High-order Sliding Mode Differentiator Research Articles

Gaussian process-based state derivative estimator with temporal input in incremental flight control design

In this paper, a state derivative estimation technique is proposed using a derivative of Gaussian process regression method and applied to an incremental dynamics-based controller. The state derivative estimates of interest are with respect to time, and to obtain them, the temporal index array is used for Gaussian process state input. Using a Gaussian process, assumptions such as numerical differentiation of the state, parametric basis functions common in adaptive approaches, and the use of upper-bound information of the state derivative commonly used in robust methods are avoided. The proposed approach is compared with the backward difference formula (BDF) differentiator, first-order low-pass filter, second-order low-pass filter, and high-order sliding mode differentiator. Monte Carlo simulation with 1000 runs is performed under four different noise level measurement scenarios. The proposed Gaussian process-based differentiator with temporal index input resulted in better robustness and adaptability in state derivative estimation with similar or better performance.

High‐order sliding‐mode functional observers for multiple‐input multiple‐output (MIMO) linear time‐invariant systems with unknown inputs

AbstractFor arbitrary multiple‐input multiple‐output linear time invariant systems with unknown inputs this article provides sufficient conditions to estimate linear functionals of the state variables. When the unknown input is uniformly bounded these conditions are strictly weaker than the classical conditions for functional unknown input observers, well‐known in the literature, and generalize previous results using discontinuous differentiators. Furthermore, a general methodology is proposed to design functional observers, that are able to estimate the functionals, when possible, exactly and in finite‐time or fixed‐time. Instead of using a cascade of Luenberger observers and high‐order sliding‐mode differentiators, standard in the literature for this problem, a bi‐homogeneous observer of reduced order is proposed in the article. Proofs of the convergence are provided using smooth Lyapunov functions and an academic example illustrates the behavior of the proposed observer for a system not tractable with the available methods.

Robust Collision Avoidance for Mobile Robots in the Presence of Moving Obstacles

This paper is concerned with the collision avoidance for mobile robots with uncertain dynamics in the presence of obstacles that can considerably change their velocities over time. To address this problem, we propose a robust collision-avoidance method based on the continuous-control-obstacles one. The proposed method uses an arbitrary-order overdamped low-pass filter to generate sufficiently smooth position commands for the robot and a high-order sliding mode differentiator to robustly estimate the obstacles’ maximum accelerations. Based on these estimates, we define a set of possible future positions for the obstacles according to how each one is changing its velocity to calculate a robust position command for the robot. The method has been numerically evaluated using a conventional quadcopter flying among moving obstacles and has been shown to be effective in providing collision avoidance and velocity constraints satisfaction.

三重タンクシステムのイベントトリガロバスト出力フィードバック制御

An event-triggered robust output feedback control method is proposed for water level tracking of an uncertain three-tank system by using only the level sensor of the target tank. The system model is transformed into a canonical form with uncertainties by coordinate transformation, and the canonical system's state variables are estimated by a higher-order sliding mode differentiator based on the measurement of the target tank's water level. Then an event-triggered robust output feedback controller is constructed. The triggering condition is derived by taking the control gains and event-triggered sampling errors into account, and the control performance is rigorously analyzed. Furthermore, it is shown that the zeno behavior is also excluded. Finally, the effectiveness of the proposed method is verified by experimental studies on a real three-tank system.

Adaptive Backstepping Control of Uncertain Sandwich-Like Nonlinear Systems With Deadzone Nonlinearity

A systematic differentiator-based adaptive backstepping control methodology is proposed for a class of sandwich-like nonlinear system with unknown state-dependent deadzone nonlinearity and parametric uncertainties. The novelty of our approach is that a high-order sliding mode differentiator is utilized to estimate the nonstrict feedback coupling term resulting from the sandwiched deadzone, and all the outputs of the differentiator are integrated into the backstepping procedure based on Lyapunov functions with flat zone recursively. By this approach, all the unknown parameters are estimated online, the discontinuity of the virtual input caused by bound estimations is avoided. It is shown that the ultimate boundedness of all the closed-loop signals is achieved and the output tracking error converges to a preset set. Simulation is performed to verify the theoretical findings.

Output feedback full-order sliding mode control for a three-tank system

A robust output feedback nonlinear control method is proposed for output tracking of an uncertain three-tank system by using only the level sensor of the target tank. By using a coordinate transformation, we first transform the system model into a canonical form with uncertainties. The canonical system’s state variables are estimated by a higher-order sliding mode differentiator based on the measurement of the target tank’s liquid level. Then based on the estimated state variables, a continuous full-order sliding mode controller is constructed which can eliminate the effects of both the additive and multiplicative uncertainties. Rigorous analysis is carried out to clarify the control performance. And the effectiveness of the proposed method is verified by experimental studies on the Inteco Multi-tank system.

HOSMD and neural network based adaptive super-twisting sliding mode control for permanent magnet synchronous generators

To solve the maximum power tracking (MPT) control problem of the direct-driven wind power system, a super-twisting integral sliding mode controller with adaptive parameter estimation is designed. A high-order sliding mode differentiator is introduced as the virtual control variate filter, which solves the difficulty of obtaining the derivative of the control variate and the controller saturation in the nonlinear system with disturbances. Since the speed loop of permanent magnet synchronous generator (PMSG) is susceptible to disturbances, a radial basis function neural network (RBFNN) approximator and its adaptive algorithm are proposed to observe the unmodeled part of the system and external disturbances. In addition, the super-twisting algorithm is introduced to improve the robust performance. An improved adaptive parameter estimation algorithm is used to obtain real-time estimated values in the circumstances of uncertain stator parameters and parameters perturbation during operation, which enhances control accuracy and reduces undesirable chatting. The errors of RBFNN approximation and parameter estimation are taken into Lyapunov functions to guarantee the stability of the whole system. The effect of the proposed scheme is verified as compared to the adaptive backstepping terminal and new reaching law sliding mode controllers.

Design and Control of a Hydraulic Hexapod Robot with a Two-Stage Supply Pressure Hydraulic System

This paper focuses on the system design and control strategies of a hydraulic hexapod robot (HHR) ZJUHEX01 with a two-stage supply pressure hydraulic system (TSS). Firstly, a brief introduction is given, including the mechanical structure, the onboard hydraulic system, and the control system architecture. Secondly, the kinematics model and hydraulic system model are built in preparation for the controller design. Then a sliding mode repetitive controller (SMRC) for the separate meter in and separate meter out (SMISMO) hydraulic system is proposed, as well as the valve configuration, to help HHR get better control performance and smaller tracking errors. Furthermore, a high order sliding mode differentiator (HOSMD) is developed to obtain the joint angular velocity and acceleration. Finally, the ADAMS and MATLAB/Simulink co-simulation model is established to verify the effectiveness of the control strategy. Also, the energy consumption of TSS is compared with that of one-stage supply pressure hydraulic system (OSS) to show a great energy-saving effect of 51.94%.

Nonlinear control of an antilock braking system in the presence of tire–road friction uncertainties

An Antilock Braking System (ABS) is characterized by nonlinear dynamics, which render more difficult the design of a controller for high performance. The problem is even harder due to the uncertainties on the parameters appearing in its dynamics. In this paper, an ABS laboratory setup is considered, which mimics a quarter car model. A super–twisting controller is proposed to overcome the problem due to parameter uncertainties. This controller is designed in order to impose a reference value of the tire slip. Two cases are considered: in the first, nominal ABS parameters are used in the controller, whereas in the second the controller embeds an estimator of the tire–road friction coefficient, which is one of most critical parameters. The friction coefficient is estimated in finite–time by means of a high–order sliding mode differentiator. The original contributions of the paper are the real–time implementation of the super–twisting controller for the laboratory setup under study, and the use of a super–twisting estimator to provide a finite–time estimation of the friction coefficient between the tire and the road, along with a comparison with classical PI–like and super–twisting controllers, available in the literature. The ABS laboratory setup allows checking experimentally the performance of the proposed nonlinear dynamic controller, showing a considerable increase of the efficiency of the control system.

Parameter estimation based robust liquid level control of quadruple tank system — Second order sliding mode approach

This article proposes a liquid level control scheme for a quadruple tank process (QTP) plant. A robust control is proposed using higher order sliding mode (HOSM) control. The four tank setup has opposite tanks actuated by a single pump and unmeasured liquid outflow from upper tanks affecting the levels in the lower tanks. This renders nonlinearity in the dynamics with uncertain behavior. A novel experimentation based estimator is proposed for identification of the unknown control valve ratios of the plant. The estimated parameter is used for robust control design and state observer model. The unmeasurable states are estimated using a sliding mode observer and the required derivatives of liquid levels in the tanks are evaluated using a HOSM differentiator. The complete schema is tested in simulation and validated experimentally on a QTP setup.

Adaptive sliding-mode controller of a lower limb mobile exoskeleton for active rehabilitation

This study describes the design, instrumentation and control of an exoskeleton for lower limb children rehabilitation with nine degrees of freedom. Three degrees of freedom in each leg exert the movements of hip, knee and ankle in the sagittal plane, and three control the drive track system composed by a caterpillar-like robot. The control scheme presents a model free decentralized output feedback adaptive high-order sliding mode control to solve the trajectory tracking problem in each degree of freedom of the exoskeleton. A high order sliding mode differentiator estimates the unmeasured states and, by means of a dynamical state extension, it approximates the unknown dynamical model of the exoskeleton. A second-order adaptive sliding mode controller based on the super-twisting algorithm drives the exoskeleton articulations to track the proposed reference trajectories, inducing an ultimate boundedness for the tracking error. Numerical and experimental simulation results demonstrate the effect of the adaptive gain on the super-twisting control design. Such evaluations confirmed the superior tracking performance forced by the adaptive law for the controller with a smaller chattering amplitude and smaller mean tracking error.

Frameworks for double hyperbolic function‐based robust sliding mode differentiator and observer for nonlinear dynamics

AbstractThis paper puts forward contemporary designs of sliding mode differentiator and state observer for evaluation of unknown nonlinear signal derivatives and unknown internal system states, respectively, by the virtue of tangential and inverse sinusoidal hyperbolic functions. The utility of the proposed frameworks is that it bestows smooth and robust estimation without inciting unacceptable oscillations (chattering), unlike high gain discontinuous function‐based preliminary observers/differentiators. Moreover, the employed double hyperbolic functions would drive the various deviations in estimations of signals or states to a very close neighborhood of origin in finite time which is substantiated via Lyapunov's energy function. To demonstrate the efficiency of the introduced techniques, two examples for estimating the derivatives of a nonlinear signal and internal states of motor are also illustrated with time varying uncertainties. At last, the attained simulation outcomes of the proposed differentiator are further compared with formerly formulated designs such as higher‐order sliding mode differentiator (HOSMD) and uniformly convergent differentiator (UCD).

On-line parameter estimation of a Lithium-Ion battery/supercapacitor storage system using filtering sliding mode differentiators

This paper introduces a new approach to obtain precise on-line estimation of the internal parameters of a hybrid energy storage system based on Lithium-Ion Batteries and Supercapacitors. Filtering high-order sliding mode differentiators and a recursive least square estimation algorithm for time varying parameters are combined to obtain the parameters of an electric model of the storage devices. Complementary, a method is proposed to validate the estimated values, in accordance with a classic identifiability condition, i.e., the persistence of excitation. Representative simulation results to show the efficiency of the proposed on-line estimation methodology are presented, considering an electric vehicle subjected to realistic power demand. To this end, a standardised test that represents city driving conditions is used. Together with the on-line parameters estimation, it is allowed to infer the State-of-Charge and the State-of-Health during vehicle operation.

Decentralized fractional-order backstepping fault-tolerant control of multi-UAVs against actuator faults and wind effects

Concurrent occurrences of actuator faults and wind effects can significantly threaten the flight safety of multiple unmanned aerial vehicles (multi-UAVs). To address this difficult control problem against actuator faults and wind effects, a composite decentralized fractional-order (FO) backstepping adaptive neural fault-tolerant control (FTC) method is presented for the attitude synchronization tracking of multi-UAVs, which is integrated with neural networks (NNs), disturbance observers (DOs), FO calculus, and high-order sliding-mode differentiators (HOSMDs). The distinctive feature of this work is addressing the attitude synchronization tracking control problem with actuator faults and wind effects in a decentralized framework and proposing a composite approximation method for multi-UAVs. It is shown that by using Lyapunov methods the synchronization tracking control is achieved even when multi-UAVs simultaneously encounter wind effects and actuator faults. Comparative simulation results illustrate the theoretical feasibility.

A Super-twisting Controller for Active Control of Ground Vehicles with Lateral Tire-road Friction Estimation and CarSim Validation

In this paper, a ground vehicle equipped with Active Front Steering (AFS) and Rear Torque Vectoring (RTV) is considered. The AFS, actuated through the front tires, adds an incremental steer angle on top of the driver’s input, whereas the RTV, actuated through the rear tires, imposes a yaw torque to the vehicle. These actuators allow the active control of the vehicle chassis, so that a feasible and safe reference trajectory can be tracked. To obtain such a feasible reference generation and an efficient control action, the lateral tire-road friction coefficient has to be estimated. To this aim, in this paper the lateral tire-road friction coefficient is estimated in finite-time by means of a high-order sliding mode differentiator. Then, based on this estimation, a high-order sliding mode controller is designed to track the desired references. The performance of the dynamic controller is evaluated using a CarSim virtual vehicle, and the simulation results highlight the characteristics of the proposed observer-based control.

Robust Control for Stable and Safe Performance of a Two Wheeled Human Transporter

A personal, electric human transportation vehicle is a promising invention for pollution and congestion free future. A two wheeled human transporter (TWHT) characterizes an inherently unstable system due to which its control for safe operation is a challenge. This paper proposes a higher order sliding mode control (HOSMC) for stable and safe performance of the TWHT system. A reduced order model is considered for control development. Unmeasurable states are estimated using higher order sliding mode differentiator. The proposals are validated in simulation and also compared with linear state feedback controller (SFC) to show its efficacy. The robust property of HOSMC is tested by considering variations in rider’s weight.

Stabilization of USVs Under Mismatched Condition Based on Fixed-Time Observer

This study investigates the problem of asymptotic stabilization of underactuated surface vehicles (USVs) in the presence of simultaneous matched and mismatched disturbances. Within the framework of back-stepping control and by introducing a periodic term in the control laws of virtual yaw angle and surge velocity, a control design method is developed to stabilize the USVs into the bounded region. By the induction method, a novel fixed-time disturbance observer based on higher-order sliding mode differentiators are introduced. Then a control law based on the proposed observers is proposed, by which the errors between the virtual and actual yaw angle and surge velocity can converge to zero, and the state of the USVs converge to a bounded region. Numerical simulations are carried out to verify the proposed control schemes.

Virtual sensing of load forces in hydraulic actuators using second- and higher-order sliding modes

External load forces are challenging for sensing or estimating in the hydraulic actuators. Once it is due to inconvenient instrumentation of the force sensors, especially on an open-end mechanical interface. The other way, the complex nonlinear system behavior aggravates reconstructing the system states in a robust and real-time suitable manner. This paper proposes a sensorless estimation of external load forces in standard hydraulic actuators by using a well-established equivalent output injection of the second-order sliding mode and also higher-order sliding mode differentiator. Only the basic inertial and frictional parameters are assumed to be known from an initial identification without external load. Afterwards, the robust exact differentiators are used in order to reconstruct the system states. Noisy signals of the cylinder chamber pressures and piston stroke are the single quantities available from the measurement. An experimental case study, accomplished on the setup of two hydraulic cylinders arranged and operated in antagonistic way, is provided. The force-cell on the rigid interface between both cylinders is used for reference measurements and evaluation of the estimation algorithms. Two estimation approaches, one of the 2nd and another of the 4th order, are assessed in performance and compared to each other along with discussion.

Kalman filter based estimation algorithm for the characterization of the spatiotemporal hemodynamic response in the brain

The characterization of the spatiotemporal hemodynamic response (stHR) in the brain is important for understanding the interaction between neighboring brain voxels and regions. In this paper, we design an identification algorithm for the characterization of the cerebral stHR which is modeled by a system of coupled hyperbolic partial differential equation (PDE) and infinite-dimensional ordinary differential equation (ODE). The proposed algorithm provides estimates of the hemodynamic variables (cerebral blood flow and mass density contributed by blood) and physiological parameters using non-invasive Blood Oxygenation Level Dependent (BOLD) data measured with functional Magnetic Resonance Imaging (fMRI) modality. The proposed solution concept follows three main steps: (i) discretization of the stHR model using Galerkin-based finite element method; (ii) estimation of the output derivative using high-order sliding mode differentiator; and (iii) estimation of the state, input, and parameters from sampled-in-space measurements using the reduced-order approximation model and a constrained extended Kalman filter with unknown input algorithm. In addition, sufficient conditions that depend on the chosen discretization scheme, and which guarantee the structural identifiability of the input and parameters, and also the observability of the system are provided. The performance of the proposed algorithm is assessed using both synthetic and real data. The set of the used real data represents the 1-D BOLD signal collected from the visual cortex and acquired in 3 Tesla fMRI scanner.

Adaptive Neural Consensus Tracking for Nonlinear Multiagent Systems Using Finite-Time Command Filtered Backstepping

This paper is concerned with the finite-time consensus tracking control problems of uncertain nonlinear multiagent systems. A neural network-based distributed adaptive finite-time control scheme is developed, which can guarantee the consensus tracking is achieved in finite time with sufficient accuracy in the presence of unknown mismatched nonlinear dynamics. Such a finite-time feature is achieved by the modified command filtered backstepping technique based on the high-order sliding mode differentiator. Moreover, the proposed control scheme is completely distributed, since the control laws only use the local information. In addition, although mismatched uncertainty nonlinear dynamics are considered, only one parameter needs to be updated for each agent in the control scheme, which will simply the computations and make the proposed scheme more effective for applications. An example is included to verify the presented method.