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

Abstract

In this paper, boundary control is designed for a Timoshenko beam system with the input dead-zone. By the Hamilton's principle, the dynamics of the Timoshenko beam system is represented by a distributed parameter model with two partial differential equations and four ordinary differential equations. The bounded part is separated from the input dead-zone and then forms the disturbance-like term together with the boundary disturbance, which finally acts on the Timoshenko beam system. Boundary control, based on the Lyapunov's direct method, is proposed to ensure the Timoshenko beam converge into a small neighbourhood of zero, where stability of the system is also analysed. Besides, the existence and uniqueness of the solution of the Timoshenko beam system are proved. Simulations are provided to reveal the applicability and effectiveness of the proposed control scheme. NomenclatureL length of the Timoshenko beam M mass of the tip payload ρ uniform mass per unit length of the Timohenko beam EI bending stiffness of the Timoshenko beam I ρ uniform mass moment of inertia of the cross section of the Timoshenko beam J inertia of the tip payload K K = kAG k a positive constant depending on the shape of the cross section of the Timoshenko beam A cross-sectional area of the Timoshenko beam G modulus of elasticity in shear w(x, t) transverse displacement of the Timoshenko beam at the position x and time t u(t) and τ (t) two control inputs with dead-zone d(t) unknown time-varying boundary disturbance on the tip payload f(x, t) disturbance of the Timonshenko beam

Dual-Loop Adaptive Iterative Learning Control for a Timoshenko Beam With Output Constraint and Input Backlash

“…As shown in [77], the Timoshenko beam system in jth iteration is described by the governing equations (11) for ∀t ∈ [0, T b ] and j ∈ N. B(τ j0 (t)) and B(u j0 (t)) are the backlash inputs and defined in (1). w j (x, t) and φ j (x, t) describe the transverse displacement and the angle displacement for the position x, the time t and the iteration j. f jw (x, t), d jw (t), and d jφ (t) express the external disturbances.…”

Disturbance observer‐based adaptive boundary iterative learning control for a rigid‐flexible manipulator with input backlash and endpoint constraint

“…The flexible structure is usually modeled into distributed parameter systems described by partial differential equation (PDE). However, the flexible part modeled by PDE with infinite dimensions is more difficult for control design comparing with the rigid part 6‐9 …”

“…To tackle the aforementioned problem, researchers investigated the PDE control of the flexible manipulator, because their models contain the infinite‐dimensional modes and overcome spillover instability 6 . Recently, boundary control strategy has been widely proposed for addressing flexible structure systems in order to avoid the spillover effects of the truncated model‐based control 6‐8 . In Reference 8, based on the Lyapunov's direct method, boundary control is designed for ensuring the stability of the Timoshenko beam system and copying with the input dead‐zone.…”

“…Recently, boundary control strategy has been widely proposed for addressing flexible structure systems in order to avoid the spillover effects of the truncated model‐based control 6‐8 . In Reference 8, based on the Lyapunov's direct method, boundary control is designed for ensuring the stability of the Timoshenko beam system and copying with the input dead‐zone. Another boundary output feedback law is proposed to reject general external disturbance for Euler‐Bernoulli beam systems in Reference 9.…”

Stabilization of exact nonlinear Timoshenko beams in space by boundary feedback

“…There are excellent works on boundary control on small (vibration) motions of Timoshenko beams, where the classical Timoshenko beam model [1] was used, (e.g., [2][3][4][5][6][7][8][9][10] based on Lyapunov's direct method or [11,12] based on the backstepping method [13]). Since the model in [1] is obtained by linearizing exact nonlinear partial differential equations (PDEs) governing motions of shear beams, the results of the above works are only valid in the neighborhood of the origin, see also Remark 2.1.…”