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Performance output tracking for coupled wave equations with unmatched boundary disturbance
Available online at www.sciencedirect.com
Journal of the Franklin Institute 356 (2019) 6280–6302 www.elsevier.com/locate/jfranklin
Performance output tracking for coupled wave equations with unmatched boundary disturbanceR Yingli Zhu, Feng-Fei Jin∗ School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China Received 12 September 2018; received in revised form 23 March 2019; accepted 29 May 2019 Available online 7 June 2019
Abstract In this paper, we consider performance output tracking for coupled wave equations with general external unmatched disturbance. An observer is designed first to estimate the state and disturbance simultaneously. Then we construct a servo system determined completely by the measured output and the reference signal which in turn gives dynamics of reference signal. In the following, an output feedback controller is designed based on this observer and servo system. It is shown that the closedloop system is well-posed and the performance output is tracking the reference signal. Finally, we present some numerical results to illustrate the effectiveness of the controller. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Output regulation problem is a classical problem in control theory. In this framework, output feedback controller needs to be designed to make output signal track the reference signal even though in the presence of disturbance. First results related to the output regulation problem can be found in linear finite-dimensional system, for example [3,4,9,10], then extended to nonlinear finite-dimensional system [2,5] and linear infinite-dimensional system [1,6,7,14,17]. Accordingly, the equivalence between solvability of the regulator problem and solvability of a pair of linear matrix Sylvester equations in finite-dimensional system [9] is also generalized R ∗
This work was supported by the National Natural Science Foundation of China (61603226). Corresponding author. E-mail address: [email protected] (F.-F. Jin).
https://doi.org/10.1016/j.jfranklin.2019.05.041 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
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to infinite-dimensional version [1]. Recently, internal model principle plays an important role in dealing with regulation problem in infinite-dimensional frame [18–20,23]. Sliding mode control is adopted to design controller to make heat and wave equation track known heat and wave dynamics in [22] while control and disturbance are distributed. Adaptive method is also introduced to such cases while the unknown parameters in disturbance can be identified [13]. In most work aforementioned reference signal and disturbance are generated by an exosystem, finite- or infinite-dimensional, which means some information of disturbance can be obtained by various methods. When the dynamics of disturbance and reference is unavailable, there is few result for output regulation of PDEs’ system [28,29]. For large disturbances or uncertainties, active disturbance rejection control (ADRC), first proposed by Han [15], has established itself as a powerful control technology in control systems. In [11], ADRC was applied to estimate the disturbance through an extended state observer (ESO) for a wave PDEs. But the derivative of the disturbance was required to be bounded and high-gain design was needed. In [8], a completely new approach to disturbance estimation by designing an infinite-dimensional disturbance estimator directly was established, which could relax the disturbance to be d ∈ L∞ (0, ∞) or d ∈ L2 (0, ∞) and high gain design was relaxed. Authors in [29] applied ADRC to estimate the state and the external disturbance for a wave equation. In the following a servo system was established through disturbance estimator and reference signal. From another point of view, this servo system can be viewed as a dynamics of reference signal. Thus output regulation was implemented by designing an output feedback control. The same problem was considered in [28], but there was internal nonlinear uncertainty on the boundary except external disturbance. In [16], we applied this method to a heat equation which suffered external disturbance on left boundary and received control on the right boundary. When disturbance and control vanished, the system has eigenvalue on imaginary axis which is the same as the one in [29]. In this paper, we are concerned with the problem of performance output tracking of coupled wave system. The external disturbance is a general bounded unknown signal and reference is a general known function. The system we considered is governed by the following PDEs: ⎧ utt (x, t ) = uxx (x, t ) x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪utt (x, t ) = uxx (x, t ), x ∈ (1, 2), t > 0, ⎪ ⎪ ⎪ ⎪ux (0, t ) = d (t ), t ≥ 0, ⎪ ⎪ ⎪ u(1− , t ) = u(1+ , t ), t ≥ 0, ⎪ ⎪ ⎪ ⎨u (1+ , t ) − u (1− , t ) = ku (1+ , t ), t ≥ 0, x x t (1.1) ⎪ u (2, t ) = −U (t ) , t ≥ 0, x ⎪ ⎪ ⎪ ⎪ ym (t ) = u(0, t ), t ≥ 0, ⎪ ⎪ ⎪ ⎪ ⎪ yc = u(2, t ), t ≥ 0, ⎪ ⎪ ⎪ ⎪ ⎪u(x, 0) = u10 (x ), ut (x , 0) = u11 (x), x ∈ [0, 1], ⎪ ⎪ ⎩ u(x, 0) = u20 (x ), ut (x , 0) = u21 (x), x ∈ [1, 2], where and henceforth u (x, t) or ux (x, t) denotes the derivative of u(x, t) with respect to x and u˙ or ut (x, t) denotes the derivative of u(x, t) with respect to t, −U (t ) is the input(control), ym (t) the measured output, yc the controlled output signal, (u10 , u11 , u20 , u21 ) the initial value; d(t) represents the general disturbance satisfying d ∈ L∞ (0, ∞), k > 0, k = 2 is a damping constant.
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The measurement signal is only one u(0, t) while there are two in [29] and [28]. For brevity, we omit initial value here and in the rest of this paper when there is no confusion. Given reference signal yref (t), we devote to design an output feedback controller by the measurement ym (t) only for system (1.1) such that yc (t ) → yre f (t ), as t → ∞,
(1.2)
despite the presence of disturbance. The boundary tracking problem is motivated by helicopter hanging in the air with suspension cable which happens in helicopter hoisting operations [24]. Even there is disturbance on cable, the helicopter needs to keep at the same position. System (1.1) is a simplification of this model in that we do not consider boundary mass and there is damping in the middle of cable. In system (1.1), the disturbance and reference signals are general bounded unknown and known ones respectively, not generated by an exosystem. Therefore, internal model principle for infinite-dimensional systems in [18–20] can not be applied to system (1.1) directly. The harmonic disturbance and reference signals considered in [13] can be represented by an exosystem, which makes adaptive method unavailable for our problem. In this paper, we design a regulator for system (1.1) based on disturbance observer. Introducing a transformation
u1 (x, t ) = u(x, t ), x ∈ (0, 1), u2 (x, t ) = u(2 − x, t ), x ∈ (0, 1),
then (u1 , u2 ) is governed by ⎧ u1tt (x, t ) = u1xx (x, t ) x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ u2tt (x, t ) = u2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ u1x (0, t ) = d (t ), t ≥ 0, ⎪ ⎪ ⎪ ⎨u (0, t ) = U (t ), t ≥ 0, 2x ⎪ u ⎪ 1 (1, t ) = u2 (1, t ), t ≥ 0, ⎪ ⎪ ⎪ ⎪u2x (1, t ) + u1x (1, t ) = −ku2t (1, t ), t ≥ 0, ⎪ ⎪ ⎪ ⎪ ⎪ ym (t ) = u1 (0, t ), t ≥ 0, ⎪ ⎪ ⎩ yc (t ) = u2 (0, t ), t ≥ 0.
(1.3)
Because the equivalence between systems (1.1) and (1.3), we only consider Eq. (1.3) only in the rest of this paper. Now Eq. (1.2) becomes u2 (0, t ) → yre f (t ), as t → ∞.
(1.4)
The rest of the paper is organized as follows. In next section, we design an observer to estimate the state and disturbance simultaneously. In Section 3, we design a servo system by measured output and reference signal yref (t) which gives a dynamics of reference signal. In Section 4, an observer based output feedback controller is designed. The tracking target is obtained in Section 5 while solution to closed-loop system is bounded. Numerical simulations are presented in Section 6 to validate the theoretical results.
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2. Observer design We design an observer for system (1.3) as follows: ⎧ uˆ1tt (x, t ) = uˆ1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ uˆ2tt (x, t ) = uˆ2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ uˆ1 (0, t ) = u1 (0, t ), t ≥ 0, ⎪uˆ2x (0, t ) = U (t ), t ≥ 0, ⎪ ⎪ ⎪ ⎪uˆ1 (1, t ) = uˆ2 (1, t ), t ≥ 0, ⎪ ⎩ uˆ2x (1, t ) + uˆ1x (1, t ) = −k uˆ2t (1, t ), t ≥ 0.
(2.1)
Let u˜ (x, t ) = uˆ (x, t ) − u(x, t ) be the error between systems (2.1) and (1.3). Then u˜ is governed by ⎧ u˜1tt (x, t ) = u˜1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ u˜2tt (x, t ) = u˜2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ u˜1 (0, t ) = 0, t ≥ 0, (2.2) u˜2x (0, t ) = 0, t ≥ 0, ⎪ ⎪ ⎪ ⎪ u˜1 (1, t ) = u˜2 (1, t ), t ≥ 0, ⎪ ⎪ ⎩ u˜2x (1, t ) + u˜1x (1, t ) = −k u˜2t (1, t ), t ≥ 0. Define state space H0 for system (2.2) H0 = {( f1 , g1 , f2 , g2 ) ∈ H 1 (0, 1) × L 2 (0, 1) × H 1 (0, 1) × L 2 (0, 1)| f1 (0) = 0, f1 (1) = f2 (1)}
(2.3) with inner product induced norm 1
( f1 , g1 , f2 , g2 ) 2H0 = [| f1 (x)|2 + |g1 (x )|2 + | f2 (x )|2 + |g2 (x )|2 ]dx , ∀ ( f1 , g1 , f2 , g2 ) ∈ H0 . 0
Define system operator A0 : D(A0 ) → H0 for Eq. (2.2) by A0 ( f1 , g1 , f2 , g2 ) = (g1 , f1 , g2 , f2 ) , ∀ ( f1 , g1 , f2 , g2 ) ∈ D(A0 ), D(A0 ) = {( f1 , g1 , f2 , g2 ) ∈ H0 |A0 ( f1 , g1 , f2 , g2 ) ∈ H0 , f2 (0) = 0, f2 (1) + f1 (1) = −kg2 (1)}. (2.4) Lemma 2.1. Suppose that (u˜1 (·, 0), u˜1t (·, 0), u˜2 (·, 0), u˜2t (·, 0)) ∈ D(A0 ), system (2.2) admits a unique solution (u˜1 , u˜1t , u˜2 , u˜2t ) ∈ C(0, ∞; D(A0 )) such that |u˜1x (0, t )| ≤ V0 LA0 e−σ t ,
(2.5)
where LA0 , V0 and σ are positive constants. Proof. It is well known that system (2.2) is well-posed and exponentially stable by [12]. So A0 generates an exponentially stable C0 -semigroup eA0 t on H0 , i.e. there exist two positive constants LA0 > 0, σ > 0 such that
eA0 t ≤ LA0 e−σ t , t ≥ 0.
(2.6)
Therefore, for any initial value (u˜10 , u˜11 , u˜20 , u˜21 ) ∈ D(A0 ), system (2.2) admits a unique solution (u˜1 , u˜1t , u˜2 , u˜2t ) ∈ C(0, ∞; D(A0 )) such that ||(u˜1 (·, t ), u˜1t (·, t ), u˜2 (·, t ), u˜2t (·, t )) ||H0 ≤ LA0 e−σ t ||(u˜1 (·, 0), u˜1t (·, 0), u˜2 (·, 0), u˜2t (·, 0)) ||H0 . (2.7)
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Let u¯ (x, t ) = u˜t (x, t ). Then u¯ (x, t ) is governed by ⎧ u¯1tt (x, t ) = u¯1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ u¯2tt (x, t ) = u¯2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ u¯1 (0, t ) = 0, t ≥ 0, u¯2x (0, t ) = 0, t ≥ 0, ⎪ ⎪ ⎪ ⎪ u¯1 (1, t ) = u¯2 (1, t ), t ≥ 0, ⎪ ⎪ ⎩ u¯2x (1, t ) + u¯1x (1, t ) = −k u¯2t (1, t ), t ≥ 0.
(2.8)
System (2.8) is the same as Eq. (2.2) and thus is exponentially stable. Therefore, system (2.8) admits a unique solution (u¯1 , u¯1t , u¯2 , u¯2t ) ∈ C(0, ∞; H0 ) such that ||(u¯1 (·, t ), u¯1t (·, t ), u¯2 (·, t ), u¯2t (·, t )) ||H0 ≤ LA0 e−σ t ||(u¯1 (·, 0), u¯1t (·, 0), u¯2 (·, 0), u¯2t (·, 0)) ||H0 .
(2.9) By the Sobolev trace-embedding, it follows from the boundary condition of Eq. (2.2) at x = 1 that |u˜1x (1, t )| ≤ |u˜2x (1, t )| + k|u˜2t (1, t )| 1 = u˜2xx (x , t )dx + k|u¯2 (1, t )| 0
≤ (1 + k) (u¯1 , u¯1t , u¯2 , u¯2t ) H0 .
(2.10)
On the other hand, by Hölder’s inequality, 1 |u˜1x (0, t )| ≤ |u˜1x (1, t )| + u˜1xx (x , t )dx 0
1 ≤ |u˜1x (1, t )| + |u˜1xx (x, t )|2 dx 0
≤ (k + 2) (u¯1 , u¯1t , u¯2 , u¯2t ) H0 .
(2.11)
Therefore, Eq. (2.5) follows from Eqs. (2.7), (2.9), (2.10) and (2.11) with V0 = (k + 2) (u¯1 , u¯1t , u¯2 , u¯2t ) H0 . We complete the proof. Remark 2.1. When initial value (u˜1 (·, 0), u˜1t (·, 0), u˜2 (·, 0), u˜2t (·, 0)) ∈ H0 , observer (2.1) is valid for Eq. (1.3) from the results in [12]. Moreover, when (u˜1 (·, 0), u˜1t (·, 0), u˜2 (·, 0), u˜2t (·, 0)) ∈ D(A0 ), we can obtain that u˜1x (0, t ) → 0 from Lemma 2.1. It equals to uˆ1x (0, t ) → u1x (0, t ) = d (t ). In this case, we treat uˆ1x (0, t ) as the estimation of disturbance d(t). Therefore, the observer (2.1) can recover the state and disturbance simultaneously for Eq.(1.3). In [25] and [27], disturbance observers were designed to recover disturbances generated by exosystems for an parabolic PDEs and Markovian jump nonlinear systems respectively. In system (1.3), the dynamics for disturbance, d(t), is assumed to be unavailable.
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3. Servomechanism design Our reference model is completely inspired from model reference adaptive control approach. Given the reference signal yref (t), we design the following reference model: ⎧ w1tt (x, t ) = w1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ w2tt (x, t ) = w2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ w1x (0, t ) = uˆ1x (0, t ), t ≥ 0, (3.1) w2 (0, t ) = yre f (t ), t ≥ 0, ⎪ ⎪ ⎪ ⎪ w1 (1, t ) = w2 (1, t ), t ≥ 0, ⎪ ⎪ ⎩ w2x (1, t ) + w1x (1, t ) = −kw2t (1, t ), t ≥ 0. It is seen that system (3.1) is completely determined by the measured output of system (1.3) and the reference signal yref (t) only. There are some advantages for system (3.1): (a) The left boundary of Eq. (3.1) produces the reference signal yref (t); (b) System (3.1) is always bounded which will be discussed later. The aim of designing Eq. (3.1) is to make system (1.3) track (3.1) and as a result, u2 (0, t) tracks w2 (0, t ) = yre f (t ). Let H1 = {( f1 , g1 , f2 , g2 ) ∈ H 1 (0, 1) × L 2 (0, 1) × H 1 (0, 1) × L 2 (0, 1)| f1 (1) = f2 (1)} with inner product induced norm
( f 1 , g1 ,
f2 , g2 ) 2H1
1
= 0
[| f1 (x)|2 + |g1 (x )|2 + | f2 (x )|2 + |g2 (x )|2 ]dx + f22 (0).
Lemma 3.1. Suppose that yre f , y˙re f , y¨re f ∈ L ∞ (0, ∞ ), d ∈ L∞ (0, ∞) or d ∈ L2 (0, ∞). For any initial value (w1 (·, 0), w1t (·, 0), w2 (·, 0), w2 (·, 0)) ∈ H1 , system (3.1) admits a unique solution (w1 (·, t ), w1t (·, t ), w2 (·, t ), w2t (·, t )) ∈ C(0, ∞; H1 ) such that sup (w1 (·, t ), w1t (·, t ), w2 (·, t ), w2t (·, t )) H1 < ∞.
(3.2)
t≥0
Proof. According to superposition principle, we divide system (3.1) into the following three systems: ⎧ v1tt (x, t ) = v1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ v2tt (x, t ) = v2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ v1x (0, t ) = d (t ), t ≥ 0, (3.3) v2 (0, t ) = 0, t ≥ 0, ⎪ ⎪ ⎪ ⎪ v2x (1, t ) + v1x (1, t ) = −kv2t (1, t ), t ≥ 0, ⎪ ⎪ ⎩ v1 (1, t ) = v2 (1, t ), t ≥ 0, ⎧ v¯1tt (x, t ) = v¯1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ v¯2tt (x, t ) = v¯2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ v¯1x (0, t ) = 0, t ≥ 0, v¯2 (0, t ) = yre f (t ), t ≥ 0, ⎪ ⎪ ⎪ ⎪ v¯2x (1, t ) + v1x (1, t ) = −k v¯2t (1, t ), t ≥ 0, ⎪ ⎪ ⎩ v¯1 (1, t ) = v¯2 (1, t ), t ≥ 0,
(3.4)
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and ⎧ vˆ1tt (x, t ) = vˆ1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ vˆ2tt (x, t ) = vˆ2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ vˆ1x (0, t ) = u˜1x (0, t ), t ≥ 0, vˆ2 (0, t ) = 0, t ≥ 0, ⎪ ⎪ ⎪ ⎪ vˆ2x (1, t ) + vˆ1x (1, t ) = −k vˆ2t (1, t ), t ≥ 0, ⎪ ⎪ ⎩ vˆ1 (1, t ) = vˆ2 (1, t ), t ≥ 0.
(3.5)
Define H1 = {( f1 , g1 , f2 , g2 ) ∈ H 1 (0, 1) × L 2 (0, 1) × H 1 (0, 1) × L 2 (0, 1)| f2 (0) = 0, f1 (1) = f2 (1)} with inner product induced norm 1 2
( f 1 , g1 , f 2 , g2 ) H1 = [| f1 (x)|2 + |g1 (x )|2 + | f2 (x )|2 + |g2 (x )|2 ]dx . 0
It is obvious that H1 is a subspace of H1 . We define system operator A1 : D(A1 ) → H1 for system (3.3) by A1 ( f1 , g1 , f2 , g2 ) = (g1 , f1 , g2 , f2 ) , ∀ ( f1 , g1 , f2 , g2 ) ∈ D(A1 ), D(A1 ) = {( f1 , g1 , f2 , g2 ) ∈ H1 |A1 ( f1 , g1 , f2 , g2 ) ∈ H1 , f1 (0) = 0, f2 (1) + f1 (1) = −kg2 (1)}. (3.6) It is readily found that ∗ A1 (φ1 , ψ1 , φ2 , ψ2 ) = (−ψ1 , −φ1 , −ψ2 , −φ2 ) , ∀ ( f1 , g1 , f2 , g2 ) ∈ D(A∗ ), D(A∗1 ) = {(φ1 , ψ1 , φ2 , ψ2 ) ∈ H1 |A∗1 (φ1 , ψ1 , φ2 , ψ2 ) ∈ H1 , φ1 (0) = 0, φ2 (1) + φ1 (1) = kψ2 (1)}.
(3.7) Then system (3.3) can be rewritten as an evolutionary equation in H1 : d (v¯1 (·, t ), v¯1t (·, t ), v¯2 (·, t ), v¯2t (·, t )) = A1 (v¯1 (·, t ), v¯1t (·, t ), v¯2 (·, t ), v¯2t (·, t )) + B1 d (t ), dt (3.8) where B1 = (0, −δ(x), 0, 0) . It is well known that the operator A1 generates an exponential stable C0 -semigroup eA1 t on H1 . Now we show that B1 is admissible for eA1 t . The dual system to Eq.(3.3) is found to be ⎧ ∗ ∗ ⎪ ⎪v1∗tt (x, t ) = v1∗xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ v2tt (x, t ) = v2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ ⎨v1∗x (0, t ) = 0, t ≥ 0, v2∗ (0, t ) = 0, t ≥ 0, (3.9) ⎪ ∗ ∗ ∗ ⎪ v (1 , t ) + v (1 , t ) = −kv (1 , t ) , t ≥ 0, ⎪ x 2x 1 2t ⎪ ⎪ ⎪ v∗ (1, t ) = v2∗ (1, t ), t ≥ 0, ⎪ ⎪ ⎩ 1∗ yo (t ) = v1∗t (0, t ). Since A1 generates a C0 -semigroup on H1 , and so does A∗1 . Hence system (3.9) associates with a C0 -semigroup solution. Define the energy functions for Eq. (3.9) as 1 1 ∗2 ∗2 ∗2 Ev∗ (t ) = (v (x, t ) + v1∗2x (x, t ) + v2t (x, t ) + v2x (x , t ))dx , 2 0 1t 1 ∗ ∗ ρv∗ (t ) = (x − 1)(v1∗t (x, t )v1∗x (x, t ) + v2t (x, t )v2x (x, t ))dx. (3.10) 0
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Differentiate Ev∗ (t ) and ρv∗ (t ) with respect to t along the solution to Eq. (3.9) to obtain ∗ E˙v∗ (t ) = −k[v2t (1, t )]2 ≤ 0, 1 ∗2 ρ˙v∗ (t ) = (v1∗2t (0, t ) + v2x (0, t )) − Ev∗ (t ). 2
(3.11)
Integrating second equation in Eq. (3.11) from 0 to T with respect to t, we have T T ∗2 v1t (0, t )dt ≤ 2(ρv∗ (t ) − ρv∗ (0)) + 2 Ev∗ (t )dt ≤ (2T + 4)Ev∗ (0). 0
(3.12)
0
On the other hand, a simple computation shows that ⎛ ⎞ ⎧ φ1 ⎪ ⎪ ⎪ ⎜ψ1 ⎟ ⎪ ⎪ ∗−1 ⎪ A ⎜ ⎟ ⎪ ⎪ ⎪ 1 ⎝φ2 ⎠ ⎪ ⎪ ⎪ ⎪ ⎪ ⎛ ψ2 ⎞ x 1 1 ⎪ ⎪ ⎪ ⎪ −kφ (1) + (1 − ξ )(ψ (ξ ) − ψ (ξ )) dξ + (ψ (ξ ) + ψ (ξ )) dξ − (x − ξ ) ψ (ξ ) dξ ⎪ 2 1 2 1 2 1 ⎪ ⎜ ⎟ ⎪ 0 0 0 ⎪ ⎜ ⎟ ⎨ ⎜ ⎟ −φ 1 ⎟, =⎜ 1 x ⎜ ⎟ ⎪ ⎪ ⎜ ⎟ ⎪ ⎝ −xkφ2 (1) + x (ψ1 (ξ ) + ψ2 (ξ ))dξ − (x − ξ )ψ2 (ξ )dξ ⎪ ⎠ ⎪ ⎪ 0 0 ⎪ ⎪ −φ2 ⎪ ⎪ ⎛ ⎞ ⎪ ⎪ ⎪ φ1 ⎪ ⎪ ⎪ ⎟ ⎪ ⎪B∗ A∗−1 ⎜ ⎜ψ1 ⎟ = φ1 (0). ⎪ ⎪ ⎪ 1 1 ⎝φ2 ⎠ ⎪ ⎩ ψ2
(3.13) Thus B∗1 A∗−1 is bounded from H1 to C. This together with Eq. (3.12) implies that 1 ∗ B∗1 is admissible for eA1 t , and so is B1 for eA1 t . By the well-posed linear infinitedimensional system theory[26], there exists a unique solution to Eq. (3.3) such that (v1 (·, t ), v1t (·, t ), v2 (·, t ), v2t (·, t )) ∈ C(0, ∞; H1 ). Now we show that the solution of Eq. (3.3) is bounded in H1 . The solution of Eq. (3.3) can be written as (v1 (·, t ), v1t (·, t ), v2 (·, t ), v2t (·, t )) =e
A1 t
(v1 (·, 0), v1t (·, 0), v2 (·, 0), v2t (·, 0)) +
t
eA1 (t−s) B1 d (s)d s.
(3.14)
0
It follows from the admissibility of B1 and Proposition 2.5 of [26] that t eA1 (t−s) B1 d (s)d s ≤ h2 d (s) , 0
(3.15)
H1
where h2 is a constant. Thus, sup (v1 (·, t ), v1t (·, t ), v2 (·, t ), v2t (·, t )) H1 t≥0
= sup (v1 (·, t ), v1t (·, t ), v2 (·, t ), v2t (·, t )) H1 < ∞. t≥0
(3.16)
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For system (3.4), we transform it into an equivalent problem by the transformation v˜1 (x, t ) = v¯1 (x , t ), v˜2 (x , t ) = v¯2 (x, t ) − (x − 1)2 yre f (t ) to obtain ⎧ v˜1tt (x, t ) = v˜1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ v˜ (x, t ) = v˜2xx (x, t ) − (x − 1)2 y¨re f (t ) + 2yre f (t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ 2tt v˜1x (0, t ) = 0, t ≥ 0, (3.17) v˜2 (0, t ) = 0, t ≥ 0, ⎪ ⎪ ⎪ ⎪ v˜ (1, t ) + v˜1x (1, t ) = −k v˜2t (1, t ), t ≥ 0, ⎪ ⎪ ⎩ 2x v˜1 (1, t ) = v˜2 (1, t ), t ≥ 0. Define f1 (x, t ) = −(x − 1)2 y¨re f (t ) + 2yre f (t ).
(3.18)
We can write Eq. (3.17) into operator form in state space H1 as follows: (v˜1 (·, t ), v˜1t (·, t ), v˜2 (·, t ), v˜2t (·, t )) = A1 (v˜1 (·, t ), v˜1t (·, t ), v˜2 (·, t ), v˜2t (·, t )) + (0, 0, 0, f1 (·, t )) .
(3.19) Then the solution of Eq. (3.17) can be written as ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ v˜1 (·, t ) v˜1 (·, 0) 0 t ⎜ v˜1t (·, t ) ⎟ ⎜ ⎟ ⎜ 0 ⎟ A1 (t−s) ⎜ ⎜ ⎟ = eA1 t ⎜v˜1t (·, 0)⎟ + ⎟ e ⎝ v˜2 (·, t ) ⎠ ⎝ v˜2 (·, 0) ⎠ ⎝ 0 ⎠ds . 0 v˜2t (·, t ) v˜2t (·, 0) f1 (·, s)
(3.20)
A simple estimation on both side of Eq. (3.20) gives ⎛ 0 t ⎜ 0 ⎜ eA1 (t−s) ⎜ 0 ⎝ 0 f1 (·, s)
⎞ ⎟ ⎟ ds ⎟ ⎠
H1
≤ 0
t
A (t−s) e 1 f1 (·, s) L2 (0,1) ds
1 A (t−s) 1 e 1 |y¨re f (s)| · (x − 1)2 dx 2 + 2|yre f (s)| ds 0 0 √ t A (t−s) 3 1 e = h1 |y¨re f (s)| + 2|yre f (s)| ds. (3.21) 3 0
t
≤ h1
Furthermore, ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ v¯1 (x, t ) v˜1 (x, t ) 0 ⎜v¯1t (x, t )⎟ ⎜v˜1t (x, t )⎟ ⎜ ⎟ 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ v¯2 (x, t ) ⎠ = ⎝ v˜2 (x, t ) ⎠ + ⎝(x − 1)2 yre f (t )⎠. v¯2t (x, t ) v˜2t (x, t ) (x − 1)2 y˙re f (t )
(3.22)
Therefore, sup (v¯1 (·, t ), v¯1t (·, t ), v¯2 (·, t ), v¯2t (·, t )) H1 t≥0
≤ sup{ (v¯1 (·, t ), v¯1t (·, t ), v¯2 (·, t ), v¯2t (·, t )) H1 + (0, 0, (· − 1)2 yre f (t ), (· − 1)2 y˙re f (t )) H1 } t≥0
4 2 1 2 2 = sup{ (v¯1 (·, t ), v¯1t (·, t ), v¯2 (·, t ), v¯2t (·, t )) H1 + yre f (t ) + y˙re f + yre f (t )} 3 5 t≥0 < ∞.
(3.23)
Y. Zhu and F.-F. Jin / Journal of the Franklin Institute 356 (2019) 6280–6302
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For system (3.5), we transform it into an equivalent problem by the transformation vˇ1 = vˆ1 − (x − 1)2 u˜1 , vˇ2 = vˆ2 to obtain ⎧ vˇ1tt (x, t ) = vˇ1xx (x, t ) + 2u˜1 (x, t ) + 4(x − 1)u˜1x (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ vˇ2tt (x, t ) = vˇ2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ vˇ1x (0, t ) = 0, t ≥ 0, (3.24) vˇ2 (0, t ) = 0, t ≥ 0, ⎪ ⎪ ⎪ ⎪ vˇ2x (1, t ) + vˇ1x (1, t ) = −k vˇ2t (1, t ), t ≥ 0, ⎪ ⎪ ⎩ vˇ1 (1, t ) = vˇ2 (1, t ), t ≥ 0. We write Eq. (3.24) into operator form of the following: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ vˇ1 vˇ1 0 ⎟ ⎜ ⎟ ⎜ ⎟ d⎜ ⎜vˇ1t ⎟ = A1 ⎜vˇ1t ⎟ + ⎜ f2 (·, t )⎟, ⎝ ⎠ ⎝ ⎠ ⎝ vˇ2 0 ⎠ dt vˇ2 vˇ2t vˇ2t 0
(3.25)
where f2 (x, t ) = 2u˜1 (x, t ) + 4(x − 1)u˜1x (x, t ).
(3.26)
The solution to Eq. (3.24) can be written as ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ vˇ1 (·, t ) vˇ1 (·, 0) 0 t ⎜vˇ1t (·, t )⎟ ⎜ ⎟ ⎜ ⎟ A1 t ⎜vˇ1t (·, 0)⎟ A1 (t−s) ⎜ f 2 (·, s)⎟ ⎜ ⎟ ⎝ vˇ2 (·, t ) ⎠ = e ⎝ vˇ2 (·, 0) ⎠ + 0 e ⎝ 0 ⎠ds. vˇ2t (·, t ) vˇ2t (·, 0) 0 ⎛ ⎞ 0 t ⎜ ⎟ eA1 (t−s) ⎜ f2 (·, s)⎟ds ⎝ ⎠ 0 0 0
≤
t
A (t−s) e 1 f2 (·, s) L2 (0,1) ds
t
A (t−s) e 1 ·
0
H1
≤ 0
=
0
(3.27)
t
A (t−s) e 1 ·
≤ h3 ≤ h3
1
| f2 (x, s)| d x 2
1 2
ds
0 1
|2u˜1 (x, s) + 4(x − 1)u˜1x (x , s)|2 dx
1 2
ds
0
t
A (t−s) e 1 ·
t
A (t−s) e 1 · (u˜1 (·, s), u˜1t (·, s), u˜2 (·, s), u˜2t (·, s)) H ds 0
0
1
|u˜1x (x , s)|2 dx ds
0
0
≤ h4 (u˜1 (·, 0), u˜1t (·, 0), u˜2 (·, 0), u˜2t (·, 0)) H0 , for some positive constants h3 , h4 . Because ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ vˆ1 (·, t ) vˇ1 (·, t ) 0 ⎜vˆ1t (·, t )⎟ ⎜vˇ1t (·, t )⎟ ⎜ f2 (·, s)⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ vˆ2 (·, t ) ⎠ = ⎝ vˇ2 (·, t ) ⎠ + ⎝ 0 ⎠, vˆ2t (·, t ) vˇ2t (·, t ) 0
(3.28)
(3.29)
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it is easy to obtain that sup (vˆ1 (·, t ), vˆ1t (·, t ), vˆ2 (·, t ), vˆ2t (·, t )) H1 < ∞.
(3.30)
t≥0
By Eqs. (3.16), (3.23) and (3.30), we have sup (w1 (·, t ), w1t (·, t ), w2 (·, t ), w2t (·, t )) H1 < ∞. t≥0
(3.31)
4. Controller design Define the error between the original system (1.3) and servo system (3.1) by ε(x, t ) = u(x, t ) − w(x, t ). Then ε(x, t) is governed by ⎧ ε1tt (x, t ) = ε1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ ε2tt (x, t ) = ε2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ ε1x (0, t ) = u1x (0, t ) − w1x (0, t ), t ≥ 0, (4.1) ε2x (0, t ) = U (t ) − w2x (0, t ), t ≥ 0, ⎪ ⎪ ⎪ ⎪ ε1 (1, t ) = ε2 (1, t ), t ≥ 0, ⎪ ⎪ ⎩ ε2x (1, t ) + ε1x (1, t ) = −kε2t (1, t ), t ≥ 0. We propose the following output feedback control: U (t ) = w2x (0, t ) + β(uˆ2t (0, t ) − w2t (0, t )) + γ (uˆ2 (0, t ) − w2 (0, t )) = w2x (0, t ) + βε2t (0, t ) + γ ε2 (0, t ) + β u˜2t (0, t ) + γ u˜2 (0, t ), where tuning parameters β, γ are some positive constants. Under this controller, system (4.1) becomes ⎧ ε1tt (x, t ) = ε1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ ε2tt (x, t ) = ε2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ ε1x (0, t ) = d (t ) − uˆ1x (0, t ), t ≥ 0, ε2x (0, t ) = βε2t (0, t ) + γ ε2 (0, t ) + β u˜2t (0, t ) + γ u˜2 (0, t ), t ≥ 0, ⎪ ⎪ ⎪ ⎪ ε1 (1, t ) = ε2 (1, t ), t ≥ 0, ⎪ ⎪ ⎩ ε2x (1, t ) + ε1x (1, t ) = −kε2t (1, t ), t ≥ 0. Let z(x, t ) = ε(x, t ) + u˜ (x, t ). Then we obtain the following system: ⎧ z1tt (x, t ) = z1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ z2tt (x, t ) = z2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ z1x (0, t ) = 0, t ≥ 0, z2x (0, t ) = βz2t (0, t ) + γ z2 (0, t ), t ≥ 0, ⎪ ⎪ ⎪ ⎪ z1 (1, t ) = z2 (1, t ), t ≥ 0, ⎪ ⎪ ⎩ z2x (1, t ) + z1x (1, t ) = −kz2t (1, t ), t ≥ 0.
(4.2)
(4.3)
(4.4)
Define state space H2 = {( f1 , g1 , f2 , g2 ) ∈ H 1 (0, 1) × L 2 (0, 1) × H 1 (0, 1) × L (0, 1)| f1 (1) = f2 (1)} with the inner product induced norm 1
( f1 , g1 , f2 , g2 ) 2H2 = [| f1 (x)|2 + |g1 (x )|2 + | f2 (x )|2 + |g2 (x )|2 ]dx 2
0
+γ f22 (0), ∀ ( f1 , g1 , f2 , g2 ) ∈ H2 .
Y. Zhu and F.-F. Jin / Journal of the Franklin Institute 356 (2019) 6280–6302
Define system operator A2 : D(A2 ) → H2 for system (4.4) by ⎧ ⎨A2 ( f1 , g1 , f2 , g2 ) = (g1 , f1 , g2 , f2 ) , ∀( f1 , g1 , f2 , g2 ) ∈ D(A2 ), D(A2 ) = {( f1 , g1 , f2 , g2 ) ∈ H2 |A2 ( f1 , g1 , f2 , g2 ) ∈ H2 , ⎩ f1 (0) = 0, f2 (0) = βg2 (0) + γ f2 (0), f2 (1) + f1 (1) = −kg2 (1)}. It is readily found that ⎧ ∗ ⎨A2 (φ1 , ψ1 , φ2 , ψ2 ) = (−ψ1 , −φ1 , −ψ2 , −φ2 ) , ∀(φ1 , ψ1 , φ2 , ψ2 ) ∈ D(A∗2 ), D(A∗ ) = {(φ1 , ψ1 , φ2 , ψ2 ) ∈ H2 |A∗2 (φ1 , ψ1 , φ2 , ψ2 ) ∈ H2 , ⎩ 2 φ1 (0) = 0, φ2 (0) = −βψ2 (0) + γ φ2 (0), φ2 (1) + φ1 (1) = kψ2 (1)}.
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(4.5)
(4.6)
Lemma 4.1. For any initial value (z1 ( · , 0), z1t ( · , 0), z2 ( · , 0), z2t ( · , 0)) ∈ H2 , there exists a unique solution (z1 ( · , t), z1t ( · , t), z2 ( · , t), z2t ( · , t)) ∈ C(0, ∞; H2 ) to Eq. (4.4) satisfies
(z1 (·, t ), z1t (·, t ), z2 (·, t ), z2t (·, t )) H2 ≤ LA2 e−σA2 t (z1 (·, 0), z1t (·, 0), z2 (·, 0), z2t (·, 0)) H2 , t ≥ 0,
(4.7)
where LA2 and σA2 are positive constants. Proof. We split the proof into three steps. Step 1. We prove that A2 generates a C0 -semigroup eA2 t on H2 by Lumer–Phillips Theorem. For given (f1 , g1 , f2 , g2 ) ∈ D(A2 ), we have Re(A2 ( f1 , g1 , f2 , g2 ) , ( f1 , g1 , f2 , g2 ) ) 1 = Re (g1 (x) f1 (x) + f1 (x)g1 (x) + g2 (x) f2 (x) + f2 (x)g2 (x) )dx + γ g2 (0) f2 (0) = Re
0
f1 (x)g1 (x)|10
+
f2 (x)g2 (x)|10
+ γ g2 (0) f2 (0) = −k|g2 (1)|2 − β|g2 (0)|2 ≤ 0,
(4.8)
which means A2 is dissipative. In addition, it is obviously that D(A2 ) is dense in H2 . Then we will show that there exists a λ0 , λ0 > 0, such that the range, R(λ0 I − A2 ), of λ0 I − A2 is H2 . Taking λ0 = 1 and any (ϕ 1 , ψ 1 , ϕ 2 , ψ 2 ) ∈ H2 , we assume there exists (f1 , g1 , f2 , g2 ) ∈ D(A2 ) such that (I − A2 )( f1 , g1 , f2 , g2 ) = (ϕ1 , ψ1 , ϕ2 , ψ2 ) , which yields to ⎧ f 1 − g1 = ϕ1 , ⎪ ⎪ ⎨ g1 − f1 = ψ1 , f 2 − g2 = ϕ2 , ⎪ ⎪ ⎩ g2 − f2 = ψ2 . Solving f1 , g1 , f2 , g2 , we have the following equations: ⎧ f − f1 = −(ϕ1 + ψ1 ), ⎪ ⎪ ⎨ 1 g1 = f 1 − ϕ1 , ⎪ ⎪ f2 − f2 = −(ϕ2 + ψ2 ), ⎩ g2 = f 2 − ϕ2 .
(4.9)
(4.10)
(4.11)
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The general solutions for f1 , g1 , f2 , g2 can be found as follows: x ⎧ ⎪ f = C cosh x + C sinh x − sinh (x − ξ )(ϕ1 (ξ ) + ψ1 (ξ ))dξ , ⎪ 1 1 2 ⎪ ⎪ 0 ⎪ ⎪ ⎨ f2 = C3 cosh x + C4 sinh x − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩g1 = f1 − ϕ1 , g2 = f 2 − ϕ2 .
x
sinh (x − ξ )(ϕ2 (ξ ) + ψ2 (ξ ))dξ ,
(4.12)
0
By the definition of A2 in Eq. (4.5) and (f1 , g1 , f2 , g2 ) ∈ D(A2 ), we can obtain C2 = 0 and ⎛ ⎞ ⎛ ⎞ C1 L1 Q1 ⎝C3 ⎠ = ⎝L2 ⎠, C4 L3 where
⎛
cosh 1 Q1 = ⎝ sinh 1 0 and
1
L1 =
− cosh 1 sinh 1 + k cosh 1 β+γ
⎞ − sinh 1 cosh 1 + k sinh 1⎠ −1
[ϕ1 (ξ ) + ψ1 (ξ ) − ϕ2 (ξ ) − ψ2 (ξ )] · sinh (1 − ξ )dξ ,
0
L2 = k
1
[ϕ2 (ξ ) + ψ2 (ξ )] · sinh (1 − ξ )dξ 1 + [ϕ1 (ξ ) + ψ1 (ξ ) + ϕ2 (ξ ) + ψ2 (ξ )] · cosh (1 − ξ )dξ + kϕ2 (1), 0
0
L3 = βϕ2 (0). Then detQ1 = − cosh 1(sinh 1 + k cosh 1) − (β + γ )(sinh 1)2 −(β + γ ) cosh 1(cosh 1 + k sinh 1) − sinh 1 cosh 1 < 0, which means that there is a unique solution for Ci , i = 1, 3, 4. (f1 , g2 , f2 , g2 ) defined by Eq. (4.12) satisfies (f1 , g2 , f2 , g2 ) ∈ D(A2 ). Furthermore, there exists positive constant Cinv such that ( f1 , g2 , f2 , g2 ) ≤ Cinv (ϕ1 , ψ2 , ϕ2 , ψ2 ) . Therefore, there exists a λ0 = 1, such that R(λ0 I − A2 ) = H2 . According to Lumer–Phillips Theorem in [21], we can obtain that A2 generates a C0 -semigroup of contractions eA2 t on H2 . So eA2 t ≤ 1. Step 2. We prove that the imaginary axis is a subset of the resolvent set of A2 , ρ(A2 ). We first show that A−1 2 exists. In fact, for a given (φ 1 , ψ 1 , φ 2 , ψ 2 ) ∈ H2 , we suppose there exits (f1 , g1 , f2 , g2 ) ∈ D(A2 ) such that A−1 2 (ϕ1 , ψ1 , ϕ2 , ψ2 ) = ( f 1 , g1 , f 2 , g2 ) ,
which yields to ⎧ g1 = ϕ1 , ⎪ ⎪ ⎨ f 1 = ψ1 , g ⎪ 2 = ϕ2 , ⎪ ⎩ f 2 = ψ2 .
(4.13)
(4.14)
Y. Zhu and F.-F. Jin / Journal of the Franklin Institute 356 (2019) 6280–6302
We will have the solutions for f1 , g1 , f2 , g2 as follows: x ⎧ ⎪ ⎪ f 1 = D1 x + D2 + (x − ξ )ψ1 (ξ )dξ , ⎪ ⎪ ⎪ 0 x ⎨ f 2 = D3 x + D4 + (x − ξ )ψ2 (ξ )dξ , ⎪ 0 ⎪ ⎪ ⎪g 1 = ϕ 1 , ⎪ ⎩ f 2 = ϕ2 .
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(4.15)
By the definition of A2 in Eq. (4.5) and (f1 , g1 , f2 , g2 ) ∈ D(A2 ), we can obtain the coefficients Di , i = 1, 2, 3, 4, as follows: ⎧ D1 = 0, ⎪ 1 ⎪ ⎪ ⎪ k β 1 x ⎪ ⎪ ϕ2 (0) + (1 − ξ )(ψ2 (ξ ) − ψ1 (ξ ))dξ − (ψ2 (ξ ) + ψ1 (ξ ))dξ , 2 (1) − ⎪ ⎨D 2 = − k + γ ϕ γ γ 0 0 x
⎪ D3 = −kϕ2 (1) − (ψ2 (ξ ) + ψ1 (ξ ))dξ , ⎪ ⎪ ⎪ 0 x ⎪ ⎪ ⎪ ⎩D4 = − k ϕ2 (1) − β ϕ2 (0) − 1 (ψ2 (ξ ) + ψ1 (ξ ))dξ . γ γ γ 0
(4.16) −1 Hence A−1 exists, where A−1 is compact. 2 2 : H2 → D(A2 ) ⊂ H2 leads to the fact that A2 Therefore the spectrum σ (A2 ) consists of only isolated eigenvalues. Furthermore, to claim ρ(A2 )⊃iR, it is sufficient to show (λI − A2 )−1 exists for any λ ∈ iR. The existence of (λI − A2 )−1 , λ ∈ iR, λ = 0 will be discussed in next step. Step 3. We prove that
M = sup{ (iωI − A2 )−1 |ω ∈ R} < ∞.
(4.17)
+
To prove it, let α ∈ R be large enough, so that Eq. (4.17) can be reached by proving the following statement: There exists M1 , M2 ∈ R+ such that M1 = sup{ (iωI − A2 )−1 H2 |ω ∈ R, ω ≤ α} < ∞,
(4.18)
M2 = sup{ (iωI − A2 )−1 H2 |ω ∈ R, ω > α} < ∞. +
(4.19) −1
For existence of M1 ∈ R , we show (iωI − A2 ) H2 is continuous with respect to ω. Suppose ω0 ∈ R. Let ω ∈ R such that 1 |ω − ω0 | < , 2[ (iωI − A2 )−1 H2 + 1] Then by representation theorem, we can get
(iωI − A2 )−1 H2 ≤ 2 (iω0 I − A2 )−1 H2 . Applying the resolvent equation we have (iωI − A2 )−1 − (iω0 I − A2 )−1 = (ω − ω0 )(iωI − A2 )−1 (iω0 I − A2 )−1 , and finally
(iωI − A2 )−1 − (iω0 I − A2 )−1 H2 ≤ |ω − ω0 | (iωI − A2 )−1 H2 (iω0 − A2 )−1 H2 ≤ 2|ω − ω0 | (iω0 I − A2 )−1 2H2 .
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For existence of M2 , considering equation (iωI − A2 )( f1 , g1 , f2 , g2 ) = (φ1 , ψ1 , φ2 , ψ2 ) ,
(4.20)
which yields to ⎧ iω f1 − g1 = φ1 , ⎪ ⎪ ⎨ iωg1 − f1 = ψ1 , iω f2 − g2 = φ2 , ⎪ ⎪ ⎩ iωg2 − f2 = ψ2 .
(4.21)
The general solutions for f1 , ⎧ ⎪ ⎪ ⎪ f1 = H1 cos ωx + H2 sin ωx + ⎪ ⎪ ⎪ ⎨ f2 = H3 cos ωx + H4 sin ωx + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩g1 = iω f1 − φ1 , g2 = iω f2 − φ2 ,
g1 , f2 , g2 can be found as follows: 1 x [−(iωφ1 (ξ ) + ψ1 (ξ ))] sin ω(x − ξ )dξ , ω 0 1 x [−(iωφ2 (ξ ) + ψ2 (ξ ))] sin ω(x − ξ )dξ , ω 0
(4.22)
where Hi , i = 1, 2, 3, 4 are constants to be determined. In order to make (f1 , g1 , f2 , g2 ) ∈ D(A2 ), we choose H2 = 0 and ⎛ ⎞ ⎛ ⎞ H1 T1 Q2 ⎝H3 ⎠ = ⎝T2 ⎠, (4.23) H4 T3 where
⎛
ω cos ω Q2 = ⎝−ω sin ω 0
−ω cos ω −ω sin ω + ikω cos ω iβω + γ
⎞ −ω sin ω ω cos ω + ikω sin ω⎠, −ω
and
1
T1 =
[iω(φ1 (ξ ) − φ2 (ξ )) + (ψ1 (ξ ) − ψ2 (ξ ))] sin ω(1 − ξ )dξ ,
0 1
T2 = 0
[iω(φ1 (ξ ) + φ2 (ξ )) + (ψ1 (ξ ) + ψ2 (ξ ))] cos ω(1 − ξ )dξ 1 +i k (iωφ2 (ξ ) + ψ2 (ξ )) sin ω(1 − ξ )dξ + kφ2 (1),
T3 = βφ2 (0). Then
0
detQ2 = ω3
sin 2ω −
and
⎛ A11 Q2∗ = ω2 ⎝A21 A31
A12 A22 A32
kβ γ kγ k k cos 2ω + sin 2ω + i − β cos 2ω − sin 2ω − cos 2ω − , ω 2 2ω 2 2
⎞ ⎛ A11 A13 A23 ⎠ = ⎝A21 A33 −1
− sin ω − cos ω − cos 2ω − ik2 sin 2ω
⎞ (iβ + ωγ ) sin ω ⎠, −(iβ + ωγ ) cos ω ik − sin 2ω + 2 (cos 2ω + 1)
Y. Zhu and F.-F. Jin / Journal of the Franklin Institute 356 (2019) 6280–6302
where
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kγ γ A11 = (1 + kβ ) sin ω − cos ω − i (k + β ) cos ω + sin ω ω ω γ sin ω − cos ω. A21 = − iβ + ω It is obviously that |Aij |, i = 1, 2, 3, are bounded. Set |Aij | ≤ Sij , where Sij , i, j = 1, 2, 3, are constants. Let S0 = min{Si j , i, j = 1, 2, 3}. Let detQ2 = ω3 h(ω), where kβ γ kγ k k . h(ω) = sin 2ω − cos 2ω + sin 2ω + i − β cos 2ω − sin 2ω − cos 2ω − ω 2 2ω 2 2
Then we have following estimation: |h(ω)|2 kβ γ kγ k k 2 2 = sin 2ω − cos 2ω + sin 2ω + − β cos 2ω − sin 2ω − cos 2ω − ω 2 2ω 2 2 2 2 γ2 k β = 1 + 2 sin2 (2ω + θ1 ) + kβ(cos 2ω + 1) + sin2 2ω + β 2 cos2 2ω ω 4 k2 γ2 k2 γ2 k2 2 1 + 2 sin (2ω + θ2 ) + + 1 + 2 sin (2ω + θ2 ) + 4 ω 2 ω 4 2 2 2 γ k β = 1 + 2 sin2 (2ω + θ1 ) + kβ(cos 2ω + 1) + sin2 2ω + β 2 cos2 2ω ω 4 k2 γ2 + 1 + 2 sin (2ω + θ2 ) + 1 2 4 ω 2 2 2 γ2 2 k 2 k β 1 − 1 + 2 + min β , ≥ 4 ω 4 2 2 k β . ≥ min β 2 , 4 Set δ02 = min{β 2 , k 4β } > 0, where δ 0 > 0. Therefore, detQ2 ≥ δ 0 ω3 , and Q2−1 = It is easy to obtain that ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ H1 T1 A11 A12 A13 T1 ∗ Q2 1 ⎝ A21 A22 A23 ⎠⎝T2 ⎠ ω⎝H3 ⎠ = · ⎝T2 ⎠ = det Q2 h(ω) A H4 T3 A32 A33 T3 31 ⎛ ⎞ A T + A12 T2 + A13 T3 1 ⎝ 11 1 A21 T1 + A22 T2 + A23 T3⎠. = h(ω) A T + A T + A T 2 2
31 1
32 2
Q2∗ detQ2
≤
Q2∗ . δ0 ω3
33 3
So |ωHi | = ≤
1 |Ai1 T1 + Ai2 T2 + Ai3 T3 | |h(ω )|
S0 (|T1 | + T2 | + |T3 | ), δ0
(4.24)
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and
1 |T1 | = [iω(φ1 (ξ ) − φ2 (ξ )) + (ψ1 (ξ ) − ψ2 (ξ ))] sin ω(1 − ξ )dξ 0 1 = −i(φ1 (0) − φ2 (0)) cos ω − i (φ1 (ξ ) − φ2 (ξ )) cos ω(1 − ξ )dξ 0 1 (ψ1 (ξ ) − ψ2 (ξ )) sin ω(1 − ξ )dξ + 0
≤ μ1 (|φ2 (0)| + φ1 L2 + φ2 L2 + ψ1 L2 + ψ2 L2 ),
(4.25)
1 |T2 | = [iω(φ1 (ξ ) + φ2 (ξ )) + (ψ1 (ξ ) + ψ2 (ξ ))] cos ω(1 − ξ )dξ 0 1 + ik (iωφ2 (ξ ) + ψ2 (ξ )) sin ω(1 − ξ )dξ + kφ2 (1) 0 1 = i (φ1 (0) + φ2 (0)) sin ω + i (φ1 (ξ ) + φ2 (ξ )) sin ω(1 − ξ )dξ 0 1 (ψ1 (ξ ) + ψ2 (ξ )) cos ω(1 − ξ )dξ + k − φ2 (1) + φ2 (0) cos ω + 0 1 + φ2 (ξ ) cos ω(1 − ξ )dξ 0 1 ψ2 (ξ ) sin ω(1 − ξ )dξ + kφ2 (1) + ik 0
≤ μ2 (|φ2 (0)| + φ1 L2 + φ2 L2 + ψ1 L2 + ψ2 L2 ),
(4.26)
|T3 | = |βφ2 (0)| ≤ μ3 (|φ2 (0)| + φ1 L2 + φ2 L2 + ψ1 L2 + ψ2 L2 ),
(4.27)
where μ1 , μ2 , μ3 are some positive constants. From Eqs. (4.24)–(4.27), we can obtain that |ωHi | ≤
S0 (μ1 δ0
+ μ2 + μ3 )(|φ2 (0)| + φ1 L2 + φ2 L2 + ψ1 L2 + ψ2 L2 ).
(4.28)
Therefore, x | f1 | = −ωH1 sin ωx + (iωφ1 (ξ ) + ψ1 (ξ )) cos ω(x − ξ )dξ 0
= −ωH1 sin ωx + iφ1 (0) sin ωx +i
0
x
φ1 (ξ ) sin ω (x
x
− ξ )dξ + 0
ψ1 (ξ ) cos ω(x − ξ )dξ
≤ ν1 (|φ2 (0)| + φ1 L2 + φ2 L2 + ψ1 L2 + ψ2 L2 ), x | f2 | = −ωH3 sin ωx + ωH4 cos ωx + (iωφ2 (ξ ) + ψ2 (ξ )) cos ω(x − ξ )dξ 0 x = ωH3 sin ωx + ωH4 cos ωx + iφ2 (0) sin ωx + i φ2 (ξ ) sin ω (x − ξ )dξ 0
(4.29)
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x
+ 0
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ψ2 (ξ ) cos ω(x − ξ )dξ
≤ ν2 (|φ2 (0)| + φ1 L2 + φ2 L2 + ψ1 L2 + ψ2 L2 ),
(4.30)
x |g1 | = iωH1 cos ωx + i [−(iωφ1 (ξ ) + ψ1 (ξ ))] sin ω(x − ξ )dξ − φ1 (x) 0 x = iωH1 cos ωx − φ1 (0) cos ωx − φ1 (ξ ) cos ω (x − ξ )dξ 0 x ψ1 (ξ ) sin ω(x − ξ )dξ −i 0
≤ ν3 (|φ2 (0)| + φ1 L2 + φ2 L2 + ψ1 L2 + ψ2 L2 ),
(4.31)
x |g2 | = iωH3 cos ωx + iωH4 sin ωx + i [−(iωφ1 (ξ ) + ψ1 (ξ ))] sin ω(x − ξ )dξ − φ2 (x) 0 x = iωH3 cos ωx + iωH4 sin ωx − φ2 (0)cosωx − φ2 (ξ ) cos ω (x − ξ )dξ 0 x ψ2 (ξ ) sin ω(x − ξ )dξ −i 0
≤ ν4 (|φ2 (0)| + φ1 L2 + φ2 L2 + ψ1 L2 + ψ2 L2 ), | f2 (0)| = |H3 | ≤
S0 (μ1 δ0 |ω|
(4.32)
+ μ2 + μ3 )(|φ2 (0)| + φ1 L2 + φ2 L2 + ψ1 L2 + ψ2 L2 ), (4.33)
where ν 1 , ν 2 , ν 3 , ν 4 are positive constants. Considering Eqs. (4.28)–(4.33), we obtain that there exists M0 > 0 independent of ω such that | f1 | + | f2 | + |g1 | + |g2 | + γ | f2 (0)| ≤ M0 (|φ2 (0)| + φ1 L2 + φ2 L2 + ψ1 L2 + ψ2 L2 ). Therefore, there exists M2 such that 1 [| f1 (x)|2 + |g1 (x )|2 + | f2 (x )|2 + |g2 (x )|2 ]dx + γ f22 (0) 0 1 ≤ M2 [|φ1 (x)|2 + |ψ1 (x )|2 + |φ2 (x )|2 + |ψ2 (x )|2 ]dx + γ φ22 (0). 0
We complete the proof of the lemma. Remark 4.1. Under the controller (4.2), the error between system (1.3) and servo system (3.1), ε(x, t ) = z(x, t ) − u˜ (x, t ), decays exponentially because of exponential stability of z(x, t) and u˜ (x, t ). From ε2 (0, t ) = u2 (0, t ) − yre f (t ) = z2 (0, t ) − u˜2 (0, t ), the decay rate of tracking error is determined by the slower decay rate of z(x, t) and u˜ (x, t ). On the one hand, the decay rate of u˜ (x, t ) is fixed because design parameters β, γ in controller do not appear in system (2.2). On the other hand, γ z2 (0, t) can remove rigid motion and βz2t (0, t) introduces damping for system (4.4). But we do not have the relations between β, γ and decay rate, because it is difficult to obtain all eigenvalues of system (4.4).
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5. Closed-loop system and main results Now we turn to the closed-loop system (1.3), (2.1) and (3.1) under the controller (4.2), that is ⎧ u1tt (x, t ) = u1xx (x , t ), u2tt (x , t ) = u2xx (x, t ), ⎪ ⎪ ⎪ ⎪ u1x (0, t ) = d (t ), ⎪ ⎪ ⎪ ⎪ u2x (0, t ) = w2x (0, t ) + β(uˆ2t (0, t ) − w2t (0, t )) + γ (uˆ2 (0, t ) − w2 (0, t )), ⎪ ⎪ ⎪ ⎪ u2x (1, t ) + u1x (1, t ) = −ku2t (1, t ), u1 (1, t ) = u2 (1, t ), ⎪ ⎪ ⎪ ⎪ ⎨uˆ1tt (x, t ) = uˆ1xx (x , t ), uˆ2tt (x , t ) = uˆ2xx (x, t ), uˆ1 (0, t ) = u1 (0, t ), (5.1) ⎪ ⎪ u ˆ (0, t ) = w (0, t ) + β( u ˆ (0, t ) − w (0, t )) + γ ( u ˆ (0, t ) − w (0, t )) , ⎪ 2x 2x 2t 2t 2 2 ⎪ ⎪ ⎪ uˆ2x (1, t ) + uˆ1x (1, t ) = −k uˆ2t (1, t ), uˆ1 (1, t ) = uˆ2 (1, t ), ⎪ ⎪ ⎪ ⎪ w1tt (x, t ) = w1xx (x , t ), w2tt (x , t ) = w2xx (x, t ), ⎪ ⎪ ⎪ ⎪ w1x (0, t ) = uˆ1x (0, t ), w2 (0, t ) = yre f (t ), ⎪ ⎪ ⎩ w2x (1, t ) + w1x (1, t ) = −kw2t (1, t ), w1 (1, t ) = w2 (1, t ). We consider system (5.1) in the state space X defined by X = {( f1 , g1 , f2 , g2 , f3 , g3 , f4 , g4 , f5 , g5 , f6 , g6 ) |( f1 , g1 , f2 , g2 , f3 , g3 , f4 , g4 ) ∈ (H 1 (0, 1) ×L 2 (0, 1))4 , ( f5 , g5 , f6 , g6 ) ∈ H1 , f1 (1) = f2 (1), f3 (1) = f4 (1), f2 (0) = f4 (0)} (5.2) with inner product induced norm
( f1 , g1 , f2 , g2 , f3 , g3 , f4 , g4 , f5 , g5 , f6 , g6 ) 2X 1 = ( f12 + g21 + f22 + g22 + f32 + g23 + f42 + g24 )dx + γ f42 (0) + ( f5 , g5 , f6 , g6 ) 2H1 , 0
∀ ( f 1 , g1 , f 2 , g2 , f 3 , g3 , f 4 , g4 , f 5 , g5 , f 6 , g6 ) ∈ X .
(5.3)
Then we have the following theorem, the main result of this paper. Theorem 5.1. Suppose that yre f , y˙re f , y¨re f ∈ L ∞ (0, ∞ ), d ∈ L∞ (0, ∞) or d ∈ L2 (0, ∞). Then for any initial value (u10 , u11 , u20 , u21 , uˆ10 , uˆ11 , uˆ20 , uˆ21 , w10 , w11 , w20 , w21 ) ∈ X , there exists a unique (weak) solution (u1 (·, t ), u1t (·, t ), u2 (·, t ), u2t (·, t ), uˆ1 (·, t ), uˆ1t (·, t ), uˆ2 (·, t ), uˆ2t (·, t ), w1 (·, t ), w1t (·, t ), w2 (·, t ), w2t (·, t )) ∈ C(0, ∞; X ) to Eq. (5.1). Moreover, this closed-loop solution has the following properties: 1 2 2 2 (i) sup [u12t (x, t ) + u12x (x, t ) + u2t (x, t ) + u2x (x, t ) + uˆ12t (x, t ) + uˆ12x (x, t ) + uˆ2t (x, t ) t≥0 0 γ 2 2 2 + uˆ2x (x, t ) + w12t (x, t ) + w12x (x, t ) + w2t (x, t ) + w2x (x, t )]dx + uˆ22 (0, t ) < ∞. 2 (ii) lim [u2 (0, t ) − yre f (t )] = 0. t→∞ 1 (iii) lim {[uˆ1t (x, t ) − u1t (x, t )]2 + [uˆ1x (x, t ) − u1x (x, t )] + [uˆ2t (x, t ) − u2t (x, t )]2 + t→0 0
[uˆ2x (x, t ) − u1x (x, t )]2 + [u1t (x, t ) − w1t (x, t )]2 + [u1x (x, t ) − w1x (x, t )]2 + [u2t (x, t ) − w2t (x, t )]2 + [u2x (x, t ) − w2x (x, t )]2 }dx = 0.
(iv) When
d (t ) = 0
→ 0 as t → ∞.
and
1
yre f (t ) = 0, 0
2 2 [u12t (x, t ) + u12x (x, t ) + u2t (x, t ) + u2x (x, t )]dx
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Proof. Define a bounded invertible operator P : X → H1 × H2 × H3 by =
(u˜1 , u˜1t , u˜2 , u˜2t , w1 , w1t , w2 , w2t , z1 , z1t , z2 , z2t ) P(u1 , u1t , u2 , u2t , uˆ1 , uˆ1t , uˆ2 , uˆ2t , w1 , w1t , w2 , w2t ) ,
where ⎛
−1 ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 P=⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎝0 0
0 −1 0 0 0 0 0 0 0 0 0 0
0 0 −1 0 0 0 0 0 0 0 0 0
0 0 0 −1 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0 −1 0 0 0
0 0 0 0 0 1 0 0 0 −1 0 0
is invertible and (u1 , u1t , u2 , u2t , uˆ1 , uˆ1t , uˆ2 , uˆ2t , w1 , w1t , w2 , w2t ) = P−1 (u˜1 , u˜1t , u˜2 , u˜2t , w1 , w1t , w2 , w2t , z1 , z1t , z2 , z2t ) .
(5.4)
0 0 0 0 0 0 1 0 0 0 −1 0
⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 1⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎠ −1
(5.5)
(5.6)
Thanks to the transformation P, closed-loop system (5.1) is equivalent to the decoupled system consisting of Eqs. (2.2), (3.1) and (4.4). It is well known that Eq. (2.2) is exponential stable by [12]. So Eq. (2.2) is bounded. The boundedness of Eqs. (3.1) and (4.4) are from Lemmas 3.1 and 4.1. And it is obviously that P−1 is bounded. According to (5.6), (i) is proved. (4.4) is exponential stable by Lemma 4.1. By transformation z = ε + u˜ = u − w + u˜ and u˜ = uˆ − u, it is easy to obtain (iii). (ii) follows from the exponential stability of Eq. (4.3). When d (t ) = 0 and yre f (t ) = 0, Eq. (1.3) is exponential stable by [12] and Eq. (3.1) is equivalent to Eq. (3.5). Therefore, (iv) is proved. 6. Numerical simulation In this section, we present some numerical simulations to illustrate the effect of proposed controller. The finite difference method is applied to compute the solution numerically and the steps of time and space are set as 0.01 and 0.05, respectively. The parameters are chosen as k = 1, β = 1, γ = 2. The disturbance and reference are set to be d (t ) = 2 + 8 sin 2t, yre f (t ) = −3 + cos 3t, respectively. The initial values are taken as ⎧ u1 (x, 0) = −3x 2 + 2 sin π x, u1t (x, t ) = −2 + 3 cos 3x, ⎪ ⎪ ⎪ ⎪ u (x, 0) = x 2 + 4 cos π x, u2t (x, t ) = −2 + 3 cos x, ⎪ ⎪ ⎨ 2 uˆ1 (x, 0) = 5x, uˆ1t (x, 0) = 2 cos π x, (6.1) uˆ2 (x, 0) = 2 + 3x, uˆ2t (x, 0) = 2 cos π x, ⎪ ⎪ ⎪ ⎪ w (x, 0) = −4 cos π x + 2 sin π x, w1t (x, t ) = 2 − 2 cos π x, ⎪ ⎪ ⎩ 1 w2 (x, 0) = 4 + 2 sin π x, w2t (x, t ) = 3 sin 2π x. Figs. 1–3 display the displacements of u, uˆ, w part of closed-loop system (5.1), respectively. We can see the boundedness of solution of (5.1) from these three figures. Fig. 4(a) presents the trajectory of controller (4.2). Fig. 4(b) gives the tracking performance of boundary signal
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20 20 10 10 0 0 -10 -10 -20 1 15 0.5
-20 1 15
10 0.5
5 0
10 5
0 0
(a) Displacement of
1 -part
0
(b) Displacement of
2 -part
Fig. 1. Displacement of u-part of closed-loop system (5.1).
20 20 10 10 0 0 -10 -10 -20 1 15 0.5
-20 1 15
10 0.5
5 0
10 5
0 0
(a) Displacement of ˆ1 -part
0
(b) Displacement of ˆ2 -part
Fig. 2. Displacement of uˆ-part of closed-loop system (5.1).
20 20 10 10 0 0 -10 -10 -20 1 15 0.5
-20 1 15
10 0.5
5 0
10 5
0 0
(a) Displacement of
1 -part
0
(b) Displacement of
Fig. 3. Displacement of w-part of closed-loop system (5.1).
2 -part
Y. Zhu and F.-F. Jin / Journal of the Franklin Institute 356 (2019) 6280–6302 50
6301
15
40 10
30 20
5
10 0
0
-10 -5
-20 -30
-10
-40 -50
-15 0
5
10
(a) Trajectory of controller
15
0
5
10
15
(b) Tracking performance
Fig. 4. Controller and boundary tracking performance of closed-loop system (5.1).
u2 (0, t) which tracks reference signal yref (t) satisfactorily after t = 10. All four figures demonstrate the effectiveness of proposed output feedback controller (4.2). 7. Concluding remark This paper investigates boundary tracking problem for coupled 1D wave equation with general external disturbance. The servomechanism is designed which gives dynamics of reference signal. Meanwhile, the disturbance observer plays a critical role in regulator design. Because of the introduction of observer, robustness seems impossible by this approach which is a disadvantage compared with internal model principle method. The merit of our method lies in that our method can deal with the system when reference and disturbance are general signal with exosystem unavailable. In the future work, we will consider robust output regulation for system with general reference and disturbance. Another future work seems interesting to consider case that the tracking signal is anti-collocated with the control. References [1] C.I. Byrnes, I.G. Lauk, D.S. Gilliam, V.I. Shubov, Output regulation problem for linear distributed parameter systems, IEEE Trans. Autom. Control 45 (2000) 2236–2252. [2] C.I. Byrnes, F.D. Priscoli, A. Isidori, Output Regulation for Uncertain Nonlinear Systems, Birkäuser, Boston, 1997. [3] F.M. Callier, C.A. Desoer, Stabilization, tracking and disturbance rejection in multivariable convolution systems, Annales de la Société scientifique de Bruxelles 94 (1980) 7–51. [4] E.J. Davison, The robust control of a servomechanism problem for linear time-invariant multivariable systems, IEEE Trans. Autom. Control 21 (1976) 25–34. [5] C.A. Desoer, C.A. Lin, Tracking and disturbance rejection of MIMO nonlinear systems with PI controller, IEEE Trans. Autom. Control 30 (1985) 861–867. [6] J. Deutscher, A backstepping approach to the output regulation of boundary controlled parabolic PDEs, Automatica 57 (2015) 56–64. [7] J. Deutscher, Output regulation for linear distributed-parameter systems using finite-dimensional dual observers, Automatica 47 (2011) 2468–2473. [8] H. Feng, B.Z. Guo, A new active disturbance rejection control to output feedback stabilization for a one-dimensional anti-stable wave equation with disturbance, IEEE Trans. Autom. Control 62 (2017) 3774–3787. [9] B.A. Francis, The linear multivariable regulator problem, SIAM J. Control Optim. 15 (1977) 486–505.
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[10] B.A. Francis, W.M. Wonham, The internal model principle for linear multivariable regulators, Appl. Math. Optim. 2 (1975) 170–194. [11] B. Z. Guo, F. F. Jin, Sliding mode and active disturbance rejection control to stabilization of one-dimensional anti-stable wave equations subject to disturbance in boundary input, IEEE Trans. Autom. Control 58 (2013) 1269–1274. [12] B.Z. Guo, W.D. Zhu, On the energy decay of two coupled strings through a joint damper, J. Sound Vib. 203 (3) (1997) 447–455. [13] W. Guo, B.Z. Guo, Performance output tracking for a wave equation subject to unmatched general boundary harmonic disturbance, Automatica 68 (2016) 194–202. [14] T. Hämäläinen, S. Pohjolainen, Robust regulation of distributed parameter systems with infinite-dimensional exosystems, SIAM J. Control Optim. 48 (2010) 4846–4873. [15] J.Q. Han, From PID to active disturbance rejection control, IEEE Trans. Ind. Electron. 56 (2009) 900–906. [16] F.F. Jin, B.Z. Guo, Performance boundary output tracking for one-dimensional heat equation with boundary unmatched disturbance, Automatica 96 (2018) 1–10. [17] H. Logemann, A. Ilchmann, An adaptive servomechanism for a class of infinite-dimensional systems, SIAM J. Control Optim. 32 (1994) 917–936. [18] L. Paunonen, Controller design for robust output regulation of regular linear systems, IEEE Trans. Autom. Control 61 (2016) 2974–2986. [19] L. Paunonen, S. Pohjolainen, Internal model theory for distributed parameter systems, SIAM J. Control Optim. 48 (2010) 4753–4775. [20] L. Paunonen, S. Pohjolainen, The internal model principle for systems with unbounded control and observation, SIAM J. Control Optim. 52 (2014) 3967–4000. [21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. [22] A. Pisano, Y. Orlov, E. Usai, Tracking control of the uncertain heat and wave equation via power-fractional and sliding-mode techniques, SIAM J. Control Optim. 49 (2011) 363–382. [23] R. Rebarber, G. Weiss, Internal model based tracking and disturbance rejection for stable well-posed systems, Automatica 39 (2003) 1555–1569. [24] Y. Ren, M. Chen, Q.X. Wu, Disturbance observer-based boundary control for a suspension cable system moving in the horizontal plane, Trans. Inst. Meas. Control 41 (2019) 340–349. [25] H.D. Wang, H.N. Wu, L. Guo, Low dimensional disturbance observer-based control for nonlinear parabolic PDE systems with spatio-temporal disturbances, Int. J. Robust Nonlinear Control 26 (2016) 2686–2707. [26] G. Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim. 27 (1989) 527–545. [27] X. Yao, L. Guo, Composite anti-disturbance control for Markovian jump nonlinear systems via disturbance observer, Automatica 49 (2013) 2538–2545. [28] H.C. Zhou, B.Z. Guo, Performance output tracking for one-dimensional wave equation with a general disturbance, Eur. J. Control 39 (2018) 39–52. [29] X. Zhang, H. Feng, S. Chai, Performance output exponential tracking for a wave equation with a general disturbance and non-collocated control, Syst. Control Lett. 98 (2016) 79–85.
Journal of the Franklin Institute 356 (2019) 6280–6302 www.elsevier.com/locate/jfranklin
Performance output tracking for coupled wave equations with unmatched boundary disturbanceR Yingli Zhu, Feng-Fei Jin∗ School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China Received 12 September 2018; received in revised form 23 March 2019; accepted 29 May 2019 Available online 7 June 2019
Abstract In this paper, we consider performance output tracking for coupled wave equations with general external unmatched disturbance. An observer is designed first to estimate the state and disturbance simultaneously. Then we construct a servo system determined completely by the measured output and the reference signal which in turn gives dynamics of reference signal. In the following, an output feedback controller is designed based on this observer and servo system. It is shown that the closedloop system is well-posed and the performance output is tracking the reference signal. Finally, we present some numerical results to illustrate the effectiveness of the controller. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Output regulation problem is a classical problem in control theory. In this framework, output feedback controller needs to be designed to make output signal track the reference signal even though in the presence of disturbance. First results related to the output regulation problem can be found in linear finite-dimensional system, for example [3,4,9,10], then extended to nonlinear finite-dimensional system [2,5] and linear infinite-dimensional system [1,6,7,14,17]. Accordingly, the equivalence between solvability of the regulator problem and solvability of a pair of linear matrix Sylvester equations in finite-dimensional system [9] is also generalized R ∗
This work was supported by the National Natural Science Foundation of China (61603226). Corresponding author. E-mail address: [email protected] (F.-F. Jin).
https://doi.org/10.1016/j.jfranklin.2019.05.041 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
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to infinite-dimensional version [1]. Recently, internal model principle plays an important role in dealing with regulation problem in infinite-dimensional frame [18–20,23]. Sliding mode control is adopted to design controller to make heat and wave equation track known heat and wave dynamics in [22] while control and disturbance are distributed. Adaptive method is also introduced to such cases while the unknown parameters in disturbance can be identified [13]. In most work aforementioned reference signal and disturbance are generated by an exosystem, finite- or infinite-dimensional, which means some information of disturbance can be obtained by various methods. When the dynamics of disturbance and reference is unavailable, there is few result for output regulation of PDEs’ system [28,29]. For large disturbances or uncertainties, active disturbance rejection control (ADRC), first proposed by Han [15], has established itself as a powerful control technology in control systems. In [11], ADRC was applied to estimate the disturbance through an extended state observer (ESO) for a wave PDEs. But the derivative of the disturbance was required to be bounded and high-gain design was needed. In [8], a completely new approach to disturbance estimation by designing an infinite-dimensional disturbance estimator directly was established, which could relax the disturbance to be d ∈ L∞ (0, ∞) or d ∈ L2 (0, ∞) and high gain design was relaxed. Authors in [29] applied ADRC to estimate the state and the external disturbance for a wave equation. In the following a servo system was established through disturbance estimator and reference signal. From another point of view, this servo system can be viewed as a dynamics of reference signal. Thus output regulation was implemented by designing an output feedback control. The same problem was considered in [28], but there was internal nonlinear uncertainty on the boundary except external disturbance. In [16], we applied this method to a heat equation which suffered external disturbance on left boundary and received control on the right boundary. When disturbance and control vanished, the system has eigenvalue on imaginary axis which is the same as the one in [29]. In this paper, we are concerned with the problem of performance output tracking of coupled wave system. The external disturbance is a general bounded unknown signal and reference is a general known function. The system we considered is governed by the following PDEs: ⎧ utt (x, t ) = uxx (x, t ) x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪utt (x, t ) = uxx (x, t ), x ∈ (1, 2), t > 0, ⎪ ⎪ ⎪ ⎪ux (0, t ) = d (t ), t ≥ 0, ⎪ ⎪ ⎪ u(1− , t ) = u(1+ , t ), t ≥ 0, ⎪ ⎪ ⎪ ⎨u (1+ , t ) − u (1− , t ) = ku (1+ , t ), t ≥ 0, x x t (1.1) ⎪ u (2, t ) = −U (t ) , t ≥ 0, x ⎪ ⎪ ⎪ ⎪ ym (t ) = u(0, t ), t ≥ 0, ⎪ ⎪ ⎪ ⎪ ⎪ yc = u(2, t ), t ≥ 0, ⎪ ⎪ ⎪ ⎪ ⎪u(x, 0) = u10 (x ), ut (x , 0) = u11 (x), x ∈ [0, 1], ⎪ ⎪ ⎩ u(x, 0) = u20 (x ), ut (x , 0) = u21 (x), x ∈ [1, 2], where and henceforth u (x, t) or ux (x, t) denotes the derivative of u(x, t) with respect to x and u˙ or ut (x, t) denotes the derivative of u(x, t) with respect to t, −U (t ) is the input(control), ym (t) the measured output, yc the controlled output signal, (u10 , u11 , u20 , u21 ) the initial value; d(t) represents the general disturbance satisfying d ∈ L∞ (0, ∞), k > 0, k = 2 is a damping constant.
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The measurement signal is only one u(0, t) while there are two in [29] and [28]. For brevity, we omit initial value here and in the rest of this paper when there is no confusion. Given reference signal yref (t), we devote to design an output feedback controller by the measurement ym (t) only for system (1.1) such that yc (t ) → yre f (t ), as t → ∞,
(1.2)
despite the presence of disturbance. The boundary tracking problem is motivated by helicopter hanging in the air with suspension cable which happens in helicopter hoisting operations [24]. Even there is disturbance on cable, the helicopter needs to keep at the same position. System (1.1) is a simplification of this model in that we do not consider boundary mass and there is damping in the middle of cable. In system (1.1), the disturbance and reference signals are general bounded unknown and known ones respectively, not generated by an exosystem. Therefore, internal model principle for infinite-dimensional systems in [18–20] can not be applied to system (1.1) directly. The harmonic disturbance and reference signals considered in [13] can be represented by an exosystem, which makes adaptive method unavailable for our problem. In this paper, we design a regulator for system (1.1) based on disturbance observer. Introducing a transformation
u1 (x, t ) = u(x, t ), x ∈ (0, 1), u2 (x, t ) = u(2 − x, t ), x ∈ (0, 1),
then (u1 , u2 ) is governed by ⎧ u1tt (x, t ) = u1xx (x, t ) x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ u2tt (x, t ) = u2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ u1x (0, t ) = d (t ), t ≥ 0, ⎪ ⎪ ⎪ ⎨u (0, t ) = U (t ), t ≥ 0, 2x ⎪ u ⎪ 1 (1, t ) = u2 (1, t ), t ≥ 0, ⎪ ⎪ ⎪ ⎪u2x (1, t ) + u1x (1, t ) = −ku2t (1, t ), t ≥ 0, ⎪ ⎪ ⎪ ⎪ ⎪ ym (t ) = u1 (0, t ), t ≥ 0, ⎪ ⎪ ⎩ yc (t ) = u2 (0, t ), t ≥ 0.
(1.3)
Because the equivalence between systems (1.1) and (1.3), we only consider Eq. (1.3) only in the rest of this paper. Now Eq. (1.2) becomes u2 (0, t ) → yre f (t ), as t → ∞.
(1.4)
The rest of the paper is organized as follows. In next section, we design an observer to estimate the state and disturbance simultaneously. In Section 3, we design a servo system by measured output and reference signal yref (t) which gives a dynamics of reference signal. In Section 4, an observer based output feedback controller is designed. The tracking target is obtained in Section 5 while solution to closed-loop system is bounded. Numerical simulations are presented in Section 6 to validate the theoretical results.
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2. Observer design We design an observer for system (1.3) as follows: ⎧ uˆ1tt (x, t ) = uˆ1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ uˆ2tt (x, t ) = uˆ2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ uˆ1 (0, t ) = u1 (0, t ), t ≥ 0, ⎪uˆ2x (0, t ) = U (t ), t ≥ 0, ⎪ ⎪ ⎪ ⎪uˆ1 (1, t ) = uˆ2 (1, t ), t ≥ 0, ⎪ ⎩ uˆ2x (1, t ) + uˆ1x (1, t ) = −k uˆ2t (1, t ), t ≥ 0.
(2.1)
Let u˜ (x, t ) = uˆ (x, t ) − u(x, t ) be the error between systems (2.1) and (1.3). Then u˜ is governed by ⎧ u˜1tt (x, t ) = u˜1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ u˜2tt (x, t ) = u˜2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ u˜1 (0, t ) = 0, t ≥ 0, (2.2) u˜2x (0, t ) = 0, t ≥ 0, ⎪ ⎪ ⎪ ⎪ u˜1 (1, t ) = u˜2 (1, t ), t ≥ 0, ⎪ ⎪ ⎩ u˜2x (1, t ) + u˜1x (1, t ) = −k u˜2t (1, t ), t ≥ 0. Define state space H0 for system (2.2) H0 = {( f1 , g1 , f2 , g2 ) ∈ H 1 (0, 1) × L 2 (0, 1) × H 1 (0, 1) × L 2 (0, 1)| f1 (0) = 0, f1 (1) = f2 (1)}
(2.3) with inner product induced norm 1
( f1 , g1 , f2 , g2 ) 2H0 = [| f1 (x)|2 + |g1 (x )|2 + | f2 (x )|2 + |g2 (x )|2 ]dx , ∀ ( f1 , g1 , f2 , g2 ) ∈ H0 . 0
Define system operator A0 : D(A0 ) → H0 for Eq. (2.2) by A0 ( f1 , g1 , f2 , g2 ) = (g1 , f1 , g2 , f2 ) , ∀ ( f1 , g1 , f2 , g2 ) ∈ D(A0 ), D(A0 ) = {( f1 , g1 , f2 , g2 ) ∈ H0 |A0 ( f1 , g1 , f2 , g2 ) ∈ H0 , f2 (0) = 0, f2 (1) + f1 (1) = −kg2 (1)}. (2.4) Lemma 2.1. Suppose that (u˜1 (·, 0), u˜1t (·, 0), u˜2 (·, 0), u˜2t (·, 0)) ∈ D(A0 ), system (2.2) admits a unique solution (u˜1 , u˜1t , u˜2 , u˜2t ) ∈ C(0, ∞; D(A0 )) such that |u˜1x (0, t )| ≤ V0 LA0 e−σ t ,
(2.5)
where LA0 , V0 and σ are positive constants. Proof. It is well known that system (2.2) is well-posed and exponentially stable by [12]. So A0 generates an exponentially stable C0 -semigroup eA0 t on H0 , i.e. there exist two positive constants LA0 > 0, σ > 0 such that
eA0 t ≤ LA0 e−σ t , t ≥ 0.
(2.6)
Therefore, for any initial value (u˜10 , u˜11 , u˜20 , u˜21 ) ∈ D(A0 ), system (2.2) admits a unique solution (u˜1 , u˜1t , u˜2 , u˜2t ) ∈ C(0, ∞; D(A0 )) such that ||(u˜1 (·, t ), u˜1t (·, t ), u˜2 (·, t ), u˜2t (·, t )) ||H0 ≤ LA0 e−σ t ||(u˜1 (·, 0), u˜1t (·, 0), u˜2 (·, 0), u˜2t (·, 0)) ||H0 . (2.7)
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Let u¯ (x, t ) = u˜t (x, t ). Then u¯ (x, t ) is governed by ⎧ u¯1tt (x, t ) = u¯1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ u¯2tt (x, t ) = u¯2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ u¯1 (0, t ) = 0, t ≥ 0, u¯2x (0, t ) = 0, t ≥ 0, ⎪ ⎪ ⎪ ⎪ u¯1 (1, t ) = u¯2 (1, t ), t ≥ 0, ⎪ ⎪ ⎩ u¯2x (1, t ) + u¯1x (1, t ) = −k u¯2t (1, t ), t ≥ 0.
(2.8)
System (2.8) is the same as Eq. (2.2) and thus is exponentially stable. Therefore, system (2.8) admits a unique solution (u¯1 , u¯1t , u¯2 , u¯2t ) ∈ C(0, ∞; H0 ) such that ||(u¯1 (·, t ), u¯1t (·, t ), u¯2 (·, t ), u¯2t (·, t )) ||H0 ≤ LA0 e−σ t ||(u¯1 (·, 0), u¯1t (·, 0), u¯2 (·, 0), u¯2t (·, 0)) ||H0 .
(2.9) By the Sobolev trace-embedding, it follows from the boundary condition of Eq. (2.2) at x = 1 that |u˜1x (1, t )| ≤ |u˜2x (1, t )| + k|u˜2t (1, t )| 1 = u˜2xx (x , t )dx + k|u¯2 (1, t )| 0
≤ (1 + k) (u¯1 , u¯1t , u¯2 , u¯2t ) H0 .
(2.10)
On the other hand, by Hölder’s inequality, 1 |u˜1x (0, t )| ≤ |u˜1x (1, t )| + u˜1xx (x , t )dx 0
1 ≤ |u˜1x (1, t )| + |u˜1xx (x, t )|2 dx 0
≤ (k + 2) (u¯1 , u¯1t , u¯2 , u¯2t ) H0 .
(2.11)
Therefore, Eq. (2.5) follows from Eqs. (2.7), (2.9), (2.10) and (2.11) with V0 = (k + 2) (u¯1 , u¯1t , u¯2 , u¯2t ) H0 . We complete the proof. Remark 2.1. When initial value (u˜1 (·, 0), u˜1t (·, 0), u˜2 (·, 0), u˜2t (·, 0)) ∈ H0 , observer (2.1) is valid for Eq. (1.3) from the results in [12]. Moreover, when (u˜1 (·, 0), u˜1t (·, 0), u˜2 (·, 0), u˜2t (·, 0)) ∈ D(A0 ), we can obtain that u˜1x (0, t ) → 0 from Lemma 2.1. It equals to uˆ1x (0, t ) → u1x (0, t ) = d (t ). In this case, we treat uˆ1x (0, t ) as the estimation of disturbance d(t). Therefore, the observer (2.1) can recover the state and disturbance simultaneously for Eq.(1.3). In [25] and [27], disturbance observers were designed to recover disturbances generated by exosystems for an parabolic PDEs and Markovian jump nonlinear systems respectively. In system (1.3), the dynamics for disturbance, d(t), is assumed to be unavailable.
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3. Servomechanism design Our reference model is completely inspired from model reference adaptive control approach. Given the reference signal yref (t), we design the following reference model: ⎧ w1tt (x, t ) = w1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ w2tt (x, t ) = w2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ w1x (0, t ) = uˆ1x (0, t ), t ≥ 0, (3.1) w2 (0, t ) = yre f (t ), t ≥ 0, ⎪ ⎪ ⎪ ⎪ w1 (1, t ) = w2 (1, t ), t ≥ 0, ⎪ ⎪ ⎩ w2x (1, t ) + w1x (1, t ) = −kw2t (1, t ), t ≥ 0. It is seen that system (3.1) is completely determined by the measured output of system (1.3) and the reference signal yref (t) only. There are some advantages for system (3.1): (a) The left boundary of Eq. (3.1) produces the reference signal yref (t); (b) System (3.1) is always bounded which will be discussed later. The aim of designing Eq. (3.1) is to make system (1.3) track (3.1) and as a result, u2 (0, t) tracks w2 (0, t ) = yre f (t ). Let H1 = {( f1 , g1 , f2 , g2 ) ∈ H 1 (0, 1) × L 2 (0, 1) × H 1 (0, 1) × L 2 (0, 1)| f1 (1) = f2 (1)} with inner product induced norm
( f 1 , g1 ,
f2 , g2 ) 2H1
1
= 0
[| f1 (x)|2 + |g1 (x )|2 + | f2 (x )|2 + |g2 (x )|2 ]dx + f22 (0).
Lemma 3.1. Suppose that yre f , y˙re f , y¨re f ∈ L ∞ (0, ∞ ), d ∈ L∞ (0, ∞) or d ∈ L2 (0, ∞). For any initial value (w1 (·, 0), w1t (·, 0), w2 (·, 0), w2 (·, 0)) ∈ H1 , system (3.1) admits a unique solution (w1 (·, t ), w1t (·, t ), w2 (·, t ), w2t (·, t )) ∈ C(0, ∞; H1 ) such that sup (w1 (·, t ), w1t (·, t ), w2 (·, t ), w2t (·, t )) H1 < ∞.
(3.2)
t≥0
Proof. According to superposition principle, we divide system (3.1) into the following three systems: ⎧ v1tt (x, t ) = v1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ v2tt (x, t ) = v2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ v1x (0, t ) = d (t ), t ≥ 0, (3.3) v2 (0, t ) = 0, t ≥ 0, ⎪ ⎪ ⎪ ⎪ v2x (1, t ) + v1x (1, t ) = −kv2t (1, t ), t ≥ 0, ⎪ ⎪ ⎩ v1 (1, t ) = v2 (1, t ), t ≥ 0, ⎧ v¯1tt (x, t ) = v¯1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ v¯2tt (x, t ) = v¯2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ v¯1x (0, t ) = 0, t ≥ 0, v¯2 (0, t ) = yre f (t ), t ≥ 0, ⎪ ⎪ ⎪ ⎪ v¯2x (1, t ) + v1x (1, t ) = −k v¯2t (1, t ), t ≥ 0, ⎪ ⎪ ⎩ v¯1 (1, t ) = v¯2 (1, t ), t ≥ 0,
(3.4)
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and ⎧ vˆ1tt (x, t ) = vˆ1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ vˆ2tt (x, t ) = vˆ2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ vˆ1x (0, t ) = u˜1x (0, t ), t ≥ 0, vˆ2 (0, t ) = 0, t ≥ 0, ⎪ ⎪ ⎪ ⎪ vˆ2x (1, t ) + vˆ1x (1, t ) = −k vˆ2t (1, t ), t ≥ 0, ⎪ ⎪ ⎩ vˆ1 (1, t ) = vˆ2 (1, t ), t ≥ 0.
(3.5)
Define H1 = {( f1 , g1 , f2 , g2 ) ∈ H 1 (0, 1) × L 2 (0, 1) × H 1 (0, 1) × L 2 (0, 1)| f2 (0) = 0, f1 (1) = f2 (1)} with inner product induced norm 1 2
( f 1 , g1 , f 2 , g2 ) H1 = [| f1 (x)|2 + |g1 (x )|2 + | f2 (x )|2 + |g2 (x )|2 ]dx . 0
It is obvious that H1 is a subspace of H1 . We define system operator A1 : D(A1 ) → H1 for system (3.3) by A1 ( f1 , g1 , f2 , g2 ) = (g1 , f1 , g2 , f2 ) , ∀ ( f1 , g1 , f2 , g2 ) ∈ D(A1 ), D(A1 ) = {( f1 , g1 , f2 , g2 ) ∈ H1 |A1 ( f1 , g1 , f2 , g2 ) ∈ H1 , f1 (0) = 0, f2 (1) + f1 (1) = −kg2 (1)}. (3.6) It is readily found that ∗ A1 (φ1 , ψ1 , φ2 , ψ2 ) = (−ψ1 , −φ1 , −ψ2 , −φ2 ) , ∀ ( f1 , g1 , f2 , g2 ) ∈ D(A∗ ), D(A∗1 ) = {(φ1 , ψ1 , φ2 , ψ2 ) ∈ H1 |A∗1 (φ1 , ψ1 , φ2 , ψ2 ) ∈ H1 , φ1 (0) = 0, φ2 (1) + φ1 (1) = kψ2 (1)}.
(3.7) Then system (3.3) can be rewritten as an evolutionary equation in H1 : d (v¯1 (·, t ), v¯1t (·, t ), v¯2 (·, t ), v¯2t (·, t )) = A1 (v¯1 (·, t ), v¯1t (·, t ), v¯2 (·, t ), v¯2t (·, t )) + B1 d (t ), dt (3.8) where B1 = (0, −δ(x), 0, 0) . It is well known that the operator A1 generates an exponential stable C0 -semigroup eA1 t on H1 . Now we show that B1 is admissible for eA1 t . The dual system to Eq.(3.3) is found to be ⎧ ∗ ∗ ⎪ ⎪v1∗tt (x, t ) = v1∗xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ v2tt (x, t ) = v2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ ⎨v1∗x (0, t ) = 0, t ≥ 0, v2∗ (0, t ) = 0, t ≥ 0, (3.9) ⎪ ∗ ∗ ∗ ⎪ v (1 , t ) + v (1 , t ) = −kv (1 , t ) , t ≥ 0, ⎪ x 2x 1 2t ⎪ ⎪ ⎪ v∗ (1, t ) = v2∗ (1, t ), t ≥ 0, ⎪ ⎪ ⎩ 1∗ yo (t ) = v1∗t (0, t ). Since A1 generates a C0 -semigroup on H1 , and so does A∗1 . Hence system (3.9) associates with a C0 -semigroup solution. Define the energy functions for Eq. (3.9) as 1 1 ∗2 ∗2 ∗2 Ev∗ (t ) = (v (x, t ) + v1∗2x (x, t ) + v2t (x, t ) + v2x (x , t ))dx , 2 0 1t 1 ∗ ∗ ρv∗ (t ) = (x − 1)(v1∗t (x, t )v1∗x (x, t ) + v2t (x, t )v2x (x, t ))dx. (3.10) 0
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Differentiate Ev∗ (t ) and ρv∗ (t ) with respect to t along the solution to Eq. (3.9) to obtain ∗ E˙v∗ (t ) = −k[v2t (1, t )]2 ≤ 0, 1 ∗2 ρ˙v∗ (t ) = (v1∗2t (0, t ) + v2x (0, t )) − Ev∗ (t ). 2
(3.11)
Integrating second equation in Eq. (3.11) from 0 to T with respect to t, we have T T ∗2 v1t (0, t )dt ≤ 2(ρv∗ (t ) − ρv∗ (0)) + 2 Ev∗ (t )dt ≤ (2T + 4)Ev∗ (0). 0
(3.12)
0
On the other hand, a simple computation shows that ⎛ ⎞ ⎧ φ1 ⎪ ⎪ ⎪ ⎜ψ1 ⎟ ⎪ ⎪ ∗−1 ⎪ A ⎜ ⎟ ⎪ ⎪ ⎪ 1 ⎝φ2 ⎠ ⎪ ⎪ ⎪ ⎪ ⎪ ⎛ ψ2 ⎞ x 1 1 ⎪ ⎪ ⎪ ⎪ −kφ (1) + (1 − ξ )(ψ (ξ ) − ψ (ξ )) dξ + (ψ (ξ ) + ψ (ξ )) dξ − (x − ξ ) ψ (ξ ) dξ ⎪ 2 1 2 1 2 1 ⎪ ⎜ ⎟ ⎪ 0 0 0 ⎪ ⎜ ⎟ ⎨ ⎜ ⎟ −φ 1 ⎟, =⎜ 1 x ⎜ ⎟ ⎪ ⎪ ⎜ ⎟ ⎪ ⎝ −xkφ2 (1) + x (ψ1 (ξ ) + ψ2 (ξ ))dξ − (x − ξ )ψ2 (ξ )dξ ⎪ ⎠ ⎪ ⎪ 0 0 ⎪ ⎪ −φ2 ⎪ ⎪ ⎛ ⎞ ⎪ ⎪ ⎪ φ1 ⎪ ⎪ ⎪ ⎟ ⎪ ⎪B∗ A∗−1 ⎜ ⎜ψ1 ⎟ = φ1 (0). ⎪ ⎪ ⎪ 1 1 ⎝φ2 ⎠ ⎪ ⎩ ψ2
(3.13) Thus B∗1 A∗−1 is bounded from H1 to C. This together with Eq. (3.12) implies that 1 ∗ B∗1 is admissible for eA1 t , and so is B1 for eA1 t . By the well-posed linear infinitedimensional system theory[26], there exists a unique solution to Eq. (3.3) such that (v1 (·, t ), v1t (·, t ), v2 (·, t ), v2t (·, t )) ∈ C(0, ∞; H1 ). Now we show that the solution of Eq. (3.3) is bounded in H1 . The solution of Eq. (3.3) can be written as (v1 (·, t ), v1t (·, t ), v2 (·, t ), v2t (·, t )) =e
A1 t
(v1 (·, 0), v1t (·, 0), v2 (·, 0), v2t (·, 0)) +
t
eA1 (t−s) B1 d (s)d s.
(3.14)
0
It follows from the admissibility of B1 and Proposition 2.5 of [26] that t eA1 (t−s) B1 d (s)d s ≤ h2 d (s) , 0
(3.15)
H1
where h2 is a constant. Thus, sup (v1 (·, t ), v1t (·, t ), v2 (·, t ), v2t (·, t )) H1 t≥0
= sup (v1 (·, t ), v1t (·, t ), v2 (·, t ), v2t (·, t )) H1 < ∞. t≥0
(3.16)
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For system (3.4), we transform it into an equivalent problem by the transformation v˜1 (x, t ) = v¯1 (x , t ), v˜2 (x , t ) = v¯2 (x, t ) − (x − 1)2 yre f (t ) to obtain ⎧ v˜1tt (x, t ) = v˜1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ v˜ (x, t ) = v˜2xx (x, t ) − (x − 1)2 y¨re f (t ) + 2yre f (t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ 2tt v˜1x (0, t ) = 0, t ≥ 0, (3.17) v˜2 (0, t ) = 0, t ≥ 0, ⎪ ⎪ ⎪ ⎪ v˜ (1, t ) + v˜1x (1, t ) = −k v˜2t (1, t ), t ≥ 0, ⎪ ⎪ ⎩ 2x v˜1 (1, t ) = v˜2 (1, t ), t ≥ 0. Define f1 (x, t ) = −(x − 1)2 y¨re f (t ) + 2yre f (t ).
(3.18)
We can write Eq. (3.17) into operator form in state space H1 as follows: (v˜1 (·, t ), v˜1t (·, t ), v˜2 (·, t ), v˜2t (·, t )) = A1 (v˜1 (·, t ), v˜1t (·, t ), v˜2 (·, t ), v˜2t (·, t )) + (0, 0, 0, f1 (·, t )) .
(3.19) Then the solution of Eq. (3.17) can be written as ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ v˜1 (·, t ) v˜1 (·, 0) 0 t ⎜ v˜1t (·, t ) ⎟ ⎜ ⎟ ⎜ 0 ⎟ A1 (t−s) ⎜ ⎜ ⎟ = eA1 t ⎜v˜1t (·, 0)⎟ + ⎟ e ⎝ v˜2 (·, t ) ⎠ ⎝ v˜2 (·, 0) ⎠ ⎝ 0 ⎠ds . 0 v˜2t (·, t ) v˜2t (·, 0) f1 (·, s)
(3.20)
A simple estimation on both side of Eq. (3.20) gives ⎛ 0 t ⎜ 0 ⎜ eA1 (t−s) ⎜ 0 ⎝ 0 f1 (·, s)
⎞ ⎟ ⎟ ds ⎟ ⎠
H1
≤ 0
t
A (t−s) e 1 f1 (·, s) L2 (0,1) ds
1 A (t−s) 1 e 1 |y¨re f (s)| · (x − 1)2 dx 2 + 2|yre f (s)| ds 0 0 √ t A (t−s) 3 1 e = h1 |y¨re f (s)| + 2|yre f (s)| ds. (3.21) 3 0
t
≤ h1
Furthermore, ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ v¯1 (x, t ) v˜1 (x, t ) 0 ⎜v¯1t (x, t )⎟ ⎜v˜1t (x, t )⎟ ⎜ ⎟ 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ v¯2 (x, t ) ⎠ = ⎝ v˜2 (x, t ) ⎠ + ⎝(x − 1)2 yre f (t )⎠. v¯2t (x, t ) v˜2t (x, t ) (x − 1)2 y˙re f (t )
(3.22)
Therefore, sup (v¯1 (·, t ), v¯1t (·, t ), v¯2 (·, t ), v¯2t (·, t )) H1 t≥0
≤ sup{ (v¯1 (·, t ), v¯1t (·, t ), v¯2 (·, t ), v¯2t (·, t )) H1 + (0, 0, (· − 1)2 yre f (t ), (· − 1)2 y˙re f (t )) H1 } t≥0
4 2 1 2 2 = sup{ (v¯1 (·, t ), v¯1t (·, t ), v¯2 (·, t ), v¯2t (·, t )) H1 + yre f (t ) + y˙re f + yre f (t )} 3 5 t≥0 < ∞.
(3.23)
Y. Zhu and F.-F. Jin / Journal of the Franklin Institute 356 (2019) 6280–6302
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For system (3.5), we transform it into an equivalent problem by the transformation vˇ1 = vˆ1 − (x − 1)2 u˜1 , vˇ2 = vˆ2 to obtain ⎧ vˇ1tt (x, t ) = vˇ1xx (x, t ) + 2u˜1 (x, t ) + 4(x − 1)u˜1x (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ vˇ2tt (x, t ) = vˇ2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ vˇ1x (0, t ) = 0, t ≥ 0, (3.24) vˇ2 (0, t ) = 0, t ≥ 0, ⎪ ⎪ ⎪ ⎪ vˇ2x (1, t ) + vˇ1x (1, t ) = −k vˇ2t (1, t ), t ≥ 0, ⎪ ⎪ ⎩ vˇ1 (1, t ) = vˇ2 (1, t ), t ≥ 0. We write Eq. (3.24) into operator form of the following: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ vˇ1 vˇ1 0 ⎟ ⎜ ⎟ ⎜ ⎟ d⎜ ⎜vˇ1t ⎟ = A1 ⎜vˇ1t ⎟ + ⎜ f2 (·, t )⎟, ⎝ ⎠ ⎝ ⎠ ⎝ vˇ2 0 ⎠ dt vˇ2 vˇ2t vˇ2t 0
(3.25)
where f2 (x, t ) = 2u˜1 (x, t ) + 4(x − 1)u˜1x (x, t ).
(3.26)
The solution to Eq. (3.24) can be written as ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ vˇ1 (·, t ) vˇ1 (·, 0) 0 t ⎜vˇ1t (·, t )⎟ ⎜ ⎟ ⎜ ⎟ A1 t ⎜vˇ1t (·, 0)⎟ A1 (t−s) ⎜ f 2 (·, s)⎟ ⎜ ⎟ ⎝ vˇ2 (·, t ) ⎠ = e ⎝ vˇ2 (·, 0) ⎠ + 0 e ⎝ 0 ⎠ds. vˇ2t (·, t ) vˇ2t (·, 0) 0 ⎛ ⎞ 0 t ⎜ ⎟ eA1 (t−s) ⎜ f2 (·, s)⎟ds ⎝ ⎠ 0 0 0
≤
t
A (t−s) e 1 f2 (·, s) L2 (0,1) ds
t
A (t−s) e 1 ·
0
H1
≤ 0
=
0
(3.27)
t
A (t−s) e 1 ·
≤ h3 ≤ h3
1
| f2 (x, s)| d x 2
1 2
ds
0 1
|2u˜1 (x, s) + 4(x − 1)u˜1x (x , s)|2 dx
1 2
ds
0
t
A (t−s) e 1 ·
t
A (t−s) e 1 · (u˜1 (·, s), u˜1t (·, s), u˜2 (·, s), u˜2t (·, s)) H ds 0
0
1
|u˜1x (x , s)|2 dx ds
0
0
≤ h4 (u˜1 (·, 0), u˜1t (·, 0), u˜2 (·, 0), u˜2t (·, 0)) H0 , for some positive constants h3 , h4 . Because ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ vˆ1 (·, t ) vˇ1 (·, t ) 0 ⎜vˆ1t (·, t )⎟ ⎜vˇ1t (·, t )⎟ ⎜ f2 (·, s)⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ vˆ2 (·, t ) ⎠ = ⎝ vˇ2 (·, t ) ⎠ + ⎝ 0 ⎠, vˆ2t (·, t ) vˇ2t (·, t ) 0
(3.28)
(3.29)
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it is easy to obtain that sup (vˆ1 (·, t ), vˆ1t (·, t ), vˆ2 (·, t ), vˆ2t (·, t )) H1 < ∞.
(3.30)
t≥0
By Eqs. (3.16), (3.23) and (3.30), we have sup (w1 (·, t ), w1t (·, t ), w2 (·, t ), w2t (·, t )) H1 < ∞. t≥0
(3.31)
4. Controller design Define the error between the original system (1.3) and servo system (3.1) by ε(x, t ) = u(x, t ) − w(x, t ). Then ε(x, t) is governed by ⎧ ε1tt (x, t ) = ε1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ ε2tt (x, t ) = ε2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ ε1x (0, t ) = u1x (0, t ) − w1x (0, t ), t ≥ 0, (4.1) ε2x (0, t ) = U (t ) − w2x (0, t ), t ≥ 0, ⎪ ⎪ ⎪ ⎪ ε1 (1, t ) = ε2 (1, t ), t ≥ 0, ⎪ ⎪ ⎩ ε2x (1, t ) + ε1x (1, t ) = −kε2t (1, t ), t ≥ 0. We propose the following output feedback control: U (t ) = w2x (0, t ) + β(uˆ2t (0, t ) − w2t (0, t )) + γ (uˆ2 (0, t ) − w2 (0, t )) = w2x (0, t ) + βε2t (0, t ) + γ ε2 (0, t ) + β u˜2t (0, t ) + γ u˜2 (0, t ), where tuning parameters β, γ are some positive constants. Under this controller, system (4.1) becomes ⎧ ε1tt (x, t ) = ε1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ ε2tt (x, t ) = ε2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ ε1x (0, t ) = d (t ) − uˆ1x (0, t ), t ≥ 0, ε2x (0, t ) = βε2t (0, t ) + γ ε2 (0, t ) + β u˜2t (0, t ) + γ u˜2 (0, t ), t ≥ 0, ⎪ ⎪ ⎪ ⎪ ε1 (1, t ) = ε2 (1, t ), t ≥ 0, ⎪ ⎪ ⎩ ε2x (1, t ) + ε1x (1, t ) = −kε2t (1, t ), t ≥ 0. Let z(x, t ) = ε(x, t ) + u˜ (x, t ). Then we obtain the following system: ⎧ z1tt (x, t ) = z1xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ z2tt (x, t ) = z2xx (x, t ), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎨ z1x (0, t ) = 0, t ≥ 0, z2x (0, t ) = βz2t (0, t ) + γ z2 (0, t ), t ≥ 0, ⎪ ⎪ ⎪ ⎪ z1 (1, t ) = z2 (1, t ), t ≥ 0, ⎪ ⎪ ⎩ z2x (1, t ) + z1x (1, t ) = −kz2t (1, t ), t ≥ 0.
(4.2)
(4.3)
(4.4)
Define state space H2 = {( f1 , g1 , f2 , g2 ) ∈ H 1 (0, 1) × L 2 (0, 1) × H 1 (0, 1) × L (0, 1)| f1 (1) = f2 (1)} with the inner product induced norm 1
( f1 , g1 , f2 , g2 ) 2H2 = [| f1 (x)|2 + |g1 (x )|2 + | f2 (x )|2 + |g2 (x )|2 ]dx 2
0
+γ f22 (0), ∀ ( f1 , g1 , f2 , g2 ) ∈ H2 .
Y. Zhu and F.-F. Jin / Journal of the Franklin Institute 356 (2019) 6280–6302
Define system operator A2 : D(A2 ) → H2 for system (4.4) by ⎧ ⎨A2 ( f1 , g1 , f2 , g2 ) = (g1 , f1 , g2 , f2 ) , ∀( f1 , g1 , f2 , g2 ) ∈ D(A2 ), D(A2 ) = {( f1 , g1 , f2 , g2 ) ∈ H2 |A2 ( f1 , g1 , f2 , g2 ) ∈ H2 , ⎩ f1 (0) = 0, f2 (0) = βg2 (0) + γ f2 (0), f2 (1) + f1 (1) = −kg2 (1)}. It is readily found that ⎧ ∗ ⎨A2 (φ1 , ψ1 , φ2 , ψ2 ) = (−ψ1 , −φ1 , −ψ2 , −φ2 ) , ∀(φ1 , ψ1 , φ2 , ψ2 ) ∈ D(A∗2 ), D(A∗ ) = {(φ1 , ψ1 , φ2 , ψ2 ) ∈ H2 |A∗2 (φ1 , ψ1 , φ2 , ψ2 ) ∈ H2 , ⎩ 2 φ1 (0) = 0, φ2 (0) = −βψ2 (0) + γ φ2 (0), φ2 (1) + φ1 (1) = kψ2 (1)}.
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(4.5)
(4.6)
Lemma 4.1. For any initial value (z1 ( · , 0), z1t ( · , 0), z2 ( · , 0), z2t ( · , 0)) ∈ H2 , there exists a unique solution (z1 ( · , t), z1t ( · , t), z2 ( · , t), z2t ( · , t)) ∈ C(0, ∞; H2 ) to Eq. (4.4) satisfies
(z1 (·, t ), z1t (·, t ), z2 (·, t ), z2t (·, t )) H2 ≤ LA2 e−σA2 t (z1 (·, 0), z1t (·, 0), z2 (·, 0), z2t (·, 0)) H2 , t ≥ 0,
(4.7)
where LA2 and σA2 are positive constants. Proof. We split the proof into three steps. Step 1. We prove that A2 generates a C0 -semigroup eA2 t on H2 by Lumer–Phillips Theorem. For given (f1 , g1 , f2 , g2 ) ∈ D(A2 ), we have Re(A2 ( f1 , g1 , f2 , g2 ) , ( f1 , g1 , f2 , g2 ) ) 1 = Re (g1 (x) f1 (x) + f1 (x)g1 (x) + g2 (x) f2 (x) + f2 (x)g2 (x) )dx + γ g2 (0) f2 (0) = Re
0
f1 (x)g1 (x)|10
+
f2 (x)g2 (x)|10
+ γ g2 (0) f2 (0) = −k|g2 (1)|2 − β|g2 (0)|2 ≤ 0,
(4.8)
which means A2 is dissipative. In addition, it is obviously that D(A2 ) is dense in H2 . Then we will show that there exists a λ0 , λ0 > 0, such that the range, R(λ0 I − A2 ), of λ0 I − A2 is H2 . Taking λ0 = 1 and any (ϕ 1 , ψ 1 , ϕ 2 , ψ 2 ) ∈ H2 , we assume there exists (f1 , g1 , f2 , g2 ) ∈ D(A2 ) such that (I − A2 )( f1 , g1 , f2 , g2 ) = (ϕ1 , ψ1 , ϕ2 , ψ2 ) , which yields to ⎧ f 1 − g1 = ϕ1 , ⎪ ⎪ ⎨ g1 − f1 = ψ1 , f 2 − g2 = ϕ2 , ⎪ ⎪ ⎩ g2 − f2 = ψ2 . Solving f1 , g1 , f2 , g2 , we have the following equations: ⎧ f − f1 = −(ϕ1 + ψ1 ), ⎪ ⎪ ⎨ 1 g1 = f 1 − ϕ1 , ⎪ ⎪ f2 − f2 = −(ϕ2 + ψ2 ), ⎩ g2 = f 2 − ϕ2 .
(4.9)
(4.10)
(4.11)
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The general solutions for f1 , g1 , f2 , g2 can be found as follows: x ⎧ ⎪ f = C cosh x + C sinh x − sinh (x − ξ )(ϕ1 (ξ ) + ψ1 (ξ ))dξ , ⎪ 1 1 2 ⎪ ⎪ 0 ⎪ ⎪ ⎨ f2 = C3 cosh x + C4 sinh x − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩g1 = f1 − ϕ1 , g2 = f 2 − ϕ2 .
x
sinh (x − ξ )(ϕ2 (ξ ) + ψ2 (ξ ))dξ ,
(4.12)
0
By the definition of A2 in Eq. (4.5) and (f1 , g1 , f2 , g2 ) ∈ D(A2 ), we can obtain C2 = 0 and ⎛ ⎞ ⎛ ⎞ C1 L1 Q1 ⎝C3 ⎠ = ⎝L2 ⎠, C4 L3 where
⎛
cosh 1 Q1 = ⎝ sinh 1 0 and
1
L1 =
− cosh 1 sinh 1 + k cosh 1 β+γ
⎞ − sinh 1 cosh 1 + k sinh 1⎠ −1
[ϕ1 (ξ ) + ψ1 (ξ ) − ϕ2 (ξ ) − ψ2 (ξ )] · sinh (1 − ξ )dξ ,
0
L2 = k
1
[ϕ2 (ξ ) + ψ2 (ξ )] · sinh (1 − ξ )dξ 1 + [ϕ1 (ξ ) + ψ1 (ξ ) + ϕ2 (ξ ) + ψ2 (ξ )] · cosh (1 − ξ )dξ + kϕ2 (1), 0
0
L3 = βϕ2 (0). Then detQ1 = − cosh 1(sinh 1 + k cosh 1) − (β + γ )(sinh 1)2 −(β + γ ) cosh 1(cosh 1 + k sinh 1) − sinh 1 cosh 1 < 0, which means that there is a unique solution for Ci , i = 1, 3, 4. (f1 , g2 , f2 , g2 ) defined by Eq. (4.12) satisfies (f1 , g2 , f2 , g2 ) ∈ D(A2 ). Furthermore, there exists positive constant Cinv such that ( f1 , g2 , f2 , g2 ) ≤ Cinv (ϕ1 , ψ2 , ϕ2 , ψ2 ) . Therefore, there exists a λ0 = 1, such that R(λ0 I − A2 ) = H2 . According to Lumer–Phillips Theorem in [21], we can obtain that A2 generates a C0 -semigroup of contractions eA2 t on H2 . So eA2 t ≤ 1. Step 2. We prove that the imaginary axis is a subset of the resolvent set of A2 , ρ(A2 ). We first show that A−1 2 exists. In fact, for a given (φ 1 , ψ 1 , φ 2 , ψ 2 ) ∈ H2 , we suppose there exits (f1 , g1 , f2 , g2 ) ∈ D(A2 ) such that A−1 2 (ϕ1 , ψ1 , ϕ2 , ψ2 ) = ( f 1 , g1 , f 2 , g2 ) ,
which yields to ⎧ g1 = ϕ1 , ⎪ ⎪ ⎨ f 1 = ψ1 , g ⎪ 2 = ϕ2 , ⎪ ⎩ f 2 = ψ2 .
(4.13)
(4.14)
Y. Zhu and F.-F. Jin / Journal of the Franklin Institute 356 (2019) 6280–6302
We will have the solutions for f1 , g1 , f2 , g2 as follows: x ⎧ ⎪ ⎪ f 1 = D1 x + D2 + (x − ξ )ψ1 (ξ )dξ , ⎪ ⎪ ⎪ 0 x ⎨ f 2 = D3 x + D4 + (x − ξ )ψ2 (ξ )dξ , ⎪ 0 ⎪ ⎪ ⎪g 1 = ϕ 1 , ⎪ ⎩ f 2 = ϕ2 .
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(4.15)
By the definition of A2 in Eq. (4.5) and (f1 , g1 , f2 , g2 ) ∈ D(A2 ), we can obtain the coefficients Di , i = 1, 2, 3, 4, as follows: ⎧ D1 = 0, ⎪ 1 ⎪ ⎪ ⎪ k β 1 x ⎪ ⎪ ϕ2 (0) + (1 − ξ )(ψ2 (ξ ) − ψ1 (ξ ))dξ − (ψ2 (ξ ) + ψ1 (ξ ))dξ , 2 (1) − ⎪ ⎨D 2 = − k + γ ϕ γ γ 0 0 x
⎪ D3 = −kϕ2 (1) − (ψ2 (ξ ) + ψ1 (ξ ))dξ , ⎪ ⎪ ⎪ 0 x ⎪ ⎪ ⎪ ⎩D4 = − k ϕ2 (1) − β ϕ2 (0) − 1 (ψ2 (ξ ) + ψ1 (ξ ))dξ . γ γ γ 0
(4.16) −1 Hence A−1 exists, where A−1 is compact. 2 2 : H2 → D(A2 ) ⊂ H2 leads to the fact that A2 Therefore the spectrum σ (A2 ) consists of only isolated eigenvalues. Furthermore, to claim ρ(A2 )⊃iR, it is sufficient to show (λI − A2 )−1 exists for any λ ∈ iR. The existence of (λI − A2 )−1 , λ ∈ iR, λ = 0 will be discussed in next step. Step 3. We prove that
M = sup{ (iωI − A2 )−1 |ω ∈ R} < ∞.
(4.17)
+
To prove it, let α ∈ R be large enough, so that Eq. (4.17) can be reached by proving the following statement: There exists M1 , M2 ∈ R+ such that M1 = sup{ (iωI − A2 )−1 H2 |ω ∈ R, ω ≤ α} < ∞,
(4.18)
M2 = sup{ (iωI − A2 )−1 H2 |ω ∈ R, ω > α} < ∞. +
(4.19) −1
For existence of M1 ∈ R , we show (iωI − A2 ) H2 is continuous with respect to ω. Suppose ω0 ∈ R. Let ω ∈ R such that 1 |ω − ω0 | < , 2[ (iωI − A2 )−1 H2 + 1] Then by representation theorem, we can get
(iωI − A2 )−1 H2 ≤ 2 (iω0 I − A2 )−1 H2 . Applying the resolvent equation we have (iωI − A2 )−1 − (iω0 I − A2 )−1 = (ω − ω0 )(iωI − A2 )−1 (iω0 I − A2 )−1 , and finally
(iωI − A2 )−1 − (iω0 I − A2 )−1 H2 ≤ |ω − ω0 | (iωI − A2 )−1 H2 (iω0 − A2 )−1 H2 ≤ 2|ω − ω0 | (iω0 I − A2 )−1 2H2 .
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For existence of M2 , considering equation (iωI − A2 )( f1 , g1 , f2 , g2 ) = (φ1 , ψ1 , φ2 , ψ2 ) ,
(4.20)
which yields to ⎧ iω f1 − g1 = φ1 , ⎪ ⎪ ⎨ iωg1 − f1 = ψ1 , iω f2 − g2 = φ2 , ⎪ ⎪ ⎩ iωg2 − f2 = ψ2 .
(4.21)
The general solutions for f1 , ⎧ ⎪ ⎪ ⎪ f1 = H1 cos ωx + H2 sin ωx + ⎪ ⎪ ⎪ ⎨ f2 = H3 cos ωx + H4 sin ωx + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩g1 = iω f1 − φ1 , g2 = iω f2 − φ2 ,
g1 , f2 , g2 can be found as follows: 1 x [−(iωφ1 (ξ ) + ψ1 (ξ ))] sin ω(x − ξ )dξ , ω 0 1 x [−(iωφ2 (ξ ) + ψ2 (ξ ))] sin ω(x − ξ )dξ , ω 0
(4.22)
where Hi , i = 1, 2, 3, 4 are constants to be determined. In order to make (f1 , g1 , f2 , g2 ) ∈ D(A2 ), we choose H2 = 0 and ⎛ ⎞ ⎛ ⎞ H1 T1 Q2 ⎝H3 ⎠ = ⎝T2 ⎠, (4.23) H4 T3 where
⎛
ω cos ω Q2 = ⎝−ω sin ω 0
−ω cos ω −ω sin ω + ikω cos ω iβω + γ
⎞ −ω sin ω ω cos ω + ikω sin ω⎠, −ω
and
1
T1 =
[iω(φ1 (ξ ) − φ2 (ξ )) + (ψ1 (ξ ) − ψ2 (ξ ))] sin ω(1 − ξ )dξ ,
0 1
T2 = 0
[iω(φ1 (ξ ) + φ2 (ξ )) + (ψ1 (ξ ) + ψ2 (ξ ))] cos ω(1 − ξ )dξ 1 +i k (iωφ2 (ξ ) + ψ2 (ξ )) sin ω(1 − ξ )dξ + kφ2 (1),
T3 = βφ2 (0). Then
0
detQ2 = ω3
sin 2ω −
and
⎛ A11 Q2∗ = ω2 ⎝A21 A31
A12 A22 A32
kβ γ kγ k k cos 2ω + sin 2ω + i − β cos 2ω − sin 2ω − cos 2ω − , ω 2 2ω 2 2
⎞ ⎛ A11 A13 A23 ⎠ = ⎝A21 A33 −1
− sin ω − cos ω − cos 2ω − ik2 sin 2ω
⎞ (iβ + ωγ ) sin ω ⎠, −(iβ + ωγ ) cos ω ik − sin 2ω + 2 (cos 2ω + 1)
Y. Zhu and F.-F. Jin / Journal of the Franklin Institute 356 (2019) 6280–6302
where
6295
kγ γ A11 = (1 + kβ ) sin ω − cos ω − i (k + β ) cos ω + sin ω ω ω γ sin ω − cos ω. A21 = − iβ + ω It is obviously that |Aij |, i = 1, 2, 3, are bounded. Set |Aij | ≤ Sij , where Sij , i, j = 1, 2, 3, are constants. Let S0 = min{Si j , i, j = 1, 2, 3}. Let detQ2 = ω3 h(ω), where kβ γ kγ k k . h(ω) = sin 2ω − cos 2ω + sin 2ω + i − β cos 2ω − sin 2ω − cos 2ω − ω 2 2ω 2 2
Then we have following estimation: |h(ω)|2 kβ γ kγ k k 2 2 = sin 2ω − cos 2ω + sin 2ω + − β cos 2ω − sin 2ω − cos 2ω − ω 2 2ω 2 2 2 2 γ2 k β = 1 + 2 sin2 (2ω + θ1 ) + kβ(cos 2ω + 1) + sin2 2ω + β 2 cos2 2ω ω 4 k2 γ2 k2 γ2 k2 2 1 + 2 sin (2ω + θ2 ) + + 1 + 2 sin (2ω + θ2 ) + 4 ω 2 ω 4 2 2 2 γ k β = 1 + 2 sin2 (2ω + θ1 ) + kβ(cos 2ω + 1) + sin2 2ω + β 2 cos2 2ω ω 4 k2 γ2 + 1 + 2 sin (2ω + θ2 ) + 1 2 4 ω 2 2 2 γ2 2 k 2 k β 1 − 1 + 2 + min β , ≥ 4 ω 4 2 2 k β . ≥ min β 2 , 4 Set δ02 = min{β 2 , k 4β } > 0, where δ 0 > 0. Therefore, detQ2 ≥ δ 0 ω3 , and Q2−1 = It is easy to obtain that ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ H1 T1 A11 A12 A13 T1 ∗ Q2 1 ⎝ A21 A22 A23 ⎠⎝T2 ⎠ ω⎝H3 ⎠ = · ⎝T2 ⎠ = det Q2 h(ω) A H4 T3 A32 A33 T3 31 ⎛ ⎞ A T + A12 T2 + A13 T3 1 ⎝ 11 1 A21 T1 + A22 T2 + A23 T3⎠. = h(ω) A T + A T + A T 2 2
31 1
32 2
Q2∗ detQ2
≤
Q2∗ . δ0 ω3
33 3
So |ωHi | = ≤
1 |Ai1 T1 + Ai2 T2 + Ai3 T3 | |h(ω )|
S0 (|T1 | + T2 | + |T3 | ), δ0
(4.24)
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and
1 |T1 | = [iω(φ1 (ξ ) − φ2 (ξ )) + (ψ1 (ξ ) − ψ2 (ξ ))] sin ω(1 − ξ )dξ 0 1 = −i(φ1 (0) − φ2 (0)) cos ω − i (φ1 (ξ ) − φ2 (ξ )) cos ω(1 − ξ )dξ 0 1 (ψ1 (ξ ) − ψ2 (ξ )) sin ω(1 − ξ )dξ + 0
≤ μ1 (|φ2 (0)| + φ1 L2 + φ2 L2 + ψ1 L2 + ψ2 L2 ),
(4.25)
1 |T2 | = [iω(φ1 (ξ ) + φ2 (ξ )) + (ψ1 (ξ ) + ψ2 (ξ ))] cos ω(1 − ξ )dξ 0 1 + ik (iωφ2 (ξ ) + ψ2 (ξ )) sin ω(1 − ξ )dξ + kφ2 (1) 0 1 = i (φ1 (0) + φ2 (0)) sin ω + i (φ1 (ξ ) + φ2 (ξ )) sin ω(1 − ξ )dξ 0 1 (ψ1 (ξ ) + ψ2 (ξ )) cos ω(1 − ξ )dξ + k − φ2 (1) + φ2 (0) cos ω + 0 1 + φ2 (ξ ) cos ω(1 − ξ )dξ 0 1 ψ2 (ξ ) sin ω(1 − ξ )dξ + kφ2 (1) + ik 0
≤ μ2 (|φ2 (0)| + φ1 L2 + φ2 L2 + ψ1 L2 + ψ2 L2 ),
(4.26)
|T3 | = |βφ2 (0)| ≤ μ3 (|φ2 (0)| + φ1 L2 + φ2 L2 + ψ1 L2 + ψ2 L2 ),
(4.27)
where μ1 , μ2 , μ3 are some positive constants. From Eqs. (4.24)–(4.27), we can obtain that |ωHi | ≤
S0 (μ1 δ0
+ μ2 + μ3 )(|φ2 (0)| + φ1 L2 + φ2 L2 + ψ1 L2 + ψ2 L2 ).
(4.28)
Therefore, x | f1 | = −ωH1 sin ωx + (iωφ1 (ξ ) + ψ1 (ξ )) cos ω(x − ξ )dξ 0
= −ωH1 sin ωx + iφ1 (0) sin ωx +i
0
x
φ1 (ξ ) sin ω (x
x
− ξ )dξ + 0
ψ1 (ξ ) cos ω(x − ξ )dξ
≤ ν1 (|φ2 (0)| + φ1 L2 + φ2 L2 + ψ1 L2 + ψ2 L2 ), x | f2 | = −ωH3 sin ωx + ωH4 cos ωx + (iωφ2 (ξ ) + ψ2 (ξ )) cos ω(x − ξ )dξ 0 x = ωH3 sin ωx + ωH4 cos ωx + iφ2 (0) sin ωx + i φ2 (ξ ) sin ω (x − ξ )dξ 0
(4.29)
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+ 0
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ψ2 (ξ ) cos ω(x − ξ )dξ
≤ ν2 (|φ2 (0)| + φ1 L2 + φ2 L2 + ψ1 L2 + ψ2 L2 ),
(4.30)
x |g1 | = iωH1 cos ωx + i [−(iωφ1 (ξ ) + ψ1 (ξ ))] sin ω(x − ξ )dξ − φ1 (x) 0 x = iωH1 cos ωx − φ1 (0) cos ωx − φ1 (ξ ) cos ω (x − ξ )dξ 0 x ψ1 (ξ ) sin ω(x − ξ )dξ −i 0
≤ ν3 (|φ2 (0)| + φ1 L2 + φ2 L2 + ψ1 L2 + ψ2 L2 ),
(4.31)
x |g2 | = iωH3 cos ωx + iωH4 sin ωx + i [−(iωφ1 (ξ ) + ψ1 (ξ ))] sin ω(x − ξ )dξ − φ2 (x) 0 x = iωH3 cos ωx + iωH4 sin ωx − φ2 (0)cosωx − φ2 (ξ ) cos ω (x − ξ )dξ 0 x ψ2 (ξ ) sin ω(x − ξ )dξ −i 0
≤ ν4 (|φ2 (0)| + φ1 L2 + φ2 L2 + ψ1 L2 + ψ2 L2 ), | f2 (0)| = |H3 | ≤
S0 (μ1 δ0 |ω|
(4.32)
+ μ2 + μ3 )(|φ2 (0)| + φ1 L2 + φ2 L2 + ψ1 L2 + ψ2 L2 ), (4.33)
where ν 1 , ν 2 , ν 3 , ν 4 are positive constants. Considering Eqs. (4.28)–(4.33), we obtain that there exists M0 > 0 independent of ω such that | f1 | + | f2 | + |g1 | + |g2 | + γ | f2 (0)| ≤ M0 (|φ2 (0)| + φ1 L2 + φ2 L2 + ψ1 L2 + ψ2 L2 ). Therefore, there exists M2 such that 1 [| f1 (x)|2 + |g1 (x )|2 + | f2 (x )|2 + |g2 (x )|2 ]dx + γ f22 (0) 0 1 ≤ M2 [|φ1 (x)|2 + |ψ1 (x )|2 + |φ2 (x )|2 + |ψ2 (x )|2 ]dx + γ φ22 (0). 0
We complete the proof of the lemma. Remark 4.1. Under the controller (4.2), the error between system (1.3) and servo system (3.1), ε(x, t ) = z(x, t ) − u˜ (x, t ), decays exponentially because of exponential stability of z(x, t) and u˜ (x, t ). From ε2 (0, t ) = u2 (0, t ) − yre f (t ) = z2 (0, t ) − u˜2 (0, t ), the decay rate of tracking error is determined by the slower decay rate of z(x, t) and u˜ (x, t ). On the one hand, the decay rate of u˜ (x, t ) is fixed because design parameters β, γ in controller do not appear in system (2.2). On the other hand, γ z2 (0, t) can remove rigid motion and βz2t (0, t) introduces damping for system (4.4). But we do not have the relations between β, γ and decay rate, because it is difficult to obtain all eigenvalues of system (4.4).
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5. Closed-loop system and main results Now we turn to the closed-loop system (1.3), (2.1) and (3.1) under the controller (4.2), that is ⎧ u1tt (x, t ) = u1xx (x , t ), u2tt (x , t ) = u2xx (x, t ), ⎪ ⎪ ⎪ ⎪ u1x (0, t ) = d (t ), ⎪ ⎪ ⎪ ⎪ u2x (0, t ) = w2x (0, t ) + β(uˆ2t (0, t ) − w2t (0, t )) + γ (uˆ2 (0, t ) − w2 (0, t )), ⎪ ⎪ ⎪ ⎪ u2x (1, t ) + u1x (1, t ) = −ku2t (1, t ), u1 (1, t ) = u2 (1, t ), ⎪ ⎪ ⎪ ⎪ ⎨uˆ1tt (x, t ) = uˆ1xx (x , t ), uˆ2tt (x , t ) = uˆ2xx (x, t ), uˆ1 (0, t ) = u1 (0, t ), (5.1) ⎪ ⎪ u ˆ (0, t ) = w (0, t ) + β( u ˆ (0, t ) − w (0, t )) + γ ( u ˆ (0, t ) − w (0, t )) , ⎪ 2x 2x 2t 2t 2 2 ⎪ ⎪ ⎪ uˆ2x (1, t ) + uˆ1x (1, t ) = −k uˆ2t (1, t ), uˆ1 (1, t ) = uˆ2 (1, t ), ⎪ ⎪ ⎪ ⎪ w1tt (x, t ) = w1xx (x , t ), w2tt (x , t ) = w2xx (x, t ), ⎪ ⎪ ⎪ ⎪ w1x (0, t ) = uˆ1x (0, t ), w2 (0, t ) = yre f (t ), ⎪ ⎪ ⎩ w2x (1, t ) + w1x (1, t ) = −kw2t (1, t ), w1 (1, t ) = w2 (1, t ). We consider system (5.1) in the state space X defined by X = {( f1 , g1 , f2 , g2 , f3 , g3 , f4 , g4 , f5 , g5 , f6 , g6 ) |( f1 , g1 , f2 , g2 , f3 , g3 , f4 , g4 ) ∈ (H 1 (0, 1) ×L 2 (0, 1))4 , ( f5 , g5 , f6 , g6 ) ∈ H1 , f1 (1) = f2 (1), f3 (1) = f4 (1), f2 (0) = f4 (0)} (5.2) with inner product induced norm
( f1 , g1 , f2 , g2 , f3 , g3 , f4 , g4 , f5 , g5 , f6 , g6 ) 2X 1 = ( f12 + g21 + f22 + g22 + f32 + g23 + f42 + g24 )dx + γ f42 (0) + ( f5 , g5 , f6 , g6 ) 2H1 , 0
∀ ( f 1 , g1 , f 2 , g2 , f 3 , g3 , f 4 , g4 , f 5 , g5 , f 6 , g6 ) ∈ X .
(5.3)
Then we have the following theorem, the main result of this paper. Theorem 5.1. Suppose that yre f , y˙re f , y¨re f ∈ L ∞ (0, ∞ ), d ∈ L∞ (0, ∞) or d ∈ L2 (0, ∞). Then for any initial value (u10 , u11 , u20 , u21 , uˆ10 , uˆ11 , uˆ20 , uˆ21 , w10 , w11 , w20 , w21 ) ∈ X , there exists a unique (weak) solution (u1 (·, t ), u1t (·, t ), u2 (·, t ), u2t (·, t ), uˆ1 (·, t ), uˆ1t (·, t ), uˆ2 (·, t ), uˆ2t (·, t ), w1 (·, t ), w1t (·, t ), w2 (·, t ), w2t (·, t )) ∈ C(0, ∞; X ) to Eq. (5.1). Moreover, this closed-loop solution has the following properties: 1 2 2 2 (i) sup [u12t (x, t ) + u12x (x, t ) + u2t (x, t ) + u2x (x, t ) + uˆ12t (x, t ) + uˆ12x (x, t ) + uˆ2t (x, t ) t≥0 0 γ 2 2 2 + uˆ2x (x, t ) + w12t (x, t ) + w12x (x, t ) + w2t (x, t ) + w2x (x, t )]dx + uˆ22 (0, t ) < ∞. 2 (ii) lim [u2 (0, t ) − yre f (t )] = 0. t→∞ 1 (iii) lim {[uˆ1t (x, t ) − u1t (x, t )]2 + [uˆ1x (x, t ) − u1x (x, t )] + [uˆ2t (x, t ) − u2t (x, t )]2 + t→0 0
[uˆ2x (x, t ) − u1x (x, t )]2 + [u1t (x, t ) − w1t (x, t )]2 + [u1x (x, t ) − w1x (x, t )]2 + [u2t (x, t ) − w2t (x, t )]2 + [u2x (x, t ) − w2x (x, t )]2 }dx = 0.
(iv) When
d (t ) = 0
→ 0 as t → ∞.
and
1
yre f (t ) = 0, 0
2 2 [u12t (x, t ) + u12x (x, t ) + u2t (x, t ) + u2x (x, t )]dx
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Proof. Define a bounded invertible operator P : X → H1 × H2 × H3 by =
(u˜1 , u˜1t , u˜2 , u˜2t , w1 , w1t , w2 , w2t , z1 , z1t , z2 , z2t ) P(u1 , u1t , u2 , u2t , uˆ1 , uˆ1t , uˆ2 , uˆ2t , w1 , w1t , w2 , w2t ) ,
where ⎛
−1 ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 P=⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎝0 0
0 −1 0 0 0 0 0 0 0 0 0 0
0 0 −1 0 0 0 0 0 0 0 0 0
0 0 0 −1 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0 −1 0 0 0
0 0 0 0 0 1 0 0 0 −1 0 0
is invertible and (u1 , u1t , u2 , u2t , uˆ1 , uˆ1t , uˆ2 , uˆ2t , w1 , w1t , w2 , w2t ) = P−1 (u˜1 , u˜1t , u˜2 , u˜2t , w1 , w1t , w2 , w2t , z1 , z1t , z2 , z2t ) .
(5.4)
0 0 0 0 0 0 1 0 0 0 −1 0
⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 1⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎠ −1
(5.5)
(5.6)
Thanks to the transformation P, closed-loop system (5.1) is equivalent to the decoupled system consisting of Eqs. (2.2), (3.1) and (4.4). It is well known that Eq. (2.2) is exponential stable by [12]. So Eq. (2.2) is bounded. The boundedness of Eqs. (3.1) and (4.4) are from Lemmas 3.1 and 4.1. And it is obviously that P−1 is bounded. According to (5.6), (i) is proved. (4.4) is exponential stable by Lemma 4.1. By transformation z = ε + u˜ = u − w + u˜ and u˜ = uˆ − u, it is easy to obtain (iii). (ii) follows from the exponential stability of Eq. (4.3). When d (t ) = 0 and yre f (t ) = 0, Eq. (1.3) is exponential stable by [12] and Eq. (3.1) is equivalent to Eq. (3.5). Therefore, (iv) is proved. 6. Numerical simulation In this section, we present some numerical simulations to illustrate the effect of proposed controller. The finite difference method is applied to compute the solution numerically and the steps of time and space are set as 0.01 and 0.05, respectively. The parameters are chosen as k = 1, β = 1, γ = 2. The disturbance and reference are set to be d (t ) = 2 + 8 sin 2t, yre f (t ) = −3 + cos 3t, respectively. The initial values are taken as ⎧ u1 (x, 0) = −3x 2 + 2 sin π x, u1t (x, t ) = −2 + 3 cos 3x, ⎪ ⎪ ⎪ ⎪ u (x, 0) = x 2 + 4 cos π x, u2t (x, t ) = −2 + 3 cos x, ⎪ ⎪ ⎨ 2 uˆ1 (x, 0) = 5x, uˆ1t (x, 0) = 2 cos π x, (6.1) uˆ2 (x, 0) = 2 + 3x, uˆ2t (x, 0) = 2 cos π x, ⎪ ⎪ ⎪ ⎪ w (x, 0) = −4 cos π x + 2 sin π x, w1t (x, t ) = 2 − 2 cos π x, ⎪ ⎪ ⎩ 1 w2 (x, 0) = 4 + 2 sin π x, w2t (x, t ) = 3 sin 2π x. Figs. 1–3 display the displacements of u, uˆ, w part of closed-loop system (5.1), respectively. We can see the boundedness of solution of (5.1) from these three figures. Fig. 4(a) presents the trajectory of controller (4.2). Fig. 4(b) gives the tracking performance of boundary signal
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20 20 10 10 0 0 -10 -10 -20 1 15 0.5
-20 1 15
10 0.5
5 0
10 5
0 0
(a) Displacement of
1 -part
0
(b) Displacement of
2 -part
Fig. 1. Displacement of u-part of closed-loop system (5.1).
20 20 10 10 0 0 -10 -10 -20 1 15 0.5
-20 1 15
10 0.5
5 0
10 5
0 0
(a) Displacement of ˆ1 -part
0
(b) Displacement of ˆ2 -part
Fig. 2. Displacement of uˆ-part of closed-loop system (5.1).
20 20 10 10 0 0 -10 -10 -20 1 15 0.5
-20 1 15
10 0.5
5 0
10 5
0 0
(a) Displacement of
1 -part
0
(b) Displacement of
Fig. 3. Displacement of w-part of closed-loop system (5.1).
2 -part
Y. Zhu and F.-F. Jin / Journal of the Franklin Institute 356 (2019) 6280–6302 50
6301
15
40 10
30 20
5
10 0
0
-10 -5
-20 -30
-10
-40 -50
-15 0
5
10
(a) Trajectory of controller
15
0
5
10
15
(b) Tracking performance
Fig. 4. Controller and boundary tracking performance of closed-loop system (5.1).
u2 (0, t) which tracks reference signal yref (t) satisfactorily after t = 10. All four figures demonstrate the effectiveness of proposed output feedback controller (4.2). 7. Concluding remark This paper investigates boundary tracking problem for coupled 1D wave equation with general external disturbance. The servomechanism is designed which gives dynamics of reference signal. Meanwhile, the disturbance observer plays a critical role in regulator design. Because of the introduction of observer, robustness seems impossible by this approach which is a disadvantage compared with internal model principle method. The merit of our method lies in that our method can deal with the system when reference and disturbance are general signal with exosystem unavailable. In the future work, we will consider robust output regulation for system with general reference and disturbance. Another future work seems interesting to consider case that the tracking signal is anti-collocated with the control. References [1] C.I. Byrnes, I.G. Lauk, D.S. Gilliam, V.I. Shubov, Output regulation problem for linear distributed parameter systems, IEEE Trans. Autom. Control 45 (2000) 2236–2252. [2] C.I. Byrnes, F.D. Priscoli, A. Isidori, Output Regulation for Uncertain Nonlinear Systems, Birkäuser, Boston, 1997. [3] F.M. Callier, C.A. Desoer, Stabilization, tracking and disturbance rejection in multivariable convolution systems, Annales de la Société scientifique de Bruxelles 94 (1980) 7–51. [4] E.J. Davison, The robust control of a servomechanism problem for linear time-invariant multivariable systems, IEEE Trans. Autom. Control 21 (1976) 25–34. [5] C.A. Desoer, C.A. Lin, Tracking and disturbance rejection of MIMO nonlinear systems with PI controller, IEEE Trans. Autom. Control 30 (1985) 861–867. [6] J. Deutscher, A backstepping approach to the output regulation of boundary controlled parabolic PDEs, Automatica 57 (2015) 56–64. [7] J. Deutscher, Output regulation for linear distributed-parameter systems using finite-dimensional dual observers, Automatica 47 (2011) 2468–2473. [8] H. Feng, B.Z. Guo, A new active disturbance rejection control to output feedback stabilization for a one-dimensional anti-stable wave equation with disturbance, IEEE Trans. Autom. Control 62 (2017) 3774–3787. [9] B.A. Francis, The linear multivariable regulator problem, SIAM J. Control Optim. 15 (1977) 486–505.
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