Time-varying Feedback Research Articles

Consensus via time-varying feedback for uncertain nonlinear multi-agent systems with rather coarse input disturbances

This paper is concerned with the leader-following consensus for a class of uncertain nonlinear multi-agent systems (NMASs) with rather coarse input disturbances, unknown control coefficients and unknown Lipschitz rate. The essential unknowns render the existing compensation strategies inapplicable and motivate us to explore a new one. First, a time-varying strategy is developed to compensate the essential unknowns, where a suitable design function of time is introduced so that the unknowns can be overtaken by the design function as time increases. Then, distributed time-varying consensus protocols are designed and the leader-following consensus is achieved. When essential time-variations exist, the designed consensus protocols are still valid if the involved design function of time is replaced by a new one. Two examples are given to illustrate the effectiveness of the proposed protocols.

Null Controllability and Finite Time Stabilization for the Heat Equations with Variable Coefficients in Space in One Dimension via Backstepping Approach

Using the backstepping approach we recover the null controllability for the heat equations with variable coefficients in space in one dimension and prove that these equations can be stabilized in finite time by means of periodic time-varying feedback laws. To this end, on the one hand, we provide a new proof of the well-posedness and the “optimal” bound with respect to damping constants for the solutions of the kernel equations; this allows one to deal with variable coefficients, even with a weak regularity of these coefficients. On the other hand, we establish the well-posedness and estimates for the heat equations with a nonlocal boundary condition at one side.

Stabilization of nonlinear stochastic systems without unforced dynamics via time-varying feedback

In this paper we give sufficient conditions under which a nonlinear stochastic differential system without unforced dynamics is globally asymptotically stabilizable in probability via time-varying smooth feedback laws. The technique developed to design explicitly the time-varying stabilizers is based on the stochastic Lyapunov technique combined with the strategy used to construct bounded smooth stabilizing feedback laws for passive nonlinear stochastic differential systems. The interest of this work is that the class of stochastic systems considered in this paper contains a lot of systems which cannot be stabilized via time-invariant feedback laws.

Optimal control of cyber physical vehicle systems

Cyber-physical vehicle systems (CPVSs) are advancing due to progress in real-time applications, control and artificial intelligence. Multi-objective design optimisation maximises CPS efficiency, capability and safety, while online regulation enables the vehicle to be responsive to disturbances, modelling errors and uncertainties. CPVS optimisation occurs at design-time and at run-time. We will survey the run-time cooperative optimisation or co-optimisation of cyber and physical systems, which have historically been considered separately. A run-time CPVS is also cooperatively regulated or co-regulated when cyber and physical resources are utilised in a manner that is responsive to both cyber and physical system requirements. This paper surveys research that considers both cyber and physical resources in co-optimisation and co-regulation schemes with applications to mobile robotic and vehicle systems. Time-varying sampling patterns, sensor scheduling, anytime control, feedback scheduling, task and motion planning and resource sharing are examined. Cyber-physical systems will transform how we interact with the physical world around us. Many grand challenges wait in the economically vital domains of transportation, healthcare, manufacturing, agriculture, energy, defence, aerospace and buildings.

Quadratic Approximation and Time-Varying Feedback Laws

We present a method to construct stabilizing time-varying feedback laws for a large class of systems. We apply our technique to several classical examples which do not satisfy the necessary Brockett condition or the Coron condition for stabilization by means of continuous stationary feedback laws.

When Optimal Feedback Control Is Not Enough: Feedforward Strategies Are Required for Optimal Control with Active Sensing

Movement planning is thought to be primarily determined by motor costs such as inaccuracy and effort. Solving for the optimal plan that minimizes these costs typically leads to specifying a time-varying feedback controller which both generates the movement and can optimally correct for errors that arise within a movement. However, the quality of the sensory feedback during a movement can depend substantially on the generated movement. We show that by incorporating such state-dependent sensory feedback, the optimal solution incorporates active sensing and is no longer a pure feedback process but includes a significant feedforward component. To examine whether people take into account such state-dependency in sensory feedback we asked people to make movements in which we controlled the reliability of sensory feedback. We made the visibility of the hand state-dependent, such that the visibility was proportional to the component of hand velocity in a particular direction. Subjects gradually adapted to such a sensory perturbation by making curved hand movements. In particular, they appeared to control the late visibility of the movement matching predictions of the optimal controller with state-dependent sensory noise. Our results show that trajectory planning is not only sensitive to motor costs but takes sensory costs into account and argues for optimal control of movement in which feedforward commands can play a significant role.

Control of non holonomic or under-actuated mechanical systems: The examples of the unicycle robot and the slider

This paper is dedicated to the control of some classes of mechanical systems, not stabilizable by means of at least continuous state feedback laws. This is the case for non holonomic mechanical systems, an example being the unicycle robot, or for under-actuated mechanical systems, an example being the slider. The similarity between these two kinds of dynamical systems will be analyzed, with respect to controllability, stabilizability and flatness properties. The control design for slider stabilization could be viewed, to some extent, as an extension of the control law used for unicycle systems, but it is not so obvious if one wants to stabilize the slider at a chosen reference position, with a chosen fixed orientation. In fact, a new fixed point stabilization result will be given for the slider, and therefore, a switched control strategy will be proposed for both systems, based on differential flatness property to track moving non singular reference trajectories, and switching to periodic time-varying feedback laws to ensure final stabilization at rest equilibrium points. The method will be illustrated with success through simulation results and also first experimental results in the case of a terrestrial quadrotor tracking non singular reference trajectories.

Output Feedback Passification of Saturated Switched Systems Under the Improved State-Dependent Switching

This paper focuses on the passivity analysis and feedback passification of discrete-time switched systems with actuator saturation. The proposed framework consists of a switching law and a bank of time-varying output feedback controllers. The switching law is depending on partial state measurements and switching-delay slack matrix, which avoid the possible too frequent switching caused by the classical state-dependent switching law. The resulting time-varying saturated controllers associated with the switching law are constructed to guarantee the passivity of the closed-loop systems. In addition, the performance index measuring the level of unknown input tolerance capacity of the systems can be also estimated. Sufficient conditions for the existence of the switching law and controllers are provided via Lyapunov---Metzler inequalities. Finally, the effectiveness of the theoretical results is illustrated through an example derived from the stirred tank reactor.

Ultimate uniform bounded-stability of inertial coupling electromechanical system via nonlinear time-varying feedback

ABSTRACTThis paper concerns with the problem of stabilisation of inertial coupling electromechanical system. The concept of inertial coupling electromechanical system is an interesting topic from the control theory point of view and its universe of applications. This is because, for stabilisation proposes, the complete model of electromechanical system significantly reduces higher frequency functions in their dynamics caused by high gain controllers. Here, the system stabilisation is achieved by applying nonlinear time-varying controllers. The reason to use a time-varying controller is to give some learning or adaptation in the control process. Moreover, to get a good result for the closed-loop system, the structure of a storage function is based on the classical quadratic function in addition to quadratic function of system total energy. So, the main goal of this paper is obtain a controller that guarantees that the system state error arrives near the required equilibrium in exponential form. Hence, the Ultimate Uniform Bounded-Stability of the system trajectories of desired equilibrium is concluded. Finally, experimental results of the control algorithm developed in this paper are presented briefly.

Master-slave synchronization for nonlinear systems via reliable control with gaussian stochastic process

In this paper, master-slave synchronization methods for nonlinear system are proposed under a reliable control scheme. In order to consider a more realistic situation in designing a reliable controller, Gaussian stochastic process is introduced to the concept of reliable control system. To confirm an effect of reliable control system, a new synchronization criterion of nonlinear systems under time-varying delay feedback and reliable control with Gaussian transition probabilities is derived within the framework of linear matrix inequalities (LMIs) based on Lyapunov method. Finally, two numerical examples are provided to show the effectiveness and applicability of the of the proposed results.

A new revised desired compensation adaptive control for enhanced tracking: application to RA-PKMs

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Optimal actuator location of minimum norm controls for heat equation with general controlled domain

In this paper, we study optimal actuator location of the minimum norm controls for a multi-dimensional heat equation with control defined in the space L2(Ω×(0,T)). The actuator domain is time-varying in the sense that it is only required to have a prescribed Lebesgue measure for any moment. We select an optimal actuator location so that the optimal control takes its minimal norm over all possible actuator domains. We build a framework of finding the Nash equilibrium so that we can develop a sufficient and necessary condition to characterize the optimal relaxed solutions for both actuator location and corresponding optimal control of the open-loop system. The existence and uniqueness of the optimal classical solutions are therefore concluded. As a result, we synthesize both optimal actuator location and corresponding optimal control into a time-varying feedbacks.

Performance analysis of transmit antenna selection for cognitive radio systems with imperfect channel estimation

In this article, we consider a spectrum sharing cognitive radio network with a transmit antenna selection scheme, subject to the peak interference constraint of primary user (PU) and the maximum transmit power limit of secondary user (SU). Due to time-varying properties or feedback latency, the channel estimation of the link from secondary transmitter to primary receiver may be imperfect, which will result in the violation of the interference constraint at PU. Under the condition of imperfect channel state information (CSI), we derive the cumulative distribution function (CDF) and probability density function (PDF) of the SU’s signal to noise ratio (SNR). Then, we obtain the closed-form expressions of average symbol error rate (SER) and capacity for the secondary system, respectively. Extensive simulations are performed with various system parameters to validate the correctness of our theoretical analysis.

A unified time-varying feedback approach and its applications in adaptive stabilization of high-order uncertain nonlinear systems

We are concerned with two key problems of adaptive control design: one is to find the feasible conditions and the valid boundary of the time-varying feedback scheme; another one lies in the improved generalization of the homogeneous idea in the global adaptive stabilization of high-order uncertain nonlinear systems. The distinguished feature of the system to be investigated is the serious coexistence between unknown time-varying parameters and unknown time-varying control coefficients. Design procedures of the continuous controller are provided based on the sign function technique and the delicate search of the time-varying function and a Lyapunov function. Finally, the control of uncertain single-link robotic manipulator demonstrates the application of design scheme.

Stabilization of a tractor-trailer wheeled robot

Wheeled mobile robots are a special class of nonholonomic mechanical systems. The mobility of such highly nonlinear systems is restricted due to the presence of nonholonomic constraints of wheels, also the system severe underactuated nature. These conditions generate major difficulties in system stabilization, i.e. to park or reach a given configuration for the overall system. This leads to a challenging control problem for research that is the focus of this paper. In this paper a new method based on time-varying feedbacks has been developed for a Tractor-trailer wheeled robot (TTWR). First kinematic model of the TTWR is obtained. Next, a novel method using timevarying feedbacks is investigated in order to stabilize the TTWR around the origin. The proposed kinematic control algorithm is developed based on switching between two finite-time controllers. Appropriate control algorithms have been designed for each step based on the stability of the closed loop system. Obtained simulation and experimental results show the effectiveness of the proposed control law.

Exponential Stabilization of Nonholonomic Systems by Means of Oscillating Controls

This paper is devoted to the stabilization problem for nonlinear driftless control systems by means of a time-varying feedback control. It is assumed that the vector fields of the system together with their first order Lie brackets span the whole tangent space at the equilibrium. A family of trigonometric open-loop controls is constructed to approximate the gradient flow associated with a Lyapunov function. These controls are applied for the derivation of a time-varying feedback law under the sampling strategy. By using Lyapunov's direct method, we prove that the controller proposed ensures exponential stability of the equilibrium. As an example, this control design procedure is applied to stabilize the Brockett integrator.

Almost periodic solutions for a competition and cooperation model of two enterprises with time-varying delays and feedback controls

In this paper, we propose and deal with a competition and cooperation model of two enterprises with time-varying delays and feedback controls, which can effectively describe the competition and cooperation of two enterprises in real economic environments. With the help of 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. We present an example to explain the economical significance of mathematical results. The paper ends with brief conclusions.

A minimum attention control law for ball catching

Digital implementations of control laws typically involve discretization with respect to both time and space, and a control law that can achieve a task at coarser levels of discretization can be said to require less control attention, and also reduced implementation costs. One means of quantitatively capturing the attention of a control law is to measure the rate of change of the control with respect to changes in state and time. In this paper we present an attention-minimizing control law for ball catching and other target tracking tasks based on Brockett’s attention criterion. We first highlight the connections between this attention criterion and some well-known principles from human motor control. Under the assumption that the optimal control law is the sum of a linear time-varying feedback term and a time-varying feedforward term, we derive an LQR-based minimum attention tracking control law that is stable, and obtained efficiently via a finite-dimensional optimization over the symmetric positive-definite matrices. Taking ball catching as our primary task, we perform numerical experiments comparing the performance of the various control strategies examined in the paper. Consistent with prevailing theories about human ball catching, our results exhibit several familiar features, e.g., the transition from open-loop to closed-loop control during the catching movement, and improved robustness to spatiotemporal discretization. The presented control laws are applicable to more general tracking problems that are subject to limited communication resources.

Lyapunov-Floquet control of satellite relative motion in elliptic orbits

This paper proposes a method for continuous-thrust control of satellite formations in elliptic orbits. A previously calculated Lyapunov-Floquet transformation relates the linearized equations of relative motion for elliptic chief orbits to the Hill-Clohessy-Wiltshire equations describing circular chiefs. Using a control law based on Lyapunov-Floquet theory, a time-varying feedback gain is computed that drives a deputy satellite toward rendezvous with an elliptic chief. This control law stabilizes the relative motion across a wide range of chief eccentricities.

Enhanced trajectory linearization control based advanced guidance and control for hypersonic reentry vehicle with multiple disturbances

In this paper, the guidance and control problem for hypersonic reentry vehicle (HRV) in the presence of control constraints and multiple disturbances is handled based on unified enhanced trajectory linearization control (TLC) framework under reference-tracking methodology. First, based on the nominal trajectory and open-loop command generated by Gauss pseudo-spectral method (GPM), a time-varying feedback guidance law with integral action is synthesized to stabilize the tracking error dynamics along the nominal trajectory under the framework of TLC. Second, to improve the robustness of attitude and angular rate loop, variations of various aerodynamic coefficients and external disturbances are considered as lumped uncertainties, reduced-order linear extended state observers (LESO) with given model information are constructed to estimate the lumped uncertainties in each loop, respectively. In addition, comparisons between the estimation efficiency of LESO and reduced-order LESO are carried out. Then augmented with the disturbance estimates and TLC control law, tracking errors of the rotational dynamics can be actively rejected without sacrificing nominal performances. More importantly, fewer control consumption and smooth transient performances are achieved by using nonlinear tracking differentiator (TD) in attitude loop. The stability of the resulting closed-loop system is well established based on Lyapunov stability theory. Finally, the effectiveness of the proposed advanced guidance and control strategy is verified through extensive simulations on the six-degree-of-freedom reentry flight.