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Hybrid Systems Containing Discrete and Continuous Systems

The hybrid systems are the dynamic systems that demonstrate both discrete as well as the continuous dynamic behaviors. Hybrid systems distinguish those systems that combine fuzzy and neutral nets in the engineering work. The system encompasses larger systems within its structure, thus, allowing more flexibility in modeling dynamic events. The hybrid systems define both values of discrete and continuous variably controlling modes. The changes of the variables can occur either continuously or in a discretely manner. In the continuous system, the flow is allowed as long as the invariants clutch. Under the discrete system, the translations occur instantly as long as the necessary conditions are satisfied. Both discrete and continuous hybrid systems are associated with events.

Under the traditional discrete event specification (DEVS), the dynamic processes utilize some piecewise segments to approximate the inputs and outputs. According to Giambiasi, Escude, and Ghosh (2000), the trajectories are organized through the piecewise segments in the Generalized Discrete Event Specification (GDVS). Both the GDVS and DEVS perform work more rapidly on host computers. This is because in the discrete system there is the occurrence of executions in the corresponding system changes unlike in the continuous system. The GDVS allows the progress of the standardized stimulation environment for the hybrid system. The experiments carried out on hybrid systems revealed that the execution speed for discrete ones is relative to the continuous system. When the two systems were modeled under the GDEVS prototype stimulator, they had an estimation of arranging factor from 2 to 4.5 (Cimatti, Mover and Tonetta, 2011).

Additionally, the DEVS exploits some piecewise trajectories of input and output in order to develop a discrete event generalization of the continuous processes. The generalization process requires a greater accuracy when modeling three continuous systems under the GDEVS. The accuracy is needed in order to produce some uniform hybrid systems. The highly dynamic complex systems require the piecewise trajectories of input and output. When modeling the accuracy of the system behavior, time interval sampling may not work. Therefore, the shortening of time in the sampling interval is necessary. This will limit the representational error, thus, achieving the acceptable accuracy (Cassandras, 2002). The major contributing factor of the GDEVS is to produce a uniform discrete event. The reason of utilizing the piecewise constant segments is to produce the higher accuracy without increasing the sampling frequency.

Powerful modeling languages, particularly the hybrid automat, permit engineers to model a wide range of physical phenomena. However, it makes it feasible to produce reasonable models either mathematically or physically. The lack of existence of solutions might cause danger in the hybrid modeling. Thus, one can construct models by admitting no solutions for the particularly initial states. The mathematical model reveals an incomplete picture of the physical authenticity, thus, the physical system can be used when the mathematical model is undefined (Castro, Kofman and Wainer, 2010). For instance, in the hybrid system model, the mathematical model is used for describing the performance of the hybrid systems. The problems of the supervisory control design for both the discrete and continuous systems are formulated. In this case, the DES plant model is used as a methodology for the supervisory control design (Koutsoukos, Antsaklis, Stiver and Lemmon, 2000).

Fuzzy Hybrid Systems

The fuzzy hybrid system contains a mixture of discrete and continuous variable components. The methods used in handling the crisp cases in both components are deterministic and subjective to the real world applications. Recently, the extension for the fuzzy discrete system was extended through the use of the fuzzy theory (Lai and Lin, 2003). The fuzzy sets are utilized in order to represent the indistinct state of variables. The researchers have developed a computational algorithm in calculating some fuzzy state transitions of the systems. According to Giambiasi, Escude and Ghosh (2000), hybrid systems emerged as a combination of the conventional real time driven events dynamics. These events provided some opportunities for frameworks as well as the methodologies that have enlarged the scope of discrete and continuous events. Moreover, many hybrid systems consist of lower level components that correspond to the time driven physical processes. Therefore, the events coordinate through switching between varied processes operating the models (Choura and Yigit, 2005).

Recently, the researchers developed a modeling and simulation driven methodology used in engineering that has embedded some real time systems (Marzban and Razzaghi, 2005). This is one of the approaches that rely on the use of the DEVS formalization in order to develop the real time systems by using the incremental development. In this case, hybrid control techniques are applied, the model used defines the discrete event controller as a time varying plant which is based on the varied model control. The discrete approach integrates some discrete events into the continuous components. Moreover, the approach enables engineers to secure, produce the reliable testing, and analyze some varied abstraction levels in the hybrid system. The most applications employed combining the discrete events and the continuous events, thus, producing a fuzzy hybrid system (Lim and Chan, 2006).

There is a need for the new language, the HyDI which is vital for modeling the hybrid systems. The purpose of recommending the language is to apply it in the symbolic model checkers in order to verify the complex entrenched system designs. The HyDI expands the typical symbolic language with harmonizing and timing aspects. Therefore, the use of the HyDI language will enable an engineer to distinguish between the continuous and discrete variables (Cimatti, Mover and Tonetta, 2011). In addition, the new language can be compiled automatically with the equivalent discrete unlimited time transition systems. The novel language which is called the HyDI is a hybrid system of the discrete interaction. It is recommendable for use because it is seen as an extension through which the standard symbolic language is able to represent the hybrid system networks. The use of the HyDI language is vital because it allows some solid representations of multifarious systems and the easy encoding of the  macro transition composition to take place easily.

Another recommendation is the need to balance between the modeling power and the tractability of the hybrid problems. It is vital to control real time discrete systems and the controllability of events. Effective computations can be done in certain cases through employing the restrictive control strategy in computation (Stiver and Antsaklis, 2003). For instance, the language generated by the subsystems is ordinary because it is being recognized by the finite automata. Thus, the hybrid parameter reorganizes the same values though the state of automation. New problems as well as difficulties may arise because the considered plant may have some numerical parameters. Therefore, it is recommendable to attempt finding the balance modeling power and tractability problems of the hybrid systems. Lastly, a novel hybrid model system can be represented, whereby the system changes behavior can be modeled though coupling the discrete and continuous events (Di and Saconi, 2008).

In conclusion, hybrid systems are the dynamic systems that demonstrate both the discrete as well as the continuous dynamic behaviors. Both discrete and continuous ones are associated with events. Under the traditional discrete event specification, the dynamic processes utilize piecewise segments to approximate the inputs and outputs. Additionally, powerful modeling languages, particularly the hybrid automata, permit the engineers to model a wide range of physical phenomena. The fuzzy hybrid system has a mixture of discrete and continuous variable components. Therefore, the DEVS formalization approach is used for developing the real time systems by using the incremental development. Lastly, the HyDI language balancing the modeling power and the tractability is recommendable in the hybrid systems.

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