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The hybrid paradigm (often referred to as Cyber-Physical-Systems) can be employed to understand (by modelling) or to manipulate (by control design) the dynamical behavior of systems. In this paper, a system of wide use in applications (

El paradigma híbrido (o de Sistemas Ciber-físicos) puede ser empleado para entender (modelar) o manipular (controlar) el comportamiento dinámico de sistemas. Este artículo aborda el modelado desde una perspectiva híbrida para describir la dinámica de un circuito convertidor de potencia bajo la acción de un controlador encendido-apagado, a través de una formulación basada en conjuntos. Este enfoque es novedoso en cuanto permite formular reglas genéricas pero formales, a partir de transiciones entre modos de operación del sistema, lo cual facilita su posterior implementación computacional en entornos de simulación. De hecho, se muestra como el sistema controlado puede explicarse en términos de la unión de los conjuntos que describen el circuito y su control. Resultados generados en un simulador comercial de circuitos replican las predicciones obtenidas tras evaluar las reglas de conjuntos propuestas, representando un paso importante en el camino hacia la comprensión del comportamiento dinámico de sistemas discontinuos más complejos.

Dynamical systems when thought as operating under realistic conditions are usually constrained to evolve inside delimited zones of the state-space, as a natural consequence of physical limitations appearing by implementations, such as the saturation of actuation components and the boundaries of motion trajectories. Moreover, different operational modes of the system define particular regions in the solution space and therefore, a conditioned representation of the dynamics. Such interaction of continuous behavior and discrete transition-rules between operation modes is defined in terms of the so-called Hybrid Dynamical Systems (HDS).

An HDS can be stated as one with solution trajectories composed of continuous paths linked by discontinuous transitions obeying a discrete rule [

Hybrid Dynamical Systems (HDS) have been a subject of increasing interest from the earlier developments in the Eastern literature till nowadays. In particular, new methodologies and techniques for analysis and control tasks appeared in an attempt to deal with the impossibility to apply the well-developed classical theory of smooth-dynamical systems. Interested readers are referred to [

As a representative example, let us consider the dynamical system described by:

with _{
F1
} and _{
F2
} representing two different modes of operation across the discontinuity boundary _{1} and S_{2}), generating respectively the flow paths _{
(1
} and _{
(2
} .

Then in order to solve (1) for an arbitrary time

In general, a dynamical system given by:

where _{0} is a set of initial conditions; can be solved as a trajectory in the state space

In an HDS such flow path Φ(

where _{
ith
} interval of the domain where

where _{
ti
} .

Hence, the total hybrid trajectory can be stated as:

Such a result is illustrated in

The mathematical representation of HDS introduced here, will be employed in the following to describe the dynamic behavior of a power converter circuit.

The equivalent circuit is depicted in _{0} represents the output voltage in the load and_{
VL
} , _{
VC
} the corresponding to inductor and capacitor devices.

In the second mode, the switch is open (_{
iL
} >0), with

On the other hand,

Then, after defining as state variables the current of the coil

where

Notice that the signum function sgn(.) includes the discontinuity to the system flow and, therefore, an HDS representation of the model can be achieved easily, shown as follows.

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.

In a more precise manner, each period T of the PWM signal is composed of an activation interval tON (for u = 1) followed by a gap of inactivity tOOF (for u = 0) i.e.

Hence, by starting from t =t0, the ith conduction instant can be defined as:

Also, the jth inactivity event can be stated by:

In accordance, the continuous set C describing the solution trajectory can be formulated as:

and equivalently, the discrete events are defined by:

In this way, it is possible to express the system flow, or solution trajectory, for the Buck-type converter circuit in

It is well known, that a proportional control action, defined by:

where e (t) represents the output error and KP constitutes the loop gain, is the most basic governing law that can be stated in a feedback control system. Moreover, if KP = 1 we have a unit gain negative feedback created naturally after closing the loop.

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 (

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

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 (20), the controller has a discontinuous fashion that can be modelled as an HDS.

In accordance with (2), the dynamical representation of an On-Off controller of the form (20) can be stated as:

for

Then, by considering that a sgn(.) function can be expressed as the following sum of step functions µ(.):

by using the fact that the derivative of a step function µ(.) is a Dirac delta function δ(.).

Then, if dynamics of the On-Off controller (21) can be solved in terms of the ordered pair:

the continuous set C describing the flow path of the system is:

and the corresponding discrete events D are given by:

Therefore, the solution trajectory describing the On-Off control action can be stated as:

The sets proposed in section 2.1 for the continuous paths and discrete events composing the solution trajectory of hybrid systems, further applied in sections 3.1 and 4.1 to describe dynamics of a Buck-type power converter circuit and an On-Off controller, can be merged here to define the hybrid controlled circuit.

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.e. by combining

and equivalently, by combining (17) and (26) the discrete events are defined by:

Finally, the flow-path of the controlled system can be stated by:

The analytic results shown in previous sections have been tested by comparing numerical evaluation of the proposed sets, with results of the commercial circuit simulation package OrCAD-Cadence®.

Source: Authors

In comparison,

As can be noticed, the qualitative behavior of the voltage and current signals is replicated in both simulation environments. In particular, a smooth shape for the capacitor voltage is achieved with a peak time around 0.5[s] and a falling slope reaching a steady state regime close to the expected value of 12 [V] near to 3 [ms].

In contrast, the behavior of the inductor current is characterized by a discontinuous shape with an initial lobe followed by a stable average of around 2 [A] starting close to 0.5 [s].

However, there are also qualitative disagreements in the results that can be explained by special features added to the commercial circuit simulator (e.g. temperature dependence, non-idealization of circuit elements), not considered in our simple approach. Nevertheless, the agreement achieved is good enough to validate the correctness of the proposed scheme to model the dynamic behavior of the

Now, in order to verify the hybrid model of the On-Off controller,

The same principle is employed in

In such a case, the error signal applied to the controller as input, has a constant frequency and duty cycle values in accordance to the nominal values illustrated in

An HDS can be represented in terms of a trajectory composed of continuous paths linked by discrete events. Those continuous and discrete intervals have been defined by sets including all the possible points in the state-space which, belonging to the system-flow solutions, can be classified as part of one of the hybrid components.

A

The proposed set-based approach has been further employed to define an On-Off controller as HDS. In such a case, the dynamics of the control signal at the output of the system was described in terms of variations on the error signal.

An approximation for the On-Off controlled

The numerical results obtained with the commercial circuit simulator were replicated by evaluating the set-based formulation of the controlled circuit, constituting a valuable tool in the path to understand the dynamical behavior of discontinuous systems.

It is interesting to notice the way it was possible to express the behavior of the controlled system, as the union of sets describing individually the circuit and the controller dynamics. This can be understood as a new framework that complements existing approaches, such as finite state machines, Petri nets and Markovian models, by defining formal rules scalable onto more complex systems. The potential of the proposed approach can be exploited by exploring the wide possibilities offered by sets theory in order to describe and to formalize continuous and discrete formulations and interactions in between.

A natural evolution for the application case considered is the study of more complex topologies of power conversion systems, including more than a simple converter with more than a simple controller. This is part of the current ongoing work on which the proposed approach is being applied as a way to model hybrid dynamic behavior.

The authors acknowledge Universidad Industrial de Santander (UIS) in Colombia for supporting this research under the project VIE-UIS 5568. We also thank Dr. Elder Jesús Villamizar, School of Mathematics-UIS, for his helpful suggestions.