Conservation of State

Consider the two parallel capacitors in Fig. 1. When the switch is open, the capacitors are independent, and, therefore, both of their charge values, q_i, are independent state variables. When the switch is closed, the voltage across the capacitors is forced to be the same, and, therefore, there is only one degree of freedom, or state. Due to the closing of the switch, an instantaneous redistribution of charge may take place to satisfy the equal voltage constraint. Because of its instantaneous character, this redistribution can be described by impulse equations.

When the switch is closed, it is clear that the voltage constraint v₁ = v₂ describes the redistribution of charge, i.e., the manifold onto which the state is projected. However, even though it is known that the voltage drop across the capacitors is the same, it is still unknown what the individual charges are. To derive this, the additional constraint of conservation of state (in this case charge) is applied [7,8,9]. This prescribes that $\Delta q_1 + \Delta q_2 = 0$, which equals $C_1 \Delta v_1 = -C_2 \Delta v_2$, or C₁ (v₁⁺ - v₁^-) = -C₂ (v₂⁺ - v₂^-), where v^- are a priori switching values (i.e., the voltages immediately before closing the switch) and v⁺ are a posteriori values (i.e., the new voltages immediately after the switch is closed). Because v₁^- and v₂^- are known, we have a system of two equations with two unknowns, and, therefore, the initial values of the charges in the new configuration can be solved.

Note that in case C₁ is replaced by a battery (Fig. 3), there is explicit interaction with the system environment (a battery is considered a source with infinite power, which models the system context), and, therefore, conservation of state does not hold. When the switch is closed, the battery enforces a voltage across the capacitor irrespective of its charge before closing, and there may be instantaneous exchange of charge. The state space projection, shown in Fig. 4, is singular along V_in = const.

  

Figure 3: Explicit interaction with the environment.

  

Figure 4: Singular projection that violates the conservation constraint.

In the case of two capacitors (Fig. 1), the conservation of state principle is easily recognized. In general, for topological models it is easily applied, e.g., in the hybrid bond graph simulator HYBRSIM [12]. However, once a system is specified by a system of DAEs, the conservation equations that define the state projection are implicit and need to be derived.