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See [ ] and references therein for a detailed description of event-driven simulators applied to systems with discontinuous flows. A set-based formulation for the hybrid automaton In general, a dynamical system given by: Hence, the total hybrid trajectory can be stated as: Such a result is illustrated in Fig.
Notice that it graphically explains, by an automaton, the situation depicted previously in Fig. Then, the system jumps from the descriptions of the vector fields and depending on the event occurring at.
Graphical representation of a hybrid dynamical system. Hybrid dynamical system as automaton. Buck type DC-DC power converter circuit. The mathematical representation of HDS introduced here, will be employed in the following to describe the dynamic behavior of a power converter circuit.
A Buck type power converter circuit Fig. The operation of the system can be split in two modes depending on the conduction state of the MOSFET transistor which operates as a switch. The equivalent circuit is depicted in Fig. As it can be seen, the source applies the power input to the load. Nevertheless, it is necessary to take into account the influence of the diode into the circuit. Conduction mode of the Buck circuit. Non-conduction mode of the Buck circuit, diode forward-biased.
On the other hand, Fig. Non-conduction mode of the Buck circuit, diode reverse-biased. Then, after defining as state variables the current of the coil and the voltage of the loadit is possible to formulate a state-space representation of the circuit according to 2 by combining expressions 6 - 11 in the following matrix representation: Notice that the signum function sgn.
A set-based formulation for the Buck circuit The dynamics of the Buck-type power converter circuit depend on the pattern of the PWM signal, given that it provides the time instants where the command input u changes among its binary values.
Also, the jth inactivity event can be stated by: In accordance, the continuous set C describing the solution trajectory can be formulated as: In this way, it is possible to express the system flow, or solution trajectory, for the Buck-type converter circuit in eq. On-Off controller as a hybrid system It is well known, that a proportional control action, defined by: If KP is increased, the disturbance effects start to vanish and steady-state errors become diminished. In practice, however, the maximum value for this loop gain is constrained by saturation boundaries and of signal levels as depicted in Fig.
In spite of this, for higher values of the loop gain KP, the controller responses are strong enough to annihilate the error, with a limiting theoretical value of.
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The latter is the special case usually termed On-Off controller, a situation illustrated in Fig. An On-Off controller delivers as control action, the maximum energy in a minimum time. There are several examples of practical applications for this kind of controller, including heat and cooling systems, level control and end-position control, among others. As can be seen from 20the controller has a discontinuous fashion that can be modelled as an HDS.
Proportional control action limited by saturation boundaries. On-Off control action as infinite loop gain of a proportional controller. A set-based formulation for the On-Off controller In accordance with 2the dynamical representation of an On-Off controller of the form 20 can be stated as: Then, by considering that a sgn.
Then, if dynamics of the On-Off controller 21 can be solved in terms of the ordered pair: Therefore, the solution trajectory describing the On-Off control action can be stated as: Hybrid formulation of the Buck circuit On-Off controlled The sets proposed in section 2.
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In essence, the two models can be coupled by considering that the time instants governing the commutation changes in the circuit, coincide with the zero crossing of the error signal, i.
Finally, the flow-path of the controlled system can be stated by: Circuit parameters used in simulations. As can be noticed, the qualitative behavior of the voltage and current signals is replicated in both simulation environments.