Static output feedback control via PDE boundary and ODE measurements in linear cascaded ODE–beam systems

Static output feedback control via PDE boundary and ODE measurements in linear cascaded ODE–beam systems

Automatica 50 (2014) 2787–2798 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Static out...

1MB Sizes 3 Downloads 51 Views

Automatica 50 (2014) 2787–2798

Contents lists available at ScienceDirect

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

Static output feedback control via PDE boundary and ODE measurements in linear cascaded ODE–beam systems✩ Huai-Ning Wu a,1 , Jun-Wei Wang b a

Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical Engineering, Beihang University (Beijing University of Aeronautics and Astronautics), Beijing 100191, PR China b

School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, PR China

article

info

Article history: Received 5 April 2012 Received in revised form 13 May 2014 Accepted 18 June 2014 Available online 29 September 2014 Keywords: Feedback control Cascade systems Distributed parameter systems Exponential stability Linear matrix inequalities (LMIs)

abstract This paper addresses the problem of feedback control design for a class of linear cascaded ordinary differential equation (ODE)–partial differential equation (PDE) systems via a boundary interconnection, where the ODE system is linear time-invariant and the PDE system is described by an Euler–Bernoulli beam (EBB) equation with variable coefficients. The objective of this paper is to design a static output feedback (SOF) controller via EBB boundary and ODE measurements such that the resulting closedloop cascaded system is exponentially stable. The Lyapunov’s direct method is employed to derive the stabilization condition for the cascaded ODE–beam system, which is provided in terms of a set of bilinear matrix inequalities (BMIs). Furthermore, in order to compute the gain matrices of SOF controllers, a twostep procedure is presented to solve the BMI feasibility problem via the existing linear matrix inequality (LMI) optimization techniques. Finally, the numerical simulation is given to illustrate the effectiveness of the proposed design method. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Cascaded dynamical systems appear in many control applications whether naturally or intentionally due to control design. Over the past decades, some effective methods for the control design of cascade-connected systems modeled by ordinary differential equations (ODEs) have been developed (Artstein, 1982; Bechlioulis & Rovithakis, 2011; Chen, 2009; Chen & Huang, 2004; Jiang & Praly, 1998; Kanellakopoulos, Kokotovic, & Morse, 1992), such as robust adaptive control (Jiang & Praly, 1998), dissipativity, stabilization and regulation (Chen & Huang, 2004), global stabilization (Chen, 2009), and robust partial-state feedback control with prescribed performance (Bechlioulis & Rovithakis, 2011). For

✩ This work was supported in part by the National Basic Research Program of China (973 Program) (2012CB720003); in part by the National Natural Science Foundations of China under Grants 61121003 and 61473011, and in part by the China Postdoctoral Science Foundation under Grant 2014M550618. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor George Weiss under the direction of Editor Miroslav Krstic. E-mail addresses: [email protected] (H.-N. Wu), [email protected] (J.-W. Wang). 1 Tel.: +86 10 82317301; fax: +86 10 82317332.

http://dx.doi.org/10.1016/j.automatica.2014.09.006 0005-1098/© 2014 Elsevier Ltd. All rights reserved.

the port-Hamiltonian systems, A. van der Schaft and collaborators have investigated the asymptotic stabilization, passivity and cyclopassivity via control by interconnection (Castanos, Ortega, Schaft, & Astolfi, 2009; Ortega, Schaft, Maschke, & Escobar, 2002). Despite these promising efforts, however, the behavior of many physical systems in nature must depend on spatial position as well as time, e.g., mechanical systems with heat flows, fluid flows, flexible structures, chemical reactor processes, to name a few. The mathematical models which describe the spatiotemporal behavior of these systems consist of partial differential equations (PDEs). Due to their infinite-dimensional nature, the existing finite-dimensional control theory and techniques for cascaded ODE systems are very difficult to be directly applied to the control design of cascaded systems including PDE dynamics. Recently, Krstic and his coworkers (Bekiaries-Liberis & Krstic, 2011; Krstic, 2009a,b,c; Krstic & Smyshlyaev, 2008; Susto & Krstic, 2010) have proposed some types of PDE–ODE cascades and developed the corresponding backstepping full-state feedback control design methods for them. In Krstic and Smyshlyaev (2008), the concept of delay-ODE cascades used in Artstein (1982) was extended to a new type of delay-ODE cascades where the actuator delay is modeled by the first-order hyperbolic PDE system. The first attempt of an exact extension of the predictor feedback and observer design was presented in Krstic (2009b), from delay-ODE

2788

H.-N. Wu, J.-W. Wang / Automatica 50 (2014) 2787–2798

cascades (Artstein, 1982; Krstic & Smyshlyaev, 2008) to diffusion PDE–ODE cascades, where the actuator dynamics modeled by heat equation are of more complex type than pure delay. In Krstic (2009c), the backstepping method was used for control and observer design in linear time-invariant (LTI) systems with a wave PDE in the actuation or sensing path. In Susto and Krstic (2010), the results of Krstic and Smyshlyaev (2008), Krstic (2009b,c) were extended to the case of input and sensor dynamics governed by diffusion PDEs with counter-convection. In Bekiaries-Liberis and Krstic (2011), a methodology for constructing control laws and observers was developed for multi-input–multi-output LTI systems with the distributed effect of actuator or sensor dynamics, governed by diffusion PDEs with counter-convection. However, it is noticed that these results require the complete information of the state variables of both PDE and ODE systems. Such a requirement results in the difficulty in implementation since the complete information of the system state is not always available in real-world applications. To overcome this difficulty, a backstepping observerbased feedback control design has been recently proposed for the linear coupled heat PDE–ODE system (Tang & Xie, 2011). Yet, the result in Tang and Xie (2011) is developed by resorting to the techniques of backstepping and observer-based design. Different from the observer-based control case, static output feedback (SOF) control represents the simplest closed-loop control system, which can be easily realized in practice. More recently, the problem of SOF control design has been considered in Wang, Wu, and Li (2014) for a class of diffusion PDE and ODE cascades, where the ODE model is used to describe the dynamics of the MIMO plant and the diffusion PDE model is employed to represent the actuator dynamics. On the other hand, the Euler–Bernoulli beam (EBB) equation is challenging due to its complex form and the fact that all of its eigenvalues (infinitely many) are on the imaginary axis. Hence, the analysis and controller synthesis for systems modeled by EBB equation are more difficult than ones for systems modeled by parabolic, hyperbolic or wave PDE models. Zhao and Weiss have presented the properties of well-posedness, regularity, and exact controllability for an EBB system coupled with an ODE system in Zhao and Weiss (2010) and developed a collocated SOF controller to suppress the vibrations of a wind turbine tower described by an EBB–ODE coupled model in Zhao and Weiss (2011a). In Zhao and Weiss (2011b), they have established the properties of controllability and observability for a well-posed infinite-dimensional system coupled with a finite-dimensional system. More recently, the problem of feedback control design with vibration suppression has been investigated in Gao, Wu, Wang, and Guo (2014) for a flexible air-breathing hypersonic vehicle, where the rigid motion dynamics is represented by an ODE and the flexible vehicle vibration is described by an EBB, respectively. But the result of Gao et al. (2014) does not consider the closed-loop well-posedness and is developed under the assumption that the full information of the rigid motion is available. To the best of authors’ knowledge, however, few results are available on the SOF control design for ODE–EBB cascades, in which only boundary information of the beam and the ODE measurement are used. Motivated by the aforementioned considerations, in this study, we will deal with the SOF control problem for a class of linear cascaded ODE–PDE systems consisting of an LTI finite-dimensional system and an EBB equation with variable coefficients via a boundary interconnection. The aim of this work is to design an SOF controller by only using the EBB boundary and ODE measurements, such that the resulting closed-loop cascaded system is exponentially stable. By employing the Lyapunov’s direct method, the SOF stabilization condition for the cascaded ODE–EBB system is derived in terms of a set of bilinear matrix inequalities (BMIs). Furthermore, a two-step procedure for the solution of the BMI feasibility problem is proposed to compute the SOF gain matrices via the existing linear matrix inequality (LMI) optimization techniques (Boyd,

Fig. 1. The cascade of an ODE system and an EBB system.

Ghaoui, Feron, & Balakrishnan, 1994; Gahinet, Nemirovskii, Laub, & Chilali, 1995). Unlike the existing results in Krstic and Smyshlyaev (2008), Krstic (2009a,b,c), Susto and Krstic (2010), and BekiariesLiberis and Krstic (2011) where full PDE and ODE states are required to be available, the proposed controller is very simple and only uses the EBB boundary and ODE measurements. Finally, the numerical simulation is provided to illustrate the effectiveness of the proposed design method. The remainder of this paper is organized as follows. Section 2 introduces the problem formulation and preliminaries. Section 3 presents a BMI-based SOF control design and gives a two-step procedure for the solution of the BMIs. In Section 4, the numerical simulation study is provided to show the effectiveness of the proposed method. Finally, Section 5 offers some concluding remarks. Notations. ℜ, ℜ+ , ℜn and ℜm×n denote the set of all real numbers, all positive scalars, n-dimensional Euclidean space and the set of all m × n matrices, respectively. Identity matrix, of appropriate dimension, will be denoted by I . For a symmetric matrix M , M > (<, ≤)0 means that it is positive definite (negative definite, negative semidefinite, respectively). The space-varying symmetric matrix function P (x), x ∈ [0, L] is positive definite (negative definite, negative semidefinite, respectively), if P (x) > (<, ≤)0 for each x ∈ [0, L]. The function f (x) of x defined on interval [0, L] is C n continuous if its n-th derivative f (n) (x) is continuous. σ (·) and σ¯ (·) denote the minimum and maximum singular values of a matrix, respectively. ρ(·) stands for the resolvent set of the operator. For an integer k ≥ 0, W k,2 (0, L) is a Sobolev space of absolutely continuous scalar functions ω( ¯ x) : [0, L] → ℜ with square integrable derivatives ω ¯ (k) (x)of the order k and with the inner L product ⟨ω ¯ 1 (·), ω¯ 2 (·)⟩W k,2 , 0 ki=0 ω¯ 1(i) (x)ω¯ 2(i) (x)dx and the inner product induced norm ∥ω ¯ 1 (·)∥2W k,2 ,

 L k  (i) 2 ¯ 1 (x) dx, i=0 ω 0

where ω ¯ 1 (·), ω¯ 2 (·) ∈ W k,2 (0, L). This space is also a Hilbert space with the corresponding inner product and denoted by H k (0, L). The superscript ‘T ’ is used for the transpose of a vector or a matrix. The symbol ‘∗’ is used as an ellipsis in matrix expressions that are induced by symmetry, e.g.,



S + [M + N + ∗]



X Y



S + [M + N + M T + NT ] XT

 ,



X . Y

2. Problem formulation and preliminaries We consider the cascade of an LTI ODE system and an EBB equation with variable coefficients (as depicted in Fig. 1) given by

 m(z )wtt (z , t ) + (EI (z )wzz (z , t ))zz = 0, 0 < z < L, t > 0    w(0, t ) = wz (z , t )|z =0 = 0, t ≥ 0 EI (z )wzz (z , t )|z =L = 0, t ≥ 0 (1)   − ( EI (z )wzz (z , t ))z |z =L = v(t ), t ≥ 0   w(z , 0) = w0 (z ), wt (z , 0) = w1 (z ), 0 ≤ z ≤ L y0 (t ) = C0 [wt (L, t ), wz (z , t )|z =L ]T

(2)

 x˙ (t ) = Ax(t ) + Bu(t ), t ≥ 0 v(t ) = cvT x(t ) + dvT u(t ) y (t ) = Cx (t ), x(0) = x0

(3)

H.-N. Wu, J.-W. Wang / Automatica 50 (2014) 2787–2798

where w(z , t ) is the transverse displacement of the EBB at the position z for time t; the subscripts z and t stand for the partial derivatives with respect to z, t, respectively; m(z ) > 0 is the mass density of the beam at position z and EI (z ) > 0 is the flexural rigidity of the beam at position z; y0 (t ) ∈ ℜq is boundary measurement of the beam which includes the velocity, angle, or their combinations at the free end of the beam, C0 ∈ ℜq×2 is a known matrix, wt (L, t ) and wz (z , t )|z =L represent the velocity and angle at the free end of the beam, respectively. L > 0 is the length of the beam, and w0 (z ) and w1 (z ) denote the initial conditions of the beam. x(t ) ∈ ℜn is the ODE state; u(t ) ∈ ℜm is the control input; y (t ) ∈ ℜl is the ODE measurement; v(t ) ∈ ℜ is the ODE output signal (boundary input of EBB equation); cv ∈ ℜn and dv ∈ ℜm are known nonzero vectors, A ∈ ℜn×n , B ∈ ℜn×m , and C ∈ ℜl×n are known matrices; x0 denotes the initial ODE state. Furthermore, we assume that the pair (A, B) is a stabilizable one. Remark 1. It is observed that the ODE system is connected in cascade with the EBB equation (1) through the ODE output signal v(t ) acting on the beam boundary (EI (z )wzz (z , t ))z |z =L . In fact, the cascaded system (1)–(3) is a special coupled one. The EBB model of the system under consideration in this paper has wide applications in practical engineering, such as helicopter rotor/blades, space structures, space aircraft, and turbine blades, etc. (Nwokah & Hurmuzlu, 2002). When x(t ) denotes the state of actuator dynamics, the system (1) and (3) can be used to describe the boundary control design of a flexible beam system with only a force actuator dynamics (Morgul, 1992). On the other hand, the Spacecraft Control Laboratory Experiment (SCOLE) model has two possible inputs: the torque and the force acting on the rigid body (Chentouf, 2003). Here the model (1) describes the non-uniform ¯ 0 wtt (L, t ) and θ1 (t ) = SCOLE model if we set v(t ) , θ2 (t ) − m ¯ 0 is the mass of inertia of the rigid body, J is J wztt (z , t )|z =L (m the moment of inertia, θ1 (t ) and θ2 (t ) are the linear moment control and force control acting on the free end of the beam, respectively). Hence, in this situation, the model (1) and (3) can be applied to describe the dynamic boundary control problem of the modified non-uniform SCOLE system, where the force actuator dynamics is described by the model (3). Moreover, for a given practical problem, the model (1) and (3) can be derived under the assumption that the influence from the beam to the finite dimensional system is neglected. Although it inevitably leads to an approximate model, this assumption is satisfied for a case when a very light beam with actuator dynamics, see Morgul (1992). Remark 2. Notice that w(L, t ) and wt (L, t ) can be respectively sensed by laser displacement and velocity sensors at the free end of the beam and wz (z , t )|z =L is measured by an inclinometer. Throughout this paper, we always assume that m and EI are C 2 continuous functions of space variable z and there exist constants m1 , m2 , α, β ∈ ℜ+ , such that the following expressions hold: m(z ) ∈ [m1 , m2 ] and

EI (z ) ∈ [α, β],

z ∈ [0, L].

Let

2789

Then H(0, L) is a Hilbert space (see Guo, 2002; Liu & Zheng, 1999, for more details) and so is H , H(0, L) × ℜn equipped with the inner product induced norm 2

∥ˆe∥2H , 0.5

L



m(z )g 2 (z )dz + 0.5

L



EI (z )fzz2 (z )dz 0

0

+ 0.5xT Qx,

eˆ , [eT1

T xT ] ∈ H.

The objective of this paper is to find a simple SOF control law for the cascaded system (1)–(3) such that the closed-loop system is exponentially stable, i.e., there exist scalars ρ¯ ∈ ℜ+ and M ∈ ℜ+ , such that the solution e(t ) , [w(·, t ) wt (·, t ) xT (t )]T to the system (1)–(3) satisfying ∥e(t )∥2H < M exp(−ρ¯ t ), where the value of M depends on the initial conditions w0 (z ), w1 (z ), and x0 . For the convenience of control design, we present the following lemmas for the subsequent development. Lemma 1 (Hardy, Littlewood, & Polya, 1952). Let w(·, t ) ∈ L2 (0, L) be an absolutely continuous scalar function with square integrable derivative wz (z , t ) and w(0, t ) = 0. Then the following inequality is achieved:

∥w(·, t )∥22 ≤ 4L2 π −2 ∥wz (·, t )∥22 . Lemma 2. Let A, C , N and E = E T be real matrices with appropriate dimensions. Then there exists a matrix P > 0 such that



AP + PAT N + CP

∗ E



<0

if and only if there exist a scalar µ > 0, matrices P > 0 and U such that

 −[U + ∗]  AU + P  N + CU U

∗ −µ−1 P

∗ ∗

0 0

E 0

 ∗ ∗  < 0. ∗  −µP

Proof. By using Lemma 1 in Wu and Cai (2006), the proof of this lemma is easily obtained and omitted.  3. SOF feedback control design In this section, we will develop a BMI-based method to design an SOF control law for the cascaded system (1)–(3). Then, a twostep procedure is presented to solve the BMIs. In order to tackle the SOF control problem in manageable steps, it is fruitful to assume that the controller has access to EBB boundary information and full ODE state rather than the ODE measurement y (t ). Furthermore, we present the well-posedness analysis of the resulting closed-loop system.

H(0, L) , HL2 (0, L) × L2 (0, L),



L





∥e1 ∥H ,

m(z )g 2 (z ) + EI (z )fzz2 (z ) dz ,



0

where e1 , [f

g ]T ∈ H(0, L), and

HL2 (0, L) , f |f ∈ H 2 (0, L), f (0) = f ′ (z )z =0 = 0 ,



 





    L   L2 (0, L) , g : [0, L] → ℜ  g 2 (z )dz < ∞ .    0

2 Considering 0 < m ≤ m(z ) ≤ m and 0 < α ≤ EI (z ) ≤ β , z ∈ 2  21  2 [0, L], we have ϑm ∥ˆe∥M ≤ ∥ˆe∥2H ≤ ϑM ∥ˆe∥M , for any eˆ ∈ H, where ϑm , − 1  , 1}, ϑM , max{m2 m −1 , 1} and ˆ −1 , α EI ˆ −1 , β EI min{m1 m  2 L 2 L 2 ∥ˆe∥ , 0.5m  fzz (z )dz + 0.5xT Qx. ˆ 0 g (z )dz + 0.5EI 0 M  2 2 Hence, the norm ∥ˆe∥ is equivalent to the one ∥ˆe∥ , where n , n1 + n2 , Q , H

2diag{P1 , P2 } and x , [w1T and x , [θe

M

w2T ]T (Morgul, 1992) or n = 2, Q , diag{IR , k2 }

θ˙e ]T (Morgul, 1991). According to the results given in Morgul (1992,  2

1991), H equipped with the norm ∥ˆe∥M is a Hilbert space. Clearly, H is also a Hilbert space with the norm ∥ˆe∥2H .

2790

H.-N. Wu, J.-W. Wang / Automatica 50 (2014) 2787–2798

3.1. The feedback case of EBB boundary and full ODE state In this subsection, instead of directly dealing with the SOF case, it is useful to begin with the feedback control design by using the EBB boundary measurement and full ODE state. The following control law is first considered for the cascaded system (1)–(3): u(t ) = Kx(t ) + Ly 0 (t )

(4)

where K ∈ ℜm×n and L ∈ ℜm×q are the control gain matrices to be determined. For convenience, define f (t ) , [wt (L, t ) wz (z , t )|z =L ]T . Substituting (4) into (1) and (3) and using (2), yields the following closed-loop system:

 m(z )wtt (z , t ) + (EI (z )wzz (z , t ))zz = 0, 0 < z < L, t > 0    w(0, t ) = wz (z , t )|z =0 = 0, t ≥ 0 EI (z )wzz (z , t )|z =L = 0, t ≥ 0 (5)  T T  ˜ | − ( EI ( z )w ( z , t )) = c x ( t ) + d LC f ( t ), t ≥ 0  zz z 0 z = L v v  w(z , 0) = w0 (z ), wt (z , 0) = w1 (z ), 0 ≤ z ≤ L ˜ (t ) + BLC 0 f (t ), x˙ (t ) = Ax

t ≥ 0, x(0) = x0

(6)

˜ , A + BK and c˜vT , cvT + dvT K . where A Let us consider the following Lyapunov candidate for the system (5) and (6): V (t ) = V1 (t ) + V2 (t ) + V3 (t )

(7)

where V1 ( t ) =

L



ξ T (z , t )P (z )ξ(z , t )dz ,

(zm(z ))z ∈ [χ1 , χ2 ] and z (EI (z ))z ∈ [α1 , β1 ],

Based on Assumption 1, we have the following lemma: Lemma 4. For the matrix functions P (z ) and Ξ (z ), z ∈ [0, L] are defined in (8) and (10), the inequalities P (z ) > 0 and Ξ (z ) < 0 hold for z ∈ [0, L], if the following LMIs are satisfied:



p11 m2

∗ 

2 p11 EI (z )wzz (z , t )dz ,

and

(13)

Proof. See Appendix C. Therefore, we can obtain the following result which shows the unique existence of classical solution of the closed-loop system (5)–(6) and presents an approach to the design of stabilizing controllers of the form (4) for the cascaded system (1)–(3). Theorem 1. For the cascaded system (1)–(3) with the feedback control law of the form (4), if there exist scalars p11 , p12 ∈ ℜ+ , and matrices X > 0, Z , L such that the inequalities in (12), (13), and the following inequality are satisfied:

p11 m(z )





p12 m(z )z p22

˜ , Θ



> 0,

z ∈ [0, L], p12 ∈ ℜ+ .

(8)

It can be easily verified that this Lyapunov function is equivalent to ∥e(t )∥2H in H (see Appendix A for more details). Lemma 3. For the system (5) and (6), the time derivative of V (t ) given by (7) satisfies the following inequality: L



ζ T (z , t )Ξ (z )ζ(z , t )dz + xˇ T (t )Θ xˇ (t )

(9)

0

(14)

(10)

(11)





[AX + BZ + ∗]

then the closed-loop system (5) and (6) has a unique classical solution initialized with [w0 (·) w1 (·) xT0 ]T ∈ D (A) (D (A) will be given in (21)) and is exponentially stable. In this case, the control gain matrix K is given by K = ZX −1 .

(15)

Proof. Assume that the inequalities in (12)–(14) hold. From Lemma 4, we can conclude that Ξ (z ) < 0 and P (z ) > 0 hold for any z ∈ [0, L]. Let and Z = KX .

(16)

Based on the matrix multiplication theory and using (16), we get ˜ diag{I , Q }. Hence, the matrix inequality Θ < 0 Θ = diag{I , Q }Θ can be easily obtained from (14) and the fact diag{I , Q } > 0. For the inequalities Θ  < 0 and Ξ (z ) < 0, z ∈ [0, L], there exists a scalar p1 , min σ (−Θ ), minz ∈[0,L] σ (−Ξ (z )) > 0 such that the following inequalities hold:

Θ + p1 I ≤ 0 and Ξ (z ) + p1 I ≤ 0 for any z ∈ [0, L].

(17)

Thus, from Lemma 3, the inequality (9) can be written as

in which p12 Lm(L) p22

Θ1 + [˜pdvT LC 0 + ∗] BLC 0 + (Xcv + Z T dv )˜pT



X = Q −1

 T where ζ(z , t ) , wt (z , t ) wzz (z , t ) ,  T xˇ (t ) , f T (t ) xT (t ) ,   −p12 (zm(z ))z −p22 Ξ (z ) , , ∗ Ξ22 (z )   Θ1 + [˜pdvT LC 0 + ∗] ∗ Θ, QBLC 0 + c˜v p˜ T [Q A˜ + ∗]

Θ1 ,

 −p22 < 0. −p12 (3α − β1 )

where

V3 (t ) = xT (t )Qx(t ),

T wz (z , t ) , 0 < Q ∈ ℜn×n , and

in which ξ(z , t ) , wt (z , t )



(12)

L





V˙ (t ) ≤



p12 m2 L > 0, p22

−p12 χ1 ∗

0

P (z ) ,

∀z ∈ [0, L].

˜ <0 Θ

0

V2 ( t ) =

Assumption 1. For the mass density m and the flexural rigidity EI of the beam, there exist constants χ1 ∈ ℜ, χ2 ∈ ℜ, α1 ∈ ℜ, β1 ∈ ℜ such that the following expressions hold:



p22 , 0

 p˜ ,



p11 , p12 L

and

Ξ22 (z ) , −p12 (3EI (z ) − z (EI (z ))z ). Proof. See Appendix B. Before going further, we need to give the following assumption, which is useful for the following analysis:

V˙ (t ) ≤ −p1 ∥ζ(·, t )∥22 + ∥x(t )∥2 .





(18)

We first show the unique existence of classical solution for the closed-loop system (5) and (6). Set the trajectory segments w(·, t ) = {w(z , t ), z ∈ [0, L]}, wt (·, t ) = {wt (z , t ), z ∈ [0, L]}, and H be the state space. The system (5) and (6) can be rewritten in the following abstract evolution equation: e˙ (t ) = Ae(t ),

e(0) ∈ H

(19)

H.-N. Wu, J.-W. Wang / Automatica 50 (2014) 2787–2798

where e(0) , [w0 (·) tor A is defined as

 

Ae¯ ,  −

w1 (·)

xT0 ]T , the linear unbounded opera-

0 ∂ 2 (EI (z )∂ 2 /∂ z 2 ) m(z )∂ z 2 0

 +

BLC 0 [g (L)

where e¯ , [f defined as

1 V (t ) ≤ V (0) exp(−p4 p− 3 t ).



Inequalities (22) and (28), imply

0

0

0  e¯

 f ′ (z )z =L ]T



(20)



x¯ ] . The domain D (A) of the operator A is T T

g

D (A) , { e¯ | f ∈ HL2 (0, L) ∩ H 4 (0, L),

g ∈ HL2 (0, L),  x¯ ∈ ℜn , EI (z )f ′′ (z )z =L = 0,   (21) (EI (z )f ′′ )′ (z )z =L + c˜vT x¯ + dvT LC 0 h = 0  ′ where h , [g (L) f (z )z =L ]T . Since V (t ) is equivalent to ∥e(t )∥2H in H, we can deduce from (18) that the operator A is dissipative. On the other hand, we can verify 0 ∈ ρ(A) (see Appendix D for detailed proof). By Theorem 1.2.4 Liu and Zheng (1999, Chapter 1, page 3), the operator A generates a C0 semigroup of contractions on H. For the terminology on the C0 semigroup theory, please refer to Pazy (1983) and Tucsnak and Weiss (2009). According to Theorem 1.3 Pazy (1983, Chapter 4, page 102), the system (19) has a unique classical solution for any initial value e(0) ∈ D (A). Then, we prove from (18) that the closed-loop system (5) and (6) is exponentially stable. Since Q > 0, EI (z ) > 0, and P (z ) > 0, z ∈ [0, L] is a spatially continuous matrix function, it is easily observed that V (t ) given by (7) satisfies p2 ∥e(t )∥ ≤ V (t ) ≤ p3 ∥ς(·, t )∥ + ∥x(t )∥ 2 H

2 2



where ς(z , t ) , [ξ T (z , t )

2



(22)

wzz (z , t )]T ,

p2 , min 2 minz ∈[0,L] σ (m−1 (z )P (z )), 2p11 , 2 > 0 and,





p3 , max maxz ∈[0,L] σ¯ (P (z )), p11 maxz ∈[0,L] EI (z ), σ¯ (Q ) > 0.





Set w( ¯ z , t ) , wz (z , t ). Since wz (z , t )|z =0 = 0, we have

w( ¯ 0, t ) = 0.

(23)

L

w ¯ 2 (z , t )dz ≤ 4L2 π −2 0

w ¯ z2 (z , t )dz .

(24)

0

We can get the following inequality from p1 ∈ ℜ+ and the inequality (24): L



2 wzz (z , t )dz ≤ −0.25L−2 π 2 p1

− p1

Hence, the closed-loop system (5) and (6) is exponentially stable. Moreover, from (16), we can obtain (15). The proof is complete.  We can see from Theorem 1 that the solvability of feedback control problem for the cascaded system (1)–(3) is formulated as the feasibility problem of the matrix inequalities (12)–(14). The desired control gain matrices in (4) are constructed via the feasible solutions of matrix inequalities (12)–(14). However, the inequality (14) is a BMI with respect to the decision variables in {p11 , p12 , p22 , X , Z , L }. Unfortunately, in contrast to LMI problems, which are convex and can be solved efficiently by polynomialtime interior-point methods (Boyd et al., 1994), BMI problems are non-convex and known to be NP-hard (Toker & Ozbay, 1995). A two-step procedure has been developed to solve the BMI problems (Tseng, Chen, & Uang, 2001). In Section 3.3, we will give a two-step procedure to solve the BMIs in (12)–(14). 3.2. The feedback case of EBB boundary and ODE measurement In this subsection, we will provide a BMI-based SOF stabilization condition for the cascaded system (1)–(3) via the EBB boundary measurement y0 (t ) and the ODE measurement y (t ). Let us consider the following SOF control law: u(t ) = Kc y (t ) + Ly 0 (t )

L



0

wz2 (z , t )dz .

(25)

0

where Kc ∈ ℜm×l and L ∈ ℜm×q are the control gain matrices to be determined. Substituting of (30) into (1) and (3) and using (2), yield the following closed-loop system:

 m(z )wtt (z , t ) + (EI (z )wzz (z , t ))zz = 0, 0 < z < L, t > 0    w(0, t ) = wz (z , t )|z =0 = 0, t ≥ 0 EI (z )wzz (z , t )|z =L = 0, t ≥ 0 (31)  T T  ¯ | − ( EI ( z )w ( z , t )) = c x ( t ) + d LC f ( t ), t ≥ 0  zz z 0 z = L v v  w(z , 0) = w0 (z ), wt (z , 0) = w1 (z ), 0 ≤ z ≤ L ¯ (t ) + BLC 0 f (t ), x˙ (t ) = Ax

¯ e(t ), e˙ (t ) = A

V˙ (t ) ≤ −p1 ∥x(t )∥ − p1 ∥wt (·, t )∥

 

¯ e¯ , − A 

2 2

≤ −p4 ∥x(t )∥2 − p4 ∥ς(·, t )∥22

(26)

m(z )∂ z 2 0

0

0

0  e¯

0



¯ A



0 0 BLC 0 [g (L)



1

f ′ (z )z =L ]T



.

¯ ) of the operator A ¯ is defined as The domain D (A ¯ ) , e¯ | f ∈ HL2 (0, L) ∩ H 4 (0, L), D (A 

where p4 , p1 min{0.5, 0.125π 2 L−2 }. Hence, from (22), the inequality (26) can be rewritten as 1 V˙ (t ) ≤ −p4 p− 3 V (t ).

0 ∂ 2 (EI (z )∂ 2 /∂ z 2 )

+ 2 −2

e(0) ∈ H



2 2

− 0.5p1 ∥wzz (·, t )∥ − 0.125π L p1 ∥wz (·, t )∥ 2 2

t ≥ 0, x(0) = x0

¯ is defined as where the linear unbounded operator A

It can be derived from (18) and (25) that 2

(30)

(32)

(21), the closed-loop coupled system (31) and (32) can be written as the following abstract evolution equation:

L



(29)

¯ , A + BK c C and c¯vT , cvT + dvT Kc C . Similar to lines of (19)– where A

Using Lemma 1 and (23), we thus get



(28)

  1 1 2 2 exp(−p4 p− ∥e(t )∥2H ≤ p− 3 t ). 2 p3 ∥ς(·, 0)∥2 + ∥x(0)∥

˜ A

0 0

By integrating the inequality (27) from 0 to t, we get



1

0

2791

(27)

g ∈ HL2 (0, L),

 x¯ ∈ ℜ , EI (z )f (z )z =L = 0,   (EI (z )f ′′ )′ (z )z =L + c¯vT x¯ + dvT LC 0 h = 0 . n

′′

2792

H.-N. Wu, J.-W. Wang / Automatica 50 (2014) 2787–2798

We give the following assumption on the matrix C : Assumption 2. The matrix C is of full row rank. This assumption implies that there exists an invertible matrix T ∈ ℜn×n such that 0 .





CT = I

(33)

Generally, the matrix T satisfying (33) is not unique for the matrix C . A special T can be given as follows: T = [C T (CC T )−1

C ⊥]

(34)

where C ⊥ denotes an orthogonal basis for the null space of C , i.e., C ⊥ C = 0. Then, we have the following result: Theorem 2. Consider the cascaded system (1)–(3) with the SOF controller (30). For some scalar µ > 0, if there exist scalars p11 ∈ ℜ+ , + + n×n p 12 ∈ ℜ , p22 ∈ ℜ and matrices X > 0 ∈ ℜ , L, W = W11 W21

0 W22

 ∈ ℜn×n , Y = Y1



0

∈ ℜm×n with W11 ∈ ℜl×l ,

W21 ∈ ℜ(n−l)×l , W22 ∈ ℜ(n−l)×(n−l) , Y1 ∈ ℜm×l , satisfying the inequalities (12), (13), and

 −[TW + ∗]  Υ21  Υ31

∗ −µ−1 X

TW

∗ ∗ Υ33

0 0

0

 ∗ ∗  <0 ∗  −µX

(35)

where T is a nonsingular matrix satisfying (33) and

Υ21 , ATW + BY + X , Υ31 , (BLC 0 )T + p˜ (cvT TW + dvT Y ), Υ33 , Θ1 + [˜pdvT LC 0 + ∗],

−1 Kc = Y1 W11 .

(36)

Proof. Assume that the inequalities (12), (13), and (35) hold. Let Y1 = Kc W11 .

(37)

By virtue of the structure of W , Y and (33), (37), we have Y = Kc CTW , then (35) can be rewritten as follows:

−[H + ∗] ¯ +X AH   (BLC 0 )T + p˜ c¯vT H

∗ −µ−1 X

∗ ∗

0 0

Θ1 + [˜pdvT LC 0 + ∗]

H

0

<0

 ∗ ∗   ∗ −µX (38)

where H = TW . It follows from Lemma 2 that (38) is equivalent to

¯ + ∗] [AX (BLC 0 )T + p˜ c¯vT X





∗ Θ1 + [˜pdvT LC 0 + ∗]

< 0.

(39)

Multiplying both sides of (39) by the matrix diag{X −1 , I } and letting Q = X −1 give

Ψ ,

[Q A¯ + ∗] (QBLC 0 )T + p˜ c¯vT



∗ Θ1 + [˜pdv LC 0 + ∗] T



< 0.

(40)

Consider the Lyapunov function of the form (7) for the closedloop system (31) and (32). Similar to Lemma 3, we can get V˙ (t ) ≤

L



ζ T (z , t )Ξ (z )ζ(z , t )dz + xˆ T (t )Ψ xˆ (t ) 0

The stabilization condition via an SOF controller (30) is presented in Theorem 2 for the cascaded system (1)–(3). The control gain matrices Kc and L can be obtained, if the feasible solution to inequalities (12), (13), and (35) is obtained. Observe that (35) is a BMI with respect to the decision variables in the set {p11 , p12 , p22 , X , L , W , Y }. As BMIs (12)–(14), the BMIs in (12), (13), and (35) can be also solved via the two-step procedure, which will be given in next subsection. Moreover, it should be noted that for the matrix C , there may exist different choices of T satisfying (34). However, similar to Theorem 2 in Dong and Yang (2007), we can easily prove that the feasibility of the condition of Theorem 2 is independent of the choices of T . Remark 3. It must be pointed out that based on the semigroup theory, the properties (e.g., well-posedness, regularity, observability, and exact controllability) have been investigated in Zhao and Weiss (2010, 2011b) for an EBB system coupled with an ODE system and a collocated SOF controller has been proposed in Zhao and Weiss (2011a) to suppress the vibrations of a wind turbine tower represented by a coupled ODE–EBB model. But, the semigroup approach is so abstract that it is difficult to grasp for control engineers. Different from the results in Zhao and Weiss (2010, 2011a,b), this paper develops a Lyapunov-based method for designing an SOF controller for ODE–EBB cascades, where the semigroup theory is only used to analyze the well-posedness of the closed-loop system. 3.3. A two-step procedure

then the closed-loop system (31) and (32) has a unique classical solu¯ ) and is exponention initialized with [w0 (·) w1 (·) xT0 ]T ∈ D (A tially stable. In this case, the control matrix Kc is given as



where ζ(z , t ), Ξ (z ) are given by (9), Ψ is given in (40), and xˆ (t ) , [xT (t ) f T (t )]T . Then, the result of this theorem follows immediately using the proof of Theorem 1 in a similar manner. Moreover, from (37), we get (36). Thus, the proof is complete. 

In this subsection, a two-step procedure is presented to solve the BMIs in (12), (13), and (14) (or (35)). We notice that inequalities (12) and (13) only contain the decision variables p11 , p12 , p22 and are LMIs with respect to them. Hence, parameters p11 , p12 , and p22 can be obtained by solving the LMIs (12) and (13). On the other hand, if p11 , p12 , and p22 satisfying (12) and (13) are fixed, then finding the matrices X > 0, Z , and L (or X > 0, W , Y , and L) satisfying (14) (or (35)) is also an LMI feasibility problem. Therefore, we can give the following two-step procedure to find the feasible solutions of the BMIs in (12), (13), and (14) (or (35)): Design procedure: Step 1: Solve the LMIs (12) and (13) for positive scalars p11 , p12 , and p22 . Step 2: With the resulting p11 , p12 , and p22 , solve the LMI (14) (or (35)) to obtain X > 0, Z , and L (or X > 0, W , Y , and L). And then calculating (15) (or (36)) gives the control gain matrix K (or Kc ). Remark 4. If the inequality (14) (or (31)) in Step 2 of the design procedure is infeasible, the feasible solutions to the BMIs in (12), (13), and (14) (or (35)) cannot be obtained via the proposed procedure. In such a case, one can resort to some other algorithms, such as global and local optimization algorithms (Goh, Turan, Safonov, Papavassilopoulos, & Ly, 1994), cone complementarity linearization algorithm (Ghaoui, Oustry, & AitRami, 1997), iterative LMI algorithm (Cao, Lam, & Sun, 1998), etc. It is worth mentioning that different from the existing algorithms (Cao et al., 1998; Ghaoui et al., 1997; Goh et al., 1994), the proposed two-step procedure is very simple and can be easily implemented via the existing LMI optimization techniques (Boyd et al., 1994; Gahinet et al., 1995), since it does not include the iterative process.

H.-N. Wu, J.-W. Wang / Automatica 50 (2014) 2787–2798

Remark 5. Notice that LMI (13) requires χ1 > 0 and 3α − β1 > 0 (because of p12 > 0) in Assumption 1. Obviously, in this situation, this assumption implies the variation in beam parameters is not sharp. Hence, the potential drawback of the design method proposed in this study lies in that it does only apply to the linear cascaded ODE–EBB system with slow-varying parameters. The flexible beam in most real applications is of slow-varying parameters.

2793

w

4. Numerical simulations In this section, numerical simulations are carried out to illustrate the effectiveness of the proposed design method. We present the simulation results for not only the feedback case of EBB boundary and full ODE state but also one of EBB boundary and ODE measurement. To this end, we utilize the finite difference method (FDM) to obtain the approximate solution of the EBB equation.

z

t

Fig. 2. Open-loop displacement profile of evolution of EBB.

4.1. FDM approximation Assuming time and space discretization interval, denoted as tsample and zsample , respectively, the following approximations assuming central differences hold for the partial derivative terms:

wik+1 − 2wik + wik−1

m(z )wtt (z , t ) ≈ mi

w

2 tsample

(EI (z )wzz (z , t ))zz ≈

EIi+1 wik+2 − 2(EIi+1 + EIi )wik+1 4 zsample

+ −

(EIi+1 + 4EIi + EIi−1 )wik 4 zsample

z

2(EIi + EIi−1 )w

− EIi−1 w

k i −1

t

k i−2

4 zsample

Fig. 3. Closed-loop displacement profile of evolution of EBB for Case 1.

evaluated at z = i · zsample and t = k · tsample , where zsample , LNz−1 . time Nz is a given positive integer. The notation wspace stands for the discretization value of w(z , t ) at point z = i · zsample and t = k · tsample . 4.2. Numerical results The parameters of the EBB equation are chosen as m(z ) = m0 (1 − 0.01z ), EI (z ) = EI0 (1 − 0.01z )3 , C0 = I , and L = 1, where m0 = 1 and EI0 = 1. The initial values of the EBB equation are set to be w0 (z ) = z 2 and w1 (z ) = 0. The parameters of the LTI system are presented as follows:





1 A= 0.1

1 , 0.02

C = cvT = 2



1 ,



  B= and

1 , 1

Fig. 4. Closed-loop state trajectory of LTI system for Case 1.

dvT = 1.



T

The ODE initial state is set to be x0 = −1 5 . Fig. 2 shows the open-loop displacement profile of evolution of the EBB, where the steady state w(z , t ) = 0 is an unstable one. Case 1: EBB boundary and full ODE state feedback Using the two-step procedure, the feasible solutions to BMIs (12)–(14) are obtained as p11 = 1.5630,

 X =

0.9974 −0.3159

 L = −0.6672

p12 = 0.6148,

−0.3159 , 1.8036  −0.3990 . 

p22 = 0.7815,

 Z = −1.2387

By (15), we can obtain the control gain matrix as

 K = −1.4377

−0.6183 . 

(41)

−0.6610 , 

(42)

Applying the controller (4) with the above control gain matrices K and L (in (42)) to the cascaded system (1)–(3), the closed-loop displacement profile of evolution of the EBB and the closed-loop state trajectory of the LTI system (3) are shown in Figs. 3 and 4, respectively. It is clear from Figs. 3 and 4 that the designed controller (4) can exponentially stabilize the cascaded system (1)– (3). Fig. 5 shows the control input u(t ). Case 2: EBB boundary and ODE feedback  measurement  By (34), we have T =

0.5 0

0.5 −1 satisfying (33). Let

µ = 0.06.

Using the two-step procedure, the feasible solutions to the BMIs in (12), (13), and (35) are obtained as (41) and 1.8731 X = 10 × −1.4739 4



 −1.4739 , 2.9804

2794

H.-N. Wu, J.-W. Wang / Automatica 50 (2014) 2787–2798

Fig. 5. Control input u(t ) for Case 1.

Fig. 8. Control input u(t ) for Case 2.

design of SOF controllers via EBB boundary and ODE measurements is developed by using the Lyapunov techniques. Furthermore, in order to compute the gain matrices of SOF controller, a two-step procedure is presented to solve the BMI feasibility problem via the existing LMI optimization techniques. Finally, the achieved simulation results illustrate the effectiveness of the proposed design method. The advantages of the control design proposed in this paper can be summarized as follows: (a) The suggested controller is easily implemented because of its simple SOF structure. (b) The proposed control design is easy to grasp for control engineers because it is developed via the Lyapunov direct method. (c) The framework of such a control design paves the way for further improvements for other different control designs for the linear cascaded ODE–beam systems, such as non-collocated control, for more complicated nonlinear coupled systems, such as nonlinear coupled ODE–beam systems, and for the analysis and control design of real coupled ODE–beam systems, such as a nonuniform SCOLE beam with a flexible shaft (Zhao & Weiss, 2011b) and a wind turbine tower moving in the plane of the turbine blades (Zhao & Weiss, 2010, 2011a), that will be addressed in future research activities.

w

z

t

Fig. 6. Closed-loop displacement profile of evolution of EBB for Case 2.

Acknowledgments The authors gratefully acknowledge the helpful comments and suggestions of the Associate Editor and anonymous reviewers, which have improved the presentation of this paper. The authors also express their thanks to some experts in distributed parameter systems modeled by Euler–Bernoulli beam equation or wave equation for their kind help toward the closed-loop wellposedness analysis of this work.

Fig. 7. Closed-loop state trajectory of LTI system for Case 2.

W = 103 ×

0.9737 0.9380



Y = −975.7801



Appendix A. Proof that the Lyapunov function V (t ) is equivalent to ∥e(t )∥2H in H



0 , −1.7868

0 ,



L = −2.4671



 −0.9701 .

(43)

By (36), we can obtain the control gain parameter as Kc = −1.0021. With the above control parameters Kc and L (in (43)), we apply the controller (30) to the cascaded system (1)–(3). Figs. 6 and 7 indicate the closed-loop profiles of evolution of the EBB displacement and velocity and the closed-loop state trajectory of the LTI system (3), respectively. It can be easily seen from Figs. 6 and 7 that the designed controller (30) can exponentially stabilize the cascaded system (1)–(3). Fig. 8 shows the control input u(t ).

Similar to (22), we can get the following two scalars such that the Lyapunov function V (t ) given by (7) satisfies p1 ∥e(t )∥2H ≤ V (t ) ≤ p5

∥e(t )∥2H +



L

m(z )wz2 (z , t )dz

 (A.1)

0

where p1 , 2 min minz ∈[0,L] σ (m−1 (z )P (z )), p11 , 1 > 0 and





p5 , 2 max maxz ∈[0,L] σ¯ (m−1 (z )P (z )), p11 , 1 > 0.





Considering 0 < m1 ≤ m(z ) ≤ m2 , z ∈ [0, L] and using the inequalities (23) and (24), we can get L



m(z )w (z , t )dz ≤ m2 2 z

5. Conclusions 0

In this paper, we have addressed the SOF control design problem for linear ODE–EBB cascades. The BMI-based approach to the



≤ 4L2 π −2 m2

L



wz2 (z , t )dz 0

L



2 wzz (z , t )dz . 0

(A.2)

H.-N. Wu, J.-W. Wang / Automatica 50 (2014) 2787–2798

We can also derive from 0 < α ≤ EI (z ) ≤ β , z ∈ [0, L]

α

L



2 wzz (z , t )dz ≤

2 EI (z )wzz (z , t )dz .

(A.3)

Substituting (A.3) into (A.2) gives L

m(z )wz2 (z , t )dz ≤ 4L2 π −2 m2 α −1

m(z )wz2 (z , t )dz ≤ 0.5

= − z wz (z , t )(EI (z )wzz (z , t ))z |z =L  L wz (z , t )(EI (z )wzz (z , t ))z dz +

L



m(z )wt2 (z , t )dz 0

0

+ 0.5(1 + 8L2 π −2 m2 α −1 )

0

L



z wz (z , t )(EI (z )wzz (z , t ))zz dz 0

L



(B.7)

L

 =−

Hence, from (A.4), we obtain

∥e(t )∥2H +

(m(z )z )z wt2 (z , t )dz , 0

0

(A.4)

0

0



m(z )z wz (z , t )wtt (z , t )dz

L 2 EI (z )wzz (z , t )dz .

2

m(z )z wt2 (z , t )z =L

L

 

2

1

L



1



0

0



m(z )z wt (z , t )wzt (z , t )dz = 0

L



2795

L



2 EI (z )wzz (z , t )dz + 0.5xT Qx. (A.5)

z wzz (z , t )(EI (z )wzz (z , t ))z dz

+

0

0

+

It can be easily derived from (A.1), (A.5), and p5 ∈ ℜ that p1 ∥e(t )∥2H ≤ V (t ) ≤ p5 (1 + 8L2 π −2 m2 α −1 )∥e(t )∥2H .

L



= z wz (z , t )|z =L (˜cvT x(t ) + dvT LC 0 f (t ))  L 2 EI (z )wzz (z , t )dz −

(A.6)

Therefore, the Lyapunov function V (t ) is equivalent to ∥e(t )∥2H in H. The proof is complete. 

0 L



z wzz (z , t )(EI (z )wzz (z , t ))z dz ,

+

(B.8)

0

Appendix B. Proof of Lemma 3

and

V˙ (t ) = V˙ 1 (t ) + V˙ 2 (t ) + V˙ 3 (t )

L



Taking the time derivative of V (t ) given in (7), yields

z wzz (z , t )(EI (z )wzz (z , t ))z dz 0

(B.1)

L



2 EI (z )wzz (z , t )dz −

=−

where

0

V˙ 1 (t ) = 2

L



p22 wz (z , t )wzt (z , t )dz

+2 0 L +2 0 L +2

z wzz (z , t )(EI (z )wzz (z , t ))z dz 0 L



2 z (EI (z ))z wzz (z , t )dz

=

p12 m(z )z wt (z , t )wzt (z , t )dz p12 m(z )z wz (z , t )wtt (z , t )dz

0



L



z wzz (z , t )EI (z )wzzz (z , t )dz ,

+

(B.2)

(B.10)

0

0

V˙ 2 (t ) = 2

(B.9)

L



L



z wzzz (z )EI (z )wzz (z , t )dz . 0

Since

p11 m(z )wt (z , t )wtt (z , t )dz 0

L



L

p11 EI (z )wzz (z , t )wzzt (z , t )dz ,

(B.3)

by adding the Eqs. (B.9) and (B.10), and then dividing both sides by 2, we can get

0

V˙ 3 (t ) = 2xT (t )Q x˙ (t ).

(B.4)

L

m(z )wt (z , t )wtt (z , t )dz = − 0



z wzz (z , t )(EI (z )wzz (z , t ))z dz = − 0

By integrating by parts and considering (5), we have that



L



+

L

wt (z , t )(EI (z )wzz (z , t ))zz dz

0

1 2

L



2 z (EI (z ))z wzz (z , t )dz .

= z wz (z , t )|z =L (˜cvT x(t ) + dvT LC 0 f (t ))  3 L 2 − EI (z )wzz (z , t )dz 2

T

+ (B.5)

0

1 2

0 L



2 z (EI (z ))z wzz (z , t )dz .

wz (z , t )wzt (z , t )dz = wt (z , t )wz (z , t )|z =L

Substituting (B.5)–(B.7), (B.12) into and rearranging (B.2), get V˙ 1 (t ) ≤ −2p11

0

L



EI (z )wtzz (z , t )wzz (z , t )dz 0

0

L

wt (z , t )wzz (z , t )dz ,

L

 (B.6)

(B.12)

0

L



(B.11)

0

m(z )z wz (z , t )wtt (z , t )dz

0



2 EI (z )wzz (z , t )dz 0

0

0



2

L



= − wt (z , t )(EI (z )wzz (z , t ))z |z =L  L − wtzz (z , t )EI (z )wzz (z , t )dz = wt (L, t )(˜cv x(t ) + dv LC 0 f (t ))  L − wtzz (z , t )EI (z )wzz (z , t )dz ,

L



Substitution of (B.11) into (B.8) results in

= − wt (z , t )(EI (z )wzz (z , t ))z |z =L  L + wtz (z , t )(EI (z )wzz (z , t ))z dz

T

1

wt (z , t )wzz (z , t )dz

− 2p22 0

2796

H.-N. Wu, J.-W. Wang / Automatica 50 (2014) 2787–2798 L



represented as follows:

2 [3EI (z ) − z (EI (z ))z ]wzz (z , t )dz

− p12 0

Ξ (z ) =

+ 2p11 wt (L, t )(˜cv x(t ) + dv LC 0 f (t ))  L (m(z )z )z wt2 (z , t )dz + 2p12 z wz (z , t )|z =L (˜cvT x(t ) − p12 T

T

2  2  2 

˜ (1) = χ1 , m  EI

 + p12 m(z )z wt2 (z , t )z =L + 2p22 wt (z , t )wz (z , t )|z =L  L ζ T (z , t )Ξ (z )ζ(z , t )dz + 2f T (t )˜pc˜vT x(t ) =

(1)

c1 (z ) = d1 (z ) = (B.13)

0

By (6), (B.4) can be rewritten as (B.14)

Substituting (B.2), (B.13), and (B.14) into (B.1), we can get (9). The proof is complete.  Appendix C. Proof of Lemma 4 By considering m(z ) ∈ [m1 , m2 ], z ∈ [0, L], we can seek the functions a1 (z ) and a2 (z ) satisfying i ∈ {1, 2}

and a1 (z ) + a2 (z ) = 1

(C.1)

for ∀z ∈ [0, L].

(C.2)

m2 − m(z ) m2



m(1)

and a2 (z ) =

m(z ) − m(1) m(2)



e1 (z ) =

m(1)

(2)

P (z ) =

2  2 

ai (z )bj (z )



p11 m

(i)



i=1 j=1



˜ (2) m



(1)

,

(2)

where m = m1 , m = m2 , z = 0, and z = L. Obviously, from (C.4) and considering ai (z ), bj (z ) ∈ [0, 1], i, j ∈ {1, 2}, ∀z ∈ [0, L], we have P (z ) > 0 for ∀z ∈ [0, L] if the following LMIs

 Pij ,

p11 m(i)



p12 m(i) z (j) p22



> 0,

,

− z (EI (z ))z , (2)  − EI (1) EI

EI (2) − EI (z ) EI (2) − EI (1)

,

 , −p12 (3EI (φ) − EI

Ξ22

c2 (z ) = d2 (z ) =

),

˜ (1) (zm(z ))z − m , ( 2 ) ( 1 ˜ −m ˜ ) m

 z (EI (z ))z − EI

e2 (z ) =

and

(τ )

 EI

(2)

 − EI

(1)

,

(1)

EI (z ) − EI (1) EI (2) − EI (1)

.

 −p22 (φ,τ ) < 0, Ξ22

σ , τ , φ ∈ {1, 2}

(C.7)

are feasible. Using the Schur complement, the LMIs (C.7) are equivalent to the following inequalities: (φ,τ )

Ξ22

1 ˜ (σ ) )−1 < 0, + p222 p− 12 (m

˜ (σ ) > 0, m

σ , τ , φ ∈ {1, 2}. (C.8)

˜ (1) ≤ m ˜ (2) , m

EI (1) ≤ EI (2) , +

 and EI

(1)

(2)

 . ≤ EI

(C.9)

+

1 ˜ (σ ) )−1 ≤ −3p12 EI (1) + p222 p− 12 (m

(2)

1 ˜ (1) )−1 + p222 p− 12 (m

which implies that the inequalities (C.8) are satisfied if the following inequality holds: (1) −1 (2) + p222 p−1 (m − 3p12 EI (1) + p12 EI ) < 0, 12 ˜

˜ (1) > 0. (C.10) m

Using the Schur complement again, the inequality (C.10) is equivalent to LMI (13). The proof is complete.

(C.4) (2)

˜ (1) m

EI (2) = β,

Appendix D. Proof that 0 ∈ ρ(A)

z ∈ [0, L], (1)

 EI



(φ,τ )

= β1 ,

(2)

˜ (σ ) −p12 m ∗

(φ,τ )

Substitution of (C.2)

(i) (j)

p12 m z p22

(2)

EI (1) = α,

the following LMIs

Ξ22

.

(C.3)

z −z (1) . z (2) −z (1)

 EI

˜ (2) − (zm(z ))z m

 + p12 EI

for ∀z ∈ [0, L]

where b1 (z ) = (z2) −(z1) and b2 (z ) = z −z and (C.3) into P (z ), yields

(C.6)

From (C.6) and considering cσ (z ), dτ (z ), eφ (z ) ∈ [0, 1], σ , τ , φ ∈ {1, 2}, ∀z ∈ [0, L], it is clear that Ξ (z ) < 0 holds for ∀z ∈ [0, L], if

Likewise, z can be represented as follows: z = b1 (z )z (1) + b2 (z )z (2)



Since p12 ∈ ℜ and p22 ∈ ℜ , we find from (C.9) that for any σ , τ , φ ∈ {1, 2}

Solving Eq. (C.1) and (C.2), gives a1 (z ) =

−p22 (φ,τ ) Ξ22

Applying Assumption 1, we have

such that m(z ) can be represented as m(z ) = a1 (z )m1 + a2 (z )m(2)

˜ (2) = χ2 , m

= α1 ,

0

ai (z ) ∈ [0, 1],

˜ (σ ) −p12 m ∗

where

+ dvT LC 0 f (t ))

˜ + ∗]x(t ) + 2xT (t )QBLC 0 f (t ). V˙ 3 (t ) ≤ xT (t )[Q A



σ =1 τ =1 φ=1

0

+ f T (t )[Θ1 + [˜pdvT LC 0 + ∗]]f (t )  L EI (z )wtzz (z , t )wzz (z , t )dz . − 2p11

cσ (z )dτ (z )eφ (z )

i, j ∈ {1, 2}

To do this, for any e˜ , [f˜ the following relation:

Aeˇ = e˜ ,



x˜ T ]T ∈ H, we need to consider

eˇ ∈ D (A)

where eˇ , [f¯

(D.1)

T T

x¯ ] . By the definition of A, the Eq. (D.1) can



be written as (C.5)

are feasible. Consider z (1) = 0 and z (2) = L, we can conclude that for each i ∈ {1, 2}, the inequality Pi2 > 0 implies Pi1 > 0. Since m1 ≤ m2 , the inequality P12 > 0 holds if the inequality P22 > 0 is fulfilled. From above analysis, the LMIs in (C.5) hold if LMI (12) is satisfied. In the same way, considering EI (z ) ∈ [α, β], z ∈ [0, L] and Assumption 1, for ∀z ∈ [0, L], the matrix Ξ (z ) given in (10) can be

 g¯ (z ) = f˜ (z ) −m−1 (z )(EI (z )f¯′′ (z ))′′ = g˜ (z ) ˜ Ax¯ + BLC0 h¯ = x˜

(D.2)

subject to



f¯ (0) = 0, f¯′ (z )z =0 = 0, EI (z )f¯′′ (z )z =L = 0 −(EI (z )f¯′′ (z ))′ z =L = c˜vT x¯ + dvT LC 0 h¯

where h¯ , [¯g (L)





f¯′ (z )z =L ]T .



(D.3)

H.-N. Wu, J.-W. Wang / Automatica 50 (2014) 2787–2798

The Eq. (D.2) subject to Eq. (D.3) can be further simplified as

 ′′ −(EI (z )f¯′′ (z )) = m(z )˜g (z )  ¯f (0) = 0, f¯′ (z ) = 0, EI (z )f¯′′ (z )z =L = 0 z =0   −(EI (z )f¯′′ (z ))′ z =L − c2 f¯′ (z )z =L = c˜vT A˜ −1 x˜ + c1 f˜ (L)

(D.4)

˜ −1 B)LC 0 h˜ , c1 f˜ (L) + c2 f¯′ (z )|z =L and where (dvT − c˜vT A f¯′ (z )z =L ]T .

h˜ , [f˜ (L)



From (17), we can verify c2 < 0.

(D.5)

Considering (D.5), we can easily deduce from (D.4) that f¯ (z ) = −d

z



EI (s)

0



z

z−y



z−s

s

 y

EI (y)

0

(s − L)ds

L

m(τ )˜g (τ )dτ dsdy L

where

 d,

L



1 + c2 0

 − c2

s−L EI (s)

L

(EI (y))−1

0

 −1 

˜ −1 x˜ + c1 f˜ (L) c˜vT A

ds

 y L

s



m(τ )˜g (τ )dτ dsdy . L

From the above analysis, the solution to the Eq. (D.1) is given as

 g¯ (z ) = f˜ (z )   z  z−s   ¯ (s − L)ds  f (z ) = −d 0 EI (s)  y s  z z−y   m(τ )˜g (τ )dτ dsdy −   EI (y) L L   0 −1 ˜ x˜ − A˜ −1 BLC0 h¯ . x¯ = A

(D.6)

It is easy to see that there exists a constant c3 > 0 independent of e˜ such that ∥ˇe∥H ≤ c3 ∥˜e∥H . Therefore, the linear operator A−1 is bounded, 0 ∈ ρ(A). The proof is complete.  References Artstein, Z. (1982). Linear systems with delayed controls: a reduction. IEEE Transactions on Automatic Control, 27(4), 869–879. Bechlioulis, C., & Rovithakis, G. (2011). Robust partial-state feedback prescribed performance control of cascade systems with unknown nonlinearities. IEEE Transactions on Automatic Control, 56(9), 2224–2230. Bekiaries-Liberis, N., & Krstic, M. (2011). Compensating the distributed effect of diffusion and counter-convection in multi-input and multi-output LTI systems. IEEE Trans. Automat. Control, 56(3), 637–643. Boyd, S., Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. PA: SIAM. Cao, Y.-Y., Lam, J., & Sun, Y.-X. (1998). Static output feedback stabilization: an ILMI approach. Automatica, 34(12), 1641–1645. Castanos, F., Ortega, R., Schaft, A., & Astolfi, A. (2009). Asymptotic stabilization via control by interconnenction of port-Hamiltonian systems. Automatica, 45(7), 1611–1618. Chen, Z. (2009). Global stabilization of nonlinear cascaded systems with a Lyapunov function in superposition form. Automatica, 45(9), 2041–2045. Chen, Z., & Huang, J. (2004). Dissipativity, stabilization, and regulation of cascadeconnected systems. IEEE Transactions on Automatic Control, 49(5), 635–650. Chentouf, B. (2003). Boundary feedback stabilization of a variant of the SCOLE model. Journal of Dynamical and Control Systems, 9(2), 201–232. Dong, J., & Yang, G.-H. (2007). Static output feedback control synthesis for linear systems with time-invariant parametric uncertainties. IEEE Transactions on Automatic Control, 52(10), 1930–1936.

2797

Gahinet, P., Nemirovskii, A., Laub, A., & Chilali, M. (1995). LMI control toolbox for use with Matlab. Natick, MA: The Math.Works Inc. Gao, Y., Wu, H.-N., Wang, J.-W., & Guo, L. (2014). Feedback control design with vibration suppression for flexible air-breathing hypersonic vehicles. Science China. Information Sciences, 57(3), 032204. 1-14. Ghaoui, L., Oustry, F., & AitRami, M. (1997). A cone complementarity linearization algorithm for static output-feedback and related problems. IEEE Transactions on Automatic Control, 42(8), 1171–1176. Goh, K., Turan, L., Safonov, M., Papavassilopoulos, G., & Ly, J. (1994). Biaffine matrix inequality properties and computational methods. in Proc. amer. control conf., Baltimore, MD, pp. 850–855. Guo, B.-Z. (2002). Riesz basis property and exponential stability of controlled Euler–Bernoulli beam equations with variable coefficients. SIAM Journal on Control and Optimization, 40(6), 1905–1923. Hardy, G. H., Littlewood, J. E., & Polya, G. (1952). Inequalities (2nd ED.). Cambridge, UK: Cambridge University Press. Jiang, Z., & Praly, L. (1998). Design of robust adaptive controllers for nonlinear systems with dynamic uncertainties. Automatica, 34(7), 825–840. Kanellakopoulos, I., Kokotovic, P., & Morse, A. (1992). A toolkit for nonlinear feedback design. Systems & Control Letters, 18(2), 83–92. Krstic, M. (2009a). Delay compensation for nonlinear, adaptive, and PDE systems. Boston: Birkhauser. Krstic, M. (2009b). Compensating actuator and sensor dynamics governed by diffusion PDEs. Systems & Control Letters, 58(5), 372–377. Krstic, M. (2009c). Compensating a string PDE in the actuation or sensing path of an unstable ODE. IEEE Trans. Automat. Control, 54(6), 1362–1368. Krstic, M., & Smyshlyaev, A. (2008). Backstepping boundary control for first order hyperbolic PDEs and application to systems with actuator and sensor delays. Systems & Control Letters, 57(9), 750–758. Liu, Z., & Zheng, S. (1999). Semigroups associated with dissipative systems. Boca Raton: Chapman and Hall/CRC. Morgul, O. (1992). Dynamic boundary control of a Euler–Bernoulli beam. IEEE Transactions on Automatic Control, 37(5), 639–642. Morgul, O. (1991). Orientation and stabilization of a flexible beam attached to a rigid body: planar motion. IEEE Trans. Automat. Control, 36(8), 953–962. Nwokah, O., & Hurmuzlu, Y. (2002). The mechanical systems design handbook: modeling measurement and control. Boca, Raton: CRC Press. Ortega, R., Schaft, A., Maschke, B., & Escobar, G. (2002). Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica, 38(4), 585–596. Pazy, A. (1983). Semigroups of linear operators and applications to partial differential equations. New York: Springer-Verlag. Susto, G., & Krstic, M. (2010). Control of PDE–ODE cascades with Neumann interconnections. Journal of the Franklin Institute, 347(1), 284–314. Tang, S., & Xie, C. (2011). State and output feedback boundary control for a coupled PDE–ODE system. Systems & Control Letters, 60(8), 540–545. Toker, O., & Ozbay, H. (1995). On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback. in Proc. amer. control conf., Seattle, WA, pp. 2525–2526. Tseng, C.-T., Chen, B.-S., & Uang, H.-J. (2001). Fuzzy tracking control design for nonlinear dynamic systems via T–S fuzzy model. IEEE Trans., Fuzzy Syst., 9(3), 381–392. Tucsnak, M., & Weiss, G. (2009). Observation and control for operator semigroups. Basel-Boston-Berlin: Birkäuser Verlag AG. Wang, J.-W., Wu, H.-N., & Li, H.-X. (2014). Static output feedback control design for linear MIMO systems with actuator dynamics governed by diffusion PDEs. International Journal of Control, 87(1), 90–100. Wu, H.-N., & Cai, K.-Y. (2006). Mode-independent robust stabilization for uncertain Markovian jump nonlinear system via fuzzy control. IEEE Transactions on Systems, Man and Cybernetics, Part B (Cybernetics), 36(3), 509–519. Zhao, X., & Weiss, G. (2010). Well-posedness, regularity and exact controllability of the SCOLE model. Mathematics of Control, Signals, and Systems, 22(2), 91–127. Zhao, X., & Weiss, G. (2011a). Suppression of the vibrations of wind turbine towers. IMA Journal of Mathematical Control and Information, 28(3), 377–389. Zhao, X., & Weiss, G. (2011b). Controllability and observability of a wellposed system coupled with a finite-dimensional system. IEEE Transactions on Automatic Control, 56(1), 88–99.

Huai-Ning Wu was born in Anhui, China, on November 15, 1972. He received the B.E. degree in automation from Shandong Institute of Building Materials Industry, Jinan, China and the Ph.D. degree in control theory and control engineering from Xi’an Jiaotong University, Xi’an, China, in 1992 and 1997, respectively. From August 1997 to July 1999, he was a Postdoctoral Researcher in the Department of Electronic Engineering at Beijing Institute of Technology, Beijing, China. In August 1999, he joined the School of Automation Science and Electrical Engineering, Beihang University (formerly Beijing University of Aeronautics and Astronautics), Beijing. From December

2798

H.-N. Wu, J.-W. Wang / Automatica 50 (2014) 2787–2798

2005 to May 2006, he was a Senior Research Associate with the Department of Manufacturing Engineering and Engineering Management (MEEM), City University of Hong Kong, Kowloon, Hong Kong. From October to December during 2006–2008 and from July to August in 2010, he was a Research Fellow with the Department of MEEM, City University of Hong Kong. From July to August in 2011 and 2013, he was a Research Fellow with the Department of Systems Engineering and Engineering Management, City University of Hong Kong. He is currently a Professor with Beihang University. His current research interests include robust control, fault-tolerant control, distributed parameter systems, and fuzzy/neural modeling and control. Dr. Wu serves as Associate Editor of the IEEE Transactions on Systems, Man & Cybernetics: Systems. He is a member of the Committee of Technical Process Failure Diagnosis and Safety, Chinese Association of Automation.

Jun-Wei Wang received the B.Sc. degree in Mathematics and Applied Mathematics and the M.Sc. degree in System Theory from Harbin Engineering University, Harbin, China, in 2007 and 2009, respectively, and the Ph.D. degree in Control Science and Engineering from Beihang University (Beijing University of Aeronautics and Astronautics), Beijing, China, 2013. In September 2013, he joined the School of Automation and Electrical Engineering, University of Science and Technology Beijing. His research interests include robust control and filtering, stochastic systems, distributed parameter systems, and fuzzy modeling and control.