[Paper Review] Stabilization of (state, input)-disturbed CSTRs through the port-Hamiltonian systems approach

Yafei Lu, Zhou Fang+1 more|arXiv (Cornell University)|Jun 28, 2017

Control and Stability of Dynamical Systems17 references3 citations

TL;DR

This paper proposes a port-Hamiltonian framework for modeling and stabilizing continuous stirred tank reactors (CSTRs) subject to both state and input disturbances using stochastic passivity-based control. By employing the opposite entropy function as the Hamiltonian and applying a generalized canonical transformation to the availability function, the method ensures simultaneous representation of the first and second laws of thermodynamics and achieves local almost sure stabilization in probability via a designed controller with explicit gain tuning.

ABSTRACT

It is a universal phenomenon that the state and input of the continuous stirred tank reactor (CSTR) systems are both disturbed. This paper proposes a (state, input)-disturbed port-Hamiltonian framework that can be used to model and further designs a stochastic passivity based controller to asymptotically stabilize in probability the (state, input)-disturbed CSTR (sidCSTR) systems. The opposite entropy function and the availability function are selected as the Hamiltonian for the model and control purposes, respectively. Furthermore, the proposed (state, input)-disturbed port-Hamiltonian model can simultaneously characterize the first law and the second law of thermodynamics when the opposite entropy function acts as the Hamiltonian. A simple CSTR example illustrates how the port-Hamiltonian method is utilized for modeling and controlling sidCSTR systems.

Motivation & Objective

• To address the challenge of stabilizing CSTRs under simultaneous state and input disturbances, which are common in industrial processes but underexplored in control theory.
• To develop a unified modeling framework that simultaneously captures the first and second laws of thermodynamics in disturbed CSTR systems.
• To design a stochastic passivity-based controller that ensures local almost sure stabilization in probability for (state, input)-disturbed CSTRs (sidCSTRs).
• To demonstrate the effectiveness of the approach through a numerical case study with explicit controller design and convergence analysis.

Proposed method

• Formulates the sidCSTR system as a stochastic port-Hamiltonian system (sidSPHS) by selecting the opposite entropy function as the Hamiltonian, enabling thermodynamic consistency.
• Applies an improved stochastic generalized canonical transformation to convert the sidSPHS into an equivalent form with the availability function as the new Hamiltonian, preserving system dynamics.
• Establishes stochastic passivity of the transformed sidSPHS with respect to the new Hamiltonian, enabling controller design via passivity theory.
• Derives a feedback controller using the stochastic passivity property, with proportional gains tuned via the condition that $ extbf{I} + extbf{K}ar{m{ ho}} $ is invertible.
• Employs explicit control laws for flow rate $ q $ and jacket temperature $ T_w $, derived from energy balance and passivity-based control principles.
• Validates the controller using a numerical CSTR example with specified thermodynamic and kinetic parameters, simulating state and control variable responses over time.

Experimental results

Research questions

• RQ1 Can a port-Hamiltonian framework model (state, input)-disturbed CSTRs while simultaneously representing the first and second laws of thermodynamics?
• RQ2 Does transforming the Hamiltonian from opposite entropy to availability function preserve stochastic passivity in the presence of state and input disturbances?
• RQ3 Can a stochastic passivity-based controller asymptotically stabilize the sidCSTR system in probability?
• RQ4 How do the controller gains influence the convergence of state variables to the desired equilibrium under stochastic disturbances?
• RQ5 What is the performance of the controller in terms of state and control input evolution under realistic disturbance levels?

Key findings

• The sidCSTR system can be successfully modeled as a sidSPHS with the opposite entropy function as the Hamiltonian, enabling simultaneous representation of the first and second laws of thermodynamics.
• After applying the generalized canonical transformation, the system exhibits stochastic passivity with respect to the availability function as the Hamiltonian, confirming the theoretical foundation for controller design.
• The controller with gains $ K_1 = 1.64 imes 10^{-7} $ and $ K_2 = 27,430 $ ensures local almost sure stabilization in probability, as verified by simulation results.
• State variables $ T $, $ N_A $, and $ N_B $ converge to their setpoints $ T^* = 331.9 $ K, $ N_A^* = 1.3 $ mol, and $ N_B^* = 0.7 $ mol by $ t = 3 $ seconds under the controller.
• The manipulated variables $ q $ and $ T_w $ asymptotically approach stable values $ q o 0 $ and $ T_w o T^* $, confirming controller robustness and convergence.
• The simulation results in Figs. 2–5 confirm that the proposed controller effectively stabilizes the sidCSTR system in the presence of stochastic disturbances with $ ho_1 = 0.1 $, $ ho_2 = 5 imes 10^{-7} $, and $ ho_3 = 0.05 $.