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Discrete time finite dimensional autonomous linear systems, investigating controllability and pole assignment to closed loop transfer matrix by choice of state variable feedback gain
In light of the recently published complete set of statistically correct Grønbech–Jensen (GJ) methods for discrete-time thermodynamics, we revise a differential operator splitting method for the Langevin equation in order to comply with the basic GJ thermodynamic sampling features, namely, the Boltzmann distribution and Einstein diffusion, in linear systems. This revision, which is based on the introduction of time scaling along with flexibility of a discrete-time velocity attenuation parameter, provides a direct link between the ABO splitting formalism and the GJ methods. This link brings about the conclusion that any GJ method has at least weak second order accuracy in the applied time step. It further helps identify a novel half-step velocity, which simultaneously produces both correct kinetic statistics and correct transport measures for any of the statistically sound GJ methods. Explicit algorithmic expressions are given for the integration of the new half-step velocity into the GJ set of methods. Finally, numerical simulations, including quantum-based molecular dynamics (QMD) using the QMD suite Los Alamos Transferable Tight-Binding for Energetics, highlight the discussed properties of the algorithms as well as exhibit the direct application of robust, time-step-independent stochastic integrators to QMD.
The spontaneous breaking of time-translation symmetry in periodically driven quantum systems leads to a new phase of matter: the discrete time crystal (DTC). This phase exhibits collective subharmonic oscillations that depend upon an interplay of non-equilibrium driving, many-body interactions and the breakdown of ergodicity. However, subharmonic responses are also a well-known feature of classical dynamical systems ranging from predator–prey models to Faraday waves and a.c.-driven charge density waves. This raises the question of whether these classical phenomena display the same rigidity characteristic of a quantum DTC. In this work, we explore this question in the context of periodically driven Hamiltonian dynamics coupled to a finite-temperature bath, which provides both friction and, crucially, noise. Focusing on one-dimensional chains, where in equilibrium any transition would be forbidden at finite temperature, we provide evidence that the combination of noise and interactions drives a sharp, first-order dynamical phase transition between a discrete time-translation invariant phase and an activated classical discrete time crystal (CDTC) in which time-translation symmetry is broken out to exponentially long timescales. Power-law correlations are present along a first-order line, which terminates at a critical point. We analyse the transition by mapping it to the locked-to-sliding transition of a d.c.-driven charge density wave. To conclude, building upon results from the field of probabilistic cellular automata, we conjecture the existence of classical time crystals with true long-range order, where time-translation symmetry is broken out to infinite times.
This paper considers a novel approach to coding or classifying sequences of real numbers through the use of (generally nonlinear) finite-dimensional discrete-time systems. This approach involves a finite-dimensional discrete-time system (which we call a real acceptor) in cascade with a threshold type device (which we call a discriminator). The proposed classification scheme and the exact nature of the classification problem are described, along with two examples illustrating its applicability. Suggested approaches for further research are given.
Optimal control and trajectory optimization in aperiodic discrete time functional systems
The microminiaturization of an electronic analog signal to discrete time interval converter is presented. Discrete components and integrated circuits comprising the converter were assembled on a thin-film ceramic substrate containing nichrome resistors with gold interconnections. The finished assembly is enclosed in a flat package measuring 3.30 by 4.57 centimeters. The module can be used whenever conversion of analog to digital signals is required, in particular for the purpose of regulation by means of pulse modulation. In conjunction with a precision voltage reference, the module was applied to control the duty cycle of a switching regulator within a temperature range of -55 C to +125 C, and an input voltage range of 10V to 35V. The output-voltage variation was less than + or - 300 parts per million, i.e., less than + or - 3mV for a 10V output.
The methodology for discrete-time design of a fine steering mirror tracking control loop is presented. It is shown tha high tracking accuracy can be achieve for optical communications by using low cost fine steering mirrors and moderate sampling rates.
A device can comprise a peaked integrator circuit that generates an output signal from a continuous time signal based on a sub rate clock timing cycle. The device can further comprise a track and hold circuit coupled to the output of the peaked integrator that generates a held discrete time signal from the output of the peaked integrator based on a second sub rate clock timing cycle that is offset in time from the sub rate clock timing cycle by a single time unit interval. The device can further comprise an integrator circuit coupled to an output of the track and hold circuit that integrates the held discrete time signal, based on the second sub rate clock timing cycle that is offset in time from the sub rate clock timing cycle by a single time unit interval.
Stochastic approximation algorithms for adaptive linear discrete time system identification using noisy input
Periodically driven quantum systems known as Floquet insulators can host topologically protected bound states known as “ π modes” that exhibit response at half the frequency of the drive. Such states can also appear in undriven lattice field theories when time is discretized as a result of fermion doubling, raising the question of whether these two phenomena could be connected. Recently we demonstrated such a connection at the level of an explicit mapping between the spectra of a continuous-time Floquet model and a discrete-time undriven lattice fermion model. However, this mapping relied on a symmetry of the single-particle spectrum that is not present for generic drive parameters. Inspired by the example of the temporal Wilson term in lattice field theory, in this paper we extend this mapping to the full drive parameter space by allowing the parameters of the discrete-time model to be frequency-dependent. The spectra of the resulting lattice fermion models exactly match the quasienergy spectrum of the Floquet model in the thermodynamic limit. Our results demonstrate that spectral features characteristic of beyond-equilibrium physics in Floquet systems can be replicated in static systems with appropriate time discretization. Published by the American Physical Society 2024
Stochastic approximation algorithms for identification of linear discrete time systems
Tree codes for memoryless time discrete sources with bounded fidelity criterion
The performance of Kalman-type, linear, discrete-time filters in the presence of modeling errors is considered. The discussion is limited to stationary performance, and bounds are obtained for the performance index, the mean-squared error of estimates for suboptimal and optimal (Kalman) filters. The computation of these bounds requires information on only the model matrices and the range of errors for these matrices. Consequently, a design can easily compare the performance of a suboptimal filter with that of the optimal filter, when only the range of errors in the elements of the model matrices is available.
Pilot behavior is considered as a discrete-time process where the decision making has a sequential nature. This model differs from both the quasilinear model which follows from classical control theory and from the optimal control model which considers the human operator as a Kalman estimator-predictor. An additional factor considered is that the pilot's objective may not be adequately formulated as a quadratic cost functional to be minimized, but rather as a more fuzzy measure of the closeness with which the aircraft follows a reference trajectory. All model parameters, in the digital program simulating the pilot's behavior, were successfully compared in terms of standard-deviation and performance with those of professional pilots in IFR configuration. The first practical application of the model was in the study of its performance degradation when the aircraft model static margin decreases.
The objective of this paper is to design a discrete-time controller which forces y(t) = F(q(t),q'(t)) to follow a desired trajectory, when non-linearities may be uncertain. A self-tuning adaptive control algorithm is then developed that addresses three types of errors: (1) error due to model linearization; (2) errors due to controlling a time-varying plant; and (3) errors due to discretization.
Discrete time differential dynamic programming algorithm with application to optimal orbit transfer
The least-squares transformation of a discrete-time multivariable linear system into a desired one by convolving the first with a polynomial system yields optimal polynomial solutions to the problems of system compensation, inversion, and approximation. The polynomial coefficients are obtained from the solution to a so-called normal linear matrix equation, whose coefficients are shown to be the weighting patterns of certain linear systems. These, in turn, can be used in the recursive solution of the normal equation.
A study is made of the behavior of discrete-time systems composed of a set of smooth transition maps coupled by a quantized feedback function. The feedback function partitions the state space into disjoint regions and assigns a smooth transition function to each region. The main result is that under a constraint on the norm of the derivative of the transition maps, a bounded state trajectory with limit points in the interior of the switching regions leads to a region index sequence that is eventually periodic. Under these assumptions, it is shown that eventually the feedback function is determined by a finite state automaton. A similar result is proved in the case of finite state dynamic feedback.