Finite-dimensional Control Research Articles

Sub‐optimal finite dimensional observer‐based boundary controller design for distributed parameter systems on a planar domain

In this paper, a certain class of distributed parameter systems is considered. We propose a three‐step design method for finding finite dimensional observer‐based boundary feedback controllers. The first step is called the boundary‐equation normalization, which transforms the boundary and system equations into a normal form. The second step is called the boundary input transformation, which integrates the boundary input equation into the system equation, and forms a type of distributed parameter system called the general boundary input system. The final step is to design the desired finite dimensional controller, based on the general boundary input system model. The design procedure utilizes the finite dimensional linear quadratic optimal control theory, so well‐developed computation tools can be applied. Though the acquired controllers are only sub‐optimal for the distributed parameter systems, an estimation of the performance degradation from that of the ideal case is derived for comparison purpose.

Control by Interconnection of the Timoshenko Beam

In this paper, the dynamical control of a mixed finite and infinite dimensional mechanical system is approached within the framework of port Hamiltonian systems. As an applicative example of the presented methodology, a flexible beam, modeled according to the Timoshenko theory, with a mass under gravity field connected to a free end, is considered. After the distributed port Hamiltonian (dpH) model of the beam is introduced, the control problem is discussed. The concept of structural invariant (Casimir function) is generalized to the infinite dimensional case and the so-called control by interconnection control technique is extended to the infinite dimensional case. In this way, finite dimensional passive controllers can stabilize distributed parameter systems by shaping their total energy, i.e. by assigning a new minimum in the desired equilibrium configuration that can be reached if a dissipation effect is introduced

Residual mode filters and H ∞ control for a class of infinite-dimensional discrete-time systems

An H ∞, control problem with measurement feedback.for infinite-dimensional discrete-time (IDDT) systems whose homogeneous parts are described by Riesz-spectra operators is considered. The aim is to construct a finite-dimensional stabilizing controller for the IDDT system that makes the H ∞ norm of the closed-loop transfer function less than a given positive number δ. For that purpose, we first formulate the IDDT system as an IDDT system in l2 and derive a finite-dimensional reduced-order system for the IDDT system in l2. A stabilizing controller that makes the H ∞ norm of the closed-loop transfer function less than another positive number is then constructed for the reduced-order model. The finite-dimensional controller together with a residual mode Jilter plays a role of a finite-dimensional stabilizing controller that makes the H ∞ norm of the closed-loop transfer function less than δ for the original IDDT system, if the order of the residual mode filter is chosen suficiently large.

Spillover Stabilization in Finite-Dimensional Control and Observer Design for Dissipative Evolution Equations

We consider the problem of global stabilization of a semilinear dissipative evolution equation by finite-dimensional control with finite-dimensional outputs. Coupling between the system modes occurs directly through the nonlinearity and also through the control influence functions. Similar modal coupling occurs in the infinite-dimensional error dynamics through the nonlinearity and measurements. For both the control and observer designs, rather than decompose the original system into Fourier modes, we consider Lyapunov functions based on the infinite-dimensional dynamics of the state and error systems, respectively. The inner product terms of the Lyapunov derivative are decomposed into Fourier modes. Upper bounds on the terms representing control and observation spillover are obtained. Linear quadratic regulator (LQR) designs are used to stabilize the state and error systems with these upper bounds. Relations between system and LQR design parameters are given to ensure global stability of the state and error dynamics with robustness with respect to control and observation spillover, respectively. It is shown that the control and observer designs can be combined to yield a globally stabilizing compensator. The control and observer designs are numerically demonstrated on the problem of controlling stall in a model of axial compressors.

Exact Finite Dimensional Feedback Control via Inertial Manifold Theory with Application to the Chafee–Infante Equation

A class of nonlinear parabolic partial differential equations is considered, and an exact finite dimensional feedback control law is designed in order to force the systems to behave in a prescribed way. The feedback law is obtained via inertial manifold theory by reducing the system to finite dimensions. The control achieved is exact, as opposed to approximate, as obtained in a previous work. The result is applied to the Chafee–Infante equation, a one-dimensional scalar reaction-diffusion equation, with distributed control.

State Constrained Feedback Stabilization

A standard finite dimensional nonlinear control system is considered, along with a state constraint set S and a target set $\Sigma$. It is proven that open loop S-constrained controllability to $\Sigma$ implies closed loop S-constrained controllability to the closed $\delta$-neighborhood of $\Sigma$, for any specified $\delta > 0$. When the S-constrained minimum time function to $\Sigma$ satisfies a local continuity condition, conclusions on closed loop S-constrained stabilizability ensue. The (necessarily discontinuous) feedback laws in question are implemented in the sample-and-hold sense and possess a robustness property with respect to state measurement errors. The feedback constructions involve the quadratic infimal convolution of a control Lyapunov function with respect to a certain modification of the original dynamics. The modified dynamics in effect provide for constraint removal, while the convolution operation provides a useful semiconcavity property.

An LMI-based controller synthesis for periodic trajectories in a class of nonlinear systems

The use of finite-dimensional linear time-invariant controllers for the stabilization of periodic solutions in sinusoidally forced nonlinear systems is investigated. By mixing results concerning absolute stability of nonlinear systems and robustness of linear systems, a linear matrix inequality-based controller synthesis technique is developed. The synthesis algorithm yields the controller maximizing a lower bound of the maximum amplitude of the forcing input, for which the corresponding periodic solutions are guaranteed to be stable. The Duffing oscillator is employed to illustrate the main features of the proposed synthesis technique.

Structured finite-dimensional controller design by convex optimization

We discuss how to solve rather general structured controller design problems by convex optimization. In our main contribution we develop, under a specific one-block hypothesis on just one subsystem in the feedback interconnection, a semi-definite programming algorithm that allows to design controllers whose McMillan degree is bounded in terms of the underlying plant description. As an additional goal, we reveal the close interplay between structural controller design and so-called multi-objective control problems in which one imposes structural performance specifications on the closed-loop system. Finally, we include various motivations for the relevance of extending our results in several directions.

From irreversible thermodynamics to a robust control theory for distributed process systems

In this paper we combine recent results that link passivity, as it is understood in system's theory, with concepts from irreversible thermodynamics to develop a robust control design methodology for distributed process systems. In this context, we show that passivity and stabilization of systems where non-dissipative phenomena are taking place is possible under very simple, finite dimensional control configurations. These include, boundary and high gain controllers, which combined with robust identification schemes should be able to provide efficient plant operation.

Low-order model identification for implementable control solutions of distributed parameter systems

Accurate solutions of the distributed parameter system (DPS) may be represented as the sum of an infinite series. Control design however, requires low-order models primarily due to implementation limitations. As such, developing low-order models of high fidelity is important if the objective is accurate control of the DPS. When an exact model (system of partial differential equations (PDEs)) of the system is known, this work presents a method to develop a low-order model that assures convergent and consistent projection to a finite space. The resulting low-order model can then be used to design finite dimensional controllers. When there is no available first-principle model of the system, this work introduces a novel system identification method, that combines the characteristics of singular value decomposition (SVD) and the Karhunen-Loève (KL) expansion for DPS to arrive at a low-order model that captures the dominant characteristics of the system. Here as well, the final model form allows for the synthesis of finite order controllers. Two non-linear reactor systems that can be described by systems of PDEs are provided to demonstrate the model identification methods. Feedback controllers are then synthesized based on these models to demonstrate their accuracy for disturbance rejection.

Low-order control-relevant models for a class of distributed parameter systems

Accurate solutions of distributed parameter systems may be represented as the sum of an infinite series. Control design however, requires low-order models primarily due to implementation limitations. As such, developing low-order models of high fidelity is important if the objective is accurate control of the DPS. This work addresses this issue by developing a method that assures a convergent and consistent projection to a finite space. The resulting model is then subsequently used to design finite dimensional state feedback controllers. The methodology is demonstrated on two quasi-linear processes under ideal and non-ideal conditions.

Formula omitted]-output feedback of infinite-dimensional systems via approximation

As in the finite-dimensional case, a state-space based controller for the infinite-dimensional H ∞ disturbance-attenuation problem may be calculated by solving two Riccati equations. These operator Riccati equations can rarely be solved exactly. We approximate the original infinite-dimensional system by a sequence of finite-dimensional systems. The solutions to the corresponding finite-dimensional Riccati equations are shown to converge to the solution of the infinite-dimensional Riccati equations. Furthermore, the corresponding finite-dimensional controllers yield performance arbitrarily close to that obtained with the infinite-dimensional controller.

Finite dimensional variable structure control design for distributed delay systems

A general design method for the variable structure control of distributed time-delay systems is studied in this paper. The proposed switching function and variable structure controller are of finite dimension. The basic idea is to construct a finite dimensional system which contains all the unstable poles of the original time-delay system and hence those unstable poles of the time delay system will be absorbed by the developed variable structure controller. The proposed method is applied to the stabilization of combustion in a liquid monopropellant rocket motor and satisfactory results are obtained by simulation study. Simulation results show that the developed variable structure controller can still stabilize the combustion when the two key parameters, pressure exponent and maximal time lag r, change from their nominal value down to 0.0 and 0.1 respectively. By this result, we can somewhat overestimate the two parameters to obtain a reliable controller for the rocket motor control systems.

On theLp-stabilization of the double integrator subject to input saturation

We consider a finite-dimensional control system , such that there exists a feedback stabilizer k that renders globally asymptotically stable. Moreover, for (H,p,q) with H an output map and , we assume that there exists a -function α such that , where x u is the maximal solution of , corresponding to u and to the initial condition x(0)=0 . Then, the gain function of (H,p,q) given by 14.5cm is well-defined. We call profile of k for (H,p,q) any -function which is of the same order of magnitude as . For the double integrator subject to input saturation and stabilized by , we determine the profiles corresponding to the main output maps. In particular, if is used to denote the standard saturation function, we show that the L 2 -gain from the output of the saturation nonlinearity to u of the system with , is finite. We also provide a class of feedback stabilizers k F that have a linear profile for (x,p,p) , . For instance, we show that the L 2 -gains from x and to u of the system with , are finite.

A semi‐discrete approach to modelling and control of the continuous casting process

Starting from a partial differential equation describing the heat transfer in the continuous casting of steel, the Kirchhoff transformation is used to obtain a more simplified governing heat equation. A boundary state and control transformation is then introduced to obtain a nonlinear heat equation with boundary control. The semi‐discrete approximation is applied in the investigation to obtain an ordinary differential equation model. The derived lumped parameter control model can be used to get an approximate solution for the system as well as for finite dimensional system control studies. Finally, local controllability is proved for the system using a linearization technique.

Finite-dimensional variable structure control design for systems with a single time delay

A design method for the variable structure control of point time-delay systems is studied in this paper. The proposed switching function and variable structure controller are of finite dimension. The basic idea is to construct a finite-dimensional system which contains all the unstable poles of the original time-delay system and hence those unstable poles of the time delay system will be absorbed by the developed variable structure controller. The proposed method is applied to the stabilization of combustion in a liquid monopropellant rocket motor and satisfactory results are obtained by simulation study. Smooth dependence of the closed-loop system response on the two parameters, the pressure exponent y and time lag ∞, are shown by simulation results.

From Irreversible Thermodynamics to a Robust Control Theory for Distributed Process Systems

Abstract In this paper we combine recent results that link passivity, as it is understood in system’s theory, with concepts from irreversible thermodynamics to develop a robust control design methodology for distributed process systems. In this context, we show that passivity and stabilization of systems where non-dissipative phenomena are taking place is possible under very simple, finite dimensional control configurations. These include, boundary and high gain controllers, which combined with robust identification schemes should be able to provide efficient plant operation.

Information states in stochastic control and filtering: a Lie algebraic theoretic approach

The purpose of the paper is two-fold: (i) to introduce the sufficient statistic algebra which is responsible for propagating the sufficient statistics, or information state, in the optimal control of stochastic systems and (ii) to apply certain Lie algebraic methods and gauge transformations, widely used in nonlinear control theory and quantum physics, to derive new results concerning finite dimensional controllers. This enhances our understanding of the role played by the sufficient statistics. The sufficient statistic algebra enables us to determine a priori whether there exist finite-dimensional controllers; it also enables us to classify all finite-dimensional controllers. Relations to minimax dynamic games are delineated.

Wave suppression by nonlinear finite-dimensional control

Korteweg–de Vries–Burgers (KdVB) and Kuramoto–Sivashinsky (KS) equations are two nonlinear partial differential equations (PDEs) which can adequately describe motion of waves in a variety of fluid flow processes. We synthesize nonlinear low-dimensional output feedback controllers for the KdVB and KS equations that enhance convergence rate and achieve stabilization to spatially uniform steady states, respectively. The approach used for controller synthesis employs nonlinear Galerkin's method to derive low-dimensional approximations of the PDEs, which are subsequently used for controller synthesis via geometric control methods. The controllers use measurements obtained by point sensors and are implemented through point control actuators. The performance of the proposed controllers is successfully tested through simulations.

A finite-dimensional robust controller for systems in the CD-algebra

Robust multivariable controllers for stable infinite-dimensional systems in the Callier-Desoer (CD) algebra are discussed. In particular, the following robust regulation problem is solved. Given reference and disturbance signals, which are linear combinations of signals of the form t/sup j/ sin(/spl omega//sub k/t+/spl phi//sub k/), j/spl ges/0, k=0, /spl middot//spl middot//spl middot/, n, find a low-order finite-dimensional controller so that the outputs asymptotically track the reference signals, asymptotically reject the disturbance signals, and the closed-loop system is stable and robust with respect to a class of perturbations in the plant. The proposed controller consists of a positive scalar gain /spl epsi/ and certain polynomial matrices K/sub k/(s) for k=0, /spl middot//spl middot//spl middot/, n. The main result of the paper shows that the matrices K/sub k/(s) must satisfy certain stability conditions involving the values of the plant transfer function only at the reference and disturbance signal frequencies /spl omega//sub k/ for k=0, /spl middot//spl middot//spl middot/, n. The controller has unstable poles on the imaginary axis. The behavior of these poles as a function of the scalar parameter /spl epsi/ in the form of a Puiseux series is given.