Optimal control of nonlinear discrete time systems by mathematical programming
Nonlinear discrete time sampled systems optimal control by dynamic programming, with application to stochastic systems
Tabak, D.↗
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Results for “nonlinear dynamical system”
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Parameter estimation in nonlinear distributed systems - Approximation theory and convergence results
An abstract approximation framework and convergence theory is described for Galerkin approximations applied to inverse problems involving nonlinear distributed parameter systems. Parameter estimation problems are considered and formulated as the minimization of a least-squares-like performance index over a compact admissible parameter set subject to state constraints given by an inhomogeneous nonlinear distributed system. The theory applies to systems whose dynamics can be described by either time-independent or nonstationary strongly maximal monotonic operators defined on a reflexive Banach space which is densely and continuously embedded in a Hilbert space. It is demonstrated that if readily verifiable conditions on the system's dependence on the unknown parameters are satisfied, and the usual Galerkin approximation assumption holds, then solutions to the approximating problems exist and approximate a solution to the original infinite-dimensional identification problem.
Banks, H. T.↗
Fuzzy logic based robotic controller
Existing Proportional-Integral-Derivative (PID) robotic controllers rely on an inverse kinematic model to convert user-specified cartesian trajectory coordinates to joint variables. These joints experience friction, stiction, and gear backlash effects. Due to lack of proper linearization of these effects, modern control theory based on state space methods cannot provide adequate control for robotic systems. In the presence of loads, the dynamic behavior of robotic systems is complex and nonlinear, especially where mathematical modeling is evaluated for real-time operators. Fuzzy Logic Control is a fast emerging alternative to conventional control systems in situations where it may not be feasible to formulate an analytical model of the complex system. Fuzzy logic techniques track a user-defined trajectory without having the host computer to explicitly solve the nonlinear inverse kinematic equations. The goal is to provide a rule-based approach, which is closer to human reasoning. The approach used expresses end-point error, location of manipulator joints, and proximity to obstacles as fuzzy variables. The resulting decisions are based upon linguistic and non-numerical information. This paper presents a solution to the conventional robot controller which is independent of computationally intensive kinematic equations. Computer simulation results of this approach as obtained from software implementation are also discussed.
Attia, F.↗
The Transient X-Ray Burster Gro J1744-28: An Unstable Accretion System?
The newly discovered burst/pulsar source GRO J1744-28 may be considered as a quasi-periodic oscillator (QPO) with a frequency in the 1-10 milli-Hz range, plus an X-ray pulsar with a period of .47 sec (IAU Circulars 6272, 6275, 6276, 6284, 6285, 6286). Since some of the observed properties of this system do not fit with the relaxation oscillators used to interpret rapid bursters, we are investigating the applicability of models that have been used for QPO in X-ray binary systems. Noting that some features of the nonlinear dynamical model called the dripping handrail are seen in the reported observations, we compare this model with the beat-frequency modulation picture of QPO.
Wood, Kent↗
Design and Stability of an On-Orbit Attitude Control System Using Reaction Control Thrusters
Basic principles for the design and stability of a spacecraft on-orbit attitude control system employing on-off Reaction Control System (RCS) thrusters are presented. Both vehicle dynamics and the control system actuators are inherently nonlinear, hence traditional linear control system design approaches are not directly applicable. This paper has two main aspects: It summarizes key RCS design principles from earlier NASA vehicles, notably the Space Shuttle and Space Station programs, and introduces advances in the linear modelling and analyses of a phase plane control system derived in the initial development of the NASA's next upper stage vehicle, the Exploration Upper Stage (EUS). Topics include thruster hardware specifications, phase plane design and stability, jet selection approaches, filter design metrics, and RCS rotational maneuver logic.
Hall, Robert A.↗
Design and Stability of an On-Orbit Attitude Control System Using Reaction Control Thrusters
Principles for the design and stability of a spacecraft on-orbit attitude control system employing on-off Reaction Control System (RCS) thrusters is presented. Both the vehicle dynamics and the control system actuators are inherently nonlinear, hence traditional linear control system design approaches are not directly applicable. This paper has three main aspects: It summarizes key RCS control System design principles from the Space Shuttle and Space Station programs, it demonstrates a new approach to develop a linear model of a phase plane control system using describing functions, and applies each of these to the initial development of the NASA's next generation of upper stage vehicles. Topics addressed include thruster hardware specifications, phase plane design and stability, jet selection approaches, filter design metrics, and automaneuver logic.
Hall, Robert A.↗
Self-Supervised Dynamical Systems
Some progress has been made in a continuing effort to develop mathematical models of the behaviors of multi-agent systems known in biology, economics, and sociology (e.g., systems ranging from single or a few biomolecules to many interacting higher organisms). Living systems can be characterized by nonlinear evolution of probability distributions over different possible choices of the next steps in their motions. One of the main challenges in mathematical modeling of living systems is to distinguish between random walks of purely physical origin (for instance, Brownian motions) and those of biological origin. Following a line of reasoning from prior research, it has been assumed, in the present development, that a biological random walk can be represented by a nonlinear mathematical model that represents coupled mental and motor dynamics incorporating the psychological concept of reflection or self-image. The nonlinear dynamics impart the lifelike ability to behave in ways and to exhibit patterns that depart from thermodynamic equilibrium. Reflection or self-image has traditionally been recognized as a basic element of intelligence. The nonlinear mathematical models of the present development are denoted self-supervised dynamical systems. They include (1) equations of classical dynamics, including random components caused by uncertainties in initial conditions and by Langevin forces, coupled with (2) the corresponding Liouville or Fokker-Planck equations that describe the evolutions of probability densities that represent the uncertainties. The coupling is effected by fictitious information-based forces, denoted supervising forces, composed of probability densities and functionals thereof. The equations of classical mechanics represent motor dynamics that is, dynamics in the traditional sense, signifying Newton s equations of motion. The evolution of the probability densities represents mental dynamics or self-image. Then the interaction between the physical and metal aspects of a monad is implemented by feedback from mental to motor dynamics, as represented by the aforementioned fictitious forces. This feedback is what makes the evolution of probability densities nonlinear. The deviation from linear evolution can be characterized, in a sense, as an expression of free will. It has been demonstrated that probability densities can approach prescribed attractors while exhibiting such patterns as shock waves, solitons, and chaos in probability space. The concept of self-supervised dynamical systems has been considered for application to diverse phenomena, including information-based neural networks, cooperation, competition, deception, games, and control of chaos. In addition, a formal similarity between the mathematical structures of self-supervised dynamical systems and of quantum-mechanical systems has been investigated.
Zak, Michail↗
Nonlinear inversion flight control for a supermaneuverable aircraft
This paper describes the use of nonlinear dynamic inversion for the design of a flight control system for a supermaneuverable aircraft. First, the dynamics to be controlled were separated into fast and slow variables. The fast variables were the angular rates and the slow variables were the attitude angles. Then a nonlinear inversion controller was designed for the fast variables. This stabilized the longitudinal short-period and improved the lateral-directional responses over a wide range of angle of attack by making use of a combination for aerodynamic surfaces and thrust vectoring control. Outer loops were then closed to allow the pilot to control the slow dynamics, the angle of attack, side-slip angle and the velocity bank angle. Nonlinear inversion was also used to design of the outer loop control laws. The dynamic inversion control laws were compared with more conventional, gain-scheduled control laws and were shown to yield much better performance.
Snell, S. Antony↗
Virtual Passive Controller for Robot Systems Using Joint Torque Sensors
This paper presents a control method based on virtual passive dynamic control that will stabilize a robot manipulator using joint torque sensors and a simple joint model. The method does not require joint position or velocity feedback for stabilization. The proposed control method is stable in the sense of Lyaponov. The control method was implemented on several joints of a laboratory robot. The controller showed good stability robustness to system parameter error and to the exclusion of nonlinear dynamic effects on the joints. The controller enhanced position tracking performance and, in the absence of position control, dissipated joint energy.
Aldridge, Hal A.↗
The determination of asymptotic and periodic behavior of dynamic systems arising in control system analysis Final report
Asymptotic and periodic behavior prediction for nonlinear control system with mathematical model of rigid body vehicle
CONTROL SYSTEM↗
Study of stability of large maneuvers of airplanes
A predictive method of nonlinear system analysis is used to investigate airplane stability and dynamic response during rolling maneuvers. The maneuver roll-rate is not assumed to be constant, and the airplane motion is represented by a set of coupled nonlinear differential equations. The general rolling maneuver is kinematically specified by its roll-rate variation p(t). A method for relating the airplane dynamic response to p(t) is developed. The method provides analytical expressions for the motion variables in terms of the maneuver descriptor p(t). A parameterized family of rolling maneuvers is considered, for which the method is used to predict specific dynamic response information, such as the dependence of the peak angle-of-attack excursion on the maneuver parameters. The stability and motion of the airplane in response to an arbitrary actuation of aileron input is considered. Analytical expressions relating motion variables to aileron input are obtained. Explicit analytical bounds on the motion variables are derived. A stability criterion which guarantees nondivergence of motion in response to aileron actuation is presented.
Haddad, E. K.↗
Control oriented model order reduction for the hydraulic system of an autonomous wheel loader
Control design for an autonomous wheel loader requires a high-fidelity and low-order dynamics model. The objective of order reduction is to decrease the number of states in the model while maintaining performance comparable to the original model. Of all the power components in a wheel loader, the model order reduction for steering and working hydraulic systems is most demanding due to its nonlinear and stiff dynamical nature. This paper describes a physics-inspired model order reduction method that can reduce the model order by close to thirty percent. It is achieved by discarding orders with fast dynamics and consolidating multiple states into less new states. Furthermore, this method can also be extended to other types of off-road vehicles such as excavators, bulldozers, etc. Simulation results demonstrate that the reduced-order model exhibits very similar performance to the full-order model under typical operating conditions of wheel loaders, with an output error of less than six percent.
42 ENGINEERING↗
Emergence of complex-regular eutectic patterns in Al–Ge: Observations from correlative nano-imaging
Eutectic solidification exemplifies nonequilibrium pattern formation, making it a well-studied moving boundary problem. Yet the mechanisms behind the formation of complex-regular microstructures – particularly in highly anisotropic systems with a significant volume fraction of a faceted phase – remain poorly understood. Our understanding of such systems is made complicated by the nonlinear interface kinetics and unique growth dynamics characteristic of faceted phases. To address these challenges, we investigate a model Al–Ge eutectic system, where the faceted Ge phase constitutes a substantial volume fraction (~0.35) and where the two solid phases arrange into so-called “fishbone” or “feather” complex-regular patterns. Using synchrotron-based x-ray nano-imaging and nanotomography with high spatial resolution (22 nm per pixel), we capture in real-time the evolution of the solid–liquid interfaces and the resulting three-dimensional microstructures in this faceted/non-faceted eutectic system. By integrating these observations with electron backscattered diffraction, we elucidate the crystallographic biases on the solidification process and the mechanisms driving the formation of such complex-regular microstructures. These findings inform a new growth model for irregular eutectics in (near-)symmetrical phase diagrams, offering insight on advanced microstructural design and processing strategies. More broadly, we demonstrate how interfacial curvature is generated in irregular eutectic alloys and how it depends on the volume fraction of the faceted phase.
36 MATERIALS SCIENCE↗
Ecosystems and the Biosphere as Complex Adaptive Systems
Ecosystems are prototypical examples of complex adaptive systems, in which patterns at higher levels emerge from localized interactions and selection processes acting at lower levels. An essential aspect of such systems is nonlinearity, leading to historical dependency and multiple possible outcomes of dynamics. Given this, it is essential to determine the degree to which system features are determined by environmental conditions, and the degree to which they are the result of self-organization. Furthermore, given the multiple levels at which dynamics become apparent and at which selection can act, central issues relate to how evolution shapes ecosystems properties, and whether ecosystems become buffered to changes (more resilient) over their ecological and evolutionary development or proceed to critical states and the edge of chaos.
Levin, Simon A.↗
Strong turbulence of plasma waves
This paper reviews recent work related to modulational instability and wave envelope self-focusing in dynamical and statistical systems. After introductory remarks pertinent to nonlinear optics realizations of these effects, the author summarizes the status of the subject in plasma physics, where it has come to be called 'strong Langmuir turbulence'. The paper treats the historical development of pertinent concepts, analytical theory, numerical simulations, laboratory experiments, and spacecraft observations. The role of self-similar self-focusing Langmuir envelope wave packets is emphasized, both in the Zakharov equation model for the wave dynamics and in a statistical theory based on this dynamical model.
Goldman, M. V.↗
Design of a helicopter autopilot by means of linearizing transformations
An automatic flight control systems design methods for aircraft that have complex characteristics and operational requirements, such as the powered lift STOL and V/STOL configurations are discussed. The method is effective for a large class of dynamic systems that require multiaxis control and that have highly coupled nonlinearities, redundant controls, and complex multidimensional operational envelopes. The method exploits the possibility of linearizing the system over its operational envelope by transforming the state and control. The linear canonical forms used in the design are described, and necessary and sufficient conditions for linearizability are stated. The control logic has the structure of an exact model follower with linear decoupled model dynamics and possibly nonlinear plant dynamics. The design method is illustrated with an application to a helicopter autopilot design.
Meyer, G.↗
Generalized fractional approach to solving partial differential equations with arbitrary dispersion relations
The dynamics of physical systems are typically modeled by partial differential equations (PDEs). Because of the complexity in physical systems, PDE models rely on approximations or limiting cases. To model the full complexity of physical systems, it is necessary to use multiscale approaches where appropriate models are used at each level. Alternatively, complexity can be directly tackled by fractional integrodifferential equations that must be derived for each system. Here, we argue that complexity can be restored in PDEs by describing them from a fractional calculus perspective. Rather than deriving a fractional integrodifferential equation, we reinterpret the dispersion relation of the system by use of the Riesz definition, which contains the required information relating the energy and momentum space of the system and thus fully describes their dynamics. The approach is demonstrated by two examples: the Landau–Lifshitz equation in a 1D ferromagnetic chain and a modified KdV equation supporting surface gravity waves or Euler dispersion. The presented approach is applicable to fluids, soft matter, and solid-state matter and can be readily generalized to higher dimensions and more complex systems. While numerical calculations are needed to determine the fractional operator, the approach is analytical and can be utilized to determine analytical solutions and investigate nonlinear problems.
97 MATHEMATICS AND COMPUTING↗
Using kernel-based statistical distance to study the dynamics of charged particle beams in particle-based simulation codes
Measures of discrepancy between probability distributions (statistical distance) are widely used in the fields of artificial intelligence and machine learning. We describe how certain measures of statistical distance can be implemented as numerical diagnostics for simulations involving charged-particle beams. Related measures of statistical dependence are also described. The resulting diagnostics provide sensitive measures of dynamical processes important for beams in nonlinear or high-intensity systems, which are otherwise difficult to characterize. Here, the focus is on kernel-based methods such as maximum mean discrepancy, which have a well-developed mathematical foundation and reasonable computational complexity. Several benchmark problems and examples involving intense beams are discussed. While the focus is on charged-particle beams, these methods may also be applied to other many-body systems such as plasmas or gravitational systems.
47 OTHER INSTRUMENTATION↗