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Buffer-Buffer Dependency

When two or more buffers become dependent, redistribution of the state variable values according to the buffer ratios is required. For the energy phase space analysis this means that a number of dimensions collapse into one. This is illustrated by the two capacitor system in Fig. 13. When the relay is open, the capacitors are independent and charge individually. When the voltage difference between the two crosses a threshold value, say 0, the relay closes and the capacitors become dependent, causing an immediate redistribution in charge (state variable) among the capacitors so both capacitors have the same effort value, tex2html_wrap_inline900 . Substitution of these values for tex2html_wrap_inline902 and tex2html_wrap_inline904 in the relay's CSPEC yields the switching condition tex2html_wrap_inline906 . This is true in the entire energy phase space (Fig. 14). Notice that a degree of freedom is lost because the buffer dependency causes the two-dimensional phase space to collapse into a one-dimensional phase space. When the two energy phase spaces are analyzed, it is apparent that divergence of time is not guaranteed. The inconsistency results from the equal sign in the switching off condition for the relay. When the relay is on, and both voltages become equal it will be switched off instantaneously and vice versa. To solve this problem, the condition for switching needs to be changed to tex2html_wrap_inline908 . Then the relay does not switch out of its on state and divergence of time is guaranteed. This example illustrates that the energy phase space analysis applies even when dependency relations among the buffers cause changes in the system model.

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Figure 13: Dependent capacitors.

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Figure 14: Dependent Capacitors: Energy phase spaces.



Pieter J. Mosterman
Mon Jul 21 19:58:19 CDT 1997