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In this paper, we pay attention to the boundary feedback stabilization of a one-dimensional flexible string system with the exogenous disturbance that acts on the control end. The active disturbance rejection control (ADRC) and disturbance observer-based control (DOBC) are introduced to reduce the vibration caused by external disturbance. By utilizing Lyapunov method and constructing a suitable Lyapunov functional, we conclude that system is uniformly bounded stable. The numerical simulations are also presented to demonstrate the validity of both methods.

We consider performance output tracking for a boundary controlled multi-dimensional heat equation with possibly unknown internal nonlinear uncertainty and external disturbance on one part and control on the rest of the boundary. The active disturbance rejection control (ADRC) approach is adopted in investigation. With partial boundary measurement only, we propose an extended state observer which is used to estimate both system state and total disturbance, design a servomechanism and an output feedback controller. Three control objectives have been achieved: a) the performance output is exponentially tracking the arbitrary given reference signal; b) all internal signals are uniformly bounded; c) the closed-loop system is internally asymptotically stable whenever both the reference signal and the disturbance vanish or belong to the spaces H1(0, ∞; L1(Γ 1 )) and L2(0, ∞; L2(Γ 0 )), respectively. In addition, when boundary measurement is suffered from the noise, it is shown that the proposed controller is robust to the measurement noise.

In this paper, we propose a new output feedback control law to exponentially stabilize an one-dimensional wave equation with unknown anti-damping coefficient subject to general external disturbance entering the control end. With the proposed control law, although the anti-damping parameter is unknown and the disturbance is presented, it is shown that the resulting closed-loop system still decays exponentially. The numerical experiments are carried out to illustrate effectiveness of the proposed control law.

In this paper we solve the tracking and disturbance rejection problem for an unstable wave equation with reference and disturbance signals that are generated by a linear exosystem whose eigenvalues are located on the imaginary axis. We consider both unmatched disturbance at the boundary and also in the domain, and also matched disturbance at the control boundary. To make the control design more easy, the in-domain disturbance is first shifted to the control boundary by introducing a new variable transformation, and the anti-stable boundary term is then transformed into a dissipative boundary term by constructing a transport equation. We find an ordinary differential equation (ODE) with delay that describes the relationship between the output tracking error and the coupled signals consisting of the disturbance and reference signal, by using Riemann variables. Based on this ODE with delay, we apply the internal model principle to find the dynamic controller that achieves the output regulation while rejecting the disturbances, and uses only the tracking error as input.

In this research we are concerned with a switched system that switches very fast between two infinite-dimensional LTI subsystems, such that the overall system spends on average an equal amount of time in both possible subsystems. We ask if it is possible to approximate the switched system with an average system that is the limit of our switched system when the switching period tends to zero. We are mainly concerned with the situation when both subsystems are described by a contraction semigroup on a Hilbert space. We examine in particular the case when both subsystems are obtained from a basic subsystem via different output feedback operators.

We solve the output tracking and disturbance rejection problem for a plant described by a one-dimensional anti-stable wave equation, with reference and disturbance signals that belong to W1,∞ (0, ∞) and L∞ (0, ∞), respectively. Generally, these signals cannot be generated by an exogenous system. We explore an approach based on proportional control. We show that the proportional low gain controller can achieve exponential output tracking while rejecting the disturbance. We solve the problem in three steps. In the first step, we convert the original system without disturbance into the two transport equations with an ordinary differential equation by using Riemann variables, then we propose a proportional control law by making use of the properties of transport systems and time delay systems. In the second step, based on our recent results on disturbance estimators, we apply the estimation/cancellation strategy to cancel to the external disturbance and to track the reference asymptotically. In the third step, we design a controller for the state observer system. Since no explicit disturbance appears in this system, the disturbance is exactly compensated, and the output signal to be controlled is exponentially tracking the reference signal. As a byproduct, we obtain a new output feedback stabilizing control law by which the resulting closed-loop system is exponentially stable using only two displacement output signals. Numerical experiments demonstrate the effectiveness of the proposed control law.

Consider a nonlinear distributed parameter system (DPS) described by ẋ (t) = Ax(t)+Bu(t)+B N N(x(t), t) for all t ≥ 0. Here A is the infinitesimal generator of a strongly continuous group of operators 𝕋 on a Hilbert space X, B and B N , defined on Hilbert spaces U and U N , respectively, are admissible control operators for 𝕋 and N : X × [0,∞) ↦ U N is continuous in t and Lipschitz in x, with Lipschitz constant L N independent of t. Thus B and B N can be unbounded as operators from U and U N to X, in which case the nonlinear term B N N(x(t), t) in the DPS is in general not a Lipschitz map from X ×[0,∞) to the state space X. Our goal is to find conditions under which this DPS is exactly controllable in some time τ, which means that for any initial state x(0) ∈ X, we can steer the final state x(τ) of the DPS to any chosen point in X by using an appropriate input function u ∈ L2([0, τ];U). We suppose that there exist linear operators F and F b such that (A, [B B N ], F) and (−A; [B B N ], F b ) are regular triples and A + BF Λ and −A + BF b,Λ are generators of strongly continuous semigroups 𝕋f and 𝕋b on X such that ∥𝕋 t f∥ · ∥𝕋 t b∥ decays to zero exponentially. We prove that if L N is sufficiently small, then the nonlinear DPS is exactly controllable in some time τ > 0. Our proof is constructive and provides a numerical algorithm for approximating the required control signal. We illustrate our approach using a simple example.

In this paper we are concerned with the output regulation problem for a system (plant) described by a boundary controlled anti-stable one-dimensional Schrodinger equation by the backstepping approach. The output to be controlled is not required to be measurable and its output operator is assumed to be admissible for a certain C 0 -semigroup that is related to the original system whereas the measurement is located at the boundary. First we show that the open-loop system is well-posed. We construct a state feedback control law to solve the output regulation problem, using the backstepping method. We design a finite-dimensional reference observer and an infinite-dimensional disturbance observer. Based on these observers, we design an output feedback controller that achieves output regulation.

This paper investigates the controllability of fractional damped dynamical systems with distributed delays. A necessary and sufficient condition for the controllability of linear fractional damped dynamical system with distributed delay in control is established, and a sufficient condition for the nonlinear system is also established. Two examples are given to illustrate the theorems.

The boundary output feedback stabilization for a multi-dimensional Kirchhoff plate with boundary observation suffered from general external disturbance is considered. The active disturbance rejection control approach is adopted in investigation. By using this approach, the disturbance is estimated by a relatively independent estimator. The disturbance is canceled in the feedback-loop. As a result, the control law can be designed almost as 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 closedloop are asymptotically stable. Some numerical simulations are provided.