Minimal Positive Realization Research Articles

Leader-Based Consensus in Directed Networks of Homogeneous Positive Agents With Multiple Inputs

Abstract In this paper, the non-zero finite positive state consensus problem of leader-following multi-agent systems (MASs) connected over a directed network is investigated. The multi-agent network consists of homogeneous linear time-invariant (LTI) positive agents whose minimal positive state-space realization is assumed to be known. In this paper, a state-feedback hierarchical control protocol is proposed where the local controller synthesizes a singular, Lyapunov stable, and Metzler system matrix with a simple dominant eigenvalue at origin. The obtained consensus vector is the positive eigenvector associated with the zero eigenvalue of the synthesized system matrix. The controller gain matrices are obtained by solving a set of necessary and sufficient conditions derived in linear programing framework. A numerical example is given to elucidate the usefulness of the proposed results.

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.

Minimal positive realizations: A survey

This survey aims to present a comprehensive and systematic synthesis of concepts and results on the minimal state space realization problem for positive, linear, time-invariant systems. Positive systems are systems for which the state and the output are always non-negative for any non-negative initial state and input. They are used to model phenomena in which the variables must take non-negative values due to the nature of the underlying physical system. Restricting the state–space realization to positive systems makes the problem extremely different and much more difficult than that for ordinary systems. Indeed, a minimal positive realization may have a dimension even much greater than the order of the transfer function it realizes. Although the problem of finding a finite-dimensional positive state–space realization of a given transfer function has been solved, the characterization of minimality for positive systems is still an open problem. This survey introduces the reader to different aspects of the problem and presents the mathematical approaches used to tackle it as well as some relevant related problems. Moreover, some partial results are presented. Finally, a comprehensive bibliography on positive systems, organized by topics, is provided.

An upper bound on the dimension of minimal positive realizations for discrete time systems

Abstract In some applications one is interested in having a state–space realization with nonnegative matrices (positive realization) of a given transfer function and it is known that such a realization may have a dimension strictly larger than the order of the transfer function itself. Moreover, in most cases, it is desirable to have a realization with minimal dimension. Unfortunately, it is not known, to date, how to determine in general the minimum dimension of a positive realization and only lower and upper bounds to it are available. This letter provides an upper bound on the dimension of a minimal positive realization for transfer functions with simple poles. This is a considerable improvement on an earlier upper bound in which only transfer functions with real poles were considered.

A lower bound on the dimension of minimal positive realizations for discrete time systems

In some applications, one is interested in having a state–space realization with nonnegative matrices (positive realization) of a given transfer function and it is known that such a realization may have a dimension strictly larger than the order of the transfer function itself. The aim of this letter is to provide a lower bound on the minimum dimension of a positive realization taking into account some spectral properties of nonnegative matrices.

Minimal Strongly Eventually Positive Realization for a Class of Externally Positive Systems

The minimal positive realization of externally positive systems is still an open problem due to various restrictions caused by the nonnegativity constraint on the state-space matrices. In this paper, the minimal strongly eventually positive realization, which relaxes the nonnegativity constraint on the state-space model and only requires that the state trajectory is nonnegative after a certain number of steps, is proposed for an externally positive system. It is shown that a discrete-time transfer function with a simple strictly dominant root is externally positive if and only if it has a minimal strongly eventually positive realization. A constructive proof is provided. The continuous-time counterpart is also addressed by transforming it into a corresponding discrete-time case.

Minimal positive continuous-time realizations of positive response maps

Positive nonlinear continuous-time local realizations of positive response maps are studied. The realizations are minimal if they are positively reachable and positively observable. Necessary and sufficient conditions for a response map to have a minimal positive local realization are provided. It is also shown that a minimal positive local realization has the minimal dimension among all positive local realizations of the given response map and that it is unique up to an isomorphism of systems.

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.

Minimal eventually positive realizations of externally positive systems

It is a well-known fact that externally positive linear systems may fail to have a minimal positive realization. In order to investigate these cases, we introduce the notion of minimal eventually positive realization, for which the state update matrix becomes positive after a certain power. Eventually positive realizations capture the idea that in the impulse response of an externally positive system the state of a minimal realization may fail to be positive, but only transiently. As a consequence, we show that in discrete-time it is possible to use downsampling to obtain minimal positive realizations matching decimated sequences of Markov coefficients of the impulse response. In continuous-time, instead, if the sampling time is chosen sufficiently long, a minimal eventually positive realization leads always to a sampled realization which is minimal and positive.

Minimal Positive Realizations of Transfer Functions With Real Poles

A standard result of linear-system theory states that a single-input-single-output (SISO) rational <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> th-order transfer function always has a state-space realization of the same order. In some applications, one is interested in having a realization with nonnegative entries (i.e., a positive system) and it is known that such constraints may lead to a minimal order positive realization of order much greater than the transfer function order <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> . In this technical note, necessary and sufficient conditions for a third-order transfer function with distinct real poles to have a third-order positive realization are given: these conditions are expressed in terms of lower bounds for the first three samples of the impulse response and therefore are very easy to check. This result is an extension of a previous result for transfer functions with distinct real positive poles.

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 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.

Minimal positive realizations of transfer functions with nonnegative multiple poles

This note concerns a particular case of the minimality problem in positive system theory. A standard result in linear system theory states that any nth-order rational transfer function of a discrete time-invariant linear single-input-single-output (SISO) system admits a realization of order n. In some applications, however, one is restricted to realizations with nonnegative entries (i.e., a positive system), and it is known that this restriction may force the order N of realizations to be strictly larger than n. A general solution to the minimality problem (i.e., determining the smallest possible value of N) is not known. In this note, we consider the case of transfer functions with nonnegative multiple poles, and give sufficient conditions for the existence of positive realizations of order N=n. With the help of our results we also give an improvement of an existing result in positive system theory.

Minimal positive realizations for a class of transfer functions

It is a standard result in linear-system theory that an nth-order rational transfer function of a single-input single-output system always admits a realization of order n. In some applications, however, one is restricted to realizations with nonnegative entries (i.e. a positive system), and it is known that this restriction may force the order N of realizations to be strictly larger than n. In this brief we present a class of transfer functions where positive realizations of order n do exist. With the help of our result we give improvements on some earlier results in positive-system theory.

Realization problem for positive linear systems with time delay

The realization problem for positive single‐input single‐output discrete‐time systems with one time delay is formulated and solved. Necessary and sufficient conditions for the solvability of the realization problem are established. A procedure for computation of a minimal positive realization of a proper rational function is presented and illustrated by an example.

A note on the existence of positive realizations

The problem of existence of positive realizations for a class of continuous and discrete time systems is addressed and solved. It is shown that the existence of a (minimal) positive realization can be decided on the basis of two sub-matrices of the (infinite) Hankel matrix associated with the transfer function to be realized. Applications of this result to the problem of existence and construction of non-minimal realizations are also discussed.

Minimal positive realizations of transfer functions with positive real poles

A standard result of linear-system theory states that a SISO rational nth-order transfer function always has an nth-order realization. In some applications, one is interested in having a realization with nonnegative entries (i.e., a positive system) and it is known that a positive system may not be minimal in the usual sense. In this paper, we give an explicit necessary and sufficient condition for a third-order transfer function with distinct real positive poles to have a third-order positive realization. The proof is constructive so that it is straightforward to obtain a minimal positive realization.

An example of how positivity may force realizations of ‘large’ dimension

A standard result of linear system theory states that an SISO proper rational transfer function of degree n always has a realization of dimension n. In some applications one is interested in having a realization with nonnegative entries and it is known that, when the dominant poles display a specific pattern, forcing nonnegativity leads to a system which is not jointly reachable and observable. In this paper, we show that the minimal dimension of a positive realization may be ‘large’ even in the case of a single dominant pole. More precisely, we provide a family of transfer functions, each of which is of degree n=3, such that for any integer N⩾3 the corresponding member of the family admits a minimal positive realization of state space dimension not smaller than N.