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.

A review of dissolved oxygen modelling techniques for lowland rivers

This review introduces the methods used to simulate the processes affecting dissolved oxygen (DO) in lowland rivers. The important processes are described and this provides a modelling framework to describe those processes in the context of a mass-balance model. The process equations that are introduced all require (reaction) rate parameters and a variety of common procedures for identifying those parameters are reviewed. This is important because there is a wide range of estimation techniques for many of the parameters. These different techniques elicit different estimates of the parameter value and so there is the potential for a significant uncertainty in the model's inputs and therefore in the output too. Finally, the data requirements for modelling DO in lowland rivers are summarised on the basis of modelling the processes described in this review using a mass-balance model. This is reviewed with regard to what data are available and from where they might be obtained. (C) 2003 Elsevier Science B.V. All rights reserved.

Modelling oil absorption during post-frying cooling - II: solution of the mathematical model, model testing and simulations

The mathematical models that describe the immersion-frying period and the post-frying cooling period of an infinite slab or an infinite cylinder were solved and tested. Results were successfully compared with those found in the literature or obtained experimentally, and were discussed in terms of the hypotheses and simplifications made. The models were used as the basis of a sensitivity analysis. Simulations showed that a decrease in slab thickness and core heat capacity resulted in faster crust development. On the other hand, an increase in oil temperature and boiling heat transfer coefficient between the oil and the surface of the food accelerated crust formation. The model for oil absorption during cooling was analysed using the tested post-frying cooling equation to determine the moment in which a positive pressure driving force, allowing oil suction within the pore, originated. It was found that as crust layer thickness, pore radius and ambient temperature decreased so did the time needed to start the absorption. On the other hand, as the effective convective heat transfer coefficient between the air and the surface of the slab increased the required cooling time decreased. In addition, it was found that the time needed to allow oil absorption during cooling was extremely sensitive to pore radius, indicating the importance of an accurate pore size determination in future studies.

Assessing the impact on chronic disease of incorporating the societal cost of greenhouse gases into the price of food: an econometric and comparative risk assessment modelling study

Objectives To model the impact on chronic disease of a tax on UK food and drink that internalises the wider costs to society of greenhouse gas (GHG) emissions and to estimate the potential revenue. Design An econometric and comparative risk assessment modelling study. Setting The UK. Participants The UK adult population. Interventions Two tax scenarios are modelled: (A) a tax of £2.72/tonne carbon dioxide equivalents (tCO2e)/100 g product applied to all food and drink groups with above average GHG emissions. (B) As with scenario (A) but food groups with emissions below average are subsidised to create a tax neutral scenario. Outcome measures Primary outcomes are change in UK population mortality from chronic diseases following the implementation of each taxation strategy, the change in the UK GHG emissions and the predicted revenue. Secondary outcomes are the changes to the micronutrient composition of the UK diet. Results Scenario (A) results in 7770 (95% credible intervals 7150 to 8390) deaths averted and a reduction in GHG emissions of 18 683 (14 665to 22 889) ktCO2e/year. Estimated annual revenue is £2.02 (£1.98 to £2.06) billion. Scenario (B) results in 2685 (1966 to 3402) extra deaths and a reduction in GHG emissions of 15 228 (11 245to 19 492) ktCO2e/year. Conclusions Incorporating the societal cost of GHG into the price of foods could save 7770 lives in the UK each year, reduce food-related GHG emissions and generate substantial tax revenue. The revenue neutral scenario (B) demonstrates that sustainability and health goals are not always aligned. Future work should focus on investigating the health impact by population subgroup and on designing fiscal strategies to promote both sustainable and healthy diets.

Body-centric modelling, identification, and acceleration tracking control of a quadrotor UAV

This paper presents the mathematical development of a body-centric nonlinear dynamic model of a quadrotor UAV that is suitable for the development of biologically inspired navigation strategies. Analytical approximations are used to find an initial guess of the parameters of the nonlinear model, then parameter estimation methods are used to refine the model parameters using the data obtained from onboard sensors during flight. Due to the unstable nature of the quadrotor model, the identification process is performed with the system in closed-loop control of attitude angles. The obtained model parameters are validated using real unseen experimental data. Based on the identified model, a Linear-Quadratic (LQ) optimal tracker is designed to stabilize the quadrotor and facilitate its translational control by tracking body accelerations. The LQ tracker is tested on an experimental quadrotor UAV and the obtained results are a further means to validate the quality of the estimated model. The unique formulation of the control problem in the body frame makes the controller better suited for bio-inspired navigation and guidance strategies than conventional attitude or position based control systems that can be found in the existing literature.

The importance of knowing the home conditions of patients receiving long-term oxygen therapy

Purpose: Long-term oxygen therapy (LTOT) is one of the main treatments for patients with chronic obstructive pulmonary disease. Patients receiving LTOT may have less than optimal home conditions and this may interfere with treatment. The objective of this study was, through home visits, to identify the characteristics of patients receiving LTOT and to develop knowledge regarding the home environments of these patients.Methods: Ninety-seven patients with a mean age of 69 plus or minus 10.5 years were evaluated. This study was a cross-sectional descriptive analysis. Data were collected during an initial home visit, using a questionnaire standardized for the study. The results were analyzed retrospectively.Results: Seventy-five percent of the patients had chronic obstructive pulmonary disease, and 11% were active smokers. The patients' mean pulse oximetry values were 85.9% plus or minus 4.7% on room air and 92% plus or minus 3.9% on the prescribed flow of oxygen. Most of the patients did not use the treatment as prescribed and most used a humidifier. The extension hose had a mean length of 5 plus or minus 3.9 m (range, 1.5-16 m). In the year prior to the visit, 26% of the patients received emergency medical care because of respiratory problems. Few patients reported engaging in leisure activities.Conclusion: The home visit allowed us to identify problems and interventions that could improve the way LTOT is used. The most common interventions related to smoking cessation, concentrator maintenance and cleaning, use of a humidifier, and adjustments of the length of the connector hose. Therefore, the home visit is a very important tool in providing comprehensive care to patients receiving LTOT, especially those who show lack of adequate progress and those who show uncertainty about the treatment method.

On non-linear dynamics and an optimal control synthesis of the action potential of membranes (ideal and non-ideal cases) of the Hodgkin-Huxley (HH) mathematical model

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Optimal design of smart structures using bonded piezoelectrics for vibration control

Smart material technology has become an area of increasing interest for the development of lighter and stronger structures which are able to incorporate actuator and sensor capabilities for collocated control. In the design of actively controlled structures, the determination of the actuator locations and the controller gains, is a very important issue. For that purpose, smart material modelling, modal analysis methods, control and optimization techniques are the most important ingredients to be taken into account. The optimization problem to be solved in this context presents two interdependent aspects. The first one is related to the discrete optimal actuator location selection problem, which is solved in this paper using genetic algorithms. The second is represented by a continuous variable optimization problem, through which the control gains are determined using classical techniques. A cantilever Euler-Bernoulli beam is used to illustrate the presented methodology.

Monitoring chronic physical stress using biomarkers, performance protocols and mathematical functions to identify physiological adaptations in rats

This study was undertaken to characterize the effects of monotonous training at lactate minimum (LM) intensity on aerobic and anaerobic performances; glycogen concentrationsin the soleus muscle, the gastrocnemius muscle and the liver; and creatine kinase (CK), free fatty acids and glucose concentrations in rats. The rats were separated into trained (n =10), baseline (n = 10) and sedentary (n=10) groups. The trained group was submitted to the following: 60 min/day, 6 day/week and intensity equivalent to LM during the 12-week training period. The training volume was reduced after four weeks according to a sigmoid function. The total CK (U/L) increased in the trained group after 12 weeks (742.0±158.5) in comparison with the baseline (319.6±40.2) and the sedentary (261.6+42.2) groups. Free fatty acids and glycogen stores (liver, soleus muscle and gastrocnemius muscle) increased after 12 weeks of monotonous training but aerobic and anaerobic performances were unchanged in relation to the sedentary group. The monotonous training at LM increased the level of energy substrates, unchanged aerobic performance, reduced anaerobic capacity and increased the serum CK concentration; however, the rats did not achieve the predicted training volume.

Forced Desynchronization of Activity Rhythms in a Model of Chronic Jet Lag in Mice

We studied locomotor activity rhythms of C57/Bl6 mice under a chronic jet lag (CJL) protocol (ChrA(6/2)), which consisted of 6-hour phase advances of the light-dark schedule (LD) every 2 days. Through periodogram analysis, we found 2 components of the activity rhythm: a short-period component (21.01 +/- 0.04 h) that was entrained by the LD schedule and a long-period component (24.68 +/- 0.26 h). We developed a mathematical model comprising 2 coupled circadian oscillators that was tested experimentally with different CJL schedules. Our simulations suggested that under CJL, the system behaves as if it were under a zeitgeber with a period determined by (24 -[phase shift size/days between shifts]). Desynchronization within the system arises according to whether this effective zeitgeber is inside or outside the range of entrainment of the oscillators. In this sense, ChrA(6/2) is interpreted as a (24 - 6/2 = 21 h) zeitgeber, and simulations predicted the behavior of mice under other CJL schedules with an effective 21-hour zeitgeber. Animals studied under an asymmetric T = 21 h zeitgeber (carried out by a 3-hour shortening of every dark phase) showed 2 activity components as observed under ChrA(6/2): an entrained short-period (21.01 +/- 0.03 h) and a long-period component (23.93 +/- 0.31 h). Internal desynchronization was lost when mice were subjected to 9-hour advances every 3 days, a possibility also contemplated by the simulations. Simulations also predicted that desynchronization should be less prevalent under delaying than under advancing CJL. Indeed, most mice subjected to 6-hour delay shifts every 2 days (an effective 27-hour zeitgeber) displayed a single entrained activity component (26.92 +/- 0.11 h). Our results demonstrate that the disruption provoked by CJL schedules is not dependent on the phase-shift magnitude or the frequency of the shifts separately but on the combination of both, through its ratio and additionally on their absolute values. In this study, we present a novel model of forced desynchronization in mice under a specific CJL schedule; in addition, our model provides theoretical tools for the evaluation of circadian disruption under CJL conditions that are currently used in circadian research.

Modelling Mathematical Reasoning in Physics Education

Many findings from research as well as reports from teachers describe students' problem solving strategies as manipulation of formulas by rote. The resulting dissatisfaction with quantitative physical textbook problems seems to influence the attitude towards the role of mathematics in physics education in general. Mathematics is often seen as a tool for calculation which hinders a conceptual understanding of physical principles. However, the role of mathematics cannot be reduced to this technical aspect. Hence, instead of putting mathematics away we delve into the nature of physical science to reveal the strong conceptual relationship between mathematics and physics. Moreover, we suggest that, for both prospective teaching and further research, a focus on deeply exploring such interdependency can significantly improve the understanding of physics. To provide a suitable basis, we develop a new model which can be used for analysing different levels of mathematical reasoning within physics. It is also a guideline for shifting the attention from technical to structural mathematical skills while teaching physics. We demonstrate its applicability for analysing physical-mathematical reasoning processes with an example.

On the mechanisms that limit oxygen uptake during exercise in acute and chronic hypoxia: role of muscle mass

[EN] Peak aerobic power in humans (VO2,peak) is markedly affected by inspired O2 tension (FIO2). The question to be answered in this study is what factor plays a major role in the limitation of muscle peak VO2 in hypoxia: arterial O2 partial pressure (Pa,O2) or O2 content (Ca,O2)? Thus, cardiac output (dye dilution with Cardio-green), leg blood flow (thermodilution), intra-arterial blood pressure and femoral arterial-to-venous differences in blood gases were determined in nine lowlanders studied during incremental exercise using a large (two-legged cycle ergometer exercise: Bike) and a small (one-legged knee extension exercise: Knee)muscle mass in normoxia, acute hypoxia (AH) (FIO2 = 0.105) and after 9 weeks of residence at 5260 m (CH). Reducing the size of the active muscle mass blunted by 62% the effect of hypoxia on VO2,peak in AH and abolished completely the effect of hypoxia on VO2,peak after altitude acclimatization. Acclimatization improved Bike peak exercise Pa,O2 from 34 +/- 1 in AH to 45 +/- 1 mmHg in CH(P <0.05) and Knee Pa,O2 from 38 +/- 1 to 55 +/- 2 mmHg(P <0.05). Peak cardiac output and leg blood flow were reduced in hypoxia only during Bike. Acute hypoxia resulted in reduction of systemic O2 delivery (46 and 21%) and leg O2 delivery (47 and 26%) during Bike and Knee, respectively, almost matching the corresponding reduction in VO2,peak. Altitude acclimatization restored fully peak systemic and leg O(2) delivery in CH (2.69 +/- 0.27 and 1.28 +/- 0.11 l min(-1), respectively) to sea level values (2.65 +/- 0.15 and 1.16 +/- 0.11 l min(-1), respectively) during Knee, but not during Bike. During Knee in CH, leg oxygen delivery was similar to normoxia and, therefore, also VO2,peak in spite of a Pa,O2 of 55 mmHg. Reducing the size of the active mass improves pulmonary gas exchange during hypoxic exercise, attenuates the Bohr effect on oxygen uploading at the lungs and preserves sea level convective O2 transport to the active muscles. Thus, the altitude-acclimatized human has potentially a similar exercising capacity as at sea level when the exercise model allows for an adequate oxygen delivery (blood flow x Ca,O2), with only a minor role of Pa,O2 per se, when Pa,O2 is more than 55 mmHg.

Outdoor production of Isochrysis galbana (T-iso) in industrial scale photobioreactors and modelling of its photosynthesis and respiration rate

Isochrysis galbana is a widely-used strain in aquaculture in spite of its low productivity. To maximize the productivity of processes based on this microalgae strain, a model was developed considering the influence of irradiance, temperature, pH and dissolved oxygen concentration on the photosynthesis and respiration rate. Results demonstrate that this strain tolerates temperatures up to 35ºC but it is highly sensitive to irradiances higher than 500 µE·m-2·s-1 and dissolved oxygen concentrations higher than 11 mg·l-1. With the researcher group of the “Universidad de Almeria”, the developed model was validated using data from an industrial-scale outdoor tubular photobioreactor demonstrating that inadequate temperature and dissolved oxygen concentrations reduce productivity to half that which is maximal, according to light availability under real outdoor conditions. The developed model is a useful tool for managing working processes, especially in the development of new processes based on this strain and to take decisions regarding optimal control strategies. Also the outdoor production of Isochrysis galbana T-iso in industrial size tubular photobioreactors (3.0 m3) has been studied. Experiments were performed modifying the dilution rate and evaluating the biomass productivity and quality, in addition to the overall performance of the system. Results confirmed that T-iso can be produced outdoor at commercial scale in continuous mode, productivities up to 20 g·m-2·day-1 of biomass rich in proteins (45%) and lipids (25%) being obtained. The utilization of this type of photobioreactors allows controlling the contamination and pH of the cultures, but daily variation of solar radiation imposes the existence of inadequate dissolved oxygen concentration and temperature at which the cells are exposed to inside the reactor. Excessive dissolved oxygen reduced the biomass productivity to 68% of maximal, whereas inadequate temperature reduces to 63% of maximal. Thus, optimally controlling these parameters the biomass productivity can be duplicated. These results confirm the potential to produce this valuable strain at commercial scale in optimally designed/operated tubular photobioreactors as a biotechnological industry.