Figure 7. Controlling and optimizing a cascade parallel RC circuit system and the corresponding network presentation.

(a) A cascade parallel R-C circuit with 3 resistors (R₁, R₂, and R₃, each of resistance 1Ω) and 3 capacitors (C₁, C₂, and C₃, each of capacitance 1F), where u(t) is the external input voltage, u₁(t), u₂(t), and u₃(t) are the voltages on the capacitors C₁, C₂, and C₃, and i₁(t), i₂(t), and i₃(t) are the currents through the resistors R₁, R₂, and R₃, respectively. (b) Network representation of the circuit in (a). (c) Circuit with an extra external current input i_e(t) into the capacitor C₂. (d) The extra external current input i_e(t) serves as a redundant control input injected into node 2 of the network in (b). There are two driver nodes (yellow) in the network: 1 and 2.