Distributed Parameter Control Theory Research Articles

Using functional gains for effective sensor location in flow control: a reduced-order modelling approach

The problem of active feedback control of fluid flows falls into a class of problems in the area of distributed parameter control. Distributed parameter systems are typically defined by partial differential equations that model the time and spatial evolution of the process. We consider the problem of locating sensors for effective feedback control of a fluid flow problem described by the Navier–Stokes equations. In this setting, the state of the system is the velocity field$\boldsymbol{v}(t,x)$, and hence all feedback laws are a function of this velocity field or, in most practical settings, a function of sensor outputs. In many designs, the feedback control law can be represented as a linear function of the state defined by an integral operator with a kernel function called the functional gain. In this paper we show that these functional gains can be used to determine effective sensor placement in complex flow control applications. The approach is to choose measurements of the state that would provide good quadrature points for the integral operator. We provide a computational validation of this approach by controlling the vortex shedding in a two-dimensional cylinder flow using a pair of fluid actuators on the cylinder surface. This model is linearized about the mean flow and a feedback control is designed by pole placement. Distributed parameter control theory yields the existence and form of the functional gains which are used to locate sensors. In particular, we use the location of the supports of the functional gains to determine two sets of four sensor locations in the wake. One of these measurement sets coincides with large magnitudes of the gain and the other set coincides with small magnitudes. Numerical experiments with a reduced-order model confirm superior performance of the closed-loop (CL) system using the former sensor set. We also show that choosing sensor locations associated with small magnitudes of the functional gains actually destabilizes the CL system.

A unified treatment on controllability/observability problems for distributed parameter systems and applications

The purpose of this paper is to present a unified treatment on controllability/observability problems for distributed parameter systems, and give its applications in distributed parameter control theory. For this purpose, from a basic weighted identity of parabolic-like partial differential operator (i.e., without elliptic condition), we will give all the known controllability/observability results for the parabolic, hyperbolic, Schrodinger and plate equations that are derived via Carleman estimate. Meanwhile, based on this weighted identity, we also give its applications in the stabilization of hyperbolic equations and the controllability/observability for the complex Ginzburg-Landau equations, etc.

Feedback Control of a Thermal Fluid Using State Estimation

In his paper we consider the problem of designing a feedback controller for a thermal fluid. Any practical feedback controller for a fluid flow system must incorporate some type of state estimator. Moreover, regardless of the approach, one must introduce approximations at some point in the analysis. The method presented here uses distributed parameter control theory to guide the design and approximation of practical slate estimators. Wc use finite clement techniques to approximate optimal infinite dimensional controllers based on linear quadratic Gaussian lpar;LQG) and MinMax theory for the Bonssincsq equations. These designs are then compared to full state feedback. We present several numerical experiments and we describe how these techniques can also be applied to sensor placement problems.

A Reduced Basis Approach to the Design of Low-Order Feedback Controllers for Nonlinear Continuous Systems

In this paper, the authors discuss a reduced basis approach to the development of low-order nonlinear feedback controllers for hybrid distributed parameter systems. This approach involves the use of distributed parameter control theory to design "optimal" infinite dimensional feedback control laws and approximation theory to design and compute low-order finite dimensional compensators. The resulting finite dimensional controller combines a nonlinear observer with a linear feedback law to produce a practical design. The authors concentrate on a weakly nonlinear distributed parameter system to illustrate the ideas.

Exponential Decay of Energy of Evolution Equations with Locally Distributed Damping

Consider an evolution equation with energy dissipation, \[ \frac{\partial }{{\partial t}}y( {x,t} ) = Ay( {x,t} ) + By( {x,t} ) \] on a bounded x-domain $\Omega $, where $By$ signifies a dissipative perturbation to an otherwise energy-conserving system. This dissipation may be due to medium impurities, viscous effects, or artificially imposed dampers and stabilizers. It is distributed over only part of the domain $\Omega $. The question of when the dissipation is effective enough to cause uniform exponential decay of energy is examined. Because of the locally distributed nature of energy dissipation, the problem lacks coercivity and is not directly solvable by energy identities. Thus, to get conditions sufficient for uniform exponential decay, a different approach needs to be taken. Provided here is a set of tight sufficient conditions in terms of the influence of the dissipative operatorBon the separated eigenmodes or clustered eigenmodes ofA. The main theorems are general enough to treat the wave and beam equations in one space dimension and the Schrödinger equation on a two-dimensional rectangle or disk. In particular, for the Schrödinger equation, it shown how such phenomena as the “whispering gallery”and “bouncing ball” eigenmodes dictate the supports of effective damping functions. Thus, the theory presented is also capable of providing valuable information about the placement and design of actuators and sensors in modern distributed parameter control theory.

Use of optimal control theory for computing optimal injection policies for enhanced oil recovery

The theory of optimal control of distributed parameter systems has been applied to the problem of determining an optimum injection policy for a chemical flooding enhanced oil recovery process. The optimization criterion is to maximize the difference between gross revenue and cost of chemicals injected. Necessary conditions for optimality are obtained through application of the calculus of variations and the distributed weak minimum principle. A gradient method is used for the computation of optimal injection policies. The performance of the algorithm was examined for the surfactant injection in a one-dimensional flooding problem. Two types of interfacial tension behaviour were considered: a type ‘A’ system in which the interfacial tension is a monotically decreasing function with solute concentration and a type ‘B’ system in which a minimum in the interfacial tension occurs at a nominal surfactant concentration. Different initial values were used in order to establish the convergence of the computational algorithm. For a type ‘A’ system, the shape of the optimum injection strategy was not unique; however, there is a unique optimum for the amount of surfactant needed. For a type ‘B’ system, the shape of the optimum injection was unique as well as the amount injected. The results of this work show that given the properties of an oil reservoir and the properties of a surfactant solution, an optimum injection policy which minimizes a specific economic objective functional can be obtained using distributed parameter control theory.

Distributed piezoelectric-polymer active vibration control of a cantilever beam

An active vibration damper for a cantilever beam was designed using a distributed-parameter actuator and distributed-parameter control theory. The distributed-parameter actuator was a piezoelectric polymer, poly (vinylidene fluoride). Lyapunov's second method for distributed-parameter systems was used to design a control algorithm for the damper. If the angular velocity of the tip of the beam is known, all modes of the beam can be controlled simultaneously. Preliminary testing of the damper was performed on the first mode of the cantilever beam. A linear constant-gain controller and a nonlinear constant-amplitude controller were compared. The baseline loss factor of the first mode was 0.003 for large-amplitude vibrations (± 2 cm tip displacement) decreasing to 0.001 for small vibrations (±0.5 mm tip displacement). The constant-gain controller provided more than a factor of two increase in the modal damping with a feedback voltage limit of 200 V rms. With the same voltage limit, the constant-amplitude controller achieved the same damping as the constant-gain controller for large vibrations, but increased the modal loss factor by more than an order of magnitude to at least 0.040 for small vibration levels.

Optimal Injection Policies for Enhanced Oil Recovery

The theory of optimal control of distributed parameter systems is presented for determining the best possible injection policies for enhanced oil recovery processes. The optimization criterion is to maximize the amount of oil recovered while minimizing injection costs. Necessary conditions for optimality are obtained through application of the calculus of variations and Pontryagin's Weak Minimum Principle . A gradient method is used for the computation of optimal injection policies.The perforrnance of the algorithm was examined for the surfactant injection in a one di mensional flooding problem. Two types of interfacial tension behavior were cons idered: a type “A” system in which the interfacial tension is a monotically decreasing function with solute concentration and a type “B” system in which a minimurn in the in terfacial tension occurs at a nominal surfactant concentration. For a type “A” system, the shape of the optimum injection strateay was not unique, however, there is a unique optimum for the amount of surfacta nt needed. For a type “B” system, the shape of the optimum injection was unique as well as the amount injected.The results of this work show that given the properties of an oil reservoir and the properties of a surfactant solution, an optimum injection policy can be obtained using distributed parameter control theory.

Some Recent Applications of Distributed Parameter Control Theory — A Survey

A survey of some recent applications of distributed parameter control theory is presented. The practical areas discussed range from process control problems in an industrial plant to the identification, monitoring and control of air and water quality in our environment. Some new, promising areas of application are discussed along with suggestions for future research emphasis.

5th Conference on Optimization Techniques.

Some aspects of urban systems of relevance to optimisation techniques.- Selection of optimal industrial clusters for regional development.- Optimal investment policies in transportation networks.- An on-line optimization procedure for an urban traffic system.- Hierarchical strategies for the on-line control of urban road traffic signals.- Application of optimization approach to the problem of land use plan design.- Combinatorial optimization and preference pattern aggregation.- A microsimulation model of the health care system in the United States: The role of the physician services sector.- A model for finite stroage message switching networks.- On constrained diameter and medium optimal spanning trees.- Simulation techniques for the study of modulated communication channels.- Gestion Optimale d'un Ordinateur Multiprogramme a Memoire Virtuelle.- State-space approach in problem-solving optimization.- Perturbation theory and the statement of inverse problems.- A model for the evaluation of alternative policies for atmospheric pollutant source emissions (Masc model).- Mathematical modelling of a nordic hydrological system, and the use of a simplified run-off model in the stochastic optimal control of a hydroelectrical power system.- A two-dimensional model for the Lagoon of Venice.- Sea level prediction models for Venice.- Optimal estuary aeration: An application of distributed parameter control theory.- Interactive simulation program for water flood routing systems.- An automatic river planning operating system (ARPOS).- On the optimal control on an infinite planning horizon of consumption, pollution, population and natural resource use.- Limited role of entropy in information economics.- On a dual control approach to the pricing policies of a trading specialist.- Problems of optimal economic investments with finite lifetime capital.- Some economic models of markets.- Utilization of heuristics in manufacturing planning and optimization.- Economic simulation of a small chemical plant.- An optimal growth model for the Hungarian national economy.- On optimization of health care systems.- Theoretical and operational problems in driving a physical model of the circulatory system.- Modeling, simulation, identification and optimal control of large biochemical systems.- Modelisation du Transfert Gazeux Pulmonaire et Calcul Automatique de la Capacite de Diffusion.- On some models of the muscle spindle.

Optimal estuary aeration: An application of distributed parameter control theory

The use of artificial induced aeration has been suggested as a means for improving water quality. The task of controlling the aeration input rate to obtain maximum improvement with least cost is addressed as an optimization problem of a distributed parameter control system. A partial differential equation model for the dissolved oxygen balance in streams and estuaries is given, and a criterion functional is proposed in which the control can be found as the solution of an optimization problem in a Hilbert space. An analytic solution for the optimal feedback control of a stream aeration system is found, and a numerical algorithm for the estuary case is applied to an example using historical data from the Delaware estuary. The sensitivity of the control to system and input variations is discussed, and the dollar cost for the example is compared with costs for other suboptimal control schemes.

Optimization of a Class of River Aeration Problems by the Use of Multivariable Distributed Parameter Control Theory

Multivariable distributed parameter control theory has been applied to the optimization of artificial in‐stream aeration of polluted rivers. The optimization was developed for a general integral type of cost functional with weighted energy constraints. Three specific types of aeration control policies were considered: distributed space and time control, distributed time control, and distributed space control. Employing typical hydrologic coefficients with the transient version of Streeter‐Phelps equation, the optimal aeration control surfaces obtained were all found to lie well within the operating characteristics of commercial aeration equipment. Distributed parameter control theory applied to water pollution systems proved to be a powerful mathematical programing method comparable to linear and dynamic programing.