Non-isothermal CSTR Research Articles

A data-driven tracking control framework using physics-informed neural networks and deep reinforcement learning for dynamical systems

This paper addresses how physical knowledge can improve machine learning in process control. A data-driven tracking control framework using physics-informed neural networks (PINNs) and deep reinforcement learning (DRL) is proposed for dynamical systems, which is particularly important when iterations or repetitions of data collection experiments are limited or even not allowed. An indirect control approach is followed to accomplish this. First, a new process model is identified using PINNs trained from a model representing the experimental data collected from the plant. Then, the DRL-based control agent utilizes the identified PINN model to deliver an offline policy. We evaluate our framework for data-driven tracking control of nonisothermal CSTR, considering measurement noise and stochastic dynamics for closed-loop control. Our approach performed comparably to an NMPC without requiring a model to predict the process dynamics.

Adaptive Control for Narrow Bandwidth Input and Output Disturbance Rejection for a Non-Isothermal CSTR System

This paper presents an adaptive control scheme to face the challenge of rejecting input and output disturbances. The research is put on a layer of the design and start-up of chemical plants. The emphasis is on handling disturbances appearing in a narrow band of frequencies, which illustrates standard forms of disturbances in the alluded kind of systems. The controller is made up of a central RS structure that stabilizes the closed-loop plant. A second layer boosts the control law performance by adding the Youla–Kucera (YK) filter or Q parametrization and taking advantage of the internal model principle (IMP). This practice aids in modeling unknown disturbances with online control adjustment. We probe the resultant compensator for three non-isothermal continuous stirred tank reactors connected in series. The plant should conduct a first-order exothermic reaction consuming reactant A, while an isothermal operation stays and the outlet concentration is close to its nominal value. The primary concerns are open-loop instability and steady-state multiplicity in the plant’s first unit. The control objective is to reject input and output disturbances in a band of frequencies of 0.0002Hz to 0.007Hz, whether there are variants or not in time. We test the controller with input signals depicting both variations in the auxiliary services and abrupt changes. We then compare the executions of the resultant control law with a model-based predictive control (MPC). We find comparable responses to multiple disturbances. However, the adaptive control offers an effortless control input. We also conclude that the adaptive controller responds well to reference changes, while the MPC fails due to input constraints.

Design of functional observers for fault detection and isolation in nonlinear systems in the presence of noises

This work deals with the problem of designing functional observers for fault diagnosis in nonlinear systems in the presence of noises. It follows up previous work of the authors on design of functional observers (residual generators) for deterministic systems from the point of view of observer error linearization. Here, we consider the effect of noises on residual generation. The effect of sensor noises on the residuals is studied analytically and the associated probability distributions are derived. Following this, a well-known statistical hypothesis testing approach, Generalized Likelihood Ratio, is used to track changes in the mean of the residual to enable robust fault detection. The approach is also extended to process noises (plant-model mismatch) numerically. Throughout the study the methods presented are tested on a non-isothermal CSTR case study. The results show that the fault diagnosis scheme is able to quickly and accurately detect faults in the presence of both sensor and process noises.

A simulation study of nonideal mixing effect on the dynamic response of an exothermic CSTR with Cholette’s model

Abstract The study of nonideal mixing effect on the dynamic behaviors of CSTRs has very rarely been published in the literature. In this work, Cholette’s model is employed to explore the nonideal mixing effect on the dynamic response of a nonisothermal CSTR. The analysis shows that the mixing parameter n (the fraction of the feed entering the zone of perfect mixing) and m (the fraction of the total volume of the reactor), indeed affect the characteristic roots of transfer function of a real CSTR, which determine the system stability. On the other hand, the inverse response and overshoot response are also affected by the nonideal mixing in a nonisothemal CSTR. These results are of much help for the design and control of a real CSTR.

Implications of dimensional analysis in bioreactor models: Parameter estimation and identifiability

The multiple-input, multiple-output nature of microbial fermentation processes are most often described by nonlinear state space models. In order to apply these models for control purposes, they require the fitting of a set of model parameters so they can provide an accurate description of the fermentation system. In order to prevent overfitting of the experimental data, the parameter estimation procedures should produce unique estimates with small confidence intervals. This is facilitated by dimensional analysis, a tool that provides proper scaling of the dynamic model, removing its dimensional units. However, it is surprising that fermentation models seldom employ dimensional analysis, although it is recognized as a tool that can reduce the number of parameters and thus facilitate a comparison between different sets of data. In this tutorial review, we develop a systematic approach for dimensionless reformulation applied to fermentation systems. The approach is demonstrated on parameter estimation studies for a non-isothermal CSTR model and two common microbial fermentation state space models. It is shown that the analysis reduces the number of parameters in the models to yield identifiable reparameterizations, increases the accuracy of the parameter estimation, facilitates sensitivity analysis, and it can be stated that model identification and dimensional analysis as two concepts/approaches are complementary to each other. These benefits are relevant for the next stage of process optimization and control implementation. This review demonstrates the benefits of the dimensional analysis procedure, especially as an essential model development step for bioreactor systems, and together with a structural identifiability analysis it corresponds to methodologies for model reparameterization and parameter reduction.

Robust data reconciliation in chemical reactors

Robust data reconciliation is an effective technique designed to minimize gross errors drawbacks over estimated process variables. This work presents a review focusing on chemical reactor problems, which generate a challenging scenario due to strongly nonlinear constraints and have not been compared in terms of several robust estimators. The main contribution is to present a comparative analysis of 16 robust estimators, including since Smith estimator, developed in the XIX century, until the newly Jin, Correntropy and Xie ones. The performance of these estimators was analyzed in three case studies under steady-state conditions, including a non-isothermal CSTR reactor with a Van de Vusse reaction system. It was used IPOPT and simulated annealing optimizers implemented in Scilab software. The results showed the efficiency and consistency of the implemented methods, and although the influence of gross errors was significant, the redescending type estimators like Correntropy, Xie and Biweight showed a better overall performance.

Constrained Uncertain System Stabilization with Enlargement of Invariant Sets

An enhanced method able to perform accurate stability of constrained uncertain systems is presented. The main objective of this method is to compute a sequence of feedback control laws which stabilizes the closed-loop system. The proposed approach is based on robust model predictive control (RMPC) and enhanced maximized sets algorithm (EMSA), which are applied to improve the performance of the closed-loop system and achieve less conservative results. In fact, the proposed approach is split into two parts. The first is a method of enhanced maximized ellipsoidal invariant sets (EMES) based on a semidefinite programming problem. The second is an enhanced maximized polyhedral set (EMPS) which consists of appending new vertices to their convex hull to minimize the distance between each new vertex and the polyhedral set vertices to ensure state constraints. Simulation results on two examples, an uncertain nonisothermal CSTR and an angular positioning system, demonstrate the effectiveness of the proposed methodology when compared to other works related to a similar subject. According to the performance evaluation, we recorded higher feedback gain provided by smallest maximized invariant sets compared to recently studied methods, which shows the best region of stability. Therefore, the proposed algorithm can achieve less conservative results.

WITHDRAWN: Tabu search based optimization of PID parameters for temperature control of non-isothermal CSTR

WITHDRAWN: Tabu search based optimization of PID parameters for temperature control of non-isothermal CSTR

Data-driven Evolution Equation Reconstruction for Parameter-Dependent Nonlinear Dynamical Systems.

When studying observations of chemical reaction dynamics, closed form equations based on a putative mechanism may not be available. Yet when sufficient data from experimental observations can be obtained, even without knowing what exactly the physical meaning of the parameter settings or recorded variables are, data-driven methods can be used to construct minimal (and in a sense, robust) realizations of the system. The approach attempts, in a sense, to circumvent physical understanding, by building intrinsic "information geometries" of the observed data, and thus enabling prediction without physical/chemical knowledge. Here we use such an approach to obtain evolution equations for a data-driven realization of the original system - in effect, allowing prediction based on the informed interrogation of the agnostically organized observation database. We illustrate the approach on observations of (a) the normal form for the cusp singularity, (b) a cusp singularity for the nonisothermal CSTR, and (c) a random invertible transformation of the nonisothermal CSTR, showing that one can predict even when the observables are not "simply explainable" physical quantities. We discuss current limitations and possible extensions of the procedure.

Effect of coolant flow rate on the dynamics of delayed recycle continuous stirred tank reactor

Abstract Numerical bifurcation analysis of delayed recycle stream in a continuous stirred tank reactor was studied using DDE-BIFTOOL. A first-order irreversible exothermic reaction s 1 → k 2 , 1 s 2 was considered for analyzing effect of delay on the stability of the delayed recycle system. The non-isothermal CSTR was operated at both infinite and finite coolant inlet flowrates. A constant delay was considered in the recycle stream of CSTR for concentration of reactant, and temperature. DDE-BIFTOOL solver was used for finding dependency of delays on the bifurcation parameters, and its stability characteristics. The bifurcation parameters considered were: (i) fresh feed flowrate, (ii) coolant temperature, and (iii) coolant inlet flowrate. In the absence of delay, the system exhibits the region of dynamic instability for both infinite and finite coolant inlet flowrates. For infinite coolant flowrate, the region of dynamic instability on the reactor temperature was modified as a result of delay by varying either fresh feed flowrate or coolant temperature. The steady-state multiplicity was observed on the coolant temperature by varying fresh feed flowrate at finite coolant inlet flowrate. In the absence of delay, the fold and Hopf bifurcations were observed on the multiple steady-states of coolant temperature. The concentration and temperature delays do not alter substantially the dynamic characteristics of coolant temperature at Tc,in = 298 K. However, delays can alter the region of dynamic instability for coolant temperature when coolant inlet temperature is higher than 298 K. These are the main result of present work.

Integration of Design and Control of Dynamic Systems under Uncertainty: A New Back-Off Approach

A new methodology for simultaneous design and control under process disturbances and parameter uncertainty is presented using power series expansions (PSE) approximations. The key idea in this methodology is to back-off from the optimal steady-state design, which might be infeasible because of process dynamics and parameter uncertainty, to obtain the optimal design parameters that result in a dynamically feasible and economically attractive process. The work focuses on calculating various optimal design and control parameters by solving a set of optimization problems in an iterative manner using mathematical expressions obtained from PSE. These approximations are used to determine the direction in the search of optimal design parameters and operating conditions that is required for an economically attractive and dynamically feasible process. The proposed method was tested on a nonisothermal CSTR, and the results were compared with the formal integration process. The effect of the methodology’s key tuning ...

Elucidation of the role of constraints in economic model predictive control

Economic model predictive control (EMPC) is a predictive feedback control methodology that unifies economic optimization and control. EMPC uses a stage cost that reflects the process/system economics. In general, the stage cost used is not a quadratic stage cost like that typically used in standard tracking model predictive control. In this paper, a brief overview of EMPC methods is provided. In particular, the role of constraints imposed in the optimization problem of EMPC for feasibility, closed-loop stability, and closed-loop performance is explained. Three main types of constraints are considered including terminal equality constraints, terminal region constraints, and constraints designed via Lyapunov-based techniques. The paper closes with a well-known chemical engineering example (a non-isothermal CSTR with a second-order reaction) to illustrate the effectiveness of time-varying operation to improve closed-loop economic performance compared to steady-state operation and to demonstrate the impact of economically motivated constraints on optimal operation.

Discontinuity analysis for the treatment of nonlinear lumped-parameter systems for singular inputs

Different approaches suggested for the treatment of nonlinear systems subjected to singular inputs are either approximate or difficult to apply. In this paper, discontinuity analysis is used for this treatment, and is found to represent a good compromise between accuracy and ease of applicability. An example of non-isothermal CSTR representing nonlinear lumped-parameter chemical engineering systems is considered for the treatment. In the treatment, the integration of a model is carried out across the time of jump discontinuity by including singularity in the cause. This allows a priori estimation of initial conditions. The estimated initial conditions are then used for the initialization of the numerical solution of the model, whose singular input variables are set at their pre-initial values. The accuracy of the procedure in providing a reliable estimation of the system behavior is quantified by comparison with the numerical solutions initialized through the application of physical balances to the initial effects of singularity on the system, and with the numerical solutions obtained through pulse approximation for impulse. The procedure has simple and uniform applicability in comparison to the application of physical balances, and is substantially accurate than the commonly used pulse approximation method.

Embedded multiple shooting methodology in a genetic algorithm framework for parameter estimation and state identification of complex systems

A novel parameter estimation and state identification algorithm for nonlinear dynamical systems from data by embedding a multiple shooting methodology in the framework of a genetic algorithm (EMSGA) is described. The advantages of EMSGA are brought out by studies with two highly nonlinear examples, viz., the chaotic dynamics of a non-isothermal CSTR and the glycolysis regulation in Lactococcus lactis for production of lactic acid. For the chaotic dynamics with extremely high sensitivity to parameter values and initial conditions, EMSGA accurately estimates all process parameters while at the same time recovering the true dynamics of monitored and unmonitored variables from limited extents of noisy dynamic data. The superiority in accuracy and computational time of EMSGA over standalone genetic or multiple shooting algorithms for parameter estimation is shown for comparison purposes. In fact, EMSGA adapts well to the use of generalized canonical models, e.g., the S-system, where use of the fundamental Taylor series method of any order for integration becomes possible. This makes EMSGA highly efficient and exemplified here by estimating all the parameters of the derived higher dimensional S-system model for the CSTR. Comparative studies with a well-known global-local enhanced scatter search method corroborate the suitability and advantages of the EMSGA methodology. For the glycolysis regulation example the numerical robustness of EMSGA is brought out by estimating a large number of model parameters when a majority of them are raised to the power of the state variables. Interestingly, we show that from noisy in vivo dynamic data it becomes possible to completely recover the noise-free dynamical behavior of unmonitored species concentrations using EMSGA.

Economic Model Predictive Control: Elucidation of the Role of Constraints

Economic model predictive control (EMPC) is a predictive feedback control methodology that unifies economic optimization and control. EMPC uses a stage cost that reects the process/system economics. In general, the stage cost used is not a quadratic stage cost like that typically used in standard tracking model predictive control. In this paper, a brief overview of EMPC methods is provided. In particular, the role of constraints imposed in the optimization problem of EMPC for feasibility, closed-loop stability, and closed-loop performance is explained. Three main types of constraints are considered including terminal equality constraints, terminal region constraints, and constraints designed via Lyapunov-based techniques. The paper closes with a well-known chemical engineering example (a non-isothermal CSTR with a second-order reaction) to illustrate the effectiveness of time-varying operation to improve closed-loop economic performance compared to steady-state operation and to demonstrate the impact of economically motivated constraints on optimal operation.

Integration of Design and Control Using Efficient PSE Approximations

A new method for the optimal design and control of chemical processes under process disturbances and parameter uncertainty is presented. The key idea is to back-off from the optimal (dynamically inoperable) steady-state design by solving a series of optimization sub-problems constructed using Power Series Expansion (PSE) functions. The PSE functions estimate the actual constraints and cost functions’ worst-case variability due to disturbances and parameter uncertainty. The proposed method was applied to address the optimal design of a non-isothermal CSTR. The results show that this method has the potential to address the optimal design of dynamic systems at low computational costs.

Nonlinear frequency response analysis of forced periodic operation of non-isothermal CSTR using single input modulations. Part II: Modulation of inlet temperature or temperature of the cooling/heating fluid

Periodic operation of a non-isothermal CSTR subject to a single input modulation is analyzed considering a nth order reaction and using the nonlinear frequency response (NFR) method, introduced in our previous publications. The method is based on deriving the asymmetrical second order frequency response function (FRF) and analyzing its sign. In Part I of this paper, the periodic modulations of inlet concentration or flow-rate of the reaction stream were investigated. In this Part II, periodic operations are analyzed for modulations of the temperature of the inlet reaction stream or the temperature of the heating/cooling fluid. Conditions that need to be fulfilled in order to achieve process improvement through periodic operation are identified. The method is applied for the same numerical example taken from literature which was used in Part I. The results obtained by the NFR method are compared with the results of numerical simulations. Good agreement is obtained, except for forcing frequencies close to the resonant frequency. In these cases, even the sign of the DC component was not predicted correctly.

Nonlinear frequency response analysis of forced periodic operation of non-isothermal CSTR using single input modulations. Part I: Modulation of inlet concentration or flow-rate

Periodic operations of a non-isothermal CSTR with n-th order reaction, subject to a single input modulation, is analysed using the nonlinear frequency response (NFR) method, introduced in our previous publications. The method is based on deriving the asymmetrical second order frequency response function (FRF) and analysing its sign. In Part I of this paper, periodic operation with modulation of the inlet concentration or flow-rate of the reaction stream is analysed. As a result, conditions regarding the reaction order, process parameters and frequency of the input modulation are identified that need to be fulfilled in order to achieve process improvement through the periodic operation compared to conventional steady state operation. The method is applied for a numerical example from literature and the results obtained by the NFR method are compared with the results of numerical simulation. Good agreement is obtained, except for imposed forcing frequencies close to the resonant frequency and high forcing amplitudes.

Evaluation of Adaptive Extended Kalman Filter Algorithms for State Estimation in Presence of Model-Plant Mismatch

The occurrence of model-plant mismatch is a common problem in dynamic model based applications such as state estimation. The use of an inaccurate model results in biased estimates of the states. Hence, conventional state estimation algorithms are modified in various ways to compensate for model-plant mismatch. In this work, the performance of four adaptive state estimation algorithms is compared in the presence of a model plant mismatch arising due to random drifts in parameter values. The comparison is carried out through simulations on a benchmark non-isothermal CSTR problem. Simulation results demonstrate that online re-identification of the parameters susceptible to drift or change is the most effective approach to minimize the effect of model-plant mismatch on the state estimates.

Modelling and control of multi-energy systems: An irreversible port-Hamiltonian approach

In recent work a class of quasi port Hamiltonian system expressing the first and second principle of thermodynamics as a structural property has been defined: Irreversible port-Hamiltonian system. These systems are very much like port-Hamiltonian systems but differ in that their structure matrices are modulated by a non-linear function that precisely expresses the irreversibility of the system. In a first instance irreversible port-Hamiltonian systems are extended to encompass coupled mechanical and thermodynamical systems, leading to the definition of reversible–irreversible port Hamiltonian systems. In a second instance, the formalism is used to suggest a class of passivity based controllers for thermodynamic systems based on interconnection and Casimir functions. However, the extension of the Casimir method to irreversible port-Hamiltonian systems is not so straightforward due to the “interconnection obstacle”. The heat exchanger, a gas-piston system and the non-isothermal CSTR are used to illustrate the formalism.