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Nonlinear perturbation theory for optimal estimation and control of time-discrete, stochastic, dynamic systems when measurement data is noisy
A new implicit algorithm for time step integration of finite element structural dynamic equations is presented. Convergence, stability and numerical damping properties are discussed. Due to the way nonlinear structural behavior is taken into account, the algorithm is expected to compare favorably with existing ones. Some simple numerical results are presented. A related explicit algorithm is also derived and shortly discussed.
Despite the apparent regularity of planetary motions, the dynamics of the solar system is in reality extremely nonlinear and replete with chaos. The difficulty in detecting this chaos is the long time scale of this dynamics as compared with human observations.
This paper describes a technique which may be used to design the flight control system for a highly maneuverable aircraft. The control system was provided to stabilize the dynamics of the aircraft model and allow it to fly simulated, poststall 'supermaneuvers'. Although the aircraft dynamics are highly nonlinear under these conditions, the gain-scheduled, flight control system was designed using basically linear techniques. A manuever generator was implemented to pilot the mathematical model through prescribed optimal trajectories. The control system design performed well while executing maneuvers involving small angular rates where the governing dynamics could be considered linear. However, the performance deteriorated once the model was subjected to high angular rates at high angle of attack.
Multistage model predictive control (MPC) provides a robust control strategy for dynamic systems with uncertainties and a setpoint tracking objective. Moreover, extending MPC to minimize an economic cost instead of tracking a pre-calculated optimal setpoint improves controller performance. This paper presents a novel multistage economic nonlinear model predictive control (E-NMPC) framework for dynamic systems operating under uncertainty, with specific application to natural gas transmission networks. A key innovation lies in the integration of cyclic steady-state (CSS) constraints within the multistage MPC formulation, enabling the controller to manage periodic operating conditions commonly observed in energy systems. A Lyapunov-based descent condition is enforced to ensure robust stability of the controller. The multistage economic MPC framework is validated on two gas pipeline case studies, where it successfully minimizes net energy consumption, respects operational constraints under uncertain demand profiles, and guides the network to optimal cyclic operation. The Lyapunov function remains bounded in both case studies, validating the robust stability of multistage E-NMPC.
Electromagnetic waves propagating through vacuum can polarize virtual electron–positron pairs; this polarization, in turn, nonlinearly modifies their propagation. A semi-classical nonlinear wave equation describing the propagation is derived from the Euler–Heisenberg Lagrangian density, which captures vacuum polarization effects up to the one-loop level. In this article, we present a reduced-actionintegral approach that enables rapid modeling of nonlinear phenomena arising from the Euler– Heisenberg Lagrangian. Application of the variational principle to the reduced action provides equations of motion for familiar light-pulse parameters, such as spot size, phase, polarization, and phase-front curvature, without requiring full-field simulations. Three examples demonstrate the utility of the approach: phase modulation, birefringence, and frequency mixing.
This short paper considers the parameter-identification problem of general discrete-time, nonlinear, multiple input-multiple output dynamic systems with Gaussian white distributed measurement errors. Knowledge of the system parameterization is assumed to be available. Regions of constrained maximum likelihood (CML) parameter identifiability are established. A computation procedure employing interval arithmetic is proposed for finding explicit regions of parameter identifiability for the case of linear systems.
A nonlinear, six degree of freedom, digital computer simulation of a vehicle which has constant mass properties and whose attitudes are controlled by both aerodynamic surfaces and reaction control system thrusters was developed. A rotating, oblate Earth model was used to describe the gravitational forces which affect long duration Earth entry trajectories. The program is executed in a nonreal time mode or connected to a simulation cockpit to conduct piloted and autopilot studies. The program guidance and control software used by the space shuttle orbiter for its descent from approximately 121.9 km to touchdown on the runway.
Consider some of the control system design methods for plants with nonlinear dynamics. If the nonlinearity is weak relative to the size of the operating region, then the linear methods apply directly. Fixed-gain design may be feasible even for significant nonlinearities. It may be possible to find a single gain which provides adequate control of the linear models at several perturbation points. If the nonlinearity is restricted to a sector, that fact may be used to obtain a fixed-gain controller. Otherwise, a gain may have to be associated with each perturbation point Pi. A gain schedule K(p(v)) is obtained by connecting the perturbation points by a function, say p(v), of the scheduling parameter v (i.e., speed). When the scheduling parameter must be multidimensional, this approach is difficult; the objective is to develop an easier procedure.
Viewgraphs and charts from a briefing summarize the accomplishments, results, conclusions, and recommendations of a feasibility study using the Pinhole Occulter Facility (POF). Accomplishments for 1986 include: (1) improved IPS Gimbal Model; (2) improved Crew Motion Disturbance Model; (3) use of existing shuttle on-orbit simulation to study the effects of orbiter attitude deadband size on POF performance; (4) increased understanding of maximum performance expected from current actuator/sensor set; (5) use of TREETOPS nonlinear time domain program to obtain system dynamics describing the complex multibody flexible structures; (6) use of HONEY-X design tool to design and evaluate multivariable compensator for stability, robustness, and performance; (7) application of state-of-the-art compensator design methodology Linear Quadratic Gaussian/Loop Transfer Recovery (LQG/LTR); and (8) examination of tolerance required on knowledge of the POF boom flexible mode frequencies to insure stability, using structure uncertainty analysis.
A model of the nonlinear dynamics of a deformable maneuvering multibody system is described, whereby elastic deformation are modeled by restoring forces and dissipative forces at point mass appendages. This model is brought into bilinear form. Estimation of deformations occasioned by rapid slewing maneuvers is carried out by a filter based on a globally equivalent linear model of the bilinear dynamics, and is shown to be an improvement over the extended Kalman filter. To further alleviate the computational burden, the estimated deformation state is propagated between observations by a low dimensional operator spline interpolator of bilinear system Volterra series, which is easily implemented.
Methods developed recently by the author to solve the problem of forward dynamics for nonlinear joint-connected multibody systems are summarized. Solution of this problem is of interest in such application areas as robotics, deploying structures, ground vehicles, and pointing of antennas and instrumented platforms. The problem is solved by the recursive filtering and smoothing techniques of state estimation theory. The filtering stage takes the applied joint moments as inputs to produce a sequence of spatial constraint forces acting at the joints of the system. The smoothing stage takes the innovations process resulting from the filter as an input and produces a set of spatial accelerations and a corresponding set of joint-angle accelerations.
There is an evident need to discover a means of establishing reliable, implementable controls for systems that are plagued by nonlinear and, or uncertain, model dynamics. The development of a generic controller design tool for tough-to-control systems is reported. The method utilizes a moving grid, time infinite element based solution of the necessary conditions that describe an optimal controller for a system. The technique produces a discrete feedback controller. Real time laboratory experiments are now being conducted to demonstrate the viability of the method. The algorithm that results is being implemented in a microprocessor environment. Critical computational tasks are accomplished using a low cost, on-board, multiprocessor (INMOS T800 Transputers) and parallel processing. Progress to date validates the methodology presented. Applications of the technique to the control of highly flexible robotic appendages are suggested.
Explicit codes are often used to simulate the nonlinear dynamics of large-scale structural systems, even for low frequency response, because the storage and CPU requirements entailed by the repeated factorizations traditionally found in implicit codes rapidly overwhelm the available computing resources. With the advent of parallel processing, this trend is accelerating because explicit schemes are also easier to parallellize than implicit ones. However, the time step restriction imposed by the Courant stability condition on all explicit schemes cannot yet and perhaps will never be offset by the speed of parallel hardware. Therefore, it is essential to develop efficient and robust alternatives to direct methods that are also amenable to massively parallel processing because implicit codes using unconditionally stable time-integration algorithms are computationally more efficient than explicit codes when simulating low-frequency dynamics. Here we present a domain decomposition method for implicit schemes that requires significantly less storage than factorization algorithms, that is several times faster than other popular direct and iterative methods, that can be easily implemented on both shared and local memory parallel processors, and that is both computationally and communication-wise efficient. The proposed transient domain decomposition method is an extension of the method of Finite Element Tearing and Interconnecting (FETI) developed by Farhat and Roux for the solution of static problems. Serial and parallel performance results on the CRAY Y-MP/8 and the iPSC-860/128 systems are reported and analyzed for realistic structural dynamics problems. These results establish the superiority of the FETI method over both the serial/parallel conjugate gradient algorithm with diagonal scaling and the serial/parallel direct method, and contrast the computational power of the iPSC-860/128 parallel processor with that of the CRAY Y-MP/8 system.
Explicit codes are often used to simulate the nonlinear dynamics of large-scale structural systems, even for low frequency response, because the storage and CPU requirements entailed by the repeated factorizations traditionally found in implicit codes rapidly overwhelm the available computing resources. With the advent of parallel processing, this trend is accelerating because explicit schemes are also easier to parallelize than implicit ones. However, the time step restriction imposed by the Courant stability condition on all explicit schemes cannot yet -- and perhaps will never -- be offset by the speed of parallel hardware. Therefore, it is essential to develop efficient and robust alternatives to direct methods that are also amenable to massively parallel processing because implicit codes using unconditionally stable time-integration algorithms are computationally more efficient when simulating low-frequency dynamics. Here we present a domain decomposition method for implicit schemes that requires significantly less storage than factorization algorithms, that is several times faster than other popular direct and iterative methods, that can be easily implemented on both shared and local memory parallel processors, and that is both computationally and communication-wise efficient. The proposed transient domain decomposition method is an extension of the method of Finite Element Tearing and Interconnecting (FETI) developed by Farhat and Roux for the solution of static problems. Serial and parallel performance results on the CRAY Y-MP/8 and the iPSC-860/128 systems are reported and analyzed for realistic structural dynamics problems. These results establish the superiority of the FETI method over both the serial/parallel conjugate gradient algorithm with diagonal scaling and the serial/parallel direct method, and contrast the computational power of the iPSC-860/128 parallel processor with that of the CRAY Y-MP/8 system.
Abstract Magnetic bearings offer significant advantages because of their noncontact operation, which can reduce maintenance. Higher speeds, no friction, no lubrication, weight reduction, precise position control, and active damping make them far superior to conventional contact bearings. However, there are technical barriers that limit the application of this technology in industry. One of them is the need for a nonlinear controller that can overcome the system nonlinearity and uncertainty inherent in magnetic bearings. This paper discusses the use of a neural network as a nonlinear controller that circumvents system nonlinearity. A neural network controller was well trained and successfully demonstrated on a small magnetic bearing rig. This work demonstrated the feasibility of using a neural network to control nonlinear magnetic bearings and systems with unknown dynamics.
Explicit codes are often used to simulate the nonlinear dynamics of large-scale structural systems, even for low frequency response, because the storage and CPU requirements entailed by the repeated factorizations traditionally found in implicit codes rapidly overwhelm the available computing resources. With the advent of parallel processing, this trend is accelerating because of the following additional facts: (a) explicit schemes are easier to parallelize than implicit ones, and (b) explicit schemes induce short range interprocessor communications that are relatively inexpensive, while the factorization methods used in most implicit schemes induce long range interprocessor communications that often ruin the sought-after speed-up. However, the time step restriction imposed by the Courant stability condition on all explicit schemes cannot yet be offset by the speed of the currently available parallel hardware. Therefore, it is essential to develop efficient alternatives to direct methods that are also amenable to massively parallel processing because implicit codes using unconditionally stable time-integration algorithms are computationally more efficient when simulating the low-frequency dynamics of aerospace structures.
Space flight beyond Low Earth Orbit requires sophisticated systems to support all aspects of the mission (life support, real-time communications, etc.). A common concern that cuts across all these systems is the selection of information technology (IT) methodology, software and hardware architectures to provide robust monitoring, diagnosis, and control support. Another dimension of the problem space is that different systems must be integrated seamlessly so that communication speed and data handling appear as a continuum (un-interrupted). One such team investigating this problem is the Advanced Integration Matrix (AIM) team whose role is to define the critical requirements expected of software and hardware to support an integrated approach to the command and control of Advanced Life Support (ALS) for future long-duration human space missions, including permanent human presence on the Moon and Mars. A goal of the AIM team is to set the foundation for testing criteria that will assist in specifying tasks, control schemes and test scenarios to validate and verify systems capabilities. This project is to contribute to the goals of the AIM team by assisting with controls planning for ALS. Control for ALS is an enormous problem it involves air revitalization, water recovery, food production, solids processing and crew. In more general terms, these systems can be characterized as involving both continuous and discrete processes, dynamic interactions among the sub-systems, nonlinear behavior due to the complex operations, and a large number of multivariable interactions due to the dimension of the state space. It is imperative that a baseline approach from which to measure performance is established especially when the expectation for the control system is complete autonomous control.