Robust Differentiator Research Articles

A survey on High-Order Sliding-Mode control design using Lyapunov functions

A survey of (some) recent Lyapunov-based methods to analyze and design High-Order Sliding-Mode (HOSM) controllers and observers is presented. This overview includes also some novel algorithms, taking advantage of a discontinuous integral action, to attain a High-Order Sliding-Mode using a continuous control signal. This account starts presenting the design based on Lyapunov functions of classical sliding-mode controllers using a discontinuous state feedback control law to impose a high-order sliding-mode of arbitrary order. In order to estimate in finite-time the required derivatives of the measured signals for the implementation of the controllers, a Lyapunov-based design of the well-known robust and exact differentiator is shown. As a mechanism to reduce the effect of the undesirable chattering effect a recent method for higher-order sliding-mode controllers has been developed, using a continuous state feedback in conjunction with a discontinuous integral action to enforce a sliding-mode by means of a continuous control signal. The analysis and design tool for this integral controller is also an explicit, strict and particularly tailored Lyapunov function. In order to make this recount accessible to a wide audience the presentation is restricted to the main results, without giving proofs or design details, and leaving aside the rigorous mathematical machinery. For this the main references are provided.

Structural Conditions for Chattering Avoidance in Implicitly Discretized Sliding Mode Differentiators

The present paper considers the implicit discretization of Levant’s arbitrary order robust exact differentiator. It is shown that an improper implicit discretization may lead to an undesired bias in the differentiation error and, surprisingly, to discretization chattering despite the implicit discretization. Necessary and sufficient structural conditions for avoiding both of these problems are presented, which define a family of chattering-free discrete-time differentiators. A guideline for selecting a representative from this family is given. Numerical simulations illustrate the results.

Robust Nonlinear Control Design for the HVAC System Based on Adaptive Sliding Mode Control

Control of Heating, Ventilating, and Air Conditioning (HVAC) aims to provide a comfortable environment for human life in terms of temperature and humidity and improve indoor air quality. The HVAC system is multi-input multi-output, where the control design of this system is challenging due to its strong nonlinearity and the coupled influence of both system controllers on the temperature within the thermal zone. The aim of this study is to design a dual-controller for the HVAC system. The first controller is a non-linear feedback controller which is devoted to control the humidity ratio of the thermal zone with the desired characteristic. While for the second one, a robust controller is designed to maintain the desired thermal zone temperature based on the adaptive sliding mode controller (ASMC). Using the ASMC enabled us to design the second controller without the need to know the uncertainty bound on the HVAC system model. Additionally, the stability of the proposed control system was verified using the Lyapunov theory. To construct the sliding variable for the temperature control, the error state which is the difference between the thermal zoon temperature and the desired value and its derivative is needed. Due to the uncertainty in the error state derivative, a robust differentiator was designed using the approximate classical sliding mode differentiator (ACSMD). Finally, the performance of the control system is confirmed via numerical simulation. The results showed the robust ability of the control system to make the humidity and temperature of the thermal area follow the required values and with high accuracy.

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.

Infrared Fast-Maneuvering Target Tracking Based on Robust Exact Differentiator with Improved Particle Filter

This paper deals with the problem of nonlinear uncertainties when tracking an infrared dim small target with fast maneuvers. The particle filter (PF)-based methods are mostly considered. The existing improvement methods for the PF can handle the infrared dim small target with the conventional maneuver, the motion state of which changes slowly and in most cases is assumed to be linear. However, when fast maneuvers appear on the target, the PF-based method will soon suffer from particle degeneracy, or even loss of the target. In this paper, a robust exact differentiator (RED)-based particle generating method is proposed to improve the PF. New birth particles are produced by the proposed method, which can keep up with the fast maneuvers, and ensure the particle diversity of the PF, so as to avoid the particle degeneracy and depletion, meanwhile the number of particles required can be decreased. Numerical simulations are conducted, showing that the proposed algorithm has more advantages than the state-of-the-art method.

Simultaneous state and parameter estimation method for a conventional ozonation system

This article presents a simultaneous state (via a nonlinear form of Luenberger observer) and parameter (using a proportional–integral least mean square form) estimator design method for a conventional ozonation system. The suggested state observer assumes that the only available output signal is the concentration of the ozone gas at the output of the reactor. The estimation of the reaction rate constants of ozonation in the presence of contaminants uses the suggested proportional–integral estimation method. The convergence proof of the developed state-parameter identification method was confirmed using a Lyapunov based stability analysis. This analysis characterizes the quality of estimation considering the presence of modeled uncertainties and external perturbations. The implementation of the super-twisting algorithm as a robust and exact differentiator allowed to perform the estimation of the reaction rate constants of the ozonation, the temporal evolution of the dissolved ozone and the evolution of contaminants concentrations. The simultaneous state and parameter estimator design method was implemented in real-time using phenol as a model contaminant. The numerically simulated and real-time implementations showed that the method provides accurate estimates of the contaminant concentration and the reaction rate coefficient in all the evaluated cases.

Robust Differentiator-Based NeuroFuzzy Sliding Mode Control Strategies for PMSG-WECS

A robust control algorithm is always needed to harvest maximum power from a Wind Energy Conversion System (WECS) by operating it consistently at a Maximum Power Point (MPP) in the presence of wind speed variations. In this work, a Maximum Power Point Tracking (MPPT) control algorithm is designed via Conventional Sliding Mode Control (CSMC), the Super Twisting Algorithm (STA), and the Real Twisting Algorithm (RTA) and is applied to a Permanent Magnet Synchronous Generator (PMSG)-based WECS. CSMC is model-based whereas the STA and RTA are model-free controllers. In practice, the unavailability of nonlinear terms and aerodynamic forces deteriorates the performance of these controllers. Thus, an offline NeuroFuzzy algorithm is incorporated to estimate the nonlinear drift and control input channel to improve the robustness of these algorithms. In addition, the generator shaft speed and its missing derivative is recovered via a Uniform Robust Exact Differentiator (URED). In order to carry out a comprehensive comparative study among the three competitors, the overall system is simulated in a closed loop under the action of these controllers at three different operating conditions, i.e., nominal, varying load and inertia, and varying wind speed, using MATLAB/Simulink. The acquired results confirm the superiority of the RTA over the STA and CSMC in terms of robustness and chatter reduction.

Adaptive proportional derivative decentralized output based controller for a biped robotic device

This paper describes the design and implementation of a novel adaptive control method to track a set of bioinspired reference trajectories. These references define anthropomorphic movements for an exoskeleton robot. The proposed controller implemented the adjustment laws for the variable gains of a state feedback (Proportional-Derivative type) structure. The method to adjust the adaptive gains was determined using a controlled Lyapunov function. The adaptation laws use velocity estimation based on a robust exact differentiator (RED) implemented as a variation of a distributed Super-Twisting algorithm. The adaptive gain controller was evaluated on a simulated exoskeleton structure. The set of simulations considered the presence of external disturbances and modeling uncertainties. The controller proved efficient in rejecting external perturbations/uncertainties affecting the exoskeleton. The proposed controller’s performance was superior to the one obtained if the standard fixed-gain proportional derivative controller was evaluated. As an additional benefit of the adaptive PD controller implementation, a controller power reduction of at least 14 [Formula: see text] concerning the non-adaptive version of the feedback controller was attained. An experimental evaluation of the proposed controller confirmed the benefits of the proposed controller with adaptive gains. The successful tracking of nine different biomechanically inspired reference trajectories justified the exoskeleton application, which could be used as a potential tool for rehabilitation purposes.

Mathematical Modeling and Robust Control of a Restricted State Suspended Biped Robot Implementing Linear Actuators for Articulation Mobilization

The aim of this study is to develop an adaptive automatic control method for solving the trajectory tracking problem for a biped robotic device (BRD) and taking into account that each articulation is mobilized by a linear actuator. Each extremity of the BRD has three articulations with a linear actuator enforcing the controlled motion for each articulation. The control problem considers the task of tracking reference trajectories that define a regular gait cycle. The suggested adaptive control form has state-dependent gains that drive the tracking error into an invariant and attractive ellipsoidal with a center at the origin; meanwhile, the articulation restrictions are satisfied permanently. The stability analysis based on a controlled Lyapunov function depending on the tracking error leads to the explicit design of the state-dependent adaptive gains. Taking into account the forward complete setting of the proposed BRD, an output feedback formulation of the given adaptive controller is also developed using a finite-time and robust convergent differentiator based on the super-twisting algorithm. A virtual dynamic representation of the BRD is used to test the proposed controller using a distributed implementation of the adaptive controller. Numerical simulations corroborate the convergence of the tracking error, while all the articulation restrictions are satisfied using the adaptive gains. With the purpose of characterizing the proposed controller, a sub-optimal tuned regular state feedback controller is used as a comparative approach for validating the suggested design. Among the compared controllers, the analysis of the convergence of the mean square error of the tracking error motivates the application of the designed adaptive variant.

Self-Triggered Adaptive NN Tracking Control for a Class of Continuous-Time Nonlinear Systems With Input Constraints

This article develops a self-triggered adaptive neural network (NN) tracking controller for a class of continuous-time nonlinear systems, that is, input constrained and with unknown drift and input dynamics. Since the drift and input dynamics are both unknown, an NN is built within a self-triggered update paradigm to approximate the unknown tracking control. The error derivative used in the weight update algorithm is derived using a robust exact differentiator technique. To address input constraints, an auxiliary compensator is designed for the unimplemented control effort. Through rigorous Lyapunov analyses, we can guarantee that all the tracking and weight errors are uniformly ultimately bounded. Finally, to show the effectiveness of the proposed control performance, simulation results of a two-link robot are provided and analyzed.

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.

Model-based dynamic sliding mode control and adaptive Kalman filter design for boiler-turbine energy conversion system

The model-based control of a boiler-turbine system (BTS) is a formidable task due to coupling in state variables, nonlinearities and constraints on the control inputs. In this paper, a model-based, multi-variable dynamic sliding mode control (DSMC) is designed for the nonlinear BTS model to maintain the drum pressure, electric power and water level at the desired levels. In DSMC, an implicit sliding manifold is designed for the water level due to its complexity and explicit dependence on the control inputs. For this purpose, an auxiliary function is computed, and the sliding mode is enforced in such a way that the system fluid density tracks the auxiliary function, and subsequently the water level tracks the desired trajectory. Owing to its complexity, the time derivatives of the auxiliary function are computed using the uniform robust exact differentiator (URED). An adaptive Kalman filter (AKF) is designed for the estimation of the unmeasurable state i.e., system fluid density. The design of AKF is based on the quasi-linear model of the BTS. Furthermore, a detailed stability analysis is carried out to ensure the boundedness of the closed-loop system. The simulation results depict that the designed control scheme exhibits the desired tracking performance in the presence of external disturbances, nonlinearities, constraints on the inputs, and measurement and process noises.

KCNQ2 R144 variants cause neurodevelopmental disability with language impairment and autistic features without neonatal seizures through a gain-of-function mechanism.

SummaryBackgroundPrior studies have revealed remarkable phenotypic heterogeneity in KCNQ2-related disorders, correlated with effects on biophysical features of heterologously expressed channels. Here, we assessed phenotypes and functional properties associated with KCNQ2 missense variants R144W, R144Q, and R144G. We also explored in vitro blockade of channels carrying R144Q mutant subunits by amitriptyline.MethodsPatients were identified using the RIKEE database and through clinical collaborators. Phenotypes were collected by a standardized questionnaire. Functional and pharmacological properties of variant subunits were analyzed by whole-cell patch-clamp recordings.FindingsDetailed clinical information on fifteen patients (14 novel and 1 previously published) was analyzed. All patients had developmental delay with prominent language impairment. R144Q patients were more severely affected than R144W patients. Infantile to childhood onset epilepsy occurred in 40%, while 67% of sleep-EEGs showed sleep-activated epileptiform activity. Ten patients (67%) showed autistic features. Activation gating of homomeric Kv7.2 R144W/Q/G channels was left-shifted, suggesting gain-of-function effects. Amitriptyline blocked channels containing Kv7.2 and Kv7.2 R144Q subunits.InterpretationPatients carrying KCNQ2 R144 gain-of-function variants have developmental delay with prominent language impairment, autistic features, often accompanied by infantile- to childhood-onset epilepsy and EEG sleep-activated epileptiform activity. The absence of neonatal seizures is a robust and important clinical differentiator between KCNQ2 gain-of-function and loss-of-function variants. The Kv7.2/7.3 channel blocker amitriptyline might represent a targeted treatment.FundingSupported by FWO, GSKE, KCNQ2-Cure, Jack Pribaz Foundation, European Joint Programme on Rare Disease 2020, the Italian Ministry for University and Research, the Italian Ministry of Health, the European Commission, the University of Antwerp, NINDS, and Chalk Family Foundation.

Robust-Exact-Differentiator-Inspired Discrete-Time Differentiation

This article proposes a discrete-time differentiation algorithm of arbitrary order inspired by the continuous-time uniform robust exact differentiator and the continuous-time arbitrary-order robust exact differentiator. As the well-known explicit Euler method is not suitable for discretizing algorithms with the fixed-time convergence property, a semi-implicit approach is proposed. The discrete-time differentiators of orders 2 and 3 are studied in detail, and it is proven that the estimation errors vanish independent of their initial condition in the unperturbed case. In the presence of perturbations, it is shown that the origin of the estimation errors is surrounded by an attractive set. Furthermore, the performance of the proposed algorithm is evaluated via simulation studies.

Finite-Time State Observer for a Class of Linear Time-Varying Systems With Unknown Inputs

A finite-time observer for a class of linear time-varying systems with bounded unknown inputs is presented in this note. The design of the observer exploits structural properties of the system and, through a linear operator, allows applying the robust exact differentiator in a block form without requiring additional stabilizing terms. The proposed observer provides an exact estimate of the states after a finite transient time despite the possible instability of the system and the effects of bounded unknown inputs.

Highly Maneuverable Target Tracking Under Glint Noise via Uniform Robust Exact Filtering Differentiator With Intrapulse Median Filter

Highly maneuverable target tracking under glint noise and nonlinear uncertainties during course changes and terminal maneuvers has been solved suboptimally for many years via the interacting multiple model algorithm with the use of the Kalman filter (KF), unscented Kalman filter, and the extended Kalman filter (EKF). Also, nonlinear KFs such as the cubature Kalman filter have been proposed to improve nonlinear tracking without being able to filter out glint noise. To this end, the particle filter and some KF based on variational Bayesian approach have been proposed with very good results in filtering out glint noise. Nonetheless, it is difficult for state-of-the-art methods to achieve efficient filtering of glint noise and nonlinear tracking at the same time. On the other hand, robust exact differentiators, based on the super-twisting algorithm, have been used for many years in output-feedback control and state observation in order to obtain the derivatives of an input signal with theoretical finite-time exactness. However, their potential for target tracking applications has not been explored sufficiently. In this article, a uniform robust exact filtering differentiator with intrapulse median filtering is proposed to filter out glint noise at the sliding manifold, while offering nonlinear tracking robustness via high-degree super-twisting terms outside the sliding manifold. Numerical simulations comparing the proposed solution to other state-of-the-art methods were conducted, showing promising results.

Design and Analysis of a Novel Generalized Continuous Tracking Differentiator

In this work, an n- th order Generalized Tracking Differentiator (GTD) is proposed based on sigmoid function with a continuous structure, including both linear and nonlinear parts, thus increasing the estimation accuracy of the input signal and its derivatives, and overcoming the inherent issues related to the classical Tracking Differentiators (TDs). Then, a 2nd-order version of the proposed GTD is derived and optimized with further improvements which are reflected in the excellent behavior in the time and frequency domains. Moreover, stability analysis using the method of Lyapunov analysis is also investigated and the performance of the GTD is proven in the time and frequency domains and revealed that the proposed GTD considerably reduces the “peaking phenomenon” and “noise” and eliminates the “chattering phenomenon” from the signal derivatives. The excellent results of the proposed GTD are demonstrated through simulations on noise-free and noisy signals and compared with the Robust Exact Uniformly Convergent Arbitrary Order Differentiator (REUCAOD).

Robust dynamic backstepping averaged sub-gradient integral sliding mode control for navigation of mobile robots

The main goal of the current investigation is to present an output feedback robust control that regulates the motion of mobile robotic autonomous devices based on the application of the dual averaged sub-gradient descendant method. Sub-gradient realization aims to solve the design of a proposed extremum seeking control to stabilize the trajectory tracking error between the three-dimensional position of mobile robotic autonomous devices and some attainable reference trajectories in the same spatial domain. The controller belongs to a class of sliding mode controller with variable gain based on a sliding surface depending on the tracking error and its time derivative. This derivative is estimated with the application of a distributed time-dependent super-twisting robust differentiator. The proposed mobile robotic autonomous device dynamics considers the dynamics of actuators (direct current motors) which drives the mobilization of mobile robotic autonomous devices. The dynamic nature of the mobile robotic autonomous devices regulated by motor actuators induces a backstepping-like controller design based on a two-stage sequential application of the sub-gradient strategy, one for the mobile robotic autonomous devices and the second for the motor dynamics. The mixed strategy appears as a novel formulation to solve the path track for reference trajectories of mobile robotic systems. The proposed control considers the optimization of the tracking error dependent functional without the full knowledge of the mobile robotic autonomous device dynamics. With the aim of proving the effectiveness of the suggested design, two numerical examples are developed. In these examples, the tracking performance, minimization of the proposed functional as well as control evolution are evaluated. The controlled tracking outcomes of the proposed integral sliding mode control are contrasted with the outcomes produced by a set of proportional–derivative and selected sliding mode (first-order type) controllers. The suggested controller shows a superior tracking of the reference trajectory than the other two comparative controllers, using quality measures such as least mean square error and its integral.

Implicit and explicit discrete-time realizations of the robust exact filtering differentiator

This paper presents explicit and implicit discrete-time realizations for the robust exact filtering differentiator, aiming to facilitate an adequate posterior implementation structure in digital devices. This paper firstly presents an analysis of an explicit discrete-time realization of the filtering differentiator based on linear systems’ exact discretization with a zero-order holder. For this case, however, high-order terms in the filter dynamics may cause instability of the estimation error for signals with unbounded derivatives. Hence, two other new discrete-time realizations of the filtering differentiator are derived by removing some high-order terms in the filter dynamics. The first one is an explicit discrete-time realization, while the second one is implicit. After a finite time, both preserve the accuracy of the continuous-time robust exact filtering differentiator in the presence of measurement noise. For each proposed discrete-time scheme, a stability analysis based on homogeneity is provided. Finally, the simulation results include comparisons between the proposed implicit and explicit discrete-time realizations with other existing schemes. These numerical studies highlight that the implicit scheme supersedes the explicit one, consistent with the implicit and explicit realizations of other continuous-time algorithms.

Virtual Sensor Using a Super Twisting Algorithm Based Uniform Robust Exact Differentiator for Electric Vehicles

The highly efficient Interior Permanent Magnet Synchronous Motor (IPMSM) is ubiquitous choice in Electric Vehicles (EVs) for today’s automotive industry. IPMSM control requires accurate knowledge of an immeasurable critical Permanent Magnet (PM) flux linkage parameter. The PM flux linkage is highly influenced by operating temperature which results in torque derating and hence power loss, unable to meet road loads and reduced life span of electrified powertrain in EVs. In this paper, novel virtual sensing scheme for estimating PM flux linkage through measured stator currents is designed for an IPMSM centric electrified powertrain. The proposed design is based on a Uniform Robust Exact Differentiator (URED) centric Super Twisting Algorithm (STA), which ensures robustness and finite-time convergence of the time derivative of the quadrature axis stator current of IPMSM. Moreover, URED is able to eliminate chattering without sacrificing robustness and precision. The proposed design detects variation in PM flux linkage due to change in operating temperature and hence is also able to establish characteristics of fault detection. The effectiveness and accuracy in different operating environments of the proposed scheme for nonlinear mathematical IPMSM model with complex EV dynamics are verified thorough extensive simulation experiments using MATLAB/Simulink.