Fig. 7: aPID control of an unstable and more complex plant.

a Plant description. The plant considered here involves L = 6 species and embeds a negative feedback from the output Z₆ to X₂ via an active degradation reaction. The underlying deterministic dynamics of the plant and the second-order aPID controller are shown in this panel. It is demonstrated that the open loop is unstable (orange response), and integral control alone cannot stabilize the dynamics since two eigenvalues carry on a positive real part for any k ≥ 0. b–d Performance of the various aPID controllers. The intensity plots show the performance index over a range of biomolecular parameter values. These plots are overlaid with contours where the PID gains K_P, K_I or K_D are constant. For the third and fourth-order aPID in (c) and (d), K_I ≈ k and thus can be tuned separately with k that is held constant throughout this figure. For the fourth-order aPID in (d), the K_P- and K_D-contours are orthogonal to the k₀- and α₂-axes, respectively, and hence can also be tuned separately. For the third-order aPID in (c), the K_D-contours are orthogonal to the δ-axis and hence K_P can be tuned separately with δ₀, whereas the inseparability of the PD components forces the K_P-contours to be oblique, and thus δ tunes both K_P and K_D simultaneously. Finally, for the second-order aPID in (b), all three contours are not orthogonal to the axes and, as a result, all three PID gains have to be mutually tuned by the biomolecular parameters. This is due to the inseparability of all PID components. Note that each set of contours is displayed on a separate intensity plot here for clarity. Observe that the optimal performance for each controller is located in the dark blue regions where the proportional gains K_P are negative. Three different examples, red, green, and purple (along with the unstable standalone aI control in gray), are picked to demonstrate the achievable high performances depicted in the response plots to the right. For the second and third-order aPID, negative K_P can be achieved by properly tuning the biomolecular parameters without having to switch the topology from N-type to P-type. However, for the (separable) fourth-order aPID controller, a hybrid design with N-type ID and P-type P can also achieve a negative K_P that is critical for controlling this plant.