Bounded Measurement Noise Research Articles

Robust Uncertainty Bounds in Reproducing Kernel Hilbert Spaces: A Convex Optimization Approach

The problem of establishing out-of-sample bounds for the values of an unknown ground-truth function is considered. Kernels and their associated Hilbert spaces are the main formalism employed herein, along with an observational model where outputs are corrupted by bounded measurement noise. The noise can originate from any compactly supported distribution, and no independent assumptions are made on the available data. In this setting, we show how computing tight, finite-sample uncertainty bounds amounts to solving parametric quadratically constrained linear programs. Next, the properties of our approach are established, and its relationship with another method is studied. Numerical experiments are presented to exemplify how the theory can be applied in various scenarios and to contrast it with other closed-form alternatives.

Robust collaborative 3D target localization using UAVs with state uncertainties

This paper addresses the problem of target localization using measurements collected by a fleet of cooperative Unmanned Aerial Vehicles (UAVs). The positions and orientations of UAVs are partly unknown. The measurement noise and other uncertainties of the system are assumed to be unknown but bounded. Moreover, some of the UAVs of the feet may produce erroneous data (outliers). Several UAVs are assumed to observe a single target, to exchange the collected information. Then they have to characterize the set of all target locations consistent with the measurements, the bounded measurement noise, and the bounded uncertainty of the position and orientation of the UAVs while being robust to possible outliers. A set-membership approach based on set inversion via interval analysis is used. Thick intervals are required to account for the uncertainty in the location and orientation of the UAVs. The robustness to outliers is obtained using q-relaxed intersection.

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.

Fixed-time parameter estimation via the discrete-time DREM method

A simple fixed-time converging estimation algorithm is presented for a linear regression using the dynamic regressor extension and mixing (DREM) method within a discrete-time setting, with a persistently exciting regressor and bounded measurement noises. The solution is based on Kreisselmeier's filters and is computationally simpler than the existing analogs.

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>

Optimal robust exact differentiation via linear adaptive techniques

The problem of differentiating a function with bounded second derivative in the presence of bounded measurement noise is considered in both continuous-time and sampled-data settings. Fundamental performance limitations of causal differentiators, in terms of the smallest achievable worst-case differentiation error, are shown. A robust exact differentiator is then constructed via the adaptation of a single parameter of a linear differentiator. It is demonstrated that the resulting differentiator exhibits a combination of properties that outperforms existing continuous-time differentiators: it is robust with respect to noise, it instantaneously converges to the exact derivative in the absence of noise, and it attains the smallest possible—hence optimal—upper bound on its differentiation error under noisy measurements. For sample-based differentiators, the concept of quasi-exactness is introduced to classify differentiators that achieve the lowest possible worst-case error based on sampled measurements in the absence of noise. A straightforward sample-based implementation of the proposed linear adaptive continuous-time differentiator is shown to achieve quasi-exactness after a single sampling step as well as a theoretically optimal differentiation error bound that, in addition, converges to the continuous-time optimal one as the sampling period becomes arbitrarily small. A numerical simulation illustrates the presented formal results.

Accelerated convergence with improved robustness for discrete-time parameter estimation

The dynamic regressor extension and mixing (DREM) method provides a fixed-time converging parameter estimator for a persistently excited regressor under bounded measurement noises. This note aims to develop this approach for cases with weaker excitation and regressor constraints. Several nonlinear estimation schemes with fixed-time convergence rates and improved measurement noise robustness properties are proposed here.

Multi‐agent consensus with stopping rules under bounded measurement noise

AbstractA multi‐agent consensus protocol is investigated over noisy undirected connected graphs. It is supposed that each agent can know her neighbors' states with bounded noise. A theoretical stopping rule having a probabilistic guarantee is established for the consensus protocol, which explicitly relates the number of iterations with the closeness of the agreement. The stopping rule is derived by utilizing Bernstein inequality that deals with a given noise bound explicitly. The sample complexity with respect to the number of agents is also investigated, where it is shown that the number of iterations required by the proposed stopping rule based on the bound of noise itself is generally smaller than that of the conventional stopping rules that utilize bound of noise variance. The results are demonstrated through numerical examples.

An Output Feedback Discrete-Time Controller for the DC-DC Buck Converter

This paper presents a discrete-time output feedback controller to regulate the output voltage of a DC-DC buck converter. The proposal’s main feature is the application of a discrete-time equivalent of the robust exact filtering differentiator. First, the document exposes a theoretical analysis of the closed-loop system, where it is considered the problem of implementing a real-time differentiator with a good relationship between exactness and noise filtration performance. Hence, secondly, the controller in a laboratory setup is presented. The first experimental results suggest that the proposed controller exhibits good robustness against noise and maintains the asymptotic accuracy, even with saturated control inputs, as in the case of the DC-DC buck converter. Consequently, aiming to verify the features of the proposed method, the controller is validated through multiple experiments, showing satisfactory voltage tracking accuracy, good suppression of instantaneous load and supply voltage disturbances, and robustness against bounded measurement noise.

Operational matrix based set-membership method for fractional order systems parameter identification

In this paper, a new method is proposed to identify the coefficients and differentiation orders of fractional order systems with measurement noise. The proposed method combines the operational matrix method and the set-membership method. First, the block pulse functions operational matrix of the fractional differentiation is used to convert the fractional order system to an algebraic system. Then, the coefficients and differentiation orders are simultaneously estimated through a nest loop optimization process, where the optimal bounding ellipsoid set-membership algorithm is utilized to estimate the system’s coefficients and the orders are estimated with the interior-point method. The proposed method can accurately estimate the coefficients and differentiation orders of fractional order systems under any bounded measurement noise with less computational effort. Experimental results demonstrate the effectiveness of the proposed method.

Adaptive Data-Driven Control for Linear Time Varying Systems

In this paper, we propose an adaptive data-driven control approach for linear time varying systems, affected by bounded measurement noise. The plant to be controlled is assumed to be unknown, and no information in regard to its time varying behaviour is exploited. First, using set-membership identification techniques, we formulate the controller design problem through a model-matching scheme, i.e., designing a controller such that the closed-loop behaviour matches that of a given reference model. The problem is then reformulated as to derive a controller that corresponds to the minimum variation bounding its parameters. Finally, a convex relaxation approach is proposed to solve the formulated controller design problem by means of linear programming. The effectiveness of the proposed scheme is demonstrated by means of two simulation examples.

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.

Set Membership Identification of Linear Systems With Guaranteed Simulation Accuracy

The problem of model identification for linear systems is considered, using a finite set of sampled data affected by a bounded measurement noise, with unknown bound. The objective is to identify one-step-ahead models and their accuracy in terms of worst-case simulation error bounds. To do so, the Set Membership identification framework is exploited. Theoretical results are derived, allowing one to estimate the noise bound and system decay rate. Then, these quantities and the data are employed to define the Feasible Parameter Set (FPS), which contains all possible models compatible with the available information. Here, the estimated decay rate is used to refine the standard FPS formulation, by adding constraints that enforce the desired converging behavior of the models' impulse response. Moreover, guaranteed simulation error bounds for an infinite future horizon are derived, improving over recent results pertaining to finite simulation horizon only. These bounds are the basis for a result and method to guarantee asymptotic stability of the identified model. Finally, the desired one-step-ahead model is identified by means of numerical optimization, and the related simulation error bounds are evaluated. Both input-output and state-space model structures are addressed. The approach is showcased on a numerical example and on real-world experimental data of the roll rate dynamics of an autonomous glider.

Robust Adaptive Model Predictive Control with Worst-Case Cost

Abstract A robust adaptive model predictive control (MPC) algorithm is presented for linear, time invariant systems with unknown dynamics and subject to bounded measurement noise. The system is characterized by an impulse response model, which is assumed to lie within a bounded set called the feasible system set. Online set-membership identification is used to reduce uncertainty in the impulse response. In the MPC scheme, robust constraints are enforced to ensure constraint satisfaction for all the models in the feasible set. The performance objective is formulated as a worst-case cost with respect to the modeling uncertainties. That is, at each time step an optimization problem is solved in which the control input is optimized for the worst-case plant in the uncertainty set. The performance of the proposed algorithm is compared to an adaptive MPC algorithm from the literature using Monte-Carlo simulations.

Consensus of second-order multi-agent systems under unknown but bounded measurement noises

In this paper, we investigate the static ϵ-consensus problem of multi-agent systems with double-integrator agent dynamics over a connected undirected communication topology. It is assumed that only noisy measurements of neighbors’ states are available. We propose two types of consensus protocols using and without using neighbors’ velocity information, respectively. With the knowledge of the bounds of measurement noises, we show that, under these consensus protocols, the velocities of all agents achieve consensus at zero and the positions of all agents achieve ϵ-consensus. Numerical examples are shown to illustrate the theoretical results.

Exponential input-to-state stability of globally Lipschitz time-delay systems under sampled-data noisy output feedback and actuation disturbances

In this paper we deal with the problem of global exponential practical stability preservation for globally Lipschitz time-delay systems, by Euler emulation of continuous-time dynamic output feedback controllers affected by measurement noises and actuation disturbances. Nonlinear time-delay systems not necessarily affine in the control input are studied. It is shown that, if the continuous-time closed-loop system at hand is globally exponentially stable and the maps describing the plant and the continuous-time dynamic output feedback controller are globally Lipschitz, then, under suitably fast sampling, the Euler emulation of the continuous-time controller at hand preserves the global exponential stability of the sampled-data closed-loop system (no matter whether periodic or aperiodic sampling is used). In the case of bounded measurement noises and bounded actuation disturbances affecting the control law, it is proved that, under suitable fast sampling, (global) exponential input-to-state stability with respect to both these external inputs is guaranteed. A generalisation of the Halanay's inequality is used as a tool in order to prove the results. The existence of a Lyapunov-Krasovskii functional for the continuous-time closed-loop system is sufficient to ensure the preservation of the global exponential practical stability. On the other hand, the explicit knowledge of a Lyapunov-Krasovskii functional allows us to compute an upper bound for the sampling period. An example is presented which validates the theoretical results.

PDα‐type iterative learning control for fractional‐order linear continuous‐time switched systems

AbstractFor a class of fractional‐order linear continuous‐time switched systems specified by an arbitrary switching rule, this paper proposes a PDα‐type fractional‐order iterative learning control algorithm. For systems disturbed by bounded measurement noise, the robustness of PDα‐type algorithm is first discussed in the iteration domain and the tracking performance is analyzed. Next, a sufficient condition for monotone convergence of the algorithm is studied when external noise is absent. The results of analysis and simulation illustrate the feasibility and effectiveness of the proposed control algorithm.

Complexity reduction in explicit MPC: A reachability approach

We propose to reduce the complexity of explicit MPC controllers by removing regions that will never be reached during the closed-loop evolution from a given set of initial conditions. The identification of such regions is done by solving a reachability analysis problem, formulated as a mixed-integer feasibility program. The procedure directly accounts for possible discrepancies between the prediction model and the actual plant dynamics by, among other things, considering a case where state measurements are affected by an unknown, but bounded measurement noise. The result of the procedure is the reduction of explicit MPC complexity without sacrificing closed-loop performance.

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 Compensation of Bounded External Disturbances and Measurement Noises for Nonlinear Systems

Abstract This paper is dedicated to the problem of compensation of bounded external disturbances in the presence of bounded measurement noises for nonlinear systems. The synthesis of a linear feedback controller minimizing the size of invariant ellipsoids of a closed-loop system is considered. Based on the method of Lyapunov functions, S-procedure and the alternating optimization method, an iterative algorithm for finding optimal controller parameters is proposed. The efficiency of the obtained algorithm is demonstrated.