Time-varying Feedback Research Articles

Robustness and dynamical features of fractional difference spacecraft model with Mittag–Leffler stability

Abstract Spacecraft models that mimic the Planck satellite’s behaviour have produced information on cosmic microwave background radiation, assisting physicists in their understanding of the composition and expansion of the universe. For achieving the intended formation, a framework for a discrete fractional difference spacecraft model is constructed by the use of a discrete nabla operator of variable order containing the Mittag–Leffler kernel. The efficacy of the suggested framework is evaluated employing a numerical simulation of the concerning dynamic systems of motion while taking into account multiple considerations such as exterior disruptions, parameterized variations, time-varying feedback delays, and actuator defects. The implementation of the Banach fixed-point approach provides sufficient requirements for the presence of the solution as well as a distinctive feature for such mechanisms Furthermore, the consistent stability is examined. With the aid of discrete nabla operators, we monitor the qualitative behavioural patterns of spacecraft systems to provide justification for structure’s chaos. We acquire the fixed points of the proposed trajectory. At each fixed point, we calculate the eigenvalue of the spacecraft system’s Jacobian matrix and check for zones of instability. The outcomes exhibit a wide range of multifaceted behaviours resulting from the interaction with various fractional orders in the offered system. To maintain stability and synchronize the system, nonlinear controllers are additionally provided. The study highlights the technique’s vulnerability to fractional-order factors, resulting in exclusive, changing trends and equilibrium frameworks. Because of its diverse and convoluted behaviour, the spacecraft chaotic model is an intriguing and crucial subject for research.

Prescribed-time and output feedback stabilization of heat equation with an intermediate-point heat source and boundary control

This paper addresses the problem of prescribed-time stabilization for the heat equation with an interior point heat source under output feedback. The study employs a two-step backstepping approach along with time-varying feedback techniques. A prescribed-time state observer and an observer-based boundary control law are developed. The proposed method extends the applicability of the prescribed time-stabilizing algorithm to heat equations with non-strict feedback terms. Numerical simulations validate its effectiveness.

Tracking control via time-varying feedback for an uncertain robotic system with both output constraint and dead-zone input

This paper is devoted to the tracking control for an uncertain robotic system with both output constraint and dead-zone input. Remarkably, the distinctive characters of the system are reflected by system uncertainties and output constraint. First, more serious uncertainties are involved since unknown nonlinear dynamic matrices, external disturbance and the dead-zone input (see unknown slopes and break points therein) are simultaneously considered, but those of the related literature are not. Second, weaker conditions on the output constraint are allowed since the constraint functions considered are only first but not more order continuously differentiable while any their time derivatives are not necessarily available for feedback. This leads to the incapability of the traditional control schemes on this topic. To solve the control problem, a novel control framework is proposed based on time-varying feedback which overcomes the serious system uncertainties while relaxes the conditions on output constraints. Specifically, a state transformation with a time-varying gain is first introduced to derive a new system. Then, by using the traditional backstepping method with the introduction of the time-varying gain in the estimations of some uncertain terms, a time-varying feedback controller is explicitly designed, which ensures that all the states of the resulting closed-loop system are bounded while system output asymptotically tracks the reference signal without any violation of the output constraint. Finally, simulation results for two practical examples are provided to validate the effectiveness of the proposed theoretical results, and moreover, a comparison with PID method is given to show the superiority of the proposed method on tracking accuracy and robustness.

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.

The impact of investor attention on green bond returns: How do market uncertainties and investment performances of clean energy and oil and gas markets affect the connectedness between investor attention and green bond?

This paper empirically investigates the spillover between investor attention and green bond returns by utilizing the connectedness approach. We will examine the impact of uncertainty in the stock, bond, and energy markets, as well as the success of clean energy, oil, and gas investments, on the relationship between green bonds and investor attention. We find that there are positive but small spillovers between investor attention and green bond returns. Besides, the connectedness between investor attention and green bond market performance is stronger in the short run than in the long run. Further, there is a time-varying feedback effect between green bond returns and investor attention. The connectedness approach also implies that the effect of investor attention on green bond returns is greatest in the COVID-19 period. This impact decreased throughout the war era. It is crucial to analyze the behaviour of green bonds in both typical and exceptional market situations. Making a general assessment may lead to insufficient information about how green bonds behave in extreme market conditions. Additionally, in instances of highly volatile market conditions, investor attention can serve as a valuable instrument for predicting the green bond market performance.

Finite-time semi-global stabilisation of polynomial growth nonlinear systems by linear time-varying feedback

The paper studied the finite-time semi-global stabilisation problems for a class of nonlinear systems with a novel polynomial growth condition. By using some key properties of a class of parametric Lyapunov equations and constructing time-varying Lyapunov-like functions, two time-varying high-gain controllers, where scalar time-varying functions are generated by differential equations, are designed. Both state feedback and observer-based output feedback are considered. It is shown that the proposed methods can achieve finite-time semi-global stabilisation for closed-loop systems and the controls are bound. Two numerical examples verify the effectiveness of the established results.

Prescribed-time stabilization of nonlinear systems with uncertainties/disturbances by improved time-varying feedback control

<p>We address the prescribed-time stability of a class of nonlinear system with uncertainty/disturbance. With the help of the parametric Lyapunov equation (PLE), we designed a state feedback control to regulate the full-state of a controlled system within prescribed time, independent of initial conditions. The result illustrated that the controlled state converges to zero as $t$ approaches the settling time and remains zero thereafter. It was further proved that the controller is bounded by a constant that depends on the system state. A numerical example is presented to verify the validity of the theoretical results.</p>

Adaptive Control With Global Exponential Stability for Parameter-Varying Nonlinear Systems Under Unknown Control Gains.

It is nontrivial to achieve exponential stability even for time-invariant nonlinear systems with matched uncertainties and persistent excitation (PE) condition. In this article, without the need for PE condition, we address the problem of global exponential stabilization of strict-feedback systems with mismatched uncertainties and unknown yet time-varying control gains. The resultant control, embedded with time-varying feedback gains, is capable of ensuring global exponential stability of parametric-strict-feedback systems in the absence of persistence of excitation. By using the enhanced Nussbaum function, the previous results are extended to more general nonlinear systems where the sign and magnitude of the time-varying control gain are unknown. In particular, the argument of the Nussbaum function is guaranteed to be always positive with the aid of nonlinear damping design, which is critical to perform a straightforward technical analysis of the boundedness of the Nussbaum function. Finally, the global exponential stability of parameter-varying strict-feedback systems, the boundedness of the control input and the update rate, and the asymptotic constancy of the parameter estimate are established. Numerical simulations are carried out to verify the effectiveness and benefits of the proposed methods.

Functional and dual observer based prescribed-time control of linear systems by periodic delayed feedback

In this paper the prescribed-time output feedback stabilization of linear systems is addressed using both a functional observer, which includes the reduced-order observer as a special case, and a dual observer. The approach is built upon smooth periodic delayed feedback which is a special case of linear periodic time-varying feedback. In contrast to existing periodic control approaches, the proposed feedback achieves prescribed-time convergence of the state to zero rather than asymptotic convergence. At the same time its synthesis requires solving algebraic equations rather than periodic differential equations. Compared with full-order observer based output feedback, the dimensions of the dynamic equations in the reduced-order observer and dual observer based output feedback are respectively n−p and n−m, where n,m and p are respectively the number of state variables, inputs and outputs, which indicates that the established controllers are less expensive than the full-order observer based controllers in implementations. A numerical example is given to verify the effectiveness of the proposed methods.

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.

Adaptive Prescribed Finite Time Control for Strict-Feedback Systems

In this paper we address the prescribed finite time control problem for a class of strict-feedback systems with unknown parameters. A backstepping based adaptive prescribed-time control algorithm is proposed for second-order strict-feedback systems, where the global stability of system is ensured, and a dynamic surface control (DSC) based adaptive prescribed-time algorithm is designed for the system in high-order case. For the DSC based method, a novel first-order filter is constructed to guarantee the boundedness of “virtual error”, which avoids the so-called “Differential Explosion” problem, however, the control result is only semi-global. Both the developed algorithms are capable of ensuring the regulation in prescribed time with a unique converging feature in that the convergence time is independent of any initial conditions and other design parameters that can be pre-assigned freely by the designer according to the control requirements. The key to achieve the objective in prescribed finite time is the introduction of a descending power time-varying feedback into the controller design. Both the theory analysis and simulation confirm the effectiveness of the proposed methods.

Prescribed-Time Tracking With Guaranteed Performance for a Class of Self-Switching Systems Under Unknown Control Directions.

This article investigates the problem of prescribed-time tracking control for a class of self-switching systems subject to nonvanishing/nonparametric uncertainties and unknown control directions. Due to the existence of the unknown inherent nonlinear dynamics and the undetectable actuation faults, the resultant control gain of the system becomes unknown and time varying, making the control impact on the system uncertain and the prescribed-time control synthesis nontrivial. The underlying problem becomes further complex as the switching is arbitrary and unknown. To circumvent the aforementioned difficulties, the following major steps are employed. First, by integrating a novel time-varying feedback gain and performance function into the control synthesis, the nonvanishing uncertainties are completely rejected and the transient performance is guaranteed. Second, to facilitate the stability analysis under arbitrarily switching, the concept of the constraining function is introduced and incorporated into a skillfully chosen common Lyapunov function. Third, to deal with the uncertain control gain, a new Nussbaum-related lemma is derived. The proposed control is shown to be capable of ensuring that the tracking error not only evolves within the prescribed bound during all the operation time but also converges to zero at the rate of convergence that can be preassigned as fast as desired, in the presence of self-switching dynamics and unknown control directions. Both theoretical analysis and numerical simulation confirm the effectiveness of the proposed method.

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.

Prescribed-time control of perturbed nonholonomic systems by time-varying feedback

This paper studies the prescribed-time control problem of nonholonomic systems. By using some properties of a class of parametric Lyapunov equations and constructing time-varying Lyapunov-like functions, a time-varying high-gain feedback controller is firstly proposed for the chained nonholonomic system such that the state of the closed-loop system converges to zero at any prescribed time. The advantage of time-varying feedback is revealed by comparison with time-invariant feedback. Then time-varying high-gain feedback controllers for a class of perturbed nonholonomic systems are designed to achieve prescribed time convergence of the states. Both state feedback and observer-based output feedback are considered and the controls are proved to be bounded in all the cases. The effectiveness of the proposed methods are verified by numerical examples.

Fixed-time bounded control of nonlinear systems without initial-state constraint

To solve the control problem of time-varying state-scale nonlinear systems whose initial state is not affected by settling time, fixed-time convergence algorithms are proposed for first-order systems and higher-order systems in this paper. First, a scalar model is used to illustrate how the time-varying feedback parameter can guarantee that the system achieves asymptotic stability while achieving finite-time convergence, and it is proved that the settling time obtained in this paper is only related to the prescribed boundary. This allows us to design the settling time with an appropriate parameter based on the prescribed boundary. To exhibit the effectiveness and extensibility of the proposed algorithm for first-order scalar systems, the results are subsequently extended to general higher-order systems based on the backstepping method. By introducing numerical simulation results, this paper verifies that the proposed algorithm will make the system achieve asymptotic stability and its output can converge to a given boundary, regardless of the system's initial states.

Practical prescribed-time seeking of a repulsive source by unicycle angular velocity tuning

All the existing source seeking algorithms for unicycle models in GPS-denied settings guarantee at best an exponential rate of convergence over an infinite interval. Using the recently introduced time-varying feedback tools for prescribed-time stabilization, we achieve source seeking in prescribed time, i.e., the convergence to a small but bounded neighborhood of the source, without the measurements of the position and velocity of the unicycle, in as short a time as the user desires, starting from an arbitrary distance from the source. The practical convergence is established using a singularly perturbed version of the Lie bracket averaging method, combined with time dilation and time contraction operations. The algorithm is robust, provably, even to bounded nonvanishing perturbations and an arbitrarily strong gradient-dependent repulsive velocity drift emanating from the source.

Control of chaos with time-delayed feedback based on deep reinforcement learning

The time-delayed feedback control method, as one of popular methods for chaos control, is noninvasive and flexible for various dynamical systems from different fields of science and technology. In the method, however, an appropriate choice of the feedback gain is challenging, which requires the explicit mathematical model of the controlled system for stability analysis. Additionally, another limitation of the method is the so-called odd number limitation. The two problems restrict its application to practical situations. Fortunately, a technique called deep reinforcement learning is capable of learning the controlled environment (the dynamical system), by continually interacting with the controlled system, which makes it possible to obtain the right feedback gain without the requirement of the accurate mathematical model of the system. Hence, in this work, a time-delayed feedback control method based on deep reinforcement learning is put forward to solve the two problems. Compared with the traditional time-delayed feedback control, the proposed method, as a data-driven method due to the combination with deep reinforcement learning, offers a time-varying feedback gain according to the well-trained policy learned by the deep reinforcement learning algorithm. It maintains the non-invasive property of the time-delayed feedback control, but expands the operating range due to the time-varying feedback gain overcoming the odd number limitation. With numerical simulations, the proposed method is successfully applied to three different kinds of systems, the discrete logistic map, the non-autonomous Duffing oscillator and the autonomous Lorenz system.

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.