Feedback control of plant-soil autotoxicity via pulse-width modulation

Abstract

Plant-soil negative feedback (PSNF) is the rise in soil of negative conditions for plant performance induced by the plants themselves, limiting the full potential yield and thus representing a loss for the agricultural industry. It has been recently shown that detrimental effects the PSNF has on the growth of plant's biomass can be mitigated by periodically intervening on the plant/soil system, for example by washing the soil. The periodic control inputs were computed by using an average model of the system and then applied in open-loop. In this paper we present two feedback control strategies, namely a PI and a MPC-based controllers, that, by adapting online the duty-cycle of the periodic control input, guarantee precise regulation of the biomass yield and at the same time robustness to unavoidable modeling errors and perturbations acting on the system. The performance of the proposed control strategies is then validated by means of extensive numerical simulations.

Figures (8)

• Figure 1: Schematic representation of self-DNA soil dynamics and interactions with plant functionality (reproduced from carteni2012negative).
• Figure 2: Block diagram of the proposed closed-loop control strategies. The feedforward model-based inversion block evaluates the reference duty-cycle $D_{\textup{ref}}$ given the desired average value of the biomass $B^{\textup{ref}}_{\textup{av}}$. At each sampling period, starting from the previously computed $D_\mathrm{ref}$, the feedback control evaluates the duty-cycle $D$ that minimize the error between the desired reference value $B_\mathrm{av}^\mathrm{ref}$ and the measured mean value of $B$ over the period $P$. Note that the control system evolves in discrete time (with sampling time $P$) while the controlled system in continuous time.
• Figure 3: Block diagrams of proposed feedback control strategies. (a) PI/PWM control strategy. Given the set-point for the average biomass $B_{\textup{ref}}$, two actions regulate the parameters of the PWM inputs that feed the system. The feedforward action consists in a model-based inversion that evaluates the duty-cycle $D_{\textup{ref}}$. A proportional-integral controller composes the feedback loop. At each time period $t_i = iP$, the error $e_i$ is corrected by a PI controller that evaluates the correction $\delta D_i$ to $D_{\textup{ref}}$. (b) MPC control strategy. At each discrete time step $i$, the MPC finds the sequence of duty-cycles $\{D^i_0, D^i_1, \dots, D^i_{N_H - 1}\}$ that minimizes the cost function $J^i$ over the prediction horizon $t_H = N_H P$. Then, only the first element of the sequence is selected ($D^i = D_0$), and the corresponding pulsatile control signal is applied to the system in the time interval $[t_i, t_i + P]$.
• Figure 4: Functional relationship between the average value of the biomass at steady state, $B_\mathrm{av}^*$ and the duty-cycle $D$ of the periodic control input $u(t, T)$. Given an initial reference value of biomass $B^{\textup{ref}}_{\textup{av}}$ on the $y$-axis, the $D_{\textup{ref}}$ is found as the corresponding $x$-axis of $\Gamma_{B^*_{\textup{av}}}(D)$. Here $B^{\textup{ref}}_{\textup{av}} = 0.9$ from which we get $D_{\textup{ref}} \approx 0.31$.
• Figure 5: Tuning of the PI controller. The red box indicates the values of PI gains $K_P = 0.1$, $K_I = 26$ that were selected as those giving the best compromise between all the metrics we considered, and used for control experiments.