Finite-dimensional Optimal Control of Poiseuille Flow
In this paper, we consider linear stabilization of plane, Poiseuille flow using linear quadratic Gaussian optimal control theory.
Poiseuille↗
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Unified Theory for Aircraft Handling Qualities and Adverse Aircraft-Pilot Coupling
A unified theory for aircraft handling qualities and adverse aircraft-pilot coupling or pilot-induced oscillations is introduced. The theory is based on a structural model of the human pilot. A methodology is presented for the prediction of (1) handling qualities levels; (2) pilot-induced oscillation rating levels; and (3) a frequency range in which pilot-induced oscillations are likely to occur. Although the dynamics of the force-feel system of the cockpit inceptor is included, the methodology will not account for effects attributable to control sensitivity and is limited to single-axis tasks and, at present, to linear vehicle models. The theory is derived from the feedback topology of the structural model and an examination of flight test results for 32 aircraft configurations simulated by the U.S. Air Force/CALSPAN NT-33A and Total In-Flight Simulator variable stability aircraft. An extension to nonlinear vehicle dynamics such as that encountered with actuator saturation is discussed.
Hess, R. A.↗
The dynamics and control of large flexible space structures, 3. Part A: Shape and orientation control of a platform in orbit using point actuators
The dynamics, attitude, and shape control of a large thin flexible square platform in orbit are studied. Attitude and shape control are assumed to result from actuators placed perpendicular to the main surface and one edge and their effect on the rigid body and elastic modes is modelled to first order. The equations of motion are linearized about three different nominal orientations: (1) the platform following the local vertical with its major surface perpendicular to the orbital plane; (2) the platform following the local horizontal with its major surface normal to the local vertical; and (3) the platform following the local vertical with its major surface perpendicular to the orbit normal. The stability of the uncontrolled system is investigated analytically. Once controllability is established for a set of actuator locations, control law development is based on decoupling, pole placement, and linear optimal control theory. Frequencies and elastic modal shape functions are obtained using a finite element computer algorithm, two different approximate analytical methods, and the results of the three methods compared.
Bainum, P. M.↗
Control of a large flexible platform in orbit
The dynamics and attitude and shape control of a large thin flexible platform in orbit are studied. Attitude and shape control is assumed to result from actuators placed perpendicular to the main surface and one edge and their effect on the rigid body and elastic modes is modelled to first order. The equations of motion are linearized about nominal orientations where the undeformed plate follows either the local vertical or local horizontal. The stability of the uncontrolled system is investigated analytically. Once controllability is established for a set of actuator locations, control law development is based on pole placement, decoupling, and linear optimal control theory.
Reddy, A. S. S. R.↗
An algorithm for the weighting matrices in the sampled-data optimal linear regulator problem
The sampled-data optimal linear regulator problem provides a means whereby a control designer can use an understanding of continuous optimal regulator design to produce a digital state variable feedback control law which satisfies continuous system performance specifications. A basic difficulty in applying the sampled-data regulator theory is the requirement that certain digital performance index weighting matrices, expressed as complicated functions of system matrices, be computed. Infinite series representations are presented for the weighting matrices of the time-invariant version of the optimal linear sampled-data regulator problem. Error bounds are given for estimating the effect of truncating the series expressions after a finite number of terms, and a method is described for their computer implementation. A numerical example is given to illustrate the results.
Armstrong, E. S.↗
Optimal discrete-time LQR problems for parabolic systems with unbounded input: Approximation and convergence
An abstract approximation and convergence theory for the closed-loop solution of discrete-time linear-quadratic regulator problems for parabolic systems with unbounded input is developed. Under relatively mild stabilizability and detectability assumptions, functional analytic, operator techniques are used to demonstrate the norm convergence of Galerkin-based approximations to the optimal feedback control gains. The application of the general theory to a class of abstract boundary control systems is considered. Two examples, one involving the Neumann boundary control of a one-dimensional heat equation, and the other, the vibration control of a cantilevered viscoelastic beam via shear input at the free end, are discussed.
Rosen, I. G.↗
The dynamics and control of the orbiting spacecraft control laboratory experiment (SCOLE) during station keeping
A mathematical model is developed to predict the dynamics of the proposed orbiting Spacecraft Control Laboratory Experiment during the station keeping phase. The Shuttle as well as the reflector are assumed to be rigid, the mast is flexible and is assumed to undergo elastic displacements very small as compared with its length. The equations of motion are derived using a Newton-Euler formulation. The model includes the effects of gravity, flexibility, and orbital dynamics. The control is assumed to be provided to the system through the Shuttle's three torquers, and through six actuators located by pairs at two points on the mast and at the mass center of the reflector. At each of the locations, an actuator acts parallel to the roll axis while the other one acts parallel to the pitch axis. It is seen that, in the presence of gravity-gradient torques in the system dynamics, the system assumes a new equilibrium position about which the equations must be linearized, primarily due to the offset in the mast attachment point to the reflector. The linear regulator theory is used to derive control laws for the linear model of the SCOLE including the first four flexible modes. Numerical results confirm the robustness of this control strategy for station keeping with maximum control efforts significantly below saturation levels.
Bainum, Peter M.↗
Project Management Using Modern Guidance, Navigation and Control Theory
The idea of control theory and its application to project management is not new, however literature on the topic and real-world applications is not as readily available and comprehensive in how all the principals of Guidance, Navigation and Control (GN&C) apply. This paper will address how the fundamental principals of modern GN&C Theory have been applied to NASA's Constellation Space Suit project and the results in the ability to manage the project within cost, schedule and budget. A s with physical systems, projects can be modeled and managed with the same guiding principles of GN&C as if it were a complex vehicle, system or software with time-varying processes, at times non-linear responses, multiple data inputs of varying accuracy and a range of operating points. With such systems the classic approach could be applied to small and well-defined projects; however with larger, multi-year projects involving multiple organizational structures, external influences and a multitude of diverse resources, then modern control theory is required to model and control the project. The fundamental principals of G N&C stated that a system is comprised of these basic core concepts: State, Behavior, Control system, Navigation system, Guidance and Planning Logic, Feedback systems. The state of a system is a definition of the aspects of the dynamics of the system that can change, such as position, velocity, acceleration, coordinate-based attitude, temperature, etc. The behavior of the system is more of what changes are possible rather than what can change, which is captured in the state of the system. The behavior of a system is captured in the system modeling and if properly done, will aid in accurate system performance prediction in the future. The Control system understands the state and behavior of the system and feedback systems to adjust the control inputs into the system. The Navigation system takes the multiple data inputs and based upon a priori knowledge of the input, will develop a statistical-based weighting of the input to determine where the system currently is located. Guidance and Planning logic of the system with the understanding of where it is (provided by the navigation system) will in turn determine where it needs to be and how to get there. Lastly, the system Feedback system is the right arm of the control system to allow it to affect change in the overall system and therefore it is critical to not only correctly identify the system feedback inputs but also the system response to the feedback inputs. And with any systems project it is critical that the objective of the system be clearly defined for not only planning but to be used to measure performance and to aid in the guidance of the system or project.
Hill, Terry↗
Optimal Output Trajectory Redesign for Invertible Systems
Given a desired output trajectory, inversion-based techniques find input-state trajectories required to exactly track the output. These inversion-based techniques have been successfully applied to the endpoint tracking control of multijoint flexible manipulators and to aircraft control. The specified output trajectory uniquely determines the required input and state trajectories that are found through inversion. These input-state trajectories exactly track the desired output; however, they might not meet acceptable performance requirements. For example, during slewing maneuvers of flexible structures, the structural deformations, which depend on the required state trajectories, may be unacceptably large. Further, the required inputs might cause actuator saturation during an exact tracking maneuver, for example, in the flight control of conventional takeoff and landing aircraft. In such situations, a compromise is desired between the tracking requirement and other goals such as reduction of internal vibrations and prevention of actuator saturation; the desired output trajectory needs to redesigned. Here, we pose the trajectory redesign problem as an optimization of a general quadratic cost function and solve it in the context of linear systems. The solution is obtained as an off-line prefilter of the desired output trajectory. An advantage of our technique is that the prefilter is independent of the particular trajectory. The prefilter can therefore be precomputed, which is a major advantage over other optimization approaches. Previous works have addressed the issue of preshaping inputs to minimize residual and in-maneuver vibrations for flexible structures; Since the command preshaping is computed off-line. Further minimization of optimal quadratic cost functions has also been previously use to preshape command inputs for disturbance rejection. All of these approaches are applicable when the inputs to the system are known a priori. Typically, outputs (not inputs) are specified in tracking problems, and hence the input trajectories have to be computed. The inputs to the system are however, difficult to determine for non-minimum phase systems like flexible structures. One approach to solve this problem is to (1) choose a tracking controller (the desired output trajectory is now an input to the closed-loop system and (2) redesign this input to the closed-loop system. Thus we effectively perform output redesign. These redesigns are however, dependent on the choice of the tracking controllers. Thus the controller optimization and trajectory redesign problems become coupled; this coupled optimization is still an open problem. In contrast, we decouple the trajectory redesign problem from the choice of feedback-based tracking controller. It is noted that our approach remains valid when a particular tracking controller is chosen. In addition, the formulation of our problem not only allows for the minimization of residual vibration as in available techniques but also allows for the optimal reduction fo vibrations during the maneuver, e.g., the altitude control of flexible spacecraft. We begin by formulating the optimal output trajectory redesign problem and then solve it in the context of general linear systems. This theory is then applied to an example flexible structure, and simulation results are provided.
Devasia, S.↗
Unified control/structure design and modeling research
To demonstrate the applicability of the control theory for distributed systems to large flexible space structures, research was focused on a model of a space antenna which consists of a rigid hub, flexible ribs, and a mesh reflecting surface. The space antenna model used is discussed along with the finite element approximation of the distributed model. The basic control problem is to design an optimal or near-optimal compensator to suppress the linear vibrations and rigid-body displacements of the structure. The application of an infinite dimensional Linear Quadratic Gaussian (LQG) control theory to flexible structure is discussed. Two basic approaches for robustness enhancement were investigated: loop transfer recovery and sensitivity optimization. A third approach synthesized from elements of these two basic approaches is currently under development. The control driven finite element approximation of flexible structures is discussed. Three sets of finite element basic vectors for computing functional control gains are compared. The possibility of constructing a finite element scheme to approximate the infinite dimensional Hamiltonian system directly, instead of indirectly is discussed.
Mingori, D. L.↗
Nonlinear control of aircraft
Transformations of nonlinear systems were used to design automatic flight controllers for vertical and short takeoff aircraft. Under the assumption that a nonlinear system can be mapped to a controllable linear system, a method using partial differential equations was constructed to approximate transformations in cases where exact ones cannot be found. An application of the design theory to a rotorcraft, the UH-1H helicopter, was presented.
Hunt, L. R.↗
An application of the quadratic penalty function criterion to the determination of a linear control for a flexible vehicle.
Optimal linear control theory to develop quadratic cost-criterion control law for flexible rocket vehicle
OPTIMAL CONTROL↗
A study of attitude control concepts for precision-pointing non-rigid spacecraft
Attitude control concepts for use onboard structurally nonrigid spacecraft that must be pointed with great precision are examined. The task of determining the eigenproperties of a system of linear time-invariant equations (in terms of hybrid coordinates) representing the attitude motion of a flexible spacecraft is discussed. Literal characteristics are developed for the associated eigenvalues and eigenvectors of the system. A method is presented for determining the poles and zeros of the transfer function describing the attitude dynamics of a flexible spacecraft characterized by hybrid coordinate equations. Alterations are made to linear regulator and observer theory to accommodate modeling errors. The results show that a model error vector, which evolves from an error system, can be added to a reduced system model, estimated by an observer, and used by the control law to render the system less sensitive to uncertain magnitudes and phase relations of truncated modes and external disturbance effects. A hybrid coordinate formulation using the provided assumed mode shapes, rather than incorporating the usual finite element approach is provided.
Likins, P. W.↗
On the Stability of Jump-Linear Systems Driven by Finite-State Machines with Markovian Inputs
This paper presents two mean-square stability tests for a jump-linear system driven by a finite-state machine with a first-order Markovian input process. The first test is based on conventional Markov jump-linear theory and avoids the use of any higher-order statistics. The second test is developed directly using the higher-order statistics of the machine s output process. The two approaches are illustrated with a simple model for a recoverable computer control system.
Patilkulkarni, Sudarshan↗
Flight control application of new stability robustness bounds for linear uncertain systems
This paper addresses the issue of obtaining bounds on the real parameter perturbations of a linear state-space model for robust stability. Based on Kronecker algebra, new, easily computable sufficient bounds are derived that are much less conservative than the existing bounds since the technique is meant for only real parameter perturbations (in contrast to specializing complex variation case to real parameter case). The proposed theory is illustrated with application to several flight control examples.
Yedavalli, Rama K.↗
A synthesis theory for self-oscillating adaptive systems /SOAS/
A quantitative synthesis theory is presented for the Self-Oscillating Adaptive System (SOAS), whose nonlinear element has a static, odd character with hard saturation. The synthesis theory is based upon the quasilinear properties of the SOAS to forced inputs, which permits the extension of quantitative linear feedback theory to the SOAS. A reasonable definition of optimum design is shown to be the minimization of the limit cycle frequency. The great advantages of the SOAS is its zero sensitivity to pure gain changes. However, quasilinearity and control of the limit cycle amplitude at the system output, impose additional constraints which partially or completely cancel this advantage, depending on the numerical values of the design parameters. By means of narrow-band filtering, an additional factor is introduced which permits trade-off between filter complexity and limit cycle frequency minimization.
Horowitz, I.↗
Multivariable control theory applied to hierarchial attitude control for planetary spacecraft
Multivariable control theory is applied to the design of a hierarchial attitude control system for the CARD space vehicle. The system selected uses reaction control jets (RCJ) and control moment gyros (CMG). The RCJ system uses linear signal mixing and a no-fire region similar to that used on the Skylab program; the y-axis and z-axis systems which are coupled use a sum and difference feedback scheme. The CMG system uses the optimum steering law and the same feedback signals as the RCJ system. When both systems are active the design is such that the torques from each system are never in opposition. A state-space analysis was made of the CMG system to determine the general structure of the input matrices (steering law) and feedback matrices that will decouple the axes. It is shown that the optimum steering law and proportional-plus-rate feedback are special cases. A derivation of the disturbing torques on the space vehicle due to the motion of the on-board television camera is presented. A procedure for computing an upper bound on these torques (given the system parameters) is included.
Boland, J. S., III↗
Design of a digital ride quality augmentation system for a commuter aircraft
Commuter aircraft with low wing loading that operate at low altitudes are particularly susceptible to unwanted accelerations caused by atmospheric gusts. This paper describes the design and analysis of a longitudinal digital Ride Quality Augmentation System (RQAS). The RQAS designs were conducted for a Cessna 402B aircraft using the flaps and the elevator as the control surfaces. The designs are generated using linear quadratic Gaussian theory and analyzed in both the time and frequency domains. Nominal designs are presented at five flight conditions that cover a total mission. Trade-off studies are conducted to investigate the effect of sample time, computational delay time, servo bandwidth and control power.
Hammond, T. A.↗