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Numerical integration methods for the solution of initial value problems for ordinary vector differential equations may be modelled as discrete time feedback systems. The stability criteria discovered in modern control theory are applied to these systems and criteria involving the routine, the step size and the differential equation are derived. Linear multistep, Runge-Kutta, and predictor-corrector methods are all investigated.
The techniques of modern optimal control theory were applied to the design of a control system for a supersonic inlet. The inlet control problem was approached as a linear stochastic optimal control problem using as the performance index the expected frequency of unstarts. The details of the formulation of the stochastic inlet control problem are presented. The computational procedures required to obtain optimal controller designs are discussed, and the analytically predicted performance of controllers designed for several different inlet conditions is tabulated. The experimental implementation of the optimal control laws is described, and the experimental results obtained in a supersonic wind tunnel are presented. The control laws were implemented with analog and digital computers. Comparisons are made between the experimental and analytically predicted performance results. Comparisons are also made between the results obtained with continuous analog computer controllers and discrete digital computer versions.
Numerical integration methods for the solution of initial value problems for ordinary vector differential equations may be modelled as discrete time feedback systems. The stability criteria discovered in modern control theory are applied to these systems and criteria involving the routine, the step size and the differential equation are derived. Linear multistep, Runge-Kutta, and predictor-corrector methods are all investigated.
Modern control theory relies upon the fidelity of the mathematical model of the system. Truncated modes, external disturbances, and parameter errors in linear system models are corrected by augmenting to the original system of equations an 'error system' which is designed to approximate the effects of such model errors. A Chebyshev error system is developed for application to the Large Space Telescope (LST).
A decentralized design procedure which combines substructural synthesis, model reduction, decentralized controller design, subcontroller synthesis, and controller reduction is proposed for the control design of flexible structures. The structure to be controlled is decomposed into several substructures, which are modeled by component mode synthesis methods. For each substructure, a subcontroller is designed by using the linear quadratic optimal control theory. Then, a controller synthesis scheme called Substructural Controller Synthesis (SCS) is used to assemble the subcontrollers into a system controller, which is to be used to control the whole structure.
A decentralized design procedure is proposed for the control design for flexible structures. The structure to be controlled is decomposed into several substructures by using a natural decomposition called substructuring decomposition. For each substructure, a subcontroller is designed by using the linear quadratic optimal control theory. Then, a controller synthesis scheme called Substructural Controller Synthesis (SCS) is used to assemble the subcontrollers into a system controller, which is to be used to control the whole structure. A plane truss example is used to illustrate the efficacy of the proposed method.
A procedure for optimizing the performance of large flexible spacecraft that require active vibration suppression to achieve required performance is presented. The procedure is to conduct on-orbit testing and system identification followed by a control system design. It is applied via simulation to a spacecraft configuration currently being considered for flight test by NASA - the Controls, Astrophysics, and Structures Experiment in Space (CASES). The system simulator is based on a NASTRAN finite element structural model. A finite number of modes is used to represent the structural dynamics. The system simulator also includes models of the electronics, actuators, sensors, the digital controller, and the internal and external disturbances. Nonlinearities caused by quantization are included in the study to examine tolerance of the procedure to modelling errors. Disturbance and sensor noise is modelled as a Gaussian process. For system identification, the system is excited using sinusoidal inputs at the resonant frequencies of the structure using each actuator. Mode shapes, frequencies, and damping ratios are identified from the unforced response sensor data after each excitation. Then, the excitation data is used to identify the actuator influence coefficients. The results of the individual parameter identification analyses are assembled into an aggregate system model. The control design is accomplished based only on the identified model using multi-input/output linear quadratic Gaussian theory. Its performance is evaluated based on time-to-damp as compared with the uncontrolled structure.
Nonlinear observers were originally developed based on the ideas of variable structure control, and for the purpose of detecting disturbances in complex systems. In this anomaly detection application, these observers were designed for estimating the distributed state of fluid flow in a pipe described by a class of advection equations. The observer algorithm uses collected data in a piping system to estimate the distributed system state (pressure and velocity along a pipe containing liquid gas propellant flow) using only boundary measurements. These estimates are then used to further estimate and localize possible anomalies such as leaks or foreign objects, and instrumentation metering problems such as incorrect flow meter orifice plate size. The observer algorithm has the following parts: a mathematical model of the fluid flow, observer control algorithm, and an anomaly identification algorithm. The main functional operation of the algorithm is in creating the sliding mode in the observer system implemented as software. Once the sliding mode starts in the system, the equivalent value of the discontinuous function in sliding mode can be obtained by filtering out the high-frequency chattering component. In control theory, "observers" are dynamic algorithms for the online estimation of the current state of a dynamic system by measurements of an output of the system. Classical linear observers can provide optimal estimates of a system state in case of uncertainty modeled by white noise. For nonlinear cases, the theory of nonlinear observers has been developed and its success is mainly due to the sliding mode approach. Using the mathematical theory of variable structure systems with sliding modes, the observer algorithm is designed in such a way that it steers the output of the model to the output of the system obtained via a variety of sensors, in spite of possible mismatches between the assumed model and actual system. The unique properties of sliding mode control allow not only control of the model internal states to the states of the real-life system, but also identification of the disturbance or anomaly that may occur.
The problem of optimal rejection of bounded persistent disturbances is solved in the case of linear discrete-time period systems. The solution consists of solving an equivalent time-invariant standard l1 optimization problem subject to an additional constraint. This constraint ensures the causality of the resulting periodic controller. By the duality theory, the problem is shown to be equivalent to a linear programming problem, which is no harder than the standard l1 problem.
The control of an airbreathing turbojet engine by an onboard digital computer is studied. The approach taken is to model the turbojet engine as a linear, multivariable system whose parameters vary with engine operating environment. From this model adaptive closed-loop or feedback control laws are designed and applied to the acceleration of the turbojet engine.
Floquet theory is applied to the problem of designing a control system for a satellite in an unstable periodic orbit. Expansion about a periodic orbit produces a time-periodic linear system, which is augmented by a time-periodic control term. It is shown that this can be done such that (1) the application of control produces only inertial accelerations, (2) positive real Poincareexponents are shifted into the left half-plane, and (3) the shift of the exponent is linear with control gain. These developments are applied to an unstable orbit near the earth-moon L(3) point pertubed by the sun. Finally, it is shown that the control theory can be extended to include first order perturbations about the periodic orbit without increase in control cost.
Global warming is posed to modify the modes of variability that control much of the climate predictability at seasonal to interannual scales. The quantification of changes in climate predictability over any given amount of time, however, remains challenging. Here we build upon recent advances in non-linear dynamical systems theory and introduce the climate community to an information entropy quantifier based on recurrence. The entropy, or complexity of a system is associated with microstates that recur over time in the time-series that define the system, and therefore to its predictability potential. A computationally fast method to evaluate the entropy is applied to the investigation of the information entropy of sea surface temperature in the tropical Pacific and Indian Oceans, focusing on boreal fall. In this season the predictability of the basins is controlled by two regularly varying non-linear oscillations, the El Niño-Southern Oscillation and the Indian Ocean Dipole. We compute and compare the entropy in simulations from the CMIP5 catalog from the historical period and RCP8.5 scenario, and in reanalysis datasets. Discrepancies are found between the models and the reanalysis, and no robust changes in predictability can be identified in future projections. The Indian Ocean and the equatorial Pacific emerge as troublesome areas where the modeled entropy differs the most from that of the reanalysis in many models. A brief investigation of the source of the bias points to a poor representation of the ocean mean state and basins' connectivity at the Indonesian Throughflow.
Stability and control characteristics of a high-performance aircraft have been examined over a wide range of maneuvering flight conditions, in order to identify general rules for the design of departure-preventing control systems. This has been accomplished using fully coupled linear dynamic models which account for nonzero mean values of aerodynamic angles and angular rates. Stability augmentation systems derived from optimal control theory are shown to maintain stable, well-damped aircraft dynamics over a wide range of maneuvering flight conditions. This design approach generates cross-feeds and control interconnects (as well as conventional feedback terms) for improved aircraft stability.
An important feature in General Aviation (GA) autopilots is the ability to intercept and hold the glideslope and localizer during a landing approach. Most General Aviation landing approach autopilots available today use analog systems. This paper presents the designs of digital landing approach autopilots for a General Aviation (NAVION) aircraft using modern linear quadratic control theory. Each constant gain, direct digitally designed autopilot operates synchronously at a slow rate (10 samples per second) and has modest memory requirements, i.e., a full state Kalman filter is not used. The autopilot is designed to track desired position trajectories constructed in an ILS or MLS terminal area using command generators. Alternative command errors for intercepting and holding the localizer beam centerline are investigated. Linear simulations and flight test results are presented demonstrating the successful application of the linear quadratic regulator approach.
Phase 3 of Project Wish saw the evolution of the Emerald City (E-City) from a collection of specialized independent analyses and ideas to a working structural design integrated with major support systems and analyses. Emphasis was placed on comparing and contrasting the closed and open cycle gas core nuclear rocket engines to further determine the optimum propulsive system for the C-City. Power and thermal control requirements were then defined and the question of how to meet these requirements was addressed. Software was developed to automate the mission/system/configuration analysis so changes dictated by various subsystems constraints could be managed efficiently and analyzed interactively. In addition, the liquid hydrogen propellant tank was statically designed for minimum mass and shape optimization using a finite element modeling package called SDRC I-DEAS while spoke and shaft cross-sectional areas were optimized on ASTROS (Automated Structural Optimization System). A structural dynamic analysis also conducted using ASTROS enabled a study of the displacements, accelerations, modes and frequencies of the C-City. Finally, the attitude control system design began with an initial mass moment of inertia analysis and was then designed and optimized using linear quadratic regulator control theory.
Phase 3 of Project WISH saw the evolution of the Emerald City (E-City) from a collection of specialized independent analyses and ideas to a working structural design integrated with major support systems and analyses. Emphasis was placed on comparing and contrasting the closed and open cycle gas core nuclear rocket engines to further determine the optimum propulsive system for the E-City. Power and thermal control requirements were then defined and the question of how to meet these requirements was addressed. Software was developed to automate the mission/system/configuration analysis so changes dictated by various subsystem constraints could be managed efficiently and analyzed interactively. In addition, the liquid hydrogen propellant tank was statically designed for minimum mass and shape optimization using a finite element modeling package called SDRC I-DEAS. Spoke and shaft cross-sectional areas were optimized on ASTROS (Automated Structural Optimization System) for mass minimization. A structural dynamic analysis of the optimal structure also conducted using ASTROS enabled a study of the modes, frequencies, displacements, and accelerations of the E-City. Finally, the attitude control system design began with an initial mass moment of inertia analysis and was then designed and optimized using linear quadratic regulator control theory.
This paper deals with the application of linear optimal control and filtering theory to control a spinning spacecraft with movable telescoping appendages. The equations of motion are linearized about the desired final state. A feedback control system is designed to maintain this final state, with plant noise and measurement noise present in the system. Analytic results are obtained for special cases and numerical results are presented for the general case.
Developments in the area of nonlinear control theory have shown how coordinate changes in the state and input spaces of a dynamical system can be used to transform certain nonlinear differential equations into equivalent linear equations. These techniques are applied to the control of a spacecraft equipped with momentum exchange devices. An optimal control problem is formulated that incorporates a nonlinear spacecraft model. An algorithm is developed for solving the optimization problem using feedback linearization to transform to an equivalent problem involving a linear dynamical constraint and a functional approximation technique to solve for the linear dynamics in terms of the control. The original problem is transformed into an unconstrained nonlinear quadratic program that yields an approximate solution to the original problem. Two examples are presented to illustrate the results.