[Paper Review] Analysis and Output Tracking Design for the Direct Contact Membrane Distillation Parabolic System

Mohamed Ghattassi, Taous‐Meriem Laleg‐Kirati+1 more|arXiv (Cornell University)|Feb 3, 2019

Stability and Controllability of Differential Equations27 references4 citations

TL;DR

This paper proposes a robust output tracking control strategy for a direct contact membrane distillation (DCMD) system using a parabolic PDE model with boundary coupling. By applying active disturbance rejection control (ADRC) with a one-side feedback law and an extended state observer, the method achieves exponential decay of tracking error for feed and permeate outlet temperatures, even under external disturbances and flux noise, as proven via semigroup theory and operator analysis in a Hilbert space framework.

ABSTRACT

This paper considers the performance output tracking for a boundary controlled Direct Contact Membrane Distillation (DCMD) system. First, the mathematical properties of a recently developed mathematical model of the DCMD system are discussed. This model consists of parabolic equations coupled at the boundary. Then, the existence and uniqueness of the solutions are analyzed, using the theory of operators. Some regularity results of the solution are also established. A particular case showing the diagonal property of the principal operator is studied. Then, based on one-side feedback law the control problem, which consists of tracking both the feed and permeate outlet temperatures of the membrane distillation system is formulated. A servomechanism and an output feedback controller are proposed to solve the control problem. In addition, an extended state observer aimed at estimating both the system state and disturbance, based on the temperature measurements of the inlet is proposed. Thus, by some regularity for the reference signal and when the disturbance vanishes, we prove the exponential decay of the output tracking error. Moreover, we show the performance of the control strategy in presence of the flux noise.

Motivation & Objective

• To analyze the well-posedness and stability of a 2D advection-diffusion PDE system modeling heat transfer in a DCMD process.
• To design a robust output feedback controller that tracks desired outlet temperatures of the feed and permeate streams.
• To ensure disturbance rejection and robustness against measurement noise and external flux disturbances.
• To establish exponential convergence of the output tracking error using semigroup theory and operator analysis.
• To extend ADRC methodology to infinite-dimensional parabolic systems with non-self-adjoint operators.

Proposed method

• Formulate the DCMD system as a coupled system of 2D parabolic PDEs with boundary conditions, modeled using semigroup theory.
• Prove the m-dissipativity and well-posedness of the system operator in a Hilbert space framework.
• Introduce a one-side feedback control law based on output measurements at the boundary to track reference temperature signals.
• Design an extended state observer to estimate both system states and external disturbances using inlet temperature measurements.
• Apply active disturbance rejection control (ADRC) to decouple and reject disturbances without requiring self-adjointness of the operator.
• Use the Dirichlet map and trace theory to analyze error dynamics and regularity of the solution under time-varying disturbances.

Experimental results

Research questions

• RQ1 Can the proposed ADRC-based control strategy ensure exponential convergence of the output tracking error in a DCMD system modeled by coupled parabolic PDEs?
• RQ2 How does the system behave under external disturbances and measurement noise, and can the controller maintain robust performance?
• RQ3 Under what conditions is the system operator diagonalizable, and how does this simplify control design?
• RQ4 Can the extended state observer accurately estimate both system states and disturbances using only boundary temperature measurements?
• RQ5 What are the sufficient conditions for well-posedness and exponential stability of the closed-loop system?

Key findings

• The system operator is proven to be m-dissipative and generates an analytic semigroup, ensuring well-posedness of the PDE system.
• In the co-current configuration, the principal operator becomes diagonalizable when thermal conductivity and flow rate satisfy specific conditions.
• The output tracking error decays exponentially over time when the reference signal is sufficiently regular and disturbances vanish.
• The closed-loop system remains robust to disturbances, with the tracking error bounded by M(e−μt + ∥σ∥W2,∞(0,∞,HΓ1)) for constants M > 0 and μ > 0 depending on initial conditions.
• Numerical simulations confirm the effectiveness of the control strategy, showing rapid convergence of the tracking error to near-zero levels within 10 seconds under realistic disturbances and noise.
• The extended state observer successfully estimates both the system state and disturbance, enabling effective decoupling and disturbance rejection in the control law.