Exponential stabilization of a clamped Timoshenko beam with actuation on a tip mass

Abstract

In this paper we consider the stabilization problem of a clamped beam with torque and force actuation on a mass in the other side of the beam. We show how to derive the model starting from the Principle of Least Action and we rewrite it as the interconnection between a 1 dimensional distributed parameter port-Hamiltonian system and a finite dimensional port-Hamiltonian system. Therefore, we propose a control law that allow to exponentially stabilise the origin of the closedloop system. In this preliminary paper we only sketch the theoretical proofs, but we give the procedure to compute the exponential bound of the system's state. Finally, we provide some numerical simulations testing the closed-loop behaviour with different choices of the control parameters.

Exponential decay rate bound of one-dimensional distributed port-Hamiltonian systems with boundary dissipation

“…Exponential stability of the class of systems described in section 2 now follows immediately, along with a bound on the decay rate. Lemma 2 implies that exists at least one multiplier function, m(x), such that condition (21) holds and so the system (1),( 5) is exponentially stable. Furthermore, Theorem 1 can be used to obtain a lower bound of the exponential decay rate for all systems with the form (1)-( 5…”

“…Set z 1 (x, t) = ρ(x) ∂w(x,t) ∂t , z 2 (x, t) = ι ρ (x) ∂φ(x,t) ∂t , z 3 (x, t) = ∂w(x,t) ∂x − φ(x, t), z 4 (x, t) = ∂φ(x,t) ∂x and z(x, t) = z 1 (x, t) z 2 (x, t) z 3 (x, t) z 4 (x, t) . System (33) can be rewritten in the port-Hamiltonian formulation as in [21] to obtain ∂z(x, t) ∂t…”

“…Similarly, defining the boundary conditions (34) can be written in the standard form ( 2)-( 4) . First, consider an inviscid beam with the same parameters as in [21]; that is γ(x) = δ(x) = 0, with ρ = 0.2kg/m, 3.9 × 10 −3 . This implies that for sufficiently small c > 0, z (x, t)A s (x)z(x, t) ≥ cz (x, t)Qz(x, t).…”

“…Then, considering a linear multiplier function, C Then, varying ξ on ( 22) and (25), we obtain the values of M and α shown in Table II. In [21] a Lyapunov approach is used for the stability analysis in the port-Hamiltonian formulation of a Timoshenko beam with viscous dissipation and unitary parameters, leading to an exponential decay rate of 0.0285 with a M = 2.783. Choosing ξ = 0.4713, from (25) we also obtain M = 2.783 and α = 0.0286.…”

“…In these works, sufficient conditions that guarantee exponential stability are obtained. An explicit lower bound for the exponential decay rate of the energy a Timoshenko beam with boundary and internal dissipation was obtained in [21] .…”