Fixed-time prescribed performance tracking for nonlinear systems with unknown time-varying input delay

Designing delay-dependent functional observers for LTI systems with multiple known time-varying state delays and unknown time-varying input delays is studied. The input delays are arbitrary, but the state delays should be upper-bounded. In addition, two scenarios of slow-varying and fast-varying state delays are investigated. The results of the paper can also be considered as one of the first contributions considering unknown-input functional observer design for linear systems with multiple time-varying state delays. Based on the Lyapunov Krasovskii approach, delay-dependent sufficient conditions of the exponential stability of the observer in each scenario are established in terms of linear matrix inequalities. Because of using effective techniques, such as the descriptor transformation and an advanced weighted integral inequality, the proposed stability criteria can result in larger stability regions compared with the other papers that study functional observers for time-varying delay systems. Furthermore, to help with the design procedure, a genetic algorithm-based scheme is proposed to adjust a weighting matrix in the established linear matrix inequalities. Two numerical examples illustrate the design procedure and demonstrate the efficacy of the proposed observer in each scenario. Copyright © 2016 John Wiley & Sons, Ltd.

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

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.

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.

In this paper, the adaptive control problem is addressed for active vehicle suspension systems with the unknown nonlinearity as well as the unknown time-varying input delay. A dynamic gain control frame is proposed to suppress vehicle vibration and maintain the required suspension performance. Firstly, a dynamic gain extended state observer (ESO) is constructed to obtain the unknown velocity information and nonlinearity for the implementation of the control scheme. Moreover, a robust control method with a dynamic gain control term and a time delay compensation term is designed to mitigate the effect of unknown input time delay and uncertainty on system stability. By resorting to the Lyapunov-Krasovkii stability theory, the ultimate uniform boundedness of all signals in the closed-loop vehicle active suspension system is guaranteed. Finally, an example is presented as an illustration of the effectiveness of the proposed method.

Sensor noise prevents the exact measurement of output, which makes it difficult to guarantee the ultimate bound of the actual output and states, which is smaller than the sensor noise amplitude. Even worse, the time-varying delay in the input does not guarantee the boundedness of the actual output and states under sensor noise. In this letter, our considered system is a chain of integrators in which time-varying delay exists in the input and there is an additive form of sensor noise in the output measurement. To guarantee the arbitrarily small ultimate bound of the actual output and states, we newly propose an adaptive output feedback controller whose gain is tuned on-line. The merits of our control method over the existing results are clearly shown in the example.

In the present work, a control method is developed to yield lower limb tracking with neuromuscular electrical stimulation (NMES), despite an unknown time-varying input delay intrinsic to NMES, uncertain nonlinear dynamics, and additive bounded disturbances. A novel filtered tracking error signal is designed to obtain a delay-free control expression in the closed-loop error system. Furthermore, the filtered tracking error signal removes the requirement of exact knowledge of the input delay in the control design. The control development is based on an approach that uses Lyapunov-Krasovskii functionals in a Lyapunov-based stability analysis to prove ultimately bounded tracking error signals. Experimental results are provided to demonstrate the performance of the developed controller.

Delayed finite-dimensional observer-based control of 1-D parabolic PDEs

In this paper, a hybrid fuzzy adaptive output feedback control design approach is proposed for a class of multiinput and multioutput strict-feedback nonlinear systems with unknown time-varying delays, unmeasured states, and input saturation. First, fuzzy logic systems are employed to approximate unknown nonlinear functions in the system. Next, a smooth function is used to approximate the input saturation and an adaptive fuzzy state observer is constructed to solve the problem of unmeasured states. Based on the designed adaptive fuzzy state observer, a serial-parallel estimation model is established. By applying adaptive fuzzy dynamic surface control technique and utilizing the prediction error between the system states observer model and the serial-parallel estimation model, a new fuzzy controller with the composite parameters adaptive laws is developed based on Lyapunov-Krasovskii functional. It is proved that all variables of the closed-loop system are bounded and the system outputs can follow the given bounded reference signals as close as possible. A simulation example is provided to further show the effectiveness of this novel control scheme.

The design of a new hybrid controller for fractional-order uncertain chaotic systems with unknown time-varying delays

This paper deals with a new method for the design of a functional filter for linear singular systems with bounded disturbance and variable time delay in the state and the known input vectors. Moreover, a disturbance unknown inputs is considered. The main objective of such design is the estimation of functional state vector, and a part of the disturbance unknown inputs for fault detection. The proposed design is based on Lyapunov-Krasovskii stability theory and, in order to minimize the disturbance effect, the designed filter satisfies a H_∞ criterion. Numerical simulation is used to illustrate the proposed design and the results were successful.

This paper addresses the problem of adaptive tracking control for a class of stochastic nonlinear systems with time-varying input delays, and the nonlinear functions of the systems are with not only the unknown parameters but also the unknown state time-varying delays, which is different from the previous work. In this paper, through a state transformation, the system can be easily transformed into a system without the time-varying input delay; the appropriate Lyapunov–Krasovskii functionals are used to compensate the unknown time-varying delay terms, and the quadratic functions instead of the quartic functions often utilized in the existing results are used as Lyapunov functions to analyze the stability of systems and the hyperbolic tangent functions are introduced to deal with the Hessian terms. Fuzzy logic systems (FLSs) in Mamdani type are used to approximate the unknown nonlinear functions. Then, based on the backstepping technique, the adaptive fuzzy controller is designed. The three main advantages of the developed scheme are that (i) unlike the existing results which deal with the nonlinearly parameterized functions by using the separation principle, the nonlinearly parameterized functions are lumped into the continuous functions which can be approximated by using the FLS; (ii) the number of the adjusted parameters only depend on the order of the investigated systems, which can reduce the computational burden greatly; and (iii) the existence of the time-varying input delay such that the controller design becomes much more difficult, and in this paper, it can be dealt with by using an appropriate state transformation. It is proven that all the signals of the closed-loop system are semi-globally uniformly ultimately bounded in probability, whereas the tracking error converges to a small neighborhood of the origin. Finally, simulation results are provided to show the effectiveness of the proposed approach.

Adaptive iterative learning control of a class of nonlinear time-delay systems with unknown backlash-like hysteresis input and control direction

In this article, the adaptive finite‐time quantized and event‐triggered dynamic surface control (DSC) schemes are proposed for a class of strict‐feedback nonlinear system with time varying input delay. In order to ensure the completeness of the DSC method, an improved dynamic surface control method based on command filtering is adopted. The designed filtering error compensation signal can not only effectively compensate the filtering error, but also deal with the discontinuous input possessing unknown input delay. The problem of finite‐time control design and stability analysis with the difficulties brought by discontinuous input with unknown time‐varying input delay is solved through the design of the new compensation signal and the corresponding Lyapunov functional. Theoretical analysis shows that all signals in the closed‐loop system are bounded in finite time. The simulation examples also verify the effectiveness of the adaptive finite time DSC schemes proposed in this article.