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Turbidity Control in Sedimentation Columns by Direction Dependent Models

Jesus-Pablo Toledo-Zucco, Daniel Sbarbaro, Joao Manoel Gomes da Silva

TL;DR

The paper addresses turbidity control in sedimentation columns where direction-dependent dynamics hinder standard linear modeling. It introduces a two-mode piecewise time-delay model for top turbidity $y_k$, switching with $\Delta u_k$ and delays $d_\sigma$, and pairs it with a PI controller tuned via LMIs to guarantee closed-loop stability through a common Lyapunov–Krasovskii functional. A practical design procedure searches stability/performance regions $\Omega_1$ and $\Omega_2$ and minimizes a guaranteed cost $J\le z_0^T(P_1+hS_2)z_0$, with validation on numerical simulations and a pilot plant showing stable regulation across scenarios. The work offers a systematic, direction-aware control framework for water recovery in mineral processing, with potential extensions to uncertainties, saturation, and anti-windup.

Abstract

Sedimentation is a crucial phenomenon in recovering water from slurries by separating solid-liquid. Thickeners and sedimentation columns are equipments widely used in the process industry to reclaim water from process slurries. This contribution addresses the problem of controlling the turbidity of the recovered water in a sedimentation column by manipulating the underflow. The phenomenological model describing the turbidity is too complex to be used in a control strategy, and it is difficult to identify its parameters using plant measurements. This work proposes an empirical piece-wise time-delay model for modeling the turbidity at the top of the column to circumvent these problems. A systematic design procedure is developed to tune a Proportional Integral controller guaranteeing closed-loop stability for systems modeled as a piece-wise time delay model. Experiments in a pilot plant validate the theoretical results and illustrate the control performance under various operational scenarios.

Turbidity Control in Sedimentation Columns by Direction Dependent Models

TL;DR

The paper addresses turbidity control in sedimentation columns where direction-dependent dynamics hinder standard linear modeling. It introduces a two-mode piecewise time-delay model for top turbidity , switching with and delays , and pairs it with a PI controller tuned via LMIs to guarantee closed-loop stability through a common Lyapunov–Krasovskii functional. A practical design procedure searches stability/performance regions and and minimizes a guaranteed cost , with validation on numerical simulations and a pilot plant showing stable regulation across scenarios. The work offers a systematic, direction-aware control framework for water recovery in mineral processing, with potential extensions to uncertainties, saturation, and anti-windup.

Abstract

Sedimentation is a crucial phenomenon in recovering water from slurries by separating solid-liquid. Thickeners and sedimentation columns are equipments widely used in the process industry to reclaim water from process slurries. This contribution addresses the problem of controlling the turbidity of the recovered water in a sedimentation column by manipulating the underflow. The phenomenological model describing the turbidity is too complex to be used in a control strategy, and it is difficult to identify its parameters using plant measurements. This work proposes an empirical piece-wise time-delay model for modeling the turbidity at the top of the column to circumvent these problems. A systematic design procedure is developed to tune a Proportional Integral controller guaranteeing closed-loop stability for systems modeled as a piece-wise time delay model. Experiments in a pilot plant validate the theoretical results and illustrate the control performance under various operational scenarios.
Paper Structure (12 sections, 2 theorems, 37 equations, 12 figures, 1 table)

This paper contains 12 sections, 2 theorems, 37 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Description of the system
  3. Empirical Model
  4. Controller design
  5. Design Procedure
  6. Numerical Simulation
  7. Experimental Results
  8. Conclusion
  9. Proof of Proposition \ref{['Prop:Main1']}
  10. Conditions for $\sigma=1$
  11. Conditions for $\sigma=2$
  12. Proof of Proposition \ref{['Prop:Main2']}

Key Result

Proposition 1

Define the following matrices for the open-loop system Eq:Model and the control law Eq:ControlLaw: The closed-loop system Eq:Model-Eq:ControlLaw is asymptotically stable if there exist $4\times 4$ matrices $P_1>0_4$, $S_1>0_4$, $S_2>0_4$, $W_1$, $W_2$, $W_3$, $M_1$, $M_2$, $P_2$, and $P_3$ such that the following LMIs are satisfied: with

Figures (12)

  • Figure 1: (a) Front view of the set up, (b) Schematic diagram of the process, and (c) Lateral view of the set up.
  • Figure 2: Output response for a step input
  • Figure 3: Comparison between the model $y_k^m$ and the real values of $y_k$, and the input signal $u_k$.
  • Figure 4: Block diagram of the closed-loop system.
  • Figure 5: Set of admissible controller gains $\Omega_1$, constrained set $\Omega_2$, and optimal pair.
  • ...and 7 more figures

Theorems & Definitions (10)

  • Proposition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 2
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8