Positive System Theory Research Articles

Heat equation as a Friedrichs system

Inspired by recent advances in the theory of (Friedrichs) symmetric positive systems, we apply newly developed results to the heat equation, by showing how the intrinsic theory of Ern, Guermond and Caplain (2007) can be used in order to get a well-posedness result for the Dirichlet initial–boundary value problem. We also demonstrate the application of the two-field theory with partial coercivity of Ern and Guermond (2008), originally developed for elliptic problems, and also discuss different possibilities for the construction of the appropriate boundary operator.

An Index Theory for Positive Dynamical Systems with an Application to Lotka–Voterra Systems

In this paper we develop an index theory of positive dynamical systems in planar case. We present a formula of index sum of isolated zeros a positive dynamical system on the positive orthant of R2. As an application, we present some sufficient conditions on existence of zeros of Lotka–Voterra systems in the positive orthant of R2.

Special issue on ‘new results on neuro‐fuzzy adaptive control systems’

Special issue on ‘new results on neuro‐fuzzy adaptive control systems’

Inverse optimal neural control for a class of discrete‐time nonlinear positive systems

SUMMARYIn this paper, a discrete‐time inverse optimal trajectory tracking for a class of nonlinear positive systems is proposed. The scheme is developed for MIMO affine discrete‐time positive nonlinear systems. This optimal controller is based on discrete time passivity and positive systems theory. The advantage of this scheme is that it avoids solving the associated Hamilton–Jacobi–Bellman equation and minimizes a meaningful cost function. The affine discrete‐time positive nonlinear system is obtained from an online neural identifier, which uses a recurrent neural network, trained with the extended Kalman filter. The applicability of the proposed approach is illustrated via simulation by trajectory tracking control of type 1 diabetes mellitus patients. Copyright © 2012 John Wiley & Sons, Ltd.

An efficient algorithm for positive realizations

We observe that successive applications of known results from the theory of positive systems lead to an efficient general algorithm for positive realizations of transfer functions. We give two examples to illustrate the algorithm, one of which complements an earlier result of [L. Benvenuti, L. Farina, An example of how positivity may force realizations of ‘large’ dimensions, Systems Control Lett. 36 (1999) 261–266]. Finally, we improve a lower-bound of [B. Nagy, M. Matolcsi, A lower-bound on the dimension of positive realizations, IEEE Trans. Circuits Syst. I 50 (2003) 782–784] to indicate that the algorithm is indeed efficient in general.

An Intrinsic Criterion for the Bijectivity of Hilbert Operators Related to Friedrich' Systems

Friedrich' theory of symmetric positive systems of first-order PDE's is revisited so as to avoid invoking traces at the boundary. Two intrinsic geometric conditions are introduced to characterize admissible boundary conditions. It is shown that the space in which admissible boundary conditions can be enforced is maximal in a positive cone associated with the differential operator. The equivalence with a formalism based on boundary operators is investigated and practical means to construct these boundary operators are presented. Finally, the link with Friedrich' formalism and applications to various PDE's are discussed.

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.

A positive linear discrete-time model of capacity planning and its controllability properties

One of the most important concepts in production planning is that of the establishment of an overall or aggregate production plan. In this paper, the problem of establishing an aggregate production plan for a manufacturing plant is considered. A new dynamic discrete-time model of capacity planning utilizing concepts arising in positive linear systems (PLS) theory is proposed and its controllability property is analyzed. Controllability is a fundamental property of the system with direct implications not only in dynamic optimization problems (such as those arising in inventory and production control) but also in feedback control problems. Some new open problems regarding controllability of stationary and nonstationary PLS with linear constraints are posed in the paper. An optimal control problem for capacity planning is formulated and discussed.

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.

A lowerbound on the dimension of positive realizations

A basic phenomenon in positive system theory is that the dimension N of an arbitrary positive realization of a given transfer function H(z) may be strictly larger than the dimension n of its minimal realizations. The aim of this brief is to provide a nontrivial lowerbound on the value of N under the assumption that there exists a time instant k/sub 0/ at which the (always nonnegative) impulse response of H(z) is 0 but the impulse response becomes strictly positive for all k>k/sub 0/. Transfer functions with this property may be regarded as extremal cases in positive system theory.

On the behavior of the recursive sequence

This paper deals with the global asymptotic behavior ef the nonlinear and nonautonomous dfference equaiion of nth order given in the title. The analysis is carried out by using methods from the theory of positive discrete dynamical systems. In particular, the recursive sequence considered defines self-mappings of a cone which are contractive with respect to the part-metric.

ON THE STABILITY OF SOME CHEMICAL PROCESSES VIA THE THEORY OF POSITIVE SYSTEMS

In this paper, an analysis of some chemical process systems is carried out using the theory of positive linear systems. Some potential simplification in the analysis of the dynamic behavior of the system is possible using this approach, and rapid qualitative predictions can be made.

Fiber-optic lattice signal processing

We discuss the implementation of fiber-optic lattice structures incorporating single-mode fibers and directional couplers. These fiber structures can be used to perform various high-speed time-domain and frequency-domain functions such as matrix operations and frequency filtering. In this paper we mainly consider systems in which the signals (optical intensities) and coupling coefficients are nonnegative quantities; these systems fit well in the theory of positive systems. We use this theory to conclude, for example, that for such systems the pole of the system transfer function with the largest magnitude is simple and positive-valued (in the Z-plane), and that the magnitude of the frequency response can nowhere exceed its value at the origin. We also discuss the effects of various noise phenomena on the performance of fiber-optic signal processors, particularly considering the effects of laser source phase fluctuations. Experimental results are presented showing that the dynamic range of the fiber systems, discussed in this paper, is limited, not by the laser source intensity noise or shot noise, but by the laser phase-induced intensity noise. Mathematical analyses of lattice structures as well as additional applications are also presented.

On partial differential equations of mixed type in N independent variables

AbstractIt is proved that a large class of 2‐nd order linear partial differential equations of mixed type in n independent variables can be solved by using the theory of positive symmetric systems of 1‐st order partial differential equations. Some quasilinear equations of mixed type can be treated in the same way.

The matrix forms of Yakubovich-Kalman-Lefschetz theorems derived from the popov theory of positive systems

New matrix forms of the Yakubovich-Kalman-Lefschetz theorems have been obtained utilizing the positive systems theory of Popov. Four propositions are included. One of them extends the Lefschetz lemma and the others correspond to the Anderson similar theorems. The results will be useful for application in the field of adaptive systems design.

Boundary Conditions and Uniqueness in Internal Gas Dynamic Flows

T importance of correct boundary conditions for establishing uniqueness in the numerical computation of internal gas dynamic flows is stressed. A procedure based on the general theory of Linear Symmetric Positive Systems is shown which generates proper boundary conditions. In particular, boundary conditions are developed for subsonic, transonic, and supersonic internal flows in steady-state and transient formulations.