Discrete-time Feedback Research Articles

Symplectic discrete-time energy-based control for nonlinear mechanical systems

We present a novel approach for discrete-time state feedback control implementation which reduces the deteriorating effects of sampling on stability and performance in digitally controlled nonlinear mechanical systems. We translate the argument of energy shaping to discrete time by using the symplectic implicit midpoint rule. The method is motivated by recent results for linear systems, where feedback imposes closed-loop behavior that exactly represents the symplectic discretization of a desired target system. For the nonlinear case, the sampled system and the target dynamics are approximated with second order accuracy using the implicit midpoint rule. The implicit nature of the resulting state feedback requires the numerical solution of an in general nonlinear system of algebraic equations in every sampling interval. For an implementation with pure position feedback, the velocities/momenta have to be approximated in the sampling instants, which gives a clear interpretation of our approach in terms of the Störmer–Verlet integration scheme on a staggered grid. Both the Hamiltonian and the Lagrangian perspective are adopted. We present discrete-time versions of impedance or energy shaping plus damping injection control as well as computed torque tracking control in the simulation examples to illustrate the performance and stability gain compared to the quasi-continuous implementation. We discuss computational aspects and show the structural advantages of the implicit midpoint rule compared to other integration schemes in the appendix.

Immersion and invariance disturbance observer-based nonlinear discrete-time control for fully actuated mechanical systems

This paper addresses the attenuation problem of input disturbances in the control of fully actuated mechanical systems in the discrete-time setting. Firstly, a discrete-time disturbance estimator design with immersion and invariance (I&I) approach is presented for the n-degrees of freedom (DOF) fully actuated mechanical systems. Then, a discrete-time combined feedback linearising and backstepping control is established that this controller uses the estimated disturbance information. Global asymptotic stability of the estimator and local asymptotic stability of the entire closed-loop system in an arbitrarily large compact set are shown utilising the Lyapunov stability theory. In order to show the effectiveness of the proposed composite observer-based discrete-time control method, it is applied to the 3-DOF robotic manipulator. Performance of proposed direct discrete-time combined feedback linearising and backstepping controller with discrete-time I&I observer is compared with a direct discrete-time conventional second-order sliding mode controller with another discrete-time nonlinear disturbance observer via simulations. The superior performance of the proposed method is demonstrated with simulation results.

Discrete-time backstepping-state feedback approach for current control of LCL grid-tied converters

Voltage source converters connected to the grid through an LCL filter effectively attenuate the high frequency harmonics generated by the PWM modulation. However, due to the intrinsic resonant peak of this filter and due to the discretization process, the controller design is commonly a challenge and it may not guarantee robustness against abnormalities of the grid. In order to overcome these problems, this paper proposes a new discrete-time current control scheme applied to grid-tied converters based on a backstepping algorithm and state feedback controller in a multi-loop approach. The tracking of converter-side current for a given reference, which is generated by the outer loop, is performed by the backstepping algorithm. In addition, the controller gain design methodology is presented taking into account an extended model, that includes the delay resulted from the discrete-time implementation. An analysis of the gain choice effect on the system closed-loop eigenvalues is demonstrated, which allows the designer to select an appropriate set of controller gains. As outer loop, a state feedback method combined with a state-space resonant controller is used for the grid-side current tracking. In this structure, the harmonic compensation is easily preformed, in addition, the feedback gains are chosen using an optimum algorithm. The robustness of the multi-loop backstepping based control structure is improved in comparison with a full state feedback controller. Simulation and experimental results are obtained to demonstrate the effectiveness of the proposed control scheme under different test conditions, as reference changes, operating during electric faults and grid impedance variation.

Estimate-Based Dynamic Event-Triggered Output Feedback Control of Networked Nonlinear Uncertain Systems

This paper develops a new estimate-based dynamic event-triggered output feedback controller for networked control systems subject to nonlinear uncertainties. Specifically, based on the sampled-data, a discrete-time output feedback controller and a discrete-time dynamic event-triggering condition are proposed by the virtue of feedback domination technique. The proposed event-triggered control method is easy to implement in digital computers due to the form of discrete time. Under the proposed dynamic event-triggered control method, the selection regions of the sampling period and the scaling gain are explicitly given to guarantee the global practical/asymptotic stability of the closed-loop system. Finally, two examples are employed to verify the efficiency of the proposed dynamic event-triggered control approach.

A Trajectory-Based Approach to Discrete-Time Flatness

For discrete-time systems, flatness is usually defined by replacing the time-derivatives of the well-known continuous-time definition by forward-shifts. With this definition, the class of flat systems corresponds exactly to the class of systems which can be linearized by a discrete-time endogenous dynamic feedback as it is proposed in the literature. Recently, verifiable necessary and sufficient differential-geometric conditions for this property have been derived. In the present contribution, we make an attempt to take into account also backward-shifts. This extended approach is motivated by the one-to-one correspondence of solutions of flat systems to solutions of a trivial system as it is known from the continuous-time case. If we transfer this idea to the discrete-time case, this leads to an approach which also allows backward-shifts. To distinguish the classical definition with forward-shifts and the approach of the present paper, we refer to the former as forward-flatness. We show that flat systems (in the extended sense with backward-shifts) still share many beneficial properties of forward-flat systems. In particular, they still are reachable/controllable, allow a straightforward planning of trajectories and can be linearized by a certain subclass of dynamic feedbacks.

Feedback Linearization of Implicit Discrete Systems

Feedback linearization of a nonlinear system is an actively employed technique to control a nonlinear system. Feedback linearizability of continuous-time systems is generally not preserved under sampling and is dependent upon the choice of discretization method. This paper modifies the set of necessary and sufficient conditions for feedback linearization given by Grizzle (1986), so they can be employed for implicitly described discrete-time control system and using them establishes an equivalence between the discrete-time feedback linearizability of the Implicit and Explicit Euler discretization schemes.

Global sampled-data stabilization for a class of nonlinear systems with arbitrarily long input delays via a multi-rate control algorithm

In this paper, the problem of global sampled-data stabilization is investigated for high-order nonlinear systems with arbitrarily long input delays. Based on the Lie algebra technique in nonlinear control theory, a discrete-time predictor-based multi-rate sampled-data state feedback control law with a series expansion form is proposed to ensure that the resulting system is globally asymptotically stable under some conditions. Compared with the existing methods, the proposed control algorithm just needs to know the approximate prediction of state variables, and the faster decrease of Lyapunov function may be provided for each subsystem. Performance of approximate versions of the proposed controller is given by theoretical analyses. It is showed that the approximate controllers achieve practical stability of the sampled-data closed-loop system. Finally, the obtained stabilization results are applied to a trajectory tracking problem for a high-order planar system.

Output feedback adaptive dynamic programming for linear differential zero-sum games

This paper addresses the problem of finding optimal output feedback strategies for solving linear differential zero-sum games using a model-free approach based on adaptive dynamic programming (ADP). In contrast to their discrete-time counterparts, differential games involve continuous-time dynamics and existing ADP approaches to their solutions require access to full measurement of the internal state. This difficulty is due to the fact that direct translation of the discrete-time output feedback ADP results requires derivatives of the input and output measurements, which is generally prohibitive in practice. This work aims to overcome this difficulty and presents a new embedded filtering based observer approach towards designing output feedback ADP algorithms for solving the differential zero-sum game problem. Two output feedback ADP algorithms based respectively on policy iteration and value iteration are developed. The proposed scheme is completely online in nature and works without requiring information of the system dynamics. In addition, this work also addresses the excitation bias problem encountered in output feedback ADP methods, which typically requires a discounting factor for its mitigation. We show that the proposed scheme is bias-free, and therefore, does not require a discounting factor. It is shown that the proposed algorithms converge to the solution obtained by solving the game algebraic Riccati equation. Two numerical examples are demonstrated to validate the proposed scheme.

Semi‐global containment of discrete‐time high‐order multi‐agent systems with input saturation via intermittent control

This study investigates the intermittent semi-global containment of discrete-time high-order multi-agent systems with input saturation. The intermittent semi-global containment protocol is proposed based on discrete-time low-gain state feedback control. Some containment conditions are obtained, and the convergence analysis is given. It is turned out that all the followers beginning from a bounded initial set will asymptotically enter into the convex hull spanned by the leaders under the proposed protocol. Moveover, intermittent semi-global leader-following consensus with input saturation is considered as a special case, and the corresponding result is introduced. Finally, numerical simulations are presented to illustrate the theoretical findings.

Feedback-controlled active brownian colloids with space-dependent rotational dynamics

The non-thermal nature of self-propelling colloids offers new insights into non-equilibrium physics. The central mathematical model to describe their trajectories is active Brownian motion, where a particle moves with a constant speed, while randomly changing direction due to rotational diffusion. While several feedback strategies exist to achieve position-dependent velocity, the possibility of spatial and temporal control over rotational diffusion, which is inherently dictated by thermal fluctuations, remains untapped. Here, we decouple rotational diffusion from thermal fluctuations. Using external magnetic fields and discrete-time feedback loops, we tune the rotational diffusivity of active colloids above and below its thermal value at will and explore a rich range of phenomena including anomalous diffusion, directed transport, and localization. These findings add a new dimension to the control of active matter, with implications for a broad range of disciplines, from optimal transport to smart materials.

Discrete output regulator design for linear distributed parameter systems

In this manuscript, we address discrete-time state and error feedback output regulator designs for a class of linear distributed parameter systems (DPS) with bounded input and output operators. By utilising the Cayley–Tustin bilinear transform, a linear infinite-dimensional discrete-time system is obtained without model spatial approximation or model order reduction. Based on the internal model principle, discrete state and error feedback regulators are designed. In particular, discrete Sylvester regulator equations are formulated, and their solvability is proved and linked to the continuous counterparts. To ensure the stability of the closed-loop system, the design of stabilising feedback gain and its dual problem of stabilising output injection gain design are provided in the discrete-time setting. Finally, three simulation examples including a first-order hyperbolic partial differential equation model and a 1-D heat equation with considerations of step-like, ramp-like and harmonic exogenous signals are shown to demonstrate the effectiveness of the proposed method.

Stabilisation of hybrid stochastic systems with Lévy noise by discrete-time feedback control

This paper is devoted to deal with the stabilisation problems of continuous-time hybrid stochastic systems (often described by stochastic differential equations with Markovian switching) with Lévy noise by feedback controls based on discrete-time state observations. There are so far few results on the stabilisation of continuous-time hybrid stochastic systems with Lévy noise based on discrete-time state observations, despite the stabilisation of stochastic systems by discrete-time feedback controls has been investigated in general. Precisely, it is discussed the stabilisation in the sense of -stability, asymptotic stability, mean-square exponential stability and almost sure exponential stability for such systems. It is also presented an upper bound on the duration τ between two consecutive state observations by means of the Lyapunov function. Two examples are given to illustrate the obtained results.

Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control

The stabilization of stochastic differential equations driven by Brownian motion (G-Brownian motion) with discrete-time feedback controls under Lipschitz conditions has been discussed by several authors. In this paper, we first give the sufficient condition for the mean square exponential instability of stochastic differential equations driven by G-Levy process with non-Lipschitz coefficients. Second, we design a discrete-time feedback control in the drift part and obtain the mean square exponential stability and quasi-sure exponential stability for the controlled systems. At last, we give an example to verify the obtained theory.

Stabilisation of highly non‐linear continuous‐time hybrid stochastic differential delay equations by discrete‐time feedback control

In this study, the authors consider how to use discrete-time state feedback to stabilise hybrid stochastic differential delay equations. The coefficients of these stochastic differential delay equations do not satisfy the conventional linear growth conditions, but are highly non-linear. Using the Lyapunov functional method, they show that a discrete feedback controller, which depends on the states of the discrete-time observations, can be designed to make the solutions of such controlled hybrid stochastic differential delay equations asymptotically stable and exponentially stable. The upper bound of the discrete observation interval is also given in this study. Finally, a numerical example is given to illustrate the proposed theory.

Discrete-Time Control Design Based on Symplectic Integration: Linear Systems

We propose a novel approach to design discrete-time state feedback controllers for sampled control systems with guaranteed stability under arbitrary sampling times that fulfill the Nyquist-Shannon condition. The key idea is borrowed from symplectic integration and backward error analysis of Hamiltonian systems: The closed-loop target system is the discretized version of a continuous-time system with appropriately shaped Hamiltonian, where a symplectic scheme is used for discretization. We adopt this argumentation for a systematic discrete-time design procedure for controllable linear systems based on the implicit midpoint rule. We motivate the approach on the example of two basic linear systems under zero order hold sampling, we show the construction of target systems with desired eigenvalues based on the discrete-time controller canonical form, and we illustrate the quality of the main result on a random sixth order system.

Stabilization of Stochastic Retarded Systems Based on Sampled-Data Feedback Control

Throughout this article, we study the mean square exponential stabilization of stochastic retarded systems described by stochastic functional differential equations (SFDEs). Different from the traditional feedback controls with continuous observation of state, we put forward a novel sampled-data feedback control depending on the discrete observation of state and time delay. It greatly enlarges the probability of finding discrete-time feedback controls for stochastic retarded systems. In addition, the novel sampled-data feedback control succeeds in handling the delays involved in SFDEs. As applications, we provide two examples with simulation figures to illustrate the main result.

Discrete-time capacitor-voltage observer and state-error feedback controller for MMC based on passive theory

At present, the capacitor voltage balancing control and the closed-loop control algorithms are two important issues in a modular multilevel converter (MMC) control system. In the published papers, the majority of the proposed balancing and closed-loop control methods require the measurement of the sub-module (SM) capacitor voltages. To reduce the cost and control complexity, firstly, this paper presents a discrete-time luenberger capacitor-voltage observer to estimate the SM capacitor voltages. Concerning the difficulty of the observation system’s stability analysis caused by the time-varying input and output switching function matrices, this paper manages to obtain the observer gain matrix design method from the point of stability analysis by the passive theory. In this process, the persistently and sufficiently exciting (PSE) condition about the switching functions to ensure the uniformly asymptotic convergence of the observation errors is obtained. This condition is also necessary for the feedback control system based on the observed state variables. Secondly, this paper presents a discrete-time state-error feedback controller using the observed capacitor voltages. According to the passive theory and separation principle, the stability of the MMC closed-loop system is analyzed. Finally, the correctness of the results obtained in this paper under different conditions, including the convergence and robustness of the observer and the state-error feedback controller using the observed capacitor voltages, are verified through the simulation.

Robust Predictive Speed Regulation of Converter-Driven DC Motors via a Discrete-Time Reduced-Order GPIO

Converter-driven direct current (dc) motors exhibit various advantages in industry, but impose several challenges to higher precision speed regulation in the presence of parametric uncertainties and exogenous, time-varying load torque disturbances. In this paper, the robust predictive speed regulation problem of a generic dc–dc buck converter-driven permanent magnet dc motors is addressed by using an output feedback discrete-time model predictive control algorithm. A new discrete-time reduced-order generalized proportional-integral observer (GPIO) is proposed to reconstruct the virtual system states as well as the lumped disturbances. The estimates of GPIO are then collected for output speed prediction. An optimized duty ratio law of the converter is obtained by solving a constrained receding horizon optimization problem, where the operational constraint on control input is explicitly taken into account. Finally, the effectiveness of the proposed new algorithm is demonstrated by various experimental testing results.

Fault-Tolerant Attitude Stabilization Incorporating Closed-Loop Control Allocation Under Actuator Failure

This paper addresses the difficult problem of fault-tolerant control and closed-loop control allocation for spacecraft attitude control system with actuator failures, actuator saturation, and external disturbances. As a fundamental step, a modified fault diagnosis observer is proposed utilizing the iterative learning methodology to reconstruct the actuator failures in real time. On the basis of the previously reconstructed failure information, a fault-tolerant control scheme incorporating with a parameter adjusting law is presented to enforce the spacecraft attitude control system to reach the real sliding mode surface in finite time. Meanwhile, the singularity of control command will be avoided using the saturation function, and the chattering problem restrained by the adjusting law. Furthermore, with the concept of control allocation, a novel on-line closed-loop constrained optimal fault-tolerant control allocation scheme is employed to distribute the signals synthesized by the baseline controller over the redundant actuators with failures and constraints. The stability of the closed-loop fault-tolerant control allocation process is guaranteed by the theory of the closed-loop discrete-time feedback control system. The key achievement of the proposed systematic strategy is that the whole closed-loop attitude control system can theoretically be guaranteed to be stable. Numerical simulations are carried out to verify the effectiveness and superiority of the developed approach.

Combined effects of sampling and dry friction on position control

This paper studies the combined effect of sampling and dry friction on the dynamics of mechanical systems subjected to position control using discrete-time state feedback. This paper aims to highlight that even in case of a single degree-of-freedom mechanical subsystem, the controlled system cannot be characterized by a model with a single dominant frequency and some of the upper harmonics become relevant resulting multi-frequency vibrations. The presented stability analysis of the frictionless system gives insight into this intricate dynamics of sampled-data systems. The numerical simulation results show that the presence of dry friction can stabilize an otherwise unstable motion due to the effect of sampling. The effect of dry friction results that decreasing vibrations occur with concave envelope, but it makes the dynamics dependent on the initial conditions.