Stabilisation of Timoshenko beam system with delay in the boundary control

Abstract

In this paper, we consider the exponential stabilisation problem of a Timoshenko beam with delay in boundary control. Suppose that the controller outputs of the forms α 1 u 1 (t) + β 1 u 1 (t − τ ) and α 2 u 2 (t) + β 2 u 2 (t − τ ) where u 1 (t) and u 2 (t) are the inputs of boundary controllers. In the past, most of the stabilisation results are required α j > β j > 0, j = 1, 2. In the present paper, we shall give a new dynamic feedback control law that makes the system exponential stabilisation ∀τ > 0 provided that |α j | = |β j |(j = 1, 2). The exponential stabilisation result is proved via test of exact observability of the system.

Output-Based Stabilization of Timoshenko Beam with the Boundary Control and Input Distributed Delay

“…Equation 2.1 is a general model that includes all models studied earlier time. For instance, if g j ≡ 0, β j = 0 and α j < 0, j = 1, 2, it is a model without delay studied in [4,10]; if g j ≡ 0, α j < 0 and β j < 0, it is a model with fixed delay studied in [24], wherein the full state is required to be known; if g j = 0, α j , β j ∈ R, and the state is known, it is just the model studied in [27]. The great difference between this paper and [27] is that the state of the system is unknown in Eq.…”

“…With this dynamic feedback control, the Euler-Bernoulli beam is stabilized exponentially provided that any real α, β satisfy |α| = |β|. Wang and Xu [23] and Xu and Wang [24] discussed the 1-d wave equation and the Timoshenko beam and proved that such a design of dynamic feedback controller also fits the wave equation and Timoshenko beam. By analyzing the proof of these papers, we see that the restriction condition |α| = |β| mainly comes from the controller zero or the characteristic equation of difference equation:…”

Stabilisation of Timoshenko beam system with a tip payload under the unknown boundary external disturbances

“…The feedback control laws were designed to stabilise system under different conditions, and many excellent results were obtained, such as the exponential stability (Kim & Renardy, 1987;Kirane, Said-Houari, & Anwar, 2011;Liu, Zhang, Han, & Xu, 2015;Morgül, 1992;Shi, Hou, & Feng, 1998;Zhang, Dawson, De Queiroz, & Vedagarbha, 1997), the spectral asymptotic expressions (Shubov, 1999;Vu, Wang, Xu, & Yung, 2005) and the Riesz basis property (Xu, 2005;Xu & Feng, 2002;Xu & Wang, 2013). These works mentioned above are based on the ideal situation without uncertainties coming from either the internal or the external disturbances.…”

Boundary control of a Timoshenko beam system with input dead-zone

“…In and Krstic, Siranosian, Smyshlyaev, and Bement (2006), the Timoshenko beam systems are efficiently stabilised by using the backstepping difference algorithm and observers. In Xu and Wang (2013), a novel boundary control design is proposed for a Timoshenko beam with input delay. In this paper, the Timoshenko beam system with the input deadzone is considered under the environmental disturbances and the boundary disturbances, which make the control design more difficult.…”