Time-varying Feedback Control Research Articles

Smooth Time-Varying Feedback Control of Nonholonomic Systems with Applications to the Attitude Control of Underactuated Spacecraft

This paper studies the control of chained nonholonomic systems by using smooth time-varying feedback. First, by using a time-varying state transformation, the chained nonholonomic system is transformed into a linear time-varying system. Second, with the aid of some properties of a class of parametric Lyapunov equations and time-varying systems theory, smooth time-varying state feedback controller is developed to ensure the exponential convergence of both the system states and controls. Different from existing results, the design parameters in the proposed control strategy are independent of the initial conditions of the system. Third, as an application of the proposed smooth time-varying feedback, the attitude control problem for underactuated axisymmetric spacecraft is addressed. Finally, a numerical simulation is presented to validate the effectiveness of the proposed method.

Finite-time stabilization of nonlinear time-varying systems involving impulsive action

This article investigates the finite-time stabilization of time-varying impulsive systems by time-varying feedback control. Moreover, the stabilizing and destabilizing effects of impulses on the stability of the system are considered, where the stabilizing impulses are used to regulate the unstable continuous dynamics. In addition, it can be found that the local and global characteristics of the finite-time stability may be affected by the stabilizing impulses. By establishing the relationship among the time-varying gain functions, the system structure and impulsive time sequences, some sufficient conditions for finite-time stability of time-varying systems involving impulsive action are proposed. Furthermore, based on the results mentioned above, a new class of finite-time controllers is designed for time-varying affine systems with external noises. The effectiveness of the proposed results is illustrated by three examples.

Optimal control problem for unbounded bilinear systems and applications

In this paper, we will investigate the optimal control problem for unbounded bilinear systems, with -admissible control operators. We will first study the case of finite time-horizon with unconstrained or constrained endpoint. This result is further applied to build the optimal control for infinite time horizon. Finally, we solve the bilinear optimal control for the transport equation. Then we consider the fractional diffusion equation, for which we prove the strong stabilisation by an optimal time-varying feedback control.

Tension Control in Unwinding System Based on Nonlinear Parameter Estimation

In this paper, two representative nonlinearities in the unwinding system are modelled. One is the rotation-disturbance caused by the rotational change of unwinding roll position; another is the time-varying radius of unwinding roll. Based on the nonlinear mechanism model, the parameters of rotation-disturbance are obtained by the Gauss-Newton algorithm, and the time-varying radius is estimated online by the Kalman filter. Then the feedforward and time-varying feedback controller are designed based on the estimated parameters to compensate for nonlinear effect in unwinding system. The experiment results show that the parameter estimation algorithm can accurately estimate parameters. And the use of estimated parameters to control can effectively solve the problem of tension fluctuation and tension deviation which caused by the nonlinearity of unwinding system.

Hyperstability of Linear Feed-Forward Time-Invariant Systems Subject to Internal and External Point Delays and Impulsive Nonlinear Time-Varying Feedback Controls

This paper investigates the asymptotic hyperstability of a single-input–single-output closed-loop system whose controlled plant is time-invariant and possesses a strongly strictly positive real transfer function that is subject to internal and external point delays. There are, in general, two controls involved, namely, the internal one that stabilizes the system with linear state feedback independent of the delay sizes and the external one that belongs to an hyperstable class and satisfies a Popov’s-type time integral inequality. Such a class of hyperstable controllers under consideration combines, in general, a regular impulse-free part with an impulsive part.

Adaptive Fixed-Time Control for Uncertain Nonlinear Cascade Systems by Dynamic Feedback

This article concerns with the adaptive fixed-time control problem for classes of nonlinear cascade systems with parametric uncertainty. The majority of existing finite/fixed-time control strategies for uncertain nonlinear systems can only make system states approach to the neighborhood of origin over a finite/fixed time. Different from these results, we put forward a time-varying gain-based control algorithm to ensure that all system states return to the origin in a fixed time. With the help of the backstepping method, a time-varying adaptive controller is designed for a high-order nonlinear systems subject to uncertain parameters. Through constructing appropriate dynamic gain-based Lyapunov functions and utilizing our proposed stability criterion, it is proved that all states of the uncertain system can be adjusted to the origin in a fixed-time interval. Moreover, the settling time is not depend on the initial conditions of system and can be preset according to the actual requirements. Finally, the simulation results are provided to illustrate the effectiveness of the developed dynamic time-varying feedback control algorithm.

Base-backstepping Continuous Time-varying Feedback Control of Automatic Magazine with Modelling Uncertainty

This paper proposes a continuous time-varying feedback control based on back-stepping for automatic magazine system, in which the control gains depend on the Lyapunov function to ensure the robustness against modelling uncertainty. The Lyapunov function, which is implicit, is differentiable functions of system error variables. Integrated with backstepping control, the performance of the automation magazine system is greatly improved, while retaining the advantage of only needing to ensure the bounds of disturbances. Lyapunov analysis shows tracking errors of the system are asymptotically stable. In addition, this proposed controller is continuous and has the characteristics of simplicity and practicality, which are beneficial to practical applications. Finally, Numerical simulations are provided to verify the high performance of the controller by comparing the simulation experimental results in automation magazine system.

Prescribed-Time Spacecraft Attitude Control Using Time-Varying State Feedback

A time-varying feedback control law based on backstepping is proposed for the prescribed arbitrary time stabilization and tracking for a chain of integrators. Prescribed arbitrary time control permits the convergence time to be chosen regardless of the initial conditions or system parameters. The sufficient condition is used to demonstrate Lyapunov stability for the system. Analytical results and explanations are provided for the influence of convergence time on the maximum needed control torque. Sliding mode control is paired with the time-varying feedback approach to make the control law robust to reject the bounded disturbance. The control law is then applied to a spacecraft’s attitude motion that is susceptible to the bounded disturbance. A quaternion-based attitude representation is utilized to avoid mathematical singularity. The control law is written using a full quaternion with four components instead of only three in the vector part. Finally, promising numerical simulation results are presented.

The Use of Hough Transform Method and Knot-Like Turning for Motion Planning and Control of an Autonomous Agricultural Vehicle

This study focuses on motion planning and reference trajectory tracking control of an autonomous agricultural vehicle to achieve precise row following and turning. The smooth time-varying feedback control method was adapted to the system to generate the required control commands. The mathematical representations for motion planning and controllers were constructed based on the car-like robot model. An algorithm to detect trees and rows of trees of an orchard was developed using the Hough transform approach. A new type of turning procedure, called knot-like turning, was proposed to perform turning from one row to another. A simulation environment was created to test and analyze the developed system. To obtain the real data from a field, the trees and rows of trees of a cherry orchard were scanned using a laser scanner rangefinder sensor. Then, the scanned data were moved to the simulation environment to generate the desired trajectory, which was followed by an autonomous agricultural vehicle. The simulation environment made it possible to determine the performance of the proposed motion planning, reference trajectory generation, tracking control and turning procedures. The results presented here indicate that the proposed methodology could be used for desired trajectory tracking tasks for agricultural operations in the case that minimum tracking errors in both straight and turning motions are needed.

Prescribed-time adaptive neural feedback control for a class of nonlinear systems

This paper studies prescribed-time stabilization for a class of unknown strict-feedback nonlinear systems. Adaptive time-varying feedback control method and non-scaling transformation strategies are employed to reduce the computation burden caused by the scaling functions. Besides, the proposed method uses a constrained function to avoid the possibility of an oversized signal when time approaches to the prescribed settling time. Based on the backstepping method, a neural prescribed-time controller is constructed to achieve finite-time regulation. Under the action of the proposed strategy, all the closed-loop signals are bounded and the system is stable in prescribed finite time. Finally, two simulation examples illustrate the effectiveness of the results.

Practical prescribed-time sampled-data control of triangular nonlinear systems satisfying a linear growth condition by linear time-varying feedback

In this paper, the practical prescribed-time sampled-data control problem of a class of triangular nonlinear systems satisfying a linear growth condition is studied. With the help of some properties of a class of parametric Lyapunov equations (PLEs) and by constructing a time-varying Lyapunov-like function, a linear time-varying feedback controller is designed to make the states of the closed-loop system converge to an arbitrarily small neighbourhood of the origin in any prescribed time, while the control siginals are guaranteed to be bounded. A numerical example validates the effectiveness of the proposed method.

Fixed-time stabilization of parabolic distributed parameter systems with spatially and temporally varying reactivity

This paper concerns the problem of boundary time-varying feedback controller for fixed-time stabilization of a linear parabolic distributed parameter system with spatially and temporally varying reactivity. By utilizing the continuous backstepping approach, the invertible Volterra integral transformation with the time-dependent gain kernel is introduced to convert the closed-loop system into a target system with a time-dependent coefficient. Meanwhile, the convergence of the target system within the prescribed time is guaranteed via the Lyapunov method. The well-posedness of the resulting kernel partial differential equations is also proven by exploiting the method of successive approximation. In addition, the growth-in-time of the kernel functions is estimated by applying the generalized Laguerre polynomials and the modified Bessel functions. Subsequently, the fixed-time stability of the closed-loop system under state feedback control within the prescribed time is proven by using the fixed-time stability of the target system and the time-varying kernel functions. Finally, a numerical example is provided to illustrate the effectiveness of the proposed control method.

Exponential Stabilization of Unstable Bilinear Systems in Finite- and Infinite-Dimensional Spaces

This article discusses the uniform exponential stabilization of bilinear systems with multiplicative control inputs. We first consider the case of finite-dimensional bilinear systems, and we present stabilization results concerning the case of open-loop stable systems using a bang-bang feedback control. For open-loop unstable systems, a normalized quadratic time-varying feedback control is proposed. These results are then applied to infinite-dimensional case. Necessary and sufficient conditions are formulated in terms of unique continuation properties and integral estimates involving the control operator and the state of the uncontrolled system. Applications to finite-dimensional systems and coupled partial differential equations (PDE) are further provided.

Continuous time-varying feedback control of a robotic manipulator with base vibration and load uncertainty

To solve precise and fast position control of a robotic manipulator with base vibration and load uncertainty, a continuous time-varying feedback control method based on the implicit Lyapunov function is studied. This method is proportional–derivative-like in the form of control law, but its proportional and differential coefficients depend on the system Lyapunov function, which are differentiable functions of system error variables. In the motion process of the robotic manipulator, the system performance is influenced by three main nonlinear factors: system friction, balance torque, and base vibration. As the former two factors are available to be modeled and identified through experiments, compensation of the two terms is added to the proposed control law to reduce the effects of system nonlinearities to a certain extent. Experimental results show that the proposed control strategy is robust to base vibration and load uncertainty. Besides, the compensation of system friction and balance torque can shorten the positioning time by 27.3%, from 1.32 s to 0.96 s. Meanwhile, the positioning precision is guaranteed, which verifies the effectiveness of the proposed control scheme.

Time varying feedback control on multi-stability in hidden attractor

We study a on-off feedback method to control the dynamics of a radio physical oscillator having hidden coexisting attractors. The proposed time-varying on-off feedback effectively controls the multi-stability which acts in an on-off manner. We show that for suitable values of feedback strength, feedback active time ratio and time period, the coexisting chaotic and periodic dynamics of the system go to a mono-stable periodic state followed by a stable steady state. This transition from multiple attractors to a single attractor is confirmed by calculating bifurcation diagram, Lyapunov exponent and the basin size as a measure. We also illustrate the generation of chaotic attractor in the system using this scheme.

Control Allocation for Wheeled Mobile Robots Subject to Input Saturation

Abstract This paper addresses the problem of input saturation in wheeled mobile robot control. Depending on the desired trajectory or on the terrain, wheel motors may be demanded to work beyond their limits. This situation can lead to undesired performance, and therefore, input saturation has to be properly managed. Although control allocation has been mainly employed in overactuated systems to enhance the control distribution, it can be also utilized for underactuated systems as well to grant performance. For this purpose, three different control allocation strategies, along with Lyapunov-based time-varying feedback controller, are applied to a differential drive mobile robot subject to input saturation. Experimental results illustrate the performance of the proposed strategies.

On Optimal Time-Varying Feedback Controllability for Probabilistic Boolean Control Networks.

This brief studies controllability for probabilistic Boolean control network (PBCN) with time-varying feedback control laws. The concept of feedback controllability with an arbitrary probability for PBCNs is formulated first, and a control problem to maximize the probability of time-varying feedback controllability is investigated afterward. By introducing semitensor product (STP) technique, an equivalent multistage decision problem is deduced, and then a novel optimization algorithm is proposed to obtain the maximum probability of controllability and the corresponding optimal feedback law simultaneously. The advantages of the time-varying optimal controller obtained by the proposed algorithm, compared to the time-invariant one, are illustrated by numerical simulations.

On the Design of Hyperstable Feedback Controllers for a Class of Parameterized Nonlinearities. Two Application Examples for Controlling Epidemic Models.

This paper studies the hyperstability and the asymptotic hyperstability of a single-input single-output controlled dynamic system whose feed-forward input-output dynamics is nonlinear and eventually time-varying consisting of a linear nominal part, a linear incremental perturbed part and a nonlinear and eventually time-varying one. The nominal linear part is described by a positive real transfer function while the linear perturbation is defined by a stable transfer function. The nonlinear and time-varying disturbance is, in general, unstructured but it is upper-bounded by the combination of three additive absolute terms depending on the input, output and input-output product, respectively. The non-linear time-varying feedback controller is any member belonging to a general class which satisfies an integral Popov’s-type inequality. This problem statement allows the study of the conditions guaranteeing the robust stability properties under a variety of the controllers designed for the controlled system and controller disturbances. In this way, set of robust hyperstability and asymptotic hyperstability of the closed-loop system are given based on the fact that the input-output energy of the feed-forward controlled system is positive and bounded for all time and any given initial conditions and controls satisfying Popov’s inequality. The importance of those hyperstability and asymptotic hyperstability properties rely on the fact that they are related to global closed-loop stability, or respectively, global closed-loop asymptotic stability of the same uncontrolled feed-forward dynamics subject to a great number of controllers under the only condition that that they satisfy such a Popov’s-type inequality. It is well-known the relevance of vaccination and treatment controls for Public Health Management at the levels of prevention and healing. Therefore, two application examples concerning the linearization of known epidemic models and their appropriate vaccination and/or treatment controls on the susceptible and infectious, respectively, are discussed in detail with the main objective in mind of being able of achieving a fast convergence of the state- trajectory solutions to the disease- free equilibrium points under a wide class of control laws under deviations of the equilibrium amounts of such populations.

Global asymptotical stability of almost periodic solutions for a non-autonomous competing model with time-varying delays and feedback controls

In this paper, we are concerned with a non-autonomous competing model with time delays and feedback controls. Applying the comparison theorem of differential equations and by constructing a suitable Lyapunov functional, some sufficient conditions which guarantee the existence of a unique globally asymptotically stable nonnegative almost periodic solution of the system are established. An example with its numerical simulations is given to illustrate the feasibility of our results.

On exponential stabilization of nonholonomic systems with time-varying drift

A class of nonlinear control-affine systems with bounded time-varying drift is considered. It is assumed that the control vector fields together with their iterated Lie brackets satisfy Hörmander’s condition in a neighborhood of the origin. Then the problem of exponential stabilization is treated by exploiting periodic time-varying feedback controls. An explicit parametrization of such controllers is proposed under a suitable non-resonance assumption. It is shown that these controllers ensure the exponential stability of the closed-loop system provided that the period is small enough. The proposed control design methodology is applied for the stabilization of an underwater vehicle model and a front-wheel drive car.