AC-DC three-phase multipulse converter with boost DC-DC stage and constant hysteresis control

This work proposes a new three-phase multipulse rectifier based on the delta autotransformer connection with DC-DC Boost stages and constant hysteresis control which has the objective of providing a reliable DC bus for on-board applications, electric motor drives and similars, always considering power quality issues. Thus, the proposal presents 0.99 power factor, 6% harmonic distortions in the currents from the mains and enhanced magnetic core utilization, which results in low weight and volume for the overall converter. The proposed control technique uses the simple constant hysteresis concept, thus leading to a low-cost but effective and reliable strategy. © 2011 IEEE.

Sliding Mode Control for Switched Mode Power Supplies

Control applications of switched mode power supplies have been widely investigated. The main objective ofresearch and development (R&D) in this field is always to find the most suitable control method to be implemented in various DC/DC converter topologies. Inother words, the goal is to select a control method capable of improving the efficiency of the converter, reducing the effect of disturbances (line and load variation), lessening the effect of EMI (electro magnetic interference), and beingless effected by component variation. The main objective of this research work is to study different control methods implemented in switched mode power supplies namely (PID control, hysteresis control, adaptive control, current programmed control, variable structure control (VSC), and sliding mode control (SMC). The advantages and drawbacks of each control method are given. Two control methods, the PID and the SMC are selected and their effects on DC/DC (Buck, Boost, and Buck-Boost) converters are examined. Matlab/SimulinkTM is used to implement PID control method in DC/DC Buck converter and SMC in DC/DC (Buck, and Buck Boost) converters. For the prototype, operational amplifiers (op-amps) are used to implement PID control in DC/DC Buck converter. For SMC op-amps are implemented in DC/DC Buck converter and dSPACETM is used to control the DC/DC Buck-Boost converter. The SMC can be applied to the DC/DC (Buck, Boost, and Buck-Boost) converters. A comparison of the effects of the PID control and the SMC on the DC/DC Buck converter response in steady state, under line variations, load variations, and different component variations is performed. Also the Conducted RF-Emissions between the PID and SMC DC/DC Buck Converter are compared. The thesis shows that, in comparison with the PID control, the SMC provides better steady-state response, better dynamic response, less EMI, inherent order reduction, robustness against system uncertainty disturbances, and an implicit stability proof. Giving a better steady-state and dynamic response, the SMC is implemented in a DC/DC resonant converter. The half-wave zero current switching (HWZCS) DC/DC Buck converter is selected as a converter topology. A general guideline to select the tank component values, needed for the designing of a HWZCS DC/DC Buck, is obtained. The implementation of the SMC to a HWZCS DC/DC Buck converter is analysed. The converter response is investigated in the steady-state region and in the dynamic region.

An AC voltage regulator with high-power-factor, and control using a FPGA device

This paper presents the analysis, design, simulation, and experimental results for a high frequency high Power-Factor (PF) AC (Alternate Current) voltage regulator, using a Sepic converter as power stage. The control technique employed to impose a sinusoidal input current waveform, with low Total Harmonic Distortion (THD), is the sinusoidal variable hysteresis control. The control technique was implemented in a FPGA (Field Programmable Gate Array) device, using a Hardware Description Language (VHDL). Through the use of the proposed control technique, the AC voltage regulator performs active power-factor correction, and low THD in the input current, for linear and non-linear loads, satisfying the requirements of the EEC61000-3-2 standards. Experimental results from an example prototype, designed for 300W of nominal output power, 50kHz (switching frequency), and 127Vrms of nominal input and output voltages, are presented in order to validate the proposed AC regulator. © 2005 IEEE.

Tehokerroinkorjatun hakkuriteholähteen toteutusvaihtoehdot

Hyötysuhteen merkitys tehoelektroniikan järjestelmissä kasvaa jatkuvasti kun käytössä olevien järjestelmien määrä lisääntyy ja toisaalta energian hinta kallistuu. Hyötysuhteen lisäksi verkkovirran käyrämuotovaatimukset asettavat omat haasteensa sähköverkkoon liitettävien tehoelektroniikkajärjestelmien suunnittelulle. Tässä työssä tutkitaan häviöiden syntyyn vaikuttavia tekijöitä hakkuriteholähteissä ja pyritään löytämään keinoja hyötysuhteen parantamiseksi. Työssä analysoidaan 300 W – 2 kW tehoalueelle soveltuvat, tehokerroinkorjattujen AC/DC-hakkuriteholähteiden topologiavaihtoehdot ja arvioidaan niiden soveltuvuutta tavanomaisen boost-topologian korvaajaksi. Tarkastelussa otetaan huomioon kustannukset, toimivuus sekä saavutettu hyötysuhteen parantuminen verrattuna perinteisellä topologialla toteutettuun teholähteeseen. Tarkasteltavat topologiat valitaan kirjallisuustutkimuksen perusteella. Valittujen topologioiden häviöt ja hyötysuhde selvitetään analyyttisin menetelmin sekä simuloimalla. Käytännön testausta varten suunnitellaan ja rakennetaan prototyyppi valitusta topologiasta.

Kestomagneettitahtikonekäyttöjen erot

Permanent magnet drives with nominal power over 10 kW were not a cost-sufficient system 25 years ago due to high material expenses. The improvements in motor drives, the rise in competition and the tightening of standards and regulations have caused that the PM-drives are more and more common in the over 10 kW nominal power range. The goal of this thesis is to research the performance in relation to nominal power of a PM-drive technique that is vastly increasing its popularity in fan related devices. The studied motor technique brushless direct current drive (BLDC) consists of a voltage source inverter, permanent motor and six-step-control. The reference drive is a brushless alternating current drive (BLAC) which consists of a VSI, PM and a hysteresis control. As a conclusion there are no major obstacles that would impede the BLDC-drive technique from expanding to larger power stages. The following factors must be taken into consideration when designing a BLDC-drive: motor’s current change rate, inverter switching frequency, motor’s nominal electric frequency, phase inductance and the current handling capability of the inverter. The fluctuating material costs create instability to the end prices of PM-motors that can in the worst case lead to diminished interest towards BLDC- and PM-drives in general.

Retificador trifásico de 18 pulsos com estágio CC controlado por histerese constante

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Novel Inductor‐less Conversion Strategy Based on Multiphase Transformer‐Coupled Converters: Analysis, Design and Applications

There are many the requirements that modern power converters should fulfill. Most of the applications where these converters are used, demand smaller converters with high efficiency, improved power density and a fast dynamic response. For instance, loads like microprocessors demand aggressive current steps with very high slew rates (100A/mus and higher); besides, during these load steps, the supply voltage of the microprocessor should be kept within tight limits in order to ensure its correct performance. The accomplishment of these requirements is not an easy task; complex solutions like advanced topologies - such as multiphase converters- as well as advanced control strategies are often needed. Besides, it is also necessary to operate the converter at high switching frequencies and to use capacitors with high capacitance and low ESR. Improving the dynamic response of power converters does not rely only on the control strategy but also the power topology should be suited to enable a fast dynamic response. Moreover, in later years, a fast dynamic response does not only mean accomplishing fast load steps but output voltage steps are gaining importance as well. At least, two applications that require fast voltage changes can be named: Low power microprocessors. In these devices, the voltage supply is changed according to the workload and the operating frequency of the microprocessor is changed at the same time. An important reduction in voltage dependent losses can be achieved with such changes. This technique is known as Dynamic Voltage Scaling (DVS). Another application where important energy savings can be achieved by means of changing the supply voltage are Radio Frequency Power Amplifiers. For example, RF architectures based on ‘Envelope Tracking’ and ‘Envelope Elimination and Restoration’ techniques can take advantage of voltage supply modulation and accomplish important energy savings in the power amplifier. However, in order to achieve these efficiency improvements, a power converter with high efficiency and high enough bandwidth (hundreds of kHz or even tens of MHz) is necessary in order to ensure an adequate supply voltage. The main objective of this Thesis is to improve the dynamic response of DC-DC converters from the point of view of the power topology. And the term dynamic response refers both to the load steps and the voltage steps; it is also interesting to modulate the output voltage of the converter with a specific bandwidth. In order to accomplish this, the question of what is it that limits the dynamic response of power converters should be answered. Analyzing this question leads to the conclusion that the dynamic response is limited by the power topology and specifically, by the filter inductance of the converter which is found in series between the input and the output of the converter. The series inductance is the one that determines the gain of the converter and provides the regulation capability. Although the energy stored in the filter inductance enables the regulation and the capability of filtering the output voltage, it imposes a limitation which is the concern of this Thesis. The series inductance stores energy and prevents the current from changing in a fast way, limiting the slew rate of the current through this inductor. Different solutions are proposed in the literature in order to reduce the limit imposed by the filter inductor. Many publications proposing new topologies and improvements to known topologies can be found in the literature. Also, complex control strategies are proposed with the objective of improving the dynamic response in power converters. In the proposed topologies, the energy stored in the series inductor is reduced; examples of these topologies are Multiphase converters, Buck converter operating at very high frequency or adding a low impedance path in parallel with the series inductance. Control techniques proposed in the literature, focus on adjusting the output voltage as fast as allowed by the power stage; examples of these control techniques are: hysteresis control, V 2 control, and minimum time control. In some of the proposed topologies, a reduction in the value of the series inductance is achieved and with this, the energy stored in this magnetic element is reduced; less stored energy means a faster dynamic response. However, in some cases (as in the high frequency Buck converter), the dynamic response is improved at the cost of worsening the efficiency. In this Thesis, a drastic solution is proposed: to completely eliminate the series inductance of the converter. This is a more radical solution when compared to those proposed in the literature. If the series inductance is eliminated, the regulation capability of the converter is limited which can make it difficult to use the topology in one-converter solutions; however, this topology is suitable for power architectures where the energy conversion is done by more than one converter. When the series inductor is eliminated from the converter, the current slew rate is no longer limited and it can be said that the dynamic response of the converter is independent from the switching frequency. This is the main advantage of eliminating the series inductor. The main objective, is to propose an energy conversion strategy that is done without series inductance. Without series inductance, no energy is stored between the input and the output of the converter and the dynamic response would be instantaneous if all the devices were ideal. If the energy transfer from the input to the output of the converter is done instantaneously when a load step occurs, conceptually it would not be necessary to store energy at the output of the converter (no output capacitor COUT would be needed) and if the input source is ideal, the input capacitor CIN would not be necessary. This last feature (no CIN with ideal VIN) is common to all power converters. However, when the concept is actually implemented, parasitic inductances such as leakage inductance of the transformer and the parasitic inductance of the PCB, cannot be avoided because they are inherent to the implementation of the converter. These parasitic elements do not affect significantly to the proposed concept. In this Thesis, it is proposed to operate the converter without series inductance in order to improve the dynamic response of the converter; however, on the other side, the continuous regulation capability of the converter is lost. It is said continuous because, as it will be explained throughout the Thesis, it is indeed possible to achieve discrete regulation; a converter without filter inductance and without energy stored in the magnetic element, is capable to achieve a limited number of output voltages. The changes between these output voltage levels are achieved in a fast way. The proposed energy conversion strategy is implemented by means of a multiphase converter where the coupling of the phases is done by discrete two-winding transformers instead of coupledinductors since transformers are, ideally, no energy storing elements. This idea is the main contribution of this Thesis. The feasibility of this energy conversion strategy is first analyzed and then verified by simulation and by the implementation of experimental prototypes. Once the strategy is proved valid, different options to implement the magnetic structure are analyzed. Three different discrete transformer arrangements are studied and implemented. A converter based on this energy conversion strategy would be designed with a different approach than the one used to design classic converters since an additional design degree of freedom is available. The switching frequency can be chosen according to the design specifications without penalizing the dynamic response or the efficiency. Low operating frequencies can be chosen in order to favor the efficiency; on the other hand, high operating frequencies (MHz) can be chosen in order to favor the size of the converter. For this reason, a particular design procedure is proposed for the ‘inductorless’ conversion strategy. Finally, applications where the features of the proposed conversion strategy (high efficiency with fast dynamic response) are advantageus, are proposed. For example, in two-stage power architectures where a high efficiency converter is needed as the first stage and there is a second stage that provides the fine regulation. Another example are RF power amplifiers where the voltage is modulated following an envelope reference in order to save power; in this application, a high efficiency converter, capable of achieving fast voltage steps is required. The main contributions of this Thesis are the following: The proposal of a conversion strategy that is done, ideally, without storing energy in the magnetic element. The validation and the implementation of the proposed energy conversion strategy. The study of different magnetic structures based on discrete transformers for the implementation of the proposed energy conversion strategy. To elaborate and validate a design procedure. To identify and validate applications for the proposed energy conversion strategy. It is important to remark that this work is done in collaboration with Intel. The particular features of the proposed conversion strategy enable the possibility of solving the problems related to microprocessor powering in a different way. For example, the high efficiency achieved with the proposed conversion strategy enables it as a good candidate to be used for power conditioning, as a first stage in a two-stage power architecture for powering microprocessors.

Winding-centre-tapped switched reluctance motor drive for multi-source charging in electric vehicle applications

Switched reluctance motors (SRMs) can provide an attractive traction drive for electric vehicle applications. To lower the investment in the off-board charging station facilities, a multi-functional switched reluctance motor topology is proposed on the basis of the traditional asymmetrical half-bridge converter. The SRM phase windings are employed as input filter inductors and centre-tapped windings are also developed to form symmetrical inductors for three-phase grid supply. Owing to the varying rotor position, phase inductors are unequal between one another. A hysteresis control scheme is therefore developed for grid-connection operation. In addition to AC supplies, the proposed topology can also supports the DC-source charging. A new current sharing strategy is employed to diminish the influence of the unequal winding inductances. The simulation and experimental tests are carried out to verify the proposed topology and control methods. Since this work eliminates the need for building charging station infrastructure, its potential economic impact on the automotive market can be significant.

Split converter-fed SRM drive for flexible charging in EV/HEV applications

Electric vehicles (EVs) and hybrid EVs are the way forward for green transportation and for establishing low-carbon economy. This paper presents a split converter-fed four-phase switched reluctance motor (SRM) drive to realize flexible integrated charging functions (dc and ac sources). The machine is featured with a central-tapped winding node, eight stator slots, and six rotor poles (8/6). In the driving mode, the developed topology has the same characteristics as the traditional asymmetric bridge topology but better fault tolerance. The proposed system supports battery energy balance and on-board dc and ac charging. When connecting with an ac power grid, the proposed topology has a merit of the multilevel converter; the charging current control can be achieved by the improved hysteresis control. The energy flow between the two batteries is balanced by the hysteresis control based on their state-of-charge conditions. Simulation results in MATLAB/Simulink and experiments on a 150-W prototype SRM validate the effectiveness of the proposed technologies, which may provide a solution to EV charging issues associated with significant infrastructure requirements.

Efficiency and power density improvement of grid-connected hybrid renewable energy systems utilizing high frequency-based power converters

High efficiency of power converters placed between renewable energy sources and the utility grid is required to maximize the utilization of these sources. Power quality is another aspect that requires large passive elements (inductors, capacitors) to be placed between these sources and the grid. The main objective is to develop higher-level high frequency-based power converter system (HFPCS) that optimizes the use of hybrid renewable power injected into the power grid. The HFPCS provides high efficiency, reduced size of passive components, higher levels of power density realization, lower harmonic distortion, higher reliability, and lower cost. The dynamic modeling for each part in this system is developed, simulated and tested. The steady-state performance of the grid-connected hybrid power system with battery storage is analyzed. Various types of simulations were performed and a number of algorithms were developed and tested to verify the effectiveness of the power conversion topologies. A modified hysteresis-control strategy for the rectifier and the battery charging/discharging system was developed and implemented. A voltage oriented control (VOC) scheme was developed to control the energy injected into the grid. The developed HFPCS was compared experimentally with other currently available power converters. The developed HFPCS was employed inside a microgrid system infrastructure, connecting it to the power grid to verify its power transfer capabilities and grid connectivity. Grid connectivity tests verified these power transfer capabilities of the developed converter in addition to its ability of serving the load in a shared manner. In order to investigate the performance of the developed system, an experimental setup for the HF-based hybrid generation system was constructed. We designed a board containing a digital signal processor chip on which the developed control system was embedded. The board was fabricated and experimentally tested. The system's high precision requirements were verified. Each component of the system was built and tested separately, and then the whole system was connected and tested. The simulation and experimental results confirm the effectiveness of the developed converter system for grid-connected hybrid renewable energy systems as well as for hybrid electric vehicles and other industrial applications.

Advanced Control for Very Fast DC-DC Converters Based on Hysteresis of the C-out Current

The bandwidth achievable by using voltage mode control or current mode control in switch-mode power supply is limited by the switching frequency. Fast transient response requires high switching frequency, although lower switching frequencies could be more suitable for higher efficiency. This paper proposes the use of hysteretic control of the output capacitor $(C_{out})$ current to improve the dynamic response of the buck converter. An external voltage loop is required to accurately regulate the output voltage. The design of the hysteretic loop and the voltage loop are presented. Besides, it is presented a non-invasive current sensor that allows measuring the current in the capacitor. This strategy has been applied for DVS (dynamic voltage scaling) on a 5 MHz buck converter. Experimental results validate the proposed control technique and show fast transient response from 1.5 V to 2.5 V in 2 $mu{rm s}$.

On the Hopf bifurcation in control systems with a bounded nonlinearity asymptotically homogeneous at infinity

The paper studies existence, uniqueness, and stability of large-amplitude periodic cycles arising in Hopf bifurcation at infinity of autonomous control systems with bounded nonlinear feedback. We consider systems with functional nonlinearities of Landesman-Lazer type and a class of systems with hysteresis nonlinearities. The method is based on the technique of parameter functionalization and methods of monotone concave and convex operators. (C) 2001 Academic Press.

Control of heart rate during thermoregulation in the heliothermic lizard Pogona barbata: importance of cholinergic and adrenergic mechanisms

During thermo regulation in the bearded dragon Pogona barbata, heart rate when heating is significantly faster than when cooling at any given body temperature (heart rate hysteresis), resulting in faster rates of heating than cooling. However, the mechanisms that control heart rate during heating and cooling are unknown. The aim of this study was to test the hypothesis that changes in cholinergic and adrenergic tone on the heart are responsible for the heart rate hysteresis during heating and cooling in P. barbata. Heating and cooling trials were conducted before and after the administration of atropine, a muscarinic antagonist, and sotalol, a beta-adrenergic antagonist. Cholinergic and beta-adrenergic blockade did not abolish the heart rate hysteresis, as the heart rate during heating was significantly faster than during cooling in all cases. Adrenergic tone was extremely high (92.3%) at the commencement of heating, and decreased to 30.7% at the end of the cooling period. Moreover, in four lizards there was an instantaneous drop in heart rate (up to 15 beats min(-1)) as the heat source was switched off, and this drop in heart rate coincided with either a drop in beta-adrenergic tone or an increase in cholinergic tone. Rates of heating were significantly faster during the cholinergic blockade, and least with a combined cholinergic and beta-adrenergic blockade. The results showed that cholinergic and beta-adrenergic systems are not the only control mechanisms acting on the heart during heating and cooling, but they do have a significant effect on heart rate and on rates of heating and cooling.

Control of bistability in a directly modulated semiconductor laser using delayed optoelectronic feedback

We show numerically that direct delayed optoelectronic feedback can suppress hysteresis and bistability in a directly modulated semiconductor laser. The simulation of a laser with feedback is performed for a considerable range of feedback strengths and delays and the corresponding values for the areas of the hysteresis loops are calculated. It is shown that the hysteresis loop completely vanishes for certain combinations of these parameters. The regimes for the disappearance of bistability are classified globally. Different dynamical states of the laser are characterized using bifurcation diagrams and time series plots.

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.