TY - JOUR

T1 - Exponential stabilization of a Rayleigh beam using collocated control

AU - Weiss, George

AU - Curtain, Ruth F.

N1 - Funding Information:
Manuscript received October 15, 2004; revised August 23, 2005. Recommended by Associate Editor P. Christofides. This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) (from the U.K.) under the Portfolio Partnership Grant GR/S61256/01.

PY - 2008/4

Y1 - 2008/4

N2 - We consider a hinged elastic beam described by the Rayleigh beam equation on the interval [0,π]. We assume the presence of two sensors: one measures the angular velocity of the beam at a point ξ ∈ [0,π] and the other measures the bending (curvature) of the beam at the same point. (If ξ = 0 or ξ = π, then the second output is not needed.) The corresponding operator semigroup is unitary on a suitable Hilbert state space. These two measurements are advantageous because they make the open-loop system exactly observable, regardless of the point ξ. We design the actuators and the feedback law in order to exponentially stabilize this system. Using the theory of collocated static output feedback developed in our recent paper [6], we design the actuators such that they are collocated, meaning that B = C*, where B is the control operator and C is the observation operator. It turns out that if ξ ∈ (0,π), then the actuators cause a discontinuity of the bending exactly at ξ (this is the price, in this example, of having collocated actuators and sensors). This obliges us to use an extension of C to define the output signal in terms of the left and right limit of the bending at ξ. We prove that, for all static output feedback gains in a suitable finite range, the closed-loop system is well posed and exponentially stable. This follows from the general theory in our paper [6], whose main points are recalled here.

AB - We consider a hinged elastic beam described by the Rayleigh beam equation on the interval [0,π]. We assume the presence of two sensors: one measures the angular velocity of the beam at a point ξ ∈ [0,π] and the other measures the bending (curvature) of the beam at the same point. (If ξ = 0 or ξ = π, then the second output is not needed.) The corresponding operator semigroup is unitary on a suitable Hilbert state space. These two measurements are advantageous because they make the open-loop system exactly observable, regardless of the point ξ. We design the actuators and the feedback law in order to exponentially stabilize this system. Using the theory of collocated static output feedback developed in our recent paper [6], we design the actuators such that they are collocated, meaning that B = C*, where B is the control operator and C is the observation operator. It turns out that if ξ ∈ (0,π), then the actuators cause a discontinuity of the bending exactly at ξ (this is the price, in this example, of having collocated actuators and sensors). This obliges us to use an extension of C to define the output signal in terms of the left and right limit of the bending at ξ. We prove that, for all static output feedback gains in a suitable finite range, the closed-loop system is well posed and exponentially stable. This follows from the general theory in our paper [6], whose main points are recalled here.

KW - Collocated sensors and actuators

KW - Exact controllability and observability

KW - Output feedback

KW - Rayleigh beam

KW - Well-posed linear system

UR - http://www.scopus.com/inward/record.url?scp=42649127091&partnerID=8YFLogxK

U2 - 10.1109/TAC.2008.919849

DO - 10.1109/TAC.2008.919849

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AN - SCOPUS:42649127091

SN - 0018-9286

VL - 53

SP - 643

EP - 654

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

IS - 3

ER -