Explicit Interaction

In case of explicitly modeled interaction with the system environment, conservation principles may be violated. Consider the electrical circuit in Fig. 6 which has a modulated voltage source V_R, i.e., the voltage drop across R is enforced on C₂. The system of equations can be compiled as $$\left.\begin{array} {c} \left[\begin{array} {cccccc} C_1 & 0... ...} \\ {i}_{C_2} \\ {i}_{C_3}\end{array}\right]\end{array}\right.$$
In this system, the first three rows contain the equations that model the constituent time-derivative behavior of the three capacitors, C₁, C₂, and C₃, and, therefore, are part of f. So, $$\dot{\bar{x}} = \left[\begin{array} {c} \dot{v}_{C_1} \\ \do... ...egin{array} {c} v_{C_1} \\ v_{C_2} \\ v_{C_3}\end{array}\right]$$
The equation on the fourth row represents the nodal equation summing the currents of C₁, C₃, and R to . The last two rows contain equations that represent the algebraic constraints v_C3 = v_C1 and v_C2 = v_C1 (due to the modulated source, V_R), respectively. The equations on the last three rows constitute h, and the last two of them have no algebraic variables so they are part of $\bar{h}$.The remaining equation in h yields $$\hat{y} = \left[\begin{array} {c} i_{C_1} \\ i_{C_3}\end{array}\right]$$
and, therefore, $$\tilde{y} = \left[\begin{array} {c} i_{C_2}\end{array}\right].$$

  

Figure 6: Explicit interaction with the environment.

This results in the matrix partitioning $$F' = \left[\begin{array} {cc} 1 & 0 \\ 0 & 0 \\ 0 & 1\end{a... ...& 0 & 0 \\ 0 & C_2 & 0 & 1 \\ 0 & 0 & C_3 & 0\end{array}\right]$$
and $$\hat{H}' = \left[\begin{array} {cc} -1 & -1\end{array}\righ... ...[\begin{array} {cccc} \frac{1}{R} & 0 & 0 & 0\end{array}\right]$$
Now, the pseude-inverse of F' is $$F'^+ = \left[\begin{array} {ccc} 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]$$
and $$\hat{H}' F'^+ F'' = \left[\begin{array} {cccc} -C_1 & 0 & -C_3 & 0\end{array}\right]$$
which yields the conservation equations $$\left[\begin{array} {cccc} \frac{1}{R} & 0 & 0 & 0\end{array... ...\ \dot{v}_{C_2} \\ \dot{v}_{C_3} \\ {i}_{C_2}\end{array}\right]$$

Integrating the conservation equations gives the projection direction $$C_1 {v}_{C_1} + C_3 {v}_{C_3} = C_1 {v}_{C_1}^0 + C_3 {v}_{C_3}^0$$
which, combined with the differentiated equations $$\left[\begin{array} {ccc} 1 & -1 & 0 \\ 1 & 0 & -1 \\ \end{a... ...ray}\right] = \left[\begin{array} {c} 0 \\ 0\end{array}\right],$$
constitutes a system of equations of rank three with three variables. The solution in terms of the conserved quantity, charge (q = C v), is

$$\begin{eqnarray} \html{eqn85}q_{C_1} = \frac{C_1}{C_1 + C_3}(q_{C_1}^0 + q_{C_3}... ... q_{C_1} \\ q_{C_3} = \frac{C_3}{C_1 + C_3}(q_{C_1}^0 + q_{C_3}^0)\end{eqnarray}$$

which shows that the charge on C₁ and C₃ is conserved, but there may be an instantaneous change of charge on C₂. This is conform the model, which shows explicit interaction with the environment between C₂ and the battery V_R.

Note that the current of C₂, i_C2, cannot be solved in terms of differentiated variables. To find the value of this variable, all $\dot{x}$ have to be eliminated from its equation. The result is an algebraic relation that poses no additional constraints on the dynamic, differential equation, behavior of the system. The only additional information is about system variable values (e.g., required for graphing purposes).