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The characterization of polynomials whose zeros lie in certain algebraic domains (and the unification of the ideas of Hermite and Lyapunov) is the basis for developing finite algorithms for the solution of linear matrix equations. Particular attention is given to equations PA + A'P = Q (the Lyapunov equation) and P - A'PA = Q the (discrete Lyapunov equation). The Lyapunov equation appears in several areas of control theory such as stability theory, optimal control (evaluation of quadratic integrals), stochastic control (evaluation of covariance matrices) and in the solution of the algebraic Riccati equation using Newton's method.
Flight and wind tunnel tests were conducted and multidiscipline computer programs were developed as part of investigations of active control technology conducted at the NASA Langley Research Center. Unsteady aerodynamics approximation, optimal control theory, optimal controller design, and the Delta wing and DC-10 models are described. The drones for aerodynamics and structural testing (DAST program) for evaluating procedures for aerodynamic loads prediction and the design of active control systems on wings with significant aeroelastic effects is described as well as the DAST model used in the wind tunnel tests.
Low-altitude wind shear is recognized as an infrequent but significant hazard to all aircraft during take-off and landing. A total energy-rate sensor, which is potentially applicable to this problem, has been developed for measuring specific total energy-rate of an airplane with respect to the air mass. This paper presents control system designs, with and without energy-rate feedback, for the approach to landing of a transport airplane through severe wind shear and gusts to evaluate application of this sensor. A system model is developed which incorporates wind shear dynamics equations with the airplance equations of motion, thus allowing the control systems to be analyzed under various wind shears. The control systems are designed using optimal output feedback and are analyzed using frequency domain control theory techniques. Control system performance is evaluated using a complete nonlinear simulation of the airplane and a severe wind shear and gust data package. The analysis and simulation results indicate very similar stability and performance characteristics for the two designs. An implementation technique for distributing the velocity gains between airspeed and ground speed in the simulation is also presented, and this technique is shown to improve the performance characteristics of both designs.
The Airspace Concepts Evaluation System is a simulation of the National Airspace System. It includes models of flights, airports, airspaces, air traffic controls, traffic flow managements, and airline operation centers operating throughout the United States. It is used to predict system delays in response to future capacity and demand scenarios and perform benefits assessments of current and future airspace technologies and operational concepts. Facilitation of these studies requires that the simulation architecture supports plug and play of different air traffic control, traffic flow management, and airline operation center models and multi-fidelity modeling of flights, airports, and airspaces. The simulation is divided into two parts that are named, borrowing from classical control theory terminology, control and plant. The control consists of air traffic control, traffic flow management, and airline operation center models, and the plant consists of flight, airport, and airspace models. The plant can run open loop, in the absence of the control. However, undesired affects, such as conflicts and over congestions in the airspaces and airports, can occur. Different controls are applied, "plug and played", to the plant. A particular control is evaluated by analyzing how well it managed conflicts and congestions. Furthermore, the terminal area plants consist of models of airports and terminal airspaces. Each model consists of a set of nodes and links which are connected by the user to form a network. Nodes model runways, fixes, taxi intersections, gates, and/or other points of interest, and links model taxiways, departure paths, and arrival paths. Metering, flow distribution, and sequencing functions can be applied at nodes. Different fidelity model of how a flight transits are can be used by links. The fidelity of the model can be adjusted by the user by either changing the complexity of the node/link network-or the way that the link models how the flights transit from one node to the other.
Product Distribution (PD) theory is a new framework for controlling Multi-Agent Systems (MAS's). First we review one motivation of PD theory, as the information-theoretic extension of conventional full-rationality game theory to the case of bounded rational agents. In this extension the equilibrium of the game is the optimizer of a Lagrangian of the (probability distribution of) the joint stare of the agents. Accordingly we can consider a team game in which the shared utility is a performance measure of the behavior of the MAS. For such a scenario the game is at equilibrium - the Lagrangian is optimized - when the joint distribution of the agents optimizes the system's expected performance. One common way to find that equilibrium is to have each agent run a reinforcement learning algorithm. Here we investigate the alternative of exploiting PD theory to run gradient descent on the Lagrangian. We present computer experiments validating some of the predictions of PD theory for how best to do that gradient descent. We also demonstrate how PD theory can improve performance even when we are not allowed to rerun the MAS from different initial conditions, a requirement implicit in some previous work.
We look at time evolution of a physical system from the point of view of dynamical control theory. Normally we solve motion equation with a given external potential and we obtain time evolution. Standard examples are the trajectories in classical mechanics or the wave functions in Quantum Mechanics. In the control theory, we have the configurational variables of a physical system, we choose a velocity field and with a suited strategy we force the physical system to have a well defined evolution. The evolution of the system is the 'premium' that the controller receives if he has adopted the right strategy. The strategy is given by well suited laboratory devices. The control mechanisms are in many cases non linear; it is necessary, namely, a feedback mechanism to retain in time the selected evolution. Our aim is to introduce a scheme to obtain Quantum wave packets by control theory. The program is to choose the characteristics of a packet, that is, the equation of evolution for its centre and a controlled dispersion, and to give a building scheme from some initial state (for example a solution of stationary Schroedinger equation). It seems natural in this view to use stochastic approach to Quantum Mechanics, that is, Stochastic Mechanics [S.M.]. It is a quantization scheme different from ordinary ones only formally. This approach introduces in quantum theory the whole mathematical apparatus of stochastic control theory. Stochastic Mechanics, in our view, is more intuitive when we want to study all the classical-like problems. We apply our scheme to build two classes of quantum packets both derived generalizing some properties of coherent states.
The role a geospace plume in influencing the efficiency of magnetopause reconnection is an open question with two contrasting theories being debated. A local-control theory suggests that a plume decreases both local and global reconnection rates, whereas a global-control theory argues that the global reconnection rate is controlled by the solar wind rather than local physics. Observationally, limited numbers of point measurements from spacecraft cannot reveal whether a local change affects the global reconnection. A distributed observatory is hence needed to assess the validity of the two theories. We use THEMIS and Los Alamos National Laboratory spacecraft to identify the occurrence of a geospace plume and its contact with the magnetopause. Global evolution and morphology of the plume is traced using GPS measurements. SuperDARN is then used to monitor the distribution and the strength of dayside reconnection. Two storm-time geospace plume events are examined and show that as the plume contacts the magnetopause, the efficiency of reconnection decreases at the contact longitude. The amount of local decrease is 81% and 68% for the two events, and both values are consistent with the mass loading effect of the plume if the plume's atomic mass is ~4 amu. Reconnection in the surrounding is enhanced, and when the solar wind driving is stable, little variation is seen in the cross polar cap potential. This study illuminates a pathway to resolve the role of cold dense plasma on solar wind-magnetosphere coupling, and the observations suggest that plumes redistribute magnetopause reconnection activity without changing the global strength substantially.
It is shown that in constructing a theory for the most elementary class of control problems defined on spheres, some results from the Lie theory play a natural role. To understand controllability, optimal control, and certain properties of stochastic equations, Lie theoretic ideas are needed. The framework considered here is the most natural departure from the usual linear system/vector space problems which have dominated control systems literature. For this reason results are compared with those previously available for the finite dimensional vector space case.
Combined optimum control and estimation theory to determine noisy measurement optimal controls on randomly disturbed plant
The benefits of narrowband control theory for simple, open-loop stable systems are illustrated, and how the approach changes for magnetic bearing systems, which are open-loop unstable, is discussed. Magnetic bearing systems are good applications for narrowband control theory. Two sources of synchronous forces, the measurement error and the magnetic unbalance, are discussed. Both the measurement error and the magnetic unbalance manifest themselves as synchronous disturbances. It is shown that narrowband control theory for simple, open-loop stable systems provides excellent performance and good stability robustness. Because magnetic bearing systems are open-loop stable, the narrowband control approach becomes more complex. Disturbance accommodating control (DAC) theory is introduced as an effective approach to reduce vibrations in magnetic bearing systems. It is used to develop a control/estimation scheme that enables the rotor to spin about its center of mass in the presence of the measurement error disturbances.
A modern control theory design package (ORACLS) for constructing controllers and optimal filters for systems modeled by linear time-invariant differential or difference equations is described. Numerical linear-algebra procedures are used to implement the linear-quadratic-Gaussian (LQG) methodology of modern control theory. Algorithms are included for computing eigensystems of real matrices, the relative stability of a matrix, factored forms for nonnegative definite matrices, the solutions and least squares approximations to the solutions of certain linear matrix algebraic equations, the controllability properties of a linear time-invariant system, and the steady state covariance matrix of an open-loop stable system forced by white noise. Subroutines are provided for solving both the continuous and discrete optimal linear regulator problems with noise free measurements and the sampled-data optimal linear regulator problem. For measurement noise, duality theory and the optimal regulator algorithms are used to solve the continuous and discrete Kalman-Bucy filter problems. Subroutines are also included which give control laws causing the output of a system to track the output of a prescribed model.
The study described here was intended to explore some ways of giving the pilot the information necessary to perform as well as an autopilot. A fixed-base flight simulator was built to study pilot/director/aircraft performance. Instrument rated pilots used the different director-displays in an approach down to touchdown, including a flare and decrab maneuver. Two well established areas of control theory are combined in the design of the cockpit display. Optimal control theory and the theory of manual control are used to find the feedback gains required to drive the display symbols. Conclusions based on the simulator are presented. The results show that the director display developed in the work provides more than adequate information for simulated landing in highly turbulent conditions.
Alternative position-error feedback modes provided. Modern control theory basis for computer algorithm used to control two-axis positioning of large antenna. Algorithm - incorporated into software of real-time control computer - enables rapid intertarget positioning as well as precise tracking (using one of two optional position-feedback modes) without need of human operator intervention. Control system for one axis of two-axis azimuth/elevation control system embodied mostly in software based on advanced control theory. System has linear properties of classical linear feedback controller. Performance described by bandwidth and linear error coefficients.
This paper describes the implementation of optimization techniques based on control theory for wing and wing-body design of supersonic configurations. The work represents an extension of our earlier research in which control theory is used to devise a design procedure that significantly reduces the computational cost by employing an adjoint equation. In previous studies it was shown that control theory could be used to~eviseransonic design methods for airfoils and wings in which the shape and the surrounding body-fitted mesh are both generated analytically, and the control is the mapping function. The method has also been implemented for both transonic potential flows and transonic flows governed by the Euler equations using an alternative formulation which employs numerically generated grids, so that it can treat more general configurations. Here results are presented for three-dimensional design cases subject to supersonic flows governed by the Euler equation.
Most of the time on this project was spent on the trajectory planning problem. The construction is equivalent to the classical spline construction in the case that the system matrix is nilpotent. If the dimension of the system is n then the spline of degree 2n-1 is constructed. This gives a new approach to the construction of splines that is more efficient than the usual construction and at the same time allows the construction of a much larger class of splines. All known classes of splines are reconstructed using the approach of linear control theory. As a numerical analysis tool control theory gives a very good tool for constructing splines. However, for the purposes of trajectory planning it is quite another story. Enclosed in this document are four reports done under this grant.
This work describes the implementation of optimization techniques based on control theory for complex aircraft configurations. Here control theory is employed to derive the adjoint differential equations, the solution of which allows for a drastic reduction in computational costs over previous design methods (13, 12, 43, 38). In our earlier studies (19, 20, 22, 23, 39, 25, 40, 41, 42) it was shown that this method could be used to devise effective optimization procedures for airfoils, wings and wing-bodies subject to either analytic or arbitrary meshes. Design formulations for both potential flows and flows governed by the Euler equations have been demonstrated, showing that such methods can be devised for various governing equations (39, 25). In our most recent works (40, 42) the method was extended to treat wing-body configurations with a large number of mesh points, verifying that significant computational savings can be gained for practical design problems. In this paper the method is extended for the Euler equations to treat complete aircraft configurations via a new multiblock implementation. New elements include a multiblock-multigrid flow solver, a multiblock-multigrid adjoint solver, and a multiblock mesh perturbation scheme. Two design examples are presented in which the new method is used for the wing redesign of a transonic business jet.
An aerodynamic shape optimization method that treats the design of complex aircraft configurations subject to high fidelity computational fluid dynamics (CFD), geometric constraints and multiple design points is described. The design process will be greatly accelerated through the use of both control theory and distributed memory computer architectures. Control theory is employed to derive the adjoint differential equations whose solution allows for the evaluation of design gradient information at a fraction of the computational cost required by previous design methods. The resulting problem is implemented on parallel distributed memory architectures using a domain decomposition approach, an optimized communication schedule, and the MPI (Message Passing Interface) standard for portability and efficiency. The final result achieves very rapid aerodynamic design based on a higher order CFD method. In order to facilitate the integration of these high fidelity CFD approaches into future multi-disciplinary optimization (NW) applications, new methods must be developed which are capable of simultaneously addressing complex geometries, multiple objective functions, and geometric design constraints. In our earlier studies, we coupled the adjoint based design formulations with unconstrained optimization algorithms and showed that the approach was effective for the aerodynamic design of airfoils, wings, wing-bodies, and complex aircraft configurations. In many of the results presented in these earlier works, geometric constraints were satisfied either by a projection into feasible space or by posing the design space parameterization such that it automatically satisfied constraints. Furthermore, with the exception of reference 9 where the second author initially explored the use of multipoint design in conjunction with adjoint formulations, our earlier works have focused on single point design efforts. Here we demonstrate that the same methodology may be extended to treat complete configuration designs subject to multiple design points and geometric constraints. Examples are presented for both transonic and supersonic configurations ranging from wing alone designs to complex configuration designs involving wing, fuselage, nacelles and pylons.