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An extension of classical methods of optimal control theory for systems described by ordinary differential equations to distributed-parameter systems described by partial differential equations is presented. An application is given involving the minimum-mass design of a simply-supported shear plate with a fixed fundamental frequency of vibration. An optimal plate thickness distribution in analytical form is found. The case of a minimum-mass design of an elastic sandwich plate whose fundamental frequency of free vibration is fixed. Under the most general conditions, the optimization problem reduces to the solution of two simultaneous partial differential equations involving the optimal thickness distribution and the modal displacement. One equation is the uniform energy distribution expression which was found by Ashley and McIntosh for the optimal design of one-dimensional structures with frequency constraints, and by Prager and Taylor for various design criteria in one and two dimensions. The second equation requires dynamic equilibrium at the preassigned vibration frequency.

The investigator surveyed a number of applications and problems motivating his current efforts on numerical techniques for parameter estimation in and optimal control of delay and partial differential equations. He then outlined two different approaches for establishing theoretical convergence results for estimation algorithms. An application of modal techniques to the investigation of transport in brain tissue is briefly explained. A sketch of a convergence theory for spline techniques for function space parameter estimation problems is given.

Although significant advances have been made in modeling and controlling flexible systems, there remains a need for improvements in model accuracy and in control performance. The finite element models of flexible systems are unduly complex and are almost intractable to optimum parameter estimation for refinement using experimental data. Distributed parameter or continuum modeling offers some advantages and some challenges in both modeling and control. Continuum models often result in a significantly reduced number of model parameters, thereby enabling optimum parameter estimation. The dynamic equations of motion of continuum models provide the advantage of allowing the embedding of the control system dynamics, thus forming a complete set of system dynamics. There is also increased insight provided by the continuum model approach.

Although significant advances have been made in modeling and controlling flexible systems, there remains a need for improvements in model accuracy and in control performance. The finite element models of flexible systems are unduly complex and are almost intractable to optimum parameter estimation for refinement using experimental data. Distributed parameter or continuum modeling offers some advantages and some challenges in both modeling and control. Continuum models often result in a significantly reduced number of model parameters, thereby enabling optimum parameter estimation. The dynamic equations of motion of continuum models provide the advantage of allowing the embedding of the control system dynamics, thus forming a complete set of system dynamics. There is also increased insight provided by the continuum model approach.

Although significant advances have been made in modeling and controlling flexible systems, there remains a need for improvements in model accuracy and in control performance. The finite element models of flexible systems are unduly complex and are almost intractable to optimum parameter estimation for refinement using experimental data. Distributed parameter or continuum modeling offers some advantages and some challenges in both modeling and control. Continuum models often result in a significantly reduced number of model parameters, thereby enabling optimum parameter estimation. The dynamic equations of motion of continuum models provide the advantage of allowing the embedding of the control system dynamics, thus forming a complete set of system dynamics. There is also increased insight provided by the continuum model approach.

Research has been conducted on nonlinear optimization in the areas of (1) dual based methods and decomposition; (2) regularization and approximation techniques; (3) nonsmooth optimization and (4) algorithms development for large- scale optimization. The methods developed are applicable to a wide range of important applications including; optimal shape design, structural optimization, and image reconstruction.

The efficient numerical treatment of optimal control problems governed by elliptic partial differential equations (PDEs) and systems of elliptic PDEs, where the control is finite dimensional is discussed. Distributed control as well as boundary control cases are discussed. The main characteristic of the new methods is that they are designed to solve the full optimization problem directly, rather than accelerating a descent method by an efficient multigrid solver for the equations involved. The methods use the adjoint state in order to achieve efficient smoother and a robust coarsening strategy. The main idea is the treatment of the control variables on appropriate scales, i.e., control variables that correspond to smooth functions are solved for on coarse grids depending on the smoothness of these functions. Solution of the control problems is achieved with the cost of solving the constraint equations about two to three times (by a multigrid solver). Numerical examples demonstrate the effectiveness of the method proposed in distributed control case, pointwise control and boundary control problems.

A Chandrasekhar-type factorization method is applied to the linear-quadratic optimal control problem for distributed parameter systems. An aeroelastic control problem is used as a model example to demonstrate that if computationally efficient algorithms, such as those of Chandrasekhar-type, are combined with the special structure often available to a particular problem, then an abstract approximation theory developed for distributed parameter control theory becomes a viable method of solution. A numerical scheme based on averaging approximations is applied to hereditary control problems. Numerical examples are given.

Parabolic partial differential equations for optimal pointwise feedback control of distributed parameter systems

The main purpose of this research is to develop a rigorous mathematical framework for the design of control laws for feedback stabilization and for controllability of the transient behavior of flexible structures based on distributed parameter models of such structures. This work has entailed deriving accurate distributed parameter models for elastic structures and understanding the implications of the various models for the controllability and stabilizability of structures. Substantial progress has been made for models of multiple-link constructions that are composed of elastic beams, plates., shells or combinations of such elastic elements. Such structures are representative of trusses, frames, robot arms, solar panels, antennae, deformable mirrors, etc., currently in use.

This paper discusses the efficient numerical treatment of optimal control problems governed by elliptic partial differential equations and systems of elliptic partial differential equations, where the control is finite dimensional. Distributed control as well as boundary control cases are discussed. The main characteristic of the new methods is that they are designed to solve the full optimization problem directly, rather than accelerating a descent method by an efficient multigrid solver for the equations involved. The methods use the adjoint state in order to achieve efficient smoother and a robust coarsening strategy. The main idea is the treatment of the control variables on appropriate scales, i.e., control variables that correspond to smooth functions are solved for on coarse grids depending on the smoothness of these functions. Solution of the control problems is achieved with the cost of solving the constraint equations about two to three times (by a multigrid solver). Numerical examples demonstrate the effectiveness of the method proposed in distributed control case, pointwise control and boundary control problem.

In this research, the issue of uniform stabilization has been investigated for large classes of conservative systems arising in elasticity. In the case of linear problems, two types of feedback have been considered: (1) dissipative feedback operators acting only on the 'velocity' (2) generally non- dissipative feedback based on Riccati operators and acting on the full pair of position and velocity. In non linear problems, feedback of type (1) have been studied. Numerical implementation for both types of feedbacks have received special attention.

The International Conference on Control and Estimation of Distributed Parameter Systems took place from July 14-20, 1996, at the Bildungshaus Chorherrenstift Vorau in Vorau (Austria). It was the seventh in a series of conferences that begun in 1982. 51 researchers from 11 states contributed to draw a broad and diverse picture of the recent developments in optimal control and parameter identification of partial differential equations, both from a theoretical and numerical viewpoint.

Here an original idea is suggested to prove the existence of optimal control for some types of non- linear problems. The obtained results can be considered as individual existence theorems (in some sense).

This research centered on the following four classes of generalized linear systems: (1) Linear continuous-time and discrete-time systems whose coefficients depend on (a priori) unknown parameters; (2) linear spatially-distributed continuous-time and discrete time systems; (3) linear continuous-time systems with commensurate or non-commensurate time delays; and (4) linear time-varying discrete-time systems.

Efforts to develop computational methods for the identification and optimal control of linear and nonlinear systems governed by distributed parameter systems are reported on. Specifically, approximation methods for determining Optimal LOG compensators (feedback control and estimator gains) and functional parameters in linear and nonlinear partial differential equations and hereditary systems were developed, analyzed and tested. The study included theoretical, experimental and numerical components. Covergence theories for spline-based and modal finite element schemes were established and extensive numerical studies on both conventional (serial) and vector supercomputers were carried out. A parameter estimation scheme was tested using experimental data taken from the RPL structure, a laboratory experiment designed to test control algorithms for the large angle slewing of spacecraft with flexible appendages, and other projects involving the identification of flexible structures based upon experimental data were initiated.

Efforts to develop computational methods for the identification and optimal control of linear and nonlinear systems governed by distributed parameter systems are reported on. Specifically, approximation methods for determining Optimal LOG compensators (feedback control and estimator gains) and functional parameters in linear and nonlinear partial differential equations and hereditary systems were developed, analyzed and tested. The study included theoretical, experimental and numerical components. Covergence theories for spline-based and modal finite element schemes were established and extensive numerical studies on both conventional (serial) and vector supercomputers were carried out. A parameter estimation scheme was tested using experimental data taken from the RPL structure, a laboratory experiment designed to test control algorithms for the large angle slewing of spacecraft with flexible appendages, and other projects involving the identification of flexible structures based upon experimental data were initiated.

The titles include: On the Dirichlet Neumann Boundary Control Problem Associated with Maxwell's Equations in a Cylindrical Region, A Floquet decomposition for Volterra Equations with Periodic Kernel and a transform Approach to Linear Recursion Equations, and Mathematical Models for the Elastic Beam and their Control Theoretic Implications, and Mathematical Models for the Elastic Beam with Frequency Proportional Damping.

We report here on several research projects funded by contract F49620-93-1-0153 have continued investigations in Bayesian analysis and empirical distribution approaches for parameter estimation problems. We are also developing computational methods for simulating subsurface transport and remediation strategies of interest at AFESC, Tyndall AFB and AL/OES Brooks AFB. (AN)

The reduced order model problem in distributed parameter systems adaptive identification and control is investigated. A comprehensive examination of real-time centralized adaptive control options for flexible spacecraft is provided.

The reduced order model problem in distributed parameter systems adaptive identification and control is investigated. A comprehensive examination of real-time centralized adaptive control options for flexible spacecraft is provided.

This instrumentation award funded the purchase of a laser vibrometer system, mass computer data storage and data acquisition equipment. This equipment used in conjunction with existing vibration testing and control facilities provides a sophisticated low frequency velocity measurement system for use in identifying the coefficients in partial differential equation models of distributed mass structures. In addition, the vibrometer system provides straightforward and direct velocity feedback for such systems. These flexible structures characteristically have very low natural frequencies which cannot be detected by accelerometers. This system has and is being used to perform tests on models and sub-assemblies of large space structures for the purpose of evaluating existing identification and control strategies as well as to stimulate new research in the area of control, observers (estimators) and identification. Several intense experiments using the laser vibrometer were performed to measure the response of a quasi isotropic cantilevered beam with a removable tip mass excited by an impulse at various locations. This data was collected, stored, and sent to AFOSR researchers at Brown University transmitted using BITNET. The data was analyzed using a spline based estimation procedure, starting with a partial differential equation model of the structure. A clear advantage over modal methods based on a finite dimensional model of the same system was observed. (JHD)

This instrumentation award funded the purchase of a laser vibrometer system, mass computer data storage and data acquisition equipment. This equipment used in conjunction with existing vibration testing and control facilities provides a sophisticated low frequency velocity measurement system for use in identifying the coefficients in partial differential equation models of distributed mass structures. In addition, the vibrometer system provides straightforward and direct velocity feedback for such systems. These flexible structures characteristically have very low natural frequencies which cannot be detected by accelerometers. This system has and is being used to perform tests on models and sub-assemblies of large space structures for the purpose of evaluating existing identification and control strategies as well as to stimulate new research in the area of control, observers (estimators) and identification. Several intense experiments using the laser vibrometer were performed to measure the response of a quasi isotropic cantilevered beam with a removable tip mass excited by an impulse at various locations. This data was collected, stored, and sent to AFOSR researchers at Brown University transmitted using BITNET. The data was analyzed using a spline based estimation procedure, starting with a partial differential equation model of the structure. A clear advantage over modal methods based on a finite dimensional model of the same system was observed. (JHD)

This paper presents an optimal control method for a class of distributed-parameter systems governed by first order, quasilinear hyperbolic partial differential equations that arise in many physical systems. Such systems are characterized by time delays since information is transported from one state to another by wave propagation. A general closed-loop hyperbolic transport model is controlled by a boundary control embedded in a periodic boundary condition. The boundary control is subject to a nonlinear differential equation constraint that models actuator dynamics of the system. The hyperbolic equation is thus coupled with the ordinary differential equation via the boundary condition. Optimality of this coupled system is investigated using variational principles to seek an adjoint formulation of the optimal control problem. The results are then applied to implement a model predictive control design for a wind tunnel to eliminate a transport delay effect that causes a poor Mach number regulation.

Bibliography: leaves 164-169

Bibliography: leaves 164-169

A brief overview and summary of results obtained in the development, analysis, and testing of approximation theory and techniques for the identification and control of distributed parameter systems is provided. The research carried out under this grant can be classified into seven sub-headings. These include (1) the identification of nonlinear distributed parameter systems, (2) the identification and control of thermoelastic systems, (3) the identification and control of degenerate distributed parameter systems, (4) discrete-time linear quadratic control of distributed parameter systems, (5) optimal LQG control of discrete time parabolic systems, (6) optimal fixed finite order compensators for infinite dimensional systems, and (7) convergence of Galerkin approximations to operator Riccati equations. Our results in each of these areas is described separately and in turn.

Large space structures exhibit distributed parameter behavior in their dynamics and thus must be described on infinite-dimensional state-spaces. However, the controller algorithm must be finite-dimensional to be implemented. The focus of this research has been to make finite-dimensional approximations of infinite-dimensional controllers which stabilize the distributed parameter system. The investigator has shown conditions under which Galerkin approximation schemes can yield finite-dimensional stabilizing controllers for linear distributed parameter systems.

The goal of this project was to put the intuitive idea of gain-scheduling on a rigorous foundation for a class of nonlinear, distributed-parameter systems. This involved a study of the existence and characterization of the ideal, infinite-dimensional, feedback control. Since in most applications the feedback function cannot be computed in closed form it was necessary to study the convergence of approximate feedback functions, based on increasingly higher order finite-dimensional approximations of the system, to the ideal function. Finally, the results were applied to Burgers' Equation, which can be viewed as a low-order approximation to a wide variety of physical phenomena, including viscous compressible flow.

This paper considers the distributed robust control problems of uncertain linear multi-agent systems with undirected communication topologies. It is assumed that the agents have identical nominal dynamics while subject to different norm-bounded parameter uncertainties, leading to weakly heterogeneous multi-agent systems. Distributed controllers are designed for both continuous- and discrete-time multi-agent systems, based on the relative states of neighboring agents and a subset of absolute states of the agents. It is shown for both the continuous- and discrete-time cases that the distributed robust control problems under such controllers in the sense of quadratic stability are equivalent to the $H_\infty$ control problems of a set of decoupled linear systems having the same dimensions as a single agent. A two-step algorithm is presented to construct the distributed controller for the continuous-time case, which does not involve any conservatism and meanwhile decouples the feedback gain design from the communication topology. Furthermore, a sufficient existence condition in terms of linear matrix inequalities is derived for the distributed discrete-time controller. Finally, the distributed robust $H_\infty$ control problems of uncertain linear multi-agent systems subject to external disturbances are discussed.

This paper considers the distributed robust control problems of uncertain linear multi-agent systems with undirected communication topologies. It is assumed that the agents have identical nominal dynamics while subject to different norm-bounded parameter uncertainties, leading to weakly heterogeneous multi-agent systems. Distributed controllers are designed for both continuous- and discrete-time multi-agent systems, based on the relative states of neighboring agents and a subset of absolute states of the agents. It is shown for both the continuous- and discrete-time cases that the distributed robust control problems under such controllers in the sense of quadratic stability are equivalent to the $H_\infty$ control problems of a set of decoupled linear systems having the same dimensions as a single agent. A two-step algorithm is presented to construct the distributed controller for the continuous-time case, which does not involve any conservatism and meanwhile decouples the feedback gain design from the communication topology. Furthermore, a sufficient existence condition in terms of linear matrix inequalities is derived for the distributed discrete-time controller. Finally, the distributed robust $H_\infty$ control problems of uncertain linear multi-agent systems subject to external disturbances are discussed.

This paper considers the distributed robust control problems of uncertain linear multi-agent systems with undirected communication topologies. It is assumed that the agents have identical nominal dynamics while subject to different norm-bounded parameter uncertainties, leading to weakly heterogeneous multi-agent systems. Distributed controllers are designed for both continuous- and discrete-time multi-agent systems, based on the relative states of neighboring agents and a subset of absolute states of the agents. It is shown for both the continuous- and discrete-time cases that the distributed robust control problems under such controllers in the sense of quadratic stability are equivalent to the $H_\infty$ control problems of a set of decoupled linear systems having the same dimensions as a single agent. A two-step algorithm is presented to construct the distributed controller for the continuous-time case, which does not involve any conservatism and meanwhile decouples the feedback gain design from the communication topology. Furthermore, a sufficient existence condition in terms of linear matrix inequalities is derived for the distributed discrete-time controller. Finally, the distributed robust $H_\infty$ control problems of uncertain linear multi-agent systems subject to external disturbances are discussed.

This research has developed a number of new results for the modeling and control of spatially continuous dynamic processes. The work includes developing numerical methods for implementing controller designs for those linear and nonlinear partial differential equations that describe the dynamics of fluid/structure systems and thermo-mechanical alloys.

A brief overview and summary of the research carried out in the area of approximation theory and computational methods for the identification and control of distributed parameter systems is provided. In particular, this final report details our efforts during the period 1 November 1990-31 October, 1993 on projects involving the adaptive control and estimation, ana on-line identification of distributed parameter systems (including a collaborative experimental-effort with Air Force personnel at Phillips Laboratory at Edwards Air Force Base), the identification and control of degenerate distributed parameter systems, Multi-grid methods for the solution of optimal LQR control problems for infinite dimensional systems, wavelet based approximation in the optimal LQ control of distributed parameter systems, the identification of nonlinear Volterra equations with application to materials with memory, the LQ control of linear and nonlinear distributed parameter systems with infinite spatial domain, and optimal control and estimation of thermoelastic systems with applications to thermo-acoustic refrigeration. Approximation control, Identification distributed parameter systems.

Research on theoretical ideas (well-posedness, feedback control, stability of inverse problems, form of damping in structures) and associated computational methods was performed. Topics include (1) fluid/structure interaction and flow control with applications to Navier-Stokes steady flow, Burgers equation, advection-diffusion, porous media flow, and acoustics/noise suppression (2) inverse problems for detection of semiconductor defects, electrical impedance tomography, damping and 'smart' material actuators in composite structures (3) linear, nonlinear and homogenization models for beams, plates, shells and PZT actuators and sensors.... Modeling, Fluid/structure interaction/flow control parameter estimation, Distributed parameter systems

A unified approximation framework for parameter estimation in general linear partial differential equators models has been completed. This framework has provided the theoretical basis for a number of identification problems on which these investigators have made significant progress. These include: (i) nondestructive evaluation techniques of composite materials using thermal probes. (ii) estimation of damping in composite material beams from vibration experiments. In connection with item (ii) it has been shown conclusively that an identification of damping mechanisms in the partial differential equation of a composite beam cannot be accomplished by the use of experimental model analysis. This is a major result in the theory of identifying damping mechanisms in flexible structures. The group has also studied questions related to the determination of irregularities (corrosion, cracks, delaminations, etc.) in composite materials using boundary observations of temperatures after known heat fluxes have been applied to the boundary. Successful efforts using experimental data with the theoretical and computational ideas developed by this group are reported. Substantial progress has been made on the development of a statistical framework (including hypothesis testing algorithms) to use in comparing the suitability of PDE models in least squares fits to data. A number of results on feedback stabilization of distributed parameter model have been obtained.

The present work is an attempt to put together the most relevant aspects of the engineering problems involving distributed parameter systems (D.P.S.'s). Simulation and optimal control are explained in detail in Chapters II and III. The original contribution of this thesis is given in Chapters V and VI, where modal control theory and a gradient subroutine that searches for the optimal reference coefficients are used. As a result, it was possible to obtain an output distribution better than the one achievable by the known methods. This technique works in situations of strongly nonlinear control and compensates the effect of having the analyzer and synthesizer approximated by low order matrices. It also makes it possible to give higher weight to some zones of the output distribution in order to have a better local fit. The necessary background for understanding Chapters V and VI is given in Chapter IV.

Finite dimensional approximation schemes that work well for distributed parameter systems are often not suitable for the analysis and implementation of feedback control systems. The relationship between approximation schemes for distributed parameter systems and their application to optimal control problems is discussed. A numerical example is given.

The efficient numerical treatment of optimal control problems governed by elliptic partial differential equations (PDEs) and systems of elliptic PDEs, where the control is finite dimensional is discussed. Distributed control as well as boundary control cases are discussed. The main characteristic of the new methods is that they are designed to solve the full optimization problem directly, rather than accelerating a descent method by an efficient multigrid solver for the equations involved. The methods use the adjoint state in order to achieve efficient smoother and a robust coarsening strategy. The main idea is the treatment of the control variables on appropriate scales, i.e., control variables that correspond to smooth functions are solved for on coarse grids depending on the smoothness of these functions. Solution of the control problems is achieved with the cost of solving the constraint equations about two to three times (by a multigrid solver). Numerical examples demonstrate the effectiveness of the method proposed in distributed control case, pointwise control and boundary control problems.

The model reference control of lumped linear systems and the model reference control of the distributed parameter system (DPS) are presented with their theory and Spacecraft Control Laboratory Experiment (SCOLE) applications.