Infinite-dimensional Observer Research Articles

Detectability and global observer design for 2D Navier–Stokes equations with uncertain inputs

We present simulation friendly detectability conditions for 2D Navier–Stokes Equation (NSE) with periodic boundary conditions, and describe a generic class of “detectable” observation operators: it includes pointwise evaluation of NSE’s solution at interpolation nodes, and spatial average measurements. For “detectable” observation operators we design a global infinite-dimensional observer for NSE with uncertain possibly destabilizing inputs: in our numerical experiments we illustrate H1-sensitivity of NSE to small perturbations of initial conditions, yet the observer converges for known and uncertain inputs.

Dynamic modeling and infinite-dimensional observer-based control for manipulation of flexible beam by a multi-link robot

This paper concerns an infinite-dimensional observer for manipulation of flexible beam by a rigid arm robot. The complex dynamic of the system is described by distributed parameter model in terms of ordinary differential equations and partial differential equation. A novel infinite-dimensional observer is proposed to estimate the vibration information of the flexible object. In addition, an observer-based independent joint controller is designed to achieve the position control and vibration suppression, which do not need end-point boundary control. The semigroup theory and LaSalle’s invariance principle are adopted to prove the asymptotic stability of the robot system. The efficiency of the observers and the proposed control strategy are demonstrated by numerical simulations.

Output synchronous control of multi‐agent beam equations with boundary and domain structure time‐varying disturbances and general disturbances

AbstractThe design of a synchronous controller for a network beam equation with structural time‐varying and general perturbations is studied. An infinite dimensional observer is established to estimate the total disturbance which is composed of structural time‐varying disturbance and unknown disturbance. Assuming that all networked partial differential equations (PDEs) are connected by communication graph, a feedback controller based on observer is proposed to solve the synchronization problem, which provides many new methods for the control of distributed systems in network space. The controller designed in this paper has three objectives: (1) asymptotically compensate structural term disturbance and general disturbance; (2) ensure the asymptotic convergence between each network state and the virtual leader state to track the virtual leader; and (3) the converged state errors are converged in the sense of synchronization. By using the analytical semigroup method, the well‐posedness of the networked closed‐loop system is proved. Three kinds of network beam equations with observer feedback control are simulated, and the simulation results show that this control method is effective for the synchronization of the network PDEs.

Robust optimization control for turbidity process with large inertia and time-delay in waterworks

To address the difficulty of optimizing turbidity process with large inertia and long time delay, a twice-optimal control and construction pruning (TOCCP) strategy is proposed, which can be used to improve the fast stability and strong robustness of industrial process. The new approach owns the advantages of twice optimal control (TOC) and construction pruning (CP) method. The TOC constructs an infinite-dimensional state observer of the system, which can eliminate the influence of time delay on the system. The CP is adopted to improve the mid-band damping of the open-loop frequency characteristics to improve the robustness of the system. Moreover, an exhaustive algorithm is designed to find the optimal time scale, the most important parameter for adjusting the system response speed. In addition, the influences of different parameter changes on the robustness of the system are analyzed, and a practical parameter tuning formula is provided. Simulation-based comparisons of TOCCP and the other two algorithms were made. The results show that the system optimized by TOCCP has strong robustness, especially in disturbances rejection and set-point tracking. Moreover, TOCCP is practical and easily implemented to set parameters, providing guidelines for industrial application of large inertia and time-delay systems.

Multi Observer-Based Sliding Mode Load Frequency Control With Input Delay Estimation

Abstract The letter proposes frequency stability in power systems with input delay. A closed loop system can be oscillatory or even unstable without the exact knowledge of delay. Therefore, it is desirable to design a control scheme which is based on the estimation of unknown delay. The proposed design consists of an infinite dimensional observer with an adaptive time delay estimation and a sliding mode controller (SMC). The merit of the proposed concept lies in the fact that the unknown delay is valued by just estimating the smallest delay segment. The controller input is obtained from a set of sequential observers that predicts the system states and ensures asymptotic stability of the closed loop system with input delay estimation. The existence of sliding mode and the closed loop system stability is proved thanks to the Lyapunov and Lyapunov–Krasovskii candidate functionals, respectively. Simulation results confirm the effectiveness of the proposed design.

Energy-based Control and Observer Design for higher-order infinite-dimensional Port-Hamiltonian Systems

Abstract In this paper, we present a control-design strategy based on the energy-Casimir method for infinite-dimensional, boundary-actuated port-Hamiltonian systems with two-dimensional spatial domain and second-order Hamiltonian. The resulting control law depends on distributed system states that cannot be measured, and therefore, we additionally design an infinite-dimensional observer by exploiting the port-Hamiltonian system representation. A Kirchhoff-Love plate serves as an example in order to demonstrate the proposed approaches.

Leader-follower synchronization controller design for a network of boundary-controlled wave PDEs with structured time-varying perturbations and general disturbances

A feedback controller based on an infinite dimensional observer is proposed to solve the synchronous control problem of network wave equations with time-varying and general perturbations at the boundary. An agent in the network as a virtual leader, and all remaining agents need to incrementally track the status of the virtual leader and each agent is controlled by the Neumann-type drive through the boundary. In this paper, by adopting the idea of the active disturbance rejection control (ADRC) technology, the design of the synchronous controller is divided into three parts and has various functions in this paper. Firstly, it compensates the total disturbance asymptotically. Secondly, it ensures the asymptotic convergence between the network state and the virtual leader state to track the virtual leader. Finally, it ensures that the paired states are synchronized. The convergence of infinite dimensional systems and the suitability of closed-loop systems are analyzed and proved. Three network wave equations with disturbance observer feedback control are simulated numerically, and the simulation results show the effectiveness of the proposed method.

Robust State Estimation for Positive Real Infinite-Dimensional Systems With Actuator and Sensor Faults

In this article, we propose a novel framework on the state estimation for a class of positive real infinite-dimensional systems, which suffer from unknown actuator and sensor faults and uncertain exogenous disturbance. Four cases of fault occurrence are taken into account: single multiplicative actuator fault, single multiplicative sensor fault, simultaneous actuator and sensor faults, and single actuator fault with a dead-zone nonlinearity. We develop new adaptive laws for the estimation of fault parameters and embed the proposed laws into the infinite dimensional state observer design. Moreover, by considering the actuator fault with a dead-zone nonlinearity, an infinite dimensional observer associated with coupled parameter update laws is developed to help to complete the fault accommodation. This is achieved by reconfiguring the actuator input based on the estimated information of parameters in the actuator fault map with a dead-zone nonlinearity.

Boundary observer design for cascaded ODE — Hyperbolic PDE systems: A matrix inequalities approach

Boundary observer design for a system of ODEs in cascade with hyperbolic PDEs is studied. An infinite dimensional observer is used to solve the state estimation problem. The interconnection of the observer and the system is written in estimation error coordinates and analyzed as an abstract dynamical system. The design of the observer is performed to achieve global exponential stability of the estimation error with respect to a suitable norm and with a tunable convergence rate. Sufficient conditions in the form matrix inequalities are given for the design of the observer. The effectiveness of the approach is shown in a numerical example.

New dynamical observer for a batch crystallization process based on solute concentration

In this paper a new observer is introduced to estimate the Crystal Size Distribution (CSD) only from the measurements of the solute concentration, temperature and a model of the growth rate. No model of the nucleation rate is needed. This approach is based on the use of a Kazantzis-Kravaris/Luenberger observer which exponentially estimates functionals of the CSD. Then, the full state is estimated by means of a Tikhonov regularization procedure. Numerical simulations are provided. Our approach relies on an infinite-dimensional observer, contrarily to the usual moment based observers.

Boundary Control for an Axially Moving System With Input Restriction Based on Disturbance Observers

The vibration offsets reduction is studied for an industrial axially moving accelerated string subjected to the external disturbances and input saturation. A boundary control strategy is put forward to reduce the vibration offsets and an auxiliary term is introduced to handle the restriction of the input saturation. In addition, both infinite-dimensional observer and finite-dimensional observer are, respectively, designed to track unknown disturbances. Under the developed control strategy, the existence, uniqueness, and convergence of the solution of the closed-loop string system are discussed without resorting to model reduction methods. By choosing the proper control parameters, the simulations are given to verify the performance of the developed control strategy.

Boundary output feedback stabilization of transport equation with non-local term

Abstract In this paper, we are concerned with boundary output feedback stabilization of a transport equation with non-local term. First, a boundary state feedback controller is designed by a backstepping approach. The closed-loop system is proved to be exponentially stable by the equivalence between original and target system. Then, we design an output feedback controller based on an infinite-dimensional observer. It is shown that the result closed-loop system is also exponentially stable. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed feedback controller.

A Backstepping Boundary Observer for a Simply Supported Beam

This paper presents the development of a full-state observer for an Euler-Bernoulli beam system with simply supported boundary conditions. This problem is motivated by the necessity of estimators in control and diagnostic applications. The observer is developed using a backstepping procedure. The full state observer cannot be completed with a single observer system, and hence this paper presents the synthesis of two observer structures that together provide the desired convergence. The necessary feedback information is limited to the slope and the sheer force of the beam at one boundary, which are practically measurable values on a physical system. The observer estimates the full infinite dimensional beam system without any discretization of the model. The result of the technique developed in this paper is that the infinite dimensional observer estimates converge to the plant system states (representing the actual beam displacement) at an arbitrary exponential rate; numerical simulation results validating this behavior are provided.

State-Observer Design of a PDE-Modeled Mining Cable Elevator With Time-Varying Sensor Delays

A time-varying sensor delay is inevitable in a mining cable elevator operating at a maximum distance which is more than 2000-m underground, when the sensor signals are transmitted from a cage at the bottom of the cable to a control center on the ground through a set of wireless devices, including emitters and receivers. This sensor delay may degrade the estimation precision of the state observer or even destroy the stability of the closed-loop system. In this brief, we design an observer which can compensate the sensor delay with a time-varying and arbitrary length to estimate the vibration states of the mining cable elevator, including the time-varying-length mining cable and the cage. The vibration dynamics of the mining cable elevator is described by a wave partial differential equation (PDE)-ordinary differential equation (ODE) coupled system on a time-varying domain, and the sensor delay dynamics is represented by a transport PDE on a time-varying domain. A full-order infinite dimensional observer is constructed to estimate the states of such a wave PDE-ODE-transport PDE coupled system where PDEs are on time-varying domains, only using available boundary values without measurements at the connection points of the PDEs and ODE. The exponential stability of the observer error system is proved via Lyapunov analysis. The effectiveness of the proposed observer is verified via numerical simulation.

Performance boundary output tracking for one-dimensional heat equation with boundary unmatched disturbance

In this paper, we consider boundary output tracking for a one-dimensional heat equation with external disturbance at the opposite end of the bar. First, an unknown input infinite-dimensional observer is designed and an estimate of disturbance is obtained from the observer. Second, with reference signal and estimate of disturbance, we design a servo system which has bounded solution given that the reference signal and its derivative are bounded. The output feedback boundary control is then designed by the states of servo system and observer. It is proved that the state of closed-loop system tracks the state of the servo system. As a result, the output tracking is included. Finally, some simulation results are presented to illustrate the effectiveness of the proposed scheme.

Backstepping Observers for Two Linearized Kermack–McKendrick Models

Abstract In this paper, we consider the problem of constructing observers for two linearized Kermack–McKendrick models. One is a first-order hyperbolic PDE (Partial Differential Equation) model without time lag in the nonlocal boundary condition, and the other contains time lag which expresses latent period. The Kermack–McKendrick model is one of SIR (Susceptible-Infected-Recovered) models, and the linearized equation is often used in early phase of infection. In this paper, we construct an infinite-dimensional observer for each PDE model by using a backstepping approach.

Boundary Stabilization of Wave Equation With Velocity Recirculation

Nonlocal terms have been the mainstay of the applications of partial differential equation (PDE) backstepping methods to parabolic PDEs. The problem of similar nonlocal terms for wave equations is still open. For wave equations, similar nonlocal terms have not been studied. In this paper, we open the topic of exploration of control of wave PDEs with nonlocal terms. This paper is concerned with the wave equation with in-domain feedback/recirculation of a boundary velocity with a spatially constant coefficient. Due to this nonlocal term, the passivity of the wave equation is destroyed. We first design an explicit state feedback controller to achieve exponential stability for the closed-loop system. Then, we design the output feedback by using infinite-dimensional observer. The backstepping approach is adopted in investigation. It is shown that by using two measurements only, the output feedback makes the closed-loop system exponentially stable.

Observer Based Temperature Control of a Tank: From Constant to Nonlinear State-Dependent Delay

The problem of temperature control of a fluid in a tank is expounded in this paper. In our problem, the delay is a an explicit nonlinear function of the state (temperature) of the system and the resulting dynamics are governed by a nonlinear State-Dependent Delay differential Equation (SD-DDE). We perform the stability analysis using Lyapunov-Razumikhin based approach. Using the new concept of delay injection with inversion, an infinite dimensional observer is designed to estimate the state of the system. We also shed some light on the state space required for characterizing the Cauchy problem and the evolution of the SD-DDE based system. A juxtaposition of fixed delay and state-dependent delay is given for a better illustration, comparison and contrast of results. Simulation results are depicted at the end which portray the effectiveness, validity and usefulness of the proposed observer scheme.

Infinite-Dimensional Observer for Process Monitoring in Managed Pressure Drilling

Utilizing flow rate and pressure data in and out of the mud circulation loop provides a driller with real-time trends for the early detection of well-control problems that impact the drilling efficiency. This paper presents state estimation for infinite-dimensional systems used in the process monitoring of oil well drilling. The objective is to monitor the key process variables associated with process safety by designing a model-based nonlinear observer that directly utilizes the available information coming from the continuous-time online process output measurements at the topside of the well. The observer consists of a copy of the plant plus output injection terms where the gain is computed analytically in terms of the Bessel function of the first kind. The design is tested using data from a real-field drilling commissioning test in the North Sea by Statoil Oil Company. The results show that the design estimates the flow and pressure accurately.

Observer based output feedback control of linear systems with input and output delays

This paper is concerned with observer based output feedback control of linear systems with both (multiple) input and output delays. Our recently developed truncated predictor feedback (TPF) approach for state feedback stabilization of time-delay systems is generalized to the design of observers. By imposing some restrictions on the open-loop system, two classes of observer based output feedback controllers, one being finite dimensional and the other infinite dimensional, are constructed. It is further shown that, the infinite dimensional observer based output feedback controllers can be generalized to linear systems with both time-varying input and output delays. It is also shown that the separation principle holds for the infinite dimensional observer based output feedback controllers, but does not hold for the finite dimensional ones. Numerical examples are worked out to illustrate the effectiveness of the proposed approaches.