Deterministic Uncertainty Research Articles

Spatial Analysis in Physical Geography and the Challenge of Deterministic Uncertainty

The application of complex and nonlinear dynamical systems (NDS) theory in physical geography and geosciences has proceeded through several stages, and has recently entered a phase where field‐testable hypotheses and historical or mechanistic explanations are being generated. However, there are some fundamental challenges. It seems clear that chaos and dynamical self‐organization are present, and may be common in earth surface systems, and that these phenomena have spatial manifestations in the landscape. However, NDS theory and methods have been formulated primarily in the temporal domain and are typically ill‐suited to real‐world spatial data. Spatial analytical methods are not generally capable of distinguishing deterministic complexity and uncertainty from noise. Thus, the detection of the signals of complex deterministic dynamics in real landscapes and spatial data is a major challenge. Entropy‐based methods of spatial analysis can be directly linked to nonlinear dynamics, and are at present the best available method to approach this problem. However, there is evidence in the spatial analysis literature suggesting that development of techniques to detect deterministic uncertainty is possible. Pending such a break‐through, three general approaches are described, based on spatial analysis of chronosequences, the characteriziation of changes in spatial structure over time, and the spatial‐domain testing of specific hypotheses relevant to deterministic uncertainty. Current trends generally suggest a shift in mathematical modeling and spatial analysis in physical geography away from traditional determinism toward approaches that incorporate locational, historical, and scale contingency.

Deterministic Uncertainty in Bed Load Transport

A methodology for detecting nonlinear dynamics in bed load transport rate time series and testing the ability of distribution functions to characterize rate variations is outlined. The Kolmogorov entropy of a time series consisting of >10,000 bed load transport measurements made in a laboratory flume indicates that >80% of the variability can be explained by the sequential passage of bed forms as modeled by Hamamori's logarithmic cumulative distribution function, and the remainder may be attributable to deterministic uncertainty (chaos). Greater understanding of the mutual adjustments between the turbulent flow, sediment transport, and bed forms might reduce the level of uncertainty. Nevertheless, the Hamamori function affords an effective characterization of the distribution of at-a-point bed load transport rates in cases where the fine temporal details of the transport process can be ignored.

A Robust Non-Linear Control Design for a Turbogenerator

Abstract In this paper is presented a new approach to robust non-linear control design, which can guarantee a prescribed decay rate of exponential stability for known system uncertainties. The proposed approach doesn’t employ matching conditions. The non-linear power system with deterministic uncertainties is chosen as demonstration example.

Direct optimal control: A hybrid approach

The study of dynamic systems without resorting to or any knowledge of differential equations is known as the direct method. In this method, algebraic equations of motion characterize the system dynamics. The algebraic optimal control laws can be derived in an explicit form for general nonlinear time-varying and time-invariant systems by minimizing an algebraic performance measure. The essence of the approach is based on using assumed-time-modes expansions of generalized coordinates and inputs in conjunction with the variational work-energy principles that govern the physical system. The resulting algebraic optimal control laws are in the form of digitally implementable generalized algebraic state feedback. However, to implement these control laws an algebraic state estimator must be designed. The development of such an estimator is proposed by utilizing neural networks within a hybrid algebraic equation of motion for general nonlinear systems. To prove the concept, numerical simulations are validated on linear systems in the deterministic, noisy and modelling uncertainty cases and a nonlinear flexible structure.

Deterministic uncertainty and complex pedogenesis in some Pleistocene dune soils

Deterministic uncertainty is a perspective on soil spatial variability that reconciles the traditional reductionist view (variability can be explained with more and better measurements) and the emerging nonlinear dynamics view (variability may be an irresolvable outcome of complex system dynamics). In the podzolized soils of the 77 ka Newport Barrier, age, parent material, climate, and general vegetation cover are constant, and topographic and drainage variables explain only about 20% of the considerable spatial variation in soil morphology. The most likely explanations for the unexplained variations are the effects of past microtopography and individual plants on moisture flux. However, the variation in soil properties such as depth to the B-horizon is disproportionately large compared to that of microtopography, and the soil lifespan is two orders of magnitude longer than that of vegetation. Deterministic complexity (chaos) associated with the unstable growth of minor perturbations could explain the observed soil landscape. A simple model of the interactions between moisture flux, soil moisture, and B-horizon formation is indeed unstable and chaotic. The Newport Barrier soil landscape is thus characterized by deterministic uncertainty — pseudo-random variation associated with unstable growth of known, but unobserved or unmeasured, pedogenic influences.

A Nonlinear Tracking Problem for Control Systems with Deterministic Uncertainty

By means of some existence results on selections of multivalued maps we solve a tracking problem for a dynamical system affected by deterministic uncertainty. The considered control system is modelled by a system of differential inclusions.AMS Subject Classification Number: 93B52, 93C15.

An Invariance Problem for Control Systems with Deterministic Uncertainty

This paper deals with a class of nonlinear control systems in $R^n$ in presence of deterministic uncertainty. The uncertainty is modelled by a multivalued map F with nonempty, closed, convex values. Given a nonempty closed set $K ⊂ R^n$ from a suitable cl

Minimax control of switching systems under sampling

We consider a general class of systems subject to two types of uncertainty: A continuous deterministic uncertainty that affects the system dynamics, and a discrete stochastic uncertainty that leads to jumps in the system structure at random times, with the latter described by a continuous-time finite state Markov chain. When only sampled values of the system state is available to the controller, along with perfect measurements on the state of the Markov chain, we obtain a characterization of minimax controllers, which involves the solutions of two finite sets of coupled PDEs, and a finite dimensional compensator. For the linear-quadratic case, a complete characterization is given in terms of coupled generalized Riccati equations, which also provides the solution to a particular H ∞ optimal control problem with randomly switching system structure and sampled state measurements.

Effect of pulse-shape uncertainty on the accuracy of deconvolved lidar profiles

The effect of random and deterministic pulse-shape uncertainties on the accuracy of the Fourier deconvolution algorithm for improving the resolution of long-pulse lidars is investigated theoretically and by computer simulations. Various cases of pulse uncertainties are considered including those that are typical of Doppler lidars. It is shown that the retrieval error is a consequence of two main effects. The first effect consists of a shift up or down (depending on the sign of the uncertainty integral area) of the lidar profile as a whole, proportionally to the ratio of the pulse uncertainty area to the true pulse area. The second effect consists of additional amplitude and phase distortions of the spectrum of the small-scale inhomogeneities of the lidar profile. The results obtained allow us to predict the order and the character of the possible distortions and to choose ways to reduce or prevent them.

Deterministic uncertainty in landscapes

AbstractDeterministic complexity (chaos) may be common in geomorphic systems, but traditional definitions may have limited practical utility for empirical geomorphology. These definitions are based on sensitivity to initial conditions, which in geomorphology are both unknown and unknowable. Further, chaos analysis depends on distinguishing deterministic complexity from stochastic complexity. This is problematic in geomorphology because some stochastic complexity is virtually always present in addition to any chaos that may be present. While it is important to recognize that some complex, apparently random patterns may derive from inherent non‐linear system dynamics, this is of limited use in explaining process–response relationships or mechanics of landscape evolution. A more general term, which subsumes chaos, is deterministic uncertainty, i.e. uncertainty associated with an identifiable but unknown or uncertain source. An analysis of landscape entropy shows that such underlying constraints produce spatial patterns which are apparently chaotic. For the case of geologic controls, the apparent contribution of deterministic chaos to the landscape entropy is a direct non‐linear function of the extent of geologic constraints. However, the underlying constraints and their contribution to observed spatial patterns can also be interpreted in non‐chaotic terms. Examples are given, involving geologic constraints on stream channel networks and parent material control of surface soil textures. Because both randomness and chaos may be more apparent than real, the concept of deterministic uncertainty is more useful in process geomorphology than that of chaos.

Experimental Implementation of a Deterministic Controller for a D.C. Motor With Uncertain Dynamics

The control of d.c. motors is often complicated by the fact that the dynamics of the system to be controlled are unknown. In this paper a deterministic controller is employed to control a d.c. motor which experiences a varying inertia load, Coriolis forces and unknown friction disturbances. All of the uncertainties in the system are assumed to be bounded. A hybrid controller, using a continuous velocity loop and a digital position loop, is developed. Experimental results verify the controller efficacy in dealing with bounded deterministic uncertainties.