Analysis and control of parabolic PDE systems with input constraints

This paper develops a general framework for the analysis and control of parabolic partial differential equations (PDE) systems with input constraints. Initially, Galerkin's method is used for the derivation of ordinary differential equation (ODE) system that capture the dominant dynamics of the PDE system. This ODE systems are then used as the basis for the synthesis, via Lyapunov techniques, of stabilizing bounded nonlinear state and output feedback control laws that provide an explicit characterization of the sets of admissible initial conditions and admissible control actuator locations that can be used to guarantee closed-loop stability in the presence of constraints. Precise conditions that guarantee stability of the constrained closed-loop parabolic PDE system are provided in terms of the separation between the fast and slow eigenmodes of the spatial differential operator. The theoretical results are used to stabilize an unstable steady-state of a diffusion-reaction process using constrained control action.     

A linear differential operator approach to flatness based tracking for linear and non-linear systems

This contribution presents a flatness based solution to the tracking for linear systems in differential operator representation. Since the differential operator representation is a flat system representation, tracking controllers can easily be designed using dynamic output feedback. Then, the differential operator approach for flatness based tracking of linear systems is extended to non-linear systems. The design of the resulting linear time varying dynamic output feedback controller is based on a linearization about the trajectory, which directly yields the differential operator representation. Different from the non-linear flatness based controller design the new approach uses linear methods, both in stabilizing the tracking and in computing the output feedback controller. The proposed design procedure assures exact tracking in the steady state when no disturbances are present. A simple example demonstrates the design of a dynamic output feedback controller for the tracking of a non-linear system.     

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 Galerkin's 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.     

A galerkin/neural-network-based design of guaranteed cost control for nonlinear distributed parameter systems.

This paper presents a Galerkin/neural-network- based guaranteed cost control (GCC) design for a class of parabolic partial differential equation (PDE) systems with unknown nonlinearities. A parabolic PDE system typically involves a spatial differential operator with eigenspectrum that can be partitioned into a finite-dimensional slow one and an infinite-dimensional stable fast complement. Motivated by this, in the proposed control scheme, Galerkin method is initially applied to the PDE system to derive an ordinary differential equation (ODE) system with unknown nonlinearities, which accurately describes the dynamics of the dominant (slow) modes of the PDE system. The resulting nonlinear ODE system is subsequently parameterized by a multilayer neural network (MNN) with one-hidden layer and zero bias terms. Then, based on the neural model and a Lure-type Lyapunov function, a linear modal feedback controller is developed to stabilize the closed-loop PDE system and provide an upper bound for the quadratic cost function associated with the finite-dimensional slow system for all admissible approximation errors of the network. The outcome of the GCC problem is formulated as a linear matrix inequality (LMI) problem. Moreover, by using the existing LMI optimization technique, a suboptimal guaranteed cost controller in the sense of minimizing the cost bound is obtained. Finally, the proposed design method is applied to the control of the temperature profile of a catalytic rod.     

A Sampled-Data Singularly Perturbed Boundary Control for a Heat Conduction System With Noncollocated Observation

This note presents a sampled-data strategy for a boundary control problem of a heat conduction system modeled by a parabolic partial differential equation (PDE). Using the zero-order-hold, the control law becomes a piecewise constant signal, in which a step change of value occurs at each sampling instant. Through the ‘lifting’ technique, the PDE is converted into a sequence of constant input problems, to be solved individually for a sampled-data formulation. The eigenspectrum of the parabolic system can be partitioned into two groups: a finite number of slow modes and an infinite number of fast modes, which is studied via the theory of singular perturbations. Controllability and observability of the sampled-data system are preserved, irrelevant to the sampling period. A noncollocated output-feedback design based upon the state observer is employed for set-point regulation. The state observer serves as an output-feedback compensator with no static feedback directly from the output, satisfying the so-called ‘low-pass property’. The feedback controller is thus robust against the observation error due to the neglected fast modes.      

Graphical analysis of a class of non-linear feedback control systems with step input†

The class of systems considered in this investigation is a cascade combination of a linear memory system and a non-linear no-memory system in the forward path of a unity feedback control system. The output of the non-linear no-memory system is assumed to be a polynomial function of the input. Regardless of the exact nature of the non-linearity, the objective of this method of analysis is to predict the behaviour of higher-order non-linear systems with different initial conditions for step inputs.

Two different cascade combinations of linear and non-linear blocks in the forward path are considered. For both configurations a similar non-linear differential equation is obtained for some variable in the system. The non-linear differential equation is further reduced to a first-order equation, explicitly independent of the independent variable, time t. Treating all other coefficients as parameters and eliminating each in turn, finally the required phase-plane trajectory is obtained.     

On sampled-data optimization in distributed parameter systems

A system of parabolic partial differential equations is transformed into ordinary differential equations in a Hilbert space, where the system operator is the infinitesimal generator of a semigroup of operators. A sampled-data problem is then formulated and converted into an equivalent discrete-time problem. The existence and uniqueness of an optimal sampled-data control is proved. The optimal control is given by a linear sampled-states feedback law where the feedback operator is shown to be the bounded seff-adjoint positive semidefinite solution of a Riccati operator difference equation.     

Finite dimensional disturbance observer based control for nonlinear parabolic PDE systems via output feedback

This paper considers the problem of finite dimensional disturbance observer based control (DOBC) via output feedback for a class of nonlinear parabolic partial differential equation (PDE) systems. The external disturbance is generated by an exosystem modeled by ordinary differential equations (ODEs), which enters into the PDE system through the control channel. Motivated by the fact that the dominant dynamic behavior of parabolic PDE systems can be characterized by a finite number of degrees of freedom, the modal decomposition technique is initially applied to the PDE system to derive a slow subsystem of finite dimensional ODEs. Subsequently, based on the slow subsystem and the exosystem, a disturbance observer (DO) and a slow mode observer (SMO) are constructed to estimate the disturbance and the slow modes. Moreover, an observation spillover observer (OSO) is also constructed to cancel approximately the effect of the observation spillover. Then, a finite dimensional DOBC design via output feedback is developed to estimate and compensate the disturbance, such that the closed-loop PDE system is exponentially stable in the presence of the disturbance. The condition for the existence of the proposed controller is given in terms of bilinear matrix inequality. Two algorithms based on the linear matrix inequality (LMI) technique are provided for solving control and observer gain matrices of the proposed controller. Finally, the developed design method is applied to the control of a one-dimensional diffusion-reaction process to illustrate its effectiveness.     

Design of periodic output feedback and fast output sampling based controllers for systems with slow and fast modes

It is well known that the straight forward application of controller design methods to two‐time‐scale systems is susceptible to numerical ill‐conditioning and stiffness problems. However, it has been demonstrated in this paper that the presence of time‐scales can be exploited for achieving better computational efficiency in case of the design of controllers based on periodic output feedback and fast output sampling principles. In particular, an approach for design of periodic output feedback and fast output sampling controllers for a two‐time‐scale system by decomposition into a slow and a fast subsystem problems, has been presented. These smaller order problems are then separately solved and the results thus obtained for the slow and fast subsystems are combined to obtain the solution of the original design problems. The procedure has been illustrated with the help of a very simplified nuclear reactor model. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Robust output feedback stabilization of uncertain dynamic systems with bounded controllers

We propose a new class of infinitely many bounded output feedback controllers for uncertain dynamic systems with bounded uncertainties. No statistical information about the uncertainties is assumed. A variable structure systems approach is employed in the synthesis of the proposed output feedback controllers. The role of the system zeros in the output feedback stabilization using the variable structure approach is discussed. We show that the proposed controllers guarantee the practical stability of the closed-loop system and give estimates of the regions of practical stability.     

On the asymptotic stability of a particular class of distributed parameter feedback control systems†

In this paper the stability problem for a particular class of non-linear feedback control systems governed by a parabolic partial differential equation is considered. A sufficient condition for global asymptotic stability of the null equilibrium state is derived by the method of comparison functions.     

Finite-dimensional constrained fuzzy control for a class of nonlinear distributed process systems.

This correspondence studies the problem of finite-dimensional constrained fuzzy control for a class of systems described by nonlinear parabolic partial differential equations (PDEs). Initially, Galerkin's method is applied to the PDE system to derive a nonlinear ordinary differential equation (ODE) system that accurately describes the dynamics of the dominant (slow) modes of the PDE system. Subsequently, a systematic modeling procedure is given to construct exactly a Takagi-Sugeno (T-S) fuzzy model for the finite-dimensional ODE system under state constraints. Then, based on the T-S fuzzy model, a sufficient condition for the existence of a stabilizing fuzzy controller is derived, which guarantees that the state constraints are satisfied and provides an upper bound on the quadratic performance function for the finite-dimensional slow system. The resulting fuzzy controllers can also guarantee the exponential stability of the closed-loop PDE system. Moreover, a local optimization algorithm based on the linear matrix inequalities is proposed to compute the feedback gain matrices of a suboptimal fuzzy controller in the sense of minimizing the quadratic performance bound. Finally, the proposed design method is applied to the control of the temperature profile of a catalytic rod.     

Static output feedback control via PDE boundary and ODE measurements in linear cascaded ODE–beam systems

This paper addresses the problem of feedback control design for a class of linear cascaded ordinary differential equation (ODE)–partial differential equation (PDE) systems via a boundary interconnection, where the ODE system is linear time-invariant and the PDE system is described by an Euler–Bernoulli beam (EBB) equation with variable coefficients. The objective of this paper is to design a static output feedback (SOF) controller via EBB boundary and ODE measurements such that the resulting closed-loop cascaded system is exponentially stable. The Lyapunov’s direct method is employed to derive the stabilization condition for the cascaded ODE–beam system, which is provided in terms of a set of bilinear matrix inequalities (BMIs). Furthermore, in order to compute the gain matrices of SOF controllers, a two-step procedure is presented to solve the BMI feasibility problem via the existing linear matrix inequality (LMI) optimization techniques. Finally, the numerical simulation is given to illustrate the effectiveness of the proposed design method.     

Robust Control for Two-Time-Scale Discrete Interval Systems

The problem of designing robust controller for discrete two-time-scale interval systems, conveniently represented using interval matrix notion, is considered. The original full order two-time-scale interval system is decomposed into slow and fast subsystems using interval arithmetic. The controllers designed independently to stabilize these two subsystems are combined to get a composite controller which also stabilizes the original full order two-time-scale interval system. It is shown that a state and output feedback control law designed to stabilize the slow interval subsystem stabilizes the original full order system provided the fast interval subsystem is asymptotically stable. The proposed design procedure is illustrated using numerical examples for establishing the efficacy of the proposed method.     

On linear modelling of non-linear systems†

In many cases of interest, systems described by non-linear differential equations are modelled by linear equations for purposes of synthesizing feedback controllers. A theorem relating the stability properties of the linear model and the non-linear system is presented and discussed.     

Robustness of dynamic output feedback control for singular perturbation systems

The problem of the robustness of dynamic output feedback control for singular perturbation systems is investigated. The solution to this problem is reduced to the simultaneous design of static output feedback controllers for the fast subsystem and the so-called auxiliary system of the slow subsystem. Some conditions are proposed to ensure the robustness of the actual closed-loop system. The exact upper bound of the parasitic parameter for the controlled system is also determined. Finally, an actual model which failed in dynamic output feedback control in [4] is reexamined successfully here.     

High-gain feedback in non-linear control systems†

High-gain state and output feedback are investigated for non-linear control systems with a single additive input by using singular perturbation techniques.

Classical approximation results (Tihonov-like theorems) in singular perturbation theory are extended to non-linear control systems by defining a composite additive control strategy, a control-dependent fast equilibrium manifold and non-linear change of coordinates.

Those tools and an appropriate change of coordinates show that high-gain state feedback and variable structure control systems can be equivalently used for approximate non-linearity compensation in feedback-linearizable systems.

Next the effect of high-gain output feedback is shown to be related to the strong invertibility property and the relative order of invertibility. For strongly invertible systems the slow reduced subsystem coincides with the dynamics of the inverse system when zero input is applied and with the unobservable dynamics when a certain input-output feedback-linearizable transformation is applied.     

Iterative solution of the nonlinear parabolic periodic boundary value problem

In this paper, the successive over-relaxation method (S.O.R.) is outlined for the numerical solution of the implicit finite difference equations derived from the Crank-Nicolson approximation to a mildly non-linear parabolic partial differential equation with periodic spatial boundary conditions. The usual serial ordering of the equations is shown to be inconsistent, thus invalidating the well known S.O.R. theory of Young (1954), but a functional relationship between the eigenvalues of the S.O.R. operator and the Jacobi operator of a closely related matrix is derived, from which the optimum over-relaxation factor, w_b, can be determined directly. Numerical experiments confirming the theory developed are given for the chosen problem.     

Accuracy and convergence of a finite element algorithm for laminar boundary layer flow

The Galerkin-weighted residuals formulation is employed to derive an implicit finite element solution algorithm for a generally non-linear initial-boundary value problem. Solution accuracy and convergence with discretization refinement are quantized in several error norms, for the non-linear parabolic partial differential equation system governing laminar boundary layer flow, using linear, quadratic and cubic functions. Richardson extrapolation is used to isolate integration truncation error in all norms, and Newton iteration is employed for all equation solutions performed in double-precision. The mathematical theory supporting accuracy and convergence concepts for linear elliptic equation appears extensible to the non-linear equations characteristic of laminar boundary layer flow.     

Exponential tracking of feedback linearizable non-linear control systems with uncertainties

This paper studies the robust output tracking problem of feedback linearizable non-linear control systems with uncertainties. Utilizing the input-output feedback linearization technique and the Lyapunov method for non-linear state feedback synthesis, a robust globally exponential output tracking controller design methodology for a broad class of non-linear control systems with uncertainties is developed. The underlying theoretical approaches are the differential geometry approach and the composite Lyapunov approach. One utilizes the parametrized coordinate transformation to transform the original non-linear system with uncertainties into a singularly perturbed model with uncertainties and the composite Lyapunov approach is then applied for output tracking. To demonstrate the practical applicability, the paper has investigated a pendulum control system.