Time-varying Feedback Research Articles

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.

Small-Time Stabilization of Homogeneous Cascaded Systems with Application to the Unicycle and the Slider Examples

This paper concerns the small-time stabilization of some classes of mechanical systems which are not stabilizable by means of time-invariant continuous state feedback laws. This is the case of nonholonomic systems, an example being the “unicycle-like” mobile robot, or for underactuated mechanical systems, an example being the slider. Explicit time-varying feedback laws leading to small-time stabilization are constructed for these two control systems. The main tools are homogeneity, backstepping, and desingularization techniques.

A new time-varying feedback RISE control for second-order nonlinear MIMO systems: theory and experiments

This study focuses on the development of a new class of the Robust Integral of the Sign of the Error (RISE) control law adequate for second-order nonlinear multi-input–multi-output systems. A revisit for the original RISE is done by altering some static feedback gains into time-varying nonlinear ones depending on the system states. The proposed controller takes advantage of both RISE control robustness towards uncertainties and the special behaviour of nonlinear feedback gains towards time-varied parameters. A Lyaponuv-based stability analysis to prove the semiglobal asymptotic tracking of the proposed new controller is included. In order to validate the relevance of the proposed controller, real-time experimental results are presented and discussed. Experiments have been conducted on a Delta parallel manipulator, in different operating conditions including payload and speed variations.

Finite-time stabilization of linear systems by bounded linear time-varying feedback

Time-varying features are generally considered to be detrimental to the analysis and design of control systems. This paper establishes methods to design bounded linear time-varying (LTV) controllers such that the control performance of a linear time-invariant (LTI) system can be improved, that is, the finite-time stability of the closed-loop system can be obtained. Specifically, for an LTI control system, by using the solution to a parametric Lyapunov equation (PLE), a bounded LTV controller containing a suitable time-varying parameter is designed. By fully exploiting properties of the solution to the PLE, it is shown that the closed-loop system is finite-time stable. Both state feedback and observer based output feedback, in which both the observer gain and the state feedback gain are time-varying, are considered. As a consequence, the finite-time semi-global stabilization and the fixed-time (prescribed finite-time) stabilization problems for linear systems by bounded controls are solved. The established method is utilized to the design of the spacecraft rendezvous control system and its effectiveness is verified by simulations.

The bifurcation analysis and optimal feedback mechanism for an SIS epidemic model on networks

It is well known that the feedback mechanism or the individual’s intuitive response to the epidemic can have a vital effect on the disease’s spreading. In this paper, we investigate the bifurcation behavior and the optimal feedback mechanism for an SIS epidemic model on heterogeneous networks. Firstly, we present the bifurcation analysis when the basic reproduction number is equal to unity. The direction of bifurcation is also determined. Secondly, different from the constant coefficient in the existing literature, we incorporate a time-varying feedback mechanism coefficient. This is more reasonable since the initiative response of people is constantly changing during different process of disease prevalence. We analyze the optimal feedback mechanism for the SIS epidemic network model by applying the optimal control theory. The existence and uniqueness of the optimal control strategy are obtained. Finally, a numerical example is presented to verify the efficiency of the obtained results. How the topology of the network affects the optimal feedback mechanism is also discussed.

On the Capacity of Symmetric M-User Gaussian Interference Channels With Feedback

A general time-varying feedback coding scheme is proposed for M-user fully connected symmetric Gaussian interference channels. Based on the analysis of the general coding scheme, we prove a theorem which gives a criterion for designing good time-varying feedback codes for Gaussian interference channels. The proposed scheme improves the Suh-Tse and Kramer inner bounds of the channel capacity for the cases of weak and not very strong interference when M = 2. This capacity improvement is more significant when the signal-to-noise ratio (SNR) is not very high. In addition, our coding scheme can be proved mathematically and numerically to outperform the Kramer code for M ≥ 2 when the SNR is equal to the interference-to-noise ratio (INR). Besides, the generalized degrees-of-freedom (GDoF) of our proposed coding scheme can be proved to be optimal in the all network situations (very weak, weak, strong, very strong) for any M. The numerical results show that our coding scheme can attain better performance than the Suh-Tse coding scheme for M = 2 or the MohajerTandon-Poor lattice coding scheme for M > 2. Furthermore, the simplicity of the encoding/decoding algorithms is another strong point of our proposed coding scheme compared with the Suh-Tse coding scheme when M = 2 and the Mohajer-TandonPoor lattice coding scheme when M > 2. More importantly, our results show that an optimal coding scheme for the symmetric Gaussian interference channels with feedback can be achieved by only using marginal posterior distributions under a better cooperation strategy between transmitters.

Velocity Observer-Based Nonlinear Self-Tuning Position Stabilizer for Ball-Beam System Applications

This brief exhibits a nonlinear self-tuning position stabilization algorithm for ball-beam system applications under the consideration of parameter uncertainties and actuator dynamics.; there are two major contributions. First, a first-order ball velocity estimator is proposed using only the ball position feedback without dependence on system parameters. Second, the resulting nonlinear controller incorporates the time-varying feedback gain driven by the proposed self-tuner. Third, a first-order disturbance observer (DOB) is introduced to remove the high-frequency disturbance attenuation performance. The various simulation results-based-on MATLAB/Simulink validate the practical merits of the proposed scheme.

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.

Sequential Predictors Under Time-Varying Feedback and Measurement Delays and Sampling

We build sequential predictors for time-varying linear systems with time-varying input delays, outputs, sampling in the control, and time-varying measurement delays. We prove global exponential stability and robustness properties. We allow the sup norm of the input delay to be arbitrarily large and aperiodic sampling. We use a new set of dynamical extensions that contain output measurements, and whose advantages over existing methods include their lack of distributed terms.

Simultaneous Stabilization and Tracking of Nonholonomic WMRs With Input Constraints: Controller Design and Experimental Validation

Considering input constraints and parameter uncertainty of the nonholonomic wheeled mobile robots, a new control law has been designed in this paper to solve the tracking and stabilization control problems simultaneously. The asymptotic convergence of the stabilization or tracking errors is achieved by a smooth controller with time-varying feedback parameters subject to set conditions. A geometric-based parameter design strategy is employed to overcome the difficulties resulting from input constraints. With the parameter design strategy, the designed parameters satisfy the set conditions and the control inputs stay in the constrained domain. Experimental studies have been conducted to verify the effectiveness of the proposed control law.

Time-variant feedback controller based on capture point and maximal output admissible set of a humanoid

ABSTRACTThis paper formulates the maximal output admissible (MOA) set of a capture point (CP) feedback controller for adaptive humanoid balancing. In previous studies where the MOA set was employed, an adaptive strategy-switching control was proposed. In general, the MOA set requires iterative computation; however, this study has proven that the MOA set of the CP feedback controller can be obtained analytically. This result was applied to a continuous switching feedback gain, which was regarded as a time-variant feedback controller realizing an adaptive response to external disturbances. The validity of the proposed time-variant controller was verified by a whole-body dynamics simulation.

A general approach to precise tracking of nonlinear systems subject to non-vanishing uncertainties

In this work we introduce a new and systematic methodology for precise tracking control of normal-form nonlinear systems subject to non-vanishing/non-parametric uncertainties. We propose a general solution to achieving robust tracking with a rate of convergence that can be pre-assigned as fast as desired (e.g. super exponential or even closer to a pre-specified finite time). The key design tool is the utilization of a time-varying feedback gain through a time-varying scaling function that satisfies certain conditions. Our method also features with complete rejection (rather than partial attenuation) of uncertainties/disturbances and regulation of the tracking error to zero. A generic and analytical procedure to construct the time-varying rate function is provided, extending and generalizing the existing results in the literature.

New Class of Attitude Controllers Guaranteed to Converge within Specified Finite-Time

This paper introduces a new class of finite-time feedback controllers for rigid-body attitude dynamics subject to full actuation. The control structure is Lyapunov-based and is designed to regulate the configuration from an arbitrary initial state to any prescribed final state within user-specified finite transfer-time. A salient feature here is that the synthesis of the control structure is explicit, i.e., given the transfer-time time, the feedback-gains are explicitly stated to satisfy the convergence specifications. A major contrast between this work and others in the literature is that instead of resorting to feedback-linearization (to get to the so-called normal form), our approach efficiently marries the process of designing time-varying feedback gains with the logarithmic Lyapunov function for attitude kinematics based on the Modified Rodrigues Parameters representation. Saliently, this finite-time solution extends nicely for accommodating trajectory tracking objectives and possesses robustness with respect to bounded external disturbance torques. Numerical simulations are performed to test and validate the performance and robustness features of the new control designs.

Event-triggered H∞ anti-synchronisation for delayed neural networks with discontinuous neuron activations via non-fragile control strategy

This paper treats of the global event-triggered anti-synchronisation issue for discontinuous neural networks with the mixed time-varying delays and random feedback gain fluctuation via non-fragile control strategy. The random gain uncertainties are described by stochastic variables satisfying the Bernoulli distribution. Firstly, the novel hybrid controllers, which are composed of the non-fragile controller and the event-triggered controller, are designed. Then, based on Clarke's non-smooth analysis theory, general free-weighting matrix method, the Lyapunov-Krasovskii functional approach and Wirtinger-based multiple integral inequality analysis technology, the global event-triggered non-fragile anti-synchronisation conditions are established in terms of linear matrix inequalities (LMIs). In addition, under the considered external disturbance, the conditions with respect to the global event-triggered non-fragile anti-synchronisation are also addressed in forms of LMIs. Finally, two illustrative examples are provided to verify the effectiveness of the designed event-triggered non-fragile control scheme and the validity of theoretical results.

Rapid Visuomotor Responses Reflect Value-Based Decisions.

Cognitive decision-making is known to be sensitive to the values of potential options, which are the probability and size of rewards associated with different choices. Here, we examine whether rapid motor responses to perturbations of visual feedback about movement, which mediate low-level and involuntary feedback control loops, reflect computations associated with high-level value-based decision-making. In three experiments involving human participants, we varied the value associated with different potential targets for reaching movements by controlling the distributions of rewards across the targets (Experiment 1), the probability with which each target could be specified (Experiment 2), or both (Experiment 3). We found that the size of rapid and involuntary feedback responses to movement perturbations was strongly influenced by the relative value between targets. A statistical model of relative value that includes a term for risk sensitivity provided the best fit to the visuomotor response data, illustrating that feedback control policies are biased to favor more frequent task success at the expense of the overall extrinsic reward accumulated through movement. Importantly however, the regulation of rapid feedback responses was associated with successful pursuit of high-value task outcomes. This implies that when we move, the brain specifies a set of feedback control gains that enable low-level motor areas not only to generate efficient and accurate movement, but also to rapidly and adaptively respond to evolving sensory information in a manner consistent with value-based decision-making.SIGNIFICANCE STATEMENT Current theories of sensorimotor control suggest that, rather than selecting and planning the details of movements in advance, the role of the brain is to set time-varying feedback gains that continuously transform sensory information into motor commands by feedback control. Here, we examine whether the fastest motor responses to perturbations of movement, which mediate low-level and involuntary feedback control loops (i.e., reflexes), reflect computations associated with high-level, value-based decision-making. We find that rapid feedback responses during reaching reflect the relative probabilities and rewards associated with target options. This suggests that low-order components of the sensorimotor control hierarchy, which generate rapid and automatic responses, can continuously evaluate evolving sensory evidence and initiate responses according to the prospect of reward.

Boundary time-varying feedbacks for fixed-time stabilization of constant-parameter reaction–diffusion systems

In this paper, the problem of fixed-time stabilization of constant-parameter reaction–diffusion partial differential equations by means of continuous boundary time-varying feedbacks is considered. Moreover, the time of convergence can be prescribed in the design. The design of time-varying feedbacks is carried out based on the backstepping approach. Using a suitable target system with a time varying-coefficient, one can state that the resulting kernel of the backstepping transformation is time-varying and rendering the control feedback to be time-varying as well. Explicit representations of the kernel solution in terms of generalized Laguerre polynomials and modified Bessel functions are derived. The fixed-time stability property is then proved. A simulation example is presented to illustrate the main results.

Some characterizations of boundary time-varying feedbacks for fixed-time stabilization of reaction-diffusion systems

This paper deals with continuous boundary time-varying feedbacks for fixed-time stabilization of constant-parameter reaction-diffusion systems. The time of convergence can be prescribed and is independent of the initial condition of the system. The design of time-varying feedbacks is carried out by using the backstepping approach for which suitable characterizations for time-varying kernels are derived. Kernel solutions are given in terms of power series involving the exponential complete Bell polynomials for which we have exploited the Faa di Bruno formula. Moreover, by particularizing the characterization, one can recover kernel solutions in terms of the generalized Laguerre polynomials and the modified Bessel functions.

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.