Discrete-time Feedback Control Research Articles

Image-based discrete-time optimal feedback control for an optomechanical derotator

Abstract An optomechanical derotator is an optical system that can be used to extend various measurement methods so they can be applied to rotating measurement objects. The function principle is to manipulate the image of the rotating measurement object by means of rotating reflective components inside the derotator. Thus it appears stationary in the measurement data. For this it is inevitably that the angular position and velocity of the derotator are controlled in such way that they always amount to half that of the measurement object. It is therefore necessary to adjust the derotator with the help of an implemented controller. Since we want to acquire the reference input of the control with a high-speed camera and image processing algorithms the sampling time of the control must be adapted to the rather long computation time of the image processing. As a result, the use of analog control techniques is prohibited so that a digital controller has to be implemented. This paper proposes an approach where an image-based discrete-time linear-quadratic regulator is used to satisfy the requirements concerning the angular position and velocity of the derotator. To improve its performance, it is supplemented by integral action and a model-based feedforward control. Furthermore, the image processing algorithms by which the reference input of the control is determined are described. Finally, it is demonstrated that this controller concept produces the desired result by generating a standing image of a rotating object with a camera and the derotator.

Gittins index based control policy for a class of pursuit‐evasion problems

In this study, the authors develop a novel approach to a class of pursuit-evasion problems modelled in the form of discrete time feedback control systems, where the opposing parties have asymmetric capability. The authors assume the control policy of the evader, described as a random variable, is unknown to the pursuer. This pursuit-evasion problem is formulated as a quadratic optimisation problem from the perspective of the pursuer. Due to the curse of dimensionality, this pursuit-evasion problem cannot be practically solved by dynamic programming. In this study, the authors reformulate it as a multi-armed bandit problem. A heuristic policy based on the Gittins index is proposed to solve this problem, which can be computed based on a forward induction. Simulation results show the proposed policy outperforms a random decision policy.

Robust Output Regulation for Continuous-Time Periodic Systems

We consider controller design for robust output tracking and disturbance rejection for continuous-time periodic linear systems with periodic reference and disturbance signals. As our main results we present four different controllers: A feedforward control law and a discrete-time dynamic error feedback controller for output tracking and disturbance rejection, a robust discrete-time feedback controller, and finally a discrete-time feedback controller that achieves approximate robust output tracking and disturbance rejection. The presented constructions are also new for time-invariant finite and infinite-dimensional systems. The results are illustrated with two examples: A periodically time-dependent system of harmonic oscillators and a nonautonomous two-dimensional heat equation with boundary disturbance.

Robust stabilization of hybrid uncertain stochastic systems with time-varying delay by discrete-time feedback control

This paper is concerned with the robust stabilization problem for a class of continuous-time hybrid uncertain stochastic systems with time-varying delay. Our attention is focused on the design of a feedback controller based on discrete-time state observations such that, for all admissible uncertainties, the closed-loop system is robustly exponentially stable in the mean square, independent of the time delay. By employing the Razumikhin technique, sufficient conditions are firstly established to guarantee the existence of the desired robust controller; they are given in terms of linear matrix inequalities (LMIs). It should be pointed out that in this paper the time-varying delay is just required to be a bounded function rather than a bounded differentiable one as in most existing results. The developed theory is illustrated by a numerical example.

P th moment exponential stabilisation of hybrid stochastic differential equations by feedback controls based on discrete‐time state observations with a time delay

The authors are concerned with the stability of hybrid stochastic differential equations by feedback controls based on discrete-time state observations. Under some reasonable conditions, they establish an upper bound on the duration τ between two consecutive state observations. Moreover, we can design the discrete-time state feedback control to stabilise the given hybrid stochastic differential equations in the sense of p th moment exponential stability by developing a new theory. In comparison to the results given in the previous literature, this study has two new characteristics: (i) the stability criterion concerns p th moment exponential stability, which is different from the existing works; (ii) discrete-time state observations depend on time delays.

On pth moment stabilization of hybrid systems by discrete-time feedback control

ABSTRACTSince Mao initiated the study of stabilization of continuous-time hybrid stochastic differential equations (SDEs) by feedback controls based on discrete-time state observations in 2013, many authors have further studied and developed it. However, so far no work on the pth moment stabilization has been reported. This paper is to investigate how to stabilize a given unstable hybrid SDE by feedback controls based on discrete-time state observations, in the sense of H∞, asymptotic and exponential stability in pth moment for all p > 1. The main techniques used are constructions of the Lyapunov functionals and generalizations of inequalities.

Synchronization of stochastic coupled systems via feedback control based on discrete-time state observations

In this paper, we formulate and investigate the synchronization of stochastic coupled systems via feedback control based on discrete-time state observations (SCSFD). The discrete-time state feedback control is used in the drift parts of response system. Combining Lyapunov method with graph theory, the upper bound of duration between two consecutive state observations is provided. And a global Lyapunov function of SCSFD is presented, which derives some sufficient criteria to guarantee the synchronization of drive–response systems in the sense of mean-square asymptotical synchronization. In addition, the theoretical results are applied to stochastic coupled oscillators and second-order Kuramoto oscillators. Finally, two numerical examples are given to verify the effectiveness of the theoretical results.

Design of Robust Controller for a Class of Uncertain Discrete-Time Systems Subject to Actuator Saturation

This technical note studies the robust output tracking controller design for a class of uncertain discrete-time systems subject to actuator saturation. To achieve this capability, a new controller is proposed by robustifying the discrete-time composite nonlinear feedback (DCNF) control law. Since, the proposed controller may be saturated; a precise analysis is done to show its robust performance in spite of presence of actuator saturation and time-varying external disturbances. Additionally, a theorem is given and proved which guarantees the disturbance attenuation via the proposed control law for three different cases of the saturation function and it is shown that even if the control signal is saturated, the proposed controller achieves steady-state error of order $O(T)$ for state regulation. Finally, computer simulations are performed for a practical system and the applicability of the proposed controller is shown.

Stabilization of Regime-Switching Processes by Feedback Control Based on Discrete Time Observations

This work aims to extend X.R. Mao's work [Automatica J. IFAC, 49 (2013), pp. 3677--3681] on stabilization of hybrid stochastic differential equations by discrete-time feedback control. In X.R. Mao's work, the feedback control depends on discrete-time observation of the state process but on continuous-time observation of the switching process, while, in this work, we study the feedback control depending on discrete-time observations of the state process and the switching process. Our criteria depend explicitly on the regular conditions of the coefficients of the stochastic differential equation and on the stationary distribution of the switching process. The sharpness of our criteria is shown through studying the stability of linear systems, which also shows explicitly that the stability of hybrid stochastic differential equations depends essentially on the long time behavior of the switching process.

<i>p</i>th moment exponential stability of hybrid stochastic functional differential equations by feedback control based on discrete-time state observations

In this paper, we discuss the \begin{document} $p$ \end{document} th moment exponential stabilization of continuous-time hybrid stochastic functional differential equations by feedback control based on discrete-time state observations. The hybrid stochastic functional differential equations are also known as stochastic functional differential equations with the Markovian switching. We follow Mao's paper to consider the auxiliary system whose control is based on continuous-time state observation. The lemma is provided that if the \begin{document} $p$ \end{document} th moment of the solution \begin{document} $y(t)$ \end{document} of the auxiliary system decays exponentially then the same with the \begin{document} $p$ \end{document} th moment of the functional \begin{document} $y_t$ \end{document} . With the help of this lemma, the criterion for \begin{document} $p$ \end{document} th moment exponential stability of the primary system is given, and the margin of the duration of the discrete-time state observation is presented. Then the special case like the linear system is considered and the discrete-time feedback control is designed.

The Control of Non-Linear Chaotic System Including Noise Using Genetic Based Algorithm

In this study, the Lorenz chaotic system including the nonlinear equations has been analyzed against the noise changes. The discrete-time state feedback control method and discrete-time PI controller method have been used for these noisy system controls. Optimization of the controller parameters was performed using a genetic algorithm method. The study was implemented in MATLAB-Simulink simulation program. In terms of noise, the Gaussian white noise was used as a system and measurement noise. Noise has emerged as a significant impact on the Lorenz chaotic system. Changes associated with this condition are given in detail on the graphics. The discrete-time PI controller method was shown to better performance when compared to discrete-time state feedback control method for the system control. Furthermore, the error performance change was given in the literature as an important parameter.

Almost Sure Exponential Stabilization by Discrete-Time Stochastic Feedback Control

Given an unstable linear scalar differential equation $\dot{x}(t)=\alpha x(t)$ $(\alpha>0)$ , we will show that the discrete-time stochastic feedback control $\sigma x([t/\tau]\tau)dB(t)$ can stabilize it. That is, we will show that the stochastically controlled system $dx(t)=\alpha x(t)dt+\sigma x([t/\tau]\tau)dB(t)$ is almost surely exponentially stable when $\sigma^{2}>2\alpha$ and $\tau>0$ is sufficiently small, where $B(t)$ is a Brownian motion and $[t/\tau]$ is the integer part of $t/\tau$ . We will also discuss the nonlinear stabilization problem by a discrete-time stochastic feedback control. The reason why we consider the discrete-time stochastic feedback control is because that the state of the given system is in fact observed only at discrete times, say $0,\tau,2\tau,\ldots$ , for example, where $\tau>0$ is the duration between two consecutive observations. Accordingly, the stochastic feedback control should be designed based on these discrete-time observations, namely the stochastic feedback control should be of the form $\sigma x([t/\tau]\tau)dB(t)$ . From the point of control cost, it is cheaper if one only needs to observe the state less frequently. It is therefore useful to give a bound on $\tau$ from below as larger as better.

Two-Step System Identification and Trajectory Tracking Control of a Small Fixed-Wing UAV

An approach for obtaining dynamically feasible reference trajectories and feedback controllers for a small unmanned aerial vehicle (UAV) based on an aerodynamic model derived from flight tests is presented. The modeling method utilizes stepwise multiple regression to determine relevant explanatory terms for the aerodynamic coefficients. A dynamically feasible trajectory is then obtained through the solution of an optimal control problem using pseudospectral optimal control software. Discrete-time feedback controllers are further designed to regulate the vehicle along the desired reference trajectory. Simulations in a realistic operational environment as well as flight testing of the feedback controllers on the aircraft platform demonstrate the capabilities of the approach.

Unified Control of a Buck Converter for Wide-Load-Range Applications

A new discrete-time state feedback controller is presented, which allows high-bandwidth voltage control of a buck converter for any load condition, whether it operates in discontinuous conduction mode (DCM), continuous conduction mode (CCM), or at the boundary of these regions. This makes the buck converter applicable for a wide range of applications. For the control design process, two large-signal models, which represent the buck converter's discrete time dynamics in CCM and DCM, are developed. A simple proportional-integral regulator is used for the voltage control of the converter. The operation mode is detected and the voltage controller is connected in cascade to a current controller in CCM or to a nonlinear state feedback decoupling structure in DCM. In this paper, the modeling and design of the proposed control topology are introduced and its performance is demonstrated in simulation and experiment.

A Simplified Two-Stage Design of Linear Discrete-Time Feedback Controllers

In this paper, we have shown how to simplify an algorithm for the two-stage design of linear feedback controllers by reducing computational requirements. The algorithm is further simplified for linear discrete-time systems with slow and fast modes (multitime scale systems or singularly perturbed systems), providing independent and accurate designs in slow and fast time scales. The simplified design procedure and its very high accuracy are demonstrated on the eigenvalue assignment problem of a steam power system.

On the Solution of a Nonlinear Semidefinite Program Arising in Discrete-Time Feedback Control Design

A sequential quadratic programming method with line search is analyzed and studied for finding the local solution of a nonlinear semidefinite programming problem resulting from the discrete-time output feedback problem. The method requires an initial feasible point with respect to two positive definite constraints. By parameterizing the optimization problem we ease that requirement. The method is tested numerically on several test problems chosen from the benchmark collection (Leibfritz, 2004).

Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control

In this paper we are concerned with the mean-square exponential stabilization of continuous-time hybrid stochastic differential equations (also known as stochastic differential equations with the Markovian switching) by discrete-time feedback controls. Although the stabilization by continuous-time feedback controls for such equations has been discussed by several authors (see e.g. Ji and Chizeck (1990); Mao, Lam, and Huang (2008); Mao, Yin, and Yuan (2007); Wu, Shi, and Gao (2010); Wu, Su, and Shi (2012)), there is so far no result on the stabilization by discrete-time feedback controls. Our aim here is to initiate the study in this area by establishing some new results.

Output Feedback H-infinity Control Design for Synchronizing of the Main Drives of the Pipe-Rolling Mill

We formulate the discrete-time output feedback controller design problem where the desired accuracy of controlled variables in the mean-square sense is guaranteed in the presence of bounded polyharmonic disturbances with a priori unknown number of harmonics, amplitudes, and frequencies. The amplitudes of the harmonics must satisfy a condition that results in the boundedness of the power of each polyharmonic component. The solution is based on the discrete-time H∞-optimization procedure by properly choosing the corresponding weighting matrices of the minimax cost criterion. Numerical solution in state-space is based on the method of linear matrix inequalities, using the LMI Control Toolbox (MATLAB application). We give the numerical example for the main drives pipe-rolling mill where the digital controller is synthesized using the LMI Control Toolbox package.

Limit cycle walker push recovery based on a receding horizon control scheme

The capability of limit cycle walking in external disturbance rejection is still a relatively unexplored field of study in comparison with zero moment point based walking, whereas the motions achievable by limit cycle walkers are highly efficient and natural looking. In this paper, a balance recovery controller is implemented for a limit cycle walker subjected to an impulsive push based on a receding horizon control scheme. The biped robot model is considered as an under-actuated hybrid planar system with five degrees of freedom having two actuators, which experiences an impulsive push event on its torso. The main idea of push recovery consists in the choice of particular trajectories for the directly controlled degrees of freedom, for which the dynamics of the indirectly controlled degrees of freedom (un-actuated coordinates) of the system tracks a desired trajectory. Particular trajectories are chosen based on a low dimensional online constrained optimization. A discrete-time feedback controller is implemented to track the obtained trajectory and control the whole system to sustain its stable cyclic walking after push. Several simulations have been done considering disturbances exerted during the walking. The results demonstrate the effectiveness of the scheme in recovery of small and large pushes occurring in single and double support phases of walking.

Robust input shaping control for multi-mode flexible structures using neuro-sliding mode output feedback control

Input shaping provides an effective method for suppressing residual vibration of flexible structure systems. However, it is not very robust to parameter uncertainties and external disturbances. In this paper, a closed-loop input shaping method is developed for suppressing residual vibration of multi-mode flexible structure systems with parameter uncertainties and external disturbances. The proposed scheme integrates both input shaping control and discrete-time neuro-sliding mode output feedback control (NSMOFC). The input shaper is designed for the reference model and implemented outside of the feedback loop to achieve the exact elimination of residual vibration. In the feedback loop, the discrete-time NSMOFC technique is employed to make the closed-loop system behave like the reference model with input shaper, where the residual vibration is suppressed. The selection of switching surface and the existence of sliding mode have been addressed. The knowledge of upper bound of uncertainties is not required. Furthermore, it is shown that increasing the robustness to parameter uncertainties does not lengthen the duration of the impulse sequence. Simulation results demonstrate the efficacy of the proposed closed-loop input shaping control scheme.