Stevan Dubljevic

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.

Boundary optimal (LQ) control of coupled hyperbolic PDEs and ODEs

This contribution addresses the development of a linear quadratic (LQ) regulator for a set of hyperbolic PDEs coupled with a set of ODEs through the boundary. The approach is based on an infinite-dimensional Hilbert state-space description of the system and the well-known operator Riccati equation (ORE). In order to solve the optimal control problem, the ORE is converted to a set of matrix Riccati equations. The feedback operator is found by solving the resulting matrix Riccati equations. The performance of the designed control policy is assessed by applying it to a system of interconnected continuous stirred tank reactor (CSTR) and a plug flow reactor (PFR) through a numerical simulation.

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

Abstract In this work an asymptotic stabilization of highly dissipative Kuramoto–Sivashinsky equation (KSE) by means of boundary model modal predictive control (MMPC) in the presence of input and state constraints is demonstrated. The KS equation is initially defined in an appropriate functional space setting and an exact transformation is used to reformulate the original boundary control problem as an abstract boundary control problem of the KSE partial differential equation (PDE). An appropriate discrete infinite-dimensional representation of the abstract boundary control problem is used for synthesis of low dimensional model modal predictive controller (MMPC) incorporating both the pointwise enforced KSE state constraints and input constraints. The proposed control problem formulation and the performance of the closed-loop system in the full state feedback controller realization have been evaluated through simulations.

LQ-boundary control of a diffusion-convection-reaction system

In this work, the boundary control of a distributed parameter system modelled by linear parabolic partial differential equations (PDEs) with spatially varying coefficients is studied. An infinite-dimensional state space setting is considered and an exact transformation of the boundary actuation is realised to obtain an evolutionary model. The evolutionary model which incorporates the spatially varying coefficients of the underlying set of the PDEs is used for subsequent linear quadratic regulator synthesis. The formulated linear quadratic-state feedback controller is applied to a nonlinear model of the reactor and its performance is studied.

Output regulation problem for a class of regular hyperbolic systems

This paper investigates the output regulation problem for a class of regular first-order hyperbolic partial differential equation (PDE) systems. A state feedback and an error feedback regulator are considered to force the output of the hyperbolic PDE plant to track a periodic reference trajectory generated by a neutrally stable exosystem. A new explanation is given to extend the results in the literature to solve the regulation problem associated with the first-order hyperbolic PDE systems. Moreover, in order to provide the closed-loop stability condition for the solvability of the regulator problems, the design of stabilising feedback gain and its dual problem design of stabilising output injection gain are considered in this paper. This paper develops an easy method to obtain an adjustable stabilising feedback gain and stabilising output injection gain with the aid of the operator Riccati equation.

Control of parabolic PDEs with time-varying spatial domain: Czochralski crystal growth process

This paper considers the optimal control problem for a class of convection-diffusion-reaction systems modelled by partial differential equations (PDEs) defined on time-varying spatial domains. The class of PDEs is characterised by the presence of a time-dependent convective-transport term which is associated with the time evolution of the spatial domain boundary. The functional analytic description of the PDE yields the representation of the initial and boundary value problem as a nonautonomous parabolic evolution equation on an appropriately defined infinite-dimensional function space. The properties of the time-varying evolution operator to guarantee existence and well posedness of the initial and boundary value problem are demonstrated which serves as the basis for the optimal control problem synthesis. An industrial application of the crystal temperature regulation problem for the Czochralski crystal growth process is considered and numerical simulation results are provided.

LQ control of coupled hyperbolic PDEs and ODEs: Application to a CSTR-PFR system

Abstract In this paper an infinite-dimensional LQR control-based design for a system containing linear hyperbolic partial differential equations coupled with linear ordinary differential equations is presented. The design is based on an infinite-dimensional Hilbert state-space representation of the coupled system. The feedback control gain is obtained by solving algebraic and differential matrix Riccati equations that result from an operator Riccati equation solution. The designed LQR control is applied to a system containing a continuous stirred tank reactor (CSTR) and a plug flow reactor (PFR) in series with the recycle-rate from PFR to CSTR as controlled variable. The LQR controllers performance is evaluated by numerical simulation of the original nonlinear system.

Boundary predictive control of parabolic PDEs

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

The state feedback servo-regulator for countercurrent heat-exchanger system modelled by system of hyperbolic PDEs

Abstract The state feedback regulator problem for a network of countercurrent heat exchangers is considered in this paper. The system is described by two sets of hyperbolic partial differential equations (PDEs) and the model is nonlinear with respect to the control input. To deal with the nonlinearity, the equilibrium temperature profile is calculated and utilized in the linearization of the original nonlinear system. Then, based on infinite-dimensional representation, the state feedback regulator problem (in particular the tracking problem) is considered, where the target is to design a controller that, while guaranteeing the stability of the closed-loop system, drives the controlled output to track a reference signal generated by an exosystem with its spectrum on the imaginary axis. Given the explicit expression of the transfer function, we provide sufficient conditions such that the resulting linearized system is causal and stable. Given that the controlled system is stable, we propose a simple and novel method to provide the stabilization feedback gain K, such that the controlled system tracks the reference signal. Finally, a numerical simulation illustrating the results is presented.

PDE backstepping control of one-dimensional heat equation with time-varying domain

In this work a PDE backstepping-based control law for one-dimensional unstable heat equation with time-varying spatial domain is developed. The underlying parabolic partial differential equation (PDE) with time-varying domain is a model emerging from process control applications such as crystal growth. The use of backstepping control methodology yields the inherent feature of a time-varying PDE describing the kernel of the associated Volterra integral. The well-posedness of PDE kernel is proven and a numerical method to compute the solution of PDE kernel augmented with the error analysis to establish the accuracy of the proposed numerical method is demonstrated. Finally, the explicit form of the full state-feedback control law is given and appropriate simulation is provided for the application of temperature regulation in the Czochralski crystal growth process.