was recently exploited as a tool for finding the solution to the existence problem and providing partial answers to the minimality problem. In this paper, the reader is carried through the key ideas which have proved to be useful in order to tackle this problem. In order to illustrate the main results, contributions and open problems, several motivating examples and a comprehensive bibliography on positive systems organized by topics are provided.     

Identifiability of linear stochastic systems operating under linear feedback

The identifiability of multiple input-multiple output stochastic systems operating in closed loop is considered for the case where the plant and the regulator are both linear and time-invariant. Two basic identification methods have been proposed for such systems: the joint input-output method, in which the input and output processes are modelled jointly as the output of a white noise driven system; and the direct method, in which a prediction error method is used on the input-output data as if the system were in open loop. Previously obtained identifiability results for the joint input-output method are extended to a number of new situations, including but extending beyond the identifiability results obtained with the direct method.     

Realization theory of discrete-time linear switched systems

The paper presents realization theory of discrete-time linear switched systems. We present necessary and sufficient conditions for an input–output map to admit a discrete-time linear switched system realization. In addition, we present a characterization of minimality of discrete-time linear switched systems in terms of reachability and observability. Further, we prove that minimal realizations are unique up to isomorphism. We also discuss algorithms for converting a linear switched system to a minimal one and for constructing a state-space representation from input–output data. The paper uses the theory of rational formal power series in non-commutative variables.     

Abduction in logic programming: A new definition and an abductive procedure based on rewriting

A long outstanding problem for abduction in logic programming has been on how minimality might be defined. Without minimality, an abductive procedure is often required to generate exponentially many subsumed explanations for a given observation. In this paper, we propose a new definition of abduction in logic programming where the set of minimal explanations can be viewed as a succinct representation of the set of all explanations. We then propose an abductive procedure where the problem of generating explanations is formalized as rewriting with confluent and terminating rewrite systems. We show that these rewrite systems are sound and complete under the partial stable model semantics, and sound and complete under the answer set semantics when the underlying program is so-called odd-loop free. We discuss an application of abduction in logic programming to a problem in reasoning about actions and provide some experimental results.     

Minimal polynomial realizations

This paper studies the problem of realizing input-output maps by polynomial continuous-time systems. This study requires a careful definition of the notions of systems and differential equations on an algebraic variety. A concept of minimality is also introduced, and a uniqueness result is proved.     

State elimination and identifiability of the delay parameter for nonlinear time-delay systems

The identifiability of the delay parameter for nonlinear systems with a single constant time delay is analyzed. We show the existence of input–output equations and relate the identifiability of the delay parameter to their form. Explicit criteria based on rank calculations are formulated. The identifiability of the delay parameter is shown not to be directly related to the well-characterized identifiability/observability of the other system parameters/states.     

Representations and structural properties of periodic systems

We consider periodic behavioral systems as introduced in [Kuijper, M., & Willems, J. C. (1997). A behavioral framework for periodically time-varying systems. In Proceedings of the 36th conference on decision & control (Vol. 3, pp. 2013-2016). San Diego, California, USA, 10-12 December 1997] and analyze two main issues: behavioral representation/controllability and autonomy. More concretely, we study the equivalence and the minimality of kernel representations, and introduce latent variable (and, in particular, image) representations. Moreover we relate the controllability of a periodic system with the controllability of an associated time-invariant system known as lifted system, and derive a controllability test. Further, we prove the existence of an autonomous/controllable decomposition similar to the time-invariant case. Finally, we introduce a new concept of free variables and inputs, which can be regarded as a generalization of the one adopted for time-invariant systems, but appears to be more adequate for the periodic case.     

Identifiability of discrete-time nonlinear systems: The local state isomorphism approach

A new theorem is provided to test the identifiability of discrete-time systems with polynomial nonlinearities. That extends to discrete-time systems the local state isomorphism approach for continuous-time systems. Two examples are provided to illustrate the approach.     

Parameter identifiability for distributed parameter systems of hyperbolic type

In this paper we study questions regarding parameter identifiability for distributed parameter systems of hyperbolic type. The unknown parameters are input distribution functions. We consider the systems with continuous-time input-output data and the systems with discrete-time inputs-output data. For these systems we present the necessary and sufficient conditions for identifiability, in the case of a finite number of sensors. We investigate the relations between the continuous-time input-output systems and the discrete-time input-output systems from the viewpoint of identifiability. Moreover we give the sets of input distribution functions which are N-step identifiable, for the discrete-time input-output systems.     

Identifiability of homogeneous systems using the state isomorphism approach

New results are presented concerning the state isomorphism approach to global identifiability analysis of parameterized classes of nonlinear state space systems with specified initial states. In particular we study the class of homogeneous systems, for which, under certain conditions, the local state isomorphism for a pair of indistinguishable parameter vectors is shown to be homogeneous of degree one. For homogeneous polynomial systems, conditions are given under which the local state isomorphism becomes linear. Here, the issue of whether or not the observability rank condition holds at the origin is shown to be of key importance. The scope of the results, which extend to the multivariable case, is discussed and illustrated by a number of worked examples. This demonstrates how the developed theory can be put to use to investigate the global identifiability properties of parameterized model classes.     

On the theory of flexible neural networks – Part I: a survey paper

Although flexible neural networks (FNNs) have been used more successfully than classical neural networks (CNNs) in many industrial applications, nothing is rigorously known about their properties. In fact they are not even well known to the systems and control community. In the first part of this paper, existing structures of and results on FNNs are surveyed. In the second part FNNs are examined in a theoretical framework. As a result, theoretical evidence is given for the superiority of FNNs over CNNs and further properties of the former are developed. More precisely, several fundamental properties of feedforward and recurrent FNNs are established. This includes the universal approximation capability, minimality, controllability, observability, and identifiability. In the broad sense, the results of this paper help that general use of FNNs in systems and control theory and applications be based on firm theoretical foundations. Theoretical analysis and synthesis of FNN-based systems thus become possible. The paper is concluded by a collection of topics for future work.     

Biomolecular Agents as Multi-behavioural Concurrent Objects

In recent years, there has been increasing interest in computational models of biological systems based on various calculi of communicating processes, such as the stochastic pi-calculus. These models make it possible to simulate and eventually visualize the dynamic evolution of complex biosystems in time and under varying environmental conditions.While the elegance of the pi-calculus lies in its minimality, this is also a drawback when it comes to modeling because much effort must be devoted to encoding high-level ideas into the low-level means that the language affords us.In this paper, we describe an on-going effort to design a new higher-level programming language that provides direct ontological support for the concepts which are used to formulate, organize and structure models of biomolecular systems.Our language has an object-oriented flavour where we view molecular components as agents with finite sets of behaviours (states). Reactions are modeled as exchanges over connected ports that may cause agents to switch states.     

A formal theory of matrix primeness

Primeness of nD polynomial matrices is of fundamental importance in multidimensional systems theory. In this paper we define a quantity which describes the “amount of primeness” of a matrix and identify it as the concept of grade in commutative algebra. This enables us to produce a theory which unifies many existing results, such as the Bézout identities and complementation laws, while placing them on a firm algebraic footing. We also present applications to autonomous systems, behavioural minimality of regular systems, and transfer matrix factorization. This work has been sponsored by EPSRC Grant No. GR/K 18504.     

Structural identifiability of linear compartmental systems

In biology and mathematics compartmental systems are frequently used. System identification of systems based on physical laws often involves parameter estimation. Before parameter estimation can take place, we have to examine whether the parameters are structurally identifiable. In this paper tests for the structural identifiability of linear compartmental systems are proposed. The method is based on the similarity transformation approach. New contributions in the theory are the conditions for structural identifiability of structured positive linear systems. In addition, structural identifiability from the Markov parameters is extended to structural identifiability from the input-output data, in which the initial condition is (partially) unknown and nonnegligible. Finally, conditions are presented for structural identifiability of a sampled continuous-time linear dynamic system     

Identifiability of uncontrolled nonlinear rational systems

In this paper an approach for the identifiability analysis of uncontrolled rational systems is provided. The method is based on the use of a local smooth state space transformation. In particular it is shown that, provided the model satisfies an observability rank condition, the state trajectories of an uncontrolled system corresponding to parameter vectors with outputs that are identical locally in time, are connected via a smooth transformation.     

Identifiability of nonlinear systems with application to HIV/AIDS models

In this note, we investigate different concepts of nonlinear identifiability in the generic sense. We work in the linear algebraic framework. Necessary and sufficient conditions are found for geometrical identifiability, algebraic identifiability and identifiability with known initial conditions. Relationships between different concepts are characterized. Constructive procedures are worked out for both generic geometrical and algebraic identifiability of nonlinear systems. As an application of the theory developed, we study the identifiability properties of a four dimensional model of HIV/AIDS. The questions answered in this study include the minimal number of measurement of the variables for a complete determination of all parameters and the best period of time to make such measurements. This information will be useful in formulating guidelines for clinical practice.     

Realization of continuous-time positive linear systems

Positive linear systems are used in biomathematics, economics, and other research areas. For discrete-time positive linear systems, part of the realization problem has been solved. In this paper the solution of the corresponding problem for continuous-time positive linear systems will be presented, which can be deduced from that of the discrete-time case by a transformation. Sufficient and necessary conditions for the existence of a positive realization are presented. To solve the problem of minimality, the solution of the factorization of positive matrices is needed.