State-dependent Uncertainty Research Articles

State-dependent and state-independent uncertainty relations for skew information and standard deviation

In this work, we derive state-dependent uncertainty relations (uncertainty equalities) in which commutators of incompatible operators (not necessarily Hermitian) are explicitly present and state-independent uncertainty relations based on the Wigner-Yanase (-Dyson) skew information. We derive uncertainty equality based on standard deviation for incompatible operators with mixed states, a generalization of previous works in which only pure states were considered. We show that for pure states, the Wigner-Yanase skew information based state-independent uncertainty relations become standard deviation based state-independent uncertainty relations which turn out to be tighter uncertainty relations for some cases than the ones given in previous works, and we generalize the previous works for arbitrary operators. As the Wigner-Yanase skew information of a quantum channel can be considered as a measure of quantum coherence of a density operator with respect to that channel, we show that there exists a state-independent uncertainty relation for the coherence measures of the density operator with respect to a collection of different channels. We show that state-dependent and state-independent uncertainty relations based on a more general version of skew information called generalized skew information which includes the Wigner-Yanase (-Dyson) skew information and the Fisher information as special cases hold. In qubits, we derive tighter state-independent uncertainty inequalities for different form of generalized skew informations and standard deviations, and state-independent uncertainty equalities involving generalized skew informations and standard deviations of spin operators along three orthogonal directions. Finally, we provide a scheme to determine the Wigner-Yanase (-Dyson) skew information of an unknown observable which can be performed in experiment using the notion of weak values.

Distributed Adaptive Synchronization in Euler–Lagrange Networks With Uncertain Interconnections

In this work we propose a new practical synchronization protocol for multiple Euler Lagrange (EL) systems without structural linear-in-the-parameters (LIP) knowledge of the uncertainty and where the agents can be interconnected before control design by unknown state-dependent interconnection terms. This setting is meant to overcome two standard a priori assumptions in the literature concerning uncertainty with LIP structure and absence of interaction among agents before designing the synchronization protocol. To overcome these assumptions, we propose an adaptive distributed control mechanism having the purpose of estimating the coefficients of the resulting state-dependent uncertainty structure.

Adaptive Integral Sliding Mode Control in the Presence of State-Dependent Uncertainty

Adaptive integral sliding mode control (AISMC) is an extension of adaptive sliding mode control which is a way to ensure sliding motion while handling system uncertainties. However, conventional AISMC formulations require to different extent <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> knowledge of the system uncertainty: either the upper bound of the uncertainty or of its time derivative are assumed to be bounded <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> , or the uncertainty is assumed to be parametrized by some structure-dependent factorization. This work proposes a variant of AISMC with reduced <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> knowledge of the system uncertainty: it is shown that Euler–Lagrange dynamics typical of sliding mode literature admit a structure-independent parametrization of the system uncertainty. This parametrization is not the result of structural knowledge, but it comes from basic properties of Euler–Lagrange dynamics, valid independently on the structure of the system. The AISMC control method arising from this parametrization is analyzed in the Lyapunov stability framework, and validated in systems with different structures: a surface vessel and an aerial vehicle.

Conic Iterative Learning Control Using Distinct Data for Constrained Systems With State-Dependent Uncertainty

In batch processes, the ability to learn from previous process data results in high-value and batch-improved products. For batch processes with constraints and state-dependent uncertainty, this article presents a conic iterative learning control (ILC) approach, which uses cone theory to incorporate historical process data into optimization-based ILC design. The proposed conic ILC approach uses rank conditioning to select distinct data samples and conic mapping to map the data to the to-be-optimized control input variables, since adding all historical process data would be computationally intensive. Our method yields a tradeoff between learning ability from historical experience and computational efficiency from solving the optimization problem. Provable constraint satisfaction and robust stability are considered separately. To demonstrate the proven properties and effectiveness of the approach, we present a case study of the injection molding process.

Nonlinear learning‐based model predictive control supporting state and input dependent model uncertainty estimates

AbstractWhile model predictive control (MPC) methods have proven their efficacy when applied to systems with safety specifications and physical limitations, their performance heavily relies on an accurate prediction model. As a consequence, a significant effort in the design of MPC controllers is dedicated to the modeling part and often requires advanced physical expertise. In order to facilitate the controller design, we present an MPC scheme supporting nonlinear learning‐based prediction models, that is, data‐driven models with probabilistic parameter uncertainties. A tube‐based MPC formulation in combination with an additional implicit state and input constraint forces the closed‐loop system to be operated in domains of sufficient model confidence, thereby ensuring asymptotic stability and constraint satisfaction at a prespecified level of probability. Furthermore, by relying on tube‐based MPC concepts, the proposed learning‐based MPC formulation offers a general framework for addressing different problem classes, such as economic MPC, while providing a general interface to probabilistic prediction models based, for example, on Bayesian regression or Gaussian processes. A design procedure is proposed for approximately linear systems and the efficiency of the method is illustrated using numerical examples.

Learning-Based SMPC for Reference Tracking Under State-Dependent Uncertainty: An Application to Atmospheric Pressure Plasma Jets for Plasma Medicine

The increasing complexity of modern technical systems can exacerbate model uncertainty in model-based control, posing a great challenge to safe and effective system operation under closed loop. Online learning of model uncertainty can enhance control performance by reducing plant–model mismatch. This article presents a learning-based stochastic model predictive control (LB-SMPC) strategy for reference tracking of stochastic linear systems with additive state-dependent uncertainty. The LB-SMPC strategy adapts the state-dependent uncertainty model online to reduce plant–model mismatch for control performance optimization. Standard reachability and statistical tools are leveraged along with the state-dependent uncertainty model to develop a chance constraint-tightening approach, which ensures state constraint satisfaction in probability. The stability and recursive feasibility of the LB-SMPC strategy are established for tracking time-varying targets, without the need to redesign the controller every time the target is changed. The performance of the LB-SMPC strategy is experimentally demonstrated on an atmospheric pressure plasma jet (APPJ) testbed with prototypical applications in plasma medicine and materials processing. Real-time control comparisons with learning-based MPC with no uncertainty handling and offset-free MPC showcase the usefulness of LB-SMPC for predictive control of safety-critical systems with hard-to-model and/or time-varying dynamics.

Insurance demand in the presence of state dependent background uncertainty

This paper considers the optimal demand for insurance in the presence of state dependent background uncertainty, that is, the uncertainty is dependent on the states which are clarified by the realization of the loss. First, we consider a situation in which the state dependent background risk is ordered by the stochastic dominance relation. When the background risk conditional on the loss state is worse (better) than the no-loss state, we determine the condition in which the insurance demand is higher (lower) compared with the state independent background risk. Second, we consider a situation in which the uncertainty is different between the loss and the no-loss state. When ambiguity is introduced into the loss (no-loss) state, introducing ambiguity is higher (lower) compared with the risk case. Last, we discuss resolving the insurance demand puzzles by the different recognition of state dependent background uncertainty.

Forecasts of Hurricanes Using Large-Ensemble Outputs

AbstractThis paper describes the development of a model framework for Forecasts of Hurricanes Using Large-Ensemble Outputs (FHLO). FHLO quantifies the forecast uncertainty of a tropical cyclone (TC) by generating probabilistic forecasts of track, intensity, and wind speed that incorporate the state-dependent uncertainty in the large-scale field. The main goal is to provide useful probabilistic forecasts of wind at fixed points in space, but these require large ensembles [O(1000)] to flesh out the tails of the distributions. FHLO accomplishes this by using a computationally inexpensive framework, which consists of three components: 1) a track model that generates synthetic tracks from the TC tracks of an ensemble numerical weather prediction (NWP) model, 2) an intensity model that predicts the intensity along each synthetic track, and 3) a TC wind field model that estimates the time-varying two-dimensional surface wind field. The intensity and wind field of a TC evolve as though the TC were embedded in a time-evolving environmental field, which is derived from the forecast fields of ensemble NWP models. Each component of the framework is evaluated using 1000-member ensembles and four years (2015–18) of TC forecasts in the Atlantic and eastern Pacific basins. We show that the synthetic track algorithm generates tracks that are statistically similar to those of the underlying global ensemble models. We show that FHLO produces competitive intensity forecasts, especially when considering probabilistic verification statistics. We also demonstrate the reliability and accuracy of the probabilistic wind forecasts. Limitations of the model framework are also discussed.

Direct experimental test of forward and reverse uncertainty relations

The canonical Robertson-Schr\"{o}dinger uncertainty relation provides a loose bound for the product of variances of two non-commuting observables. Recently, several tight forward and reverse uncertainty relations have been proved which go beyond the traditional uncertainty relations. Here we experimentally test multifold of state-dependent uncertainty relations for the product as well as the sum of variances of two incompatible observables for photonic qutrits. These uncertainty relations are independent of any optimization and still tighter than the Robertson-Schr\"{o}dinger uncertainty relation and other ones found in current literatures. For the first time, we also test the state-dependent reverse uncertainty relations for the sum of variances of two incompatible observables, which implies another unique feature of preparation uncertainty in quantum mechanics. Our experimental results not only foster insight into a fundamental limitation on preparation uncertainty but also may contribute to the study of upper limit of precession measurements for incompatible observables.

The problem of reliable design of vector-field path following in the presence of uncertain course dynamics

Abstract Reliable guidance of fixed-wing Unmanned Aerial Vehicles (UAVs) is challenging, as their high maneuverability exposes them to several dynamical changes and parametric uncertainties. Reliability of state-of-the-art guidance methods is often at stake, as these methods heavily rely on precise UAV course dynamics, assumed in a decoupled first-order form with known time constant. To improve reliability of guidance for fixed-wing UAVs, this work proposes a novel vector field law that can handle uncertain course time constant and state-dependent uncertainty in the course dynamics arising from coupling. Stability is studied in the Lyapunov framework, while reliability of the proposed method is tested on a software-in-the loop UAV simulator. The simulations show that, in the presence of such uncertainty, the proposed method outperforms the standard vector field approaches.

Robust Data-driven Estimation of Wave Excitation Force for Wave Energy Converters

This paper proposes a data-driven technique to estimate the wave excitation force (WEF) which is an essential signal for wave forecasting and implementing power efficiency maximization control of Wave Energy Converters (WECs). A Bayesian probabilistic model of a WEC hydrodynamic system is described to generate robust WEF estimates. Specifically, the WEF uncertainty can be estimated based on observations through Gaussian Process (GP) modeling. It is shown that this modern way of incorporating the first principle modelling into a probabilistic framework has stronger robustness properties than the alternative of calculating estimates of a parametric function representation. Unlike the sample-based non-linear Kalman Filter, the means and covariances of the joint probabilities can be directly computed based on analytic moment matching that allow for reliable state-dependent uncertainty propagation. The results presented demonstrate the accuracy and robustness of the proposed data-driven wave excitation force estimator.

Asset Pricing With Data Revisions

This paper documents the asset pricing implications of the data release process of National Income and Product Accounts (NIPA) consumption expenditure. We show that initial consumption data releases are more suitable for asset pricing than final revised releases. This is because most revisions smooth out important short-term consumption growth fluctuations. This also explains the documented success of non-revised proposals like garbage and electricity and unfiltered consumption which recover these lost short-term fluctuations. Yet, first revisions specifically incorporate novel information and their magnitude is strongly linked to uncertainty about consumption growth. On this basis, we formulate a new consumption-based model, the Revised CCAPM, which incorporates these effects from the release process using only NIPA vintage data. It explains a striking 75% of the cross-sectional variation in average returns on the 25 size-value portfolios, including a large part of the value premium, when featuring the first data release and state-dependent uncertainty coming from first revisions. These effects are consistent with investors' state-dependent ambiguity attitudes and are priced in the financial market as such.

High-Precision Trajectory Tracking Control for Space Manipulator With Neutral Uncertainty and Deadzone Nonlinearity

This brief investigates the high-precision trajectory tracking control of space manipulator after capturing an unknown target. The neutral uncertainty induced by parameter variation poses great challenges to the controller design. Moreover, the existence of actuator deadzone nonlinearity makes the situation more complicated. In order to address the aforementioned difficulties, a novel control scheme is proposed in this brief. First, to reduce the conservativeness in the descriptions of uncertainties, the dynamic of space manipulator is described by an uncertain system with neutral uncertainty, state-dependent uncertainty, and norm-bounded uncertainty for the first time. Then, by incorporating a novel deadzone compensation control effort, a composite controller is proposed to attenuate these uncertainties and compensate the deadzone nonlinearity. Finally, stability and robustness analyses are carried out, and simulation results are given to demonstrate the effectiveness of the proposed methods.

Valuing portfolios of interdependent real options under exogenous and endogenous uncertainties

Although the value of portfolios of real options is often affected by both exogenous and endogenous sources of uncertainty, most existing valuation approaches consider only the former and neglect the latter. In this paper, we introduce an approach for valuing portfolios of interdependent real options under both types of uncertainty. In particular, we study a large portfolio of options (deferment, staging, mothballing, abandonment) under conditions of four underlying uncertainties. Two of the uncertainties, decision-dependent cost to completion and state-dependent salvage value, are endogenous, the other two, operating revenues and their growth rate, are exogenous. Assuming that endogenous uncertainties can be exogenised, we formulate the valuation problem as a discrete stochastic dynamic program. To approximate the value of this optimisation problem, we apply a simulation-and-regression-based approach and present an efficient valuation algorithm. The key feature of our algorithm is that it exploits the problem structure to explicitly account for reachability – that is the sample paths in which resource states can be reached. The applicability of the approach is illustrated by valuing an urban infrastructure investment. We conduct a reachability analysis and show that the presence of the decision-dependent uncertainty has adverse computational effects as it increases algorithmic complexity and reduces simulation efficiency. We investigate the way in which the value of the portfolio and its individual options are affected by the initial operating revenues, and by the degrees of exogenous and endogenous uncertainty. The results demonstrate that ignoring endogenous, decision- and state-dependent uncertainty can lead to substantial over- and under-valuation, respectively.

New AFSMC Method for Nonlinear System with State-dependent Uncertainty: Application to Hexapod Robot Position Control

Conventional Adaptive Fuzzy Sliding Mode Control (AFSMC) method is extended for nonlinear affine systems with state-dependent upper bound of uncertainties. More general affine model of the system with state-dependent uncertainties is proposed where such a model is more applicable in robotics. Position control of a Stewart Manipulator (SM) is then considered as a challenging case study to experimentally verify the effectiveness of the proposed Extended AFSMC (E-AFSMC) method. The proposed method is encompassed of a fuzzy system for estimation of a nonlinear system, a robust controller for compensation of uncertainties and some appropriate adaptation laws for optimization of performance. The second Lyapunov theorem and Barbalat lemma are used to prove the closed-loop asymptotic stability. Furthermore, numerical simulations depict the robustness of the proposed controller and in particular, under the very critical situation of actuator saturation and unexpected uncertainties. The effectiveness of the proposed control method is validated through experimental results.

Tighter uncertainty and reverse uncertainty relations

We prove a few state-dependent uncertainty relations for the product as well as the sum of variances of two incompatible observables. These uncertainty relations are shown to be tighter than the Robertson-Schr\"odinger uncertainty relation and other ones existing in the current literature. Also, we derive a state-dependent upper bound to the sum and the product of variances using the reverse Cauchy-Schwarz inequality and the Dunkl-Williams inequality. Our results suggest that not only can we not prepare quantum states for which two incompatible observables can have sharp values, but also we have both the lower and the upper limits on the variances of quantum mechanical observables at a fundamental level.

Aperiodic disturbance rejection in a modified repetitive‐control system with non‐linear uncertainty

This study deals with the problem of aperiodic-disturbance rejection for a class of plants with non-linear state-dependent uncertainty in a modified repetitive-control system (MRCS). Since both the non-linear uncertainty and the disturbances are often unknown, an equivalent-input-disturbance (EID) estimator is constructed using a full-order state observer with variant system matrix to estimate them. The EID estimate, which exhibits the overall effect on the output of the non-linear uncertainty and all types of disturbances, is incorporated into a linear repetitive control law to compensate for the non-linear uncertainty and the disturbances. A continuous-discrete two-dimensional model of the EID-based MRCS is built that enables the preferential adjustment of the control and learning actions. A sufficient condition for the robust stability for the EID-based MRCS is given in terms of a linear matrix inequality. It yields the parameters of the repetitive controller and the EID estimator. Finally, a numerical example demonstrates the design procedures and illustrates the effectiveness of the method.

New state dependent uncertainty principle and design of objective function for optimisation: utility of kinetic and potential energy uncertainties

In this work a new state dependent Uncertainty Principle between kinetic and potential energy is formulated using Time independent Schrodinger Equation and Correspondence Principle. Validity of the above said Uncertainty Principle is also demonstrated by linear variational calculation of a one-dimensional model potential. An objective function is also been designed, to support excited state non-linear variational description of bound stationary state of any type of system. This newly designed function will serve as an alternative of ΔH in excited state nonlinear variational purpose. The utility of these objective functions is demonstrated by already evolved non-linear variational technique by using a one-dimensional model potential where crux of the problem lies on the determination of the node of excited states.

Coupling ensemble Kalman filter with four-dimensional variational data assimilation

This study examines the performance of coupling the deterministic four-dimensional variational assimilation system (4DVAR) with an ensemble Kalman filter (EnKF) to produce a superior hybrid approach for data assimilation. The coupled assimilation scheme (E4DVAR) benefits from using the state-dependent uncertainty provided by EnKF while taking advantage of 4DVAR in preventing filter divergence: the 4DVAR analysis produces posterior maximum likelihood solutions through minimization of a cost function about which the ensemble perturbations are transformed, and the resulting ensemble analysis can be propagated forward both for the next assimilation cycle and as a basis for ensemble forecasting. The feasibility and effectiveness of this coupled approach are demonstrated in an idealized model with simulated observations. It is found that the E4DVAR is capable of outperforming both 4DVAR and the EnKF under both perfect- and imperfect-model scenarios. The performance of the coupled scheme is also less sensitive to either the ensemble size or the assimilation window length than those for standard EnKF or 4DVAR implementations.

BOUNDED ROBUST CONTROLLER DESIGN FOR MISMATCHED STATE-DEPENDENT UNCERTAINTY

A bounded robust control is presented for dealing with matched external disturbance and mismatched state-dependent uncertainty in linear time-invariart systems. By using the famous pole-assignment method, an important switching vector is selected first before the controller is designed. Based on the Lyapunov theory, system stability is proven, and the upper bound of the control input is discussed. It is found that the system initial state has to be located in a suitable region determined by the upper bounds on the control input, the matched disturbance and the mismatched uncertainty. A numerical example is used to demonstrate use of the bounded robust controller in simulation.