Discrete Dynamical Systems

Exercises and solutions for a third or fourth year maths course. Diagrams for the questions are all together in the support.zip file, as .eps files

Modelling, control and supervision for a class of hybrid systems

The aim of this thesis is to narrow the gap between two different control techniques: the continuous control and the discrete event control techniques DES. This gap can be reduced by the study of Hybrid systems, and by interpreting as Hybrid systems the majority of large-scale systems. In particular, when looking deeply into a process, it is often possible to identify interaction between discrete and continuous signals. Hybrid systems are systems that have both continuous, and discrete signals. Continuous signals are generally supposed continuous and differentiable in time, since discrete signals are neither continuous nor differentiable in time due to their abrupt changes in time. Continuous signals often represent the measure of natural physical magnitudes such as temperature, pressure etc. The discrete signals are normally artificial signals, operated by human artefacts as current, voltage, light etc. Typical processes modelled as Hybrid systems are production systems, chemical process, or continuos production when time and continuous measures interacts with the transport, and stock inventory system. Complex systems as manufacturing lines are hybrid in a global sense. They can be decomposed into several subsystems, and their links. Another motivation for the study of Hybrid systems is the tools developed by other research domains. These tools benefit from the use of temporal logic for the analysis of several properties of Hybrid systems model, and use it to design systems and controllers, which satisfies physical or imposed restrictions. This thesis is focused in particular types of systems with discrete and continuous signals in interaction. That can be modelled hard non-linealities, such as hysteresis, jumps in the state, limit cycles, etc. and their possible non-deterministic future behaviour expressed by an interpretable model description. The Hybrid systems treated in this work are systems with several discrete states, always less than thirty states (it can arrive to NP hard problem), and continuous dynamics evolving with expression: with Ki ¡ Rn constant vectors or matrices for X components vector. In several states the continuous evolution can be several of them Ki = 0. In this formulation, the mathematics can express Time invariant linear system. By the use of this expression for a local part, the combination of several local linear models is possible to represent non-linear systems. And with the interaction with discrete events of the system the model can compose non-linear Hybrid systems. Especially multistage processes with high continuous dynamics are well represented by the proposed methodology. Sate vectors with more than two components, as third order models or higher is well approximated by the proposed approximation. Flexible belt transmission, chemical reactions with initial start-up and mobile robots with important friction are several physical systems, which profits from the benefits of proposed methodology (accuracy). The motivation of this thesis is to obtain a solution that can control and drive the Hybrid systems from the origin or starting point to the goal. How to obtain this solution, and which is the best solution in terms of one cost function subject to the physical restrictions and control actions is analysed. Hybrid systems that have several possible states, different ways to drive the system to the goal and different continuous control signals are problems that motivate this research. The requirements of the system on which we work is: a model that can represent the behaviour of the non-linear systems, and that possibilities the prediction of possible future behaviour for the model, in order to apply an supervisor which decides the optimal and secure action to drive the system toward the goal. Specific problems can be determined by the use of this kind of hybrid models are: - The unity of order. - Control the system along a reachable path. - Control the system in a safe path. - Optimise the cost function. - Modularity of control The proposed model solves the specified problems in the switching models problem, the initial condition calculus and the unity of the order models. Continuous and discrete phenomena are represented in Linear hybrid models, defined with defined eighth-tuple parameters to model different types of hybrid phenomena. Applying a transformation over the state vector : for LTI system we obtain from a two-dimensional SS a single parameter, alpha, which still maintains the dynamical information. Combining this parameter with the system output, a complete description of the system is obtained in a form of a graph in polar representation. Using Tagaki-Sugeno type III is a fuzzy model which include linear time invariant LTI models for each local model, the fuzzyfication of different LTI local model gives as a result a non-linear time invariant model. In our case the output and the alpha measure govern the membership function. Hybrid systems control is a huge task, the processes need to be guided from the Starting point to the desired End point, passing a through of different specific states and points in the trajectory. The system can be structured in different levels of abstraction and the control in three layers for the Hybrid systems from planning the process to produce the actions, these are the planning, the process and control layer. In this case the algorithms will be applied to robotics ¡V a domain where improvements are well accepted ¡V it is expected to find a simple repetitive processes for which the extra effort in complexity can be compensated by some cost reductions. It may be also interesting to implement some control optimisation to processes such as fuel injection, DC-DC converters etc. In order to apply the RW theory of discrete event systems on a Hybrid system, we must abstract the continuous signals and to project the events generated for these signals, to obtain new sets of observable and controllable events. Ramadge & Wonham¡¦s theory along with the TCT software give a Controllable Sublanguage of the legal language generated for a Discrete Event System (DES). Continuous abstraction transforms predicates over continuous variables into controllable or uncontrollable events, and modifies the set of uncontrollable, controllable observable and unobservable events. Continuous signals produce into the system virtual events, when this crosses the bound limits. If this event is deterministic, they can be projected. It is necessary to determine the controllability of this event, in order to assign this to the corresponding set, , controllable, uncontrollable, observable and unobservable set of events. Find optimal trajectories in order to minimise some cost function is the goal of the modelling procedure. Mathematical model for the system allows the user to apply mathematical techniques over this expression. These possibilities are, to minimise a specific cost function, to obtain optimal controllers and to approximate a specific trajectory. The combination of the Dynamic Programming with Bellman Principle of optimality, give us the procedure to solve the minimum time trajectory for Hybrid systems. The problem is greater when there exists interaction between adjacent states. In Hybrid systems the problem is to determine the partial set points to be applied at the local models. Optimal controller can be implemented in each local model in order to assure the minimisation of the local costs. The solution of this problem needs to give us the trajectory to follow the system. Trajectory marked by a set of set points to force the system to passing over them. Several ways are possible to drive the system from the Starting point Xi to the End point Xf. Different ways are interesting in: dynamic sense, minimum states, approximation at set points, etc. These ways need to be safe and viable and RchW. And only one of them must to be applied, normally the best, which minimises the proposed cost function. A Reachable Way, this means the controllable way and safe, will be evaluated in order to obtain which one minimises the cost function. Contribution of this work is a complete framework to work with the majority Hybrid systems, the procedures to model, control and supervise are defined and explained and its use is demonstrated. Also explained is the procedure to model the systems to be analysed for automatic verification. Great improvements were obtained by using this methodology in comparison to using other piecewise linear approximations. It is demonstrated in particular cases this methodology can provide best approximation. The most important contribution of this work, is the Alpha approximation for non-linear systems with high dynamics While this kind of process is not typical, but in this case the Alpha approximation is the best linear approximation to use, and give a compact representation.

Studies of dynamics of physical agent ecosystems

This thesis addresses the problem of learning in physical heterogeneous multi-agent systems (MAS) and the analysis of the benefits of using heterogeneous MAS with respect to homogeneous ones. An algorithm is developed for this task; building on a previous work on stability in distributed systems by Tad Hogg and Bernardo Huberman, and combining two phenomena observed in natural systems, task partition and hierarchical dominance. This algorithm is devised for allowing agents to learn which are the best tasks to perform on the basis of each agent's skills and the contribution to the team global performance. Agents learn by interacting with the environment and other teammates, and get rewards from the result of the actions they perform. This algorithm is specially designed for problems where all robots have to co-operate and work simultaneously towards the same goal. One example of such a problem is role distribution in a team of heterogeneous robots that form a soccer team, where all members take decisions and co-operate simultaneously. Soccer offers the possibility of conducting research in MAS, where co-operation plays a very important role in a dynamical and changing environment. For these reasons and the experience of the University of Girona in this domain, soccer has been selected as the test-bed for this research. In the case of soccer, tasks are grouped by means of roles. One of the most interesting features of this algorithm is that it endows MAS with a high adaptability to changes in the environment. It allows the team to perform their tasks, while adapting to the environment. This is studied in several cases, for changes in the environment and in the robot's body. Other features are also analysed, especially a parameter that defines the fitness (biological concept) of each agent in the system, which contributes to performance and team adaptability. The algorithm is applied later to allow agents to learn in teams of homogeneous and heterogeneous robots which roles they have to select, in order to maximise team performance. The teams are compared and the performance is evaluated in the games against three hand-coded teams and against the different homogeneous and heterogeneous teams built in this thesis. This section focuses on the analysis of performance and task partition, in order to study the benefits of heterogeneity in physical MAS. In order to study heterogeneity from a rigorous point of view, a diversity measure is developed building on the hierarchic social entropy defined by Tucker Balch. This is adapted to quantify physical diversity in robot teams. This tool presents very interesting features, as it can be used in the future to design heterogeneous teams on the basis of the knowledge on other teams.

Solar influences on dynamical coupling between the stratosphere and troposphere

We use a simplified atmospheric general circulation model (AGCM) to investigate the response of the lower atmosphere to thermal perturbations in the lower stratosphere. The results show that generic heating of the lower stratosphere tends to weaken the sub-tropical jets and the tropospheric mean meridional circulations. The positions of the jets, and the extent of the Hadley cells, respond to the distribution of the stratospheric heating, with low latitude heating displacing them poleward, and uniform heating displacing them equatorward. The patterns of response to the low latitude heating are similar to those found to be associated with solar variability in previous observational data analysis, and to the effects of varying solar UV radiation in sophisticated AGCMs. In order to investigate the chain of causality involved in converting the stratospheric thermal forcing to a tropospheric climate signal we conduct an experiment which uses an ensemble of model spin-ups to analyse the time development of the response to an applied stratospheric perturbation. We find that the initial effect of the change in static stability at the tropopause is to reduce the eddy momentum flux convergence in this region. This is followed by a vertical transfer of the momentum forcing anomaly by an anomalous mean circulation to the surface, where it is partly balanced by surface stress anomalies. The unbalanced part drives the evolution of the vertically integrated zonal flow. We conclude that solar heating of the stratosphere may produce changes in the circulation of the troposphere even without any direct forcing below the tropopause. We suggest that the impact of the stratospheric changes on wave propagation is key to the mechanisms involved.

Classifying dynamical forcing mechanisms using a climatology of extratropical cyclones

A climatology of almost 700 extratropical cyclones is compiled by applying an automated feature tracking algorithm to a database of objectively identified cyclonic features. Cyclones are classified according to the relative contributions to the midlevel vertical motion of the forcing from upper and lower levels averaged over the cyclone intensification period (average U/L ratio) and also by the horizontal separation between their upper-level trough and low-level cyclone (tilt). The frequency distribution of the average U/L ratio of the cyclones contains two significant peaks and a long tail at high U/L ratio. Although discrete categories of cyclones have not been identified, the cyclones comprising the peaks and tail have characteristics that have been shown to be consistent with the type A, B, and C cyclones of the threefold classification scheme. Using the thresholds in average U/L ratio determined from the frequency distribution, type A, B, and C cyclones account for 30\%, 38\%, and 32\% of the total number of cyclones respectively. Cyclones with small average U/L ratio are more likely to be developing cyclones (attain a relative vorticity $\ge 1.2 \times 10^{-4} \mbox{s}^{-1}$) whereas cyclones with large average U/L ratio are more likely to be nondeveloping cyclones (60\% of type A cyclones develop whereas 31\% of type C cyclones develop). Type A cyclogenesis dominates in the development region East of the Rockies and over the gulf stream, type B cyclogenesis dominates in the region off the East coast of the USA, and type C cyclogenesis is more common over the oceans in regions of weaker low-level baroclinicity.

An objective climatology of the dynamical forcing of polar lows in the Nordic seas

A new objective climatology of polar lows in the Nordic (Norwegian and Barents) seas has been derived from a database of diagnostics of objectively identified cyclones spanning the period January 2000 to April 2004. There are two distinct parts to this study: the development of the objective climatology and a characterization of the dynamical forcing of the polar lows identified. Polar lows are an intense subset of polar mesocyclones. Polar mesocyclones are distinguished from other cyclones in the database as those that occur in cold air outbreaks over the open ocean. The difference between the wet-bulb potential temperature at 700 hPa and the sea surface temperature (SST) is found to be an effective discriminator between the atmospheric conditions associated with polar lows and other cyclones in the Nordic seas. A verification study shows that the objective identification method is reliable in the Nordic seas region. After demonstrating success at identifying polar lows using the above method, the dynamical forcing of the polar lows in the Nordic seas is characterized. Diagnostics of the ratio of mid-level vertical motion attributable to quasi-geostrophic forcing from upper and lower levels (U/L ratio) are used to determine the prevalence of a recently proposed category of extratropical cyclogenesis, type C, for which latent heat release is crucial to development. Thirty-one percent of the objectively identified polar low events (36 from 115) exceeded the U/L ratio of 4.0, previously identified as a threshold for type C cyclones. There is a contrast between polar lows to the north and south of the Nordic seas. In the southern Norwegian Sea, the population of polar low events is dominated by type C cyclones. These possess strong convection and weak low-level baroclinicity. Over the Barents and northern Norwegian seas, the well-known cyclogenesis types A and B dominate. These possess stronger low-level baroclinicity and weaker convection.

Dynamical downscaling: Assessment of model system dependent retained and added variability for two different regional climate models

The Splitting of the Stratospheric Polar Vortex in the Southern Hemisphere, September 2002: Dynamical Evolution

The polar vortex of the Southern Hemisphere (SH) split dramatically during September 2002. The large-scale dynamical effects were manifest throughout the stratosphere and upper troposphere, corresponding to two distinct cyclonic centers in the upper troposphere–stratosphere system. High-resolution (T511) ECMWF analyses, supplemented by analyses from the Met Office, are used to present a detailed dynamical analysis of the event. First, the anomalous evolution of the SH polar vortex is placed in the context of the evolution that is usually witnessed during spring. Then high-resolution fields of potential vorticity (PV) from ECMWF are used to reveal several dynamical features of the split. Vortex fragments are rapidly sheared out into sheets of high (modulus) PV, which subsequently roll up into distinct synoptic-scale vortices. It is proposed that the stratospheric circulation becomes hydrodynamically unstable through a significant depth of the troposphere–stratosphere system as the polar vortex elongates.

Can the dynamical impact of the stratosphere on the troposphere be described by large-scale adjustment to the stratospheric PV distribution?

Recent numerical experiments have demonstrated that the state of the stratosphere has a dynamical impact on the state of the troposphere. To account for such an effect, a number of mechanisms have been proposed in the literature, all of which amount to a large-scale adjustment of the troposphere to potential vorticity (PV) anomalies in the stratosphere. This paper analyses whether a simple PV adjustment suffices to explain the actual dynamical response of the troposphere to the state of the stratosphere, the actual response being determined by ensembles of numerical experiments run with an atmospheric general-circulation model. For this purpose, a new PV inverter is developed. It is shown that the simple PV adjustment hypothesis is inadequate. PV anomalies in the stratosphere induce, by inversion, flow anomalies in the troposphere that do not coincide spatially with the tropospheric changes determined by the numerical experiments. Moreover, the tropospheric anomalies induced by PV inversion are on a larger scale than the changes found in the numerical experiments, which are linked to the Atlantic and Pacific storm-tracks. These findings imply that the impact of the stratospheric state on the troposphere is manifested through the impact on individual synoptic-scale systems and their self-organization in the storm-tracks. Changes in these weather systems in the troposphere are not merely synoptic-scale noise on a larger scale tropospheric response, but an integral part of the mechanism by which the state of the stratosphere impacts that of the troposphere.

Variational data assimilation for parameter estimation: application to a simple morphodynamic model

Data assimilation is a sophisticated mathematical technique for combining observational data with model predictions to produce state and parameter estimates that most accurately approximate the current and future states of the true system. The technique is commonly used in atmospheric and oceanic modelling, combining empirical observations with model predictions to produce more accurate and well-calibrated forecasts. Here, we consider a novel application within a coastal environment and describe how the method can also be used to deliver improved estimates of uncertain morphodynamic model parameters. This is achieved using a technique known as state augmentation. Earlier applications of state augmentation have typically employed the 4D-Var, Kalman filter or ensemble Kalman filter assimilation schemes. Our new method is based on a computationally inexpensive 3D-Var scheme, where the specification of the error covariance matrices is crucial for success. A simple 1D model of bed-form propagation is used to demonstrate the method. The scheme is capable of recovering near-perfect parameter values and, therefore, improves the capability of our model to predict future bathymetry. Such positive results suggest the potential for application to more complex morphodynamic models.