Dynamic Feedback Control Law Research Articles

Finite‐time local piecewise control for parabolic PDEs with ODE output feedback

This paper is devoted to the finite-time local piecewise control for parabolic partial differential equations (PDEs) by using dynamic output feedback control strategy, where the controller is designed as an ordinary differential equation (ODE). This makes the closed-loop system PDE-ODE coupled, which is employed to accurately describe the dynamics of the PDE system. According to the constructed PDE-ODE coupled model, a local piecewise dynamic feedback control law is first proposed. Sufficient conditions on finite-time stabilisation of the parabolic PDE-ODE coupled system by the suggested feedback controller are then developed in the sense of both complete spatial measurement and incomplete spatial measurement of the observed output of the PDE system, respectively. Finally, the issues regarding the finite-time stabilisation of the closed-loop system is converted into the feasibility of matrix inequalities, and some simulation studies are provided to verify the effectiveness of the proposed results.

The H∞ Model Following Control: An LMI Approach

The aim of this paper is to develop a new approach for a solution of the model following control (MFC) problem with a dynamic compensator by using linear matrix inequalities (LMIs). TheH1 model following control problem is derived following LMI formulation. First, the H1 optimal control problem is revisited by referring to Lemmas assuring all admissible controllers minimizing the H1 norm of the transfer function between the exogenous inputs and the outputs. Then, the solvability condition and a design procedure for a two degrees of freedom (2 DOF) dynamic feedback control law is introduced. The existence of a 2 DOF dynamic output feedback controller for the model following control is proven and the stability of the closed-loop system is satisfied by assuring the Hurwitz condition. The benchmark thermal process (PT-326) as the first order process with timedelay is regulated by the presented 2 DOF dynamic output feedback controller. The simulation results illustrate that the presented controller regulates a system with dead-time as a large set of generic industrial systems and the H1 norm of the closed-loop system is assured less than the H1 norm of the desired model system.

Compensation of Wave PDEs in Actuator Dynamics for Extremum Seeking Feedback

Gradient extremum seeking for compensating wave actuator dynamics in cascade with static scalar maps is addressed in the present paper. This class of Partial Differential Equations (PDEs) for extremum seeking has not been studied yet. A dynamic feedback control law based on distributed parameters is proposed by employing backstepping transformation with an appropriate target system and an adequate formulation using Neumann interconnections. Local stability and convergence to a small neighborhood of the desired (but unknown) extremum is proved by means of a Lyapunov functional and the theory of averaging in infinite dimensions. Numerical simulations illustrate the theoretical results.

Cooperative preview tracking problem of discrete-time linear multi-agent systems: A distributed output regulation approach

We are concerned with the cooperative preview tracking problem of discrete-time linear multi-agent systems under the fixed directed acyclic communication topology. First, by introducing a preview-information-based auxiliary system and a type of distributed internal model, we establish an augmented system for each follower and formulate the cooperative preview tracking problem as a distributed output regulation (DOR) problem. Then, from the solvability of the DOR problem and the stabilizability of the augmented system, we present the sufficient conditions that guarantee the achievement of the cooperative tracking consensus. With this procedure, we finally propose a distributed dynamic feedback control law for the original problem with virtual regulated output integral and preview feedforward compensation.

Asymptotic Model Matching for LPV Systems

Abstract We consider the problem of forcing the output of a given Linear Parameter Varying (LPV) plant to match asymptotically that of a given LPV model using a static or a dynamic feedback control law that also quadratically stabilizes the plant. The problem is investigated from a structural point of view and solvability conditions are expressed in geometric terms. Under suitable hypotheses, checkable sufficient conditions and viable procedures for constructing solutions are provided.

Design of a feedback control law for steering steady-states of biochemical reaction networks in quasi-polynomial framework

This paper presents a design of stabilizing dynamic feedback control law for steering steady-states of biochemical reaction networks. The design is carried out on a structurally similar reduced linear system model, obtained at the neighborhood of equilibrium point of the nonlinear system. The biochemical reaction network is modeled as a quasi-polynomial system with a translation into the positive orthant retaining all qualitative behaviors of the original system. The control law design is formulated as a bilinear matrix inequality (BMI) problem and an iterative algorithm is presented to find the controller gains. Two examples are illustrated to show the effectiveness of the proposed method.

Stabilisation of a thermo-elastic system under nonuniformly bounded disturbances in the boundary input

ABSTRACTWe consider the exponential stabilisation for a thermo-elastic system of type II with nonuniformly bounded external disturbances in the boundary control. We begin with finding feedback controls for the system without disturbances by Lyapunov function approach. Then, based on the resulting feedback control, we give new dynamic feedback control laws that make the system exponentially stable by tracking disturbances exponentially through a time-varying high gain observer. We apply the linear operator semi-group theory to prove the well-posedness of the resulting closed system. Finally, we prove that the solution of the closed-loop system can approach zero polynomially or exponentially by the Lyapunov function approach.

Stabilization of the wave equation with interior input delay and mixed Neumann-Dirichlet boundary

This paper considers the stabilization of a wave equationwith interior input delay: $\mu_1u(x,t)+\mu_2u(x,t-\tau)$,where $u(x,t)$ is the control input. A new dynamic feedbackcontrol law is obtained to stabilize the closed-loop systemexponentially for any time delay $\tau>0$ provided that$|\mu_1|\neq|\mu_2|$. Moreover, some sufficient conditionsare given for discriminating the asymptotic stabilityand instability of the closed-loop system.

Distributed stabilized region regulator for synchronization of a class of multi-agent systems

In this paper, a stabilized-region regulator method is presented to solve the output synchronization problem for discrete-time multi-agent systems. The topology structure for the information communication of the agents contains a spanning tree. The innovation of our result is that a stabilized region is designed for discrete-time multi-agent systems and the synchronization problem could be solved by choosing the appropriate parameters. A distributed dynamic feedback control law is designed such that the distribution of the eigenvalues of Laplacian matrix could be regulated into the specified region, then some sufficient conditions used to guarantee output synchronization are presented. At last, the result is extended to the uncertain multi-agent systems. Finally, a numerical simulation is given to demonstrate the correctness of obtained results.

Flocking with connectivity preservation of multiple double integrator systems subject to external disturbances by a distributed control law

This paper studies the flocking with connectivity preservation problem of multiple double integrator systems subject to a class of external disturbances. We solve the problem by a distributed dynamic position feedback control law under the standard assumptions. Our approach is a combination of the potential function technique and observer design. Our result also includes some existing results as special cases.

Output-based stabilization of Euler-Bernoulli beam with time-delay in boundary input

In this paper, we consider the output feedback stabilization problem of a Euler–Bernoulli beam with timedelay in boundary control. A similar question was studied in Shang and Xu (2012, Syst. Control Lett., 61, 1069–1078), wherein the full state of the system was used in the design of the dynamic feedback law. In the present paper, we consider mainly an output-based stabilization problem for the same model. Using the tricks of the Luenberger observer and partial state predictor, we deduce a kind of dynamic feedback control law that exponentially stabilizes the delayed system.

Stabilisation of Timoshenko beam system with delay in the boundary control

In this paper, we consider the exponential stabilisation problem of a Timoshenko beam with delay in boundary control. Suppose that the controller outputs of the forms α1u1(t) + β1u1(t − τ) and α2u2(t) + β2u2(t − τ) where u1(t) and u2(t) are the inputs of boundary controllers. In the past, most of the stabilisation results are required αj > βj > 0, j = 1, 2. In the present paper, we shall give a new dynamic feedback control law that makes the system exponential stabilisation ∀τ > 0 provided that |αj| ≠ |βj|(j = 1, 2). The exponential stabilisation result is proved via test of exact observability of the system.

Anti-control of Hopf bifurcation for Chen's system through washout filters

In this paper, we consider the problem of anti-controlling bifurcations for Chen's system, that is, a certain bifurcation is created at a desired location with preferred properties by appropriate control. Washout-filter-aided dynamic feedback control laws are developed for the creation of Hopf bifurcations. By choosing appropriate control parameter, we investigate the existence of Hopf bifurcation. Conditions are given to determine the directions and stabilities of the bifurcating periodic solutions. We also find that this control laws can be applied to Chen's system for the purpose of control and anti-control of chaotic attractor. Finally, some numerical simulations are given to illustrate the effectiveness of the results found.

Nonlinear Dynamics and Bifurcations of a Supercavitating Vehicle

In this effort, a numerical study of the bifurcation behavior of a supercavitating vehicle is conducted. The vehicle model is nonsmooth; this is a result of the planing force acting on the vehicle. With a focus on dive-plane dynamics, bifurcations with respect to a quasi-static variation of the cavitation number are studied. The system is found to exhibit rich and complex nonlinear dynamics including nonsmooth bifurcations such as the grazing bifurcation; smooth bifurcations such as Hopf bifurcations, cyclic fold bifurcations, and period-doubling bifurcations; and aperiodic behaviors such as transient chaotic motions and chaotic crises. The tailslap phenomenon of the supercavitating vehicle is identified as the consequence of a Hopf bifurcation followed by a grazing event. It is shown that the occurrences of these bifurcations can be delayed or triggered earlier by using dynamic linear feedback control laws employing washout filters.

Load carrying capacity of flexible joint manipulators with feedback linearization

A computational technique for obtaining the maximum load carrying capacity of robotic manipulators with joint elasticity, subject to accuracy and actuators constraints, is described herein. A feedback linearization technique is used to minimize end-effector deflection. An inversion algorithm is employed for the synthesis of a dynamic feedback control law that provides input-output decoupling and full state linearization. The linearizing input transformations and the corresponding state diffeomorphisms are presented. The proposed technique is then applied to a flexible joint robot. Linearizing control law is been expressed in terms of different sets of model variables and their derivatives. As a result, different tracking errors and torques are introduced in the robot-given trajectory and different load carrying capacities are obtained.

A general modelling and control framework for electrostatically actuated mechanical systems

AbstractThis paper presents a geometric framework for the stabilization and control of a general class of electrostatically actuated mechanical systems. Microelectromechanical systems (MEMS), such as micromirrors, are one motivating application for this work. There wavelengths of applications of interest lead to positioning requirements on the order of 40–100 nm. Furthermore, electrostatic actuation is poised to be the method of choice for the emerging field of nanoelectromechanical systems (NEMS), and the approach presented should be applicable there as well. The class of devices under study consists of a movable, rigid, grounded electrode, with a variety of allowable rotational and/or translational degrees of freedom, and a set of multiple, fixed, independently addressable, drive electrodes. A key contribution of this paper places general electrostatic forces in a framework suitable for passivity‐based control. The configuration space of the movable body is assumed to have the structure of a simple mechanical system on a Lie group, and stabilizing static and dynamic feedback control laws are derived in terms of co‐ordinate‐independent geometric formulas. To obtain controllers for a specific device it is then necessary only to evaluate these formulas. Appropriate approximations may be applied to make the computations more tractable. The static output feedback controller requires only measurement of the charge and voltage on each drive electrode to provide almost‐global stabilization of a desired feasible configuration, but performance is limited by the natural dynamics of the mechanical subsystem. Performance may be improved using dynamic output feedback, but additional information is needed, typically in the form of a model relating electrode capacitances to the system configuration. We demonstrate the controller computations on a representative MEMS, and validate performance using ANSYS simulations. Copyright © 2005 John Wiley & Sons, Ltd.

Observer-based Dynamic Visual Feedback Control of Robotic Systems

This paper considers vision-based motion control with the manipulator dynamics using position measurements and visual information, which we term dynamic visual feedback control. Firstly the visual feedback system of rigid body motion is described in order to derive the dynamic visual feedback system. Secondly we propose a dynamic visual feedback control law which guarantees local asymptotic stability of the overall closed-loop system using a Lyapunov function. L2-gain performance analysis for the proposed control law has been discussed using the energy function which plays the role of a storage function. Next, we show that the control law is based on passivity and the dynamic visual feedback system is constructed from two passive systems. Finally simulation results confirm the effectiveness of the dynamic visual feedback control law.

Anti-control of Hopf bifurcations

Bifurcation control generally means to design a controller that is capable of modifying the bifurcation characteristics of a bifurcating nonlinear system, thereby achieving some desirable dynamical behaviors. A typical objective is to delay and/or stabilize an existing bifurcation. In this paper, we consider the problem of anti-controlling bifurcations, that is, a certain bifurcation is created at a desired location with preferred properties by appropriate control. Washout-filter-aided dynamic feedback control laws are developed for the creation of Hopf bifurcations. As Hopf bifurcations give rise to limit cycles, anti-control of Hopf bifurcations suggests a new approach for designing limit cycles with specified oscillatory behaviors into a system via feedback control when such dynamical behaviors are desirable,.

Active vibration control of doubly tapered thin-walled beams using piezoelectric actuation

This paper deals with a study of the modeling and open/closed-loop vibrational behavior of cantilevered beams of non-uniform cross-sections featuring non-classical properties such as transverse shear, anisotropy and heterogeneity of their constituent materials. The active feedback control capability is based upon the implementation of adaptive materials, consisting of piezoelectric layers that are bonded or embedded into the host structure and fully distributed along the entire beam span. As a result of the converse piezoelectric effect considered in conjunction with the out-of-phase activation, a boundary moment is induced at the beam tip. A combined dynamic feedback control law relating the piezoelectrically induced boundary bending moment to the various kinematical response quantities is implemented, and its effects upon the closed-loop eigenfrequencies are investigated. The results obtained reveal the high efficiency of the control methodology adopted towards enhancement of the vibrational behavior of these structures, and emphasize, in this context, the implications of the non-uniformity of the beam cross-sections considered in conjunction with other non-classical effects.

Boundary control of free and forced oscillation of shearable thin-walled beam cantilevers

This paper deals with the problem of controlling the bending oscillations of cantilevers subjected to harmonic time-dependent excitations and with that of enhancing their eigenvibration response characteristics. The structural model consisting of a thin-walled beam of arbitrary cross-section includes a number of nonclassical effects, such as transverse shear, secondary warping and heterogeneity. The control is achieved via the action of a bending moment applied at the tip of the structure. A dynamic feedback control law relating the boundary bending moment to the angular velocity at the tip of the structure is implemented and its implications upon the closed-loop eigenvalues and dynamic response to harmonic excitations are revealed. A computational methodology based on the extended Galerkin's technique aimed at determining the closed-loop response characteristics is used and numerical results revealing the efficiency of the adopted control methodology are displayed.