Active control for flexible mechanical systems with mixed deadzone-saturation input nonlinearities and output constraint

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Journal of the Franklin Institute 356 (2019) 4749–4772 www.elsevier.com/locate/jfranklin

Active control for flexible mechanical systems with mixed deadzone-saturation input nonlinearities and output constraint Xiuyu He a,b, Zhijia Zhao c,∗, Yuhua Song a,b a School

of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China b Institute of Artificial Intelligence, University of Science and Technology Beijing, Beijing 100083, China c School of Mechanical and Electrical Engineering, Guangzhou University, Guangzhou 510006, China Received 19 January 2018; received in revised form 28 September 2018; accepted 1 November 2018 Available online 11 January 2019

Abstract There exist mixed deadzone-saturation input nonlinearities and output constraint in the practical implementation environment for flexible mechanical systems, and they have crucial influences on the performance of flexible systems. In this paper, two class of flexible structures are investigated and analyzed by designing the active boundary vibration control with auxiliary systems. Based on the infinite dimensional dynamic model of flexible mechanical systems, the barrier logarithmic terms are brought into the Lyapunov function and boundary vibration control laws for maintaining the output signals within the constrained region. Besides, the auxiliary terms are designed in the control laws to compensate for mixed nonlinear inputs which integrate the deadzone and saturation characteristics. With the simulation results, the theoretical analysis for the flexible mechanical systems is verified to be correct and the designed control laws are effective. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

∗

Corresponding author. E-mail address: [email protected] (Z. Zhao).

https://doi.org/10.1016/j.jfranklin.2018.11.028 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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1. Introduction Constraints including nonsmooth input nonlinearities (i.e., saturation, backlash, hysteresis and deadzone) [1–4] and safety and performance specifications of the system output (i.e., deflection, angle and tension constraint) [5,6] ubiquitously exist in industrial control systems, and generally arise from the influence of mechanical structures or production security considerations. If they are neglected in the context of controller design, it will generate system performance deterioration, give rise to system instability or lead to serious production accidents [7,8]. Consequently, it is of necessity to incorporate both input and output constrains in the course of control design. Flexible structures [9–14] are deemed as significant constituent parts of many mechanical systems, and become increasingly important in mechanical engineering field. However, flexible structures are subject to the structure vibration due to their physical characteristic and external disturbances. How to develop active vibration control schemes [14–16] to suppress the desired vibration is a crucial and hot problem for the research of infinite-dimensional flexible systems, and many control methods have been developed including modal control [17–19] and boundary control [20–22]. Compared with modal control, which is to be conducted depending on discretization of infinite dimensional distributed parameter dynamics [23–26], boundary control is generally regarded to be physically more realistic due to nonintrusive actuation and sensing [27]. Recently, some research on boundary control design of flexible mechanical systems with input nonlinearities or output constraints has launched in [6,28–32]. To mention a few, in [6,28], the barrier Lyapunov function (BLF) was introduced to construct a stabilizing boundary control law for weakening the vibration and guaranteeing tension constraints satisfaction for flexible mechanical systems. In [29], the auxiliary system and high-gain observers were employed to develop the output feedback control for the global stabilization of flexible systems subject to input saturation constraint. The constrained boundary control was presented to regulate the elastic deflection, and restrict the input of flexible hoses with the aid of backstepping method and Nussbaum function in [30]. In [31], the flexible manipulator was stabilized constructing the boundary control law, and the disturbance-like term was designed to tackle the nonlinear input backlash. In [32], the deflection of flexible beam was restrained proposing the adaptive boundary iterative learning control, and hyperbolic tangent functions and saturation functions were utilized to tackle the input constraint. The aforesaid control methods settled the input or output constraint problems for flexible systems, which also sets up a theoretical framework to cope with the flexible systems with both input and output constraints [33–35]. In [33], the input backlash and output constraint issues of uncertain flexible Timoshenko robotic manipulator were simultaneously tackled via constructing the disturbance observers, BLF and adaptive laws. In [34], the authors developed adaptive neural network control for coping with the input dead-zone and output constraint of the string system.In [35], the BLF and auxiliary system were respectively employed to restrict the boundary output in the desired region and eliminate the nonlinear input saturation. However, it is worthy of noticing that the control design of literatures [33–35] was limited to handle the input saturation, backlash or deadzone and the output constraint for flexible systems, and the synthetical effects of input nonlinearities including saturation and dead-zone, and the output constraint are not considered. To our best knowledge, in spite of the great progress of boundary control for flexible systems having been achieved, few attempts are made to develop the boundary control for handling the simultaneous effects of input saturation and dead-zone, and output constraint, which inspires this study.

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In this paper, we aim at constructing a vibration boundary control scheme for flexible mechanical systems to cope with the input saturation and dead-zone nonlinearities, and output constraint effects. The main contributions of this study are threefold. First, the auxiliary system and variable are introduced to compensate for the input nonlinearities including saturation and dead-zone effects. Second, the BLF is exploited to establish a stabilizing boundary control law for restraining the vibration and keeping the boundary output restricted in the specified region. Third, the uniformly bounded stability of the controlled system is achieved through rigorous analysis without resort to discretizing the system infinite-dimensional distributed parameter dynamics.  Notations are defined as follows: ( )(t ) = ( ), ( ) = ∂ ( )/∂x , (˙ ) = ∂ ( )/∂t , (˙ ) =  ∂ 2 ( )/∂ x∂ t , ( ) = ∂ 2 ( )/∂x 2 , ( ) = ∂ 3 ( )/∂x 3 , (˙ ) = ∂ 4 ( )/∂ t∂ x 3 , ( ) = ∂ 4 ( )/∂x 4 , and (¨ ) = ∂ 2 ( )/∂t 2 . The arrangement of this paper is listed as below. Dynamics analysis and preliminaries are introduced in Section 2. The procedure of control scheme design and stability analysis is presented in Section 3. Numerical simulations are completed in Section 4, and we reach a conclusion in Section 5. 2. Problem statement and preliminaries 2.1. Problem statement Consider the input saturation of the actuator, and the saturation nonlinearity is described as follows ⎧ ζ (t ) ≥ ζmax , ⎨ζmax , λ(t ) = sat (ζ (t )) = ζ (t ), ζmin < ζ (t ) < ζmax , (1) ⎩ ζmin , ζ (t ) ≤ ζmin , where ζ max > 0 and ζ min < 0 are the saturation limits. Consider the input dead-zone of the actuator, and the dead-zone nonlinearity is defined as follows ⎧ λ(t ) ≥ br , ⎨mr (λ(t ) − br ), 0, −bl < λ(t ) < br , u(t ) = D(λ(t )) = (2) ⎩ ml (λ(t ) + bl ), λ(t ) ≤ −bl , where br = [bsr , bbr ] > 0 and bl = [bsl , bbl ] > 0 are unknown parameters of the dead-zone, and mr = [msr , mbr ] and ml = [msl , mbl ] are slope of the dead-zone and they are positive constants. From the expression of input saturation and dead-zone, we easily learn that nonlinear characteristics are fairly sophisticated making them difficult to directly handle. According to [36], the input saturation and dead-zone nonlinearities can be transformed and expressed by a virtual input saturation. Therefore, to describe the problem of virtual input nonlinearity, we define the right inverse D+ of D in the following ⎧ ⎨τ (t )/mr + br , τ (t ) > 0, 0, τ (t ) = 0, ζ (t ) = D+ (τ (t )) = (3) ⎩ τ (t )/ml − bl , τ (t ) < 0, with τ (t ) = [τs (t ), τb (t )] being the control law to be designed.

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Fig. 1. Nonlinear input characteristic.

From Yang and Chen [36], the nonsymmetric saturation and dead-zone control input can be described in the following form ⎧ τ (t ) ≥ mr (ζmax − br ), ⎨mr (ζmax − br ), τ (t ), ml (ζmin − br ) < τ (t ) < mr (ζmax − br ), (4) u(t ) = D(sat (D+ (τ (t )))) = ⎩ ml (ζmin − bl ), τ (t ) ≤ ml (ζmin − bl ), It is evident that Eq. (4) means u(t ) = D(sat (D+ (τ (t )))). From the expression of u(t ) = [us (t ), ub (t )], it can be seen that the input saturation and dead-zone problem can be considered as an input saturation problem to handle, and the nonlinear input characteristic is shown in Fig. 1. 2.2. Preliminaries Before proceeding further, we put forward the following necessary lemmas and assumptions for subsequent development. Lemma 1. Let ψ1 (x, t ), ψ2 (x, t ) ∈ R, σ > 0 with (x, t ) ∈ [0, L] × [0, +∞ ), then the following inequality holds [37,38]   √ 1 ψ1 (x, t )ψ2 (x, t ) ≤ | ψ1 (x, t )ψ2 (x, t ) |=| √ ψ1 (x, t ) [ σ ψ2 (x, t )] | σ 1 2 ≤ ψ1 (x, t ) + σ ψ22 (x, t ) (5) σ Lemma 2. Let ψ (x, t ) ∈ R be a function defined on (x, t ) ∈ [0, L] × [0, +∞ ) satisfying ψ (0, t ) = 0, ∀t ∈ [0, +∞ ), then we have the following inequality [39]  L ψ 2 (x, t ) ≤ L ψ 2 (x, t )dx (6) 0

Assumption 1 [40]. For the new expression of input saturation Eq. (4), we assume there exists a positive constant ι such that | u(t)| ≤ ι, in which u(t ) = u(t ) − τ (t ).

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Fig. 2. Control design procedure for flexible mechanical systems.

Fig. 3. A flexible string system.

3. Control design In this paper, two classical mechanical systems are studied considering the influences of input nonlinearities and output constraint, and boundary vibration control strategies are developed to deal with the vibration problem and constraint problem. The control design procedure is shown in Fig. 2. For the control design, dynamical models of two classical mechanical systems are given directly, and revelent system parameters are given as L m M T EI Ys Yb u(t )

Length of the flexible structures Mass per unit length of the structures Mass of the tip payload Tension of the structures Bending stiffness of the E-beam system Boundary output constraint for the string system Boundary output constraint for the E-beam system Boundary control input applied on the tip payload

3.1. Flexible string system In this study, the dynamics of the studied string system shown in Fig. 3 is presented as follows ms y¨s (x, t ) − Ts ys (x, t ) = 0,

0
(7)

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ys (0, t ) = 0,

(8)

Ts ys (L, t ) = us (t ) − Ms y¨s (L, t )

(9)

Then, in order to eliminate the input nonlinearities effects, we introduce the following auxiliary system 1 (−ks s (t ) − us (t ) + Ts ys (L, t )) Ms

˙s (t ) =

(10)

with ϱs (t) and ks being the state variable of the auxiliary system to be designed and a positive constant, respectively. For convenience of the stability analysis of closed-loop system, we define an auxiliary variable as ηs (t ) = y˙s (L, t ) + ys (L, t ) + s (t )

(11)

Differentiating (11) and substituting Eqs. (9) and (10), we have η˙s (t ) = y¨s (L, t ) + y˙s (L, t ) + ˙s (t ) 1 1 = (us (t ) − Ts ys (L, t ) + Ms y˙s (L, t )) + (−ks s (t ) − us (t ) + Ts ys (L, t )) Ms Ms 1 = (τs (t ) − ks s (t ) + Ms y˙s (L, t )) Ms

(12)

According to the above analysis, we design the control input τ (t) as τs (t ) = −ks1 ηs (t ) − ks2 ηs (t )/ ln +ks s (t ) − Ms y˙s (L, t )

Yc2 ys (L , t )y˙s (L , t ) Yc2 − M η (t ) / ln s s Yc2 − ys2 (L, t ) Yc2 − ys2 (L, t ) Yc2 − ys2 (L, t ) (13)

where ks1 , ks2 > 0. Then define the Lyapunov candidate function as Vs (t ) = Vsm (t ) + Vsn (t ) + Vso (t ) where Vsm (t ) =

χs m s 2

 0



y˙s2 (x, t )dx +

χs Ts 2

 0

L

ys2 (x, t )dx

(15)

x ys (x , t )y˙s (x , t )dx

(16)

Ms 2 Ms 2 Yc2

s (t ) + ηs (t ) ln 2 2 2 Yc − ys2 (L, t )

(17)

Vsn (t ) = ςs ms 0

Vso (t ) =

L

L

(14)

with χ s , ς s > 0.

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Lemma 3. The Lyapunov function candidate (14) is a positive function with the following form 0 ≤ γs1 (Vsm (t ) + Vso (t ) ) ≤ Vs (t ) ≤ γs2 (Vsm (t ) + Vso (t ) )

(18)

where γ s1 , γ s2 > 0. Proof. We can see Vsn (t) satisfies   ςs ms L L 2 ςs ms L L 2 |Vsn (t )| ≤ ys (x, t )dx + y˙s (x, t )dx ≤ φsVsm (t ) 2 2 0 0

(19)

ςs ms L where φs = χs min . (ms ,Ts ) Further, we then obtain

−φsVsm (t ) ≤ Vsn (t ) ≤ φsVsm (t )

(20)

Then, we have −φs1Vsm (t ) ≤ Vsm (t ) + Vsn (t ) ≤ φs2Vsm (t ) where, φs1 = 1 − φs and φs2 = 1 + φs . Assuming that ς s satisfies 0 < ςs <

χs min (ms ,Ts ) , ms L

(21) we can obtain 0 < φ s < 1, and

0 ≤ γs1 (Vsm (t ) + Vso (t ) ) ≤ Vs (t ) ≤ γs2 (Vsm (t ) + Vso (t ) )

(22)

where γs1 = min (φs1 , 1 ) = φs1 and γs2 = max (φs2 , 1 ) = φs2 .  Lemma 4. The time derivative of the Lyapunov function candidate (14) can be upper bounded with V˙s (t ) ≤ −γsVs (t ) + μs

(23)

where γ s , μs > 0. Proof. Taking the differentiation of Vs (t) yields V˙s (t ) = V˙sm (t ) + V˙sn (t ) + V˙so (t )

(24)

Differentiating Vsm (t) and substituting Eq. (7) results in V˙sm (t ) = χs T ys (L , t )y˙s (L , t )

(25)

Combining Eq. (11), we derive ηs2 (t ) = y˙s2 (L, t ) + ys2 (L, t ) + s2 (t ) + 2y˙s (L , t )ys (L , t ) + 2y˙s (L, t ) s (t ) + 2ys (L, t ) s (t ) (26) Then, we further obtain from the above equation V˙sm (t ) =

χs Ts 2 χs Ts 2 χs Ts 2 χs Ts 2 ηs (t ) − y˙s (L, t ) − ys (l, t ) −

(t ) 2 2 2 2 s −χs Ts y˙s (L, t ) s (t ) − χs Ts ys (L, t ) s (t )

(27)

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Applying Lemma 1 to Eq. (27), we get V˙sm (t ) ≤

χs Ts 2 χs Ts 2 χs Ts 2 χs Ts 2 η (t ) − y˙ (L, t ) − y (l, t ) −

(t ) 2 s 2 s 2 s 2 s χs Ts 2 χs Ts 2 + y˙ (L, t ) + χs Ts s1 s2 (t ) + y (l, t ) + χs Ts s2 s2 (t ) s1 s s2 s

(28)

where ϖs1 , ϖs2 > 0. Differentiating Vsn (t) and substituting Eq. (7), we have   ςs Ts L 2 ςs ms L 2 ςs Ts L 2 ςs ms L 2 V˙sn (t ) = − ys (x, t )dx + ys (L , t ) + y˙s (L , t ) − y˙s (x , t )dx (29) 2 0 2 2 2 0 Differentiating Vso (t), substituting Eqs. (10)–(13), and then applying Lemma 1, we get 1 2 Ts 2

(t ) + s3 us2 (t ) +

(t ) + Ts s4 ys2 (L, t ) s3 s s4 s Yc2 −ks1 ηs2 (t ) ln 2 − ks2 ηs2 (t ) Yc − ys2 (L, t )

V˙so (t ) ≤ −ks s2 (t ) +

(30)

where ϖs3 , ϖs4 > 0. Substituting Eqs. (28)–(30) into Eq. (24) leads to   χs Ts χs Ts 2 1 Ts ˙ ηs (t ) − ks +

2 (t ) Vs (t ) ≤ − ks2 − − χs Ts s1 − χs Ts s2 − − 2 2 s3 s4 s   χs Ts χs Ts ςs ms L 2 χs Ts χs Ts ςs Ts L − − − y˙s (L, t ) − − − − Ts s4 ys2 (L, t ) 2 s1 2 2 s2 2   Yc2 ςs Ts L 2 ςs ms L 2 −ks1 ηs2 (t ) ln 2 − y (x, t ) dx − y˙s (x, t )dx Yc − ys2 (L, t ) 2 0 s 2 0 +s3 us2 (t )

(31)

where the parameters χ s , ς s , ks , ks1 , ks2 , and ϖsi , for i = 1, 2, 3, 4, are chosen to satisfy the following conditions χs Ts χs Ts ςs ms L − − ≥0 2 s1 2

(32)

χs Ts χs Ts ςs Ts L − − − Ts s4 ≥ 0 2 s2 2

(33)

ks2 −

χs Ts ≥0 2

αs = ks +

χs Ts 1 Ts − χs Ts s1 − χs Ts s2 − − >0 2 s3 s4

μs = s3 ι2 < +∞

(34)

(35)

(36)

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Combining Eqs. (32)–(36), Eq. (31) can be rewritten as   ςs Ts L 2 ςs ms L 2 ˙ Vs (t ) ≤ − y (x, t )dx − y˙s (x, t )dx − αs s2 (t ) 2 0 s 2 0 Yc2 −ks1 ηs2 (t ) ln 2 + μs Yc − ys2 (L, t )

4757

(37)

Then, we further obtain V˙s (t ) ≤ −γs3 [Vsm (t ) + Vso (t )] + μs

(38)



where γs3 = min . Combining Eqs. (22) and (38) yields ςs ςs 2αs 2ks1 χs , χs , Ms , Ms

V˙s (t ) ≤ −γsVs (t ) + μs

(39)

where γs = γs3 /γs2 .  Theorem 1. For the studied string system described by Eqs. (7)–(9), under the developed control (13) , provided that the initial conditions are bounded, and the parameters designed χ s , ς s , ks , ks1 , ks2 , and ϖsi , for i = 1, 2, 3, 4, are chosen such that the constraints specified in Eqs. (32)–(36) hold, then we have the following conclusions. (1) The closed-loop system is proven to be uniformly bounded, that is, ys (x, t) satisfies

 2L |ys (x,t )|≤ γ χs Ts Vs (0)e−γs t + μγss ,∀(x ,t )∈[0,L ]×[0,∞ ). s1 (2) The closed-loop system

is also proven to be uniformly ultimately bounded, that is, ys (x, s t) satisfies lim |ys (x, t )| ≤ γs T2Lμ , ∀x ∈ [0, L]. s γs1 χs t→∞

Proof. Multiplying Eq. (23) by eαs t , and integrating the resulting equation, we get  μs  1 − e−γs t Vs (t ) ≤ Vs (0)e−γs t + γs Combining Vsm (t), Eq. (22) and Lemma 2 results in  χs Ts 2 χs Ts L 2 1 ys (x, t ) ≤ ys (x, t )dx ≤ Vsm (t ) ≤ Vs (t ) 2L 2 0 γs1 Substituting Eq. (40) into Eq. (41) yields       2L μs 2L μs −γ t −γ t Vs (0)e s + Vs (0) + | ys (x, t ) |≤ (1 − e s ) ≤ χs γs1 Ts γs χs γs1 Ts γs

(40)

(41)

(42)

∀(x, t ) ∈ [0, L] × [0, +∞ ).

s We finally obtain, lim |ys (x, t )| ≤ γs T2Lμ ,∀x ∈[0,L ], therefore ys (x, t) is uniformly ultimately s γs 1 χs t→∞ bounded.  3.2. Flexible beam system In this study, the dynamics of the studied beam system shown in Fig. 4 is presented as follows mb y¨b (x, t ) − Tb yb (x, t ) + E I yb (x, t ) = 0, 0 < x < L

(43)

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Fig. 4. A flexible beam system.

yb (0, t ) = yb (0, t ) = yb (L, t ) = 0

(44)

Tb yb (L, t ) − E I yb (L, t ) = ub (t ) − Mb y¨b (L, t )

(45)

For eliminating the input nonlinearities effects, we introduce the following auxiliary system

˙b (t ) =

1 (−kb b (t ) − ub (t ) + Tb yb (L, t ) − E I yb (L, t )) Mb

(46)

with ϱb (t) and kb being the state variable of the auxiliary system to be designed and a positive constant, respectively. For convenience of analyzing the stability of closed-loop system, we define an auxiliary variable as ηb (t ) = y˙b (L, t ) + yb (L, t ) − yb (L, t ) + b (t )

(47)

Differentiating Eq. (11) and substituting Eqs. (9) and (10), we arrive at η˙b (t ) = y¨b (L, t ) + y˙b (L, t ) + ˙b (t ) 1 = (ub (t ) − Tb yb (L, t ) + E I yb (L, t ) + Mb y˙b (L, t ) − Mb y˙b (L, t )) Mb 1 + (−kb b (t ) − ub (t ) + Tb yb (L, t ) − E I yb (L, t )) Mb 1 = (τb (t ) − kb b (t ) + Mb y˙b (L, t ) − Mb y˙b (L, t )) Mb

(48)

According to the aforementioned analysis, we design the control input τ (t) as τb (t ) = −kb1 ηb (t ) − kb2 ηb (t )/ ln

Yb2 Yb2 yb (L , t )y˙b (L , t ) − M η (t ) / ln b b Yb2 − yb2 (L, t ) Yb2 − yb2 (L, t ) Yb2 − yb2 (L, t )

+kb b (t ) − Mb y˙b (L, t ) + Mb y˙b (L, t )

(49)

where kb1 , kb2 > 0. Then define the Lyapunov candidate function as Vb (t ) = Vbm (t ) + Vbn (t ) + Vbo (t )

(50)

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where χb m b Vbm (t ) = 2



L 0



L

χb Tb + 2



L 0

yb2 (x, t )dx

χb E I + 2

 0

L

yb2 (x, t )dx

(51)

x yb (x , t )y˙b (x , t )dx

(52)

Yb2 Mb 2 Mb 2

b (t ) + ηb (t ) ln 2 2 2 Yb − yb2 (L, t )

(53)

Vbn (t ) = ςb mb 0

Vbo (t ) =

y˙b2 (x, t )dx

4759

with χ b , ς b > 0. Lemma 5. The Lyapunov function candidate (50) is a positive function with the following form 0 ≤ γb1 (Vbm (t ) + Vbo (t ) ) ≤ Vb (t ) ≤ γb2 (Vbm (t ) + Vbo (t ) )

(54)

where γ b1 , γ b2 > 0. Proof. We can see Vbn (t) satisfies   ςb mb L L 2 ςb mb L L 2 |Vbn (t )| ≤ yb (x, t )dx + y˙b (x, t )dx ≤ φbVbm (t ) 2 2 0 0

(55)

ςb mb L where φb = χb min . (mb ,Tb ) We then obtain

−φb1Vbm (t ) ≤ Vbn (t ) ≤ φb2Vbm (t ) where φ b1 , φ b2 > 0. Assuming that ς b satisfies 0 < ςb <

(56) χb min (mb ,Tb ) , mb L

we can obtain 0 < φ b < 1, and

0 ≤ γb1 (Vbm (t ) + Vbo (t ) ) ≤ Vb (t ) ≤ γb2 (Vbm (t ) + Vbo (t ) )

(57)

where γb1 = min (1 − φb1 , 1 ) = 1 − φb1 and γb2 = max (1 + φb2 , 1 ) = 1 + φb2 .  Lemma 6. The time derivative of the Lyapunov function candidate (50) can be upper bounded with V˙b (t ) ≤ −γbVb (t ) + μb

(58)

where γ b , μb > 0. Proof. Taking the differentiation of Vb (t) gives V˙b (t ) = V˙bm (t ) + V˙bn (t ) + V˙bo (t )

(59)

Differentiating Vbm (t) and substituting Eq. (43) yields V˙bm (t ) = χb Tb yb (L , t )y˙b (L , t ) − χb E I yb (L , t )y˙b (L , t )

(60)

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Combining Eq. (47), we get ηb2 (t ) = y˙b2 (L, t ) + yb2 (L, t ) + yb2 (L, t ) + b2 (t ) + 2y˙b (L , t )yb (L , t ) +2y˙b (L, t ) b (t ) + 2yb (L, t ) b (t ) −2y˙b (L, t )yb (L, t ) − 2yb (L, t )yb (L, t ) − 2 b (t )yb (L, t )

(61)

Then, we further obtain from Eq. (61) V˙bm (t ) =

χb E I 2 χb E I 2 χb E I  2 χb E I 2 χb E I 2 η (t ) − y˙ (L, t ) − y (L, t ) − y (L, t ) −

(t ) 2 b 2 b 2 b 2 b 2 b   +χb (Tb − E I )yb (L , t )y˙b (L , t ) − χb E I y˙b (L, t ) b (t ) − χb E I yb (L, t ) b (t ) +χb E I yb (L, t )yb (L, t ) + χb E I b (t )yb (L, t )

(62)

Differentiating Vn (t) and substituting Eq. (43), we derive   ςb Tb L 2 ςb mb L 2 ςb Tb L 2 ςb mb L 2 V˙bn (t ) = − yb (x, t )dx + yb (L , t ) + y˙b (L , t ) − y˙b (x , t )dx 2 0 2 2 2 0  3ςb E I L 2 − yb (x, t )dx − ςb E I Lyb (L, t )yb (L, t ) (63) 2 0 Differentiating Vbo (t), substituting Eqs. (10)–(13), and then applying Lemma 1, we have 1 2 T 2

(t ) + b1 ub2 (t ) +

(t ) + Tb b2 yb2 (L, t ) b1 b b2 b Yb2 EI 2 +

b (t ) + E I b3 yb2 (L, t ) − kb1 ηb2 (t ) ln 2 − kb2 ηb2 (t ), b3 Yb − yb2 (L, t )

V˙bo (t ) ≤ −kb b2 (t ) +

(64)

where ϖb1 , ϖb2 , ϖb3 > 0. Substituting Eqs. (62)–(64) into Eq. (59) and using Lemma 1 leads to   χb E I χb E I 2 χb E I b4 χb E I b5 χb E I b6 1 ˙ ηb (t ) − kb + Vb (t ) ≤ − kb2 − − − − − 2 2 2 2 2 b1  E I E I | T − E I |  m L Tb EI χ χ ς χ b b b b b7 b b

2 (t ) − − − − − − y˙b2 (L, t ) b2 b3 b 2 2b4 2 2    ςb Tb L 2 ςb mb L 2 χb E I χb |Tb − E I | χb E I − yb (x, t )dx − y˙b (x, t )dx − − − 2 0 2 2 2b5 2b7 0  E I E I |χb − ςb L| ςb Tb L χb E I χ b − − − Tb b2 yb2 (L, t ) + b1 ub2 (t ) − − 2b8 2 2 2b6 2 Yb E I |χb − ςb L|b8 EI y2 (L, t ) − kb1 ηb2 (t ) ln 2 − − 2 b3 b Yb − yb2 (L, t )  3ςb E I L 2 − yb (x, t )d (65) 2 0 where ϖb4 ∼ ϖb8 > 0 and the parameters χ b , ς b , kb , kb1 , kb2 , and ϖbi , for i = 1 . . . 8, are chosen to satisfy the following conditions χb E I χb E I χb |Tb − E I |b7 ςb mb L − − − ≥0 2 2b4 2 2

(66)

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χb E I χb E I χb |Tb − E I | E I |χb − ςb L| ςb Tb L − − − − − Tb b2 ≥ 0 2 2b5 2b7 2b8 2

(67)

χb E I χb E I E I |χb − ςb L|b8 EI − − − ≥0 2 2b6 2 b3

(68)

kb2 −

χb E I ≥0 2

αb = kb +

χb E I χb E I b4 χb E I b5 χb E I b6 1 Tb EI − − − − − − >0 2 2 2 2 b1 b2 b3

μb = b3 ι2 < +∞ Combining Eqs. (66)–(71), Eq. (65) can be rewritten as    ςb Tb L 2 ςb mb L 2 3ςb E I L 2 ˙ Vb (t ) ≤ − y (x, t )dx − y˙b (x, t )dx − yb (x, t )dx 2 0 b 2 2 0 0 Yb2 −αb b2 (t ) − kb1 ηb2 (t ) ln 2 + μb Yb − yb2 (L, t )

(69)

(70)

(71)

(72)

Then we further get V˙b (t ) ≤ −γb3 [Vbm (t ) + Vbo (t )] + μb

 b where γb3 = min χςbb , χςbb , 3χςbb , 2α , 2kb1 . Mb Mb Combining Eqs. (57) and (73) leads to

(73)

V˙b (t ) ≤ −γbV (t ) + μb

(74)

where γb = γb3 /γb2 .  Theorem 2. For the studied beam system described by Eqs. (43)–(45), under the developed control (49), provided that the initial conditions are bounded, and the parameters designed χ b , ς b , kb , kb1 , kb2 , and ϖbi , for i = 1 · · · 8, are chosen such that the constraints specified in Eqs. (66)–(71) hold, then we have the following conclusions. (1) The closed-loop system is proven to be uniformly bounded, that is, yb (x, t) satisfies

 |yb (x, t )| ≤ γb12Lχb Tb Vb (0)e−γbt + μγbb ,∀(x,t )∈[0,L]×[0,∞ ). (2) The closed-loop system is also proven to be uniformly ultimately bounded, that is, yb (x,

b t) satisfies lim |yb (x, t )| ≤ γb T2Lμ ,∀x ∈[0,L ]. b γb1 χb t→∞

Proof. Proof process of Theorem 2 is the same as that in Theorem 1 and thus is omitted.  Remark 1. For the string system, we derive that the designed Lyapunov function is bounded from the deduction of (40). However, the value of the barrier Lyapunov function Eq. (17) will grow to infinity whenever ys (L, t) approaches some limits Yc . Therefore, we can conclude that

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Table 1 Parameters of the flexible string system. Parameter

Description

Value

L Ms Mb ms mb Ts Tb EI Yc Yb

Length of the flexible system Mass of the tip load in flexible string system Mass of the tip load in E-beam system Uniform mass per unit length of the flexible string system Uniform mass per unit length of the E-beam system Tension of the flexible string system Tension of the E-beam system Bending stiffness of the E-beam system Output constraint of the flexible string system Output constraint of the E-beam system

1.0 m 1 kg 5 kg 1.0 kg/m 1.0 kg/m 6N 10 N 10 Nm2 0.1 m 0.07 m

the output signal satisfies |ys (L, t)| = Yc . Given that the initial condition |ys (L, 0)| < Yc , we infer that ys (L, t) will remains in the sets |ys (L, t)| < Yc , ∀t ∈ [0, ∞) which implies that the desired output constraint is ensured. In addition, the similar BLF is constructed in Eq. (53) to keep output state yb (L, t) in the constraint Yb and the output constraint problem in the E-beam system is handled. Remark 2. Active vibration controls are designed for two typical flexible mechanical systems with input and output constraints, and the determined bounded signals which are composed of the control laws can be measured or calculated with the sensors mounted at the end of the flexible structures. Remark 3. In this paper, the uniform ultimate boundedness of flexible mechanical systems is ensured considering the input nonlinearities, and the convergent bound of the system states is related to the parameters μs and μb which are generated by the input errors between the actual nonlinear inputs and the desired designed inputs. Nevertheless, the result of uniform ultimate boundedness is a little weak, and we can refer to the approaches in [41,42] for achieving the exponential stability of the string and beam with constraints in the future. Remark 4. In this study, we simultaneously consider the vibration suppression problem of the flexible structures and the influences of the input nonlinearities and the output constraints. For the physical characteristic of the flexible system, there always exist external disturbances, and many methods can be used to eliminate their effect, e.g., sign function, hyperbolic tangent function, disturbance observer, UDE control, etc. In the previous works [37,38], the disturbance of the flexible systems is considered in the dynamic analysis and is handled in the control design. If the considered string and beam system is subjected to external disturbances, we can use the approaches of handling disturbances as presented in [37,38]. 4. Numerical simulation In this paper, two class of flexible systems with input nonlinearities are studied, and the simulation experiments are made by using the finite difference method to analyze the effectiveness of the designed control laws. The system parameters of the flexible systems are presented in Table 1. In addition, the initial conditions of the closed-loop system are given as ys (x, 0) = yb (x, 0) = xL/20 m, y˙s (x, 0) = 1/3 m/s, y˙b (x, 0) = 0.5 m/s.

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Fig. 5. Displacement y(x, t) of the flexible string without any control.

Fig. 6. Displacement y(x, t) of the E-beam without any control.

The effectiveness of the designed boundary control laws will be represented by comparing with three cases: Without control: In this case, the dynamic characteristic of the flexible systems without any controller is analyzed. Without the external disturbances and damping, it can be seen that the flexible system is vibrating freely. Figs. 5 and 6 illustrate the vibration of the flexible string

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Fig. 7. End point and center point displacements of the flexible string without any control.

Fig. 8. End point displacements of the E-beam without any control.

and beam system respectively with 3D graphs. In order to clarify the vibration of flexible systems, the time-varying curves of point displacements of the flexible string system ys (L, t) and ys (0.5L, t) are given in Fig. 7. Besides, the time-varying curves of the point displacement of the flexible beam system yb (L, t) is given in Fig. 8. Through analyzing the above figures, we can know that flexible systems will suffer from unattenuated oscillations under no control.

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Fig. 9. Displacement y(x, t) of the flexible string under the proposed boundary control.

Fig. 10. Displacement y(x, t) of the E-beam under the proposed boundary control.

Under the proposed boundary control: Compared with the above case, boundary control laws with auxiliary system are used into these flexible systems. With boundary control laws (10), (11) and (13), the control performances for the flexible string system is illustrated with a 3D graph (Fig. 9) and a time-varying point displacement curve (Fig. 13) with determined control gains ks = 10, ks1 = 1 and ks2 = 15. Similarly, the control performances of the flexible beam system with control laws (46), (47) and (49) are illustrated in Figs. 9 and 14 by choosing

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Fig. 11. Displacement y(x, t) of the flexible string under the boundary control without auxiliary system and barrier term.

Fig. 12. Displacement y(x, t) of the E-beam under the boundary control without auxiliary system and barrier term.

control gains kb = 0.1, kb1 = 1000, and kb2 = 0.5. The nonlinear inputs uc (t) and ub (t) are given in Figs. 17 and 18, where, us max = 2 N, us min = −2 N, bsl = bsr = 0.05, msl = msr = 1, and ub max = 100 N, ub min = −100 N, bbl = bbr = 0.05, mbl = mbr = 1. It is easy to know that the vibrations of flexible systems with input nonlinearities are reduced with the designed control laws.

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Fig. 13. End point and center point displacements of the flexible string under the proposed boundary control.

Fig. 14. End point displacement of the E-beam under the proposed boundary control.

Under the boundary control without auxiliary system and barrier term: In the previous works [37,38], two boundary controllers are designed to improve the performance of the flexible systems with ignoring the input nonlinearities and the output constraint, and they are given as τs = Ts ys (L, t ) − Ms y˙s (L, t ) − 5[y˙s (L, t ) + ys (L , t )] and τb = −E I yb (L , t ) + Tb ys (L, t ) − Mb [y˙b (L, t ) − y˙b (L, t )] − 10[y˙b (L, t ) − yb (L, t ) + yb (L, t )]. For comparison of the control

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Fig. 15. End point and center point displacements of the flexible string under the boundary control without auxiliary system and barrier term.

Fig. 16. End point displacement of the E-beam under the boundary control without auxiliary system and barrier term.

performance, we put these two boundary control strategies without auxiliary system into these two flexible systems which consider the effect of input nonlinearities in this case. Consistent with the above cases, corresponding simulation results are made and illustrated in Figs. 11, 12, 15 and 16. From obtained simulation results, we know that the convergence of

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Fig. 17. Boundary control with input nonlinearity of the flexible string system us (t ).

Fig. 18. Boundary control with input nonlinearity of the E-beam system ub (t ).

system states is not guaranteed with these two boundary controllers, and the input nonlinearities have great effect on the flexible systems. With simulation results in above cases, it is seen that the designed boundary controllers have good control performance on handling the vibration problem for flexible systems

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and stabilizing these two flexible mechanical systems. Besides, there exist good control performances for the flexible systems under the proposed boundary control law with the auxiliary system. In addition, the effectiveness of the boundary control laws are validated by the theoretical analysis and simulation experiments. 5. Conclusion In this paper, we dealt with the vibration control and global stabilization issue of flexible mechanical systems preceded by input deadzone and saturation nonlinearities and output constraint. The boundary barrier-based control laws with auxiliary system and variable are developed to attenuate the vibrations of the flexible systems, to satisfy the output constraints, and to eliminate the input nonlinearities effects. The uniformly bounded stability of the controlled system is analyzed and demonstrated through rigorous Lyapunov analysis without any model reduction. Finally, simulation results are demonstrated for control performance verification. Future interests lie in neural or learning approaches [43–49] for flexible systems and the implementation of the proposed control. Acknowledgment The authors would like to thank the Editor-In-Chief, the Associate Editor and the anonymous reviewers for their constructive comments which helped improve the quality and presentation of this paper. This work was supported in part by the National Natural Science Foundation of China (61803109, 11832009, 61803111) and by the Innovative School Project of Education Department of Guangdong (2017KQNCX153, 2017KZDXM060). References [1] Z.J. Zhao, X.G. Wang, C.L. Zhang, Z.J. Liu, J.F. Yang, Neural network based boundary control of a vibrating string system with input deadzone, Neurocomputing 275 (2018) 1021–1027. [2] J. Zhou, C. Wen, Adaptive Backstepping Control of Uncertain Systems: Nonsmooth Nonlinearities, Interactions or Time-Variations, Springer, Germany, 2008. [3] W. He, T.T. Meng, X.Y. He, S. Ge, Unified iterative learning control for flexible structures with input constraints, Automatica 86 (2018) 326–336. [4] H. Li, J. Wang, P. Shi, Output-feedback based sliding mode control for fuzzy systems with actuator saturation, IEEE Trans. Fuzzy Syst. 24 (6) (2016) 1282–1293. [5] X.Y. He, W. He, J. Shi, C.Y. Sun, Boundary vibration control of variable length crane systems in two dimensional space with output constraints, IEEE/ASME Trans. Mechatron. 22 (5) (2017) 1952–1962. [6] W. He, S.S. Ge, Cooperative control of a nonuniform gantry crane with constrained tension, Automatica 66 (4) (2016) 146–154. [7] Z.J. Zhao, Y. Liu, W. He, F. Luo, Adaptive boundary control of an axially moving belt system with high acceleration/deceleration, IET Control Theory Appl. 10 (11) (2016) 1299–1306. [8] Z.J. Liu, J.K. Liu, W. He, Dynamic modeling and vibration control for a nonlinear 3-dimensional flexible manipulator, Int. J. Rob. Nonlinear Control 28 (13) (2018) 3927–3945. [9] Z.J. Zhao, Y. Liu, F. Guo, Y. Fu, Modelling and control for a class of axially moving nonuniform system, Int. J. Syst. Sci. 48 (4) (2017) 849–861. [10] A. Tavasoli, Adaptive robust boundary control of shaft vibrations under perturbations with unknown upper bounds, Int. J. Adapt. Control Signal Process. 29 (5) (2015) 537–562. [11] K.D. Do, Stochastic boundary control design for extensible marine risers in three dimensional space, Automatica 77 (2017) 184–197. [12] K.D. Do, Boundary control design for extensible marine risers in three dimensional space, J. Sound Vib. 388 (3) (2017) 1–19.

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