Positive Realization Problem Research Articles

Minimal positive realizations for linear systems with two-different-pole transfer functions

The minimal positive realization problem for linear systems is to find positive systems with a state space of minimal dimensions while keeping their transfer functions unchanged. Here the necessary and sufficient condition is established for the linear systems with third-order two-different-pole transfer functions to have a three-dimensional minimal positive realization by developing a new approach. The approach is to identify a canonical form of positive realizations with the help of polyhedral cone theory, and the identification consists of a sequential edge transformations with some invariant feature. The analyses are then successfully extended to the linear systems with general nth-order two-different-pole transfer functions. At the same time, the approach can also be used to generate infinitely many distinct minimal positive realizations for some linear systems.

Internally Hankel $k$-Positive Systems

There has been an increased interest in the variation diminishing properties of controlled linear time-invariant (LTI) systems and time-varying linear systems without inputs. In controlled LTI systems, these properties have recently been studied from the external perspective of $k$-positive Hankel operators. Such systems have Hankel operators that diminish the number of sign changes (the variation) from past input to future output if the input variation is at most $k-1$. For $k=1$, this coincides with the classical class of externally positive systems. For linear systems without inputs, the focus has been on the internal perspective of $k$-positive state-transition matrices, which diminish the variation of the initial system state. In the LTI case and for $k=1$, this corresponds to the classical class of (unforced) positive systems. This paper bridges the gap between the internal and external perspectives of $k$-positivity by analyzing internally Hankel $k$-positive systems, which we define as state-space LTI systems where controllability and observability operators as well as the state-transition matrix are $k$-positive. We show that the existing notions of external Hankel and internal $k$-positivity are subsumed under internal Hankel $k$-positivity, and we derive tractable conditions for verifying this property in the form of internal positivity of the first $k$ compound systems. As such, this class provides new means to verify external Hankel $k$-positivity, and lays the foundation for future investigations of variation diminishing controlled linear systems. As an application, we use our framework to derive new bounds for the number of over- and undershoots in the step responses of LTI systems. Since our characterization defines a new positive realization problem, we also discuss geometric conditions for the existence of minimal internally Hankel $k$-positive realizations.

The NIEP and the positive realization problem

The nonnegative inverse eigenvalue problem is the problem of determining necessary and sufficient conditions for a multiset of complex numbers to be the spectrum of a nonnegative real matrix of size equal to the cardinality of the multiset itself. The problem is longstanding and proved to be very difficult so that several variations have been defined by considering particular classes of multisets and nonnegative real matrices. In this paper, a novel variation of the problem is proposed. This variation is motivated by a practical application in the positive realization problem, that is the problem of characterizing existence and minimality of a positive state--space representation of a given transfer function.

A direct method to obtain a realization of a polynomial matrix and its applications

In this paper we present a Silverman-Ho algorithm-based method to obtain a realization of a polynomial matrix. This method provides the final formulation of a minimal realization directly from a full rank factorization of a specific given matrix. Also, some classical problems in control theory such as model reduction in singular systems or the positive realization problem in standard systems are solved with this method.

Minimal positive realizations of linear continuous-time fractional descriptor systems: Two cases of an input-output digraph structure

Abstract In the last two decades, fractional calculus has become a subject of great interest in various areas of physics, biology, economics and other sciences. The idea of such a generalization was mentioned by Leibniz and L’Hospital. Fractional calculus has been found to be a very useful tool for modeling linear systems. In this paper, a method for computation of a set of a minimal positive realization of a given transfer function of linear fractional continuous-time descriptor systems has been presented. The proposed method is based on digraph theory. Also, two cases of a possible input-output digraph structure are investigated and discussed. It should be noted that a digraph mask is introduced and used for the first time to solve a minimal positive realization problem. For the presented method, an algorithm was also constructed. The proposed solution allows minimal digraph construction for any one-dimensional fractional positive system. The proposed method is discussed and illustrated in detail with some numerical examples.

Quantum learning of classical stochastic processes: The completely positive realization problem

Among several tasks in Machine Learning, a specially important one is the problem of inferring the latent variables of a system and their causal relations with the observed behavior. A paradigmatic instance of this is the task of inferring the hidden Markov model underlying a given stochastic process. This is known as the positive realization problem (PRP), [L. Benvenuti and L. Farina, IEEE Trans. Autom. Control 49(5), 651–664 (2004)] and constitutes a central problem in machine learning. The PRP and its solutions have far-reaching consequences in many areas of systems and control theory, and is nowadays an important piece in the broad field of positive systems theory. We consider the scenario where the latent variables are quantum (i.e., quantum states of a finite-dimensional system) and the system dynamics is constrained only by physical transformations on the quantum system. The observable dynamics is then described by a quantum instrument, and the task is to determine which quantum instrument — if any — yields the process at hand by iterative application. We take as a starting point the theory of quasi-realizations, whence a description of the dynamics of the process is given in terms of linear maps on state vectors and probabilities are given by linear functionals on the state vectors. This description, despite its remarkable resemblance with the hidden Markov model, or the iterated quantum instrument, is however devoid of any stochastic or quantum mechanical interpretation, as said maps fail to satisfy any positivity conditions. The completely positive realization problem then consists in determining whether an equivalent quantum mechanical description of the same process exists. We generalize some key results of stochastic realization theory, and show that the problem has deep connections with operator systems theory, giving possible insight to the lifting problem in quotient operator systems. Our results have potential applications in quantum machine learning, device-independent characterization and reverse-engineering of stochastic processes and quantum processors, and more generally, of dynamical processes with quantum memory [M. Guţă, Phys. Rev. A 83(6), 062324 (2011); M. Guţă and N. Yamamoto, e-print arXiv:1303.3771(2013)].

Positive Realization of Stable Fractional Discrete‐Time Linear Systems

AbstractThe positive stable realization problem of fractional discrete‐time linear systems is formulated and a method for finding a positive stable realization of a given proper transfer matrix is proposed. Sufficient conditions for the existence of a positive stable realization of this class of linear systems are established. A procedure for computation of a positive stable realization is proposed and illustrated by a numerical example.

POSITIVE MINIMAL REALIZATION OF CONTINUOUS-DISCRETE LINEAR SYSTEMS WITH ALL-POLE AND ALL-ZERO TRANSFER FUNCTION

Abstract The positive and minimal realization problem for continuous-discrete linear single-input and single-outputs (SISO) systems is formulated. Two special case of the continuous-discrete systems are given. Method based on the state variable diagram for finding a positive and minimal realization of a given proper transfer function is proposed. Sufficient conditions for the existence of a positive minimal realization of a given proper transfer function of all-pole and all-zero systems are established. Two procedures for computation of a positive minimal realization are proposed and illustrated by a numerical examples.

A construction method for positive realizations with an order bound

This paper presents an efficient construction method to address the positive realization problem with an order bound for primitive transfer functions with multiple real or complex conjugate pairs of poles using polyhedral cone concepts. Through a down-sampling step, the non-primitive transfer function is decomposed into a sum of primitive transfer functions. A construction method for the positive realization of a primitive transfer function with multiple poles is proposed by using a regular polygon in the complex plane. A sufficient condition for the existence of the positive realization with an order bound is analyzed. Numerical examples are provided to demonstrate the efficiency of the proposed method for obtaining positive realizations.

THE DIMENSION REDUCTION ALGORITHM FOR THE POSITIVE REALIZATION OF DISCRETE PHASE-TYPE DISTRIBUTIONS

This paper provides an efficient dimension reduction algorithm of the positive realization of discrete phase type(DPH) distributions. The relationship between the representation of DPH distributions and the positive realization of the positive system is explained. The dimension of the positive realization of a discrete phase-type realization may be larger than its McMillan degree of probability generating functions. The positive realization with sufficient large dimension bound can be obtained easily but generally, the minimal positive realization problem is not solved yet. We propose an efficient dimension reduction algorithm to make the positive realization with tighter upper bound from a given probability generating functions in terms of convex cone problem and linear programming.

Positive Partial Realization Problem for Linear Discrete-Time Systems

Positive Partial Realization Problem for Linear Discrete-Time SystemsA partial realization problem for positive linear discrete-time systems is addressed. Sufficient conditions for the existence of its solution are established. A procedure for the computation of a positive partial realization for a given finite sequence of the values of the impulse response is proposed. The procedure is illustrated by four numerical examples.

Positive Realizations of Transfer Matrices With Real Poles

In this brief, we prove that any normal transfer matrix with nonnegative impulse response and simple real poles of second order in discrete-time linear systems admits a minimal positive realization. This result can be used to obtain a minimal positive realization of some transfer matrices of any order with simple real poles. In addition, several results obtained for single-input single-output systems are adapted to solve the positive realization problem of transfer matrices with multiple real poles in multiple-input multiple-output systems.

Minimal positive realizations for discrete‐time systems with state time‐delays

PurposeAims to formulate and solve the minimal positive realization problem for discrete‐time linear single‐input, single‐output systems with many state time‐delays.Design/methodology/approachA conceptual discussion and approach are taken.FindingsSufficient conditions for the existence of positive realizations of a given proper rational function are established. A procedure for computation of minimal positive realizations is presented and illustrated by an example.Originality/valueExtends an earlier method for positive discrete‐time systems with many state time‐delays.

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.

Phase-type distributions and representations: Some results and open problems for system theory

In this paper we consider phase-type distributions. These distributions correspond to the random hitting time of an absorbing Markov chain. They are used for modelling various random times, in particular, those which appear in manufacturing systems as processing times, times to failure, repair times, etc. The Markovian nature of these distributions allows the use of very efficient matrix based computer methods for performance evaluation. In this paper we give a system theory oriented introduction to phase-type distributions. We concentrate mainly on the representation problem which consists of finding a Markov chain associated with some phase-type distribution. This is a realization problem in the sense of system theory with a lot of links with the classical linear system theory but also with a number of constraints which make the problem harder but more interesting. Indeed this problem has strong connections with the positive realization problem in control theory. The paper recalls known results, gives some new results, and points out the main remaining problems.

A Behavioural Approach to Positive Real Interpolation

We study interpolation by positive real functions from a behavioural point of view. In particular, by considering the notion of mirror image data, the interpolation problem with passivity constraint is transformed into an unconstrained behavioural modeling one. The main conclusion is that the generating system for this problem has to be unitary with respect to an indefinite matrix. In addition we investigate the multiple-point positive real interpolation problem. This leads to a solution of the positive real realization problem where the data are Markov parameters.

ON THE POSITIVE REALIZATION OF CONTROLLABLE BEHAVIORS

Positive linear systems, traditionally investigated within the state-space framework, have been recently analyzed within the behavioral setting, by focusing the attention on the autonomous case. Also, the positive realization problem has been fully explored in the special case of autonomous behaviors.In this contribution, we focus our attention on controllable behaviors. We first address the general realization problem by means of driving variable state-space representations and later analyze the possibility of realizing a controllable behavior by means of a positive driving variable representation. Several necessary and sufficient conditions for problem solvability are presented.

The Realization Problem for Hidden Markov Models

If {X t} is a finite-state Markov process, and {Y t} is a finite-valued output process with Y t+1 depending (possibly probabilistically) on X t, then the process pair is said to constitute a hidden Markov model. This paper considers the realization question: given the probabilities of all finite-length output strings, under what circumstances and how can one construct a finite-state Markov process and a state-to-output mapping which generates an output process whose finite-length strings have the given probabilities? After reviewing known results dealing with this problem involving Hankel matrices and polyhedral cones, we develop new theory on the existence and construction of the cones in question, which effectively provides a solution to the realization problem. This theory is an extension of recent theoretical developments on the positive realization problem of linear system theory.

Positive realization of discrete-time systems by geometric approach

The positive realization problem treated in this paper is that of finding a linear state realization from the given transfer function, such that all the coefficients of the realization are nonnegative. The specific problems treated here are realizability conditions and the way of getting positive realizations. These problems are discussed in detail through convex cone analysis and downsampling technique. It is shown that the problem of positive realizability is reduced to a set of the problems of positive realizability corresponding to the downsampled impulse responses.

A counterexample for positive realization problem

Gives a counterexample for positive realization of impulse response. The positive realization problem is to find a state space model x(k+1)=Ax(k)+bu(k), y(k)=c/sup T/ x(k) from a given impulse response H/sub k/, k=1, 2, ... of transfer function G(z), in that all the elements of A, b, and c are restricted to be nonnegative real numbers. Such problem occurs in modeling compartmental systems and in the hidden Markov process.