Boundary Output Feedback Stabilization Research Articles

Acoustic Boundary Output Feedback Stabilization of Dynamic $n-\tau$ Model Flame via Duct with Variable Cross Section

Acoustic Boundary Output Feedback Stabilization of Dynamic $n-\tau$ Model Flame via Duct with Variable Cross Section

Boundary output feedback stabilization of state delayed reaction–diffusion PDEs

This paper addresses the problem of output feedback stabilization of a class of 1-D reaction–diffusion PDEs in the presence of a state delay in the reaction term. The control input applies through a Robin boundary condition. The system output is selected as either a Dirichlet or Neumann boundary trace. The developed control design procedure does not aim to compensate, but rather to dominate the state delayed term. This is achieved by introducing an observer that only estimates a finite number of modes of the PDE while applying a suitable pole shifting-based procedure on a truncated model that captures the unstable part of the plant. We show that the reported output feedback control strategy always achieves, for an arbitrarily given value of the state delay, the exponential stabilization of the plant in H1-norm. A notable feature of the domination-based approach is that the tuning of the controller and observer gains, as well as the subsequent stability conditions, appears to be independent of the value of the state delay.

Boundary output feedback stabilization for a cascaded wave PDE-ODE system with velocity recirculation and matched disturbance

In this paper, we are concerned with a boundary stabilization problem of a cascaded wave PDE-ODE (partial differential equation-ordinary differential equation) system with velocity recirculation and matched disturbance. First, an auxiliary PDE system is proposed to estimate the boundary disturbance by active disturbance rejection control (ADRC) method. Next with the estimation of the disturbance in hand, a state observer is constructed to recover the state of the PDE-ODE system. Then we design an output feedback controller under which the solution of the resulting closed-loop system is bounded and the state of the cascaded wave PDE-ODE system decays exponentially. Finally, some simulation examples are given to illustrate the effectiveness of the designed controller.

Nonlinear boundary output feedback stabilization of reaction–diffusion equations

This paper studies the design of a finite-dimensional output feedback controller for the stabilization of a reaction-diffusion equation in the presence of a sector nonlinearity in the boundary input. Due to the input nonlinearity, classical approaches relying on the transfer of the control from the boundary into the domain with explicit occurrence of the time-derivative of the control cannot be applied. In this context, we first demonstrate using Lyapunov direct method how a finite-dimensional observer-based controller can be designed, without using the time derivative of the boundary input as an auxiliary command, in order to achieve the boundary stabilization of general 1-D reaction-diffusion equations with Robin boundary conditions and a measurement selected as a Dirichlet trace. We extend this approach to the case of a control applying at the boundary through a sector nonlinearity. We show from the derived stability conditions the existence of a size of the sector (in which the nonlinearity is confined) so that the stability of the closed-loop system is achieved when selecting the dimension of the observer to be large enough.

Observer-based stabilization of an Orr-Sommerfeld equation cascaded by both the Squire equation and ODE with boundary control

In this paper, we study the boundary output feedback stabilization for an Orr-Sommerfeld equation cascaded by both the Squire equation and ordinary differential equations. The observer is proposed and the gain functions are found by the backstepping method. The observer-based controller is designed by using the certainty-equivalent principle. The existence and exponential stability of the closed-loop system are proved by calculating the system operator to generate a C0-semigroup and its eigenfunctions to form the Riesz basis approach.

Boundary Output Feedback Stabilization for a Cascaded-Wave PDE-ODE System with Velocity Recirculation

This paper considers the output feedback stabilization for a cascaded-wave PDE-ODE system with velocity recirculation by boundary control. First, we choose a well-known exponentially stable system as its target system and find a backstepping transformation to design a state feedback controller for the original system. Second, we attempt to give an output feedback controller for the original system by introducing the observer. The resulting closed-loop system admits a unique solution which is proved to be exponentially stable. Finally, we give some numerical examples to prove the validity for the theoretical results.

Boundary Output Feedback Stabilization of the Linearized Schrödinger Equation with Nonlocal Term

In this paper, we are concerned with the boundary output feedback stabilization problem of a Schrodinger equation with a nonlocal term. Firstly, we design an explicit boundary state feedback controller by backstepping approach. Under this controller, the closed-loop system is proved to be exponentially stable from the equivalence between the original system and the target system. Then, we propose an observer-based output feedback controller by replacing the state in state feedback controller with its estimation. The resulting closed-loop system admits a unique solution which is exponentially stable. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed feedback controller.

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.

Output Feedback Stabilization for Multi-Dimensional Wave Equation with Boundary Control Matched Disturbance * *This work was supported by the National Natural Science Foundation of China, the National Research Foundation of South Africa, and the Israel Science Foundation (grant no. 800/14).

Abstract In this paper, we consider boundary output feedback stabilization for a multidimensional wave equation with boundary control matched unknown nonlinear internal uncertainty and external disturbance. A new unknown input type extended state observer is proposed to recover both state and total disturbance which consists of internal uncertainty and external disturbance by using the hidden regularity of the wave equation. By the active disturbance rejection control (ADRC) strategy, the total disturbance is compensated (canceled) in the feedback loop, which together with a collocated stabilizing controller without uncertainty, leads to an output feedback stabilizing feedback control. It is shown that the resulting closed-loop system is well-posed and asymptotically stable under weak assumption on internal uncertainty and external disturbance. The numerical experiments are carried out to show the effectiveness of the proposed scheme.

Boundary output feedback stabilization for a class of coupled parabolic systems

In this paper, the boundary output feedback stabilization problem is addressed for a class of coupled nonlinear parabolic systems. An output feedback controller is presented by introducing a Luenberger‐type observer based on the measured outputs. To determine observer gains, a backstepping transform is introduced by choosing a suitable target system with nonlinearity. Furthermore, based on the state observer, a backstepping boundary control scheme is presented. With rigorous analysis, it is proved that the states of nonlinear closed‐loop system including state estimation and estimation error of plant system are locally exponentially stable in the L2norm. Finally, a numerical example is proposed to illustrate the effectiveness of the presented scheme.

Unknown input observer design and output feedback stabilization for multi-dimensional wave equation with boundary control matched uncertainty

In this paper, we consider boundary output feedback stabilization for a multi-dimensional wave equation with boundary control matched unknown nonlinear internal uncertainty and external disturbance. A new unknown input type extended state observer is proposed to recover both state and total disturbance which consists of internal uncertainty and external disturbance. A key feature of the proposed observer in this paper is that we do not use the high-gain to estimate the disturbance. By the active disturbance rejection control (ADRC) strategy, the total disturbance is compensated (canceled) in the feedback loop, which together with a collocated stabilizing controller without uncertainty, leads to an output feedback stabilizing feedback control. It is shown that the resulting closed-loop system is well-posed and asymptotically stable under weak assumption on internal uncertainty and external disturbance. The numerical experiments are carried out to show the effectiveness of the proposed scheme.

Output feedback stabilization for a Kirchhoff‐type nonlinear beam with general corrupted boundary observation

SummaryThis paper is concerned with the boundary output feedback stabilization for a Kirchhoff‐type nonlinear beam equation with boundary observation subject to a general disturbance. The active disturbance rejection control approach is adopted to design the controller and the disturbance estimator. By this approach, the disturbance is estimated by a relatively independent estimator and then canceled in the feedback loop. The existence and uniqueness of the classical solution to the closed‐loop system are proved. The asymptotical convergence of the closed‐loop system is obtained. Copyright © 2017 John Wiley & Sons, Ltd.

Output feedback stabilization for multi-dimensional Kirchhoff plate with general corrupted boundary observation

We consider boundary output feedback stabilization for a multi-dimensional Kirchhoff plate with boundary observation suffered from a general external disturbance. We adopt for the first time the active disturbance rejection control approach to stabilization of multi-dimensional partial differential equations under corrupted output feedback. In terms of this approach, the disturbance is estimated by a relatively independent estimator, based on (possibly) an infinite number of ordinary differential equations reduced from the original PDEs by infinitely many time-dependent test functions. This gives a state observer, an additional result via this approach. The disturbance is compensated in the feedback-loop. As a result, the control law can be designed almost as the same as that for the system without disturbance. We show that with a time varying gain properly designed, the observer driven by the disturbance estimator is convergent; and that all subsystems in the closed-loop are asymptotically stable. We also provide numerical simulations which demonstrate the convergence results and underline the effect of the time varying high gain estimator.

Output Feedback Stabilization for One-Dimensional Wave Equation Subject to Boundary Disturbance

We consider boundary output feedback stabilization for a one-dimensional anti-stable wave equation subject to general control matched disturbance. The active disturbance rejection control (ADRC) approach is adopted in investigation. Using the output of the system, we first design a variable structure unknown input type state observer which is shown to be exponentially convergent. The disturbance is estimated, in real time, through an extended state observer for an ODE reduced from the PDE observer. The disturbance is then canceled in the feedback loop by its approximated value. The stability of the resulting closed-loop system is proven. Simulation results are presented to validate the theoretical conclusions and to exhibit the peaking value reduction by time varying gain instead of constant high gain.

Output feedback stabilization of an unstable wave equation with general corrupted boundary observation

We consider boundary output feedback stabilization for an unstable wave equation with boundary observation subject to a general disturbance. We adopt for the first time the active disturbance rejection control approach to stabilization for a system described by the partial differential equation with corrupted output feedback. By the approach, the disturbance is first estimated by a relatively independent estimator; it is then canceled in the feedback loop. As a result, the control law can be designed almost as that for the system without disturbance. We show that with a time varying gain properly designed, the observer driven by the disturbance estimator is convergent, and that all subsystems in the closed-loop are asymptotically stable in the energy state space. We also provide numerical simulations which demonstrate the convergence results and underline the effect of the time varying gain estimator on peaking value reduction.

Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation

The stabilization with time delay in observation or control represents difficult mathemat- ical challenges in the control of distributed parameter systems. It is well-known that the stability of closed-loop system achieved by some stabilizing output feedback laws may be destroyed by what- ever small time delay there exists in observation. In this paper, we are concerned with a particularly interesting case: Boundary output feedback stabilization of a one-dimensional wave equation system for which the boundary observation suffers from an arbitrary long time delay. We use the observer and predictor to solve the problem: The state is estimated in the time span where the observation is available; and the state is predicted in the time interval where the observation is not available. It is shown that the estimator/predictor based state feedback law stabilizes the delay system asymptoti- cally or exponentially, respectively, relying on the initial data being non-smooth or smooth. Numerical simulations are presented to illustrate the effect of the stabilizing controller.

Boundary Output Feedback Stabilization of A One-Dimensional Wave Equation System With Time Delay

The stabilization with time delay in observation and control represents extremely difficult mathematical challenges in distributed parameter systems control. It is well-known that the closed-loop stability achieved by some stabilizing output feedback laws is not robust for any small time delay. When this happens, it necessary to reconsider the stabilizing feedback control laws. We are concerned with a particularly interesting case: boundary output feedback stabilization of one-dimensional wave equation system for which the boundary observation suffers a time delay. This stabilization problem has been unsolved for over two decades. We construct an infinite-dimensional observer such that the estimation error converges exponentially to zero as time goes to infinity and we design a stabilizing state feedback law. Using the separation principle as in the finite dimensional cases, we show that the delay system is exponentially stabilized by the feedback law based on the estimated state.