Absence Of Measurement Noise Research Articles

Worst-case error bounds for the super-twisting differentiator in presence of measurement noise

The super-twisting differentiator, also known as the first-order robust exact differentiator, is a well known sliding mode differentiator. In the absence of measurement noise, it achieves exact reconstruction of the time derivative of a function with bounded second derivative. This note proposes an upper bound for its worst-case differentiation error in the presence of bounded measurement noise, based on a novel Lipschitz continuous Lyapunov function. It is shown that the bound can be made arbitrarily tight and never exceeds the true worst-case differentiation error by more than a factor of two. A numerical simulation illustrates the results and also demonstrates the non-conservativeness of the proposed bound.

Asymptotically stable optimal multi-rate rigid body attitude estimation based on lagrange-d'alembert principle

<abstract><p>The rigid body attitude estimation problem is treated using the discrete-time Lagrange-d'Alembert principle. Three different possibilities are considered for the multi-rate relation between angular velocity measurements and direction vector measurements for attitude: 1) integer relation between sampling rates, 2) time-varying sampling rates, 3) non-integer relation between sampling rates. In all cases, it is assumed that angular velocity measurements are sampled at a higher rate compared to the inertial vectors. The attitude determination problem from two or more vector measurements in the body-fixed frame is formulated as Wahba's problem. At instants when direction vector measurements are absent, a discrete-time model for attitude kinematics is used to propagate past measurements. A discrete-time Lagrangian is constructed as the difference between a kinetic energy-like term that is quadratic in the angular velocity estimation error and an artificial potential energy-like term obtained from Wahba's cost function. An additional dissipation term is introduced and the discrete-time Lagrange-d'Alembert principle is applied to the Lagrangian with this dissipation to obtain an optimal filtering scheme. A discrete-time Lyapunov analysis is carried out to show that the optimal filtering scheme is asymptotically stable in the absence of measurement noise and the domain of convergence is almost global. For a realistic evaluation of the scheme, numerical experiments are conducted with inputs corrupted by bounded measurement noise. These numerical simulations exhibit convergence of the estimated states to a bounded neighborhood of the actual states.</p></abstract>

Computing a projection operator onto the null space of a linear imaging operator: tutorial.

Many imaging systems can be approximately described by a linear operator that maps an object property to a collection of discrete measurements. However, even in the absence of measurement noise, such operators are generally "blind" to certain components of the object, and hence information is lost in the imaging process. Mathematically, this is explained by the fact that the imaging operator can possess a null space. All objects in the null space, by definition, are mapped to a collection of identically zero measurements and are hence invisible to the imaging system. As such, characterizing the null space of an imaging operator is of fundamental importance when comparing and/or designing imaging systems. A characterization of the null space can also facilitate the design of regularization strategies for image reconstruction methods. Characterizing the null space via an associated projection operator is, in general, a computationally demanding task. In this tutorial, computational procedures for establishing projection operators that map an object to the null space of a discrete-to-discrete imaging operator are surveyed. A new machine-learning-based approach that employs a linear autoencoder is also presented. The procedures are demonstrated by use of biomedical imaging examples, and their computational complexities and memory requirements are compared.

A globally Mittag-Leffler bounded high-gain observer for systems with unknown dynamics and noisy measurements

In this work, we present a globally Mittag-Leffler bounded high-gain observer for fractional order nonlinear systems with unmodeled dynamics and additive measurement noise at the output. Our proposal starts from an alternative representation of the fractional order system, whose output does not depend on the additive measurement noise and in which the original system’s output is treated as an additional state variable. This representation allows us two things: 1) to simultaneously estimate the state variables and the uncertain term and 2) to incorporate into the design scheme a fractional integral-type contribution, which is useful to give robustness against the measurement noise and the unmodeled dynamics, as well as to attenuate the noise amplification, typical of any high-gain observer. Through the corresponding mathematical analysis, we prove that the estimation error of the proposed observer is uniformly bounded and converges asymptotically to a globally Mittag-Leffler compact attractive set, this is, the proposed observer is globally Mittag-Leffler bounded. Additionally, we show that under certain conditions, such as an integer derivation order or the absence of measurement noise, the proposed observer exhibits some particular properties. Finally, we consider a continuously stirred biochemical reactor to exemplify our design methodology. The numerical results confirm that the observer is able to accurately estimate the state variables as well as the uncertainty term of the fractional model. In other words, the globally Mittag-Leffler bounded high-gain observer is robust against measurement noise and uncertainties.

System identification approach for inverse optimal control of finite-horizon linear quadratic regulators

The inverse optimal control for finite-horizon discrete-time linear quadratic regulators is investigated in this paper, which is to estimate the parameters in the objective function using noisy measurements of partial optimal states only. By the Pontryagin’s minimum principle, the concerned inverse optimal control problem is recast as the identification of a parameterized causal-and-anticausal mixed system excited by boundary conditions. Sufficient identifiability conditions for the unknown parameters are provided in terms of the system model itself, rather than relying on the exact values of optimal states or control inputs. In addition, an elegant algebraic solution is provided for the concerned identification problem that is inherently a challenging optimization problem with trilinear equality constraints, and it can recover the true parameters (up to a scalar ambiguity) in the absence of measurement noise or can consistently identify the parameters (up to a scalar ambiguity) in the presence of white measurement noise. The presented algebraic solution relies on recursive matrix calculations so that its computational burden is much less than directly solving a high-dimensional non-convex optimization problem as done in many existing works. The effectiveness of the proposed method as well as its noise sensitivity issue is illustrated by simulation examples.

Resolving nonuniform temperature distributions with single-beam absorption spectroscopy. Part I: Theoretical capabilities and limitations

Absorption spectroscopy is traditionally used to determine the average gas temperature and species concentration along the laser line-of-sight by measuring the magnitude of two or more absorption transitions with different temperature dependence. Previous work has shown that the nonlinear temperature dependence of the absorption strength of each transition, set by the lower-state energy, E″, can be used to infer temperature variations along the laser line-of-sight. In principle, measuring more absorption transitions with broader bandwidth light sources improves the ability to resolve temperature variations. Here, we introduce a singular value decomposition framework in order to explore the theoretical limits to resolving temperature distributions with single-beam line-of-sight absorption measurements. We show that in the absence of measurement noise or error, only the first ∼14 well-selected absorption features improve the temperature resolution, and a Tikhonov regularization method improves the accuracy of the temperature inversion, particularly for recovery of the maximum gas temperature along the laser beam. We use inversion simulations to demonstrate that one can resolve a selection of temperature distributions along a laser beam line-of-sight to within 3% for the sample cases analyzed. In part II of this work, we explore the influence of measurement noise and error, and experimentally demonstrate the technique to show that there is benefit to measuring additional absorption transitions under real conditions.

Uncertainty Estimation in Extremum Seeking Control of Unknown Static Maps

This letter proposes a design of extremum seeking controllers that guarantees precise convergence of the control system to the unknown optimizer of a measured unknown cost function. The approach introduces an uncertainty estimation technique that provides an estimate of the worst case uncertainty associated with the unknown optimizer. An uncertainty set update algorithm is proposed to reduce the radius of the uncertain set as new data is received. The radius of the uncertainty set is used to reduce the required dither amplitude. The convergence to the unknown optimum and the removal of the dither signal are achieved simultaneously. Asymptotic convergence of the extremum seeking control system to the unknown minimizer is achieved in the absence of measurement noise. A simulation example is provided to demonstrate the effectiveness of the approach.

Robustness analysis of nonlinear observers for the slow variables of singularly perturbed systems

SummaryEstimation of unmeasured variables is a crucial objective in a broad range of applications. However, the estimation process turns into a challenging problem when the underlying model is nonlinear and even more so when additionally it exhibits multiple time scales. The existing results on estimation for systems with two time scales apply to a limited class of nonlinear plants and observers. We focus on analyzing nonlinear observers designed for the slow state variables of nonlinear singularly perturbed systems. Moreover, we consider the presence of bounded measurement noise in the system. We generalize current results by considering broader classes of plants and estimators to cover reduced‐order, full‐order, and higher‐order observers. First, we show that the singularly perturbed system has bounded solutions under an appropriate set of assumptions on the corresponding boundary layer and reduced systems. We then exploit this property to prove that, under reasonable assumptions, the error dynamics of the observer designed for the reduced system are semiglobally input‐to‐state practically stable when the observer is implemented on the original plant. We also conclude stability results when the measurement noise belongs to . In the absence of measurement noise, we state results on semiglobal practical asymptotical stability for the error dynamics. We illustrate the generality of our main results through three classes of systems with corresponding observers and one numerical example.

Necessary and Sufficient Conditions for Leader-Following Bipartite Consensus With Measurement Noise

This paper considers leader-following bipartite consensus of single-integrator multiagent systems in the presence of measurement noise. To attenuate the noise, a time-varying consensus gain ${q}$ ( ${t}$ ) is introduced into the stochastic approximation-type protocol. Necessary and sufficient conditions for ensuring a strong mean square leader-following bipartite consensus are given. In particular, in the absence of measurement noise, the convergence speed of error dynamics is dependent on the eigenvalues of Laplacian and the rate of ${\int ^{{t}}_{0}{q}({s})\text {d}{s}}$ approaching infinity. By appropriately choosing ${q}$ ( ${t}$ ), the speed of leader-following bipartite consensus convergence can be improved in a fixed communication topology. It is proven that conditions for the signed digraph to be structurally balanced and having a spanning tree are necessary and sufficient to ensure leader-following bipartite consensus, regardless of measurement noise.

Enhancement of low-voltage ride-through capability of permanent magnet synchronous generator wind turbine by applying state-estimation technique

Purpose The purpose of this study is to propose a control scheme based on state estimation algorithm to improve zero or low-voltage ride-through capability of permanent magnet synchronous generator (PMSG) wind turbine. Design/methodology/approach Based on the updated grid codes, during and after faults, it is necessary to ensure wind energy generation in the network. PMSG is a type of wind energy technology that is growing rapidly in the network. The control scheme based on extended Kalman filter (EKF) is proposed to improve the low voltage ride-through (LVRT) capability of the PMSG. In the control scheme, because the state estimation algorithm is applied, the requirement of DC link voltage measurement device and generator speed sensor is removed. Furthermore, by applying this technique, the extent of possible noise on measurement tools is reduced. Findings In the proposed control scheme, zero or low-voltage ride-through capability of PMSG is enhanced. Furthermore, the requirement of DC link voltage measurement device and generator speed sensor is removed and the amount of possible noise on the measurement tools is minimized. To evaluate the ability of the proposed method, four different cases, including short and long duration short circuit fault close to PMSG in the presence and absence of measurement noise are studied. The results confirm the superiority of the proposed method. Originality/value This study introduces EKF to enhance LVRT capability of a PMSG wind turbine.

Sampled-data observers for 1-D parabolic PDEs with non-local outputs

The present work provides a systematic approach for the design of sampled-data observers to a wide class of 1-D, parabolic PDEs with non-local outputs. The studied class of parabolic PDEs allows the presence of globally Lipschitz nonlinear and non-local terms in the PDE. Two different sampled-data observers are presented: one with an inter-sample predictor for the unavailable continuous measurement signal and one without an inter-sample predictor. Explicit conditions on the upper diameter of the (uncertain) sampling schedule for both designs are derived for exponential convergence of the observer error to zero in the absence of measurement noise and modeling errors. Moreover, explicit estimates of the convergence rate can be deduced based on the knowledge of the upper diameter of the sampling schedule. When measurement noise and/or modeling errors are present, Input-to-Output Stability (IOS) estimates of the observer error hold for both designs with respect to noise and modeling errors. The main results are illustrated by two examples which show how the proposed methodology can be extended to other cases (e.g., boundary point measurements).

Predictive control of unstable time delay series cascade processes with measurement noise

In this paper, generalized predictive control strategy is applied for unstable time delay series cascade processes. The control structure consists of a secondary loop with a secondary controller designed based on simple internal model control (IMC) method and the primary loop consists of a set-point achievement controller and is designed using synthesis method. Further, a generalized predictor consisting of two filters that deals with time delay, disturbance rejection and noise rejection capabilities is also designed in the primary loop. The filters are designed to ensure internal stability as well. For the predictor, only one tuning parameter is involved, which can be tuned in such a way that a trade-off is attained between disturbances attenuation and performance under uncertainties. In this work, various examples of cascade processes with unstable dynamics and time delay are considered for simulation. Results obtained from simulation reveals that with the suggested predictive control scheme, enhanced closed loop control performances are obtained for both nominal and perturbed conditions both in the presence and absence of measurement noise. The comparison is also verified quantitatively with the literature methods using integral absolute error (IAE) and total variation (TV) measures.

Distributed Kalman–Bucy Filter With Embedded Dynamic Averaging Algorithm

This paper focuses on distributed estimation using networked sensor agents with local measurements and local communication. A distributed Kalman–Bucy filter implementation is proposed and analyzed for the general case of continuous-time linear time-varying systems as compared to linear time-invariant systems. The sensor agents employ a distributed average tracking algorithm and its extension that accounts for bounded noise to estimate the averages of certain time-varying signals by communicating with their local neighbors in the network. Such estimates are then used to recover certain information in the implementation of the distributed Kalman–Bucy filter. It is shown that using the proposed distributed filter, in the absence of measurement noise, the distributed local estimates approach the centralized filter's estimate asymptotically. In the presence of bounded measurement noise, the distributed local estimates asymptotically approach the centralized filter's estimate within some bound. Simulation results illustrate the good performance of the distributed filter.

Optimal estimation of sensor biases for asynchronous multi-sensor data fusion

An important step in a multi-sensor surveillance system is to estimate sensor biases from their noisy asynchronous measurements. This estimation problem is computationally challenging due to the highly nonlinear transformation between the global and local coordinate systems as well as the measurement asynchrony from different sensors. In this paper, we propose a novel nonlinear least squares (LS) formulation for the problem by assuming the existence of a reference target moving with an (unknown) constant velocity. We also propose an efficient block coordinate decent (BCD) optimization algorithm, with a judicious initialization, to solve the problem. The proposed BCD algorithm alternately updates the range and azimuth bias estimates by solving linear least squares problems and semidefinite programs (SDPs). In the absence of measurement noise, the proposed algorithm is guaranteed to find the global solution of the problem and the true biases. Simulation results show that the proposed algorithm significantly outperforms the existing approaches in terms of the root mean square error (RMSE).

Actuator and sensor fault detection and isolation for nonlinear systems subject to uncertainty

SummaryThis work addresses the problem of simultaneous actuator and sensor fault detection and isolation (FDI) for control affine nonlinear uncertain systems in the absence of measurement noise. The FDI is achieved by using a bank of filters, which utilize a subset of the measurements along with prescribed values of the control actuators to estimate states and compute expected process behavior. Residuals are next defined as the difference between the observed and expected behavior. Detectability conditions are developed, which, upon satisfaction, ensure that each residual remains sensitive to a subset of fault scenarios in the presence of uncertainty. To this end, first the ability of observers in providing bounded estimation error for a generalized class of nonlinear uncertain systems is rigorously established. These bounds allow determining thresholds that account for the impact of uncertainty on each residual. Finally, the ability of the proposed framework to achieve FDI by ensuring a unique residual breaching pattern for each fault scenario is established. The efficacy of the FDI framework subject to uncertainty and measurement noise is illustrated using a chemical reactor example.

Accurate and Effective Localization of an Object in Large Equal Radius Scenario

This paper develops an attractive estimator for locating an object using time differences of arrival with inaccurate sensor positions in the large equal radius (LER) scenario. The estimator is developed through a geometric approach without the need of introducing an auxiliary variable. It is shown analytically to be able to reach the Cramer-Rao Lower Bound (CRLB) accuracy under Gaussian noise over the small error region when the LER condition is strong. The estimation bias resulted from the LER modeling in the absence of measurement noise is derived and it can be compensated if it is significant when the LER condition is weak. The proposed estimator is more computationally efficient than the other LER positioning estimators from the literature. To complete the study, a detector for the LER scenario is included.

Rigid body pose estimation based on the Lagrange–d’Alembert principle

Stable estimation of rigid body pose and velocities from noisy measurements, without any knowledge of the dynamics model, is treated using the Lagrange–d’Alembert principle from variational mechanics. With body-fixed vision and inertial sensor measurements, a Lagrangian is obtained as the difference between a kinetic energy-like term that is quadratic in velocity estimation error and the sum of two artificial potential functions; one obtained from a generalization of Wahba’s function for attitude estimation and another which is quadratic in the position estimate error. An additional dissipation term that is linear in the velocity estimation error is introduced, and the Lagrange–d’Alembert principle is applied to the Lagrangian with this dissipation. A Lyapunov analysis shows that the state estimation scheme so obtained provides stable asymptotic convergence of state estimates to actual states in the absence of measurement noise, with an almost global domain of attraction. This estimation scheme is discretized for computer implementation using discrete variational mechanics, as a first order Lie group variational integrator. The discrete estimation scheme can also estimate velocities from such onboard sensor measurements. Moreover, all states can be estimated during time periods when measurements of only two inertial vectors, the angular velocity vector, and one feature point position vector are available in body frame. In the presence of bounded measurement noise in the vector measurements, numerical simulations show that the estimated states converge to a bounded neighborhood of the true states.

Adaptive Control for Regulation of a Quadratic Function of the State

We propose an adaptive controller for a dynamical system, where we need a quadratic function of the state to track a given reference signal. This problem appears in the control of generators in weak-grid conditions for example. While the controller can measure the tracking error, the main difficulty arises from the fact that the parameters of the quadratic function itself are not known to the controller. Our approach consists of simultaneously estimating the quadratic function while tracking the reference signal, similar to the approach employed in adaptive control. The quadratic structure of the tracking function necessitates, however, a new adaptive law for estimating the parameters. Even though estimation and control are in general two contradicting requirements, using this new adaptive law and a multilevel controller we prove that the tracking error converges to zero in the absence of measurement noise. In the presence of bounded noise we show that the tracking error can be driven to a neighborhood around the origin.

Rigid body attitude estimation based on the Lagrange–d’Alembert principle

Estimation of rigid body attitude and angular velocity without any knowledge of the attitude dynamics model is treated using the Lagrange–d’Alembert principle from variational mechanics. It is shown that Wahba’s cost function for attitude determination from two or more non-collinear vector measurements can be generalized and represented as a Morse function of the attitude estimation error on the Lie group of rigid body rotations. With body-fixed sensor measurements of direction vectors and angular velocity, a Lagrangian is obtained as the difference between a kinetic energy-like term that is quadratic in the angular velocity estimation error and an artificial potential obtained from Wahba’s function. An additional dissipation term that depends on the angular velocity estimation error is introduced, and the Lagrange–d’Alembert principle is applied to the Lagrangian with this dissipation. A Lyapunov analysis shows that the state estimation scheme so obtained provides stable asymptotic convergence of state estimates to actual states in the absence of measurement noise, with an almost global domain of attraction. These estimation schemes are discretized for computer implementation using discrete variational mechanics. A first order Lie group variational integrator is obtained as a discrete-time implementation. In the presence of bounded measurement noise, numerical simulations show that the estimated states converge to a bounded neighborhood of the actual states.

Vectorial Phase Retrieval of 1-D Signals

Reconstruction of signals from measurements of their spectral intensities, also known as the phase retrieval problem, is of fundamental importance in many scientific fields. In this paper we present a novel framework, denoted as vectorial phase retrieval, for reconstruction of pairs of signals from spectral intensity measurements of the two signals and of their interference. We show that this new framework can alleviate some of the theoretical and computational challenges associated with classical phase retrieval from a single signal. First, we prove that for compactly supported signals, in the absence of measurement noise, this new setup admits a unique solution. Next, we present a statistical analysis of vectorial phase retrieval and derive a computationally efficient algorithm to solve it. Finally, we illustrate via simulations, that our algorithm can accurately reconstruct signals even at considerable noise levels.