Predictive control of transport-reaction processes

This work focuses on the development of computationally efficient predictive control algorithms for nonlinear parabolic and hyperbolic PDEs with state and control constraints arising in the context of transport-reaction processes. We first consider a diffusion-reaction process described by a nonlinear parabolic PDE and address the problem of stabilization of an unstable steady-state subject to input and state constraints. Galerkin’s method is used to derive finite-dimensional systems that capture the dominant dynamics of the parabolic PDE, which are subsequently used for controller design. Various model predictive control (MPC) formulations are constructed on the basis of the finite dimensional approximations and are demonstrated, through simulation, to achieve the control objectives. We then consider a convection-reaction process example described by a set of hyperbolic PDEs and address the problem of stabilization of the desired steady-state subject to input and state constraints, in the presence of disturbances. An easily implementable predictive controller based on a finite dimensional approximation of the PDE obtained by the finite difference method is derived and demonstrated, via simulation, to achieve the control objective.     

Predictive control of parabolic PDEs with boundary control actuation

This work focuses on predictive control of linear parabolic partial differential equations (PDEs) with boundary control actuation subject to input and state constraints. Under the assumption that measurements of the PDE state are available, various finite-dimensional and infinite-dimensional predictive control formulations are presented and their ability to enforce stability and constraint satisfaction in the infinite-dimensional closed-loop system is analyzed. A numerical example of a linear parabolic PDE with unstable steady state and flux boundary control subject to state and control constraints is used to demonstrate the implementation and effectiveness of the predictive controllers.     

Linear model predictive control for transport‐reaction processes

The article deals with systematic development of linear model predictive control algorithms for linear transport‐reaction models emerging from chemical engineering practice. The finite‐horizon constrained optimal control problems are addressed for the systems varying from the convection dominated models described by hyperbolic partial differential equations (PDEs) to the diffusion models described by parabolic PDEs. The novelty of the design procedure lies in the fact that spatial discretization and/or any other type of spatial approximation of the process model plant is not considered and the system is completely captured with the proposed Cayley‐Tustin transformation, which maps a plant model from a continuous to a discrete state space setting. The issues of optimality and constrained stabilization are addressed within the controller design setting leading to the finite constrained quadratic regulator problem, which is easily realized and is no more computationally intensive than the existing algorithms. The methodology is demonstrated for examples of hyperbolic/parabolic PDEs. © 2017 American Institute of Chemical Engineers AIChE J, 63: 2644–2659, 2017     

Feedback control of dissipative PDE systems using adaptive model reduction

The problem of feedback control of spatially distributed processes described by highly dissipative partial differential equations (PDEs) is considered. Typically, this problem is addressed through model reduction, where finite dimensional approximations to the original infinite dimensional PDE system are derived and used for controller design. The key step in this approach is the computation of basis functions that are subsequently utilized to obtain finite dimensional ordinary differential equation (ODE) models using the method of weighted residuals. A common approach to this task is the Karhunen‐Loève expansion combined with the method of snapshots. To circumvent the issue of a priori availability of a sufficiently large ensemble of PDE solution data, the focus is on the recursive computation of eigenfunctions as additional data from the process becomes available. Initially, an ensemble of eigenfunctions is constructed based on a relatively small number of snapshots, and the covariance matrix is computed. The dominant eigenspace of this matrix is then utilized to compute the empirical eigenfunctions required for model reduction. This dominant eigenspace is recomputed with the addition of each snapshot with possible increase or decrease in its dimensionality; due to its small dimensionality the computational burden is relatively small. The proposed approach is applied to representative examples of dissipative PDEs, with both linear and nonlinear spatial differential operators, to demonstrate its effectiveness of the proposed methodology. © 2009 American Institute of Chemical Engineers AIChE J, 2009     

Dynamic modeling and model predictive control of a continuous pulp digester

This work explores the design of a model predictive controller of the continuous pulp digester process consisting of the co-current zone and counter-current zone modeled by a set of nonlinear coupled hyperbolic partial differential equations (PDEs). The distributed parameter system of interest is not spectral, and slow–fast dynamic separation does not hold. To address this challenge, the nonlinear continuous-time model is linearized and discretized in time utilizing the Cayley–Tustin discretization framework, which ensures system theoretic properties and structure preservation without spatial discretization or model reduction. The discrete model is used in the full state model predictive controller design, which is augmented by the Luenberger observer design to achieve the output constrained regulation. Finally, a numerical example is provided to demonstrate the feasibility and applicability of the proposed controller designs.     

Guaranteed cost distributed fuzzy observer‐based control for a class of nonlinear spatially distributed processes

The guaranteed cost distributed fuzzy (GCDF) observer‐based control design is proposed for a class of nonlinear spatially distributed processes described by first‐order hyperbolic partial differential equations (PDEs). Initially, a T–S fuzzy hyperbolic PDE model is proposed to accurately represent the nonlinear PDE system. Then, based on the fuzzy PDE model, the GCDF observer‐based control design is developed in terms of a set of space‐dependent linear matrix inequalities. In the proposed control scheme, a distributed fuzzy observer is used to estimate the state of the PDE system. The designed fuzzy controller can not only ensure the exponential stability of the closed‐loop PDE system but also provide an upper bound of quadratic cost function. Moreover, a suboptimal fuzzy control design is addressed in the sense of minimizing an upper bound of the cost function. The finite difference method in space and the existing linear matrix inequality optimization techniques are used to approximately solve the suboptimal control design problem. Finally, the proposed design method is applied to the control of a nonisothermal plug‐flow reactor. © 2013 American Institute of Chemical Engineers AIChE J, 59: 2366–2378, 2013     

Dynamic output feedback covariance control of stochastic dissipative partial differential equations

In this work, we develop a method for dynamic output feedback covariance control of the state covariance of linear dissipative stochastic partial differential equations (PDEs) using spatially distributed control actuation and sensing with noise. Such stochastic PDEs arise naturally in the modeling of surface height profile evolution in thin film growth and sputtering processes. We begin with the formulation of the stochastic PDE into a system of infinite stochastic ordinary differential equations (ODEs) by using modal decomposition. A finite-dimensional approximation is then obtained to capture the dominant mode contribution to the surface roughness profile (i.e., the covariance of the surface height profile). Subsequently, a state feedback controller and a Kalman-Bucy filter are designed on the basis of the finite-dimensional approximation. The dynamic output feedback covariance controller is subsequently obtained by combining the state feedback controller and the state estimator. The steady-state expected surface covariance under the dynamic output feedback controller is then estimated on the basis of the closed-loop finite-dimensional system. An analysis is performed to obtain a theoretical estimate of the expected surface covariance of the closed-loop infinite-dimensional system. Applications of the linear dynamic output feedback controller to both the linearized and the nonlinear stochastic Kuramoto-Sivashinsky equations (KSEs) are presented. Finally, nonlinear state feedback controller and nonlinear output feedback controller designs are also presented and applied to the nonlinear stochastic KSE.     

Robust model predictive control of nonlinear process systems: Handling rate constraints

This work considers the problem of stabilization of control affine nonlinear process systems subject to constraints on the rate of change and magnitude of control inputs in the presence of uncertainty. We first handle rate constraints within a soft constraints framework. A new robust predictive controller formulation that minimizes rate constraint violation while guaranteeing stabilization and input constraint satisfaction from an explicitly characterized stability region is designed. We then derive conditions that allow for guaranteed satisfaction of hard rate constraints. Subsequently, a predictive controller is designed that ensures rate constraints satisfaction when the required conditions are satisfied, relaxing them otherwise to preserve feasibility and robust stability. The implementation of the proposed predictive controllers is illustrated via a chemical reactor example.     

Model predictive control of Kuramoto–Sivashinsky equation with state and input constraints

This work focuses on the model predictive control design methodology that successfully accounts for the state and input constraints applied in the context of highly dissipative Kuramoto–Sivashinsky (KS) partial differential equation (PDE) describing stability of a thin film thickness in the two-phase annular flow in vertical pipes. The evolution of a linear dissipative KSE PDE state is given by an abstract evolution equation in an appropriate functional space. The proposed constrained predictive control law utilizes a low order modal representation in the optimization functional, while higher modes are included only in the PDE state constraints. Simulation results demonstrate a successful application of the proposed predictive control technique that achieves optimal stabilization of a spatially-uniform unstable steady state of Kuramoto–Sivashinsky equation in the presence of input and state constraints.     

Model predictive control of a second-order hyperbolic transport-reaction process

The model predictive controller (MPC) design is developed for a tubular chemical reactor, considering a second-order hyperbolic partial differential equation as the model of the transport-reaction process with boundary actuation. Without loss of generality, closed–closed boundary conditions and relaxed total flux are assumed. At the same time, the model is discretized in time by the Cayley–Tustin method, and, under the assumption that only the reactor's output is measurable, the observer design for the state reconstruction is addressed and integrated with the MPC design. The Luenberger observer gain is obtained by solving the operator Ricatti equation in the discrete-time setting, while the MPC accounts for constrained and optimal control. The simulations show that the output-based MPC design stabilizes the system under the input and output constraints satisfaction. In addition, to address the models' disparities, the results for both parabolic and hyperbolic equations are presented and discussed.     

Optimal mechano-electric stabilization of cardiac alternans

Alternation of normal action-potential morphology in the myocardium is a condition with a beat-to-beat oscillation in the length of the electric wave which is linked through electromechanical coupling to the cardiac muscle contraction, and is believed to be the first manifestation of the onset of life threatening ventricular arrhythmias and sudden cardiac death. In this work, the effects of electrical and mechanical stimuli are utilized in alternans annihilation problem. Electrical stimuli that alter the action-potential morphology are represented by a pacer located at the domain's boundary, while mechanical stimuli are distributed within the spatial domain and affect the action potential by altering intracellular calcium kinetics. Alternation of action potential is described by the small amplitude of alternans parabolic partial differential equation (PDE). Spatially uniform unstable steady state of the alternans amplitude PDE is stabilized by optimal control methods through boundary and spatially distributed actuation. Mixed boundary and spatially distributed actuation is manipulated by a linear quadratic regulator (LQR) in the full-state-feedback control structure and in a compensator design with a finite-dimensional Luenberger-type observer, and it achieves exponential stabilization in a finite size tissue cable length. The proposed control problem formulation and the performance and robustness of the closed-loop system under the proposed linear controller are evaluated through simulations.     

Safe-parking of nonlinear process systems: Handling uncertainty and unavailability of measurements

This work considers the problem of handling actuator faults in nonlinear process systems subject to input constraints, uncertainty and availability of limited measurements. A framework is developed to handle faults that preclude the possibility of continued operating at the nominal equilibrium point using the existing robust or reconfiguration-based fault-tolerant control approaches. The key consideration is to operate the plant using the depleted control action at an appropriate ‘safe-park’ point to prevent onset of hazardous situations as well as enable smooth resumption of nominal operation upon fault-repair. First, we consider the presence of constraints and uncertainty and develop a robust Lyapunov-based model predictive controller that enhances the set of initial conditions from which closed-loop stability is achieved. The stability region characterization provided by the robust predictive controller is subsequently utilized in a safe-parking algorithm that appropriately selects ‘safe-park’ points from the safe-park candidates (equilibrium points subject to failed actuators) to preserve closed-loop stability upon fault-repair. Specifically, a candidate parking point is termed a safe-park point if (1) the process state at the time of failure resides in the stability region of the safe-park candidate (subject to depleted control action and uncertainty) and (2) the safe-park candidate resides within the stability region of the nominal control configuration. Then we consider the problem of availability of limited measurements. An output feedback Lyapunov-based model predictive controller, utilizing an appropriately designed state observer (to estimate the unmeasured states), is formulated and its stability region explicitly characterized. An algorithm is then presented that accounts for the estimation errors in the implementation of the safe-parking framework. The proposed framework is illustrated using a chemical reactor example and demonstrated on a styrene polymerization process.     

Robust model predictive control and fault handling of batch processes

This work considers the control of batch processes subject to input constraints and model uncertainty with the objective of achieving a desired product quality. First, a computationally efficient nonlinear robust Model Predictive Control (MPC) is designed. The robust MPC scheme uses robust reverse‐time reachability regions (RTRRs), which we define as the set of process states that can be driven to a desired neighborhood of the target end‐point subject to input constraints and model uncertainty. A multilevel optimization‐based algorithm to generate robust RTRRs for specified uncertainty bounds is presented. We then consider the problem of uncertain batch processes subject to finite duration faults in the control actuators. Using the robust RTRR‐based MPC as the main tool, a robust safe‐steering framework is developed to address the problem of how to operate the functioning inputs during the fault repair period to ensure that the desired end‐point neighborhood can be reached upon recovery of the full control effort. The applicability of the proposed robust RTRR‐based controller and safe‐steering framework subject to limited availability of measurements and sensor noise are illustrated using a fed‐batch reactor system. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

改进的模糊预测控制算法及其在CSTR过程中的应用

A finite horizon predictive control algorithm,which applies a saturated feedback control law as its local control law,is presented for nonlinear systems with time-delay subject to input constraints.In the algorithm,N free control moves,a saturated local control law and the terminal weighting matrices are solved by a minimization problem based on linear matrix inequality(LMI) constraints online.Compared with the algorithm with a nonsaturated local law,the presented algorithm improves the performances of the closed-loop systems such as feasibility and optimality.This model predictive control(MPC) algorithm is applied to an industrial continuous stirred tank reactor(CSTR) with explicit input constraint.The simulation results demonstrate that the presented algorithm is effective.     

Fault-tolerant control of fluid dynamic systems via coordinated feedback and switching

This work addresses the problem of designing a fault-tolerant control system for fluid dynamic systems modeled by highly-dissipative partial differential equations (PDEs) with constrained control actuators. The proposed approach is predicated upon the idea of coordinating feedback controller synthesis and switching between multiple, spatially-distributed control actuator configurations. Using appropriate finite-dimensional approximations of the PDE system, a stabilizing feedback controller is designed for a given actuator configuration, and an explicit characterization of the constrained stability region is obtained. Switching laws are then derived, on the basis of these stability regions, to orchestrate the switching between the control actuator configurations, in a way that guarantees constraint satisfaction and preserves closed-loop stability of the infinite-dimensional system in the event of actuator failures. The results are demonstrated through an application of the proposed methodology to the suppression of wave formation in falling liquid films via the stabilization of the zero solution of the one-dimensional Kuramoto–Sivashinsky equation (KSE), with periodic boundary conditions, subject to actuator constraints and failures.     

Heat exchanger system boundary regulation

The boundary feedback regulator design for heat exchangers with delayed feedback is developed. Counter-flow/parallel-flow heat exchanger systems described by a pair of coupled transport hyperbolic partial differential equations (PDEs) with delayed boundary feedback loop modeled by the boundary time lag are considered. The coupled transport hyperbolic PDEs and boundary delay by application of boundary transformation are transformed in the corresponding linear infinite-dimensional system utilized in the regulator design. The regulator design initially addresses a full state feedback controller realization augmented by the observer design to achieve simultaneously output exponential stabilization as well as tracking and disturbance rejection of polynomial and/or harmonic type of reference signals. The simulations studies demonstrate the proposed design for counter-flow and parallel-flow heat exchangers, two common configurations present in industrial practice.     

Boundary control synthesis for a lithium‐ion battery thermal regulation problem

The thermal regulation problem for a lithium ion (Li‐ion) battery with boundary control actuation is considered. The model of the transient temperature dynamics of the battery is given by a nonhomogeneous parabolic partial differential equation (PDE) on a two‐dimensional spatial domain which accounts for the time‐varying heat generation during the battery discharge cycle. The spatial domain is given as a disk with radial and angular coordinates which captures the nonradially symmetric heat‐transfer phenomena due to the application of the control input along a portion of the spatial domain boundary. The Li‐ion battery model is formulated within an appropriately defined infinite‐dimensional function space setting which is suitable for spectral controller synthesis. The key challenges in the output feedback model‐based controller design addressed in this work are: the dependence of the state on time‐varying system parameters, the restriction of the input along a portion of the battery domain boundary, the observer‐based optimal boundary control design where the separation principle is utilized to demonstrate the stability of the closed loop system, and the realization of the outback feedback control problem based on state measurement and interpolation of the temperature field. Numerical results for simulation case studies are presented. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3782–3796, 2013     

Nonlinear control of rapid thermal chemical vapor deposition under uncertainty

This article focuses on nonlinear control of a rapid thermal chemical vapor deposition (RTCV'D) process in the presence of significant model uncertainty and disturbances. Initially, a detailed mathematical model of the RTCVD process is presented consisting of a nonlinear parabolic partial differential equation (PDE) which describes the time evolution of the wafer temperature across the radius of the wafer, coupled with a set of nonlinear ordinary differential equations (ODEs), which describe the time evolution of the concentrations of the various species. Then, the synthesis of a nonlinearoutput feedback controller based on the RTCVD process model by following a control methodology for nonlinear parabolic PDE systems introduced in (Baker and Christofides, 1998) is discussed. The controller uses measurements of wafer temperature at four locations to manipulate the power of the top lamps in order to achieve uniform temperature, and thus, uniform deposition of the thin film on the wafer over the entire process cycle. The nonlinearoutput feedback controller is successfully implemented through computer simulations and is shown to attenuate significant model uncertainty end disturbances and to outperform a proportional integral (PI) control scheme.     

Backstepping control of chemical tubular reactors

In this paper, a globally stabilizing boundary feedback control law for an arbitrarily fine discretization of a nonlinear PDE model of a chemical tubular reactor is presented. A model that assumes no radial velocity and concentration gradients in the reactor, the temperature gradient described by use of a proper value of the effective radial conductivity, a homogeneous reaction, the properties of the reaction mixture characterized by average values, the mechanism of axial mixing described by a single parameter model, and the kinetics of the first order is considered. Depending on the values of the nondimensional Peclet numbers, Damköhler number, the dimensionless adiabatic temperature rise, and the dimensionless activation energy, the coupled PDE equations for the temperature and concentration can have multiple equilibria that can be either stable or unstable. The objective is to stabilize an unstable steady state of the system using boundary control of temperature and concentration on the inlet side of the reactor. We discretize the original nonlinear PDE model in space using finite difference approximation and get a high order system of coupled nonlinear ODEs. Then, using backstepping design for parabolic PDEs we transform the original coupled system into two uncoupled target systems that are asymptotically stable in l²-norm with appropriate homogeneous boundary conditions. In the real system, the designed control laws would be implemented through small variations of the prescribed inlet temperature and prescribed inlet concentration. The control design is accompanied by a simulation study that shows the feedback control law designed with sensing only on a very coarse grid (using just a few measurements of the temperature and concentration fields) can successfully stabilize the actual system for a variety of different simulation settings (on a fine grid).