Boundary Actuation Research Articles

Effect of interfilament hydrodynamic interaction on swimming performance of two-filament microswimmers.

The motion of two-filament artificial swimmers is modeled by assuming interfilament coupling via hydrodynamic viscous drag. The filaments are assumed to be in parallel and attached to a rigid spherical head. The boundary actuation is assumed to occur at the head-filament joint through an external oscillatory magnetic field and the filament motion is taken to be confined to the flexural plane. The hydrodynamic coupling modifies the viscous drag on one filament due to motion of the other. Assuming in-phase, small amplitude, low frequency actuation the swimmer performance metrics (propulsive thrust, propulsion speed and energy efficiency) are calculated using Lauga's formulation for the swimmer kinematics coupled with filament dynamics. The results are compared with the performance of a single-filament and an uncoupled two-filament swimmer. The hydrodynamic coupling is found to enhance the performance measures in a parametric window. Also, it is found that there occurs an optimum combination of head size and swimmer length that can maximize the microswimmer performance. The findings are in agreement with the experimental observations on multi-filament artificial microswimming.

Stabilization of infinite dimensional port-Hamiltonian systems by nonlinear dynamic boundary control

The conditions for existence of solutions and stability, asymptotic and exponential, of a large class of boundary controlled systems on a 1D spatial domain subject to nonlinear dynamic boundary actuation are given. The consideration of such class of control systems is motivated by the use of actuators and sensors with nonlinear behavior in many engineering applications. These nonlinearities are usually associated to large deformations or the use of smart materials such as piezo actuators and memory shape alloys. Including them in the controller model results in passive dynamic controllers with nonlinear potential energy function and/or nonlinear damping forces. First it is shown that under very natural assumptions the solutions of the partial differential equation with the nonlinear dynamic boundary conditions exist globally. Secondly, when energy dissipation is present in the controller, then it globally asymptotically stabilizes the partial differential equation. Finally, it is shown that assuming some additional conditions on the interconnection and on the passivity properties of the controller (consistent with physical applications) global exponential stability of the closed-loop system is achieved.

UDE-based Robust Boundary Control of Heat Equation with Unknown Input Disturbance

In this paper, a novel robust control strategy named uncertainty and disturbance estimator (UDE) is applied to the stabilization of an unstable heat equation with Dirichlet type boundary actuator and unknown time-varying input disturbance. The system is stabilized by the backstepping approach and the unknown input disturbance is compensated by the UDE-based method which constructs an estimation of the disturbance through filtering the system input U(t) and boundary state u(1, t). Compared to other existing disturbance compensation methods, the UDE-based method only requires the spectrum information of the disturbance signal. Furthermore, the output feedback version of the proposed control is also derived for the practical implementation purpose. Stability analysis of the closed-loop system for both state feedback and output feedback cases are carried out and simulation examples are also provided to verify the proposed method.

Optimal continuous-time state estimation for linear finite and infinite-dimensional chemical process systems with state constraints

This work addresses optimal constrained state estimation problem for finite and infinite-dimensional chemical process systems. We consider cases when the prior information, in addition to the model parameters and the measurements, is available in the form of an inequality constraint with respect to the system's state. In the latest developments of the optimal state estimation theory, considerations of the state constraints have been often neglected since constraints do not fit easily in the structure of the optimal state estimator. Therefore, the issue of the state constraints being present needs to be addressed adequately, in particular, nonnegativity of concentration. Motivated by this, we developed a sequential, algorithmic optimal constrained state estimator for both finite and infinite-dimensional process systems commonly found in chemical process engineering (CSTR, tubular reactor). In this paper, we also designed an optimal constrained state estimator for a large class of dissipative infinite-dimensional systems which involve boundary actuation and point observation. Finally, illustrative examples of chemical process systems and proposed optimal state constrained estimation are presented.

Optimal boundary control for water hammer suppression in fluid transmission pipelines

When fluid flow in a pipeline is suddenly halted, a pressure surge or wave is created within the pipeline. This phenomenon, called water hammer, can cause major damage to pipelines, including pipeline ruptures. In this paper, we model the problem of mitigating water hammer during valve closure by an optimal boundary control problem involving a nonlinear hyperbolic PDE system that describes the fluid flow along the pipeline. The control variable in this system represents the valve boundary actuation implemented at the pipeline terminus. To solve the boundary control problem, we first use the method of lines to obtain a finite-dimensional ODE model based on the original PDE system. Then, for the boundary control design, we apply the control parameterization method to obtain an approximate optimal parameter selection problem that can be solved using nonlinear optimization techniques such as Sequential Quadratic Programming (SQP). We conclude the paper with simulation results demonstrating the capability of optimal boundary control to significantly reduce flow fluctuation.

Backstepping output-feedback control of moving boundary parabolic PDEs

This paper extends the backstepping-based observer design to the state estimation of parabolic PDEs with time-dependent spatial domain. The design is developed for the stabilization of a collocated boundary measurement and actuation of an unstable 1D heat equation with the application to the temperature distribution regulation in Czochralski crystal growth process. The PDE system that describes the estimation error dynamics is transformed to an exponentially stable target system through invertible transformations to obtain the time-varying kernel PDE defined on the time-varying triangular-shape domain. The exponential stability of the closed-loop system with an observer-based output-feedback controller is established by the use of a Lyapunov function. Finally, numerical solutions to the kernel PDEs and simulations are given to demonstrate successful stabilization of the unstable system.

Low‐order optimal regulation of parabolic PDEs with time‐dependent domain

Observer and optimal boundary control design for the objective of output tracking of a linear distributed parameter system given by a two‐dimensional (2‐D) parabolic partial differential equation with time‐varying domain is realized in this work. The transformation of boundary actuation to distributed control setting allows to represent the system's model in a standard evolutionary form. By exploring dynamical model evolution and generating data, a set of time‐varying empirical eigenfunctions that capture the dominant dynamics of the distributed system is found. This basis is used in Galerkin's method to accurately represent the distributed system as a finite‐dimensional plant in terms of a linear time‐varying system. This reduced‐order model enables synthesis of a linear optimal output tracking controller, as well as design of a state observer. Finally, numerical results are prepared for the optimal output tracking of a 2‐D model of the temperature distribution in Czochralski crystal growth process which has nontrivial geometry. © 2014 American Institute of Chemical Engineers AIChE J, 61: 494–502, 2015

Fuzzy Boundary Control Design for a Class of Nonlinear Parabolic Distributed Parameter Systems

This paper deals with the problem of fuzzy boundary control design for a class of nonlinear distributed parameter systems which are described by semilinear parabolic partial differential equations (PDEs). Both distributed measurement form and collocated boundary measurement form are considered. A Takagi–Sugeno (T–S) fuzzy PDE model is first applied to accurately represent the semilinear parabolic PDE system. Based on the T–S fuzzy PDE model, two types of fuzzy boundary controllers, which are easily implemented since only boundary actuators are used, are proposed to ensure the exponential stability of the resulting closed-loop system. Sufficient conditions of exponential stabilization are established by employing the Lyapunov direct method and the vector-valued Wirtinger's inequality and presented in terms of standard linear matrix inequalities. Finally, the advantages and effectiveness of the proposed control methodology are demonstrated by the simulation results of two examples.

On the damping effect of gas rarefaction on propagation of acoustic waves in a microchannel

We consider the response of a gas in a microchannel to instantaneous (small-amplitude) non-periodic motion of its boundaries in the normal direction. The problem is formulated for an ideal monatomic gas using the Bhatnagar, Gross, and Krook (BGK) kinetic model, and solved for the entire range of Knudsen (Kn) numbers. Analysis combines analytical (collisionless and continuum-limit) solutions with numerical (low-variance Monte Carlo and linearized BGK) calculations. Gas flow, driven by motion of the boundaries, consists of a sequence of propagating and reflected pressure waves, decaying in time towards a final equilibrium state. Gas rarefaction is shown to have a “damping effect” on equilibration process, with the time required for equilibrium shortening with increasing Kn. Oscillations in hydrodynamic quantities, characterizing gas response in the continuum limit, vanish in collisionless conditions. The effect of having two moving boundaries, compared to only one considered in previous studies of time-periodic systems, is investigated. Comparison between analytical and numerical solutions indicates that the collisionless description predicts the system behavior exceptionally well for all systems of the size of the mean free path and somewhat larger, in cases where boundary actuation acts along times shorter than the ballistic time scale. The continuum-limit solution, however, should be considered with care at early times near the location of acoustic wavefronts, where relatively sharp flow-field variations result in effective increase in the value of local Knudsen number.

Single phase boundary actuation of a ferromagnetic shape memory foil

The novel mechanism of temperature-gradient-induced single phase boundary actuation is presented for a single crystalline ferromagnetic shape memory alloy (FSMA) foil. It is shown that applying a temperature gradient along the FSMA foil specimen results in the formation and propagation of a martensite–austenite phase boundary from the hot to the cold side, allowing for reproducible strain–temperature characteristics. The selection of martensite variants upon phase transformation is controlled by simultaneously applying a bias magnetic field, which determines the maximum strain response. Single phase boundary actuation is demonstrated for a Ni–Mn–Ga foil of 100μm thickness with 10M martensite structure at room temperature. A small temperature gradient of 5Kmm–1 and a bias field along the temperature gradient of 120mT are sufficient to achieve the maximum possible strain of 4.1%, corresponding to the length difference of the short c-axis of tetragonal martensite and the axis of cubic austenite. For a bias magnetic field in the perpendicular direction, the maximum strain change is −1.9%.

Travelling waves in boundary-controlled, non-uniform, cascaded lumped systems

A companion paper considers travelling and standing waves in cascaded, lumped, mass-spring systems, controlled by two boundary actuators, one at each end, when the system is uniform. It first proposes definitions of waves in finite lumped systems. It then shows how to control the actuators to establish desired waves from rest, and to maintain them despite disturbances. The present paper extends this work to the more general, non-uniform case, when mass and spring values can be arbitrary. A special “bi-uniform” case is first studied, consisting of two different uniform cascaded systems in series, with an obvious, uncontrolled, impedance mismatch where they meet. The paper shows how boundary actuator control systems can be designed to establish, and robustly maintain, apparently pure travelling waves of constant amplitude in either the first or the second uniform section, in each case with an appropriate, partial, standing wave pattern in the other section. Then a more general non-uniform case is studied. A definition of a “pure travelling wave” in non-uniform systems is proposed. Curiously, it does not imply constant amplitude motion. It does however yield maximum power transfer between boundary actuators. The definition, and its implementation in a control system, involves extending the notions of “pure” travelling waves, of standing waves, and of input and output impedances of sources and loads, when applied to non-uniform lumped systems. Practical, robust control strategies are presented for all cases.

Boundary-controlled travelling and standing waves in cascaded lumped systems

This paper shows how pure travelling waves in cascaded, lumped, uniform, mass–spring systems can be defined, established, and maintained, by controlling two boundary actuators, one at each end. In most cases the control system for each actuator requires identifying and measuring the notional component waves propagating in opposite directions at the actuator–system interfaces. These measured component waves are then used to form the control inputs to the actuators. The paper also shows how the boundaries can be actively controlled to establish and maintain standing waves of arbitrary standing wave ratio, including those corresponding to the classical modes of vibration of such systems with textbook boundary conditions. These vibration modes are achieved and maintained by controlled reflection of the pure travelling wave components. The proposed control systems are also robust to system disturbances: they react to overcome external disturbances quickly and so to re-establish the desired steady motion.

LQ-boundary control of a diffusion-convection-reaction system

In this work, the boundary control of a distributed parameter system modelled by linear parabolic partial differential equations (PDEs) with spatially varying coefficients is studied. An infinite-dimensional state space setting is considered and an exact transformation of the boundary actuation is realised to obtain an evolutionary model. The evolutionary model which incorporates the spatially varying coefficients of the underlying set of the PDEs is used for subsequent linear quadratic regulator synthesis. The formulated linear quadratic-state feedback controller is applied to a nonlinear model of the reactor and its performance is studied.

Cargo‐Towing Fuel‐Free Magnetic Nanoswimmers for Targeted Drug Delivery

Fuel-free nanomotors are essential for future in-vivo biomedical transport and drug-delivery applications. Herein, the first example of directed delivery of drug-loaded magnetic polymeric particles using magnetically driven flexible nanoswimmers is described. It is demonstrated that flexible magnetic nickel-silver nanoswimmers (5-6 μm in length and 200 nm in diameter) are able to transport micrometer particles at high speeds of more than 10 μm s(-1) (more than 0.2 body lengths per revolution in dimensionless speed). The fundamental mechanism of the cargo-towing ability of these magnetic (fuel-free) nanowire motors is modelled, and the hydrodynamic features of these cargo-loaded motors discussed. The effect of the cargo size on swimming performance is evaluated experimentally and compared to a theoretical model, emphasizing the interplay between hydrodynamic drag forces and boundary actuation. The latter leads to an unusual increase of the propulsion speed at an intermediate particle size. Potential applications of these cargo-towing nanoswimmers are demonstrated by using the directed delivery of drug-loaded microparticles to HeLa cancer cells in biological media. Transport of the drug carriers through a microchannel from the pick-up zone to the release microwell is further illustrated. It is expected that magnetically driven nanoswimmers will provide a new approach for the rapid delivery of target-specific drug carriers to predetermined destinations.

Growth, geometry, and mechanics of a blooming lily

Despite the common use of the blooming metaphor, its floral inspiration remains poorly understood. Here we study the physical process of blooming in the asiatic lily Lilium casablanca. Our observations show that the edges of the petals wrinkle as the flower opens, suggesting that differential growth drives the deployment of these laminar shell-like structures. We use a combination of surgical manipulations and quantitative measurements to confirm this hypothesis and provide a simple theory for this change in the shape of a doubly curved thin elastic shell subject to differential growth across its planform. Our experiments and theory overturn previous hypotheses that suggest that blooming is driven by differential growth of the inner layer of the petals and in the midrib by providing a qualitatively different paradigm that highlights the role of edge growth. This functional morphology suggests new biomimetic designs for deployable structures using boundary or edge actuation rather than the usual bulk or surface actuation.

Optimal Tracking Control of Current Profile in Tokamaks

Setting up a suitable current spatial profile in tokamak plasmas has been demonstrated to be a key condition for one possible advanced scenario with improved confinement and possible steady-state operation. Experiments at the DIII-D tokamak focus on creating the desired current profile during the plasma current ramp-up and early flattop phases with the aim of maintaining this target profile during the subsequent phases of the discharge. The evolution in time of the current profile is related to the evolution of the poloidal magnetic flux, which is modeled in normalized cylindrical coordinates using a parabolic partial differential equation usually referred to as the magnetic diffusion equation. We propose a framework to solve a finite-time, optimal tracking control problem for the current profile evolution via diffusivity, interior, and boundary actuation during the ramp-up and early flattop phases of the discharge. The proposed approach is based on reduced order modeling via proper orthogonal decomposition and successive optimal control computation for a bilinear system. Simulation results illustrate the performance of the proposed controller.

Boundary Controllers and Observers for the Linearized Schrödinger Equation

We consider a problem of stabilization of the linearized Schrodinger equation using boundary actuation and measurements. We propose two different control designs. First, a simple proportional collocated boundary controller is shown to exponentially stabilize the system. How- ever, the decay rate of the closed-loop system cannot be prescribed. The second, full-state feedback boundary control design, achieves an arbitrary decay rate. We formally view the Schrodinger equa- tion as a heat equation in complex variables and apply the backstepping method recently developed for boundary control of reaction-advection-diffusion equations. The resulting controller is then sup- plied with the backstepping observer to obtain an output-feedback compensator. The designs are illustrated with simulations.

Boundary Actuation in Galerkin-Models of Linear Distributed Parameter Systems

AbstractProper orthogonal decomposition and subsequent Galerkin-projection is a common technique to obtain low-dimensional models for distributed parameter systems. In this paper a transformation of boundary conditions into equivalent source terms for linear partial differential equations is used to set up such reduced-order models. This approach provides the means to consider boundary actuation in Galerkin-models, which is essential for the design of feedback controllers. Using the example of a one-dimensional convection-diffusion equation, the transient dynamics between the non-actuated and actuated steady state are described with such a reduced-order-model. Dynamic range and model accuracy are investigated in connection with the employed number of POD modes and the dynamics of actuation.

Linear quadratic optimal boundary control of a diffusion-convection-reaction system

AbstractIn this work, the boundary control of a distributed parameter system (DPS) modeled by parabolic partial differential equations with spatially varying coefficients is studied. An infinite dimensional state space setting is formulated and an exact transformation of the boundary actuation is realized to obtain an evolutionary model. The evolutionary model is used for subsequent linear quadratic regulator synthesis which incorporates the spatially varying coefficients of the underlying set of the PDEs. The formulated LQR controller is applied to the nonlinear model of the system and its performance is studied.

Receding-horizon optimal control of the current profile evolution during the ramp-up phase of a tokamak discharge

The control of the toroidal current density spatial profile in tokamak plasmas will be absolutely critical in future commercial-grade reactors to enable high fusion gain, non-inductive sustainment of the plasma current for steady-state operation, and magnetohydrodynamic (MHD) instability-free performance. The evolution in time of the current profile is related to the evolution of the poloidal magnetic flux, which is modeled in normalized cylindrical coordinates using a partial differential equation (PDE) usually referred to as the magnetic flux diffusion equation. The control objective during the ramp-up phase is to drive an arbitrary initial profile to approximately match, in a short time windows during the early flattop phase, a predefined target profile that will be maintained during the subsequent phases of the discharge. Thus, such a matching problem can be treated as an optimal control problem for a PDE system. A distinctive characteristic of the current profile control problem in tokamaks is that it admits interior, boundary and diffusivity actuation. A receding-horizon control scheme is proposed in this work to exploit this unique characteristic and to solve the associated open-loop finite-time optimal control problem using different optimization techniques. The efficiency of the proposed scheme is shown in simulations.