Panagiotis D. Christofides

Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes

Preface List of Figures List of Tables Introduction Feedback Control of Hyperbolic PDE Systems Robust Control of Hyperbolic PDE Systems Feedback Control of Hyperbolic PDE Systems Feedback Control of Parabolic PDE Systems Robust Control of Parabolic PDE Systems Nonlinear and Robust Control of Parabolic PDE Systems with Time-Dependent Spatial Domains Case Studies Proofs of Chapters (1-6) Karhunen-Loeve Expansion References Index

Nonlinear and Robust Control of PDE Systems

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Distributed model predictive control: A tutorial review and future research directions

Abstract In this paper, we provide a tutorial review of recent results in the design of distributed model predictive control systems. Our goal is to not only conceptually review the results in this area but also to provide enough algorithmic details so that the advantages and disadvantages of the various approaches can become quite clear. In this sense, our hope is that this paper would complement a series of recent review papers and catalyze future research in this rapidly evolving area. We conclude discussing our viewpoint on future research directions in this area.

Predictive control of switched nonlinear systems with scheduled mode transitions

In this work, a predictive control framework is proposed for the constrained stabilization of switched nonlinear systems that transit between their constituent modes at prescribed switching times. The main idea is to design a Lyapunov-based predictive controller for each constituent mode in which the switched system operates and incorporate constraints in the predictive controller design which upon satisfaction ensure that the prescribed transitions between the modes occur in a way that guarantees stability of the switched closed-loop system. This is achieved as follows: For each constituent mode, a Lyapunov-based model predictive controller (MPC) is designed, and an analytic bounded controller, using the same Lyapunov function, is used to explicitly characterize a set of initial conditions for which the MPC, irrespective of the controller parameters, is guaranteed to be feasible, and hence stabilizing. Then, constraints are incorporated in the MPC design which, upon satisfaction, ensure that: 1) the state of the closed-loop system, at the time of the transition, resides in the stability region of the mode that the system is switched into, and 2) the Lyapunov function for each mode is nonincreasing wherever the mode is reactivated, thereby guaranteeing stability. The proposed control method is demonstrated through application to a chemical process example.

Stabilization of nonlinear systems with state and control constraints using Lyapunov-based predictive control ☆

This work considers the problem of stabilization of nonlinear systems subject to state and control constraints. We propose a Lyapunov-based predictive control design that guarantees stabilization and state and input constraint satisfaction from an explicitly characterized set of initial conditions. An auxiliary Lyapunov-based analytical bounded control design is used to characterize the stability region of the predictive controller and also provide a feasible initial guess to the optimization problem in the predictive controller formulation. For the case when the state constraints are soft, we propose a switched predictive control strategy that reduces the time for which state constraints are violated, driving the states into the state and input constraints feasibility region of the Lyapunov-based predictive controller. We demonstrate the application of the Lyapunov-based predictive controller designs through a chemical process example.

Lyapunov-Based Model Predictive Control of Nonlinear Systems Subject to Data Losses

In this work, we focus on model predictive control of nonlinear systems subject to data losses. The motivation for considering this problem is provided by wireless networked control systems and control of nonlinear systems under asynchronous measurement sampling. In order to regulate the state of the system towards an equilibrium point while minimizing a given performance index, we propose a Lyapunov-based model predictive controller which is designed taking data losses explicitly into account, both in the optimization problem formulation and in the controller implementation. The proposed controller allows for an explicit characterization of the stability region and guarantees that this region is an invariant set for the closed-loop system under data losses, if the maximum time in which the loop is open is shorter than a given constant that depends on the parameters of the system and the Lyapunov-based controller that is used to formulate the optimization problem. The theoretical results are demonstrated through a chemical process example.

Isolation and handling of actuator faults in nonlinear systems

This work considers the problem of control actuator fault detection and isolation and fault-tolerant control for a multi-input multi-output nonlinear system subject to constraints on the manipulated inputs and proposes a fault detection and isolation filter and controller reconfiguration design. The implementation of the fault detection and isolation filters and reconfiguration strategy are demonstrated via a chemical process example.

Finite-dimensional approximation and control of non-linear parabolic PDE systems

This article proposes a rigorous and practical methodology for the derivation of accurate finite-dimensional approximations and the synthesis of non-linear output feedback controllers for non-linear parabolic PDE systems for which the manipulated inputs, the controlled and measured outputs are distributed in space. The method consists of three steps: first, the Karhunen-Loeve expansion is used to derive empirical eigenfunctions of the non-linear parabolic PDE system, then the empirical eigenfunctions are used as basis functions within a Galerkins and approximate inertial manifold model reduction framework to derive low-order ODE systems that accurately describe the dominant dynamics of the PDE system, and finally, these ODE systems are used for the synthesis of non-linear output feedback controllers that guarantee stability and enforce output tracking in the closed-loop system. The proposed method is used to perform model reduction and synthesize a non-linear dynamic output feedback controller for a rapid thermal chemical vapour deposition process. The controller uses measurements of wafer temperature at five locations to manipulate the power of the top lamps in order to achieve spatially uniform temperature, and thus, uniform deposition of the thin film on the wafer over the entire process cycle. The performance of the non-linear controller is successfully tested through simulations and is shown to be superior to the one of a linear controller.

Dynamic optimization of dissipative PDE systems using nonlinear order reduction

Abstract This article presents computationally efficient methods for the solution of dynamic constraint optimization problems arising in the context of spatially distributed processes governed by highly dissipative nonlinear partial differential equations (PDEs). The methods are based on spatial discretization using the method of weighted residuals with spatially global basis functions (i.e., functions that cover the entire domain of definition of the process and satisfy the boundary conditions). More specifically, we perform spatial discretization of the optimization problems using the method of weighted residuals with analytical or empirical (obtained via Karhunen–Loeve expansion) eigenfunctions as basis functions, and combination of the method of weighted residuals with approximate inertial manifolds. The proposed methods account for the fact that the dominant dynamics of highly dissipative PDE systems are low dimensional in nature and lead to approximate optimization problems that are of significantly lower order compared to the ones obtained from spatial discretization using finite-difference and finite-element techniques, and thus, they can be solved with significantly smaller computational demand. The resulting dynamic nonlinear programs include equality constraints that constitute a low-order system of coupled ordinary differential equations and algebraic equations, which can then be solved with combination of standard temporal discretization and nonlinear programming techniques. We employ backward finite differences (implicit Euler) to perform temporal discretization and solve the nonlinear programs resulting from temporal and spatial discretization using reduced gradient techniques (MINOS). We use two representative examples of dissipative PDEs, a diffusion-reaction process with constant and spatially varying coefficients, and the Kuramoto–Sivashinsky equation, a model that describes incipient instabilities in a variety of physical and chemical systems, to demonstrate the implementation and evaluate the effectiveness of the proposed optimization algorithms.

Bounded robust control of constrained multivariable nonlinear processes

Abstract This work focuses on control of multi-input multi-output (MIMO) nonlinear processes with uncertain dynamics and actuator constraints. A Lyapunov-based nonlinear controller design approach that accounts explicitly and simultaneously for process nonlinearities, plant-model mismatch, and input constraints, is proposed. Under the assumption that all process states are accessible for measurement, the approach leads to the explicit synthesis of bounded robust multivariable nonlinear state feedback controllers with well-characterized stability and performance properties. The controllers enforce stability and robust asymptotic reference-input tracking in the constrained uncertain closed-loop system and provide, at the same time, an explicit characterization of the region of guaranteed closed-loop stability. When full state measurements are not available, a combination of the state feedback controllers with high-gain state observes and appropriate saturation filters, is employed to synthesize bounded robust multivariable output feedback controllers that require only measurements of the outputs for practical implementation. The resulting output feedback design is shown to inherit the same closed-loop stability and performance properties of the state feedback controllers and, in addition, recover the closed-loop stability region obtained under state feedback, provided that the observer gain is sufficiently large. The developed state and output feedback controllers are applied successfully to non-isothermal chemical reactor examples with uncertainty, input constraints, and incomplete state measurements. Finally, we conclude the paper with a discussion that attempts to put in perspective the proposed Lyapunov-based control approach with respect to the nonlinear model predictive control (MPC) approach and discuss the implications of our results for the practical implementation of MPC, in control of uncertain nonlinear processes with input constraints.