On the existence of periodic motion and the maximum sector bounds for absolute stability of nonlinear nonstationary systems

In this contribution we consider the absolute stability problem. We derive necessary and sufficient conditions for the existence of periodic motion using an operator approach. The results yield an efficient algorithm to approximate the maximum sector bounds for absolute stability numerically.     

On Linear Copositive Lyapunov Functions and the Stability of Switched Positive Linear Systems

We consider the problem of common linear copositive Lyapunov function existence for positive switched linear systems. In particular, we present a necessary and sufficient condition for the existence of such a function for switched systems with two constituent linear time-invariant systems. Several applications of this result are also given.     

Signalling and obfuscation for congestion control

We aim to reduce the social cost of congestion in many smart city applications. In our model of congestion, agents interact over limited resources after receiving signals from a central agent that observes the state of congestion in real time. Under natural models of agent populations, we develop new signalling schemes and show that by introducing a non-trivial amount of uncertainty in the signals, we reduce the social cost of congestion, i.e., improve social welfare. The signalling schemes are efficient in terms of both communication and computation, and are consistent with past observations of the congestion. Moreover, the resulting population dynamics converge under reasonable assumptions.     

On interconnected systems,passivity and some generalisations

A sufficient condition for the stability of large-scale interconnections of N linear time-variant systems is presented. Such a condition represents important extensions to passivity criteria and ensures stability by means of the existence of a positive definite (full-block) matrix P which is a common solution to Lyapunov equations involving a diagonal stacking of the N systems and the interconnection structure matrix. An experimental methodology for the verification of the sufficient condition also is proposed, based on evolutionary computation techniques. Applications of the new stability results are provided through illustrative examples, which are developed using particle swarm optimisation and genetic algorithms.     

On the design and synthesis of limit cycles using switching linear systems

This article deals with the design and synthesis of limit cycles for a class of switched linear systems. This work is motivated by an application in the context of electrical networks, but the methodologies can be applied in other engineering fields as well. Several methods for the design and synthesis of limit cycles are presented. Based on a monodromy matrix associated with a periodically switched system, a design method and two feedback strategies are developed. The first strategy is based on linear feedback design using pole placement. The second strategy is based on the observation that certain state-dependent switching strategies can be implemented by means of a simple nonlinear output feedback controller. Advantages of this latter strategy are not only the ease with which the switching strategy can be implemented, but also the fact that classical techniques may also be used to ascertain the stability of the resulting limit cycle. Our next contribution is the development of a novel frequency domain-based approach to limit cycle design. This approach is based on the observation that the existence of certain limit cycles can be deduced from an infinity of circle criteria generated by a family of periodic systems. By making use of recent results, this observation can be used to develop a one-parameter spectral search to deduce the approximate frequencies of feasible limit cycles. Having selected a frequency of oscillation, one may then make use of the aforementioned nonlinear elements to realise a switching strategy that generates a stable limit cycle with given frequency and amplitude. Examples are given to illustrate the effectiveness of the approaches presented.     

On time-domain multiplier criteria for single-input single-output systems

In this paper we consider the problem of obtaining time-domain multiplier stability criteria. We show that in the case of single-input single-output systems, time-domain versions of classical frequency domain multiplier criteria can be obtained in a straightforward manner. In many cases, the time-domain criteria are more compact and more readily verified than the original criteria. Our results show that conventional constraints on the frequency response of a transfer function become simple eigenvalue evaluations in the time-domain.     

Modeling the interaction of milking frequency and nutrition on mammary gland growth and lactation

A mathematical model of biological mechanisms regulating lactation is constructed. In particular, the model allows prediction of the effect of milking frequency on milk yield and mammary regression, and the interaction of nutrition and milking frequency in determining yield. Possible interactions of nutrition with milking frequency on alveolar dynamics are highlighted. The model is based upon the association of prolonged engorgement (as a consequence of milk accumulation) of active secretory alveoli with changes in gene expression that result in impairment and, ultimately, cessation of milk secretion. The emptying of alveoli at milking, following alveolar contraction induced by oxytocin, prevents this process and also allows quiescent alveoli to reactivate. Prolonged engorgement results in apoptosis of the secretory cells and, hence, regression of the mammary gland. Milk yield is linked to alveolar populations, with secretion rates being modulated by nutrition and udder fill effects. The model was used to investigate different management scenarios, and is in agreement with experimental results. The model shows that while milking frequency drives alveolar population, and therefore potential milk production, actual production varies considerably with nutrition. A significant portion of the loss associated with once-daily milking was due to udder fill rather than loss of secretory tissue. The model showed qualitative agreement with experimental data, on the acute and chronic effects of temporary once-daily milking.     

A positive systems model of TCP-like congestion control: asymptotic results

We study communication networks that employ drop-tail queueing and Additive-Increase Multiplicative-Decrease (AIMD) congestion control algorithms. It is shown that the theory of nonnegative matrices may be employed to model such networks. In particular, important network properties, such as: 1) fairness; 2) rate of convergence; and 3) throughput, can be characterized by certain nonnegative matrices. We demonstrate that these results can be used to develop tools for analyzing the behavior of AIMD communication networks. The accuracy of the models is demonstrated by several NS studies.     

On analysis and design of discrete-time constrained switched systems*

This paper presents novel design techniques for discrete-time constrained switched systems. Two problems are considered: the first one is centred at determining a set of switching rules for which the system is unconditionally stable; the second one focuses on the design of a stabilising state-dependent switching function that is subject to structural constraints. Convex optimisation is the kernel of the devised procedures and, thus, these techniques are computationally viable.     

A mathematical model for mammary fatty acid synthesis and triglyceride assembly: the role of stearoyl CoA desaturase (SCD)

An increase in the proportion of unsaturated fatty acids in milk is considered desirable for human health. A prerequisite for the manipulation of milk fat composition is a co-ordinated understanding of the complex interactions in its biosynthesis. It has been suggested that an increase in the expression of mammary stearoyl-CoA-desaturase (SCD) would enrich mono-unsaturated fatty acids in milk, and therefore improve its nutritional properties. To investigate the potential effects of changes in expression of mammary enzymes and substrate availability on milk fat composition, we constructed, parameterized and evaluated a mechanistic mathematical model of fatty acid biosynthesis and milk-fat triglyceride assembly. The objective was to describe changes in the amount and composition of milk fat produced by bovine mammary cells due to changes in nutrition. Using the model we found that a 50% up-regulation in SCD activity increased the molar fraction of milk triglyceride 18:1 from 0.30 to 0.33 and 16:1 from 0.04 to 0.06. Up-regulation of SCD therefore did not appear to be the optimal method for increasing the content of unsaturated fatty acids in milk fat. The model was also used to determine the likely rate-limiting processes for the incorporation of unsaturated fatty acids into milk fat. Halving the concentration of glycerol 3-phosphate increased the molar fraction of milk triglyceride 18:1 from 0.30 to 0.35 and decreased the molar fraction of milk triglyceride 16:0 from 0.30 to 0.22. This achieved the desirable outcome of producing more unsaturated low-fat milk. Our model also predicted that a K232A mutation in the bovine mammary DGAT1 gene that is linked with an increase in milk fat yield would be consistent with a 120% increase in the DGAT acylation rate and also would be associated with a decrease in milk mono-unsaturated fatty acids.