Boundary Control Law Research Articles

Boundary stabilization of non-classical micro-scale beams

In this paper, the problem of boundary stabilization of a vibrating non-classical micro-scale Euler–Bernoulli beam is considered. In non-classical micro-beams, the governing Partial Differential Equation (PDE) of motion is obtained based on the non-classical continuum mechanics which introduces material length scale parameters. In this research, linear boundary control laws are constructed to stabilize the free vibration of non-classical micro-beams which its governing PDE is derived based on the modified strain gradient theory as one of the most inclusive non-classical continuum theories. Well-posedness and asymptotic stabilization of the closed loop system are investigated for both cases of complete and incomplete boundary control inputs. To illustrate the performance of the designed controllers, the closed loop PDE model of the system is simulated via Finite Element Method (FEM). To this end, new strain gradient beam element stiffness and mass matrices are derived in this work.

Biomimetic-based output feedback for attitude stabilization of a flapping-wing micro aerial vehicle

SUMMARYThe paper deals with the development of a bounded control law for Flapping-wing Micro Aerial Vehicles that mimics a strategy adopted by animal flapping flyers to stabilize their orientation. The control consists on generating torques about the body's principal axes by means of a modulation of the wing angle amplitudes. It is known that flapping flyers orient their body without any numerical computation or estimation of their current attitude. Therefore, the proposed control law is computed using the direct measurements of onboard sensors mimicking animal sensitive organs, more specifically the halteres, legs sensilla, and magnetic sense. The technological equivalents of these biological sensors are three rate gyros, a tri-axis accelerometer, and a tri-axis magnetometer, respectively. Besides, the control signal is bounded to keep the wing angle amplitudes below the maximal values. Owing to its simplicity, this control law is suitable for applications where onboard computational resources are limited. The stability of the closed-loop system is proved based on Lyapunov analysis and averaging theory. The effectiveness of the proposed control law is shown in simulations. The robustness with respect to external disturbances is also shown emphasizing the importance and need of the bounded control.

Dirac Structures on Hilbert Spaces and Boundary Control of Distributed Port-Hamiltonian Systems

The scope of this paper is to show how to exploit the properties of the Stokes–Dirac structure in the development of energy-based boundary control laws for distributed port-Hamiltonian systems. Usually, stabilisation of non-zero equilibria has been achieved by shaping the open-loop Hamiltonian function by interconnecting a finite-dimensional port-Hamiltonian controller to the boundary of the distributed parameter system. The procedure is based on the presence of structural invariants, namely Casimir functions, in closed-loop, but this approach fails when a non-zero power flow from the controller is required at the equilibrium (dissipation obstacle). This paper illustrates that the class of stabilising controllers can be enlarged by relying on the parametrisation of the system dynamics provided by the image representation of the Stokes–Dirac structure, that is able to show the effects of the boundary inputs on the state evolution in a simple and effective way.

Finite-time consensus of second-order leader-following multi-agent systems without velocity measurements

This Letter investigates the finite-time consensus problems of second-order multi-agent systems in the presence of one and multiple leaders under a directed graph. Specifically, we propose two bounded control laws, which are independent of velocity information, to deal with the finite-time consensus tracking problem with one leader and the finite-time containment control problem with multiple leaders, respectively. With the aid of homogeneous theory, some sufficient conditions are established for the achievement of the finite-time tracking control problem of second-order multi-agent systems. Numerical examples are finally provided to illustrate the theoretical results.

Bounded control for preserving connectivity of multi-agent systems using the constraint function approach

In multi-agent control systems, coordination of agents often relies critically on the connectivity of the underlying communication network. This study analyses the problem of preserving connectivity with bounded control inputs for multi-agent systems. Firstly, a general potential-based control framework is investigated and sufficient conditions for achieving rendezvous while preserving connectivity are presented. Then, based on the modification of the potential-based control, a novel bounded control law with very mild design requirements is proposed and the issues of preserving connectivity, the agent group rendezvous, and the input bound of the controller are discussed. The effectiveness of the proposed control law and the boundedness of the inputs are illustrated by numerical examples.

Normal Feedback Stabilization of Periodic Flows in a Three-Dimensional Channel

We present an extension from two dimensions to three dimensions of a boundary control law, which stabilizes the parabolic profile of an infinite channel flow. The controller acts on the normal component of the velocity only. The stability is achieved without any a priori condition on the viscosity coefficient, that is on Reynolds number.

Output tracking control of a parallel-flow heat exchange process

In this paper, we consider an output tracking problem of a parallel-flow heat exchange process with distributed and boundary inputs. As the distributed inputs to the system, the output feedback control is first applied. Under zero boundary inputs, it is shown that the C 0 -semigroup describing the closed-loop system satisfies the spectrum determined growth condition. Next, we apply a backstepping method to the design of the boundary inputs for output tracking. Our main result shows that the output of the system reaches a reference signal in finite time under both the output feedback control and the boundary control law derived by the backstepping method.

Pattern transition in spacecraft formation flying using bifurcating potential fields

Many new and exciting space mission concepts have developed around spacecraft formation flying, allowing for autonomous distributed systems that can be robust, scalable and flexible. This paper considers the development of a new methodology for the control of multiple spacecraft. Based on the artificial potential function method, research in this area is extended by considering the new approach of using bifurcation theory as a means of controlling the transition between different formations. For real, safety or mission critical applications it is important to ensure that desired behaviours will occur. Through dynamical systems theory, this paper also aims to provide a step in replacing traditional algorithm validation with mathematical proof, supported through simulation. This is achieved by determining the non-linear stability properties of the system, thus proving the existence or not of desired behaviours. Practical considerations such as the issue of actuator saturation and communication limitations are addressed, with the development of a new bounded control law based on bifurcating potential fields providing the key contribution of this paper. To illustrate spacecraft formation flying using the new methodology formation patterns are considered in low-Earth-orbit utilising the Clohessy–Wiltshire relative linearised equations of motion. It is shown that a formation of spacecraft can be driven safely onto equally spaced projected circular orbits, autonomously reconfiguring between them, whilst satisfying constraints made regarding each spacecraft.

Global Stabilization of The Generalized Burgers-Korteweg-de Vries Equation by Boundary Control

AbstractThe generalized Korteweg-de Vries (gKdV) equation is a nonlinear partial differential equation describing the propagation of nonlinear dispersive waves. For higher-order non-linearities, the equation has unstable solitary wave solutions. To take account for dissipative mechanisms, a Burgers-type term is often added which yields the generalized Burgers-Korteweg-de Vries (gBKdV) equation. In this paper, stabilizing boundary control laws are constructed for the gBKdV equation using the Lyapunov method. The control laws achieve global exponential stability in the L2-sense. Adaptive control laws are also constructed for the case of an unknown dissipation coefficient. Other control laws where the dissipation coefficient ε = 0 and the dispersion coefficient δ = 0 are also treated and stability is proved. Finally, an observer for the control laws where measurements inside the domain is needed is derived in the last section. Again closed loop exponential stability is shown.

$H^\infty$ Feedback Boundary Stabilization of the Two-Dimensional Navier–Stokes Equations

We study the robust or $H^\infty$ exponential stabilization of the linearized Navier-Stokes equations around an unstable stationary solution in a two-dimensional domain $\Omega$. The disturbance is an unknown perturbation in the boundary condition of the fluid flow. We determine a feedback boundary control law, robust with respect to boundary perturbations, by solving a max-min linear quadratic control problem. Next we show that this feedback law locally stabilizes the Navier-Stokes system. Similar problems have been studied in the literature in the case of distributed controls and disturbances. To the authors' knowledge, it is the first time that the robust stabilization of the Navier-Stokes equations is studied for boundary controls and boundary disturbances.

Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties

This paper considers the stabilization problem of a one-dimensional unstable heat conduction system (rod) modeled by a parabolic partial differential equation (PDE), powered with a Dirichlet type actuator from one of the boundaries. By applying the Volterra integral transformation, a stabilizing boundary control law is obtained to achieve exponential stability in the ideal situation when there are no system uncertainties. The associated Lyapunov function is used for designing an infinite-dimensional sliding manifold, on which the system exhibits the same type of stability and robustness against certain types of parameter variations and boundary disturbances. It is observed that the relative degree of the chosen sliding function with respect to the boundary control input is zero. A continuous control law satisfying the reaching condition is obtained by passing a discontinuous (signum) signal through an integrator.

On consensus algorithms for double-integrator dynamics without velocity measurements and with input constraints

This note deals with consensus strategy design for double-integrator dynamics. Specifically, we consider the case where the control inputs are required to be a priori bounded and the velocity (second state) is not available for feedback. Two different design methods are proposed. First, based on the auxiliary system approach, we propose a consensus algorithm that extends some of the existing results in the literature to account for actuator saturations and the lack of velocity measurement. The proposed velocity-free control scheme, using local information exchange, achieves consensus among the team members with an a priori bounded control law, whose upper bound depends on the number of neighbors of the vehicle. Second, we propose another approach based on the use of a high order dynamic auxiliary system such that the upper bound of the control law is independent of the number of neighbors of the vehicle, and the performance of the closed loop system is improved in terms of the response damping. Finally, simulation results are provided to illustrate the effectiveness of the proposed algorithms.

Hamiltonian approach to the stabilization of systems of two conservation laws

This paper aims at providing some synthesis between two alternative representations of systems of two conservation laws and interprets different conditions on stabilizing boundary control laws. The first is the representation in Riemann invariants coordinates whose representation has been applied successfully for the stabilization of linear and non-linear of such hyperbolic systems. The second representation is based on physical modelling and leads to port Hamiltonian systems which are extensions of infinite-dimensional Hamiltonian systems defined on Dirac structure. The stability conditions on the boundary feedback relations derived with respect to the Riemann invariants are interpreted in terms of the dissipation inequality of the Hamiltonian functional.

Asymptotic stabilization of vibrating composite plates

This paper presents a solution to the boundary stabilization problem of free vibration of composite plates. It is assumed that an object such as a flange which has mass and mass moment of inertia can be attached to the free portion of the boundary of the plate. Both cases of with and without boundary object have been analyzed for other types of plates by many researches. In the presence of boundary mass and mass moment of inertia, two ordinary differential equations govern their motion. For both cases, the plate dynamics is presented by a linear fourth order partial differential equation (PDE). Linear boundary control laws are constructed to stabilize the composite plates asymptotically. The control forces and moments consist of feedbacks of the velocity and normal derivative of the velocity at the boundaries of the plate. Asymptotic stabilization of free vibration of composite plates in both cases (i.e. with and without boundary object) is achieved via boundary actions. Finally, it is shown that exponentially stability of the hybrid system cannot be fulfilled (in the other words the asymptotical stability is the best). Our main tools in this article are semigroup techniques, spectral theory of the linear operators and Lyapunov stability method.

Boundary Observer for Output-Feedback Stabilization of Thermal-Fluid Convection Loop

In this paper, we consider a 2-D model of thermal fluid convection that exhibits the prototypical Rayleigh-Bernard convective instability. The fluid is enclosed between two cylinders, heated from above, and cooled from below, which makes its motion unstable for a large enough Rayleigh number. We design an stabilizing output feedback boundary control law for a realistic collocated setup, with actuation and measurements located at the outer boundary. Actuation is through rotation (direct velocity actuation) and heat flux (heating or cooling) of the outer cylinder, while measurements of friction and temperature are obtained at the same boundary. Though only a linearized version of the plant is considered in the design, an extensive closed loop simulation study of the nonlinear model shows that our design works for reasonably large initial conditions. A highly accurate approximation to the control kernels and observer output injection gains is found in closed form.

Stabilization for a class of complex interlaced systems using asymptotical gain

AbstractThis paper addresses the stabilization problem of a class of interlaced systems that are not in a strict‐feedforward form and contain some severe nonlinear terms. Bounded control laws in a fractional form are explicitly constructed. The feature of stability analysis allows the closed‐loop system, when it is proven to have no finite escape time, to employ the asymptotical gain, which is obtained from an input‐to‐state stable (ISS) subsystem, to calculate the severe nonlinear terms, and the related estimate in turn guides the assignment of small controls. Together with the use of the passivity theory and the ISS‐based stability criterion, the proposed method requires no small gain analysis although the asymptotical gain is used, and differs from the state‐dependent saturation scheme since the controls do not include saturation functions. As an application, a new stabilizing control law is presented for the well‐known friction ball‐and‐beam system. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

G1000-5-3 Active Boundary Controlを基調とした閉空間場における静粛場生成(機械力学・計測制御部門一般講演(5):音響・波動,社会変革を技術で廻す機械工学)

This paper presents an acitve boundary control (ABC) method of a three dimensional rectangular enclosure and to clarify fundamental properties of the control system. Firstly, a transfer matrix method for the target system is introduced to describe the dynamics in the enclosure. This is followed by the derivation of an active boundary control laws and its comparison with active wave control laws. Then, from a viewpoint of numerical analyses, basic properties of the proposed method is verified. It is found that the proposed method enables the generation of a silent zone

Exponential Stabilization of Transverse Vibration and Trajectory Tracking for General In-Plane Motion of an Euler–Bernoulli Beam Via Two-Time Scale and Boundary Control Methods

In this paper, exponential stabilization of vibration of a flexible beam together with its general in-plane trajectory tracking is presented. Coupled beam dynamics including beam vibration (flexible/fast subsystem) and its rigid in-plane motion (rigid/slow subsystem) takes place in two different time domains. Therefore, to have the beam track a desired trajectory while suppressing its vibration by an exponential rate of decay, a composite control scheme is elaborated by two-time scale (TTS) control theory. This control law has two parts: one is a tracking controller designed for the rigid subsystem based on inverse dynamic law, and the other one is an exponential stabilizing controller for the flexible subsystem based on boundary control (BC) laws. Exponential stabilization is proved by using a metric containing kinetic and potential energies of the fast subsystem and by feedback of the rate of deflection and the slope at one end of the beam. Simulation results show that fast BC is able to remove undesirable vibration of the flexible beam and together with the slow inverse controller is able to provide very good trajectory tracking with acceptable actuating forces/moments. Also, they illustrate that tracking errors and the vibration amplitude are decreased versus time by the fast exponential stabilizing control law compared with an asymptotic stabilizing control law.

Boundary control of container cranes from the perspective of controlling an axially moving string system

The control objectives in this paper are to move the load of a container crane to its target position and to suppress the transverse vibration of the load. Owing to the fact that the load is hoisted up and down, the crane is modeled as an axially moving string system. The dynamics of the moving string are derived using Hamilton’s principle for systems with changing mass. The Lyapunov function method is used in deriving a boundary control law, where the Lyapunov function candidate takes the form of the total mechanical energy of the system. The boundary control law utilizes the hoisting speed as well as the sway angle of the rope at the gantry side. Through experiment, the effectiveness of the proposed control law is demonstrated particularly in the lift-up process of the load.

Robust stabilization the Korteweg–de Vries–Burgers equation by boundary control

The problem of robust global stabilization by nonlinear boundary feedback control for the Korteweg–de Vries–Burgers equation on the domain [0,1] is considered. The main purpose of this paper is to derive nonlinear robust boundary control laws which make the system robustly globally asymptotically stable in spite of uncertainty in the system parameters. Furthermore, we show that the proposed boundary controllers guarantee L 2-robust exponential stability, L ∞-robust asymptotic stability and boundedness in terms of both L 2 and L ∞.