Delayed Output Feedback Control Research Articles

Arbitrary coefficient assignment to the characteristic function of the scalar type for linear systems with state delays by linear static delayed output feedback

We consider a control system defined by a system of linear differential equations in real or complex space with multiple lumped and distributed delays in the state variable. For this system, we study the problem of arbitrary coefficient assignment for the characteristic function of the scalar type by linear static delayed output feedback with lumped and distributed delays. We obtain necessary and sufficient conditions for solvability of the arbitrary coefficient assignment problem by the static delayed output feedback controller when the system coefficients have a special form. Corollaries on arbitrary finite spectrum assignment and on stabilization of the system are obtained. We prove that the obtained results generalize the corresponding theorem proved earlier for a control system defined by a linear delay-differential equation of nth order. Examples are presented illustrating the results obtained.

Parametric Delayed Output Feedback Control for Versatile Human-Exoskeleton Interactions During Walking and Running

This paper presents a parametric delayed output feedback controller for various types of human-exoskeleton interactions during walking or running gait. The parametric delayed-feedback controller operates reliably during various gait pattern changes, utilizing three adjustable control parameters: smoothing factor, time delay, and feedback gain. We show that various human-exoskeleton interaction torques can be generated using these three parameters, such as assistance with running or walking and different types of resistances. Feasibility tests were performed to demonstrate the efficacy of the proposed interaction controller. Hip exoskeleton assistance reduces the metabolic cost by 12.3% during running at 7 km/h compared with running without the exoskeleton. The generalizability of the controller performance was also demonstrated through various gait speed tests. Furthermore, we measured the electromyography signal data and the human lower limb joint kinematics to examine the feasibility of providing different gait resistances using different time delays. The results show that resistance with a delay of 0.3 s (Late mode) instead of a delay of 0.1 s (Early mode) is possible for walking resistance training, focusing on the hip flexors compared to the zero-torque mode.

Exercise with a wearable hip-assist robot improved physical function and walking efficiency in older adults

Wearable assistive robotics has emerged as a promising technology to supplement or replace motor functions and to retrain people recovering from an injury or living with reduced mobility. We developed delayed output feedback control for a wearable hip-assistive robot, the EX1, to provide gait assistance. Our purpose in this study was to investigate the effects of long-term exercise with EX1 on gait, physical function, and cardiopulmonary metabolic energy efficiency in elderly people. This study used parallel experimental (exercise with EX1) and control groups (exercise without EX1). A total of 60 community-dwelling elderly persons participated in 18 exercise intervention sessions during 6 weeks, and all participants were assessed at 5 time points: before exercise, after 9 exercise sessions, after 18 sessions, and 1 month and 3 months after the last session. The spatiotemporal gait parameters, kinematics, kinetics, and muscle strength of the trunk and lower extremities improved more after exercise with EX1 than in that without EX1. Furthermore, the effort of muscles over the trunk and lower extremities throughout the total gait cycle (100%) significantly decreased after exercise with EX1. The net metabolic energy costs during walking significantly improved, and functional assessment scores improved more in the experimental group than in the control group. Our findings provide evidence supporting the application of EX1 in physical activity and gait exercise is effective to improve age-related declines in gait, physical function, and cardiopulmonary metabolic efficiency among older adults.

LMI-Based Delayed Output Feedback Controller Design for a Class of Fractional-Order Neutral-Type Delay Systems Using Guaranteed Cost Control Approach

In this research work, we deal with the stabilization of uncertain fractional-order neutral systems with delayed input. To tackle this problem, the guaranteed cost control method is considered. The purpose is to design a proportional–differential output feedback controller to obtain a satisfactory performance. The stability of the overall system is described in terms of matrix inequalities, and the corresponding analysis is performed in the perspective of Lyapunov’s theory. Two application examples verify the analytic findings.

Sampled-Data Delayed Output Feedback Control of a Class of Nonlinear Systems

This letter presents a sampled-data control strategy for robust stabilization of a family of uncertain nonlinear systems via delayed output signals. Instead of building dynamic observers,we use an output signal sampled at the current time and delayed outputmeasurements at the previous n -1 sampling points to design sampleddata controllers, which are easier for implementation by computer. Under a linear growth condition, it is proved that global robust stabilization is achievable by sampleddata delayed output feedback. The robust stability analysis of the hybrid closed-loop system with delays is conducted by the Lyapunov-Krasovskii functional method.

Delayed Output Feedback Control for Gait Assistance and Resistance Using a Robotic Exoskeleton

In this study, we propose an interaction control framework for gait assistance and resistance using a robotic exoskeleton. We define a smoothed state variable that represents joint angle movements while walking. Furthermore, a self-feedback controller is designed with the delayed output state. By applying an appropriate time-delay and positive or negative feedback gain to the state variable, we can generate assistive or resistive torque stably without any gait phase or environment recognition. The time-delayed selffeedback controller reflects the movement of the wearer's joints at every moment of control, thereby stably coping with sudden task transitions (e.g., walk-stop-walk, forward-backward walking) as well as walking speed or environment changes. Case studies involved gait assistance with a knee exoskeleton and gait assistance and resistance with a hip exoskeleton. We performed various preliminary tests including metabolic energy measurements and a comparison of the positive or negative power of the generated torque profiles. The results show the flexibility and effectiveness of the proposed interaction control method for gait assistive or resistive training.

Delayed Output Feedback Control for Gait Assistance With a Robotic Hip Exoskeleton

In this paper, we propose a new and simple control strategy for gait assistance with a hip exoskeleton robot. This controller is based on the time delayed, self-feedback known for stabilizing oscillatory systems under certain conditions. In this controller, there are no separate estimators for the gait phase nor the environment, yet the controller can be generalized to operate under various walking conditions (e.g., stair and ramp walking). We first define a state variable representing the current leg's movement with hip joint angles. A simple assistance control can be described in closed-loop form with the delayed state feedback. By assigning the appropriate time-delay and self-feedback gain, we can generate assistive torques stably under the interaction between human and exoskeleton. The controller provides immediate and smooth assistance to user movement by reflecting the change of leg motion at every control period. The proposed joint-angle-based delayed-feedback assistance controller can operate under various walking speeds and environmental changes (e.g., stairs and ramps) without the need for additional sensors, computational processing, and parameter adjustment. Using a simple leg swing model, we perform a stability analysis under a simplified condition to provide insights into the effects of the time-delayed feedback in oscillatory systems. Then, we experimentally validate the efficacy of the proposed assistance controller by measuring the metabolic energy expenditure for level treadmill walking. We also test and analyze the generated assistive torques and power under the different walking conditions to show the generalizability of the controller.

Delayed output feedback of discrete‐time time‐delay systems with applications to spacecraft rendezvous

This work studies the delayed output feedback (DOF) of discrete-time systems with input and output delays that are arbitrarily large, bounded, and exactly known. The significance of the DOF is that only the present and delayed inputs and outputs are used for feedback. For systems with a single input delay and multiple output delays, the idea of state prediction is utilised to construct the DOF controller. While for systems with multiple input and output delays, the DOF controller is designed by firstly reducing the original system to a delay-free one. It is shown in both cases that the closed-loop systems controlled by DOF are asymptotically stable and act like delay-free systems with the same dimensions as open-loop systems. Compared with the continuous-time case, the DOF controllers for discrete-time systems do not contain distributed delay terms and the implementation problems can be avoided. The DOF is used to stabilise the spacecraft rendezvous system with input and output delays and simulations show the effectiveness of the proposed approach.

Implementation of delayed output feedback for linear systems with multiple input delays

This study is concerned with the safe implementation of delayed output feedback (DOF) for linear systems with multiple input delays. It is observed that the numerical implementation of the distributed delay terms involved in the DOF controller may lead to instability of the closed-loop system. To solve this problem, a generalised low-pass filter (LPF) is introduced into the control loop. It is shown that the stability of the closed-loop system can be maintained if the numerical integration is sufficiently precise. An augmented system is then constructed to design the LPF by applying the traditional pole assignment algorithm for linear systems without delay. Two illustrative examples are worked out to show the effectiveness of the proposed approach.

Observer Design and Exponential Stabilization for Wave Equation in Energy Space by Boundary Displacement Measurement Only

In this technical note, we consider finite-time and exponential stabilization for one-dimensional wave equations by boundary displacement measurement only. We limit ourselves in the energy state space where the usual observability inequality is not valid anymore. However, in the optimal state space, the boundary displacement is indeed exactly observable. This motivates us to design observer via displacement output only for these systems. We first discuss a simple case as a motivation. An observer is designed and an output feedback control is then synthesized to make system finite-time stable in energy space. In this same spirit, we consider the same problem for an unstable wave equation and the finite-time stability is also achieved by displacement output feedback. Finally, we consider an anti-stable wave equation. A direct delayed output feedback control can achieve exponential stability with arbitrary decay rate. Simulation results are presented to validate the theoretical conclusions.

Design of Maximum Decay Rate for SISO Systems with Delayed Output Feedback Using Elimination Theory

In this paper we study the delayed output feedback control of a class of LTI systems. We present an analytical methodology based on discriminant operations by which the maximum decay rate of the closed-loop is designed as an explicit function of the system parameters. More precisely, this approach uses elimination theory for the tuning of the controller in an optimal fashion such that the system is not only stable but also responds to control commands as fast as possible. A case study is presented to demonstrate the utility of the approach.

Delayed Output Feedback Control for Nonlinear Systems With Two-Layer Interval Fuzzy Observers

In this paper, the design problem of a delayed output feedback control scheme using two-layer interval fuzzy observers for a class of nonlinear systems with state and output delays is investigated. The Takagi-Sugeno-type fuzzy linear model with an online update law is used to approximate the nonlinear system. Based on the fuzzy model, a two-layer interval fuzzy observer is used to reconstruct the system states according to equal interval output time delay slices. Subsequently, a delayed output feedback adaptive fuzzy controller is developed to overcome the nonlinearities, time delays, and external disturbances such that H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> tracking performance is achieved. The linguistic information is developed by setting the membership functions of the fuzzy logic system and the adaptation parameters to estimate the model uncertainties directly using linear analytical results instead of estimating nonlinear system functions. The tracking error dynamics are designed to enable our adaptive controller to avoid the filtering of the basis vectors whose dimension is much larger than that of the state vector of the controlled system. This is achieved by not imposing the strictly positive real condition. Based on the Lyapunov stability criterion and linear matrix inequalities, some sufficient conditions are derived so that all the states of the system are uniformly ultimately bounded. Therefore, the effect of the external disturbances on the tracking error can be attenuated to any prescribed level, and H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> tracking control is achieved. Building on our previous work in this area, the proposed control scheme is extended to handle a class of uncertain nonlinear systems with state and output delays and external disturbances. This is achieved through the use of a robust variable structure scheme and H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> control techniques. Finally, a numerical example of a two-link robot manipulator is studied to illustrate the effectiveness of the proposed control scheme.

LMI-based H ∞ synchronization of second-order neutral master-slave systems using delayed output feedback control

The H ∞ synchronization problem of the master and slave structure of a second-order neutral master-slave systems with time-varying delays is presented in this paper. Delay-dependent sufficient conditions for the design of a delayed output-feedback control are given by Lyapunov-Krasovskii method in terms of a linear matrix inequality (LMI). A controller, which guarantees H ∞ synchronization of the master and slave structure using some free weighting matrices, is then developed. A numerical example has been given to show the effectiveness of the method. The simulation results illustrate the effectiveness of the proposed methodology.

A separation principle for dynamical delayed output feedback control of chaos

In this Letter, a dynamical delayed output-feedback (DDOF) control strategy is proposed for stabilizing unstable periodic orbits (UPOs) of chaotic systems. Using the Floquet theory, a separation principle is established which gives a necessary and sufficient stability condition for DDOF UPO stabilizing control systems. The new principle shows that the so-called “odd number limitation” for delayed state-feedback control systems also applies to DDOF control.