Periodic Reference Input Research Articles

Disturbance rejection design for repetitive control system with state delay based on equivalent input disturbance compensation

A modified equivalent-input-disturbance compensation method is investigated for repetitive control systems with state time delays, which achieve effective rejection of unknown disturbance and effective tracking control of periodic reference input. By incorporating the sliding mode observer into the equivalent-input-disturbance theory, the equivalent-input-disturbance estimator structure is determined. The structure presented in this work can effectively estimate the unknown disturbance without making use of the inverse model of the system. Moreover, an asymptotic stability condition is devised to ensure that the repetitive control systems are stable, as well as the corresponding controller gains are designed with the aim to guarantee the performance of the system to suppress unknown disturbances. Numerical example simulations demonstrate the effectiveness and advantages of the devise scheme.

Model-free continuation of periodic orbits in certain nonlinear systems using continuous-time adaptive control

This paper generalizes recent results by the authors on noninvasive model-reference adaptive control designs for control-based continuation of periodic orbits in periodically excited linear systems with matched uncertainties to a larger class of periodically excited nonlinear systems with matched uncertainties and known structure. A candidate adaptive feedback design is also proposed in the case of scalar problems with unmodeled nonlinearities. In the former case, rigorous analysis shows guaranteed performance bounds for the associated prediction and estimation errors. Together with an assumption of persistent excitation, there follows asymptotic convergence to periodic responses determined uniquely by an a priori unknown periodic reference input and independent of initial conditions, as required by the control-based continuation paradigm. In particular, when the reference input equals the sought periodic response, the steady-state control input vanishes. Identical conclusions follow for the case of scalar dynamics with unmodeled nonlinearities, albeit with slow rates of convergence. Numerical simulations validate the theoretical predictions for individual parameter values. Integration with the software package coco demonstrates successful continuation along families of stable and unstable periodic orbits with a minimum of parameter tuning. The results expand the envelope of known noninvasive feedback strategies for use in experimental model validation and engineering design.

Inversion-Based Hysteresis Compensation Using Adaptive Conditional Servocompensator for Nanopositioning Systems

Abstract Nanopositioning stages are widely used in high-precision positioning applications. However, they suffer from an intrinsic hysteretic behavior, which deteriorates their tracking performance. This study proposes an adaptive conditional servocompensator (ACS) to compensate the effect of the hysteresis when tracking periodic references. The nanopositioning system is modeled as a linear system cascaded with hysteresis at the input side. The hysteresis is modeled with a modified Prandtl–Ishlinskii (MPI) operator. With an approximate inverse MPI operator placed before the system hysteresis operator, the resulting system takes a semi-affine form. The design of the ACS consists of two stages: first, we design a continuously implemented sliding mode control (SMC) law. The hysteresis inversion error is treated as a matched disturbance, and an analytical bound on the inversion error is used to minimize the conservativeness of the SMC design. The second part of the controller is the ACS. Under mild assumptions, we establish the well-posedness and periodic stability of the closed-loop system. In particular, the solution of the closed-loop error system will converge exponentially to a unique periodic solution in the neighborhood of zero. The efficacy of the proposed controller is verified experimentally on a commercial nanopositioning device under different types of periodic reference inputs, via comparison with multiple inversion-based and inversion-free approaches.

Tracking control with disturbance rejection of nonlinear fractional order fuzzy systems: Modified repetitive control approach

This paper addresses the problem of tracking control and disturbance rejection for nonlinear fractional order systems. To this end, based on Takagi-Sugeno (T-S) fuzzy modeling approach and a fractional order modified repetitive controller, an output feedback strategy is constructed for a class of nonlinear fractional order systems. T-S fuzzy model represents an accurate model of nonlinear fractional order systems and simplifies the procedure of control system design. By introducing a novel Fuzzy Lyapunov Function (FLF), sufficient conditions are established in the framework of Linear Matrix Inequalities (LMIs) to ensure the stability of the closed-loop system. Based on the achieved conditions, it is shown that the external disturbance is attenuated and the output of the system tracks the periodic reference input with an ignorable error. Lastly, three examples including fractional order Chua’s circuit, Lorenz-like and Chen systems are performed to show the efficiency of the method proposed in this paper.

Relative control of an ion beam shepherd satellite in eccentric orbits

The “ion-beam shepherd” is a contactless space debris removal concept, which implies that the shepherd satellite (SS) must be controlled to move at a certain small distance in front of a space debris object during the de-orbiting phase. As shown in recent studies, the strategy to control the relative motion of this formation using only one compensation thruster of the SS can significantly reduce propellant mass of the actuators. However, such a control system maintains a required relative position of the underactuated SS with a periodic error. The amplitude of this error is small for quasi-circular orbits, but it can be significant for eccentric orbits due to the time-varying behavior of the plant. Given that there are debris in non-circular orbits that need to be removed, this article investigates the feasibility of using only one compensation thruster to control relative position of the SS in eccentric orbits. Both time-invariant and periodic linear-quadratic regulators are examined for using as a controller of the SS. A periodic reference input of the regulator is designed to reduce the relative position errors and improve the efficiency of the ion beam impact on the space debris object.

Generalized extended state observer–based repetitive control for systems with mismatched disturbances

SummaryA generalized extended state observer (GESO) is devised to improve the disturbances rejection performance in a repetitive‐control system (RCS) for a class of single‐input, single‐output nonlinear plants with nonintegral chain form and mismatched disturbances. By appropriately choosing a disturbance compensation gain and incorporating the disturbance estimate into a repetitive control law, a GESO‐based RCS is established. In this system, the repetitive controller ensures tracking of a periodic reference input, and the incorporation of the disturbance compensation into the control input enables attenuating the lumped disturbance from the system output. Stability criteria and design algorithms have been developed for the system. A case study on the speed control of a rotational control system exhibits that the GESO‐based RCS delivers not only a promising disturbance rejection performance but also a superior property of tracking performance.

Retracted: SCRIM drive system using adaptive backstepping control and mended recurrent Romanovski polynomials neural network with reformed particle swarm optimization

SummaryDue to air‐gap field harmonic, cogging torque, stator's current time harmonic, and the influence of flux saturation, a six‐phase copper rotor induction motor (SCRIM) drive system has highly nonlinear uncertainties. Thus, the linear control method for the SCRIM drive system is difficult to achieved good performance under the nonlinear uncertainty action. To obtain better control performance, the adaptive backstepping control system using switching function is firstly proposed for controlling the SCRIM drive system to overcome the uncertainty influence. With the proposed control system, the SCRIM drive system holds in robustness to these uncertainties for the tracking of periodic reference trajectories. To enhance the robustness of the SCRIM drive system, the adaptive backstepping control system using adaptive law is proposed for estimating the required lumped uncertainty to reduce chattering phenomenon. When the inertia of the counterweight is varying, this proposed method can perform well in general situations but cannot get a satisfactory performance. The adaptive backstepping control system using mended recurrent Romanovski polynomials neural network with reformed particle swarm optimization (PSO) is thus proposed to estimate the lumped uncertainty and to compensate estimated error for obtaining better control performance. Furthermore, two variable learning rates of the weights in the mended recurrent Romanovski polynomials neural network are adopted by using reformed PSO to speed up parameter's convergence. Finally, some experimental results with comparative control performances are demonstrated, and then, the effectiveness of proposed control system with better control performance is verified for the position tracking of periodic reference inputs.

Integrated Modified Repetitive Control With Disturbance Observer of Piezoelectric Nanopositioning Stages for High-Speed and Precision Motion

The tracking performance of piezoelectric nanopositioning stages is vital in many applications, such as scanning probe microscopes (SPMs). Although modified repetitive control (MRC) can improve tracking performance for commonly used periodic reference input, it is sensitive to unexpected disturbances that deteriorate tracking precision, especially for high-speed motion. In order to achieve high-speed and precision motion, in this paper, a new composite control scheme by integrating MRC with disturbance observer (DOB) is developed. To simplify controller implementation, the hysteresis nonlinearity is treated as external disturbance and the proposed method is designed in frequency domain. The stability and robust stability are analyzed, and an optimization procedure to calculate the controller parameters is employed to enhance the performance to the maximum extent. To validate the effectiveness of the proposed method, comparative experiments are performed on a piezoelectric nanopositioning stage. Experimental results indicate that the hysteresis is suppressed effectively and the proposed method achieves high-speed and precision tracking with triangular waves references up to 25 Hz and improves the disturbance rejection ability with disturbances under different frequencies and robustness to model uncertainty through comparing with feedback controllers and MRC, respectively.

Periodic output regulation for general linear heterodirectional hyperbolic systems

This paper presents a backstepping solution of the periodic output regulation problem for general linear heterodirectional hyperbolic systems. By utilizing a time-varying signal model, regulators are determined for a general class of periodic reference inputs and disturbances. A systematic design procedure is developed for the state feedback regulator by solving the periodic regulator equations. In order to obtain an output feedback regulator, a time-varying finite-dimensional reference observer and a time-varying infinite-dimensional disturbance observer are determined. For this, a systematic backstepping approach is provided. The results of the paper are illustrated for a 4 × 4 hyperbolic system with periodic triangle reference inputs and sinusoidal disturbances subject to time-varying frequencies.

Novel Nonlinear Backstepping Control of Synchronous Reluctance Motor Drive System for Position Tracking of Periodic Reference Inputs with Torque Ripple Consideration

Owing to air-gap field harmonic, cogging torque, stator’s current time harmonic and the influence of flux saturation, a synchronous reluctance motor (SynRM) drive system has highly nonlinear uncertainties. Thus the linear control method for the SynRM drive system is difficult to achieved good performance under the nonlinear uncertainty action. To obtain better control performance the novel nonlinear backstepping control system using upper bound with switching function is firstly proposed for controlling the SynRM drive system to prevail the lumped uncertainty. With the proposed control system, the SynRM servo-drive system holds in robustness to uncertainties for the tracking of periodic reference trajectories. To enhance the robustness of the SynRM drive system, the novel nonlinear backstepping control system using adaptive law is proposed for estimating the required lumped uncertainty to reduce chattering phenomenon. When the inertia of the counterweight is varying, this proposed method can perform well in general situations, but cannot get a satisfactory performance. The novel nonlinear backstepping control system using reformed recurrent Hermite polynomial neural network with adaptive law and error estimated law is thus proposed to estimate the lumped uncertainty and compensate estimated error for obtaining better control performance. Furthermore, two varied learning rates of the reformed recurrent Hermite polynomial neural network is derived according to increment type Lyapunov function to speed-up parameter’s convergence. Finally, some experimental results with comparative control performances are demonstrated, and then the effectiveness of the proposed control system with better control performance is verified for the position tracking of periodic reference inputs with torque ripple consideration.

Robust H∞ control of an observer-based repetitive-control system

This paper concerns the problem of designing a robust observer-based modified repetitive-control system with a prescribed H∞ disturbance rejection level for a class of strictly proper linear plants with unknown aperiodic disturbances and time-varying structural uncertainties. A correction to the amount of the delay in the repetitive controller is introduced that leads to a significant improvement in tracking performance. An integrated performance index is defined to quantify the overall effect of rejecting the aperiodic disturbances and tracking the periodic reference input. A Lyapunov functional with two tuning parameters is used to derive a linear-matrix-inequality based robust stability condition for the system with a prescribed disturbance-rejection bound. Combining the performance indices, an optimization algorithm that searches for the best combination of state-observer gain and the feedback control gains is developed. A numerical example illustrates the design procedure and demonstrates the effectiveness of the method.

Tracking of periodic reference signal: A parameterized finite dimensional repetitive control approach

This paper is concerned with the development of a Modified Finite Dimensional Repetitive Control (MFDRC) system. Conventional FDRC is modified through parameterization of the controller using co-prime factorization. The proposed method of controller design specifies the input output characteristics beforehand, ensuring highly accurate tracking and disturbance rejection property of the system subjected to periodic reference input. Another problem is the stabilization issues in the input–output and the disturbance rejection characteristics of a conventional repetitive control system with respect to the periodic reference input owing to have infinite number of poles in the transfer function of it. It is desirable that the transfer functions from both the reference input and the disturbance to the output have a finite number of poles and that will be taken care of in the proposed scheme of MFDRC design depicted in this paper. The performance of designed controller has been validated in steps applied to a laboratory based servo system. Comparisons are done using conventional PID controller, FDRC-based PID controller and MFDRC to show advantage, disadvantage or limitation of applied control scheme.

Compensation for state‐dependent nonlinearity in a modified repetitive control system

SummaryThis paper presents an estimation and compensation of state‐dependent nonlinearity for a modified repetitive control system. It is based on the equivalent‐input‐disturbance (EID) approach. The nonlinearity is estimated by an EID estimator and compensated by incorporation of the estimate into the repetitive control input. A two‐dimensional model of the EID‐based modified repetitive control system is established that enables the preferential adjustment of control and learning actions by means of 2 tuning parameters. The singular‐value‐decomposition technique and Lyapunov stability theory are used to derive a linear‐matrix‐inequality–based asymptotic stability condition. Exploiting the stability condition and an overall performance evaluation index, a design algorithm is developed. Simulation results for the tracking control of a chuck‐workpiece system show that the method not only compensates state‐dependent nonlinearity but also improves the tracking performance for the periodic reference input, thereby demonstrating the validity of the method.

EID estimator-based modified repetitive control for singular systems with time-varying delay

This paper investigates the disturbance rejection for a modified repetitive control system (MRCS) that is described by a class of linear singular systems in the presence of external disturbances and time-varying delay. In particular, an equivalent-input-disturbance (EID)-based estimator is included in the MRCS to compensate both periodic and aperiodic disturbances which yields an EID-based MRCS. More precisely, the incorporation of the EID-based estimator into the control input enables rejection of all types of disturbances in MRCS and tracking of a periodic reference input is archived via a repetitive controller. Attention is focused on the state-feedback repetitive controller design which not only guarantees the regular, impulse free, and asymptotic stability of the closed-loop singular MRCS, but also provides an optimized upper bound of the time-varying delay. Based on Lyapunov stability theory and utilizing some advanced mathematical techniques, a new set of delay-dependent sufficient conditions is presented in terms of linear matrix inequalities for obtaining the required result. Then, an explicit expression for the desired state-feedback repetitive control law is developed. Further, the obtained results are validated through two numerical examples in the simulation section.

Compensation for Input Nonlinearities in Repetitive Control Systems Based on Improved Equivalent-Input-Disturbance Approach

The equivalent-input-disturbance (EID) approach is used to deal with the problem of tracking period signals for a plant with input actuator nonlinearities in a repetitive control system. First, an EID estimator is constructed by taking the full advantage of an extended state observer (ESO), the design of the ESO is explained. Next, an EID estimate, which represents the synthetic effect of the nonlinearities, is incorporated into a repetitive control law to compensate for the effect caused by the nonlinearities. This method does not require any prior information about the nonlinearities. It guarantees perfect tracking for periodic reference input and satisfactory compensation for input nonlinearities at the same time. Finally, simulation and experimental results show the effectiveness of the method.

Repetitive controller: an advanced servomechanism for periodic reference input

The presented work envisaged to explore the possibility of implementing a repetitive control strategy along with a proportional (P) controller for a liquid level system (LLS) in real time to track a periodic reference command. The objective of this paper is to validate the servo mechanism (accurate tracking) and periodic disturbance attenuating property of repetitive control (RC) along with P control system. The whole effort is divided in to two distinct parts. In the first part, the controller and plant are modeled, developed and experimented in software. The tracking performance and stability measure of the plant has been studied with inclusion of repetitive controller (RC) in presence of P controller and compared with conventional P and PI controller in both time and frequency domain subjected to periodic command to establish the usefulness of the RC-P control using simulation. The limitation of the first part is due to the consideration of all ideal condition of experimentation like ideal source, ideal plant etc. In the second part of this work the limitation of the first part of taking all ideal condition is avoided through the development of a real time experimentation setup. In this section, software model of LLS is replaced by its analogous electrical circuit. Emulation of the actual hardware plant is carried out interfaced through DAC NI6221 with PC using LabVIEW for better and flexible experimenting environment. The deployment of RC in Controller In Loop is done to conduct all experiments with actual source of reference and actual hardware plant. The effectiveness of RC with P controller for periodic reference input is shown through simulation and validated through real time experimentation.

Servomechanism for periodic reference input: Discrete wavelet transform-based repetitive controller

The presented work explored the possibility of developing a discrete wavelet transform-based repetitive control (DWTRC) strategy along with a proportional controller for a liquid level system (LLS) to track a periodic reference signal. The modified DWTRC with a P control system is a type of servo mechanism, effective for periodic reference input. The entire effort derives from two distinct parts. First, the tracking performance and stability measure of the system with inclusion of a repetitive controller (RC) in presence of P controller that has been studied and compared with conventional P and PI controllers in both time and frequency domains, subjected to periodic command to establish the usefulness of the former. The most obvious disadvantage of the RC with P control is the requirement of large memory for the presence of RC, and the memory requirement linearly increases with both repetition time and sampling frequency. Second, to overcome this difficulty, deployment of the discrete wavelet transform (DWT) in the RC loop is done to reduce the problem of memory requirement, and ensures efficient and high-speed computation due to the filtering and data reduction property of DWT. Finally, the effectiveness of DWTRC is validated through saving memory and optimizing the tracking error with variation of the wavelet order and its decomposition level.

Robust disturbance rejection in modified repetitive control system

This study concerns disturbance rejection for a modified repetitive control system (MRCS) that contains a strictly proper plant with time-varying uncertainties. Since an MRCS is affected by both periodic and aperiodic disturbances, and since the disturbances are often unknown, an equivalent-input-disturbance (EID)-based estimator was added to an MRCS to yield an EID-based MRCS that compensates for all types of disturbances. In this system, the repetitive controller ensures tracking of a periodic reference input, and the incorporation of an EID estimate into the control input enables rejection of unknown periodic and aperiodic disturbances. A robust stability condition for the MRCS was established in the form of a linear matrix inequality, and the condition was used to design the parameters of the controller. This design method handles uncertainties and enables the preferential adjustment of the tracking and control performance of the MRCS. Simulation results demonstrate the validity of the method.

Robust stabilizing simple multi-period repetitive controllers for multiple-input/multiple-output time-delay plants

The simple multi-period repetitive control system proposed by Yamada and Takenaga is a type of servomechanism for a periodic reference input. That is, the simple multi-period repetitive control system follows the periodic reference input with a small steady state error, even if a periodic disturbance or uncertainty exists in the plant. In addition, simple multi-period repetitive control systems ensure that the transfer functions from the periodic reference input to the output and from the disturbance to the output have a finite number of poles. Yamada et al. clarified the parameterization of all stabilizing simple multi-period repetitive controllers. Recently, Sakanushi et al. proposed the parameterization of all robust stabilizing simple multi-period repetitive controllers for multiple-input/multiple-output plants with uncertainty. However, their method cannot be applied to time-delay plants. In this paper, we propose the parameterization of all robust stabilizing simple multi-period repetitive controllers for multiple-input/multiple-output time-delay plants.

Compensation in Repetitive Control System for Aperiodic Disturbances and Input Dead Zone

This paper extends the equivalent-input-disturbance (EID) approach to deal with the problem of aperiodic disturbance rejection for a plant with an input dead zone in a repetitive-control system so as to improve control performance for the tracking of a periodic signal. Since a dead zone greatly degrades control performance, we apply the EID approach to design a compensator for the nonlinearity by treating it as an input-dependent disturbance. An EID estimator is constructed by making the best use of a full-order generalized state observer (GSO). And a method of designing the GSO is explained. The EID estimate, which exhibits the synthetic effect of the nonlinearity and the aperiodic disturbance, is incorporated into a repetitive control law to compensate for the nonlinearity and the aperiodic disturbance. This method does not require any information about the dead zone. It guarantees perfect tracking for periodic reference input and satisfactory compensation of input dead zone and aperiodic disturbance at the same time. Simulation results demonstrate the effectiveness of this method.