An infinite-dimensional Luenberger-like observer for vibrating membranes

International audienceThe main objective of this paper consists in studying the dynamic and observation of a wave equation [1] in a bounded domain in the plan. This work is inscribed in the field of control of systems governed by partial differential equations (PDE). We consider the wave equation system with Dirichlet boundary condition whose dynamic evolves in an infinite-dimensional Hilbert space. We assume that velocity is measured on some subdomain along the boundary. An infinite-dimensional exponentially convergent Luenberger-like observer is presented to estimate the system state: displacement and velocity on the whole domain. The main contribution of the work consists in building a reliable numerical simulator based on the finite element method (FEM). We examine the influence of the gain on the convergence rate of the observer

Feedback control on the velocity field and source term of a normal flow equation

open4openA. Alessandri; P. Bagnerini; M. Gaggero; A. RossiAlessandri, A.; Bagnerini, P.; Gaggero, M.; Rossi, A

Modeling and Estimation of Thermal Flows Based on Transport and Balance Equations

Heat transfer in counterflow heat exchangers is modeled by using transport and balance equations with the temperatures of cold fluid, hot fluid, and metal pipe as state variables distributed along the entire pipe length. Using such models, boundary value problems can be solved to estimate the temperatures over all the length by means of measurements taken only at the boundaries. Conditions for the stability of the estimation error given by the difference between the temperatures and their estimates are established by using a Lyapunov approach. Toward this end, a method to construct nonlinear Lyapunov functionals is addressed by relying on a polynomial diagonal structure. This stability analysis is extended in case of the presence of bounded modeling uncertainty. The theoretical findings are illustrated with numerical results, which show the effectiveness of the proposed approach

An observer-based attitude and nutation control and flexible dynamic analysis for the NASA Magnetospheric Multiscale Mission

Current research with the NASA Goddard Space Flight Center (GSFC) involves the dynamic modeling and control of the NASA Magnetospheric Multiscale (MMS) Mission, a. Solar-Terrestrial Probe mission to study Earth\u27s magnetosphere. Four observer-based attitude and nutrition controllers are designed and evaluated to determine the most effective feedback control system as it applies to MMS. Also, a dynamic analysis of each of the four identical satellites\u27 two Axial Double Probe (ADP) booms is performed to provide an understanding of flexible boom dynamics. The Finite Element method is used in evaluating boom modes of vibration for confirmation of NASA GSFC theoretical analysis and use in flexible model development. The dynamic transient and modal extraction technique are investigated for vibration analysis of constrained and unconstrained bodies. A fully flexible boom and rigid spacecraft model is also developed for vibrational analysis under steady-state rotation and thruster loads. Results indicate, however, the need for future research in numerical analysis of propagating systems through finite element methods and in the stability of the observer-based control system. Linear and nonlinear observers are developed through simulations to estimate satellite attitude and angular body rates without the use of rate sensors. Control systems are then developed assuming perfect state measurements. Euler angles are used to describe satellite attitude in this research. Finally, linear and nonlinear (Sliding Mode Control) techniques are implemented in conjunction with the nonlinear observers to complete the observer-based control system. The results of this research show that, of the methods analyzed, both the Extended Kalman Filter and Sliding Mode Observer implemented with Sliding Mode Control yield the most satisfactory performance. These observer-based control systems both meet NASA design requirements while reducing thruster control effort and reducing the effects of measurement noise and spacecraft uncertainties/disturbances. More simulations, however, are needed to verify performance of the proposed observer-based control system over all possible ranges of operation

Non-Linear Robust Observers For Systems With Non-Collocated Sensors And Actuators

Challenges in controlling highly nonlinear systems are not limited to the development of sophisticated control algorithms that are tolerant to significant modeling imprecision and external disturbances. Additional challenges stem from the implementation of the control algorithm such as the availability of the state variables needed for the computation of the control signals, and the adverse effects induced by non-collocated sensors and actuators. The present work investigates the adverse effects of non-collocated sensors and actuators on the phase characteristics of flexible structures and the ensuing implications on the performance of structural controllers. Two closed-loop systems are considered and their phase angle contours have been generated as functions of the normalized sensor location and the excitation frequency. These contours were instrumental in the development of remedial actions for rendering structural controllers immune to the detrimental effects of non-collocated sensors and actuators. Moreover, the current work has focused on providing experimental validation for the robust performances of a self-tuning observer and a sliding mode observer. The observers are designed based on the variable structure systems theory and the self-tuning fuzzy logic scheme. Their robustness and self-tuning characteristics allow one to use an imprecise model of the system and eliminate the need for the extensive tuning associated with a fixed rule-based expert fuzzy inference system. The first phase of the experimental work was conducted in a controlled environment on a flexible spherical robotic manipulator whose natural frequencies are configuration-dependent. Both controllers have yielded accurate estimates of the required state variables in spite of significant modeling imprecision. The observers were also tested under a completely uncontrolled environment, which involves a 16-ft boat operating in open-water under different sea states. Such an experimental work necessitates the development of a supervisory control algorithm to perform PTP tasks, prescribed throttle arm and steering tasks, surge speed and heading tracking tasks, or recovery maneuvers. This system has been implemented herein to perform prescribed throttle arm and steering control tasks based on estimated rather than measured state variables. These experiments served to validate the observers in a completely uncontrolled environment and proved their viability as reliable techniques for providing accurate estimates for the required state variables

Improving convergence in numerical analysis using observers - The wave-like equation case

International audienceWe propose an observer-based approach to circumvent the issue of unbounded approximation errors -- with respect to the length of the time window considered -- in the discretization of wave-like equations in bounded domains, which covers the cases of the wave equation per se and of linear elasticity as well as beam, plate and shell formulations, and so on. Namely, taking advantage of some measurements available on the system over time, we adopt a strategy inspired from sequential data assimilation and by which the discrete system is dynamically corrected using the discrepancy between the solution and the measurements. In addition to the classical cornerstones of numerical analysis made up by stability and consistency, we are thus led to incorporating a third crucial requirement pertaining to observability -- to be preserved through discretization. The latter property warrants exponential stability for the corrected dynamics, hence provides bounded approximation errors over time. Special care is needed to establish the required observability at the discrete level, in particular due to the fact that we focus on an original observer method adapted to measurements of the main variable, whereas measurements of the time-derivative -- admissible, of course, albeit less frequent in practical systems -- lead to a stability analysis in which existing results can be more directly applied. We also provide some detailed application examples with several such wave-like equations, and the corresponding numerical assessments illustrate the performance of our approach

Observation and control of a ball on a tilting

The ball and plate system is a nonlinear MIMO system that has interesting characteristics which are also present in aerospace and industrial systems, such as: instability, subactuation, nonlinearities such as friction, backlash, and delays in the measurements. In this work, the modeling of the system is based on the Lagrange approach. Then it is represented in the state-space form with plate accelerations as inputs to the system. These have a similar effect as applying torques. In addition, the use of an internal loop of the servo system is considered. From the obtained model, we proceed to carry out the analysis of controllability and observability resulting in that the system is globally weak observable and locally controllable in the operating range. Then, the Jacobi linearization is performed to use the linearized model in the design of linear controllers for stabilization. On the other hand, analyzing the internal dynamics of the ball and plate system turns out to be a non-minimum phase system, which makes it difficult to design the tracking control using the exact model. This is the reason why we proceed to make approximations. Using the approximate model, nonlinear controllers are designed for tracking using different approaches as: feedback linearization for tracking with and without integral action, backstepping and sliding mode. In addition, linear and nonlinear observers are designed to provide full state information to the controller. Simulation tests are performed comparing the different control and observation approaches. Moreover, the effect of the delay in the measurement is analyzed, where it is seen that the greater the frequency of the reference signal the more the error is increased. Then, adding the Smith predictor compensates the delay and reduces the tracking error. Finally, tests performed with the real system. The system was successfully controlled for stabilization and tracking using the designed controllers. However, it is noticed that the effect of the friction, the spring oscillation and other non-modeled characteristics significantly affect the performance of the control.Tesi

Robust state estimation for the control of flexible robotic manipulators

In this thesis, a novel robust estimation strategy for observing the system state variables of robotic manipulators with distributed flexibility is established. Motivation for the derived approach stems from the observation that lightweight, high speed, and large workspace robotic manipulators often suffer performance degradation because of inherent structural compliance. This flexibility often results in persistent residual vibration, which must be damped before useful work can resume. Inherent flexibility in robotic manipulators, then, increases cycle times and shortens the operational lives of the robots. Traditional compensation techniques, those which are commonly used for the control of rigid manipulators, can only approach a fraction of the open-loop system bandwidth without inducing significant excitation of the resonant dynamics. To improve the performance of these systems, the structural flexibility cannot simply be ignored, as it is when the links are significantly stiff and approximate rigid bodies. One thus needs a model to design a suitable compensator for the vibration, but any model developed to correct this problem will contain parametric error. And in the case of very lightly damped systems, like flexible robotic manipulators, this error can lead to instability of the control system for even small errors in system parameters. This work presents a systematic solution for the problem of robust state estimation for flexible manipulators in the presence of parametric modeling error. The solution includes: 1) a modeling strategy, 2) sensor selection and placement, and 3) a novel, multiple model estimator. Modeling of the FLASHMan flexible gantry manipulator is accomplished using a developed hybrid transfer matrix / assumed modes method (TMM/AMM) approach to determine an accurate low-order state space representation of the system dynamics. This model is utilized in a genetic algorithm optimization in determining the placement of MEMs accelerometers for robust estimation and observability of the system’s flexible state variables. The initial estimation method applied to the task of determining robust state estimates under conditions of parametric modeling error was of a sliding mode observer type. Evaluation of the method through analysis, simulations and experiments showed that the state estimates produced were inadequate. This led to the development of a novel, multiple model adaptive estimator. This estimator utilizes a bank of similarly designed sub-estimators and a selection algorithm to choose the true value from a given set of possible system parameter values as well as the correct state vector estimate. Simulation and experimental results are presented which demonstrate the applicability and effectiveness of the derived method for the task of state variable estimation for flexible robotic manipulators.Ph.D