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In this chapter we investigate the role of Lyapunov equations in the stability related analysis of differential linear repetitive processes. As for the discrete processes of the previous chapter, there are two types of Lyapunov equation, again termed 1D and 2D respectively. Also it is shown that the Lyapunov equation approach can be extended to deal with one case of dynamic boundary conditions. [ABSTRACT FROM AUTHOR]

In this chapter we first review progress in terms of the results given in this monograph. Then we proceed to discuss areas for further work. These include further development/extension of the results given here and new applications areas. [ABSTRACT FROM AUTHOR]

In this chapter, the application of repetitive process theory to iterative learning control is considered. The first part uses the stability theory of Chap. 2 to establish a fundamental link between convergence of a powerful class of such schemes and the resulting error dynamics. Following this, the norm optimal approach is developed and an experimental verification on a chain conveyor system described. Finally, some highly promising results on robust control are obtained and experimentally verified on a gantry robot system. [ABSTRACT FROM AUTHOR]

In this chapter the theme is again control law (or controller) design but here we allow uncertainty in the model structure. The uncertainty allowed is described by particular model structures, some of which have already been introduced in Chap. 5. [ABSTRACT FROM AUTHOR]

In this chapter a substantial volume of results on the control of differential and discrete linear repetitive processes are developed and illustrated, where applicable, by numerical examples. The control laws (or controllers) used make use of both current and previous pass (or passes in the non-unit memory case) and the resulting design algorithms fall, in the main, into two general classes. The first of these execute the design using Linear Matrix Inequalities and the second by minimizing a suitably formulated cost function. [ABSTRACT FROM AUTHOR]

In this chapter, we first investigate controllability and observability of discrete and differential linear repetitive processes. As with the 2D linear systems case, it will be shown that more than one physically relevant controllability property can be defined in the discrete case and also that clear differences exist with 2D discrete linear systems. Observability is only addressed for differential processes since for the discrete case this property is almost identical to that for 2D discrete linear systems described by the Roesser or Fornasini-Marchesini state-space models. The next topic covered in this chapter is the extension of the 1D theory of system equivalence to linear repetitive processes using the 2D transfer-function matrix descriptions of Chap. 1. Finally, a behavioral approach is used to define poles and zeros for these processes which, in contrast to those in the 2D/nD systems literature, have interpretations in terms of the existence of exponential trajectories in the dynamics, i.e. a natural generalization of the well known 1D linear systems case. [ABSTRACT FROM AUTHOR]

In this chapter we study the stability robustness of discrete unit memory linear repetitive processes of the form defined by Example 1.2.10 and their differential counterparts of Example 1.2.3. In the former case, a range of techniques are considered which draw on results already known in 2D systems theory. For the differential case, it is only an LMI approach which can be applied. In subsequent chapters, it will be shown that (of those currently available) only the LMI approach can be extended to permit control law design, including the case when there is uncertainty associated with the process model. [ABSTRACT FROM AUTHOR]

This chapter presents the stability theory for the Banach space based abstract model of Definition 1.2.1, together with the results of applying it to the various forms of differential and discrete linear repetitive processes introduced in the previous chapter. Also the question of testing the resulting stability conditions is addressed by a number of routes. These lead in most cases to computationally feasible tests and, in one case of applications interest, computable bounds on expected performance. [ABSTRACT FROM AUTHOR]

This chapter first introduces the unique features and control problems for repetitive processes by reference to two physical examples - long-wall coal cutting and metal rolling. Two so-called algorithmic examples are considered next, i.e. problem areas where adopting a repetitive process approach to modelling and analysis has clear advantages over alternatives. All these examples are shown to be special cases of the general abstract model in a Banach space setting on which the stability theory for linear repetitive processes is based. Finally, the links at the modelling/structural level with well known 2D discrete and standard, termed 1D in this setting, linear systems are detailed. [ABSTRACT FROM AUTHOR]

We model the behaviour of a surfer who makes a walk on the webgraph favouring webpages with a high ranking. [ABSTRACT FROM AUTHOR]

This paper presents new results to verify the robust absolute stability property of the Lur'e systems. As it is known, to solve the problem of robust absolute stability, the strict positive realness property (SPR) of a ficticious transfer function needs to be verified. Therefore, the aim of this work is determine sufficient conditions to verify the robust absolute stability property of Lur'e systems using a sign decomposition technique that guarantees the positivity of multivariable polynomials. [ABSTRACT FROM AUTHOR]

Iterated means of neighbourhoods are used to define finite condensing systems of particles in euclidean N-space. Synchronized systems are proved to converge. Necessary and sufficient conditions are discussed for the limit to be a singleton (consensus). [ABSTRACT FROM AUTHOR]

Some systems display repetitive crisis which perturb their periodic evolution. Several examples are demonstrated in such diverse fields as social sciences, physiology, population dynamics, ecology and economics. For instance, the so-called resilience property, which has received much interest from specialists of complex systems (see [3]), has some connection with such repetitive crisis. In this article, we address the question to describe and explain this type of behaviour for a tritrophic slow-fast system. Of course, such a system should incorporate two different time scales: one for the normal periodic state and another for the periodic crisis. The crucial point consists in understanding the "matching" between slow and fast motions. Thus, we focus on the analysis of bifurcations of the fast dynamics. The first part of this article introduces the system which extends Lotka-Volterra dynamics (see [6]) by adding a superpredator. Second part shows the analysis of singular points and bifurcations undergone by fast dynamics. Third part is devoted to the analysis of the various behaviours of the full system depending on the parameters. Fourth part is concerned with the main results relating to the existence and unicity of periodic bursting orbit in those different cases. [ABSTRACT FROM AUTHOR]

System reduction is applied to a mathematical model of glycolysis in yeast and to a chain network without and with feedback. The method of system reduction used is linearization of a rational positive system at a steady state, balancing of the local linear system, and truncation of the balanced linear system. For a model of glycolysis in yeast with glucose as input and pyruvate as output, it is shown that a third order linear system locally approximates well the original thirteenth order nonlinear system. [ABSTRACT FROM AUTHOR]

This paper deals with global convergence to equilibria, and in particular Hirsch's generic convergence theorem for strongly monotone systems, for singular perturbations of monotone systems. [ABSTRACT FROM AUTHOR]

In this paper, the positive invariance of sets defined by nonlinear inequalities with respect to nonlinear maps from a partially ordered set into itself is investigated. The results of this paper extend former results relative to the positive invariance of polyhedral sets with respect to linear systems defined in ordered semirings as well as former results relative to nonlinear dynamical systems defined in the real n-space. The v-monotonicity of nonlinear maps is also been investigated. It is shown that, opposite to the case of linear maps, the v-monotonicity of a nonlinear map is not a necessary condition for a set defined by nonlinear inequalities to be positively invariant. [ABSTRACT FROM AUTHOR]

In this paper several recent results concerning the dynamics of order preserving (sub)homogeneous maps on polyhedral cones are reviewed. These results were obtained by the author in collaboration with Marianne Akian, Stéphane Gaubert, Roger Nussbaum, Michael Scheutzow and Colin Sparrow in [2], [13] and [15], and are new nonlinear extensions of the Perron-Frobenius theory. [ABSTRACT FROM AUTHOR]

P-temporal Petri Nets are convenient tools to model manufacturing systems whose activities times are included between a minimum and a maximum value. The typical feature of these nets is that a control of the firing dates of the transitions is required for the controlled system to behave as close as possible to the specified outputs and to ensure the liveness of tokens, when possible. This paper aims at designing a control law which is obtained by using residuated and dual residuated mappings. An example is given to illustrate the proposed approach. [ABSTRACT FROM AUTHOR]

In this paper we deal with the problem of controlling timed continuous Petri nets in order to reach a final (steady state) configuration while minimizing a quadratic performance index. The formulation of a discrete-time linear positive model with dynamic (or state-based) constraints on the control input, enables us to design a state-feedback control law based on explicit model predictive control (eMPC). The eMPC partitions the state space into polytopes: an affine state-feedback control law is uniquely associated to each polytope, while the on-line phase of the approach consists in evaluating the current region and consequently the optimal control law. [ABSTRACT FROM AUTHOR]

Networks can support Quality of Service (QoS) architectures for a wide variety of applications. The Network Calculus (NC) appeared as a set of rules and results for computing bounds for QoS parameters. The goal of this paper is to improve the systematic use of the Max-Plus algebra in the formulation and derivation of NC results. To illustrate the presented methodology, we analyze the problem of optimizing the playback delay and the buffer size of an optimal video decoder with some look-ahead. [ABSTRACT FROM AUTHOR]

Biologists are increasingly conscious of the critical role that noise plays in cellular functions such as genetic regulation, often in connection with fluctuations in small numbers of key regulatory molecules. This has inspired the development of models that capture this fundamentally discrete and stochastic nature of cellular biology — most notably the Gillespie stochastic simulation algorithm (SSA). The SSA simulates a temporally homogeneous, discrete-state, continuous-time Markov process, and of course the corresponding probabilities and numbers of each molecular species must all remain positive. While accurately serving this purpose, the SSA can be computationally inefficient due to very small time stepping so faster approximations such as the Poisson and Binomial τ-leap methods have been suggested. This work places these leap methods in the context of numerical methods for the solution of stochastic differential equations (SDEs) driven by Poisson noise. This allows analogues of Euler-Maruyuma, Milstein and even higher order methods to be developed through the Itô-Taylor expansions as well as similar derivative-free Runge-Kutta approaches. Numerical results demonstrate that these novel methods compare favourably with existing techniques for simulating biochemical reactions by more accurately capturing crucial properties such as the mean and variance than existing methods. [ABSTRACT FROM AUTHOR]

In this paper we consider a finite state Markov chain with two outputs, an observed output and a to-be-estimated output, and derive a recursive estimator for the to-be-estimated output from an observed output string. The main point of this article is to illustrate that for this kind of filtering problem, it is not needed to have a positive hidden Markov realization of the joint process, but it suffices to have a quasi-realization. We also present an approximate quasi-realization algorithm. We perform a simulation comparing the behavior of the exact, experimental and approximate quasi-realizations and checking the performance of the estimator. [ABSTRACT FROM AUTHOR]

A direct adaptive control framework for stochastic nonnegative and compartmental dynamical systems is developed. In the face of stochastic variations of system coefficients, we present analysis and adaptive control methodology that guarantees global ultimate boundedness of the second moment of the nonlinear stochastic nonnegative systems. [ABSTRACT FROM AUTHOR]

This paper investigates the relation between the spectrum of the system matrix of a positive linear system and the existence of positive linear observers. We only consider single output case. It is shown that a positive linear system admits a Luenberger type positive linear observer only if the number of nonnegative real eigenvalues of the system matrix is at most one. Moreover, we provide a necessary condition on the existence of positive linear observers which employ a coordinates transformation and whose system matrices are diagonal. [ABSTRACT FROM AUTHOR]

We present new results on decomposing the transfer function t(z) of a linear, asymptotically stable, discrete-time SISO system as a difference t(z) = t1(z) − t2(z) of two positive linear systems. We extend the results of [4] to a class of transfer functions t(z) with multiple poles. One of the appearing positive systems is always 1-dimensional, while the other has dimension corresponding to the location and order of the poles of t(z). Recently, in [11], a universal approach was found, providing a decomposition for any asymptotically stable t(z). Our approach here gives lower dimensions than [11] in certain cases but, unfortunately, at present it can only be applied to a relatively small class of transfer functions, and it does not yield a general algorithm. [ABSTRACT FROM AUTHOR]

In this paper, the conditions are discussed for the third-order systems with the complex poles to have third-order positive realizations. These systems are divided into two classes by the distribution of the complex poles. For one class, a sufficient condition is presented, and for the other one, a necessary and sufficient condition is presented. Finally some examples are given to show the contribution and limitation of these results. [ABSTRACT FROM AUTHOR]

A non-linear model for the description of packet-switched communication networks is presented. This model is combined with a model of an additive increase, multiplicative decrease end-to-end controller to obtain a global description of a controlled packet switched network with buffering delays. The global stability of this general model is then analysed using singular perturbation analysis. It is shown that under sufficiently small propagation delays, this general model is globally and exponentially stable. [ABSTRACT FROM AUTHOR]

We deal with problems of maximizing the norm of the transfer matrix function of linear dynamical systems. It is shown that if a linear system is positive and (exponentially) asymptotically stable then the norm of its transfer matrix function attains the maximum at a specific point on the boundary of the stable region. For sake of space, exposure is kept to minimum in this paper. Two examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper a global stabilizing control law for a PVTOL aircraft is developed. It considers the positiveness of the thrust and the boundedness of the control inputs. The performance of the proposed controller is compared with other existing approaches. [ABSTRACT FROM AUTHOR]

Sufficient conditions are reported for the existence of a solution to the positive linear quadratic (LQ) problem. These conditions are established by using a Newton-type iterative scheme converging to the unique stabilizing positive semidefinite solution of the algebraic Riccati equation. These results are obtained for stable and detectable linear continuous time-invariant positive systems. [ABSTRACT FROM AUTHOR]

In the paper we give sufficient conditions for the controllability of nonlinear control systems to the arbitrary point of a given interval Q contained in ℝn. [ABSTRACT FROM AUTHOR]

An approach that uses tools from systems and control theory is presented. Such technique enables the determination of the essential interactions in the autoregulatory process that describes the cell cycle of Xenopus frog eggs. The results make possible an assessment of the effect of each protein on the biochemical network stability. The proposed method is based on bifurcation diagram and linearization near the operating point of interest. A perturbation analysis makes possible the determination of the critical proteins responsible for complex behavior in the system, emphasizing how system theory can play an essential role in the study of these systems. [ABSTRACT FROM AUTHOR]

In this paper, the optimal drug injection problem arising in cancer treatment by cell-cycle-specific chemotherapy is investigated. It is shown that this problem can be approximately solved up to any desired precision by using an indexed family of state-unconstrained optimal control problems. The existence of solutions is proved and the resulting approximation is characterized by appropriate two-sides inequalities. Simulations are provided to show the efficiency and relevance of the proposed formulation. [ABSTRACT FROM AUTHOR]

In the course of his work on the analysis of genetic regulatory networks represented by signed directed graphs, R. Thomas has proposed that the occurrence of a positive regulatory circuit is a necessary condition for the existence of multiple stable states, whereas a negative circuit is necessary to generate stable oscillations. Here, we formulate and prove a theorem establishing these rules in a multilevel discrete framework. [ABSTRACT FROM AUTHOR]

In this paper, a nonuniform Space-Time Conservation Element-Solution Element (CE-SE) method is developed in order to improve the relation accuracy-computational cost in the study of unsteady flow with chemical transport. We analyze this improvement by means of an illustrative example. [ABSTRACT FROM AUTHOR]

This papers analyzes a class of non-linear additive-increase multiplicative-decrease (AIMD) protocols that are widely deployed in communication networks. It is demonstrated that the use of these protocols guarantee that the system has a unique stable outcome to which it converges geometrically under all starting points. The development is based on a contraction argument and the derivation of explicit bounds on the contraction coefficient of corresponding operators in terms of the network parameters. In particular, bounds on the corresponding rate of convergence are obtained, improving upon known bounds for standard (linear) AIMD networks. [ABSTRACT FROM AUTHOR]

We introduce the definition of Social Leader that gives a local centrality measure for each node in a graph. A node u is a Social Leader if the number of cycles of length 3 passing through u is greater than the corresponding number for its neighbors. This concept is used to visualize large graphs, identify influent agents in social networks (word of mouth effect) and find communities. [ABSTRACT FROM AUTHOR]

Recently, a number of articles in scientific and popular journals dealt with the fact that there are losing gambling games which can be combined in a suitable stochastic or deterministic way to a winning game - the so-called Parrondo's paradox. Here we investigate a generalization of Parrondo's games introduced by E. Behrends in 2002 which can be analyzed by means of discrete Markov chains. We show that Behrends' requirements for such a chain's transition matrix can be weakened and give some new results concerning the behavior of combinations of these games. [ABSTRACT FROM AUTHOR]

The interaction of agents seeking an agreement or consensus is modeled in higher dimensional space as a time-dependent process of forming convex combinations. It is proved that convergence to a consensus is still possible provided the strength of communication does not fade away "too fast" and the structure of communication becomes not "too loose". The relationship to results obtained in [1] on the agreement algorithm, in [3] on opinion dynamics and in [4] on particle interaction will be discussed. [ABSTRACT FROM AUTHOR]

We present a convergence result for infinite products of stochastic matrices with positive diagonals. We regard infinity of the product to the left. Such a product converges partly to a fixed matrix if the minimal positive entry of each matrix does not converge too fast to zero and if either zero-entries are symmetric in each matrix or the length of subproducts which reach the maximal achievable connectivity is bounded. Variations of this result have been achieved independently in [1], [2] and [3]. We present briefly the opinion dynamics context, discuss the relations to infinite products where infinity is to the right (inhomogeneous Markov processes) and present a small improvement and sketch another. [ABSTRACT FROM AUTHOR]

For a class of positive matrices A + K with a stable positive nominal part A and a structured positive perturbation part K, we address the problem of finding the largest admissible perturbation such that the global matrix remains stable. Theoretical bounds on the size of this set are derived and an algorithm for constructing a set of admissible perturbation is presented. [ABSTRACT FROM AUTHOR]

In this paper, controllability and reachability properties of continuoustime positive switched systems are investigated. It is first shown that such systems are never controllable. On the other hand, monomial reachability, which represents a necessary condition for reachability, is fully characterized. Such a condition is also sufficient for reachability when dealing either with switched systems of size 2 or with positive (non-switched) systems. [ABSTRACT FROM AUTHOR]

It has been shown that a particular class of time-homogeneous Markov chains, called class C, possesses a closed form steady state solution. In this paper, we are interested in the transient analysis of this class of Markov chains. Unlike steady state analysis, transient analysis for continuous time (CTMC) and discrete time (DTMC) Markov chains have to be treated differently. First we show that transient distributions of class C Markov chains can be also computed through closed form solutions both for the cases of DTMC and CTMC. By means of the stochastic comparison approach, we construct bounding chains belonging to this class thus bounding transient distributions of the underlying Markov chain can be easily computed through the closed form solution. In this work we propose bounds both in the sense of the strong stochastic order, and the increasing convex stochastic order. [ABSTRACT FROM AUTHOR]

For the numerous applications of Markov chains (in particular MCMC methods), the problem of detecting an instant at which the convergence takes place is crucial. The ‘cut-off phenomenon', or abrupt convergence, provides an answer to this problem. When a sample of Markov chains, or more generally of exponentially converging processes, is simulated in parallel, it remains far from its stationary distribution until a deterministic instant, and approaches it exponentially fast afterwards. The cut-off instant is explicitly known, and can be detected algorithmically using appropriate stopping times. The technique is illustrated on the Ornstein-Uhlenbeck diffusion. [ABSTRACT FROM AUTHOR]

Deflection Routing is proposed for all optical core switches because of the lack of optical memory. In Shortest-path Deflection Routing, switches attempt to forward packets along a shortest hop path to their destinations. Each link can send a finite number of packets per time-slot (the link capacity). Incoming packets have to be sent immediately to their next switch along the path. If the number of packets which require a link is larger than the link capacity, only some of them will use the link they ask for and the other ones have to be misdirected or deflected. We build the Markov chain which models a packet routing in an odd torus. We prove that this matrix is ≼st-monotone. The proof is based on increasing sets as we consider a partial ordering on the state space based on the network topology. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the comparison of expected rewards for discrete-time reward Markov chains with different state spaces. Necessary and sufficient conditions for such a comparison are derived. Due to the special nature of the introduced binary relation, a criterion may be formulated in terms of an inclusion of polyhedral sets. Then, algebraic and geometric forms are easily obtained from Haar's Lemma. Our results allow us to discuss some earlier results on the stochastic comparison of functions of Markov chains. [ABSTRACT FROM AUTHOR]

A reduction in blood volume can reduce the stroke volume of the heart by affecting the Frank-Starling mechanism and reducing the filling pressure of the heart, leading to a drop in arterial blood pressure. A severe drop in pressure, not compensated by the cardiovascular system (CVS) control system, may lead to a self-reinforcing deterioration of CVS function. We will consider a CVS model with a single feedback loop to study the impact of increased heart rate on stabilizing arterial blood pressure after an acute hemorrhage and using various options of fluid replacement. Our simulations show that the application of a saline solution is stabilizing in the short term but in the long run only marginally better than the situation with no infusion. However, for the use of whole blood the infused volume stays in the blood vessel compartment and is able to improve the cardiovascular performance. [ABSTRACT FROM AUTHOR]

Nonlinear parameter estimation is usually achieved via the minimization of some possibly nonconvex cost function. We show how interval analysis can be used to characterize the set of all global minimizers of this cost function in a guaranteed way. Positivity allows us to take advantage of Müller's theorems to make such a guaranteed search more efficient in the case of models described by ordinary differential equations. [ABSTRACT FROM AUTHOR]

Mathematical modelling is widely used for estimating the carbon stocks and fluxes of forest soils, for example to meet the reporting requirements set in the Kyoto protocol. An essential prerequisite for numerical parameter estimation from experimental data is the a priori global identifiability of the model, which tells whether the available data contain enough information to unequivocally estimate all the unknown parameters of the postulated model. For soil models especially, this is a justified question, since typically there is a lack of experimentally verifiable counterparts to the conceptual compartments they contain. In this work, the a priori global identifiability analysis of a simple soil carbon model Yasso, with respect to the currently measurable soil carbon data, was performed for the first time. To do this, a recently proposed method based on differential algebraic concepts was used. The above soil carbon model, as it is, was not globally identifiable. The requirements for both soil measurements and model structure to meet the global identifiability of the model were proposed and discussed. [ABSTRACT FROM AUTHOR]

Compartmental models are useful, explanatory tools, that have been widely used to represent the dynamic behavior of complex physiological systems. In particular, multi-compartmental models of ever-increasing complexity have recently been developed to describe the postprandial distribution of dietary nitrogen in humans, from its ingestion to its elimination or retention in the different metabolic pools of the body according to a multi-tissue scheme. This paper briefly reviews the entire development process of such models, laying stress on two limiting but crucial steps: the a priori identifiability of the model (concerning the theoretical uniqueness of model parameters) and parameter estimation (dealing with the numerical identification of parameter values). We rapidly review the classic methods used to solve both these challenging mathematical problems, while the newest conceptual and computational methods are applied in the case of a linear 13-compartment, 21-parameter model of dietary nitrogen postprandial metabolism in humans. [ABSTRACT FROM AUTHOR]