Mathematical control of complex systems

Copyright © 2013 ZidongWang et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

A categorical approach to open and interconnected dynamical systems

In his 1986 Automatica paper Willems introduced the influential behavioural approach to control theory with an investigation of linear time-invariant (LTI) discrete dynamical systems. The behavioural approach places open systems at its centre, modelling by tearing, zooming, and linking. We show that these ideas are naturally expressed in the language of symmetric monoidal categories.Our main result gives an intuitive sound and fully complete string diagram algebra for reasoning about LTI systems. These string diagrams are closely related to the classical notion of signal flow graphs, endowed with semantics as multi-input multi-output transducers that process discrete streams with an infinite past as well as an infinite future. At the categorical level, the algebraic characterisation is that of the prop of corelations.Using this framework, we derive a novel structural characterisation of controllability, and consequently provide a methodology for analysing controllability of networked and interconnected systems. We argue that this has the potential of providing elegant, simple, and efficient solutions to problems arising in the analysis of systems over networks, a vibrant research area at the crossing of control theory and computer science.<br/

Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)

Oxford, UK, 26 August 200

Moving from phenomenological to predictive modelling: Progress and pitfalls of modelling brain stimulation in-silico

Brain stimulation is an increasingly popular neuromodulatory tool used in both clinical and research settings; however, the effects of brain stimulation, particularly those of non-invasive stimulation, are variable. This variability can be partially explained by an incomplete mechanistic understanding, coupled with a combinatorial explosion of possible stimulation parameters. Computational models constitute a useful tool to explore the vast sea of stimulation parameters and characterise their effects on brain activity. Yet the utility of modelling stimulation in-silico relies on its biophysical relevance, which needs to account for the dynamics of large and diverse neural populations and how underlying networks shape those collective dynamics. The large number of parameters to consider when constructing a model is no less than those needed to consider when planning empirical studies. This piece is centred on the application of phenomenological and biophysical models in non-invasive brain stimulation. We first introduce common forms of brain stimulation and computational models, and provide typical construction choices made when building phenomenological and biophysical models. Through the lens of four case studies, we provide an account of the questions these models can address, commonalities, and limitations across studies. We conclude by proposing future directions to fully realise the potential of computational models of brain stimulation for the design of personalized, efficient, and effective stimulation strategies

Optimal anticipatory control as a theory of motor preparation

Supported by a decade of primate electrophysiological experiments, the prevailing theory of neural motor control holds that movement generation is accomplished by a preparatory process that progressively steers the state of the motor cortex into a movement-specific optimal subspace prior to movement onset. The state of the cortex then evolves from these optimal subspaces, producing patterns of neural activity that serve as control inputs to the musculature. This theory, however, does not address the following questions: what characterizes the optimal subspace and what are the neural mechanisms that underlie the preparatory process? We address these questions with a circuit model of movement preparation and control. Specifically, we propose that preparation can be achieved by optimal feedback control (OFC) of the cortical state via a thalamo-cortical loop. Under OFC, the state of the cortex is selectively controlled along state-space directions that have future motor consequences, and not in other inconsequential ones. We show that OFC enables fast movement preparation and explains the observed orthogonality between preparatory and movement-related monkey motor cortex activity. This illustrates the importance of constraining new theories of neural function with experimental data. However, as recording technologies continue to improve, a key challenge is to extract meaningful insights from increasingly large-scale neural recordings. Latent variable models (LVMs) are powerful tools for addressing this challenge due to their ability to identify the low-dimensional latent variables that best explain these large data sets. One shortcoming of most LVMs, however, is that they assume a Euclidean latent space, while many kinematic variables, such as head rotations and the configuration of an arm, are naturally described by variables that live on non-Euclidean latent spaces (e.g., SO(3) and tori). To address this shortcoming, we propose the Manifold Gaussian Process Latent Variable Model, a method for simultaneously inferring nonparametric tuning curves and latent variables on non-Euclidean latent spaces. We show that our method is able to correctly infer the latent ring topology of the fly and mouse head direction circuits.This work was supported by a Trinity-Henry Barlow scholarship and a scholarship from the Ministry of Education, ROC Taiwan

What can systems and control theory do for agricultural science?

Abstract: While many professionals with a background in agricultural and bio-resource sciences work with models, only few have been exposed to systems and control theory. The purpose of this paper is to elucidate a selection of methods from systems theory that can be beneficial to quantitative agricultural science. The state space representation of a dynamical system is the corner stone in the mainstream of systems theory. It is not well known in agro-modelling that linearization followed by evaluation of eigenvalues and eigenvectors of the system matrix is useful to obtain dominant time constants and dominant directions in state space, and offers opportunities for science-based model reduction. The continuous state space description is also useful in deriving truly equivalent discrete time models, and clearly shows that parameters obtained with discrete models must be interpreted with care when transferred to another model code environment. Sensitivity analysis of dynamic models reveals that sensitivity is time and input dependent. Identifiability and sensitivity are essential notions in the design of informative experiments, and the idea of persistent excitation, leading to dynamic experiments rather than the usual static experiments can be very beneficial. A special branch of systems theory is control theory. Obviously, control plays an important part in agricultural and bio-systems engineering, but it is argued that also agronomists can profit from notions from the world of control, even if practical control options are restricted to alleviating growth limiting conditions, rather than true crop control. The most important is the idea of reducing uncertainty via feed-back. On the other hand, the systems and control community is challenged to do more to address the problems of real life, such as spatial variability, measurement delays, lacking data, environmental stochasticity, parameter variability, unavoidable model uncertainty, discrete phenomena, variable system structures, the interaction of technical ad living systems, and, indeed, the study of the functioning of life itself

The Algebra of Open and Interconnected Systems

Herein we develop category-theoretic tools for understanding network-style diagrammatic languages. The archetypal network-style diagrammatic language is that of electric circuits; other examples include signal flow graphs, Markov processes, automata, Petri nets, chemical reaction networks, and so on. The key feature is that the language is comprised of a number of components with multiple (input/output) terminals, each possibly labelled with some type, that may then be connected together along these terminals to form a larger network. The components form hyperedges between labelled vertices, and so a diagram in this language forms a hypergraph. We formalise the compositional structure by introducing the notion of a hypergraph category. Network-style diagrammatic languages and their semantics thus form hypergraph categories, and semantic interpretation gives a hypergraph functor. The first part of this thesis develops the theory of hypergraph categories. In particular, we introduce the tools of decorated cospans and corelations. Decorated cospans allow straightforward construction of hypergraph categories from diagrammatic languages: the inputs, outputs, and their composition are modelled by the cospans, while the 'decorations' specify the components themselves. Not all hypergraph categories can be constructed, however, through decorated cospans. Decorated corelations are a more powerful version that permits construction of all hypergraph categories and hypergraph functors. These are often useful for constructing the semantic categories of diagrammatic languages and functors from diagrams to the semantics. To illustrate these principles, the second part of this thesis details applications to linear time-invariant dynamical systems and passive linear networks.Comment: 230 pages. University of Oxford DPhil Thesi

Continuization of Timed Petri Nets: From Performance Evaluation to Observation and Control

Abstract. State explosion is a fundamental problem in the analysis and synthesis of discrete event systems. Continuous Petri nets can be seen as a relaxation of discrete models allowing more efficient (in some cases polynomial time) analysis and synthesis algorithms. Nevertheless computational costs can be reduced at the expense of the analyzability of some properties. Even more, some net systems do not allow any kind of continuization. The present work first considers these aspects and some of the alternative formalisms usable for continuous relaxations of discrete systems. Particular emphasis is done later on the presentation of some results concerning performance evaluation, parametric design and marking (i.e., state) observation and control. Even if a significant amount of results are available today for continuous net systems, many essential issues are still not solved. A list of some of these are given in the introduction as an invitation to work on them.