Unknown Time-varying Input Delay Research Articles

Performance Guaranteed Robust Tracking Control of MIMO Nonlinear Systems With Input Delays: A Global and Low-Complexity Solution.

This article presents a global performance guaranteed tracking control method for a class of general strict-feedback multi-input and multi-output (MIMO) nonlinear systems with unknown nonlinearities and unknown time-varying input delays. By introducing a novel error transformation embedded with the Lyapunov-Krasovskii functional (LKF), the developed control scheme exhibits several appealing features: 1) it is able to achieve global prescribed performance tracking for uncertain MIMO systems with delayed inputs, while at the same time eliminating the constraint conditions imposing on initial values between the tracking/virtual error and the performance function; 2) there is no need for any a priori knowledge regarding the nonlinearities of the system nor a prior knowledge of time derivatives of the desired trajectory, making the resultant controller simpler in structure and less expensive in computation; 3) the control scheme includes a new differentiable time-varying feedback term, which gracefully compensates the unknown input delays and unknown control gain coefficient matrices; and 4) the controllability condition is relaxed, which enlarges the applicability of the proposed strategy. Finally, a two-link robotic manipulator example is provided to demonstrate the reliability of the theoretical results.

Adaptive path tracking control for autonomous vehicles with sideslip effects and unknown input delays

This paper presents an adaptive path-tracking control of autonomous ground vehicles with sideslip effects and time-varying input delays. To handle the problem of unknown time-varying input delays, the vehicle kinematic model is transformed into a novel chain model, and a delay-based auxiliary integral term is introduced for the estimation of the unknown terms caused by time-varying delays. To avoid modeling errors caused by existing linear approximation methods, a new adaptive path-tracking control strategy is proposed, such that the optimal tracking performance is achieved by appropriately adjusting certain design parameters. In addition, the inherent complexity explosion issue is avoided with the aid of a boundary estimation strategy. All signals of the closed-loop systems are guaranteed to be bounded. Simulation results illustrate the effectiveness of the method proposed.

Fixed-time tracking control for high-order nonlinear systems with unknown time-varying input delay

Fixed time tracking control for nonstrict feedback high-order nonlinear systems with unknown time-varying external disturbance and unknown time-varying input delay has been studied in this research. Firstly, the obstacle caused by unknown time-varying input delay has been tackled by introducing a compensation signal. The unknown nonlinear functions are identified by neural networks and the algebraic loop problem for nonstrict feedback systems has been overcome by using the properties of neural networks. Further, a bound estimation mechanism is employed to estimate the identify error and unknown time-varying external disturbance without using the priori knowledge of the boundary. Regarding the obstacle caused by high-order terms and the singularity problem have been tackled by using adding a power integrator mechanism and the inequality transformation technique, respectively. The boundedness of all the signals in the closed-loop systems has been proved by using the Lyapunov stability theorem, and the convergence time is independent of the initial value of the system state. Finally, the simulation results show the effectiveness of the proposed method.

Robust Cadence and Power Tracking on a Switched FES Cycle With an Unknown Electromechanical Delay

Functional electrical stimulation (FES) is commonly used to facilitate cycling tasks for people with lower-limb movement disorders. In this work, FES and motor controllers are designed to track a desired power and cadence, respectively, and a Lyapunov-based switched systems analysis is performed to guarantee uniformly ultimately bounded power tracking and global exponential cadence tracking for a switched, delayed, nonlinear, and uncertain FES-cycling system. A unique challenge in this problem is that there is an unknown time-varying input delay to produce force, and a different unknown time-varying residual input delay where force is still produced after stimulation is removed. These delays impact the dwell-time conditions that dictate stimulation timing, and if not properly accounted for can lead to undesired effects such as antagonistic muscles exerting force at the same time or potential instabilities. The proposed controllers were validated by experimental analysis of four participants with neurological conditions (NCs) and five able-bodied participants, and yielded average power and cadence tracking errors of 0.01 ± 0.09 W and −0.05 ± 0.65 revolutions per minute (RPM), respectively, for the able-bodied participants and 0.01 ± 1.11 W and −0.07 ± 1.17 RPM, respectively, for the participants with NCs.

Adaptive delay dependent sliding mode fault-tolerant controller design for nonlinear systems with unknown time-varying input and state delays

This paper presents a novel adaptive delay-dependent fault-tolerant sliding mode control (SMC) strategy for the class of nonlinear Lipschitz systems with time-varying unknown delays in system states and inputs. A modified integral sliding surface (ISS) based on which the SMC approach is developed using a linear matrix inequality. Lyapunov-Krasovskii stability theory is employed to guarantee asymptotic stability of the closed-loop system such that system states starting from any arbitrary initial conditions with time-varying delays reach the predefined sliding surface in a finite time. It is also proved that states stay on the surface for all subsequent time, and the effects of the actuator’s faults are simultaneously attenuated. Adaptive tuning laws are used to adjust controller parameters and estimate actuators’ faults. The controllers’ structures are more straightforward than the most existing recent fault-tolerant control methods. Simulation results of practical nonlinear inverted pendulum system verified the outstanding merits and capabilities of the proposed scheme in the presence of actuators’ faults and multiple delays in the system states and inputs.

Robust tracking for functional electrical stimulation cycling with unknown time-varying input delays: A switched systems approach.

Motorized functional electrical stimulation (FES) cycling has been demonstrated to have numerous health benefits for individuals suffering from neurological disorders. FES-cycling is usually designed to track the desired trajectories in real time. However, there are input delays between the exertion of the stimulation and the corresponding muscle contraction that potentially destabilize the system and undermine training efforts. Meanwhile, muscle fatigue gives rise to a time-varying input delay and decreased force. Moreover, switching between FES and motor control can be chattering and destabilizing owing to the high frequency. This article constructs Lyapunov-Krasovskii functionals to analyze the stability and robustness of the nonlinear cycling system with time-varying input delay. A new average dwell time condition is then provided to ensure the input-to-state stability of the considered systems. Finally, numerical simulations are illustrated to verify the effectiveness of the developed controller.

Event-Triggered Global Regulation of an Uncertain Chain of Integrators under Unknown Time-Varying Input Delay

Event-Triggered Global Regulation of an Uncertain Chain of Integrators under Unknown Time-Varying Input Delay

Observer-based fault-tolerant control for non-infinitely observable descriptor systems with unknown time-varying state and input delays

This paper presents a fault-tolerant control (FTC) scheme for a class of non-infinitely observable descriptor systems (NIODS) that is affected by unknown time-varying state and input delays. The NIODS is first transformed into a form facilitating the manipulation of design freedom inherent in its structure. Some of the system states are then treated as unknown inputs to form an infinitely observable reduced-order system. A sliding mode observer (SMO) is applied to estimate the states and faults based on measurable signals. Next, a memoryless controller is designed using these estimates. Design conditions for the SMO and controller are derived in the form of linear matrix inequalities. In the case where the state and input delays are different, the FTC scheme is designed to limit the effect of the mismatch in delays on the output. In the case of equivalent state and input delays, the FTC scheme is designed so that the system output asymptotically converges to zero. A simulated example is lastly performed to verify the scheme’s effectiveness.

Position and cadence tracking of a motorized FES-cycle with an unknown time-varying input delay using saturated FES control

Neurological conditions (NCs) can have debilitating effects on an individual and can impede a person’s ability to perform activities of daily living. A common rehabilitation for individuals with NCs is the use of functional electrical stimulation (FES) to induce muscle contractions to perform a functional task, such as cycling. In this paper, we develop a closed-loop FES cycling controller that yields exponential cadence tracking. Contributions of the work result from the design and analysis innovations to compensate for switching between the different muscles and motor control inputs, compensating for the inherent uncertain nonlinear dynamics, compensating for the time-varying unknown input delay resulting from the complex electrochemical FES muscle torque production process, and compensating for saturation in the evoked torque due to fatigue, available muscle mass, or stimulation sensitivity. The performance of the developed control system was examined through a series of experiments on nine participants (five able-bodied and four with NCs). The experiments resulted in a cadence tracking error of −0.03 ± 1.69 RPM for non-informed passive able-bodied participants and −0.04 ± 1.98 RPM for the participants with NCs, for a desired cadence of 50 RPM.

Robust Cadence Tracking for Switched FES-Cycling With an Unknown Time-Varying Input Delay

For an individual affected by a neuromuscular condition (NC), functional electrical stimulation (FES)-induced cycling provides a means of functional restoration and therapeutic exercise. Although FES-cycling has been shown to have numerous benefits, there are challenges to implementing closed-loop FES control for coordinated motion. For example, there exists a potentially destabilizing input delay between the application (or removal) of stimulation and the resulting muscle force. Moreover, switching between multiple actuators (such as FES or motor control) can also be destabilizing. This brief develops delay-dependent switching conditions and a robust control method to account for an unknown time-varying input delay of a switched system. A Lyapunov-like analysis is performed to yield semiglobal exponential cadence tracking to an ultimate bound. Experiments were performed on six able-bodied participants and four participants with NCs to validate the developed controller. The proposed controller resulted in an average cadence error of 0.01 ± 2.00 revolutions per minute (RPM) for the able-bodied participants and 0.01 ± 2.72 RPM for participants with NCs.

Robust backstepping control of uncertain nonlinear systems with unknown time-varying input delay

A robust compensator is developed for a class of strict-feedback uncertain nonlinear systems with additive disturbance and unknown time-varying input delay. The compensator is composed of a Proportional-Integral (PI) control and delay compensation term based on a finite integral of the past control values. The sufficient inequality conditions on controller gains and upper bound of input delay are derived using a Lyapunov-based stability analysis by choosing suitable L-K functionals, which guarantee a global uniformly ultimately bounded (GUUB) tracking result. Simulation results show the performance and robustness of controller for different values of time-varying input delay.

Saturated Control of a Switched FES-Cycle with an Unknown Time-Varying Input Delay

Abstract A common rehabilitation for those with lower limb movement disorders is motorized functional electrical stimulation (FES) induced cycling. Motorized FES-cycling is a switched system with uncertain dynamics, unknown disturbances, and there exists an unknown time-varying input delay between the application/removal of stimulation and the onset/removal of muscle force. This is further complicated by the fact that each participant has varying levels of sensitivity to the FES input, and the stimulation must be bounded to ensure comfort and safety. In this paper, saturated FES and motor controllers are developed for an FES-cycle that ensure safety and comfort of the participant, while likewise being robust to uncertain parameters in the dynamics, unknown disturbances, and an unknown time-varying input delay. A Lyapunov-based stability analysis is performed to ensure uniformly ultimately bounded cadence tracking.

Robust Cadence Tracking for Switched FES-Cycling with an Unknown Time-Varying Input Delay Using a Time-Varying Estimate

Abstract For an individual affected by a lower limb movement disorder, motorized functional electrical stimulation (FES) induced cycling provides a means of functional restoration and therapeutic exercise. However, there exists a potentially destabilizing input delay between the application (and removal) of the stimulation and the production of muscle force. Exacerbating the problem, fatigue results in decreased force production and a time-varying input delay. Moreover, switching between FES and motor control can be destabilizing. This paper implements a time-varying estimate of the delay and develops a control method and switching conditions to account for the time-varying input delayed response of muscle. The controller is shown to yield semi-global uniformly ultimately bounded tracking for the uncertain switched nonlinear dynamic system with input delays.

Cascade Integral Predictors and Feedback Control for Nonlinear Systems with Unknown Time-varying Input-delays

In this paper, we consider the problem of predictor design for nonlinear systems in the presence of unknown time-varying input-delays. A cascade integral high-gain predictor is proposed to estimate the future state. With a distinctive structure, the predictor can handle unknown delays and eliminate the “peaking phenomenon” during the transient period. Then, a predictor-based output feedback control is designed to guarantee the boundedness of system states. Lyapunov-Krasovskii functional and perturbation theories are used to prove the convergence of the estimation error and the closed-loop system. Finally, simulation results illustrate the superior performance of the cascade integral predictor compared to the standard high-gain predictor.

A novel observer-predictor control for uncertain systems with unknown time-varying input and output delays

In this paper, a novel observer-predictor control is proposed to stabilise uncertain systems subject to unknown but bounded arbitrarily-fast time-varying input and output delays. Due to the presence of time-varying model uncertainties and delay mismatches, the separation principle cannot be used to design the observer and control gains separately, which becomes a challenging problem. By applying small gain theory and Lyapunov-Krasovskii functionals (LKF), in combination with Bessel-Legendre inequalities and the Projection Lemma, we develop sufficient conditions based on Linear Matrix Inequalities (LMIs) for control design. As a result, the closed-loop robust performance can be improved whilst satisfying a prescribed decay rate in the closed-loop dynamics at the expense of introducing more LMI decision variables without increasing the complexity of the control scheme. Also, the Extended State Observer (ESO) is integrated in the proposed control scheme to improve the rejection performance of mismatched disturbances. The effectiveness of the proposed method is illustrated by simulation results.

Closed-Loop Neuromuscular Electrical Stimulation Method Provides Robustness to Unknown Time-Varying Input Delay in Muscle Dynamics

Neuromuscular electrical stimulation (NMES) is commonly used to rehabilitate people with motor impairment (e.g., following stroke or spinal cord injury). Closed-loop NMES holds the promise to facilitate coordinated limb motion, but technical challenges remain. In particular, there is a potentially destabilizing delay between the application of the electrical stimulation and the ensuing muscle contraction, which changes as muscle fatigues. In this brief, a closed-loop NMES method is developed to yield lower limb tracking, despite an unknown time-varying input delay, uncertain nonlinear limb dynamics, and additive bounded disturbances. A novel filtered error signal is designed using the past states in a finite integral over a constant estimated delay interval. The control development is based on an approach that uses Lyapunov–Krasovskii functionals in a Lyapunov-based stability analysis to prove ultimately bounded tracking. Experimental results in healthy individuals and participants with neurological conditions are provided to demonstrate the performance of the developed controller.

A Robust Delay Compensator for Uncertain Nonlinear Systems With Unknown Time-Varying Input Delay

This paper proposes a robust compensator for a class of uncertain nonlinear systems subjected to unknown time-varying input delay. The proposed control law is based on the integral of past values of control and a novel filtered tracking error. Sufficient gain conditions dependent on the known bound of the delay are derived using a Lyapunov-based stability analysis, where Lyapunov–Krasovskii (LK) functionals are used to achieve a global uniformly ultimately bounded (GUUB) tracking result. Simulation results for a nonlinear system are used to evaluate the performance and robustness of the controller for different values of time-varying input delay.

Enhanced Predictor-Based Control Synthesis for Discrete-Time TS Fuzzy Descriptor Systems With Time-Varying Input Delays

We present a novel predictor-feedback gain-scheduled control synthesis method for discrete-time Takagi-Sugeno (TS) fuzzy descriptor systems with unknown time-varying input delays. The goal is to reduce the computational complexity in the control design by keeping the descriptor form whilst improving the closed-loop robust performance. It is worth mentioning the following aspects: 1) our approach easily allows dealing with parametric uncertainties on the descriptor matrix, which are generally difficult to handle via matrix inversion; and 2) the control design problem is casted into linear matrix inequality conditions by using nonquadratic Lyapunov functions, the Projection Lemma and small gain theory. Finally, three examples are provided to show the effectiveness of the proposed method.

Control scheme for LTI systems with Lipschitz non‐linearity and unknown time‐varying input delay

In this study, the authors propose a control structure for a class of linear time-invariant (LTI) systems with Lipschitz non-linearity and unknown time-varying input delay. This scheme considers the worst-case scenario in control design with truncated prediction feedback approach, and takes into account the information of the lower bound of delay in the stability analysis. A finite-dimensional controller is constructed, requiring neither the non-linear function nor the exact delay function. The truncated prediction deviation is minimised by employing the delay range, and then bounded by integral construction and related techniques. Within the framework of Lyapunov–Krasovskii functionals, sufficient delay–range-dependent conditions are derived for the closed-loop system to guarantee the global stability. Two numerical examples are given to validate the proposed control design.

An online identification algorithm of unknown time-varying delay and internal multimodel control for discrete non-linear systems

ABSTRACTIn this paper, an online algorithm is proposed for the identification of unknown time-varying input delay in the case of discrete non-linear systems described by decoupled multimodel. This method relies on the minimization of a performance index based on the error between the real system and the partial internal models outputs. In addition, a decoupled internal multimodel control is proposed for the compensation of discrete non-linear systems with time-varying delay. This control scheme incorporates partial internal model controls. Each partial controller is associated to a specified operating zone of the non-linear system. The switching between these controllers is ensured by a supervisor that contains a set of local predictors. A simulation example is carried out to illustrate the significance of the proposed time-varying delay identification algorithm and the proposed internal multimodel control scheme.