Uncontrolled Boundary Research Articles

Responsiveness and Precision of Digital IMUs under Linear and Curvilinear Motion Conditions for Local Navigation and Positioning in Advanced Smart Mobility.

Sensors based on MEMS technology, in particular Inertial Measurement Units (IMUs), when installed on vehicles, provide a real-time full estimation of vehicles' state vector (e.g., position, velocity, yaw angle, angular rate, acceleration), which is required for the planning and control of cars' trajectories, as well as managing the in-car local navigation and positioning tasks. Moreover, data provided by the IMUs, integrated with the data of multiple inputs from other sensing systems (such as Lidar, cameras, and GPS) within the vehicle, and with the surrounding information exchanged in real time (vehicle to vehicle, vehicle to infrastructure, or vehicle to other entities), can be exploited to actualize the full implementation of "smart mobility" on a large scale. On the other hand, "smart mobility" (which is expected to improve road safety, reduce traffic congestion and environmental burden, and enhance the sustainability of mobility as a whole), to be safe and functional on a large scale, should be supported by highly accurate and trustworthy technologies based on precise and reliable sensors and systems. It is known that the accuracy and precision of data supplied by appropriately in-lab-calibrated IMUs (with respect to the primary or secondary standard in order to provide traceability to the International System of Units) allow guaranteeing high quality, reliable information managed by processing systems, since they are reproducible, repeatable, and traceable. In this work, the effective responsiveness and the related precision of digital IMUs, under sinusoidal linear and curvilinear motion conditions at 5 Hz, 10 Hz, and 20 Hz, are investigated on the basis of metrological approaches in laboratory standard conditions only. As a first step, in-lab calibrations allow one to reduce the variables of uncontrolled boundary conditions (e.g., occurring in vehicles in on-site tests) in order to identify the IMUs' sensitivity in a stable and reproducible environment. For this purpose, a new calibration system, based on an oscillating rotating table was developed to reproduce the dynamic conditions of use in the field, and the results are compared with calibration data obtained on linear calibration benches.

Adaptive stabilization for a class of wave equations with uncertain nonlinear boundary actuator dynamics

This paper investigates the stabilization of a wave equation with uncertain boundary actuator dynamics. The remarkable characters of the system under investigation are revealed by the essential nonlinearities involved in the boundary actuator and the serious parametric unknowns coming from anti-damping coefficient at uncontrolled boundary and those in the actuator, which result into the incapability of the traditional control methods on this topic. For this, a novel adaptive control strategy is proposed in this paper. Specifically, a Riemann variables based state transformation is first introduced to change the original system into a new one whose stabilization implies that of the original system. Then, a state-feedback controller joint with some adaptive laws which are smartly chosen for the compensation of unknown parameters is explicitly designed by a finite-dimensional backstepping procedure. Owing to the compensation of boundary actuator dynamics, the performance analysis of the closed-loop system is more complicated than that of the related literature. For this, a rigorous recursive step is given to show the stability of the resulting closed-loop system. Finally, a numerical example is provided to verify the effectiveness of the proposed theoretical results.

Rapid Stabilization of Timoshenko Beam by PDE Backstepping

In this paper, we present rapid boundary stabilization of a Timoshenko beam with anti-damping and anti-stiffness at the uncontrolled boundary, by using infinite-dimensional backstepping. We introduce a Riemann transformation to map the Timoshenko beam states into a set of coordinates that verify a 1-D hyperbolic PIDE-ODE system. Then backstepping is applied to obtain a control law guaranteeing closed-loop stability of the origin in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^{2}$</tex-math></inline-formula> sense. Arbitrarily rapid stabilization can be achieved by adjusting control parameters, and has not been achieved in previous results. Finally, a numerical simulation shows the effectiveness of the proposed controller. This result extends a previous work which considered a slender Timoshenko beam with Kelvin-Voigt damping, by allowing destabilizing boundary conditions at the uncontrolled boundary and attaining an arbitrarily rapid convergence rate.

Adaptive Control of Hyperbolic PDEs Coupled With a Disturbed and Highly Uncertain ODE

In this article, we address adaptive output-feedback boundary control of coupled hyperbolic partial differential equations (PDEs) with spatially varying coefficients and on a time-varying domain, whose uncontrolled boundary is coupled with a disturbed ordinary differential equation (ODE), where multiple parameters in the state matrix and the amplitudes of the harmonic disturbance are unknown. The asymptotic convergence to zero of the ODE state and the boundedness of the PDE states are ensured. This article is motivated by lateral vibration suppression of a mining cable elevator, where the interaction dynamics between the cage and the flexible guide is approximated as a viscoelastic system, including spring and damping, with unknown stiffness and damping coefficients. The performance of the proposed controller is tested in the application of the mining cable elevator by numerical simulation.

Velocity Stabilization of a Wave Equation With a Nonlinear Dynamic Boundary Condition

This paper deals with a one-dimensional wave equation with a nonlinear dynamic boundary condition and a Neumann-type boundary control acting on the other extremity. We consider a class of nonlinear stabilizing feedbacks that only depend on the velocity at the controlled extremity. The uncontrolled boundary is subject to a nonlinear first-order term, which may represent nonlinear boundary anti-damping. Initial data is taken in the optimal energy space associated with the problem. Exponential decay of the mechanical energy is investigated in different cases. Stability and attractivity of suitable invariant sets are established.

Backstepping-Based Exponential Stabilization of Timoshenko Beam with Prescribed Decay Rate

In this paper, we present a rapid boundary stabilization of a Timoshenko beam with anti-damping and anti-stiffness at the uncontrolled boundary, by using PDE backstepping. We introduce a transformation to map the Timoshenko beam states into a (2+2) × (2+2) hyperbolic PIDE-ODE system. Then backstepping is applied to obtain a control law guaranteeing closed-loop stability of the origin in the H1 sense. Arbitrarily rapid stabilization can be achieved by adjusting control parameters. Finally, a numerical simulation shows that the proposed controller can rapidly stabilize the Timoshenko beam. This result extends a previous work which considered a slender Timoshenko beam with Kelvin-Voigt damping, allowing destabilizing boundary conditions at the uncontrolled boundary and attaining an arbitrarily rapid convergence rate.

Boundary Feedback Control of 2×2 Quasilinear Hyperbolic Systems: Predictive Synthesis and Robustness Analysis

We present a predictive feedback control method for a class of quasilinear hyperbolic systems with one boundary control input. Assuming exact model knowledge, convergence to the origin, or tracking at the uncontrolled boundary, are achieved in finite time. A robustness certificate is provided, showing that at least under more restrictive assumptions on the system coefficients, the control method has inherent robustness properties with respect to small errors in the model, measurements and control input. Rigorous, although conservative conditions on the time derivative of the initial condition and on the design parameter controlling the convergence speed are given to ensure global existence of the solution for initial conditions with arbitrary infinity-norm.

Adaptive event-triggered PDE control for load-moving cable systems

Motivated by lateral vibration suppression of a mining cable elevator, which is a load-moving cable system, where the load moves along a viscoelastic guideway whose stiffness and damping coefficients are unknown, we present event-triggered adaptive output-feedback boundary control design of a hyperbolic PDE–ODE coupled system using the measurements at the PDE actuated boundary and the ODE, where the PDE subsystem is a class of 2 × 2 coupled hyperbolic PDEs with spatially-varying coefficients and on a time-varying domain, and a high uncertainty exist in the system matrix of the ODE subsystem at the uncontrolled boundary of the PDE. A continuous-in-time observer-based adaptive backstepping control law is designed where the control gains can be self-tuned to adjust the system matrix of the ODE into a given target system matrix, based on which an observer-based dynamic event-triggering mechanism is built and the existence of a minimal dwell-time is proved. The asymptotic stability of the overall adaptive event-based output-feedback closed-loop system is proved via Lyapunov analysis. In numerical simulation, the performance of the proposed controller is verified in lateral vibration suppression of a mining cable elevator.

Backstepping control of ODE/wave PDE cascaded systems with dynamic boundary conditions

We consider backstepping control for ODE/wave PDE cascaded systems with dynamic boundary conditions. The ODE state is driven by the uncontrolled boundary of the wave PDE. A backstepping compensator is designed such that the closed-loop system is globally asymptotically stable. The controller design and stability analysis are based on coordinate and backstepping transformations. Changes of coordinate recast the original system into a coupled first-order hyperbolic system with spatially-varying coefficients cascading into an ODE. Backstepping transformations are exploited to design a compensator for the resulting coupled system. The validity of the proposed design is illustrated by controlling a bit of the oil well drilling system to operate at a constant uniform rotary speed.

Boundary Control of Nonlinear ODE/Wave PDE Systems With a Spatially Varying Propagation Speed

We consider the boundary control of a nonlinear ordinary differential equation (ODE) actuated through a wave equation whose propagation speed is spatially varying. The ODE state is driven by the uncontrolled boundary of the wave equation. We design a nonlinear backstepping compensator to enable global asymptotic stability of the closed-loop system. We deduce the controller design and the stability proof by introducing a two-step backstepping transformation. The first transformation recasts the original system into a coupled 2×2 first-order hyperbolic system with spatially varying coefficients cascading into a nonlinear ODE. The second transformation is used in the design of a compensator for the resulting cascaded system. Our design offers a global stability result that is guaranteed assuming that the spatially varying propagation speed is continuously differentiable and positive. Moreover, for nonlinear systems, our result is the first contribution enabling actual compensation of actuator delays governed by a coupled first-order hyperbolic partial differential equation (PDE) induced by a wave PDE dynamics with a spatially varying propagation speed. The validity of the proposed controller is illustrated by the benchmark system controlled via a cable.

Flatness based trajectory planning and open-loop control of shallow-water waves in a tube

Open-loop control design for a fluid in a pipe is considered. The position (resp. the velocity) of a piston moving within the tube acts as control input. The system dynamics are described by the so-called Saint-Venant equations with a moving boundary condition. Rest-to-rest transitions are parameterized by means of the so called flatness-based approach to the control of dynamical systems the flat output being the water level at the uncontrolled boundary. The control algorithms are implemented on a test bench and verified in numerical and experimental studies.

Output feedback boundary control of heterodirectional semilinear hyperbolic systems

We solve the problem of stabilizing a general class of 1-d semilinear hyperbolic systems with an arbitrary number of states convecting in each direction and with the actuation and sensing restricted to one boundary. The control design is based on the dynamics on the characteristic lines along which the inputs propagate through the domain and the predictability of states in the interior of the domain up to the time they are affected by the inputs. In the context of broad solutions, the state-feedback controller drives systems with globally Lipschitz nonlinearities from an arbitrary initial condition to the origin in minimum time. Alternatively, it is possible to satisfy a tracking objective at the uncontrolled boundary or, for systems with C1-coefficients and initial conditions, to design the control inputs to obtain classical C1-solutions that also reach the origin in finite time. Further, we design an observer that estimates the distributed state from boundary measurements only. The observer combined with the state-feedback controller solves the output-feedback control problem.

Predicting droplet velocity in a Hele-Shaw cell

We study the motion of low viscous non-wetting droplet in a Hele-Shaw cell, while it is pushed by an external phase of imposed flow rate, at low capillary numbers. In this regime, the droplet's mobility, defined as the ratio between the droplet velocity and the external phase mean velocity, evolves non linearly with the capillary number, a signature of the different dissipation mechanisms at play. Experiments are performed with surfactant free air bubbles in fluorinated oil, and with surfactant laden fluorinated oil droplets in water. We propose a model based on a power balance which takes into account the dissipation in the thin wetting film trapped between the bubble (or the drop) and the channel wall. The full topography of this thin film is obtained theoretically for the bubble case. By contrast, the presence of surfactants in the drop case induces uncontrolled boundary conditions at the interface, thus imposing to use the experimental topography measured in the previous paper [Reichert et al., J. Fluid. Mech., 850 p.708 (2018)]. Remarkably, the model reproduces the experimental velocities and shows that the velocity can be strongly affected by a stagnant cap effect at the rear of the drop, even if localized in less than a few percents of the total film area.

Adaptive output-feedback control of torsional vibration in off-shore rotary oil drilling systems

Motivated by an engineering application of torsional vibration suppression of off-shore oil drilling, we design an adaptive output-feedback controller for a one-dimensional wave partial differential equation (PDE) system, where an anti-damping term with an unknown coefficient and a general harmonic disturbance with unknown amplitudes exist in the bit, which is modeled as a second-order-in-time boundary. The control input anti-collocated with this boundary subject to uncertainty, is designed by using the adaptive control method and infinite-dimensional backstepping technique. The asymptotic convergence to zero of the uncontrolled boundary states, i.e., the oscillations of the angular displacement and velocity at the bit, and the boundedness of all states in the closed-loop system, are proved via Lyapunov analysis. The effectiveness of the proposed adaptive controller is verified via numerical simulation. The results also can be applied to other applications, such as vibration control of cable elevators with uncertain cage-guide friction and cage disturbances.

Characterization of giant magnetostrictive materials under static stress: influence of loading boundary conditions

Giant magnetostrictive materials (GMM) can be integrated in actuator or sensor applications. The design of these systems is optimized based on a good knowledge of the material properties and conditions of use. Terfenol-D exhibits the greatest room temperature strain among commercially available GMM, however, its magneto-elastic behavior is very sensitive to pre-stress level. In this work, the design of an experimental setup dedicated to the characterization of GMM magneto-mechanical behavior under constant stress is described. A major difficulty is to master the mechanical boundary conditions while the sample is subjected to dynamic magnetic loading. The dynamic stress experienced by the sample is connected to the magnitude of the magnetostriction strain, the stiffness of the sample and the stiffness of the characterization setup. Results show that an appropriate setup is able to reduce the dynamic stress variations induced by magnetic excitation variations below 0.1 MPa, while this dynamic stress can reach up to 20 times the magnitude of the applied stress when the control system is not used. With the boundary conditions being controlled, magnetic and magnetostrictive behavior of Terfenol-D are characterized under various uniaxial compressive stress levels, from the stress-free conditions to 90 MPa. By comparing the results obtained under controlled and non-controlled stress conditions, it is shown that uncontrolled boundary conditions can be responsible for errors of several percent on the magnetic induction measurement. The measurement of strain is even more sensitive to the boundary conditions, with errors up to 40% and 30% on the longitudinal and transverse strains, respectively. This work highlights the utmost importance to control the boundary conditions in order to characterize the magneto-mechanical behavior of GMM.

Flatness-based control design for the Saint-Venant equations with experimental results

Control design for a fluid in a pipe with arbitrary cross-sectional area is considered. The position (resp. the velocity) of a piston moving within the tube acts as control input. The system dynamics are described by the so-called Saint-Venant equations defined on a varying spatial domain. The flatness-based approach is utilized in order to parameterize rest-to-rest transitions of the system by means of the water level at the uncontrolled boundary. This way the contribution extends previous results to systems with moving boundary. The proposed control scheme is validated in experimental studies.

Exponential regulation of the anti-collocatedly disturbed cage in a wave PDE-modeled ascending cable elevator

In cable elevators, large axial vibrations appear when a cage subject to disturbance is lifted up via a compliant cable. The axial vibration dynamics can be described by a wave partial differential equation (PDE) on a time-varying spatial interval with an unknown boundary disturbance. In this paper, we design an output feedback controller actuating at the boundary anti-collocated with the disturbance to regulate the state on the uncontrolled boundary of the wave PDE based on the backstepping idea and the active disturbance rejection control (ADRC) approach. The control law uses the state and disturbance information recovered from the state observer and the disturbance estimator, respectively, which are constructed via limited boundary measurements. The exponential convergence of the state on the uncontrolled boundary and uniform boundedness of all states in the closed-loop system are proved by Lyapunov analysis. Effective vibration suppression in the cable elevator with the designed controller is verified via numerical simulation.

Output feedback control of general linear heterodirectional hyperbolic PDE-ODE systems with spatially-varying coefficients

ABSTRACTThis paper presents a backstepping solution for the output feedback control of general linear heterodirectional hyperbolic PDE-ODE systems with spatially varying coefficients. Thereby, the ODE is coupled to the PDE in-domain and at the uncontrolled boundary, whereas the ODE is coupled with the latter boundary. For the state feedback design, a two-step backstepping approach is developed, which yields the conventional kernel equations and additional decoupling equations of simple form. In order to implement the state feedback controller, the design of observers for the PDE-ODE systems in question is considered, whereby anti-collocated measurements are assumed. Exponential stability with a prescribed convergence rate is verified for the closed-system pointwise in space. The resulting compensator design is illustrated for a 4 × 4 heterodirectional hyperbolic system coupled with a third-order ODE modelling a dynamic boundary condition.

Comparison of measured and simulated performance of natural displacement ventilation systems for classrooms

Children spend the majority of their weekdays in classrooms that often have low indoor air quality and limited financial resources for the initial and running costs of mechanical ventilation systems. Designing effective natural ventilation (NV) systems in schools is difficult due to the intense use of the classroom spaces and the dependence of NV on building geometry and outdoor conditions. Building thermal and airflow simulation tools are fundamental to predict NV system performance in the design phase. These predictions of these tools must be validated (preferably with data from real buildings). This paper presents a set of detailed measurements of buoyancy driven natural DV systems of three classrooms located in two buildings in Lisbon (Portugal). The rooms are located in two educational buildings, a kindergarten and a university, and have different buoyancy driven natural DV systems (with and without chimneys). The experimental measurements are used to validate a three-node DV model implemented on the open-source thermal building simulation software EnergyPlus. The validation results show that the building thermal simulation model tested is able to predict bulk airflow rate with an average error of 16%. In addition, a good agreement is also obtained for the vertical temperature prediction: an average error of 4% corresponding (average deviation of 0.7°C). Analysis of the kindergarten rooms results revealed, that as expected, increasing chimney height from 1 to 4m has a significant positive impact in NV system performance. The performance of natural DV systems depends on the number of thermal plumes in the room. For the same sensible heat load, increasing the number of plumes lowers the average occupied zone air temperature and increases the bulk airflow rate. In light of the complexity of the cases tested, NV with uncontrolled boundary conditions, the results of the comparisons performed between measurements and simulations should contribute to increase confidence in the use of EnergyPlus to simulate buoyancy driven natural DV systems.

Global effect of local skin friction drag reduction in spatially developing turbulent boundary layer

A numerical investigation of two locally applied drag-reducing control schemes is carried out in the configuration of a spatially developing turbulent boundary layer (TBL). One control is designed to damp near-wall turbulence and the other induces constant mass flux in the wall-normal direction. Both control schemes yield similar local drag reduction rates within the control region. However, the flow development downstream of the control significantly differs: persistent drag reduction is found for the uniform blowing case, whereas drag increase is found for the turbulence damping case. In order to account for this difference, the formulation of a global drag reduction rate is suggested. It represents the reduction of the streamwise force exerted by the fluid on a plate of finite length. Furthermore, it is shown that the far-downstream development of the TBL after the control region can be described by a single quantity, namely a streamwise shift of the uncontrolled boundary layer, i.e. a changed virtual origin. Based on this result, a simple model is developed that allows the local drag reduction rate to be related to the global one without the need to conduct expensive simulations or measurements far downstream of the control region.