Multidimensional Multiplicative Combinatorial Properties of Dynamical Syndetic Sets

In this paper, it is shown that for a minimal system (X, G), if H is a normal subgroup of G with finite index n, then X can be decomposed into n components of closed sets such that each component is minimal under H-action. Meanwhile, we prove that for a residual set of points in a minimal system with finitely many commuting homeomorphisms, the set of return times to any non-empty open set contains arbitrarily long geometric progressions in multidimension, extending a previous result by Glasscock, Koutsogiannis and Richter.

On topological properties of the set of maldistributed sequences

The real sequence (xn) is maldistributed if for any non-empty interval I, the set {n ∈𝕅 : xn ∈I} has upper asymptotic density 1. The main result of this note is that the set of all maldistributed real sequences is a residual set in the set of all real sequences (i.e., the maldistribution is a typical property in the sense of Baire categories). We also generalize this result.

A context in which finite or unique ergodicity is generic

We show that for good measures, the set of homeomorphisms of Cantor space which preserve that measure and which have no invariant clopen sets contains a residual set of homeomorphisms which are uniquely ergodic. Additionally, we show that for refinable Bernoulli trial measures, the same set of homeomorphisms contains a residual set of homeomorphisms which admit only finitely many ergodic measures.

Health Assessment Methods for Wind Turbines Based on Power Prediction and Mahalanobis Distance

The output power of wind turbine has great relation with its health state, and the health status assessment for wind turbines influences operational maintenance and economic benefit of wind farm. Aiming at the current problem that the health status for the whole machine in wind farm is hard to get accurately, in this paper, we propose a health status assessment method in order to assess and predict the health status of the whole wind turbine, which is based on the power prediction and Mahalanobis distance (MD). Firstly, on the basis of Bates theory, the scientific analysis for historical data from SCADA system in wind farm explains the relation between wind power and running states of wind turbines. Secondly, the active power prediction model is utilized to obtain the power forecasting value under the health status of wind turbines. And the difference between the forecasting value and actual value constructs the standard residual set which is seen as the benchmark of health status assessment for wind turbines. In the process of assessment, the test set residual is gained by network model. The MD is calculated by the test residual set and normal residual set and then normalized as the health status assessment value of wind turbines. This method innovatively constructs evaluation index which can reflect the electricity generating performance of wind turbines rapidly and precisely. So it effectively avoids the defect that the existing methods are generally and easily influenced by subjective consciousness. Finally, SCADA system data in one wind farm of Fujian province has been used to verify this method. The results indicate that this new method can make effective assessment for the health status variation trend of wind turbines and provide new means for fault warning of wind turbines.

A New Robust Controller with Applications to Bioreactors

In this work an anaerobic digester is controlled using input-output linearization and Lyapunov-like function methods. It is assumed that model parameters are unknown, time-varying, and bounded, and upper or lower bounds are also unknown. To tackle the effect of input saturation, a state observer is designed. The tracking and observer errors are defined in terms of the noisy measured output instead of ideal output, given by the mathematical model. The design of the observer mechanism and the update laws is based on the Lyapunov-like function technique, whereas the design of the control law is based on the input-output linearization method. In this paper two important properties of the controlled system are proven. First, the observer error converges asymptotically to a residual set whose size is user-defined, and such convergence is not disrupted, neither by the input saturation nor by the parameter uncertainties. Second, when the control input is nonsaturated the tracking error converges to a residual set whose size is user-defined. The model parameter uncertainties are included to prove the convergence of errors. Finally, a numerical example to validate the developed control is presented.

Prevalence of Non-degenerate Periodic Solutions in the Forced Pendulum Equation

Many known properties of the forced pendulum equation are generic. This means that they hold on a residual set of forcings. The classical duality category/measure inspires the main question in the paper: are these properties also valid for a prevalent set of forcings? This is the case for a key property: the non-degeneracy of periodic solutions. The main tool in the proof is a prevalent version of the parametric transversality theorem.

Two-Valued Control for a Second-Order Plant with Additive External Disturbance

In this work a two-valued state feedback control for a plant of second order with known constant coefficients and an additive bounded disturbance is designed. In this controller the control signal can take only two possible values. The controller design is based on Lyapunov-like function method, achieving the convergence of the tracking error to a user-defined residual set. A boundedness condition for the user-defined reference signal is defined, which is necessary to allow out-put tracking. The developed scheme avoids large commutation rate of the control input. The controller design and stability analysis have important contributions with respect to closely related controllers based on the direct Lyapunov method, namely, (i) conditions to guarantee the expected convergence of the tracking error are established. These conditions are imposed on the reference signal and the extreme values of the control input. The stability analysis is developed by means of the Lyapunov-like function method and the Barbalat's Lemma and includes (ii) the bounded nature of the Lyapunov function, (iii) the monotonic convergence of the Lyapunov function to a residual set, and (iv) the asymptotic convergence of the tracking error to a residual set of user-defined size.

Transitivity of Euclidean-type extensions of hyperbolic systems

Letf:X→Xbe the restriction to a hyperbolic basic set of a smooth diffeomorphism. We show that in the class ofCr(r>0) cocycles with fiber the special Euclidean group SE(n), those that are transitive form a residual set (countable intersection of open dense sets). This result is new for odd values ofn≥3. More generally, we consider Euclidean-type groupsG⋉ℝnwhereGis a compact connected Lie group acting linearly on ℝn. When Fix G={0}, it is again the case that the transitive cocycles are residual. When Fix G≠{0}, the same result holds upon restriction to the subset of cocycles that avoid an obvious and explicit obstruction to transitivity.