Regularization of Descriptor Systems by Derivative and Proportional State Feedback

For linear multivariable time-invariant continuous or discrete-time singular systems it is customary to use a proportional feedback control in order to achieve a desired closed loop behaviour. Derivative feedback is rarely considered. This paper examines how derivative feedback in descriptor systems can be used to alter the structure of the system pencil under various controllability conditions. It is shown that derivative and proportional feedback controls can be constructed such that the closed loop system has a given form and is also regular and has index at most 1. This property ensures the solvability of the resulting system of dynamic-algebraic equations. The construction procedures used to establish the theory are based only on orthogonal matrix decompositions and can therefore be implemented in a numerically stable way. The problem of pole placement with derivative feedback alone and in combination with proportional state feedback is also investigated. A computational algorithm for improving the “conditioning” of the regularized closed loop system is derived.

Regularization of time-varying descriptor systems by the asvd

We study linear variable coefficient control problems in descriptor form. Based on a behaviour approach and the general theory for linear differential algebraic systems we give the theoretical analysis and describe numerically stable methods to determine the structural properties of the system.

Inverse problems in neural field theory

We study inverse problems in neural field theory, i.e., the construction of synaptic weight kernels yielding a prescribed neural field dynamics. We address the issues of existence, uniqueness, and stability of solutions to the inverse problem for the Amari neural field equation as a special case, and prove that these problems are generally ill-posed. In order to construct solutions to the inverse problem, we first recast the Amari equation into a linear perceptron equation in an infinite-dimensional Banach or Hilbert space. In a second step, we construct sets of biorthogonal function systems allowing the approximation of synaptic weight kernels by a generalized Hebbian learning rule. Numerically, this construction is implemented by the Moore–Penrose pseudoinverse method. We demonstrate the instability of these solutions and use the Tikhonov regularization method for stabilization and to prevent numerical overfitting. We illustrate the stable construction of kernels by means of three instructive examples.

Steering without Circe : attending to reinforcing loops in social systems

Relating system dynamics to the broad systems movement, the key notion is that reinforcing loops deserve no less attention than balancing loops. Three specific propositions follow. First, since reinforcing loops arise in surprising places, investigations of complex systems must consider their possible existence and potential impact. Second, because the strength of reinforcing loops can be misinferred - we include an example from the field of servomechanisms - computer simulation can be essential. Be it project management, corporate growth or inventory oscillation, simulation helps to assess consequences of reinforcing loops and options for interventions. Third, in social systems the consequences of reinforcing loops are not inevitable. Examples concerning globalization illustrate how difficult it might be to challenge such assumptions. However, system dynamics and ideas from contemporary social theory help to show that even the most complex social systems are, in principle, subject to human influence. In conclusion, by employing these ideas, by attending to reinforcing as well as balancing loops, system dynamics work can improve the understanding of social systems and illuminate our choices when attempting to steer them.

Formal theory building for the avalanche game: explaining counter-intuitive behaviour of a complex system using geometrical and human behavioural/physiological effects

In 'Avalanche', an object is lowered, players staying in contact throughout. Normally the task is easily accomplished. However, with larger groups counter-intuitive behaviours appear. The paper proposes a formal theory for the underlying causal mechanisms. The aim is to not only provide an explicit, testable hypothesis for the source of the observed modes of behaviour-but also to exemplify the contribution that formal theory building can make to understanding complex social phenomena. Mapping reveals the importance of geometry to the Avalanche game; each player has a pair of balancing loops, one involved in lowering the object, the other ensuring contact. For more players, sets of balancing loops interact and these can allow dominance by reinforcing loops, causing the system to chase upwards towards an ever-increasing goal. However, a series of other effects concerning human physiology and behaviour (HPB) is posited as playing a role. The hypothesis is therefore rigorously tested using simulation. For simplicity a 'One Degree of Freedom' case is examined, allowing all of the effects to be included whilst rendering the analysis more transparent. Formulation and experimentation with the model gives insight into the behaviours. Multi-dimensional rate/level analysis indicates that there is only a narrow region in which the system is able to move downwards. Model runs reproduce the single 'desired' mode of behaviour and all three of the observed 'problematic' ones. Sensitivity analysis gives further insight into the system's modes and their causes. Behaviour is seen to arise only when the geometric effects apply (number of players greater than degrees of freedom of object) in combination with a range of HPB effects. An analogy exists between the co-operative behaviour required here and various examples: conflicting strategic objectives in organizations; Prisoners' Dilemma and integrated bargaining situations. Additionally, the game may be relatable in more direct algebraic terms to situations involving companies in which the resulting behaviours are mediated by market regulations. Finally, comment is offered on the inadequacy of some forms of theory building and the case is made for formal theory building involving the use of models, analysis and plausible explanations to create deep understanding of social phenomena.

Theory building with system dynamics: topic and research contributions

The article discusses various reports published within the issue, including the articles "Closing the Loop: Promoting Synergies with other Theory Building Approaches to Improve System Dynamics Practice," by Birgit Kopainsky and Luis Luna-Reyes, and "On improving dynamic decision-making: Implications from multiple-process cognitive theory," by Bent Bakken.

A mathematical approach to chemical equilibrium theory for gaseous systems—I: theory

Equilibrium theory occupies an important position in chemistry and it is traditionally based on thermodynamics. A novel mathematical approach to chemical equilibrium theory for gaseous systems at constant temperature and pressure is developed. Six theorems are presented logically which illustrate the power of mathematics to explain chemical observations and these are combined logically to create a coherent system. This mathematical treatment provides more insight into chemical equilibrium and creates more tools that can be used to investigate complex situations. Although some of the issues covered have previously been given in the literature, new mathematical representations are provided. Compared to traditional treatments, the new approach relies on straightforward mathematics and less on thermodynamics, thus, giving a new and complementary perspective on equilibrium theory. It provides a new theoretical basis for a thorough and deep presentation of traditional chemical equilibrium. This work demonstrates that new research in a traditional field such as equilibrium theory, generally thought to have been completed many years ago, can still offer new insights and that more efficient ways to present the contents can be established. The work presented here can be considered appropriate as part of a mathematical chemistry course at University level.

Enriching enterprise resource planning systems for strategic advantage: a semiotic motivated approach

Enterprise Resource Planning is often endorsed as a means to facilitate strategic advantage for businesses. The scarcity of resources is the method by which some businesses maintain their position. However, the ubiquitous trend towards the adoption of Enterprise Resourcing Planning systems coupled with market saturation makes the promise of advantage less compelling. Reported in this paper is a proposed solution based upon semiotic theory that takes a typical Enterprise Resource Planning deployment scenario and shapes it according to the needs of people in post-implementation contexts to leverage strategic advantage in different ways.

A unified theory of balance in the extratropics

Many physical systems exhibit dynamics with vastly different time scales. Often the different motions interact only weakly and the slow dynamics is naturally constrained to a subspace of phase space, in the vicinity of a slow manifold. In geophysical fluid dynamics this reduction in phase space is called balance. Classically, balance is understood by way of the Rossby number R or the Froude number F; either R ≪ 1 or F ≪ 1. We examined the shallow-water equations and Boussinesq equations on an f -plane and determined a dimensionless parameter _, small values of which imply a time-scale separation. In terms of R and F, ∈= RF/√(R^2+R^2 ) We then developed a unified theory of (extratropical) balance based on _ that includes all cases of small R and/or small F. The leading-order systems are ensured to be Hamiltonian and turn out to be governed by the quasi-geostrophic potential-vorticity equation. However, the height field is not necessarily in geostrophic balance, so the leading-order dynamics are more general than in quasi-geostrophy. Thus the quasi-geostrophic potential-vorticity equation (as distinct from the quasi-geostrophic dynamics) is valid more generally than its traditional derivation would suggest. In the case of the Boussinesq equations, we have found that balanced dynamics generally implies hydrostatic balance without any assumption on the aspect ratio; only when the Froude number is not small and it is the Rossby number that guarantees a timescale separation must we impose the requirement of a small aspect ratio to ensure hydrostatic balance.

A generalized collectively compact operator theory with an application to integral equations on unbounded domains

In this paper a generalization of collectively compact operator theory in Banach spaces is developed. A feature of the new theory is that the operators involved are no longer required to be compact in the norm topology. Instead it is required that the image of a bounded set under the operator family is sequentially compact in a weaker topology. As an application, the theory developed is used to establish solvability results for a class of systems of second kind integral equations on unbounded domains, this class including in particular systems of Wiener-Hopf integral equations with L1 convolutions kernels