Flat Output Research Articles

Compact Q-switched Tm:YLF laser with high slope efficiency, high power stability and high beam quality

This paper presents a high-efficiency and actively Q-switched Tm:YLF laser with a compact design by arranging the optical elements closest to each other in a linear cavity. With a flat output coupler (OC), slope efficiency of 74.7% and output power of 12.41 W were achieved in continuous-wave operation. Using a volume Bragg grating (VBG) as the OC and wavelength selector, the output power and slope efficiency of the 1.908 μm laser were 10.65 W and 67.3%, respectively. The corresponding beam quality M2 was about 1.5 at the maximum power. The Tm:YLF laser with VBG revealed high stability with a power fluctuation of 0.45% and central wavelength variation of 0.23 nm for one-hour continuous testing. In pulsed operation at 10 kHz, average power of 10.07 W and slope efficiency of 66.0% were obtained. This actively Q-switched Tm:YLF laser of 10-W level will be an ideal pump source for compact and high-efficiency mid-infrared optical parametrical oscillators for practical applications.

A High Efficiency 100-W E-Band Continuous Wave Traveling Wave Tube

A continuous wave (CW) E-band traveling wave tube (TWT) is developed to meet the demand of broadband, high power and high efficiency amplifiers for future wireless communications. Several methods are combined to improve the TWT performance, in particular the efficiency, including: increase of the interaction impedance by setting operation point near the cutoff frequency, use of the folded waveguide (FWG) slow wave structure (SWS) with modified circular bends (MCBs) and the phase velocity jumping (PVJ) technique. A novel dual-mode asymmetry multistage depressed collector is used to significantly improve the recovery efficiency without the return of electrons. The first prototype tube has been assembled and tested. A total efficiency of 43% with an electronic efficiency of 9.8%, output power of 120 W and gain of 41 dB is achieved in 81-86GHz where the beam voltage is 15.8kV and the beam current is 77mA. The tube shows very flat output characteristics in saturation state and provides high linear power and efficiency. The -1dB compression point output is over 50W with the linear (output back off 4dB) total efficiency up to 26.4%.

Are all parental experiences equal?: Cluster analysis of salivary cortisol and perception of parental stress in mothers of children with autism spectrum disorder

Many parents of children with autism spectrum disorder (ASD) report high stress levels related to their parenting role. The experience of stress is reflected in physical symptoms and well-being, as observed by abnormal patterns of cortisol regulation. However, it may be a mistake to conceive parenthood as unilaterally stressful as diverse experiences are possible. Mothers of children with ASD provided salivary cortisol samples and self-rated parental stress. The area under the curve with respect to ground was calculated in regard to three daily collection time points. As a group, mothers reported average levels of parenting stress and flat daily cortisol output. The current age of the child and age at the time of diagnosis moderately predicted overall daily cortisol. Hierarchical clusters analysis identified four distinct profiles of stress regulation based on daily cortisol regulation and perception of parental stress. Groups did not differ based on the severity of symptoms related to autism or demographic information. We suggest that other variables, such as stress mediators and secondary stressors, may explain variability in stress regulation. Future research and intervention should perceive the parental experience as heterogeneous and focus on individualizing the support offered in light of diverse experiences.

Ground vehicle lane-keeping assistance system via differential flatness output feedback control and algebraic derivative estimation

Vehicle run-off-road is one of the most frequent and fatal traffic accidents in the United States. Various lane-keeping assistance (LKA) systems have been developed in the last decade to help drivers stay on the road. Most of them are built upon linear driver–vehicle–road (DVR) models and treat road curvature as a disturbance. Albeit effective, their control performance would degrade if road curvature varies rapidly. This paper proposes a novel nonlinear DVR model by integrating a driver steering model into vehicle–road kinematics, which explicitly considers road curvature. Particularly, this nonlinear DVR system has been proven to be differentially flat, and a flatness-based LKA system is designed. Additionally, Model-Free Control is introduced to compensate for system modeling errors. Hardware-in-the-loop simulations and diver-in-the-loop experiments validate the proposed control framework and demonstrate the performance enhancement with respect to a representative linear robust LKA system.

Robust trajectory tracking controller for an octocopter UAV

This work proposes an Input-Output linearization (IOL) trajectory tracking controller for an actuated octocopter unmanned aerial vehicle (UAV). The underactuated octocopter dynamics is converted into actuated one using the differential flat outputs. The proposed controller for actuated octocopter UAV is robustified using α Uncertainty and Disturbance Estimator (αUDE). The Lyapunov theory is used to ascertain closed-loop stability in achieving the desired trajectory. A comparative study is carried out to show the better performance of the proposed controller. Results show the efficacy of the proposed controller in handling the effects of slow as well as fast varying disturbances, parametric uncertainties, and measurement noise. The quantitative analysis based on performance indices and root mean square error validated the results.

Spherical Inverted Pendulum on a Quadrotor UAV: A Flatness and Discontinuous Extended State Observer Approach

This article addresses the problem of balancing an inverted spherical pendulum on a quadrotor. The full dynamic model is obtained via the Euler-Lagrange formalism, where the dynamics of the pendulum is coupled to the dynamics of the quadrotor, taking as control inputs the torques associated with the yaw, roll, and pitch dynamics, and a control input for the vertical displacement in height. A trajectory tracking control scheme is proposed by means of an active disturbance rejection control based on a discontinuous extended state observer (ADRC-DESO) that allows controlling the system in the translational dynamics of the quadrotor including the rotational dynamics and the inverted pendulum dynamics. To address this problem, the dynamic model is linearized around an equilibrium point, taking into consideration that the system operates in close vicinity of the equilibrium points, thus considerably simplifying the dynamic model. Proving that the linear model is controllable and therefore differentiable flat, flat outputs are proposed around the displacements associated with the three cartesian axes of the Euclidean space, including a dynamic associated with the yaw dynamics of the quadrotor allowing to parameterize the full linear system. Simulation results as well as a convergence analysis validate the performance of the strategy.

Game Theoretic Approach for Robust Trajectory Tracking by a Multirotor

In this article, the problem of precisely tracking a nominal trajectory by an under-actuated multirotor platform is formulated as a game against bounded external disturbances, measurement noise, and initial condition uncertainty. The nominal trajectory is composed of desired position and yaw angle, which is known to be a differentially flat output for the multirotor state dynamics. Using this property, a linearized error dynamics is obtained in the vicinity of the nominal trajectory. Using the linearized error dynamics, a linear-quadratic differential game approach is used to propose an integrated estimation and control method for trajectory tracking. A salient feature of this approach is that the drag coefficients are estimated online from the information of the tracking error. Moreover, there are only six parameters to be tuned for both estimation and control loops. The intuitive rationale behind the tuning of these parameters is also discussed. Simulations as well as experimental validations are presented to demonstrate the performance and applicability of the controllers.

Differential Flatness-Based Parameter Estimation for Suspended Load Drones

The transportation of goods by drones using cable towing has recently attracted considerable attention. When flying a suspended load drone, any discrepancy between the mathematical model and the actual drone deteriorates control performance. However, because some physical parameters are difficult to measure, creating an accurate mathematical model is extremely difficult. Therefore, we propose a parameter estimation method using differential flatness that can be extended for application to suspended load drones. This method overcomes the problem of dealing with higher-order derivatives of flat outputs and enables the estimation of physical parameters. In this study, we experimentally show that the proposed method improves trajectory tracking performance.

Efficient GaN-on-Si Power Amplifier Design Using Analytical and Numerical Optimization Methods for 24–30 GHz 5G Applications

This paper presents the design procedure of an efficient compact monolithic microwave integrated circuit power amplifier (MMIC PA) in a 0.1 μm GaN-on-Si process for 5G millimeter-wave communication. Load/source-pull simulations were conducted to correctly create equivalent large-signal matching models for stabilized power cells and to determine the optimal impedance domain. The shorted stub with bypass capacitors minimizes the transistor’s output reactance, simplifying the matching objective to an approximate real impedance transformation (IT). With miniaturization as the implementation guide, explicit formulas and tabulated methods based on mathematical analysis were applied to synthesize the filtering matching networks (MNs) for the input and output stages. In addition, a CAD-dependent numerical optimization approach was used for the interstage MN that needs to cope with high IT ratio and complex loads. The continuous-wave (CW) characterization for the proposed two-stage PA demonstrated 19.8 ± 0.7 dB of small-signal gain, very flat output power (Pout) and power-added efficiency (PAE) at 4 dB gain compression of 32–32.4 dBm and 34–34.6%, respectively, over 24–30 GHz, with 37.1% of peak PAE at mid-frequency.

Analysis of DC/DC Boost Converter–Full-Bridge Buck Inverter System for AC Generation

This paper presents an analysis and simulation of the mathematical model associated to the DC/DC Boost converter–full-bridge Buck inverter system to regulate the voltage output of the DC/DC Boost converter allowing bipolar voltages higher than the input voltage via the full-bridge Buck inverter. To validate the model, the differential flatness property is applied via the flat outputs of the system (energy for the DC/DC Boost converter and the voltage of the full-bridge Buck inverter) considering the complete dynamics, in conjunction with fixed and time-variant trajectory planning. In the simulation results, it is observed that the error signals of the states versus the reference trajectories are acceptable. Regarding the validation of the model, this is performed with open-loop simulations at the circuit level using the SimPowerSystems Toolbox of Matlab-Simulink. The simulation results validate the good performance of the system under study. In this way, the main contribution of this work is that for the first time in the literature, the analysis of a complete dynamics for a conversion system from DC to AC without the use of a transformer and taking advantage of differential flatness is reported; thus, the system analyzed could be represented as an alternative in applications of renewable energies that require conversion from DC to AC.

Necessary and Sufficient Conditions for Difference Flatness

We show that the flatness of a nonlinear discrete-time system can be checked by computing a unique sequence of involutive distributions. The well-known test for static feedback linearizability is included as a special case. Since the computation of the sequence of distributions requires only the solution of algebraic equations, it allows an efficient implementation in a computer algebra program. In case of a positive result, a flat output can be obtained by straightening out the involutive distributions with the Frobenius theorem.

A Differential Flatness-Based Model Predictive Control Strategy for a Nonlinear Quarter-Car Active Suspension System

Controlling an automotive suspension system using an actuator is a complex nonlinear problem that requires both fast and precise solutions in order to achieve optimal performance. In this work, the nonlinear model of a quarter-car active suspension is expressed in terms of a flat output and its derivatives in order to embed the nonlinearities of the system in the flat output. Afterward, a Model Predictive Controller based on the differential flatness derivation (MPC-DF) of the quarter-car is proposed in order to achieve optimal control performance in both passenger comfort and road holding without diminishing the lifespan of the wheel. This formulation results in a linear optimization problem while maintaining the nonlinear behavior of the active suspension system. Afterward, the optimization problem is solved by means of Quadratic Programming (QP), enabling real-time implementation. Simulation results are presented using a realistic road disturbance to show the effectiveness of the proposed control strategy.

Trajectory Tracking of Autonomous Vehicle: A Differential Flatness Approach With Disturbance-Observer-Based Control

In this article, the problem of trajectory tracking control is studied for autonomous vehicle with consideration of the nonlinearity and coupling characteristics. To achieve accurately trajectory tracking in longitudinal and lateral motions simultaneously, a flatness disturbance-observer-based control scheme is developed to address the kinematic coupling issue of the vehicle, which can avoid extra modeling errors introduced by linearization. The main idea is that, derived from a suitable selection of flat outputs, the complex kinodynamic vehicle model is transformed into three fully-actuated subsystems with same structure, which will provide convenience for controller design. Then, in order to compensate the lumped disturbances caused by modeling and transformation, a disturbance-observer-based control strategy is designed via backstepping. Trajectory tracking control is accomplished through longitudinal, lateral and yaw motions simultaneously. The stability of the closed-loop dynamics is guaranteed in the framework of Lyapunov and input-to-state stability theories. Finally, the CarSim simulation tests demonstrate that the proposed integrated control method can successfully promote the trajectory tracking performance in different manoeuvres.

Flatness-Based Backstepping Antisway Control of Underactuated Crane Systems under Wind Disturbance

A control method that combines trajectory planning and backstepping is proposed for the antisway problem of underactuated overhead cranes under wind disturbance. First, a set of flat outputs is proposed so that the crane system dynamics can be represented by each order of flat outputs. Sufficient relevant constraints are given to ensure that the trolley can arrive at the desired position in a limited time under variable rope lengths, and that the swing angle can be suppressed when the payload is lifted or lowered during operation. The planned trajectory is obtained by solving for the optimal parameters of the flat output. Next, to reduce the deviation caused by wind disturbance on the actual control of the trajectory, a tracking controller is designed. Because the system output space and flat output space are differentiable homeomorphisms, the backstepping controller constructed based on the flat output can indirectly control the system output, which makes the backstepping method applicable to underactuated cranes. The simulation results show that the proposed method is effective and has strong robustness.

A Flat System Possessing no (x, u)-Flat Output

In general, flat outputs of a nonlinear system may depend on the system's state and input as well as on an arbitrary number of time derivatives of the latter. If a flat output which also depends on time derivatives of the input is known, one may pose the question whether there also exists a flat output which is independent of these time derivatives, i.e., an (x,u)-flat output. Until now, the question whether every flat system also possesses an (x,u)-flat output has been open. In this contribution, this conjecture is disproved by means of a counterexample. We present a two-input system which is differentially flat with a flat output depending on the state, the input and first-order time derivatives of the input, but which does not possess any (x,u)-flat output. The proof relies on the fact that every (x,u)-flat two-input system can be exactly linearized after an at most dim(x)-fold prolongation of one of its (new) inputs after a suitable input transformation has been applied.

Discrete-time Flatness and Linearization along Trajectories

The paper studies the relation between a nonlinear time-varying flat discrete-time system and the corresponding linear time-varying systems which are obtained by a linearization along trajectories. It is motivated by the continuous-time case, where it is well-known that the linearization of flat systems along trajectories results in linear time-varying systems which are again flat. Since flatness implies controllability, this constitutes an important verifiable necessary condition for flatness. In the present contribution, it is shown that this is also true in the discrete-time case: We prove that the linearized system is again flat, and that a possible flat output is given by the linearization of a flat output of the nonlinear system. Analogously, the map that describes the parameterization of the system variables of the linear system by this flat output coincides with the linearization of the corresponding map of the nonlinear system. The results are illustrated by two examples.

Flatness Analysis for the Sampled-data Model of a Single Mast Stacker Crane

We show that the Euler-discretization of the nonlinear continuous-time model of a single mast stacker crane is flat. The construction of the flat output is based on a transformation of a subsystem into the linear time-variant discrete-time controller canonical form. Based on the derived flat output, which is also a function of backward-shifts of the system variables, we discuss the planning of trajectories to achieve a transition between two rest positions and compute the corresponding discrete-time feedforward control.

Continuous-Time Trajectory Estimation for Differentially Flat Systems

Continuous-time estimation using splines on Lie groups has been gaining traction in the literature due to the ability to incorporate high-frequency sensor data without introducing new optimization parameters. However, evaluating time derivatives and Jacobians of Lie group splines is computationally expensive, limiting their use mainly to offline applications. Motivated by the trajectory planning literature, we develop a new estimation technique that leverages the differential flatness property of many dynamic systems to define the spline in the system's flat output space, which is often Euclidean. Doing so has the added benefit of providing a simple and effective way to include system inputs in the estimation process. We show an example of flatness-based estimation for the unicycle dynamic model. We then show that this new method can achieve similar performance as Lie group spline estimation with significantly less computation time, and validate its use in hardware using a differential-drive robot.

On the exact linearisation and control of flat discrete-time systems

The paper addresses the exact linearisation of flat nonlinear discrete-time systems by generalised static or dynamic feedbacks which may also depend on forward-shifts of the new input. We first investigate the question which forward-shifts of a given flat output can be chosen in principle as a new input, and subsequently how to actually introduce the new input by a suitable feedback. With respect to the choice of a feasible input, easily verifiable conditions are derived. Introducing such a new input requires a feedback which may in general depend not only on this new input itself but also on its forward-shifts. This is similar to the continuous-time case, where feedbacks which depend on time derivatives of the closed-loop input – and in particular quasi-static ones – have already been used successfully for the exact linearisation of flat systems since the nineties of the last century. For systems with a flat output that does not depend on forward-shifts of the input, it is shown how to systematically construct a new input such that the total number of the corresponding forward-shifts of the flat output is minimal. Furthermore, it is shown that in this case the calculation of a linearising feedback is particularly simple, and the subsequent design of a discrete-time flatness-based tracking control is discussed. The presented theory is illustrated by the discretised models of a wheeled mobile robot and a 3DOF helicopter.

Estimated Flatness-Based Active Disturbance Rejection Control for Load Frequency Control of Power Systems

An estimated flatness-based active disturbance rejection control (EF-ADRC) is proposed for the load frequency control (LFC) of power systems in this paper. Different from the traditional ADRC design, the selections of the order and the gain of the flatness-based ADRC controller are based on the flatness output, which is a linear combination of the system states, rather than the nominal integral cascade model. In addition, an extra extended state observer (ESO) is adopted to estimate the system states and the unknown external disturbance. The former is used to replace the actual system states in the flat output, and the latter is used in trajectory planning of the estimated flat output to achieve the unbiased tracking. The proposed method is applied to single-area systems with reheat turbine and hydro turbine, as well as a four-area system, and the simulation results show that the proposed method can achieve satisfactory performance.