ON HARMONIC DISTURBANCE REJECTION OF AN UNDAMPED EULER-BERNOULLI BEAM WITH RIGID TIP BODY

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1 ESAIM: COCV October 2004, Vol. 0, DOI: 0.05/cocv: ESAIM: Control, Optimisation and Calculus of Variations ON HARMONIC DISTURBANCE REJECTION OF AN UNDAMPED EULER-BERNOULLI BEAM WITH RIGID TIP BODY Bao-Zhu Guo, 2 and Qiong Zhang Abstract. A hybrid flexible beam equation ith harmonic disturbance at the end here a rigid tip body is attached is considered. A simple motor torque feedback control is designed for hich only the measured time-dependent angle of rotation and its velocity are utilized. It is shon that this control can impel the amplitude of the attached rigid tip body tending to zero as time goes to infinity. Mathematics Subject Classification. 47F, 35K. Received July 22, Revised November 4, Introduction In this paper, e consider a flexible beam rotated by a motor in a horizontal plane at one end and a tip body rigidly attached at the free end (see Fig. belo). This model fits a large class of real applications such as links of robot systems and space-shuttle arms in hich high speed manipulation and long and slender geometrical dimensions are the major factors causing mechanical vibration. To achieve simultaneously the high speed, precise end point position and robustness of the flexible beam, the boundary control is one of the major strategies in production and space applications. Let l be the length of the beam, ρ the uniform mass density per unit length, EI the uniform flexural rigidity, M the mass of the tip body attached, Ĩm the moment of inertia of the motor and J the moment of inertia associated ith the tip body. Let θ(t) be the angle of rotation of motor at time t and (x, t) be the deflection at position x along moving axis and time t. Suppose that a) the deflection (x, t) is small and any extension is neglected; b) all terms θ 2 are negligible; c) the tist of the beam is ignored; d) both the distance beteen the beam s tip point and the intersection of the beam s tip tangent ith a perpendicular plane passing through the tip body s center and the damping effects are ignored. Then (x, t) satisfies the folloing Euler-Bernoulli beam equation and the Neton-Euler rigid-body equations [2]: ρ tt (x, t)+ei xxxx (x, t) = x θ(t), 0 <x<l,t>0, (0,t)= x (0,t)=0, EI xxx (l, t) = M[l θ(t)+ tt (l, t)], EI xx (l, t) = J[ θ(t)+ xtt (l, t)]. Keyords and phrases. Beam equation, disturbance rejection, noncollocated control, strong stability. Supported by the National Natural Science Foundation of China. Academy of Mathematics and System Sciences, Academia Sinica, Beijing 00080, China; bzguo@iss03.iss.ac.cn 2 Departement of Computational and Applied Mathematics, University of the Witatersrand, Private Bag-3, Wits-2050, Johannesburg, South Africa. c EDP Sciences, SMAI 2004 (.)

2 66 B.-Z. GUO AND Q. ZHANG Figure. The vibration control of a flexible link ith a rigid tip body. The equation of motion of rotation motor is [2] Ĩ m θ(t) =ũ(t)+eixx (0,t) (.2) here ũ(t) is the torque developed by the motor and applied at the root of the beam. If e define ỹ(x, t) = (x, t)+xθ(t) as the total deflection of the beam []. Then ỹ satisfies [4, 5, 5] ρỹ tt (x, t)+eiỹ xxxx (x, t) =0, 0 <x<l,t>0, ỹ(0,t)=0, EIỹ xx (0,t) Ĩmỹ xtt (0,t)+ũ(t) =0, EIỹ xxx (l, t) Mỹ tt (l, t) =0, EIỹ xx (l, t)+ Jỹ xtt (l, t) =0. The model (.3) as established again in [5] based on [7]. The stability by using the feedback of measured time-dependent angular and its velocity at x = 0 as studied in [4, 5]. Most existing boundary control design methods for flexible system are using the collocated actuators and sensors based on passive principle because the collocated system ith rate feedback is inherently stable (see e.g. [2, 9], etc.). Hoever, as it as indicated in [] that ith such collocated measurements, the vibration of the system are not controlled ell enough. On the other hand, some effort has been made to design the noncollocated control for the flexible systems, for instance [3] and [3] here some alternative noncollocated measurements are proposed to deal ith the difficulty of presence of the right half plane zeros in the transfer function and the finite dimensional compensator as used to stabilize the damped flexible systems. In this paper, e consider a different problem: disturbance rejection through noncollocated control by hich e mean that the control is at the one end and the output of concern is at the another. For notational simplicity, e make the folloing transformation ( ρl 4 y(x, t) =ỹ lx, EI t ) (,u(t) = l2 ρl 4 EI ũ EI t ),I m = Ĩm l 3 ρ,m = M lρ,j = and consider the harmonic disturbance at the end here the rigid tip bogy is attached. After that y satisfies y tt (x, t)+y xxxx (x, t) =0, 0 <x<,t>0, y(0,t)=0, I m y xtt (0,t) y xx (0,t)=u(t), (.4) Jy xtt (,t)+y xx (,t)=ρ J sin(ω J t + ψ J ), My tt (,t) y xxx (,t)=ρ M sin(ω M t + ψ M ) J l 3 ρ (.3)

3 DISTURBANCE REJECTION OF EULER-BERNOULLI BEAM 67 Figure 2. The block of disturbance rejection by the measured feedback. here in the input harmonic disturbances ρ J sin(ω J t + ψ J )andρ M sin(ω M t + ψ M ), the frequencies ω J and ω M are assumed to be knon but both the amplitudes ρ J, ρ M and the phases ψ J, ψ M are uncertain constants. Suppose that y out (t) =y(,t), t 0 (.5) is the output to be concerned and the measured output is given by y mea (t) =(y x (0,t),y xt (0,t)), t 0. (.6) The objective of this paper is to design the control input u(t) by using only the measured output so that lim y out(t) = lim y(,t)=0. (.7) Disturbance rejection or attenuation is one of the most important and idely studied areas in control theory. One of the main principles is the internal model principle, that is, making the transfer function of controller have poles at the frequencies of the disturbances. We refer this approach to [6] and the references therein here the transfer function is required to be in Callier-Desoer algebra. The results of [6] ere extended in [0] that for the ell-posed system [4] ith disturbance, the output y out L 2 γ(0, ) forsomeγ>0. The result of [0] is then applied to the rejection of external noise in the structural acoustics model and to coupled beams. Very different to internal model principle, in this paper, e propose the folloing simple controller by using only measured output: ξ (t) =ωξ 2(t), ξ 2(t) = ωξ (t) kξ 2 (t)+ky tx (0,t), u(t) = αy x (0,t)+kξ 2 ky tx (0,t) here the nonzero constants α, k and ω can be turned in practical design. We ill sho that controller (.8) can stabilize not only the system (.4) ithout disturbance but also impel the output concerned tending to standstill in the presence of disturbances as in the equation (.4). The process can be illustrated by a typical block in H control frameork as in Figure 2 above. 2. Well-posedness and stability ithout disturbance Suppose there are no disturbances at the rigid tip body attached end. Then the closed-loop system of (.4) ith control la (.8) becomes y tt (x, t)+y xxxx (x, t) =0, 0 <x<, t 0, y(0,t)=0, I m y ttx (0,t) y xx (0,t)+αy x (0,t)=kξ 2 ky tx (0,t), Jy ttx (,t)+y xx (,t)=0, My tt (,t) y xxx (,t)=0, ξ (t) =ωξ 2(t), ξ 2 (t) = ωξ (t) kξ 2 (t)+ky tx (0,t). (.8) (2.)

4 68 B.-Z. GUO AND Q. ZHANG Consider (2.) in the state Hilbert space H = V L 2 (0, ) C 5,V = {z H 2 (0, ) z(0) = 0} endoed ith the inner product norm (y,, v 0,v,v 2,ξ,ξ 2 ) 2 H = y xx (x) 2 dx + α y x (0) (x) 2 dx + I m v J v 2 + M v ξ 2 + ξ 2 2. Then the system (2.) can be ritten as an evolution equation in H Ẏ (t) =AY (t) (2.2) here Y (t) =(y(,t),y t (,t),y tx (0,t),y tx (,t),y t (,t),ξ,ξ 2 ) is the state and the system operator A :(H ) D(A) H is defined as follos: A y v 0 v v 2 ξ ξ 2 D(A) = = y xxxx I m (y xx (0) αy x (0) + kξ 2 k x (0)) J y xx() M y xxx() ωξ 2 ωξ kξ 2 + k x (0) { (y,, v 0,v,v 2,ξ,ξ 2 ) H, y H4 (0, ), V,v 0 = x (0), v = x (),v 2 = () } (2.3) Theorem 2.. Let A be given by (2.3). Then A generates a C 0 -semigroup T (t) of contractions on H. Moreover, 0 ρ(a) and A is compact. Therefore, A is discrete and σ(a) only consists of isolated eigenvalues. Proof. For any Y =(y,, v 0,v,v 2,ξ,ξ 2 ) D(A), it has Re AY, Y H [ = Re xx y xx dx + α x (0)y x (0) y xxxx dx 0 0 (y xx (0) αy x (0) + kξ 2 k x (0)) x (0) y xx () x () ] +y xxx ()() + ωξ 2 ξ +( ωξ kξ 2 + k x (0))ξ 2 = k x (0) ξ Hence A is dissipative. Furthermore, for any fixed but arbitrary element f =(f,f 2,f 3,f 4,f 5,f 6,f 7 )inh, solving equation y f y xxxx f 2 v 0 I m (y xx (0) αy x (0) + kξ 2 k x (0)) A v = J v 2 y f 3 xx() M ξ y = f 4 xxx() f 5 ωξ 2 f 6 ξ 2 ωξ kξ 2 + k x (0) f 7

5 DISTURBANCE REJECTION OF EULER-BERNOULLI BEAM 69 gives and y(x) x s ( 2 x2 ) 6 x3 f 5 = f 2 (s)dsds 3 ds 2 ds 0 0 s 2 s 3 2 Jx2 f 4 M + ( Jf α x 4 Mf 5 + f 2 (s)dsds + k ) 0 s ω f 6 kf x (0) I m f 3 = f,v 0 = f x (0), v = f x (), v 2 = f (), ξ 2 = ω f 6,ξ = k ω f x(0) k ω 2 f 6 ω f 7 and the compactness follos from the Sobolev s embedding theorem applied to A. When the control is assigned to be u(t) = αy x (0,t) only, that is to say ξ 2 = y tx (0,t) in equation (2.), e obtain a conservative system associated hich the system operator, denoted by A 0 ith D(A 0 )=D(A), is also discrete and moreover, ske-adjoint in H: A 0 y v 0 v v 2 ξ ξ 2 = y xxxx I m (y xx (0) αy x (0)) J y xx() M y xxx() ωξ 2 ωξ It follos from a ell-knon fact in functional analysis that all eigenvalues of A 0 lie in the imaginary axis hich are symmetric ith respect to the real axis. So, e may assume σ(a 0 )={(iτ 2, iτ 2 ) τ are positive}. A direct computation shos that τ satisfies the folloing characteristic equation:. ( I m τ 4 + α)[ + MJτ 4 +(Mτ Jτ 3 )sinhτ cos τ (Mτ + Jτ 3 )coshτ sin τ +( MJτ 4 )coshτ cos τ] +τ[ 2Jτ 3 cosh τ cos τ +( +MJτ 4 )coshτ sin τ +( MJτ 4 )sinhτ cos τ 2Mτ sinh τ sin τ] =0. (2.4) Since τ is positive, a close examination of characteristic equation (2.4) reveals that cos τ = sin τ Mτ + O(τ 2 ) hose solution can asymptotically found to be [5] τ = τ n = ( n ) π + O(n ) for sufficiently large n. (2.5) 2 As for those zeros of (2.4) closed to the origin one can easily find them through numerical methods. Theorem 2.2. Choose τ = ω not to be the root of (2.4). Then the semigroup T (t) generated by A is strongly stable. Proof. Since T (t) is bounded and A is discrete, by a ell-knon sufficient characterization condition on strong stability of bounded C 0 -semigroup (see e.g. Th of [8]), the proof ill be accomplished by shoing that there is no eigenvalue of A on the imaginary axis. Suppose no that 0 λ R. Solving eigenvalue problem of the folloing AY = iλy, Y =(y,, v 0,v,v 2,ξ,ξ 2 ) D(A), (2.6)

6 620 B.-Z. GUO AND Q. ZHANG e find that y, ξ and ξ 2 satisfy λ 2 y + y xxxx =0, y(0) = 0, λ 2 I m y x (0) y xx (0) + αy x (0) kξ 2 + iλky x (0) = 0, λ 2 Jy x () + y xx () = 0, λ 2 My() y xxx () = 0, iλξ ωξ 2 =0, iλξ 2 + ωξ + kξ 2 kiλy x (0) = 0. (2.7) In light of the fact that 0 = Re AY, Y H = k iλy x (0) ξ 2 2 as in the proof of Theorem 2., e obtain that iλy x (0) = ξ 2. Hence (2.7) becomes λ 2 y + y xxxx =0, y(0) = 0, λ 2 I m y x (0) y xx (0) + αy x (0) = 0, λ 2 Jy x () + y xx () = 0, λ 2 My() y xxx () = 0, iλξ ωξ 2 =0, iλξ 2 + ωξ =0. It is clear that if one of ξ and ξ 2 is not identically zero, then the last to equalities of (2.8) imply that λ =0. Suppose first that both ξ and ξ 2 are not identically zero. Then λ 2 = ω 2 and since the first five equalities of (2.8) are nothing but the eigen equation of A 0, it follos that ω is a root of (2.4). This contradicts the hypothesis. While both ξ and ξ 2 are identically zero, then y x (0) = 0 from the proved fact that iλy x (0) = ξ 2. In this case, (2.8) becomes λ 2 y + y xxxx =0, y(0) = y x (0) = y xx (0) = 0, λ 2 Jy x () + y xx () = 0, λ 2 My() y xxx () = 0. Put λ = τ 2,τ > 0. Then the solution to the equation above takes the form y(x) =sinhτx sin τx ith τ satisfying τ 3 J(cosh τ cos τ)+(sinhτ +sinτ) =0, τm(sinh τ sin τ) (cosh τ +cosτ) =0 from hich e can readily obtain τ 2 J(cosh 2 τ cos 2 τ)=m(sinh 2 τ +sin 2 τ) (2.8) and hence λ must be identically zero. We have once again had a contradiction. The result follos. Remark 2.. Theorem 2.2 is the best stability result e can hope by the assigned output feedback for the exponentially stability needs additional measured signals, for hich e refer to [5] for more details. 3. Disturbance rejection In this section, e turn to our main object of disturbance rejection. Obviously, the condition of Theorem 2.2 is necessary for this purpose. We ill discuss to different cases: ρ J =0andρ M = 0. First e investigate the first case of ρ J =0.

7 DISTURBANCE REJECTION OF EULER-BERNOULLI BEAM 62 Theorem 3.. Suppose ρ J =0and ω = ω M. If τ = ω M is not the root of (2.4) but one of the folloing equation [2τ sinh τ +(α I m τ 4 )(cosh τ cos τ)][ 2Jτ 4 cos τ 2τ sin τ +(α I m τ 4 )( Jτ 3 sinh τ +coshτ Jτ 3 sin τ +cosτ)] [2τ sin τ +(α I m τ 4 )(cosh τ cos τ)][ 2Jτ 4 (3.) cosh τ +2τ sinh τ +(α I m τ 4 )( Jτ 3 sinh τ +coshτ Jτ 3 sin τ +cosτ)] = 0 by properly choosing the parameter α, then the output of the system (.4) and (.8) satisfies lim y out (t) = lim y(,t)=0. Remark 3.. Similar to that of (2.4), the positive solutions of (3.) are of the folloing asymptotic expansion: cos τ = ( + ) sin τ I m J τ 3 + O(τ 4 ), and hence τ = τ n = ( n ) π + O(n 3 )forn sufficiently large. (3.2) 2 Proof of Theorem 3.. We begin by constructing a particular solution ỹ(x, t) =φ(x)sin(ω M t + ψ M )forthe folloing equation: ỹ tt (x, t)+ỹ xxxx (x, t) =0, 0 <x<, t 0, ỹ(0,t)=0, I m ỹ ttx (0,t) ỹ xx (0,t)+αỹ x (0,t)=0, (3.3) Jỹ ttx (,t)+ỹ xx (,t)=0, ỹ(,t)=0. This is possible henever φ satisfies φ (4) (x) ωm 2 φ(x) =0, 0 <x<, φ(0) = 0, φ (0) = (α I m ωm 2 )φ (0), φ () = JωM 2 φ (), φ() = 0. Solving equation (3.4) shos that all is ell there does exist a nonzero solution to (3.4) under the hypothesis, hich is justified by observing that (3.) is nothing but the characteristic equation hich guarantees the existence of a nonzero solution to the equation (3.4). Finding this solution is a direct computation, and since it poses no special difficulty, e just rite don it explicitly: (3.4) φ(x) = [2τ sin τ +(α I m τ 4 )(cosh τ cos τ)] sinh τx +[2τ sinh τ +(α I m τ 4 )(cosh τ cos τ)] sin τx +(α I m τ 4 )(sinh τ sin τ)(cosh τx cos τx). (3.5) Therefore, φ () = τ 3 [ 2τ sin τ cosh τ 2τ sinh τ cos τ 2(α I m τ 4 )sinhτ sin τ] =2τ 7 e τ [ τ 3 cos τ +(I m τ 3 ατ 4 )sinτ + O(e τ ) ] (3.6) hich is not identically zero henever τ large enough by comparing ith (3.2).

8 622 B.-Z. GUO AND Q. ZHANG No that define y d (x, t) = ρ Mφ(x)sin(ω M t + ψ M ) φ, () ξ d (t) = ρ Mω M φ (0) sin(ω M t + ψ M ) φ, () ξ 2d (t) = ρ Mω M φ (0) cos(ω M t + ψ M ) φ () (3.7) Then a direct computation shos that (y, ξ,ξ 2 )=(y d,ξ d,ξ 2d ) given by (3.7) satisfy (.4) and (.8) as ρ J =0 and the most importantly, y d (,t)=0. (3.8) Decompose the solution of (.4) and (.8) as y(x, t) =y s (x, t)+y d (x, t),ξ = ξ s + ξ d,ξ 2 = ξ 2s + ξ 2d. It is obviously that (y, ξ,ξ 2 )=(y s,ξ s,ξ 2s ) is a solution of (2.) hich is strongly stable by virtue of Theorem 2.2, in particular lim y s(,t)=0. This together ith (3.8) finally verifies that lim y(,t)=0. Along the same line of proof of Theorem 3., e have the disturbance rejection result for the case of ρ M =0. Theorem 3.2. Suppose ρ M =0and ω = ω J. If τ = ω J is not the root of (2.4) but that of the folloing equation [2τ sinh τ +(α I m τ 4 )(cosh τ cos τ)][2mτ 2 sin τ 2τ cos τ +(α I m τ 4 )(Mτ cosh τ +sinhτ Mτ cos τ sin τ)] [2τ sin τ +(α I m τ 4 )(cosh τ cos τ)][2mτ 2 (3.9) sinh τ +2τ cosh τ +(α I m τ 4 )(Mτ cosh τ +sinhτ Mτ cos τ sin τ)] = 0, by properly choosing the parameter α, then the output of the system (.4) and (.8) satisfies lim y out (t) = lim y(,t)=0. Remark 3.2. It should be pointed out that e use lo order controller (.8) to stabilize the displacement of the tip body. If e choose the output as y out (t) =(y(,t),y x (0,t)), e need design a high order controller of the folloing form so that additional constants β or m can be regulated to ensure the particular solution of (3.3) satisfying ỹ x (,t)=0. ξ (t) =ωξ 2(t), ξ 2 (t) = ωξ (t) kξ 2 (t)+ky tx (0,t), η (t) =η 2 (t), η 2 (t) = (α + β)/mη (t)+α/my(0,t), u(t) = αy x (0,t)+kξ 2 ky tx (0,t)+αη (t). (3.0) Unfortunately, under this controller, the corresponding system operator A of the closed-loop system is not dissipative and hence it is not easy to determine its asymptotic stability. The related ork is still going on.

9 DISTURBANCE REJECTION OF EULER-BERNOULLI BEAM Concluding remarks In this paper, e start directly from a partial differential equation model of beam equation and propose a simple integrator type controller here only measured angular of rotation of motor and its velocity at one end of the beam are utilized in the design of feedback control la to stabilize the output of concern in the presence of the harmonic disturbance at the another end here the rigid tip body is attached. The key ideas lie in a) the controller must stabilize the system ithout disturbance; b) constructing a particular solution ith fixed end position through spectral analysis. The rigorous mathematical proof is presented for the PDE model ithout any damping. References [] R.H. Cannon, Jr. and E. Schmitz, Inital experiments on the end-point control of a flexible one-link Robot. Inter. J. Robotics Res. 3 (984) [2] G. Chen, M.C. Delfour, A.M. Krall and G. Payre, Modeling, stabilization and control of serially connected beam. SIAM J. Control Optim. 25 (987) [3] P.A. Chodavarapu and M.W. Spong, On noncolocted control of a single flexible link, in Proc. of the 996 IEEE International Conference on Robotics and Automation, Minneapolis, Minnesota (April 996) [4] B.Z. Guo and Q. Song, Tracking control of a flexible beam by nonlinear boundary feedback. J. Appl. Math. Stochastic Anal. 8 (995) [5] B.Z. Guo, Riesz basis approach to the tracking control of a flexible beam ith a tip rigid body ithout dissipativity. Optim. Methods Soft. 7 (2002) [6] T. Hämäläinen and S. Pohjolainen, A finite dimensional robust controller for systems in the CD-algebra. IEEE Trans. Autom. Control 45 (2000) [7] S. Lee and J.L. Junkins, Explicit generalization of Langrange s equations for hybrid coordinate dynamic systems, J. Guid. Control Dyn. 5 (992) [8] Z.H. Luo, B.Z. Guo and Ö. Morgül, Stability and Stabilization of Linear Dimensional Systems ith Applications. Springer- Verlag, London (999). [9] J.C. Oostveen and R.F. Curtain, Robustly stabilizing controllers for dissipative infinite-dimensional systems ith collocated actuators and sensors. Automatica 36 (2000) [0] R. Rebarber and G. Weiss, Internal model based tracking and disturbance rejection for stable ell-posed systems. Automatica, in press. [] Y. Sakaa, F. Matsuno and S. Fukushima, Modeling and feedback control of a flexible arm. J. Rob. Syst. 2 (985) [2] Y. Sakaa and Z.H. Luo, Modeling and control of coupled bending and torsional vibrations of flexible beam. IEEE Trans. Autom. Control 34 (989) [3] V.A. Spetor and H. Flashner, Modeling and design implification of noncollocated control in flexible systems. J. Dyn. Syst. Meas. Control 2 (990) [4] G. Weiss and R. Rebarber, Optimizability and estimatability for infinite-dimensional linear systems. SIAM J. Control Optim. 39 (200) [5] K. Yuan and L. Liu, Achieving mininmum phase transfer function for a noncollocated single-link flexible manipulator. Asian J. Control 2 (2000) 79-9.