Boundary Controllers for Euler-Bernoulli Beam with Arbitrary Decay Rate

Transcription

1 Proceedings of the 7th IEEE Conference on Decision and Control Cancun, Meico, Dec. 9-11, TuA7.1 Boundar Controllers for Euler-Bernoulli Beam with Arbitrar Deca Rate Andre Smshlaev, Bao-Zhu Guo, and Miroslav Krstic Abstract We consider a problem of stabilization of the Euler-Bernoulli beam. The beam is controlled at one end (using position and moment actuators) and has the sliding boundar condition at the opposite end. We design the controllers that achieve an prescribed deca rate of the closed loop sstem, improving upon the eisting boundar damper controllers. The idea of the control design is to use the well-known representation of the Euler-Bernoulli beam model through the Schrödinger equation, and then adapt recentl developed backstepping designs for the latter in order to stabilize the beam. We derive the eplicit integral transformation (and its inverse) of the closed-loop sstem into an eponentiall stable target sstem. The transformation is of a novel Volterra/Fredholm tpe. The design is illustrated with simulations. I. INTRODUCTION In this paper we present a novel control design for the Euler-Bernoulli beam, which achieves arbitrar deca rate of the closed-loop sstem b boundar feedback. The beam is controlled at one boundar using position and moment actuators and has the sliding boundar condition at the uncontrolled boundar (Fig. 1). We assume that the full state (displacement and velocit) measurements are available. The idea of our method is to use the well-known representation of the Euler-Bernoulli beam model through the Schrödinger equation [13]. For the Schrödinger equation, recentl developed controllers [6] based on the backstepping method [16] improved on the common passive damper controllers b moving all open-loop eigenvalues arbitraril to the left in the comple plane. In this paper we adapt the design from [6] to the Euler-Bernoulli beam. We should note that the design does not carr over triviall from one sstem to another because the boundar conditions of the beam do not directl correspond to the boundar conditions of the Schrödinger equation. We derive the invertible integral transformation which, together with boundar feedbacks, converts the beam into an eponentiall stable target sstem. The kernels of the transformation and control gains are given eplicitl, epressed in terms of Kelvin functions. In contrast to other backstepping designs, that have been developed for more compicated models of the beams, such as the shear beam model [5] and the Timoshenko beam model [7], [], here Andre Smshlaev and Miroslav Krstic are with the Department of Mechanical and Aerospace Engineering, Universit of California at San Diego, La Jolla, CA 993, USA asmshl@ucsd.edu and krstic@ucsd.edu Bao-Zhu Guo is with Institute of Sstems Science, Academ of Mathematics and Sstems Sciences Academia Sinica, Beijing 1, P. R. China bzguo@iss.ac.cn w(,t) Fig. 1. Uncontrolled Euler-Bernoulli beam with the sliding boundar condition at = and the hinged boundar condition at = 1. the transformation is not of a strict-feedback form. Instead, it contains both Volterra- and Fredholm-tpe integrals. The eisting literature on the control of the Euler-Bernoulli beam is etensive (see, e.g. [1], [], [3], [1], [11], [9], [1] and references therein), however, unlike for the wave equation, the popular passive damper controllers provide onl a limited damping for the beam. Two constructive approaches have been used to achieve an arbitrar deca rate for beams. In [], and later in [17], b choosing a special cost function the authors design the controllers that do not require a solution of the Riccati equation. A pole assignment (or Riesz basis) approach was used in [15], with an application to Euler-Bernoulli beam with structural damping. In these three papers, the unbounded control operator is assumed to be admissible, which is not satisfied at least for the moment control for Euler-Bernoulli beam considered in this paper. An interesting etension of Riesz basis approach, which removes the admissibilit assumption, was presented in [1], a necessar and sufficient condition was given to assign the poles b bounded feedback. However, the resulting feedback is not as eplicit as the controllers presented in this paper, as it is represented as an infinite sum of infinite products. II. PROBLEM FORMULATION Consider the Euler-Bernoulli beam model w tt (, t) w (, t) =, < < 1. (1) We assume the sliding end boundar conditions at = w (, t) = w (, t) =. () and the following boundar conditions at = 1: w(1, t) = u 1 (t), w (1, t) = u (t). (3) Here w is a beam displacement and u 1, u are the position and moment control inputs. The open-loop case u 1 = u corresponds to the hinged end (Fig. 1). The objective is to stabilize the zero equilibrium of the beam //$5. IEEE 15

2 7th IEEE CDC, Cancun, Meico, Dec. 9-11, TuA7.1 Let us introduce a new comple variable v = w t jw, () j is the imaginar unit. The direct substitution shows that v defined in this wa satisfies the Schrödinger equation v t (, t) = jv (, t) (5) v (, t) =, v(1, t) = u(t). (6) In a recent paper [6], the backstepping controllers were designed that achieve an arbitrar deca rate for the closed-loop sstem (5) (6). This gives an idea to adapt the control design from [6] to the Euler-Bernoulli beam (1) (3). However, the design is not going to be trivial because, as can be seen from (), regulation of v to zero does not necessaril impl the regulation of w to zero. Before we proceed, we summarize the backstepping control design for (5) (6). III. SUMMARY OF BACKSTEPPING DESIGN FOR THE SCHRÖDINGER EQUATION As shown in [6], the controller v(1, t) = and the transformation ψ(, t) = v(, t) k(1, )v(, t)d, (7) k(, )v(, t)d, () k(, ) is a comple-valued function that satisfies k (, ) = k (, ) cjk(, ) (9) k (, ) =, k(, ) = cj (1) with c >, map (5), (6) into the following eponentiall stable target sstem ψ t (, t) = jψ (, t) cψ(, t) (11) ψ (, t) = ψ(1, t) =. (1) The eigenvalues of this sstem are σ = c j π (n1), n =, 1,,..., therefore the parameter c allows to move them arbitraril to the left in the comple plane. The solution to the PDE (9) (1) is ( cj( ) I 1 ) k(, ) = cj cj( ) = [ ( ) (j 1)ber 1 c ( ) ( )] (1 j)bei 1 c ( ). (13) c ( ) Here I 1 ( ) is the modified Bessel function and ber 1 ( ) and bei 1 ( ) are the Kelvin functions, which are defined in terms of I 1 as { ber 1 () = Im bei 1 () = Re ( 1 j I 1 { ( 1 j I 1 )} )} (1). (15) The inverse transformation with v(, t) = ψ(, t) l(, ) = cj l(, )ψ(, t)d (16) ( cj( ) J 1 ) cj( ) maps (11) (1) back into (5), (6), (7). IV. TARGET SYSTEM (17) In this section we choose the target sstem which sets the desired behavior of the beam. Let us define α(, t) = Im {ψ(ξ, t)} dξd, (1) ψ is the state of the target sstem (11) (1) for the Schrödinger equation. It is straightforward to verif that α satisfies the following fourth-order PDE: with boundar conditions α tt cα t c α α = (19) α (, t) = α (, t) = () α (1, t) = α(1, t) =. (1) That this target sstem is eponentiall stable is easil seen from the definition (1) and the fact that (11) (1) is eponentiall stable. With a straightforward computation we obtain the eigenvalues of (19) (1): σ n = c ± j π (n 1), for n =, 1,,... () To state the precise result, we define the energ state space H α = H L (, 1) L (, 1), with H L(, 1) = {f H (, 1) f () = f(1) = } (3) and with the inner product induced norm (f, g) H α = [f () g() ] d () for all (f, g) H α. The sstem (19) (1) can be written as d dt (α, α t) = C(α, α t ) D(α, α t ), (5) C is a skew-adjoint operator given b C(f, g) = (g, f () ), (f, g) D(C), (6) D(C) = {(f, g) H α f H (, 1), g HL (, 1), f () = f (1) = }, (7) and D is a bounded operator given b D(f, g) = (, cg c f), (f, g) H α. () Theorem 1: Let C, D be defined b (7)-(). Then: 16

3 7th IEEE CDC, Cancun, Meico, Dec. 9-11, TuA7.1 (i) There is a famil of eigenfunctions of C D, which form a Riesz basis for H α. Hence C D generates a C - semigroup e (CD)t on H α. For an (α(, ), α t (, )) H α, there eists a unique (mild) solution to (5): (α(, t), α t (, t)) = e (CD)t (α(, ), α t (, )) C(, ; H α ), and if (α(, ), α t (, )) D(C), then (α(, t), α t (, t)) C 1 (, ; D(C)). (ii) The spectrum-determined growth condition holds for the semigroup e (CD)t : ω(c D) = S(C D) = c, ω(c D) is the growth bound of e (CD)t and S(C D) is the spectral bound of C D. (iii). The sstem (5) is eponentiall stable: for an given ε >, there eists C ε > such that E α (t) C ε e (cε)t E(), (9) E α (t) = 1 Proof. Omitted. [α (, t) α t (, t)]d. (3) V. CONTROL LAWS THAT STABILIZE THE BEAM TO A CONSTANT PROFILE From (1) and (11) (1) it follows that the state ψ is epressed through α in the following wa: ψ = α t cα jα. (31) Taking the real and imaginar parts of the transformation (), we get α t (, t) cα(, t) = w t (, t) α (, t) = w (, t) r(, )w t (, t)d s(, )w (, t)d (3) s(, )w t (, t)d r(, )w (, t)d, (33) the gains r(, ) and s(, ), defined correspondingl as the real and imaginar part of k(, ) = r(, )js(, ), satisf two coupled PDEs and r (, ) = r (, ) cs(, ) (3) r (, ) = r(, ) = (35) s (, ) = s (, ) cr(, ) (36) s (, ) = s(, ) = c. (37) The solutions to these PDEs are obtained from (13): r(, ) = s(, ) = c ( ) c ( ) [ ber 1 ( c ( ) ) ( )] bei 1 c ( ) [ ( ) ber 1 c ( ) (3) ( )] bei 1 c ( ). (39) The control laws are obtained b setting = 1 in (3), (33): w t (1, t) = w (1, t) = r(1, )w t (, t)d r(1, )w (, t)d s(1, )w (, t)d () s(1, )w t (, t)d. (1) Note that the feedback () would be implemented as integral, not proportional, control. The control laws (), (1) stabilize the beam to a constant profile. To see this, we use the equations (), (31), and the inverse transformation (16) to get w t (, t) = α t (, t) cα(, t) w (, t) = α (, t) s(, )α (, t)d r(, )(α t (, t) cα(, t))d () s(, )(α t (, t) cα(, t))d r(, )α (, t)d. (3) In deriving (), (3) we used the fact that l(, ) = r(, )js(, ), which can be shown from (17). We can see that when α converges to zero, w t and w converge to zero. Since w () =, this implies that w converges to a constant. Therefore, the straightforward application of the control design for the Schrödinger equation to the Euler- Bernoulli beam equation results in controls that suppress oscillations without necessaril bringing the beam to the zero position. VI. CONTROL LAWS THAT GUARANTEE REGULATION TO ZERO To achieve regulation to zero, we are going to modif the control law () to make it proportional, not integral, control. To this end, we want to epress w in () through the time derivatives w t and w tt and then integrate () with respect to time. A. Control Laws First, let us calculate (integrating b parts twice) s(1, )w (, t)d ( = γ 1 w (1, t) γ 1 = Since z z s(1, )d = sinh s(1, ξ)dξ dz = (1 ) ) s(1, ξ)dξ dz u (, t)d, () ( ) ( ) c c sin. (5) s(1, ξ)dξ s(1, ξ)(ξ )dξ (6) 17

4 7th IEEE CDC, Cancun, Meico, Dec. 9-11, TuA7.1 and w = w tt, from () and () we get w t (1, t) = In a similar wa, we get r(1, )w t (, t)d γ 1 w (1, t) r(1, )w (, t)d w tt (, t) = (1 γ )w (1, t) γ = 1 w tt (, t) s(1, ξ)(ξ )dξ d γ 1 (1 )w tt (, t)d. (7) r(1, )d = cosh Substituting () into (1) gives w (1, t) = 1 s(1, )w t (, t)d γ (γ 1)(1 )w tt (, t)d r(1, ξ)(ξ )dξ d, () ( ) ( ) c c cos. (9) 1 (γ 1)(1 )w tt (, t)d γ 1 w tt (, t) γ r(1, ξ)(ξ )dξ d. Substituting (5) into (7), after simplifications we get w t (1, t) = (r(1, ) γs(1, )w t (, t)d w tt (, t) (5) (γr(1, ξ) s(1, ξ))(ξ )dξd γ(1 )w tt (, t)d, (51) γ = γ 1 γ = tanh ( ) c tan ( ) c. (5) The control gains in (51) involve a division b γ, which ma become zero for certain values of c. Therefore, c should satisf the condition c π (n 1), n =, 1,,..., (53) which is easil achievable because c is the designer s choice. We now integrate (51) with respect to time to get the controller w(1, t) = (r(1, ) γs(1, )w(, t)d w t (, t) (γr(1, ξ) s(1, ξ))(ξ )dξ d γ(1 )w t (, t)d, (5) the constant of integration is chosen to be zero since this choice ensures the regulation of w to zero. To see this, note from the transformation (), (3), and the boundar condition w (, t) = that w converges to a constant. Suppose w(, ) A, then passing to the limit t in (5) we get A = A (r(1, ) γs(1, )) d. (55) Computing the integral on the right hand side of (55), we obtain = A cosh(a) sin(a) cosh(a)cos(a), (56) a = c/. Note that cosh(a) sin(a) > 1 for all c >, and cos(a) due to the condition (53). Therefore, A =. The other controller (33) can also be represented in terms of w and w t as follows w (1, t) = B. Transformation s(1, )w t (, t)d c w(1, t) r (1, )w(, t)d. (57) To find out what the transformation from w to α is, we start with the definition (1) and note that { } Im {ψ(, t)} = Im v(, t) k(, )v(, t)d = w (, t) Substituting this into (1), we get α(, t) = w(, t) w(1, t) z z r(, )w (, t)d s(, )w t (, t)d. (5) r(z, ξ)w (ξ, t)dξdzd s(z, ξ)w t (ξ, t)dξdzd. (59) Integrating b parts the term with w, changing the order of integration in both integral terms, and using (5), we obtain the final form of the transformation: α(, t) = w(, t) (r(, ) cs(, ))w(, t)d [ w t (, t) S(1, ) γ(1 ) ] (s(1, ξ) γr(1, ξ))(ξ )dξ d (cs(1, ) γs(1, ))w(, t)d S(, )w t (, t)d, (6) 1

5 7th IEEE CDC, Cancun, Meico, Dec. 9-11, TuA7.1 S(, ) = ( ξ)s(ξ, )dξ. (61) Note that this integral transformation is not strict-feedback, it is of a mied Volterra/Fredholm tpe. VII. INVERSE TRANSFORMATION To prove stabilit of the closed-loop sstem through the stabilit of the target sstem, we need to derive the transformation which is inverse to (6). It is natural to assume that the inverse transformation has the same structure as the direct one, consisting of two Volterra and two Fredholm integrals of the state of the target sstem and its time derivative. Therefore, we look for it in the form w(, t) = α(, t) A(, )α(, t)d B(, )α t (, t)d C()α(, t)d D()α t (, t)d, (6) A, B, C, D are the gains to be determined. Differentiating (6) w.r.t. time and space (twice) and matching the result to the equations () and (3) and to the boundar conditions w (, t) = w (, t) =, one can show that (we omit these straightforward calculations) A(, ) = r(, ) cs(, ), (63) B(, ) = S(, ), C() = cd(), (6) and D() satisfies the ODE D () = c D() (65) D () = D(1) = D (1) =, D () = c, (66) which has the solution sinh(a)cos(a( )) cos(a)sinh(a( )) D() = a(cosh(a) sin(a) ) sin(a)cosh(a( )) cosh(a)sin(a( )) a(cosh(a) sin(a). ) Note that the eplicit form of the above transformation allows one to write the solution of the closed-loop sstem in closed form using the eplicit solution of the target sstem. VIII. MAIN RESULT The design procedure presented in previous sections makes it clear wh the sstem (1) () with the controllers (5), (57) is eponentiall stable. In this section we give the precise statement of well-posedness and stabilit of the closed-loop sstem. First, we define the state space H = { (f, g) H (, 1) L (, 1) f () =, f(1) = (r(1, ) γs(1, ))f()d γ g() (γr(1, ξ) s(1, ξ))(ξ )dξ d } (1 )g()d. (67) It is eas to check that [r(1, ) γs(1, )]d 1. Therefore, we can define the inner product induced norm of H as the energ of the sstem: (f, g) H = [ f () g() ] d (6) for all (f, g) H. The sstem (1) (), (5), (57) can be written as d dt (w(, t), w t(, t)) = A(w(, t), w t (, t)), (69) A(f, g) = (g, f () ), (f, g) D(A), (7) D(A) = {(f, g) H A(f, g) H, f () = f (1) = c f(1) 1 } [r (1, )f() s(1, )g()] d Net two lemmas establish the eistence and boundedness of A 1 and the eistence and uniqueness of a classical solution. The proofs are straightforward and we omit them. Lemma : Let A be defined b (7) and let the condition (53) hold. Then ρ(a), the resolvent set of A, is not empt. In fact, ρ(a). Lemma 3: Let A be defined b (7) and let the condition (53) hold. Then for an (w(, ), w t (, )) D(A) there eists a unique classical solution to (69). Now we are read to state the main result of the paper. Theorem : Let A be defined b (7) and let the condition (53) hold. Then: (i) A generates a C -semigroup on H. For an initial value (w(, ), w t (, )) H, there eists a unique (mild) solution to (69): (w(, t), w t (, t)) = e At (w(, ), w t (, )) C(, ; H), (ii) The sstem (69) is eponentiall stable at the origin: for an given ε >, there eists M ε >, which depends onl on ε, such that for all initial conditions (w(, ), w t (, )) H, E(t) M ε e (cε)t E() E(t) = 1 [wt (, t) w (, t)] d. (71) Proof. Statement (i) follows from Lemmas, 3 and Theorem 1.3 of [1] on p.1. Statement (ii) follows from the densit of D(A) in H and (), (3) and Theorem 1. 19

6 7th IEEE CDC, Cancun, Meico, Dec. 9-11, TuA7.1 1 Gains 5 5 displacement Fig.. Control gains for c = 9: from w() to w(1) (solid), from w t() to w(1) (dashed), from w() to w (1) (dash-dotted), from w t() to w (1) (thin solid). Fig t Open-loop response of the Euler-Bernoulli beam. IX. SIMULATION RESULTS The results of simulation of the Euler-Bernoulli beam with the controllers (5), (57) are presented in Figs.. In Fig. the control gains are shown for c = 9. In Fig. 3 we can see the oscillations of the uncontrolled beam. With control, the beam is quckl brought to the zero equilibrium (Fig. ). X. FUTURE WORK In future work there are two etensions of the result of the paper to pursue. First, one would like to control beams with other tpes of boundar conditions. From the design procedure presented in the paper it is clear that an etension to a hinged tpe of the uncontrolled end should not pose an difficulties. One would just change the tpe of the uncontrolled boundar condition in the Schrödinger equation from Neumann to Dirichlet. However, it is not clear how to eploit the connection of the Euler-Bernoulli beam with the Schrödinger equation in case of the beam with a free uncontrolled end, the most important case from practical point of view. One would also like to etend the results of the paper to the output-feedback case. For the Schrödinger equation, successful observer-based output-feedback design was developed in [6]. It seems that there are no conceptual obstacles in adapting this design to the Euler-Bernoulli beam. REFERENCES [1] G. Chen, M. C. Delfour, A. M. Krall, and G. Pare, Modeling, stabilization and control of seriall connected beam, SIAM J. Control & Optim., vol. 5, pp , 197. [] B. Z. Guo and R. Yu, The Riesz basis propert of discrete operators and application to an Euler-Bernoulli beam equation with boundar linear feedback control, IMA J. Math. Control Inform., vol. 1, pp. 1 51, 1. [3] B. Z. Guo, Riesz basis propert and eponential stabilit of controlled Euler-Bernoulli beam equations with variable coefficients, SIAM J.Control Optim., vol., pp ,. [] V. Komornik, Rapid boundar stabilization of linear distributed sstems, SIAM J.Control Optim., vol. 35, pp , [5] M. Krstic, A. Balogh, and A. Smshlaev, Backstepping boundar controller and observer for the undamped shear beam, 17th Int. Smposium on Mathematical Theor of Networks and Sstems, 6. [6] M. Krstic, B. Z. Guo, and A. Smshlaev, Boundar controllers and observers for Schrödinger equation, IEEE Conference on Decision and Control, 7. displacement Fig Closed-loop response of the Euler-Bernoulli beam. [7] M. Krstic, A. Smshlaev, and A. Siranosian, Backstepping boundar controllers and observers for the slender Timoshenko beam: Part I Design, American Control Conference, 6. [] M. Krstic, A. Siranosian, A. Smshlaev, and M. Bement, Backtepping boundar controllers and observers for the slender Timoshenko beam: Part II Stabilit and simulations, IEEE Conference on Decision and Control, 6. [9] I. Lasiecka and R. Triggiani, Eact Controllabilit of Semilinear Abstract Sstems with Application to Waves and Plates Boundar Control Problems, Appl. Math. Optim., vol. 3, pp , [1] K. S. Liu and Z. Liu, Eponential deca of energ of the Euler- Bernoulli beam with locall distributed Kelvin-Voigt damping, SIAM J. Control & Optim., vol. 36, pp , 199. [11] Z. H. Luo, B. Z. Guo, and O. Morgul, Stabilit and Stabilization of Infinite Dimensional Sstems with Applications, Springer-Verlag, London, [1] K. A. Morris, ed., Control of Fleible Structures, AMS, [13] E. Machtngier and E. Zuazua, Stabilization of the Schrödinger equation, Portugaliae Mathematica, vol. 51, pp. 3 56, 199. [1] A. Paz, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 193. [15] R. Rebarber, Spectral assignabilit for distributed parameter sstems with unbounded scalar control, SIAM J.Control & Optim., vol.7, pp , 199. [16] A. Smshlaev and M. Krstic, Closed form boundar state feedbacks for a class of 1D partial integro-differential equations, IEEE Trans. on Automatic Control, vol. 9, pp. 15,. [17] J. M. Urquiza, Rapid eponential feedback stabilization with unbounded control operators, SIAM J. Control Optim., vol. 3, pp. 33, 5. [1] C. Z. Xu and G. Sallet, On spectrum and Riesz basis assignment of infinite-dimensional linear sstems b bounded linear feedbacks, SIAM J.Control Optim., vol. 3, pp , t 3 19