Describing function

About: Describing function is a research topic. Over the lifetime, 1742 publications have been published within this topic receiving 26702 citations.

Stable haptic interaction using passive and active actuators

Abstract: This paper presents a stable control method for a hybrid haptic device comprising a brake and a motor. A review of stability condition via describing function analysis is first presented. The results show that while brakes are intrinsically stable, an active device is limited in terms of stiffness. The stability is however improved if the brake simulates a physical damping. Subsequently, the stability condition is obtained via passivity condition analysis. The results demonstrate that the stiffness is improved by engaging both actuators to create resistive forces and the passivity is respected assuming a passive virtual environment. An energy and a stiffness-bounding algorithms have been developed in order to assure the stability of the coupled system in this case. It has been tested and validated using a 1-DOF hybrid haptic device by the simulation of an unstable and an active virtual environments respectively . Experimental results show that the displayable stiffness is improved under stability conditions using the control method. Furthermore, it allows the hybrid system to simulate nonlinear and unstable virtual environments and the controller remains independent of the virtual environment model.

A note on the describing function of an element with coulomb, static, and viscous friction

Abstract: IT IS NOW a standard procedure to use the sinusoidal-analysis, or describing-function, method in the investigation of quasilinear servomechanisms. The Nyquist criterion is applied to determine the stability conditions, and the Nyquist diagram provides some information as to the type of equalization that should be employed.1 These techniques may be used even when the describing-function of the nonlinear element is both frequency and amplitude sensitive, but this requires the graphing of a family of describing functions.2 This complication may be avoided by isolating the frequency-sensitive and amplitude-sensitive portions of the describing function into separate terms. In the trivial case, the describing function Ho? may be represented as $H_o{\prime} = H_{n}H_{o}\eqno{\hbox{(1)}}$ where Ho is amphtude dependent and frequency independent, and Hn is amplitude independent and frequency dependent. Unfortunately, this easy separation of terms canot be achieved for the describing function of an element with coulomb, static, and viscous friction, a type of element present in many servo systems. This paper with show, however, that an equivalent result is achieved by means of an equivalent, block-diagram representation of Ho?.

Closed-loop frequency analyses of reset systems.

Abstract: Today, linear controllers cannot satisfy requirements of high-tech industry due to fundamental limitations like the waterbed effect. This is one of the reasons why nonlinear controllers, such as reset elements, are receiving increased attention. To analyze reset elements in the frequency domain, researchers use the Describing Function (DF) method. However, it cannot accurately predict the closed-loop frequency responses of the system because it neglects high order harmonics. To overcome these barriers, this paper proposes a mathematical framework to model the closed-loop frequency responses of reset systems including the closed-loop high order harmonics. Furthermore, pseudo-sensitivities for reset systems are defined to make their analyses more straight-forward. In addition, a user-friendly toolbox is developed based on the proposed approach to facilitate frequency analyses of reset systems. To show the effectiveness of the method, multiple illustrative examples on a high-tech precision positioning stage are used to compare the results of the closed-loop frequency responses obtained using our proposed method with DF method. The results demonstrated that the proposed method is significantly more precise than the DF method. Indeed, this developed toolbox can enable reset controllers to be widely-used in industry and academia.

Adaptive control of uncertain Hammerstein systems with hysteretic nonlinearities

Abstract: We numerically investigate the sense in which an adaptive control law achieves internal model control of Hammerstein plants with Prandtl-Ishlinskii hysteresis We apply retrospective cost adaptive control (RCAC) to a command-following problem for uncertain Hammerstein systems with hysteretic input nonlinearities The only required modeling information of the linear plant is a single Markov parameter Describing functions are used to determine whether the adaptive controller inverts the plant at the exogenous frequencies

Tuning PID controllers with symmetric send-on-delta sampling strategy

Abstract: A procedure for tuning PID controllers with SSOD sampling for FOPTD systems is proposed. It is based on the definition of a new robustness measure to avoid limit cycle oscillations, called the Tsypkin margin (MT). This margin is based on the Tsypkin method and does not rely on the attenuation of high order harmonics, as the describing function approaches require. Therefore, the avoidance of limit cycle oscillations can be guaranteed for any system, as a difference with the describing function based procedures. The procedure allows to obtain the PID controller that minimizes the disturbance IAE while fulfilling constraints on robustness to oscillations and on control action bumps due to the SSOD sampling. A freely available Java tool has been developed in order to simplify the application of the tuning procedure. In case of a non FOPTD system, it first calculates an approximate FOPTD model. The paper shows that the derivative filter parameter, N is a critical tuning parameter in order to find a compromise between performance and control action bumps.