Figure 3. Illustration of parallel R-C circuit and optimization of control energy.

(a) A cascade parallel R-C circuit with L = 7 resistors (R₁, R₂, …, and R_L, each of resistance 1Ω) and 7 capacitors (C₁, C₂, …, and C_L, each of capacitance 1F). External voltage input u(t) is applied from the left side of the circuit, and the voltage of capacitor C_i is u_i(t)(1 ≤ i ≤ L). An extra external current input i_e(t) serves as a redundant control input injected into the capacitor C₃, where i₃ and i₄ denote the currents through resistors R₃ and R₄, respectively. In absence of the extra current input, i₃(t) − i₄(t) is the current through the branch of C₃. (b) Network representation of the circuit in (a) as a bidirectional 1D chain network of seven nodes, where the external voltage input u(t) is injected into node 1 (yellow driver node, the controller). The dynamical state of node i is described by the voltage of its capacitor, u_i(t). Links (blue) between nodes are bidirectional and have uniform weight 1 in either direction. Each node has a self-link (red) of weight −2, except the ending node (node 7) whose self-link has weight −1. The extra external current input i_e(t) serves as a redundant control input injected into node 3 of the network in (b). Now there are two driver nodes (yellow) in the network, nodes 1 and 3. (c) Energy required for controlling a unidirectional chain (red circle) and the corresponding circuit (blue square) as well as the dissipated energy (green triangle) of the circuit calculated from Eq. (4) versus chain length L. (d) Control and dissipated energies in presence of a redundant control signal to node i (i > 1), which breaks the chain into two subchains of lengths i and L − i, respectively.