Lyapunov approach to output feedback stabilization for the Euler–Bernoulli beam equation with boundary input disturbance

Automatica 52 (2015) 95–102

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Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Lyapunov approach to output feedback stabilization for the Euler–Bernoulli beam equation with boundary input disturbance✩ Feng-Fei Jin a,c , Bao-Zhu Guo b,c a

School of Mathematics Science, Qingdao University, Qingdao 266071, China

b

Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, China

c

School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa

article

info

Article history: Received 17 March 2014 Received in revised form 24 September 2014 Accepted 18 October 2014

abstract We propose a boundary output feedback control law for a one-dimensional Euler–Bernoulli beam equation with general external disturbance entering the control end. A Galerkin approximation scheme is constructed to show the existence of solution to the closed-loop system. The exponential stability of the closed-loop system is obtained by the Lyapunov functional method. Numerical simulations are presented for illustration. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Euler–Bernoulli beam Disturbance Output feedback Boundary control

1. Introduction Spurred by the outer space applications and flexible arm manipulators, the vibration control for the Euler–Bernoulli beam equation has attracted much attention in the last three decades (Chen, Delfour, Krall, & Payre, 1987; Guo & Yu, 2001; Luo, Guo, & Morgul, 1999; Luo, Kitamura, & Guo, 1995; Smyshlyaev, Guo, & Krstic, 2009). We refer Han, Benaroya, and Wei (1999) for derivation and engineering interpretation of the Euler–Bernoulli beam equation. Since 1980s, the boundary pointwise control and observation have been dominating stabilization for the Euler–Bernoulli beam owing to its simplicity and easy implementation in engineering practice. However, most of the works are restricted to those beams that have no uncertainty. This is not the case particularly in space applications. In the last few years, there are some works attributed to control of infinite-dimensional systems with uncertainty. In Cheng, Radisavljevic, and Su (2011), a boundary variable structure control is designed to stabilize a heat equation subject to control

✩ This work was partially supported by the National Natural Science Foundation of China, the National Basic Research Program of China (2011CB808002), and the National Research Foundation of South Africa. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Thomas Meurer under the direction of Editor Miroslav Krstic. E-mail addresses: [email protected] (F.-F. Jin), [email protected] (B.-Z. Guo).

http://dx.doi.org/10.1016/j.automatica.2014.10.123 0005-1098/© 2014 Elsevier Ltd. All rights reserved.

matched external disturbance. Stabilizations for one-dimensional wave equations with harmonic disturbances suffered from input and output are considered in Guo and Guo (2013) and Guo, Guo, and Shao (2011). In Pisano, Orlov, and Usai (2011), a distributed control is designed for one-dimensional heat and wave equations with distributed control matched external disturbance. The Lyapunov functional approach is applied in Ge, Zhang, and He (2001) to attenuate external distributed and boundary disturbance for a beam equation. However, it seems that the external distributed disturbance in Ge et al. (2001) is not attenuated by the boundary feedback only. In our previous work (Guo & Jin, 2013), we design a state feedback control for a conservative Euler–Bernoulli beam equation with control matched boundary disturbance via both sliding mode control approach and active disturbance rejection control approach. The same system is considered in Guo and Kang (2014) where a variable structure output feedback control is designed via Lyapunov functional method yet the disturbance is limited to finite sum of harmonic disturbances. Other interesting works on stabilization for PDEs with uncertainty can be found in Krstic and Smyshlyaev (2008) where it introduces adaptive design for the first time for handling the parabolic PDEs with disturbance and anti-damping; Immonen and Pohjolainen (2006), Jayawardhana and Weiss (2009), Ke, Logemann, and Rebarber (2009) and Rebarber and Weiss (2003) where the internal model principle is generalized to infinite-dimensional systems. In this paper, we are concerned with output feedback stabilization for a one-dimensional Euler–Bernoulli beam equation with

96

F.-F. Jin, B.-Z. Guo / Automatica 52 (2015) 95–102

disturbance flowing from boundary control channel. The system is described by the following PDE:

u (x, t ) + u (x, t ) = 0, x ∈ (0, 1), t > 0, tt xxxx    u(0, t ) = ux (0, t ) = 0, t > 0,   uxx (1, t ) = 0, t > 0, uxxx (1, t ) = U (t ) + d(t ), t > 0,     ut (x, 0) = u1 (x), x ∈ [0, 1], u(x, 0) = u0 (x), yo (t ) = {ut (1, t ), ux (1, t )}, t > 0,

(1)

Under control (2), the closed-loop of system (1) is

u (x, t ) + u (x, t ) = 0, x ∈ (0, 1), t > 0, tt xxxx    u(0, t ) = ux (0, t ) = 0, t > 0,   uxx (1, t ) = 0, t > 0, uxxx (1, t ) ∈ kut (1, t ) + M sign(ut (1, t )     + δ ux (1, t )) + d(t ), t > 0,  u(x, 0) = u0 (x), ut (x, 0) = u1 (x), x ∈ [0, 1]. Let

where u(x, t ) is the transverse displacement of the beam at time t and position x, U is the input (control) through shear force, d is 1 the external disturbance that is assumed to satisfy d ∈ Hloc (0, ∞), and bounded: |d(t )| 6 M for some M > 0 and all t > 0, yo is the output (measurement). It is well known that when there is no disturbance, the collocated feedback control U (t ) = kut (1, t ), k > 0 exponentially stabilizes system (1) (Chen et al., 1987). However, this stabilizing controller is not robust to external disturbance, for instance, when d = −6, system (1) under U (t ) = kut (1, t ) has a solution u(x, t ) = 3x2 − x3 . So, in the presence of disturbance, the control must be re-designed. The remaining of the paper is organized as follows. Section 2 designs an output feedback control. The existence of the Filippov solution to the closed-loop system is shown in Section 3 by the Galerkin approximation approach. The exponential stability of the closed-loop system is concluded. Section 4 presents some numerical simulations to illustrate the effectiveness of the control law.

E (t ) =

ρ(t ) =

1,

[−1, 1], −1,

=

(6)

xut (x, t )ux (x, t )dx,

2

− M |ut (1, t ) + δ ux (1, t )| − d(t )(ut (1, t )  δ 1 2 [ut (x, t ) + 3u2xx (x, t )]dx + δ ux (1, t )) − 2 0   k2 δ δ u2t (1, t ) 6 − k− − 2

− 6 −

δ 2

2

1



[u2t (x, t ) + u2xx (x, t )]dx 0

δ V (t ), 1+δ

(7)

where we choose δ > 0 sufficiently small so that k − 2δ − k2δ > 0. Thus, the energy of system (5) decays exponentially. However, all aforementioned computations are formal since we do not know whether the derivatives involved make sense. The existence of the Filippov solution to system (5) is discussed in next section. 2

3. Existence of solution to a closed-loop system

 g (t )| g (t )



1



− δ xuxxx (x, t )ux (x, t )|10 + δ uxx (x, t )ux (x, t )|10 1  1 u2 ( x , t )   −δ + δx t [u2t (x, t ) + 3u2xx (x, t )]dx  2 2 0   0 δ 6 − k− u2t (1, t ) − kδ ut (1, t )ux (1, t )

(3)

which is crucial in the proof of existence of the solution in Theorem 3.1. In order to the function on the right-hand side of (2) to be measurable, for any T > 0 and f ∈ L2 (0, T ), we restrict the setvalued composition of the symbolic function as follows: sign(f (t )) =

[u2xx (x, t ) + u2t (x, t )]dx, 0

V˙ (t ) = −uxxx (x, t )ut (x, t )|10 + uxx (x, t )uxt (x, t )|10

(2)

x > 0, x = 0, x < 0,

2

It is obvious that (1 − δ)E (t ) 6 V (t ) 6 (1 + δ)E (t ) for all t > 0. Differentiate V formally with respect to t along the solution of (5) to obtain

where k > 0, 0 < δ < 1 are design parameters and the symbolic function is a multi-valued function defined by sign(x) =

1



0

We design the following output feedback control:



1

V (t ) = E (t ) + δρ(t ).

2. Output feedback control design

U (t ) = kut (1, t ) + M sign(ut (1, t ) + δ ux (1, t )),

(5)

sign(f (t )), f (t ) ∈ L2 (Z ), |f (t ) | ≤ 1,



f (t ) ̸= 0, , f (t ) = 0,

(4)

Z = {τ | f (τ ) = 0}. Our control law (2) is different to that designed in Guo and Kang (2014) where only the velocity signal ut (1, t ) that is a commonly used signal to stabilize the Euler–Bernoulli beam is used but the disturbance rejection is limited to very special finite sum of harmonic disturbance. The result of Guo and Kang (2014) shows that in order to reject very general disturbance, the additional angular signal ux (1, t ) seems necessary. This can also be seen from the fact that when d = −6, system (1) has zero dynamic u(x, t ) = 3x2 − x3 that gives rise to the velocity at the right end identical to zero: ut (1, t ) ≡ 0. In addition, we can make ut (1, t ) and ux (1, t ) equidimensional in (2) by properly choosing δ . Similar design can also be found in Cheng et al. (2011) and Ge et al. (2001).

Let L2 (0, 1) be the usual Hilbert space with the inner product ⟨·, ·⟩ and induced norm ∥ · ∥. Define  H = {f ∈ H 4 (0, 1)| f (0) = f ′ (0) = f ′′ (1) = 0},    ∥f ∥ = ∥f ′′′ ∥ + ∥f (4) ∥, ∀ f ∈ H ,  H

H0 = {φ ∈ H 2 (0, 1)| φ(0) = φ ′ (0) = 0},

   

∥f ∥H0 = ∥f ′′ ∥,

(8)

∀ f ∈ H0 .

It is obvious that both H and H0 are closed, dense, subspaces of H 4 (0, 1) and H 2 (0, 1) respectively. For notational simplicity, we also use u˙ to denote ut in what follows. Theorem 3.1. Given T > 0. Assume that the initial value (u0 , u1 ) ∈ H × H satisfies the compatible condition: ′ u′′′ 0 (1) ∈ ku1 (1) + M sign(u1 (1) + δ u0 (1)) + d(0),

(9)

where the left side of (9) is specified to be a fixed value of the right side and d˙ ∈ L2loc (0, ∞). Then (5) admits a Filippov solution. Moreover, the

F.-F. Jin, B.-Z. Guo / Automatica 52 (2015) 95–102

97

solution is exponentially stable:

which satisfies

1+δ − δ t E (t ) 6 e 1+δ E (0), 1−δ

 N ⟨u (·, t ), φ⟩ + ⟨uNxx (·, t ), φxx ⟩    tt ∈ −φ(1)[kuNt (1, t ) + M sign(uNt (1, t ) +δ uNx (1, t )) + d(t )], ∀ φ ∈ VN ,    N u (x, 0) = u0 (x), uNt (x, 0) = u1 (x).

∀ t > 0,

(10)

where E is defined in (6). Proof. We first claim that the classical solution must be unique locally provided that ut (1, t ) + δ ux (1, t ) ̸= 0 locally for t. Suppose otherwise, there are two solutions (u, ut ) and (w, wt ) to (5) with the same initial value and ut (1, t ) + δ ux (1, t ) ̸= 0, wt (1, t ) + δwx (1, t ) ̸= 0. Let p(x, t ) = u(x, t ) − w(x, t ). Then p satisfies

 ptt (x, t ) + pxxxx (x, t ) = 0,    p(0, t ) = px (0, 1) = pxx (1, t ) = 0, pxxx (1, t ) = kpt (1, t ) + M [sign(ut (1, t )   + δ ux (1, t )) − sign(wt (1, t ) + δwx (1, t ))],   p(x, 0) = pt (x, 0) = 0.

(11)

V1 ( t ) =

2

X (t ) = (a1N (t ), . . . , aNN (t ), a˙ 1N (t ), . . . , a˙ NN (t ))⊤ .

(19) where A1 , A2 are matrices and c , c1 are vectors in R identity matrix of RN ):

[p2t (x, t ) + p2xx (x, t )]dx 1



xpt (x, t )pxx (x, t )dx,

(12)

0

 A1 =

IN 0





0

({⟨φi , φj ⟩}

)

N ⊤ i,j=1



A2 =

V˙ 1 (t ) = −pt (1, t )pxxx (1, t ) − δ px (1, t )pxxx (1, t )

c = (0, . . . , 0, φ1 (1), . . . , φN (1))⊤ ,

δ

1



2

2

∈ − (1, t ) − kδ pt (1, t )px (1, t ) − M × [sign(ut (1, t ) + δ ux (1, t )) − sign(wt (1, t ) + δwx (1, t ))][ut (1, t ) + δ ux (1, t ) − wt (1, t )  δ 1 2 [pt (x, t ) + 3p2xx (x, t )]dx − δwx (1, t )] −

1 −1 − M sign(⟨c + δ c1 , X (t )⟩)A− 1 c − d(t )A1 c .

where we used the following fact that is valid for multi-valued symbolic function defined by (3):

[sign(f (t )) − sign(g (t ))][f (t ) − g (t )] ≥ 0,

∀ f , g ∈ L2 (0, T ). (14)

This shows that V1 (t ) ≡ V1 (0) = 0, that is, (u, ut ) = (w, wt ). Next, we prove the existence of the Filippov solution by the Galerkin approximation method motivated by Guo and Guo (2007). Multiply the first equation of (5) by φ ∈ H0 and integrate over [0, 1] with respect to x to obtain

⟨utt (·, t ), φ⟩ + ⟨uxx (·, t ), φxx ⟩ ∈ −φ(1)[kut (1, t ) + M sign(ut (1, t ) + δ ux (1, t )) + d(t )].

(15)

+ Suppose that {φn }∞ 1 is an orthonormal basis for H . For any N ∈ Z , define a finite-dimensional subspace of H0 by VN = span{φ1 , φ2 , . . . , φN }. Since (u0 , u1 ) ∈ H × H , we may assume without loss of generality that u0 , u1 ∈ span{φ1 , φ2 , . . . , φN }. A Galerkin approximation solution to (5) is constructed as follows: N

uN ( x , t ) =

 n =1

anN (t )φn (x),

(20)

1 ⊤ X˙ (t ) ∈ A− 1 (A2 − ckc )X (t )

(13)

2

,

{φi }Ni=1 for L2 (0, 1) owing to the density of H in L2 (0, 1), we can 1 rewrite (19) as (noticing c1⊤ A− 1 c = 0)

0

δ + p2t (1, t ) 6 0,

IN 0

Since {⟨φi , φj ⟩}Ni,j=1 is the Gramian matrix composed of the basis

kp2t

2

−({⟨φi′′ , φj′′ ⟩}Ni,j=1 )⊤

(IN is the

c1 = (φ1′ (1), . . . , φN′ (1), 0, . . . , 0)⊤ .

δ [p2t (x, t ) + 3p2xx (x, t )]dx + p2t (1, t )

0

0

2N

,

where δ is the same as that in (6). Computing the derivative of V1 along the solution of (11) gives

−

(18)

A1 X˙ (t ) ∈ (A2 − ckc ⊤ )X (t ) − M sign(⟨c + δ c1 , X (t )⟩)c − d(t )c ,

0

+δ

 N ⟨u (·, t ), φn ⟩ + ⟨uNxx (·, t ), φn′′ ⟩    tt ∈ −φn (1)[kuNt (1, t ) + M sign(uNt (1, t ) + δ uNx (1, t )) + d(t )], ∀ φn , n = 1, 2, . . . , N ,    N u (x, 0) = u0 (x), uNt (x, 0) = u1 (x).

Then (18) can be written as an ODE system of the following:

1



Eq. (17) is equivalent to

Set

Define a Lyapunov functional 1

(17)

(16)

(21)

Owing to the multi-valued nature for symbolic function, the solution of (21) is understood in the sense of Filippov that (Filippov, 1998, p. 48) 1 −1 −1 ⊤ X˙ (t ) ∈ A− 1 (A2 − ckc )X (t ) − MU (X (t ))A1 c − d(t )A1 c ,

(22)

where U (X ) = sign(⟨c + δ c1 , X ⟩). When c = 0, (22) admits only 1 classical solution: X˙ (t ) = A− 1 A2 X (t ). When c ̸= 0, set S (t ) = ⟨c + δ c1 , X (t )⟩. Then U (X (t )) is discontinuous only on S (t ) = 0. Now we use the equivalent control method to find the sliding mode solution (locally) for (22). That is, we need to find a continuous function U eq , called the equivalent control, so that 1 −1 −1 ⊤ eq X˙ (t ) = A− 1 (A2 − ckc )X (t ) − MU (t )A1 c − d(t )A1 c .

(23)

Since the equivalent control is needed only when S (t ) = 0 and hence S˙ (t ) = ⟨c + δ c1 , X˙ (t )⟩ = 0, we find

 eq −1 −1 U (t ) = M −1 (c ⊤ A1 c )−1 [(c ⊤ + δ c1⊤ )A1 ⊤ ⊤ ⊤ −1  eq × (A2 − ckc )X (t ) − d(t )(c + δ c1 )A1 c ], U (t ) ∈ [−1, 1].

(24)

Returning back to (17), the sliding mode is S (t ) = uNt (1, t ) + δ (1, t ) = 0 on which we have uNx

 N ⟨utt (·, t ), φ⟩ + ⟨uNxx (·, t ), φxx ⟩ ∈ −φ(1) × [kuNt (1, t ) + MU eq (t ) − d(t )], ∀ φ ∈ VN ,  N ut (1, t ) + δ uNx (t )(1, t ) ≡ 0.

(25)

98

F.-F. Jin, B.-Z. Guo / Automatica 52 (2015) 95–102

The rest of proof is split into the following four lemmas where the proofs for Lemmas 3.1–3.3 are presented in the Appendix. These lemmas are uniform estimations for discrete solutions that are used in the proof of Theorem 3.1 on the convergence of discrete solutions. Lemma 3.1. For either the classical solution or the Filippov solution of (17), u˙ N (1, t ) ∈ L2 (0, ∞), sup max[∥˙uN (·, t )∥ + ∥uNxx (·, t )∥] < ∞.



(26)

t ≥0

N

Lemma 3.2. For either the classical solution or the Filippov solution of (17), sup ∥¨uN (·, 0)∥ < ∞.

(27)

N

Lemma 3.3. For either the classical solution or the Filippov solution of (17), sup[∥¨uN (·, t )∥ + ∥˙uNxx (·, t )∥] < ∞,

∀ t ∈ [0, T ] a.e.

(28)

N

Lemma 3.4. For any T > 0, suppose that fi → f in L2 (0, T ) strongly as i → ∞ and g ∈ L2 (0, T ). Then under definition (4), there exists a ∞ subsequence {fij }∞ j=1 of {fi }i=1 such that T



sign(fij (t ))g (t )dt ⊂

lim

j→∞

0

T



sign(f (t ))g (t )dt ,

(29)

0

which is understood in the following sense:

integer N > 0 such that for all j ≥ N,

 1 1  f ≥ min f > 0 in P2+ε ,  fij > f − min + +

(30)

Continuation of proof of Theorem 3.1. From Lemmas 3.1 to 3.3, we can extract a subsequence, still denoted by {N } without confusion, such that

 N u ⇀ u in L∞ (0, T ; H 2 (0, 1)) weak∗ , u˙ N ⇀ u˙ in L∞ (0, T ; H 2 (0, 1)) weak∗ ,  N u¨ ⇀ u¨ in L∞ (0, T ; L2 (0, 1)) weak∗ .

∪

To this end, we need to show that (a) {uNt (1, ·)} is bounded in L2 (0, T ); (b) {uNt (1, ·)} is equicontinuous, that is, ∥uNt (1, t + ξ ) − N ut (1, t )∥L2 (0,T ) → 0 as ξ → 0 uniformly in N. The claim (a) comes directly from Lemma 3.1. We need only show the claim (b). From (C.6) in the proof of Lemma 3.3, we have

 k−

δ 2

−

2 u˙ (1, t + ξ ) − u˙ (1, t ) dt

k2 δ

 T

2

0

N

N

≤ −[V (T , ξ ) − V N (0, ξ )]   −1  T δ k2 δ 1 + k− − +δ [d(t + ξ ) − d(t )]2 dt . (36) 2

2

2

0

The second term on the right-hand side of (36) obviously tends to zero as ξ → 0. To show the first term on the right-hand side of (36) tending to zero as ξ → 0, it suffices, by definition of V N (t , ξ ) in (C.5), that for τ = 0, T ,

∥˙uN (·, τ + ξ ) − u˙ N (·, τ )∥2 + ∥uNxx (·, τ + ξ ) − uNxx (·, τ )∥2 → 0 as ξ → 0,

P2−ε .

(31) P2+ε

2ε

(35)

N

Since and are closed and f is continuous on ∪ P2−ε , mint ∈P + f > 0 and maxt ∈P − f < 0. By (31), there exists a positive 2ε

(34)

We first show that there exists a subsequence of {uNt (1, ·)}, still denote by itself without confusion, such that

t ∈ [0, T ] a.e. as j → ∞.

P2−ε

(33)

This proves the first part of (30). Notice that in derivation of (33), we used the estimation meas((P + \ P2+ε ) ∪ (P − \ P2−ε )) ≤ 4ε . 

2

By Lusin’s theorem (Royden & Fitzpatrick, 2010, p. 66), there exist closed sets P1+ε ⊂ P + and P1−ε ⊂ P − with meas(P + \ P1+ε ) < ε and meas(P − \ P1−ε ) < ε such that f is continuous over P1+ε ∪ P1−ε . By Egoroff’s theorem (Royden & Fitzpatrick, 2010, p. 64), there exist closed sets P2+ε ⊂ P1+ε and P2−ε ⊂ P1−ε satisfying meas(P1+ε \ P2+ε ) < ε , meas(P1−ε \ P2−ε ) < ε such that

P2+ε

2 t ∈P2ε

      sign(fi (t ))g (t )dt − sign(f (t ))g (t )dt  j   P P     [sign(fij (t )) − sign(f (t ))]g (t )dt  =  (P + \P2+ε )∪(P − \P2−ε ) √ ≤ 4 ε∥g ∥L2 ((P + \P + )∪(P − \P − )) 2ε 2ε √ ≤ 4 ε∥g ∥L2 (0,T ) .

1

Proof. The second part comes from (4) that ∪∞ j=1 sign(fij ) ⊂ sign(f ). We need only show the first part of (30). Let P + = {t ∈ [0, T ]| f (t ) > 0} and P − = {t ∈ [0, T ]| f (t ) < 0}; P = P + ∪ P − . Since fi → f in L2 (0, T ) strongly, by the Riesz–Fischer theorem (Royden & Fitzpatrick, 2010, p. 148), there exists {fij }∞ j=1 so that

fij (t ) → f (t ) uniformly for t ∈

2 t ∈P2ε

(32)

Therefore, for j ≥ N,

where P = {t ∈ [0, T ]| f (t ) ̸= 0} and Q = {t ∈ [0, T ]| f (t ) = 0}.

P2+ε

2 t ∈P2ε

uNt (1, ·) → ut (1, ·) in L2 (0, T ) strongly.

    sign(fij (t ))g (t )dt = sign(f (t ))g (t )dt ,  jlim   P  →∞ P     sign(fij (t ))g (t )dt = λ(t )g (t )dt | λ(t )    Q Q        ∈ sign(fij (t ))|Q , j = 1, 2, . . . ,         ⊂ sign(f (t ))g (t )dt = λ(t )g (t )dt | λ(t )    Q Q           ∈ sign(f (t ))|Q ,

fij (t ) → f (t ),

2 x∈P2ε

1 1   f ≤ max f < 0 in P2−ε . fij < f − max − −

(37)

uniformly in N. To this end, by Lemma 3.3, we have, for τ = 0, T , that there exists a constant C3 > 0 such that



∥˙uN (·, τ + ξ ) − u˙ N (·, τ )∥2 ≤ ∥¨uN (·, τ1ξ )∥2 |ξ | ≤ C3 |ξ |, ∥uNxx (·, τ + ξ ) − uNxx (·, τ )∥2 ≤ ∥˙uNxx (·, τ2ξ )∥2 |ξ | ≤ C3 |ξ |,

(38)

where τ1ξ , τ2ξ ∈ (t , t + ξ ). This leads to (37) and hence (35). Second, we show that there exists a subsequence of {uNx (1, ·)}, still denote by itself without confusion, such that uNx (1, ·) → ux (1, ·) in L2 (0, T ) strongly.

(39)

F.-F. Jin, B.-Z. Guo / Automatica 52 (2015) 95–102

(a) The displacement of the beam.

99

(b) The velocity of the beam.

Fig. 1. Displacement and velocity of the beam without control (U = 0).

To this end, by Lemmas 3.1 and 3.3, sup max [∥ t ∈[0,T ]

N

uNxx

We thus obtain that

(·, t )∥ + ∥˙ (·, t )∥] < ∞.

Since uNx (1, t ) = have

uNxx

1 0

uNxx (x, t )dx and u˙ Nx (1, t ) =

1 0

u˙ Nxx (x, t )dx, we

 N ∥u (1, ·)∥L2 (0,T ) ≤ sup max ∥uNxx (·, t )∥,    x N t ∈[0,T ] (40) |uNx (1, t1 ) − uNx (1, t2 )|    ≤ sup max ∥˙uNxx (·, t )∥ |t1 − t2 |, ∀ ti ∈ [0, T ], i = 1, 2. N

t ∈[0,T ]

Therefore, {uNx (1, ·)} is uniformly bounded and equicontinuous in L2 (0, T ). This gives (39). Since for any ψ ∈ C0∞ (0, T ) and φ ∈ H , T



⟨uNtt (·, t ), φ⟩ψ(t )dt + 0

T



⟨uNxx (·, t ), φxx ⟩ 0

× ψ(t )dt ∈ −φ(1)

T



[kuNt (1, t ) 0

+ M sign(uNt (1, t ) + δ uNx (1, t )) + d(t )]ψ(t )dt ,

(41)

by (35), (39), and Lemma 3.4, T



sign(uNt (1, t ) + δ uNx (1, t ))ψ(t )dt

lim

N →∞

utt + uxxxx = 0, u(0, t ) = ux (0, 1) = uxx (1, t ) = 0, (47) uxxx (1, t ) ∈ kut (1, t ) + M sign(ut (1, t ) + δ ux (1, t )) + d(t ).



This proves the existence of the solution to closed-loop system (5). The exponential stability (10) is then a consequence of (7). Remark 3.1. As a consequence of Theorem 3.1, we can construct an unknown input type observer for system (1), which is an independent work in control theory. The observer is designed as follows:

uˆ (x, t ) + uˆ (x, t ) = 0, x ∈ (0, 1), t > 0, tt xxxx    uˆ (0, t ) = uˆ x (0, t ) = 0, t > 0,   uˆ xx (1, t ) = 0, t > 0, ˆ xxx (1, t ) ∈ k[ˆut (1, t ) − ut (1, t )] u     + M sign(ˆut (1, t ) − ut (1, t ) + δ uˆ x (1, t )  − δ ux (1, t )) + U (t ), k > 0, t > 0.

(48)

It is easily seen that the error variable (˜u, u˜ t ) = (ˆu − u, uˆ t − ut ) satisfies (5) and hence is exponentially stable. Therefore, under some regularity assumption, the observer (48) is exponentially convergent. We believe that this is the first unknown input type observer designed for Euler–Bernoulli beam with external disturbance.

0

4. Numerical simulation

T



sign(ut (1, t ) + δ ux (1, t ))ψ(t )dt .

⊂

(42)

0

By (34) and (42), passing to the limit as N → ∞ in (41), we obtain

⟨utt , φ⟩ + ⟨uxx , φ ′′ ⟩ ∈ −φ(1)[kut (1, t ) + M sign(ut (1, t ) + δ ux (1, t )) + d(t )], t ∈ [0, T ] a.e.

(43)

Taking φ ∈ C0 (0, 1) in (43) gives ∞

⟨utt , φ⟩ + ⟨uxx , φ ′′ ⟩ = 0,

∀ φ ∈ C0∞ (0, 1).

(44)

This shows that the generalized derivative uxxxx exists and utt + uxxxx = 0 for almost all t ∈ [0, T ]. In particular, u(·, t ) ∈ H 4 (0, 1). Performing integration by parts for (43) over [0, 1] with respect to x gives

⟨utt , φ⟩ + uxx (1, t )φ ′ (1) − uxx (0, t )φ ′ (0) − uxxx (1, t )φ(1) + uxxx (0, t )φ(0) + ⟨uxxxx , φ⟩ = ⟨utt , φ⟩ + uxx (1, t )φ ′ (1) − uxxx (1, t )φ(1) + ⟨uxxxx , φ⟩ ∈ −φ(1)[kut (1, t ) + M sign(ut (1, t ) + δ ux (1, t )) + d(t )], (45) which implies that uxx (1, t ) = 0, uxxx (1, t ) ∈ kut (1, t ) + M sign(ut (1, t ) + δ ux (1, t )) + d(t ).

(46)

In this section, a finite difference method is applied to compute numerical solution for the displacement and velocity of closedloop system (5) to illustrate the effect of the control law and for observer (48) to demonstrate the estimation effect. Since the exact value of zero is never managed in computer simulations, in our numerical simulations we set sign(0) = 0, similar to that of the standard MATLAB function sign. However, we do not know if under the single valued condition sign(0) = 0, there exists solution to the closed-loop system (5). We take the steps for space and time as 0.05 and 0.0005, respectively, and choose k = 2, M = 8, δ = 3/4, d(t ) = 3(−1 + cos t ), and the initial value: u0 = −8x3 + 4x4 ,

u1 = 15x2 − 5x3 .

(49)

It is easy to show that (u0 , u1 ) ∈ H × H satisfies condition (9) in Theorem 3.1. Fig. 1 displays the displacement (Fig. 1(a)) and velocity (Fig. 1(b)) without control (i.e., U = 0). It is seen that both displacement and velocity are vibrating. Fig. 2 displays the displacement (Fig. 2(a)) and velocity (Fig. 2(b)) with control law (2). It is seen that the control law (2) is very effective to make both displacement and velocity convergent.

100

F.-F. Jin, B.-Z. Guo / Automatica 52 (2015) 95–102

(a) The displacement.

(b) The velocity.

Fig. 2. Displacement and velocity of the beam with the control law (2).

(a) The displacement.

(b) The velocity.

(c) Square wave disturbance.

Fig. 3. Displacement and velocity of the beam with the control law (2) and square wave disturbance.

(a) Displacement of system (1).

(b) Displacement of observer (48).

(c) Displacement of error system (5).

Fig. 4. Displacements for system (1), observer (48) with control U (t ) = 3 sin t, and error (5).

Fig. 3 demonstrates the displacement (Fig. 3(a)) and velocity (Fig. 3(b)) with control law (2) and the disturbance (Fig. 3(c)). It is seen that even for the discontinuous disturbance (square wave disturbance) that does not satisfy the smooth condition required in mathematical proof, the control law (2) can also stabilize both displacement and velocity satisfactorily. Fig. 4 demonstrates the convergence of observer (48), where Fig. 4(a) displays the displacement of system (1) with control U = 3 sin t and d(t ) = 3(−1 + cos t ) whereas, Fig. 4(b) shows the displacement of system (48) with the same control. The initial values for Fig. 2 are chosen as u0 = −8x3 + 4x4 , uˆ 0 = 5 sin x,

u1 = 15x2 − 5x3 ,

uˆ 1 = 4x2 .

(50)

It is seen that the initial value of uˆ system is actually not compatible. However, from Fig. 4(c), after t = 3 around, the displacement of the observer recovers the displacement of the control plant satisfactorily. Since the error system is the same as that for Fig. 2 with only different initial condition, Fig. 4(c) shows that system (5) may

converge with incompatible initial condition (note that the compatible condition is only used for mathematical proof for the existence of the solution). 5. Concluding remarks In this paper, we design a variable structure output feedback control to stabilize an Euler–Bernoulli beam with general control matched disturbance. The Filippov solution to the closed-loop system is proved to exist by a Galerkin approximation scheme. The closed-loop is shown to be exponentially stable. This improves the disturbance rejection presented in Guo and Kang (2014) where the disturbance is a finite sum of harmonic series. We need an additional output signal, the angular of the control end, except the velocity to remove the zero dynamics in the presence of disturbance. The Lyapunov functional approach is applied for both design and stability analysis. The numerical simulations validate the theoretical results. Compared with our previous work (Guo & Jin, 2013) by state feedback, the present paper improves the design to output feedback.

F.-F. Jin, B.-Z. Guo / Automatica 52 (2015) 95–102

101

Appendix A. Proof of Lemma 3.1

By compatible condition (9), we finally obtain that

Take φ(x) = u˙ N (x, t ) and φ(x) = xuNx (x, t ) in (17) separately to give

∥¨uN (·, 0)∥2 = −⟨u(04) , u¨ N (·, 0)⟩,



1 2

d dt

∥˙uN (·, t )∥2 +

d dt

∥uNxx (·, t )∥2



The result follows.

∈ −˙uN (1, t )[ku˙ N (1, t ) + M sign(˙uN (1, t ) (A.1)

For fixed t , ξ > 0 with ξ < T − t, replace t by t + ξ in (17) and subtract the first equation of (17) to obtain

and

⟨uNtt (·, t ), xuNx (·, t )⟩

⟨¨uN (·, t + ξ ) − u¨ N (·, t ), φ⟩ + ⟨uNxx (·, t + ξ ) − uNxx (·, t ), φ ′′ ⟩

∈ −⟨uNxx (·, t ), (xuNx (·, t ))xx ⟩ − uNx (1)[ku˙ N (1, t )

∈ −φ(1)[ku˙ N (1, t + ξ ) + M sign(˙uN (1, t + ξ ) + δ uNx (1, t + ξ )) + d(t + ξ )] + φ(1)[ku˙ N (1, t )

+ M sign(˙uN (1, t ) + δ uNx (1, t )) + d(t )] 3

1

2

= − ∥uNxx (·, t )∥2 − (uNxx (1, t ))2 − uNx (1) × [ku˙ (1, t ) + M sign(˙u (1, t ) + δ N

N

uNx

(1, t )) + d(t )].

+ M sign(˙uN (1, t ) + δ uNx (1, t )) + d(t )]. (A.2)

Define a Lyapunov functional E N (t )

=

[∥˙u (·, t )∥ + ∥ 2 ≤ D2 E N (t ),

D1 E (t ) ≤

N

2

uNxx

2 dt

(A.3)

E˙ N (t ) 6 − k −

δ

δ 2

−

k2 δ



2 2

2

uNxx

(·, t )∥ ] (A.4)

− uNx (·, t )]⟩

× [ku˙ N (1, t + ξ ) + M sign(˙uN (1, t + ξ ) + δ uNx (1, t + ξ )) + d(t + ξ )] + [uNx (1, t + ξ ) + δ uNx (1, t )) + d(t )]

2

3

= − ∥uNxx (·, t + ξ ) − uNxx (·, t )∥2 2 1

− [uNxx (1, t + ξ ) − uNxx (1, t )]2 − [uNx (1, t + ξ ) 2

− uNx (1, t )][ku˙ N (1, t + ξ ) + M sign(˙uN (1, t + ξ ) + δ uNx (1, t + ξ )) + d(t + ξ )] + [uNx (1, t + ξ )

Set t = 0 in (17) to give

⟨¨uN (·, 0), φ⟩ + ⟨u′′0 , φ ′′ ⟩ ∈ −φ(1)[ku1 (1)

− uNx (1, t )][ku˙ N (1, t ) + M sign(˙uN (1, t )

+ M sign(u1 (1) + δ u′0 (1)) + d(0)]. Take φ(x) = u¨ N (x, 0) in (B.1) to give ∥¨uN (·, 0)∥2 ∈ −⟨u′′0 , u¨ Nxx (·, 0)⟩ − u¨ N (1, 0) × [ku1 (1) + M sign(u1 (1) + δ u′0 (1)) + d(0)]

(B.1)

(4)

V N (t , ξ ) =

N

1 2

[∥˙uN (·, t + ξ ) − u˙ N (·, t )∥2

+ δ⟨˙uN (·, t + ξ ) − u˙ N (·, t ),

− u¨ (1, 0)[ku1 (1) + M sign(u1 (1) + δ u0 (1)) + d(0)] ′

x[uNx (·, t + ξ ) − uNx (·, t )]⟩,

= u′′′ uN (1, 0) − u¨ N (1, 0)[ku1 (1) 0 (1)¨ + M sign(u1 (1) + δ u′0 (1)) + d(0)] − ⟨u(04) , u¨ N (·, 0)⟩.

(C.4)

+ ∥uNxx (·, t + ξ ) − uNxx (·, t )∥2 ]

× u¨ (1, 0) − u0 (0)¨u (0, 0) − ⟨u0 , u¨ (·, 0)⟩ N

+ δ uNx (1, t )) + d(t )]. Define a Lyapunov functional

= −u′′0 (1)¨uNx (1, 0) + u′′0 (0)¨uNx (0, 0) + u′′′ 0 (1) N

Taking φ(x) = x[uNx (x, t + ξ ) − uNx (x, t )] in (C.1), we obtain that

− uNx (1, t )][ku˙ N (1, t ) + M sign(˙uN (1, t )

Appendix B. Proof of Lemma 3.2

′′′

− d(t )][˙uN (1, t + ξ ) − u˙ N (1, t )].

− uNx (·, t )])xx ⟩ − [uNx (1, t + ξ ) − uNx (1, t )]

which is the first inequality of (26). The second inequality of (26) comes from the equivalence between E N (t ) and 12 [∥˙uN (·, t )∥2 + ∥uNxx (·, t )∥2 ]. 

N

(C.3)

∈ −⟨uNxx (·, t + ξ ) − uNxx (·, t ), (x[uNx (·, t + ξ )

2

δ E N (t ). 1+δ This shows that E N (t ) 6 E N (0) for all t > 0 and   −1  ∞ δ k2 δ N 2 (˙u (1, t )) dt ≤ k − − E N (0), 2

+ d(t + ξ ) − M sign(˙uN (1, t ) + δ uNx (1, t ))

⟨¨uN (·, t + ξ ) − u¨ N (·, t ), x[uNx (·, t + ξ )

6 −

0

where

I2 = −[M sign(˙uN (1, t + ξ ) + δ uNx (1, t + ξ ))

(˙uN (1, t ))2

− [∥˙u (·, t )∥ + ∥ N

(C.2)

I1 = −k[˙uN (1, t + ξ ) − u˙ N (1, t )]2 6 0,

(·, t )∥ ] 2

where D1 and D2 are constants independent of N. The last fact comes from (16) by uNx (0, t ) = 0. The derivative of E N along the solution (classical or Filippov solution) of (17) is found to be



N

∥˙uN (·, t + ξ ) − u˙ N (·, t )∥2  + ∥uNxx (·, t + ξ ) − uNxx (·, t )∥2 ∈ I1 + I2 ,

[∥˙uN (·, t )∥2 + ∥uNxx (·, t )∥2 ] 2  1 +δ xu˙ N (x, t )uNx (x, t )dx, 1

(C.1)

Take φ(x) = u˙ (x, t + ξ ) − u˙ (x, t ) in (C.1) to yield N

1 d 

1

0 N



Appendix C. Proof of Lemma 3.3

+ δ uNx (1, t )) + d(t )],

2

(B.3)

∥¨uN (·, 0)∥ 6 ∥u(04) ∥.

(B.2)

(C.5)

which is the same type of (A.3). Differentiate (C.5) with respect to t along the solution of (17), by taking (14), (C.2), and (C.4) into

102

F.-F. Jin, B.-Z. Guo / Automatica 52 (2015) 95–102

account, to obtain



V˙ N (t , ξ ) ≤ − k −

δ 2

k δ 2

−



2

[˙uN (1, t + ξ ) − u˙ N (1, t )]2

− [d(t + ξ ) − d(t )][˙uN (1, t + ξ ) − u˙ N (1, t )] δ + [d(t + ξ ) − d(t )]2 2   −1 1 δ k2 δ ≤ k− − + δ [d(t + ξ ) − d(t )]2 , (C.6) 2

2

2

1

where we used the fact [uNx (1, t + ξ ) − uNx (1, t )]2 ≤ 0 [uNxx (x, t + ξ ) − uNxx (x, t )]2 dx since uN (·, t ) ∈ H0 . Integrating (C.6) over [0, t ] with respect to t, we obtain 1

V (t , ξ ) − V (0, ξ ) ≤ N

N

 k−

2

δ 2

−

k2 δ

 −1

2

 +δ

t



[d(s + ξ ) − d(s)]2 ds.

×

(C.7)

0

Divide (C.7) on both sides by ξ 2 and pass to the limit as ξ → 0, to give 1 2

[∥¨uN (·, t )∥2 + ∥˙uNxx (·, t )∥2 ]  1  6 −δ xu¨ N (x, t )˙uNx (x, t )dx − 0

1

1



xu¨ N (x, 0)˙uNx (x, 0)dx 0

+ [∥¨u (·, 0)∥ + ∥˙ (·, 0)∥ ] 2    −1 t 1 δ k2 δ + k− − +δ |d˙ (s)|2 ds, N

2

2

uNxx

2

2

2

(C.8)

0

which is equivalent to

∥¨uN (·, t )∥2 + ∥˙uNxx (·, t )∥2  1+δ ∥¨uN (·, 0)∥2 + ∥˙uNxx (·, 0)∥2 + ∥˙uNx (·, 0)∥2 6 1−δ    −1 t k2 δ δ 2 ˙ |d(s)| ds . +δ + k− − 2

2

(C.9)

0

By Lemma 3.2, we obtain that sup max [∥¨uN (·, t )∥ + ∥˙uNxx (·, t )∥] < ∞.  N

t ∈[0,T ]

Ge, S. Sam, Zhang, S., & He, W. (2001). Vibration control of an Euler–Bernoulli beam under unknown spatiotemporally varying disturbance. International Journal of Control, 84, 947–960. Guo, B. Z., & Guo, W. (2007). Adaptive stabilization for a Kirchhoff-type nonlinear beam under boundary output feedback control. Nonlinear Analysis, 66, 427–441. Guo, W., & Guo, B. Z. (2013). Stabilization and regulator design for a onedimensional unstable wave equation with input harmonic disturbance. International Journal of Robust and Nonlinear Control, 23, 514–533. Guo, W., Guo, B. Z., & Shao, Z. C. (2011). Parameter estimation and stabilization for a wave equation with boundary output harmonic disturbance and noncollocated control. International Journal of Robust and Nonlinear Control, 21, 1297–1321. Guo, B. Z., & Jin, F. F. (2013). The active disturbance rejection and sliding mode control approach to the stabilization of Euler–Bernoulli beam equation with boundary input disturbance. Automatica, 49, 2911–2918. Guo, B. Z., & Kang, W. (2014). Lyapunov approach to the boundary stabilization of a beam equation with boundary disturbance. International Journal of Control, 87, 925–939. Guo, B. Z., & Yu, R. (2001). The Riesz basis property of discrete operators and application to a Euler–Bernoulli beam equation with boundary linear feedback control. IMA Journal of Mathematical Control and Information, 18, 241–251. Han, S. M., Benaroya, H., & Wei, T. (1999). Dynamics of transversely vibrating beams using four engineering theories. Journal of Sound and Vibration, 225, 935–988. Immonen, E., & Pohjolainen, S. (2006). Feedback and feedforward output regulation of bounded uniformly continuous signals for infinite-dimensional systems. SIAM Journal on Control and Optimization, 45, 1714–1735. Jayawardhana, B., & Weiss, G. (2009). State convergence of passive nonlinear systems with an L2 input. IEEE Transactions on Automatic Control, 54, 1723–1727. Ke, Z., Logemann, H., & Rebarber, R. (2009). Approximate tracking and disturbance rejection for stable infinite-dimensional systems using sampled-data low-gain control. SIAM Journal on Control and Optimization, 48, 641–671. Krstic, M., & Smyshlyaev, A. (2008). Adaptive boundary control for unstable parabolic PDEs—part I: Lyapunov design. IEEE Transactions on Automatic Control, 53, 1575–1591. Luo, Z. H., Guo, B. Z., & Morgul, O. (1999). Stability and stabilization of infinite dimensional systems with applications. London: Springer-Verlag. Luo, Z. H., Kitamura, N., & Guo, B. Z. (1995). Shear force feedback control of flexible robot arms. IEEE Transactions on Robotics and Automation, 11, 760–765. Pisano, A., Orlov, Y., & Usai, E. (2011). Tracking control of the uncertain heat and wave equation via power-fractional and sliding-mode techniques. SIAM Journal on Control and Optimization, 49, 363–382. Rebarber, R., & Weiss, G. (2003). Internal model based tracking and disturbance rejection for stable well-posed systems. Automatica, 39, 1555–1569. Royden, H. L., & Fitzpatrick, P. M. (2010). Real analysis (4th ed.). Boston: Prentice Hall. Smyshlyaev, A., Guo, B. Z., & Krstic, M. (2009). Arbitrary decay rate for Euler–Bernoulli beam by backstepping boundary feedback. IEEE Transactions on Automatic Control, 54, 1134–1140.

Feng-Fei Jin received the M.Sc. degree from the Shandong Normal University, Jinan, China in 2008, and the Ph.D. degree from the Academy of Mathematics and Systems Science, Academia Sinica, Beijing, China in 2011. He was a postdoctoral fellow at University of the Witwatersrand, South Africa from 2011 to 2012. He is currently a postdoctoral fellow at University of the Witwatersrand, South Africa. His research interests focus on distributed parameter systems control.

(C.10)

References Chen, G., Delfour, M. C., Krall, A. M., & Payre, G. (1987). Modeling, stabilization and control of serially connected beam. SIAM Journal on Control and Optimization, 25, 526–546. Cheng, M. B., Radisavljevic, V., & Su, W. C. (2011). Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties. Automatica, 47, 381–387. Filippov, A. F. (1998). Differential equations with discontinuous righthand sides. Dordrecht: Kluwer Academic Publishers.

Bao-Zhu Guo received the Ph.D. degree from the Chinese University of Hong Kong in applied mathematics in 1991. From 1985 to 1987, he was a Research Assistant at the Beijing Institute of Information and Control, China. During the period 1993–2000, he was with the Beijing Institute of Technology, first as an associate professor (1993–1998) and subsequently a professor (1998–2000). Since 2000, he has been with the Academy of Mathematics and Systems Science, the Chinese Academy of Sciences, where he is a research professor in mathematical system theory. His research interests include the theory of control and application of infinite-dimensional systems.