Differential Flatness Theory Research Articles

Event-triggered trajectory tracking control for USV with prescribed performance and time delay based on differential flatness

This paper presents a trajectory tracking control scheme for underactuated surface vessels (USVs) with input delay. Firstly, the underactuated surface vessel system is transformed into a fully actuated system using differential flatness theory. To estimate the unknown nonlinear terms introduced in the transformation process, a fuzzy neural network (FNN) is employed. Secondly, to conserve control resources and communication bandwidth, the controller of the system under prescribed performance is designed using the backstepping method. This method updates the controller according to an event-triggered condition that is designed using a Lyapunov function. Finally, theoretical proof and simulation experiments are conducted to demonstrate the convergence and effectiveness of the proposed method.

Differential flatness properties and multivariable adaptive fuzzy control of hormonal system dynamics

Computer-based control for hormonal (ovarian) system dynamics and for administering in a computationally optimized manner the associated pharmaceutical treatment has been little investigated up to now. Although there has been prior work on the modeling of the ovarian system dynamics and its stability features, there have been hardly any results for applying to the ovarian system exogenous computer-based control. This is due to the complexity, nonlinearity and the high dimensionality of this system. Actually, the ovarian system exhibits nonlinear dynamics which is modeled by a set of thirteen coupled nonlinear differential equations, while receiving only two control inputs. Besides, the fact that the parameters of the model cannot be precisely identified and the fact that any mathematical model for such a biosystem is likely to be subjected to uncertainties and external perturbations, make the application of model-based control schemes for this system be questionable. To overcome this, the paper proposes model-free adaptive fuzzy control based on differential flatness theory for the complex dynamics of the ovarian system. It is proven that the dynamic model of the ovarian system, having as state variables the LH and the FSH hormones and their derivatives, is a differentially flat one. This means that all its state variables and its control inputs can be written as differential functions of the flat output and its derivatives. By exploiting differential flatness properties the system's dynamic model is written in the multivariable linear canonical (Brunovsky) form, for which the design of a state feedback controller becomes possible. After this transformation, the new control inputs of the system contain unknown nonlinear parts, which are identified with the use of neurofuzzy approximators. Lyapunov stability analysis shows that H-infinity tracking performance is achieved for the feedback control loop and this assures improved robustness to the aforementioned model uncertainty as well as to external perturbations.

Quadrotor UAV Dynamic Visual Servoing Based on Differential Flatness Theory

In this paper, we propose 2D dynamic visual servoing (Dynamic IBVS), where a quadrotor UAV tries to track a moving target using a single facing-down perspective camera. As an application, we propose the tracking of a car-type vehicle. In this case, data related to the altitude and the lateral angles have no importance for the visual system. Indeed, to perform the tracking, we only need to know the longitudinal displacements (along the x and y axes) and the orientation along the z-axis. However, those data are necessary for the quadrotor’s guidance problem. Thanks to the concept of differential flatness, we demonstrate that if we manage to extract the displacements according to the three axes and the orientation according to the yaw angle (the vertical axis) of the quadrotor, we can control all the other variables of the system. For this, we consider a camera equipped with a vertical stabilizer that keeps it in a vertical position during its movement (a gimbaled camera). Other specialized sensors measure information regarding altitude and lateral angles. In the case of classic 2D visual servoing, the elaboration of the kinematic torsor of the quadrotor in no way guarantees the physical realization of instructions, given that the quadrotor is an under-actuated system. Indeed, the setpoint has a dimension equal to six, while the quadrotor is controlled only by four inputs. In addition, the dynamics of a quadrotor are generally very fast, which requires a high-frequency control law. Furthermore, the complexity of the image processing stage can cause delays in motion control, which can lead to target loss. A new dynamic 2D visual servoing method (Dynamic IBVS) is proposed. This method makes it possible to generate in real time the necessary movements for the quadrotor in order to carry out the tracking of the target (vehicle) using a single point of this target as visual information. This point can represent the center of gravity of the target or any other part of it. A control by flatness has been proposed, which guarantees the controllability of the system and ensures the asymptotic convergence of the generated trajectory in the image plane. Numerical simulations are presented to show the effectiveness of the proposed control strategy.

Run-to-Run Adaptive Nonlinear Feedforward Control of Electromechanical Switching Devices

Feedforward control can greatly improve the response time and control accuracy of any mechatronic system. However, in order to compensate for the effects of modeling errors or disturbances, it is imperative that this type of control works in conjunction with some form of feedback. In this paper, we present a new adaptive feedforward control scheme for electromechanical systems in which real-time measurements or estimates of the position and its derivatives are not technically or economically feasible. This is the case, for example, of commercial electromechanical switching devices such as solenoid actuators. Our proposal consists of two blocks: on the one hand, a feedforward controller based on differential flatness theory; on the other, an iterative adaptation law that exploits the repetitive operation of these devices to modify the controller parameters cycle by cycle. As shown, this law can be fed with any available measurement of the system, with the only requirement that it can be processed and converted into an indicator of the performance of any given operation. Simulated and experimental results show that our proposal is effective in dealing with a long-standing control problem in electromechanics: the soft-landing control of electromechanical switching devices.

Safety Control for Quadrotor UAV Against Ground Effect and Blade Damage

This article proposes a safety control scheme for quadrotor unmanned aerial vehicle (UAV) to counteract ground effect and blade damage. The impacts of ground effect and blade damage are quantitatively analyzed in the established model. The proposed safety control scheme consists of position and attitude loops. Specifically, a proportional and derivative controller is designed as a baseline control with acceleration feedforward in the position loop. A disturbance observer (DO) using the measured Euler angles is exploited to estimate the lumped disturbance forces. The developed DO can effectively mitigate the estimation deviation caused by the angle tracking error. Subsequently, differential flatness theory is used to obtain the Euler angle references and the derivatives. In the attitude loop, sliding mode control and fixed-time sliding mode observer (SMO) are integrated to deal with ground effect and blade damage promptly. To demonstrate the effectiveness of the proposed safety control scheme, flight experiments are carried out, where the quadrotor UAV flies close to the rugged ground in the presence of blade damage.

Sensorless Control of Induction Motor Based on Differential Flatness Theory and Reduced MRAS Observer

In this paper, we propose a hybrid sensorless control method of IM, This method consists of using the differential flatness theory and reduced MRAS observer. The control design proceeds by showing that each input of the motor model stands for a differentially flat system, where the flat output is chosen to be the associated state variable. Next for each regulation loop is a virtual control input is computed, that can invert the loop’s dynamics and can eliminate the system’s tracking error. The reduced MRAS observer is used to estimate the rotor speed and flux. Simulation and experimental results are presented to illustrate the effectiveness of the proposed approach for sensorless control of induction motor.

Trajectory-tracking control of an underactuated unmanned surface vehicle based on quasi-infinite horizon model predictive control algorithm

The method of a designed trajectory tracking for an underactuated unmanned surface vehicle (USV) in the presence of ocean disturbances is addressed in this paper; the differential flatness theory is applied to get the reference inputs and speed states at the reference position trajectory. Second, a transition process is arranged for the reference trajectory to reduce the overshoot of the actuator, which caused by the large deviation in the initial tracking. Third, the nonlinear disturbance observer is designed to obtain the estimated values of unknown disturbances in the ocean. Then, a controller-based model predictive control (MPC) and terminal cost function is designed for the nominal system. The inherent robustness of the controller and estimates of the observer are used to resist and compensate disturbances. Finally, the simulation experiments of linear trajectory and sinusoidal trajectories are carried out to prove the effectiveness and reliability of the control algorithm designed.

Robust Control and Stable Performance of a Grid-Tied Dumbbell-Type Multilevel Converter Interfaced DG Unit Using Differential Flatness Theory

Robust Control and Stable Performance of a Grid-Tied Dumbbell-Type Multilevel Converter Interfaced DG Unit Using Differential Flatness Theory

A derivative-free nonlinear Kalman filtering approach using flat inputs

ABSTRACT Differential flatness theory has been widely used for both tracking control and state estimation of nonlinear systems. Recently, the concept of flat inputs has been proved to be a valuable tool for control design if the system is non-differentially flat. However, flat inputs have not been used yet to solve state estimation problems for these systems. By constructing flat inputs, we introduced a novel state estimation-based control strategy for both observable non-differentially flat nonlinear systems and differentially flat nonlinear systems whose flat output vector is not a measurable variable, as long as the internal dynamics of the system are stable. The efficiency of the proposed approach is illustrated through two numerical examples. Our findings indicated that we increased the class of systems to which the derivative-free nonlinear Kalman filtering based on differential flatness theory can be applied.

Differential Flatness-Based, Full-Order Nonlinear Control of a Modular Multilevel Converter (MMC)

Modular multilevel converter (MMC) is an attractive topology in industrial applications such as high-voltage direct current (HVdc) transmission system where fast power control is required. The most conventional control method for MMC uses cascaded structures, where the power transmission and internal energy dynamic (outer loops) is limited by the “frequency separation” constraints since the dynamic choice of the currents (inner loops) must consider the tradeoff between rapidity and robustness. In this article, we present a new control method based on a differential flatness theory. The main interest of the proposed control is the possibility to obtain a very high dynamic performance of the MMC power flow, even under noisy measurements and parametric disturbances. Using the well-known average model of the MMC, a flat output is proposed, proving the differential flatness property of the MMC. Trajectory tracking controllers of ac grid active and reactive power are proposed using input-state full-order linearization. Simulation results, including losses corresponding to a realistic case of on-shore, point-to-point HVdc interconnection, are presented to show the performance of the proposed control scheme even with strong measurement noise.

Linear Active Disturbance Rejection Control for Double-Pendulum Overhead Cranes

Because of the complexity, non-linearity, and under-actuated features of double pendulum overhead cranes, a control method based on linear active disturbance rejection control (LADRC) and differential flatness theory is proposed to realize accurate trolley positioning and effective swing eliminating. Specifically, in order to simplify the system model, we introduce the differential flatness theory to construct system output. Based on this method, the uncertainty and system external disturbance become total disturbance. The control method of the crane is designed according to LADRC. Then, we use the bird swarm algorithm (BSA) to optimize controller’s parameters. Finally, the double-pendulum crane control method based on the LADRC can better accomplish the crane’s anti-swing and location in the real environment according to simulation and experimental results, which confirms its effectiveness and robustness in existing system.

Design and control of multiphase interleaved boost converters-based on differential flatness theory for PEM fuel cell multi-stack applications

This article is focused on the development of an energy management algorithm applied to a multi-stack fuel cell (FC) system for DC microgrid applications. To guarantee the performance of the FC stacks, the current ripple is reduced by employing multiphase interleaved boost converters. A proposed advanced control technique of the multi-stack with multiphase converters for the proton exchange membrane (PEM) FCs is estimated based on a differential flatness approach, in which it can track the power demand in real-time. Furthermore, the differential flatness based-control can ensure the balance of the DC bus voltage of the DC microgrid when load disturbance occurs. The flatness-based energy management strategy is based on both inner current loops (control of the multi-stack PEMFC through their multiphase interleaved boost converters) and outer voltage loop (DC bus voltage regulation). Compared to classic PI controllers mainly based on the linearization of the system to obtain the transfer function (making complex its application), the flatness-based theory leans on time-domain making it easier its use for various applications while ensuring good performances. To validate the proposed control structure, an FC converter system (5 kW) is realized and validated in the laboratory. For hydrogen production, the methanol FC system has consisted of a reformer engine that changes water mixed methanol liquid into hydrogen to supply FC stacks (ME2Power Fuel Cell System: 50 V, 5 kW). The proposed control algorithm is tested experimentally by using a dSPACE controller board platform. Simulation and test bench results authenticate the excellent performance during load cycles in DC microgrid.

Differential flatness theory-based control and filtering for a mobile manipulator

The article proposes a differential flatness theory based control and filtering method for the model of a mobile manipulator. This is a difficult control and robotics problem due to the system’s strong nonlinearities and due to its underactuation. Using the Euler-Lagrange approach, the dynamic model of the mobile manipulator is obtained. This is proven to be a differentially flat one, thus confirming that it can be transformed into an input-output linearized form. Through a change of state and control inputs variables the dynamic model of the manipulator is finally written into the linear canonical (Brunovsky) form. For the latter representation of the system’s dynamics the solution of both the control and filtering problems becomes possible. The global asymptotic stability properties of the control loop are proven. Moreover, a differential flatness theory-based state estimator, under the name of Derivative-free nonlinear Kalman Filter, is developed. This comprises (i) the standard Kalman Filter recursion on the linearized equivalent model of the mobile manipulator and (ii) an inverse transformation, relying on the differential flatness properties of the system which allows for estimating the state variables of the initial nonlinear model. Finally, by redesigning the aforementioned Kalman Filter as a disturbance observer one can achieve estimation and compensation of the disturbance inputs that affect the model of the mobile manipulator.

Differential Flatness-Based Cascade Energy/Current Control of Battery/Supercapacitor Hybrid Source for Modern e–Vehicle Applications

This article proposes a new control law for an embedded DC distributed network supplied by a supercapacitor module (as a supplementary source) and a battery module (as the main generator) for transportation applications. A novel control algorithm based on the nonlinear differential flatness approach is studied and implemented in the laboratory. Using the differential flatness theory, straightforward solutions to nonlinear system stability problems and energy management have been developed. To evaluate the performance of the studied control technique, a hardware power electronics system is designed and implemented with a fully digital calculation (real-time system) realized with a MicroLabBox dSPACE platform (dual-core processor and FPGA). Obtained test bench results with a small scale prototype platform (a supercapacitor module of 160 V, 6 F and a battery module of 120 V, 40 Ah) corroborate the excellent control structure during drive cycles: steady-state and dynamics.

Flatness-Based Grey Wolf Control for Load Voltage Unbalance Mitigation in Three-Phase Four-Leg Voltage Source Inverters

Standalone power-supply systems become a key solution to effectively address the load demand in remote locations, wherein the voltage asymmetry raises as a particular concern in view of a large number of single-phase household and communities loads. Over these loading conditions, the use of the four-leg voltage source inverters (FL-VSIs) appears as a suitable topology to provide clean power with symmetrical voltage waveforms. The reported work employs the differential flatness theory associated with grey wolf optimization in order to guarantee proper operations of the FL-VSIs in different loading conditions. In fact, the differential flatness theory is applied to check the flatness of the FL-VSIs, which allows implementing a reduced control model. Then, a grey wolf algorithm, which acts as a tracking controller [grey wolf controller (GWC)], is developed. The GWC generates an optimal control signal used by the differentially flat model so as to ensure adequate control performance, even under severe load disturbances and model inaccuracies. A comprehensive experimental test is carried out to validate the effectiveness of the proposed flatness-based grey wolf controller.

Flatness-Based Active Disturbance Rejection Control For a Wheeled Mobile Robot Subject To Slips and External Environmental Disturbances

Abstract This work suggests flatness-based Active Disturbance Rejection Control (ADRC) to deal with the problem of trajectory tracking for Wheeled Mobile Robot (WMR). Based on the differential flatness theory, the nonlinear WMR system is transformed into two integral-chains, which makes the creation of a state feedback controller easier. In order to improve the WMR tracking, slip and external environmental disturbance must be considered in the controller design. Therefore, an Extended State Observer (ESO) is created to estimate the obtained linearized system state and the extended state known as lumped uncertainties. The latter represent the total effects of slip and the external environmental disturbances to WMR. After that, according to the ESO results, a complementary element is added to the state feedback controller to compensate the effects of lumped disturbances. Simulation results are introduced to demonstrate the advantages of combining ADRC with flatness control.

Stabilization of a stock‐loan valuation PDE process using differential flatness theory

AbstractThe article proposes flatness‐based control for stabilization of a stock‐loan valuation process that is described by a partial differential equation. By applying semi‐discretization and the finite differences method, the state‐space model of the stock loan has been obtained. It has been proven that the individual rows of this state‐space model are nonlinear ODEs which can be viewed as differentially flat subsystems. For the local subsystems, into which the stock‐loan PDE is decomposed, it becomes possible to apply boundary feedback control. The controller design proceeds by showing that the state‐space model of the stock loan PDE stands for a differentially flat system. Next, for each subsystem which is related to a nonlinear ODE, a virtual control input is computed, that can invert the subsystem's dynamics and can eliminate the subsystem's tracking error. From the last row of the state‐space description, the control input (boundary condition) that is actually applied to the stock loan PDE system is found. This control input contains recursively all virtual control inputs which were computed for the individual ODE subsystems associated with the previous rows of the state‐space description. Thus, by tracing the rows of the state‐space model backwards, at each iteration of the control algorithm, one can finally obtain the control input that should be applied to the stock loan PDE system so as to assure that all its state variables will converge to the desirable setpoints.By showing the feasibility of such a control method it is also proven that through selected modification of the PDE boundary conditions the value of the stock loan can be made to converge and stabilize at specific reference values.

Flatness-based finite-time leader–follower formation control of multiple quadrotors with external disturbances

This study considers the leader–follower formation control problem for multiple quadrotors in the presence of external disturbances. Based on the differential flatness theory, the underactuated quadrotor system is transformed into a fully actuated one with four degrees of freedom and four control inputs. Based on this model, a distributed finite-time observer is developed to reconstruct the leader's states for each follower. Then, an observer-based finite-time controller is proposed based on an adaptive disturbance rejection approach, which is independent of the upper bounds of the disturbances, to address the formation control problem for multiple quadrotors. The stability analysis indicates that the closed-loop system theoretically achieves the finite-time stability and robustness against the disturbances. Finally, a numerical simulation is provided to verify the performance of the proposed formation control scheme.

Flatness‐based control approach to networked control under sensor delays

A new control method based on differential flatness theory is developed in this article, aiming at solving the problem of networked control under sensor delays. A sensor network comprising N sensors and N control inputs is considered. Time delays between the control inputs and the system's outputs (sensor readings) are taken into account. Using the principle of dynamic extension, which means that by considering certain control inputs and their derivatives as additional state variables, a state-space description for the networked control system is obtained. It is proven that the dynamic model of the networked control system is a differentially flat one. This enables its transformation into a linear canonical and decoupled form, for which the design of a stabilising feedback controller becomes possible. The proposed feedback controller is of proven stability and assures fast and accurate tracking of the reference setpoints by the outputs of the network control system. Moreover, by using a Kalman-filter-based disturbances' estimator, it becomes possible to estimate in real time and compensate for parametric uncertainty, external perturbation inputs, and variation in sensor delays that may appear in the networked control system.

Fixed-time trajectory tracking for a quadrotor with external disturbances: A flatness-based sliding mode control approach

This study proposes a flatness-based fixed-time sliding mode control strategy to address the trajectory tracking problem for a quadrotor with external disturbances. Based on the differential flatness theory, the underactuated dynamics of the quadrotor is converted to four fully-actuated subsystems. As the velocity information of the desired trajectory cannot be obtained directly in some practical applications, a fixed-time state observer is presented to estimate the velocity of the desired trajectory. In order to enhance the system's robustness, a fixed-time disturbance observer is also constructed to estimate and compensate the lumped disturbances. Based on the fixed-time state observer and disturbance observer, the backstepping technique is utilized to perform the sliding mode controller design process, where a fixed-time differentiator is presented to estimate the virtual control inputs to avoid the problem of “explosion of complexity” suffered by the typical backstepping. The stability analysis is conducted to demonstrate the uniformly ultimately bounded stabilization of the closed-loop system with fixed-time convergence regardless of the initial conditions. Finally, the simulation results validate the superior performance of the proposed control scheme.