Periodic Reference Trajectories Research Articles

Adaptive Feedforward Neural Network Control With an Optimized Hidden Node Distribution

Composite adaptive radial basis function neural network (RBFNN) control with a lattice distribution of hidden nodes has three inherent demerits: 1) the approximation domain of adaptive RBFNNs is difficult to be determined a priori; 2) only a partial persistence of excitation (PE) condition can be guaranteed; and 3) in general, the required number of hidden nodes of RBFNNs is enormous. This paper proposes an adaptive feedforward RBFNN controller with an optimized distribution of hidden nodes to suitably address the above demerits. The distribution of the hidden nodes calculated by a K-means algorithm is optimally distributed along the desired state trajectory. The adaptive RBFNN satisfies the PE condition for the periodic reference trajectory. The weights of all hidden nodes will converge to the optimal values. This proposed method considerably reduces the number of hidden nodes, while achieving a better approximation ability. The proposed control scheme shares a similar rationality to that of the classical PID control in two special cases, which can thus be seen as an enhanced PID scheme with a better approximation ability. For the controller implemented by digital devices,the proposed method, for a manipulator with unknown dynamics, potentially achieves better control performance than model-based schemes with accurate dynamics.Simulation results demonstrate the effectiveness of the proposed scheme. This result provides a deeper insight into the coordination of the adaptive neural network control and the deterministic learning theory.

Design of Repetitive Control Systems Using a Delayed Control Input and a State Error

This paper presents a modified repetitive control scheme comprising of a state error and a control input via delayed feedback to track periodic reference trajectories and/or attenuate disturbances. The closed-loop state error dynamics can be represented using a typical neutral delay system with an exogenous input to be attenuated. The sufficient conditions to achieve overall stability and H∞ performance to minimize state error are derived by applying a Lyapunov-Krasovskii functional and a Hamiltonian, which are expressed as an algebraic Riccati inequality (ARI) and a linear matrix inequality (LMI). Based on the derived conditions, it is shown that the repetitive controller design problem can be reformulated as an optimization problem with an LMI constraint to determine the state error feedback gain. Finally, a numerical example is presented to demonstrate the feasibility of the proposed method.

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.

Minimizing Energy Consumption and Peak Power of Series Elastic Actuators: A Convex Optimization Framework for Elastic Element Design.

Compared to rigid actuators, series elastic actuators (SEAs) offer a potential reduction of motor energy consumption and peak power, though these benefits are highly dependent on the design of the torque-elongation profile of the elastic element. In the case of linear springs, natural dynamics is a traditional method for this design, but it has two major limitations-arbitrary load trajectories are difficult or impossible to analyze and it does not consider actuator constraints. Parametric optimization is also a popular design method that addresses these limitations, but solutions are only optimal within the space of the parameters. To overcome these limitations, we propose a nonparametric convex optimization program for the design of the nonlinear elastic element that minimizes energy consumption and peak power for an arbitrary periodic reference trajectory. To obtain convexity, we introduce a convex approximation to the expression of peak power; energy consumption is shown to be convex without approximation. The combination of peak power and energy consumption in the cost function leads to a multiobjective convex optimization framework that comprises the main contribution of this paper. As a case study, we recover the elongation-torque profile of a cubic spring, given its natural oscillation as the reference load. We then design nonlinear SEAs for an ankle prosthesis that minimize energy consumption and peak power for different trajectories and extend the range of achievable tasks when subject to actuator constraints.

Two-degrees-of-freedom PI[formula omitted]D controller for precise nanopositioning in the presence of hardware-induced constant time delay

The fast and accurate tracking of periodic and arbitrary reference trajectories is the principal goal in many nanopositioning applications. Flexure-based piezoelectric stack driven nanopositioners are widely employed in applications where accurate mechanical displacements at these nanometer scales are required. The performance of these nanopositioners is limited by the presence of lightly damped resonances in their dynamic response and actuator nonlinearities. Closed-loop control techniques incorporating both damping and tracking are typically used to address these limitations. However, most tracking schemes employed use a first-order integrator where a triangular trajectory commonly used in nanopositioning applications necessitates a double integral for zero-error tracking. The phase margin of the damped system combined with the hardware-induced delay deem the implementation of a double-integrator unstable. To overcome this limitation, this paper presents the design, analysis and application of a new control scheme based on the structure of the traditional Two-Degrees-of-Freedom PID controller (2DOF-PID). The proposed controller replaces the integral action of the traditional 2DOF-PID with a double integral action (2DOF-PI2D). Despite its simplicity, the proposed controller delivers superior tracking performance compared to traditional combined damping and tracking control schemes based on well-reported designs such as positive position feedback (PPF), Integral resonant control (IRC), and Positive Velocity and Position Feedback (PVPF). The stability of the control system is analyzed in the presence of a time delay in the system. Experimental results validating the efficacy of the proposed chattering-free control of a piezo-driven nanopositioning system are included.

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.

Distributed cooperative learning for a group of uncertain systems via output feedback and neural networks

This paper studies the problem of adaptive neural network (NN) output-feedback control for a group of uncertain nonlinear multi-agent systems (MASs) from the viewpoint of cooperative learning. It is assumed that all MASs have identical unknown nonlinear dynamic models but carry out different periodic control tasks, i.e., each agent system has its own periodic reference trajectory. By establishing a network topology among systems, we propose a new consensus-based distributed cooperative learning (DCL) law for the unknown weights of radial basis function (RBF) neural networks appearing in output-feedback control laws. The main advantage of such a learning scheme is that all estimated weights converge to a small neighborhood of the optimal value over the union of all system estimated state orbits. Thus, the learned NN weights have better generalization ability than those obtained by traditional NN learning laws. Our control approach also guarantees the convergence of tracking errors and the stability of closed-loop system. Under the assumption that the network topology is undirected and connected, we give a strict proof by verifying the cooperative persisting excitation condition of RBF regression vectors. This condition is defined in our recent work and plays a key role in analyzing the convergence of adaptive parameters. Finally, two simulation examples are provided to verify the effectiveness and advantages of the control scheme proposed in this paper.

A GENERAL FRAMEWORK FOR MINIMIZING ENERGY CONSUMPTION OF SERIES ELASTIC ACTUATORS WITH REGENERATION.

The use of actuators with inherent compliance, such as series elastic actuators (SEAs), has become traditional for robotic systems working in close contact with humans. SEAs can reduce the energy consumption for a given task compared to rigid actuators, but this reduction is highly dependent on the design of the SEA's elastic element. This design is often based on natural dynamics or a parameterized optimization, but both approaches have limitations. The natural dynamics approach cannot consider actuator constraints or arbitrary reference trajectories, and a parameterized elastic element can only be optimized within the given parameter space. In this work, we propose a solution to these limitations by formulating the design of the SEA's elastic element as a non-parametric convex optimization problem, which yields a globally optimal conservative elastic element while respecting actuator constraints. Convexity is proven for the case of an arbitrary periodic reference trajectory with a SEA capable of energy regeneration. We discuss the optimization results for the tasks defined by the human ankle motion during level-ground walking and the natural motion of a single mass-spring system with a nonlinear spring. For all these tasks, the designed SEA reduces energy consumption and satisfies the actuator's constraints.

Underactuated Spacecraft Switching Law for Two Reaction Wheels and Constant Angular Momentum

This paper develops a switching feedback controller for the attitude of an underactuated spacecraft that exploits two internal control torques provided by reaction wheels. The problem is challenging; for example, even in the zero total angular momentum case, no smooth or even continuous time-invariant feedback law for stabilizing a desired orientation exists. The method introduced here exploits the separation of the system states into inner-loop base variables and outer-loop fiber variables. The base variables track periodic reference trajectories, the amplitude of which is governed by parameters that are adjusted to induce an appropriate change in the fiber variables. Under suitable assumptions on the total angular momentum, this controller stabilizes an equilibrium that corresponds to a desired inertially fixed orientation. If the desired attitude violates the assumption on angular momentum, then controlled oscillations in a neighborhood around the target orientation are induced by the switching controller. The control scheme is based on several approximations and is designed for relatively small maneuvers close to the desired attitude in a vicinity which may be achieved by thruster-based control schemes. Simulation results demonstrate that the switching feedback law provides good performance in controlling the attitude of an underactuated spacecraft.

Supervised Multi‐Rejector of Periodic Disturbances for Nonlinear Systems Using Decoupled State Multimodel

AbstractIn this paper, a multi‐rejector of periodic disturbances is proposed for discrete‐time nonlinear systems represented by a decoupled state multimodel. We report a decoupled state multimodel repetitive‐predictive control based on a supervised algorithm to ensure reference trajectory tracking and periodic disturbances rejection. Partial predictors associated to the local controllers make the best choice of the most valid partial controller that meets the desired closed loop performances. The effectiveness of the supervised multi‐rejector is shown via a simulation example. The obtained results are satisfactory and show a good rejection of periodic disturbances and reference trajectory tracking.

Output regulation problem for a class of regular hyperbolic systems

This paper investigates the output regulation problem for a class of regular first-order hyperbolic partial differential equation (PDE) systems. A state feedback and an error feedback regulator are considered to force the output of the hyperbolic PDE plant to track a periodic reference trajectory generated by a neutrally stable exosystem. A new explanation is given to extend the results in the literature to solve the regulation problem associated with the first-order hyperbolic PDE systems. Moreover, in order to provide the closed-loop stability condition for the solvability of the regulator problems, the design of stabilising feedback gain and its dual problem design of stabilising output injection gain are considered in this paper. This paper develops an easy method to obtain an adjustable stabilising feedback gain and stabilising output injection gain with the aid of the operator Riccati equation.

Indirect Intelligent Sliding Mode Control of Antagonistic Shape Memory Alloy Actuators Using Hysteretic Recurrent Neural Networks

This paper presents the development of an indirect intelligent sliding mode control (IISMC) system for use with antagonistic shape memory alloy (SMA) actuators. The controller manipulates input voltages to a pair of offset antagonistic SMA tendons, resulting in heat-induced bending of a flexible beam. Hysteresis compensation is achieved using a pair of hysteretic recurrent neural networks, which map the nonlinear, hysteretic relationships between SMA temperatures and bending angle for each tendon. The sliding mode control law regulates tendon temperatures based on reference and measured bending angle. Additionally, the IISMC incorporates an antislack component to increase tendon responsiveness. Experimental results demonstrate precise tracking of periodic reference trajectories. The controller is robust to model uncertainty, disturbances, and parameter variations, with bending angle tracking results superior to those of an optimized proportional + integral controller.

Total sliding-mode-based particle swarm optimization control for linear induction motor

In this study, a total sliding-mode-based particle swarm optimization control (TSPSOC) scheme is designed for the periodic motion control of an indirect field-oriented linear induction motor (LIM) drive. First, an indirect field-oriented mechanism for a LIM drive is introduced to preserve the decoupling control characteristic. Then, the concept of total sliding-mode control (TSC) is incorporated into particle swarm optimization (PSO) to form an on-line TSPSOC framework for preserving the robust control characteristics and reducing the chattering control phenomena of TSC. Moreover, an adaptive inertial weight is devised to accelerate the searching speed effectively. In this control scheme, a PSO control system is utilized to be the major controller, and the stability can be indirectly ensured by the concept of TSC without strict constraint and detailed system knowledge. With the proposed TSPSOC system, the mover position of the controlled LIM drive possesses the advantages of favorable robust characteristic, control effort without chattering, and simple control framework. Numerical simulations and experimental results are given to verify the effectiveness of the proposed control scheme for the tracking of periodic reference trajectories. In addition, the superiority of the proposed TSPSOC scheme is indicated in comparison with the TSC, Petri fuzzy-neural-network control (PFNNC) and traditional fuzzy-neural-network control (TFNNC) systems.

Analog Robust Repetitive Control for Nanopositioning Using Bucket Brigade Devices

In many applications of nanopositioning, such as scanning probe microscopy, tracking fast periodic reference trajectories with high accuracy is highly desirable. Repetitive control is a simple and effective control scheme to obtain good tracking of such reference trajectories. In order to implement repetitive control, a method for introducing time-delay is necessary. This can easily be implemented using a memory buffer with digital signal processing equipment. To achieve fast, high accuracy, and low noise performance, fast microcontrollers or field-programmable gate array hardware with fast high-resolution analog-to-digital and digital-to-analog converters are needed. As an inexpensive alternative to digital signal processing, the use of an analog bucket brigade device to implement the time-delay is investigated in this paper. Bucket brigade devices use switching to carry the input voltage over an array of capacitors, achieving a specified time-delay. Low-noise bucket brigade devices can achieve a signal-to-noise ratio around 80 dB, comparable to the actual performance when using 16-bit analog-to-digital converters. In this paper, the proposed control scheme utilizes a modified integral control law in conjunction with the repetitive control law. The overall control scheme ensures robustness towards plant uncertainty. Experimental results demonstrate the effectiveness of the overall control scheme and the analog implementation.

Fixed-Structure, Low-Order Damping and Tracking Control Schemes for Nanopositioning

Fast and accurate tracking of periodic reference trajectories is highly desirable in many nanopositioning applications, including scanning probe microscopy. Performance in common positioning stage designs is limited by the presence of lightly damped resonances, and actuator nonlinearities such as hysteresis and creep. To improve the tracking performance in such systems, several damping and tracking control schemes have been presented in the literature. In this paper, five different control schemes are presented and applied to a nanopositioning system for experimental comparison. They include schemes applying damping control in the form of positive position feedback, integral resonant control, integral force feedback, and passive shunt-damping. Also, a control scheme requiring only a combination of a low-pass filter and an integrator is presented. The control schemes are fixed-structure, low-order control laws, for which few results exist in the literature with regards to optimal tuning. A practical tuning procedure for obtaining good tracking performance for all of the presented control schemes is therefore presented. The schemes provide similar performance, and the main differences are due to the specific implementation of each scheme.

Adaptive feed-forward hysteresis compensation for piezoelectric actuators

Piezoelectric actuators are often employed for high-resolution positioning tasks. Hysteresis and creep nonlinearities inherent in such actuators deteriorate positioning accuracy. An online adaptive nonlinear hysteresis compensation scheme for the case of symmetric hysteretic responses and certain periodic reference trajectories is presented. The method has low complexity and is well suited for real-time implementation. Experimental results are presented in order to verify the method, and it is seen that the error due to hysteresis is reduced by more than 90% compared to when assuming a linear response.

Accounting for hysteresis in repetitive control design: Nanopositioning example

This paper deals with designing a repetitive controller (RC) for tracking periodic reference trajectories for systems that exhibit hysteresis, such as piezoelectric actuators used in nanopositioning systems. Hysteresis can drastically limit the performance of an RC designed around a linear dynamics model, and thus the effect of hysteresis on the closed-loop stability of RC is analyzed and the allowable size of the hysteresis nonlinearity for a stable RC is quantified. But when the hysteresis effect exceeds the maximum bound, an inverse-hysteresis feedforward controller based on the Prandtl–Ishlinskii hysteresis model is used to compensate for the nonlinearity. The control method is implemented on a custom-designed nanopositioning stage. Experimental results show that by incorporating hysteresis compensation the stability margin and the rate of error reduction improve. Likewise, the maximum tracking error reduces by 71%, from 13.7% (using industry-standard integral control) to 3.9% (using RC with hysteresis compensation), underscoring the benefits of RC with hysteresis compensation.

Integral Backstepping Control for a PMSLM Using Adaptive RNNUO

Due to uncertainties exist in the applications of the a permanent magnet synchronous linear motor (PMSLM) servo drive which seriously influence the control performance, thus, an integral backstepping control system using adaptive recurrent neural network uncertainty observer (RNNUO) is proposed to increase the robustness of the PMSLM drive. First, the field-oriented mechanism is applied to formulate the dynamic equation of the PMSLM servo drive. Then, an integral backstepping approach is proposed to control the motion of PMSLM drive system. With proposed integral backstepping control system, the mover position of the PMSLM drive possesses the advantages of good transient control performance and robustness to uncertainties for the tracking of periodic reference trajectories. Moreover, to further increase the robustness of the PMSLM drive, an adaptive RNN uncertainty observer is proposed to estimate the required lumped uncertainty. The effectiveness of the proposed control scheme is verified by experimental results.

Dual-stage repetitive control with Prandtl–Ishlinskii hysteresis inversion for piezo-based nanopositioning

The positioning performance of piezo-based nanopositioning systems is limited by dynamic and hysteresis effects in the piezoactuator. Herein, a high-performance, dual-stage repetitive controller (dual-RC) with a feedforward hysteresis compensator is proposed for tracking periodic trajectories, such as the scanning-type motion, in nanopositioning systems. Firstly, a discrete-time dual-RC is created by cascading a conventional RC with an odd-harmonic RC. The favorable gain characteristics of the dual-RC coincide with the odd harmonics of the scanning-type periodic reference trajectory, thus offering good robustness and low tracking error. Secondly, a new inverse-hysteresis compensator is developed based on the Prandtl–Ishlinskii hysteresis model. The structure of the inverse model mimics the structure of the forward model, where the parameters of the inverse model can be easily identified from measured input–output data. Finally, the controllers are applied to a custom-designed high-speed nanopositioner, and simulations and experimental results are provided to illustrate the performance improvement of the proposed control scheme compared to industry-standard PID control and conventional RC. High-speed positioning results (tracking of triangle scan trajectories) at rates of 1kHz, 1.5kHz, and 2kHz are shown. Compared to a conventional RC, the tracking error of the dual-RC is 48% lower at 1kHz and 33% lower at 2kHz scanning frequency. It is also shown that by compensating for hysteresis, the performance of the RC system designed based on the linear dynamics can be enhanced.

Robust intelligent backstepping control system using RCMAC for tracking periodic trajectories

Since the backstepping control technique is a systematic and recursive design methodology for nonlinear systems, a robust intelligent backstepping control (RIBC) system is proposed in this study to control a linear ultrasonic motor (LUSM) for the tracking of periodic reference trajectories. In the proposed control system, an adaptive recurrent cerebellar model articulation controller (RCMAC) is used to mimic an ideal backstepping control and a robust controller is designed to attenuate the effects caused by unmodeled dynamics, disturbances and approximate errors. The proposed adaptive RCMAC has superior capability to the conventional cerebellar model articulation controller (CMAC) in efficient learning mechanism and dynamic response. Moreover, the Taylor linearization technique is employed to derive the linearized model of the RCMAC. The online adaptation laws of the RIBC system are derived in the sense of the Lyapunov function, the Taylor linearization technique and the H ∞ control approach, so that the stability of the system and robust tracking performance can be guaranteed. Finally, the effectiveness of the proposed control system is indicated in comparison with an integral–proportional (IP) position control system. Experimental results show that high-precision tracking performance can be achieved by using the proposed RIBC system.