Preprints of the 19th World Congress The International Federation of Automatic Control Cape Town, South Africa. August 24-29, 2014
Active Disturbance Rejection Control for a 2×2 Hyperbolic System with an Input Disturbance Shuxia Tang ∗ Bao-Zhu Guo ∗∗ Miroslav Krstic ∗∗∗ ∗ Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093 USA (e-mail:
[email protected]) ∗∗ Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190 China (e-mail:
[email protected]), School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa and Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O.Box 80203, Jeddah 21589, Saudi Arabia ∗∗∗ Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093 USA (e-mail:
[email protected])
Abstract: In this paper, active disturbance rejection control (ADRC) approach is used to stabilize a 2×2 system of first-order linear hyperbolic partial differential equations (PDEs) subject to a boundary input disturbance. Disturbance attenuation is achieved with the designed controller, and the resulting closed-loop control system admits a unique solution, which could tend to any arbitrary vicinity of zero. Keywords: Linear 2×2 hyperbolic systems; Active disturbance rejection control; Boundary control. performances of both SMC and ADRC are shown from the simulation results in Section 5.
1. INTRODUCTION This paper considers a 2×2 system of first-order linear hyperbolic PDEs with a boundary input disturbance. The control objective is to stabilize the system while attenuating the disturbance, and the control method is active disturbance rejection boundary control. 2×2 hyperbolic systems have wide physical backgrounds, such as oil wells (Landet et al. (2013)), gas flow (Gugat and Dick (2011)), transmission lines (Curr´o et al. (2011)), road traffic (Goatin (2006)), open channels (Gugat and Leugering (2003)), and so on. And stabilization of these systems has attracted many researchers, see, e.g., Aamo (2013), Bastin and Coron (2010), Vazquez et al. (2011). Different methods have been employed to deal with particular types of boundary input disturbances, see, e. g., Guo and Guo (2013a), Guo and Guo (2013b). Disturbance rejection/attenuation is usually desirable in system control designs. Application of the ADRC method, firstly proposed by Han in 1990s (see, Han (2009)), has been studied for decades. Recently, this approach has been generalized to distributed parameter systems. For example, it is used to attenuate the more general boundary input disturbance in wave equation (Guo and Jin (2013b)), Euler-Bernoulli beam equation (Guo and Jin (2013a)) and Schrodinger ¨ equation (Guo and Liu (2013)). This paper is organized as follows. In Section 2, the system to be considered is introduced. In Section 3, previous results from the sliding mode control (SMC) design in Tang and Krstic (2014) are presented. Inspired by the SMC, an active disturbance rejection boundary controller is designed in Section 4. Existence and uniqueness of solutions to the resulting closed-loop system are also proved. Moreover, the solution could tend to any arbitrary vicinity of zero. Effectiveness and Copyright © 2014 IFAC
2. PROBLEM STATEMENT In this paper, we intend to stabilize the following 2×2 system of coupled hyperbolic PDEs: ut (x,t) = − ε1 (x)ux (x,t) + c1 (x)v(x,t) vt (x,t) =ε2 (x)vx (x,t) + c2 (x)u(x,t) u(0,t) =qv(0,t) v(1,t) =U(t) + d(t),
(1) (2) (3) (4)
where u(x,t), v(x,t) are system states with x ∈ [0, 1], t > 0; U(t) is control input; d(t) is external disturbance at the control end. Here are some assumptions: 1. ε1 (x), ε2 (x) ∈ C1 [0, 1], ε1 (x), ε2 (x) > 0, 2. c1 (x), c2 (x) ∈ C[0, 1], 3. q 6= 0, ˙ are uniformly bounded measurable, 4. d(t) and d(t) 5. Initial data u0 (x), v0 (x) ∈ L2 [0, 1]. Following Vazquez et al. (2011), we introduce a backstepping transformation α(x,t) =u(x,t) − −
Z x
K uv (x, ξ )v(ξ ,t)dξ
β (x,t) =v(x,t) − Z x
K uu (x, ξ )u(ξ ,t)dξ
0
0
−
Z x
Z x
(5)
K vu (x, ξ )u(ξ ,t)dξ
0
K vv (x, ξ )v(ξ ,t)dξ ,
(6)
0
in which the continuous kernel functions are uniquely determined by the following system of coupled PDEs:
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19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014
ε1 (x)Kxuu + ε1 (ξ )Kξuu = −ε10 (ξ )K uu − c2 (ξ )K uv
(7)
ε1 (x)Kxuv − ε2 (ξ )Kξuv = ε20 (ξ )K uv − c1 (ξ )K uu
(8)
ε2 (x)Kxvu − ε1 (ξ )Kξvu = ε10 (ξ )K vu + c2 (ξ )K vv ε2 (x)Kxvv + ε2 (ξ )Kξvv = −ε20 (ξ )K vv + c1 (ξ )K vu
(9) (10)
with boundary conditions ε2 (0) uv c1 (x) K uu (x, 0) = K (x, 0), K uv (x, x) = , (11) qε1 (0) ε1 (x) + ε2 (x) c2 (x) qε1 (0) vu K vu (x, x) = − , K vv (x, 0) = K (x, 0). ε1 (x) + ε2 (x) ε2 (0) (12) The transformation (5) − (6) is invertible and the inverse is: Z x
u(x,t) =α(x,t) + Z x
+
Lαβ (x, ξ )β (ξ ,t)dξ Z x
v(x,t) =β (x,t) + Z x
+
Lαα (x, ξ )α(ξ ,t)dξ
0
0
Z 1 2−x 2q2 (1 + x) = f1 (x) f2 (x) + g1 (x)g2 (x) dx, ε1 (x) ε2 (x) 0 ∀ ( f1 , g1 )T , ( f2 , g2 )T ∈ H. (25) In Tang and Krstic (2014), firstly, a sliding surface for the system (21) − (24) is chosen as follows: (26) S(α,β )T (t) = β (1,t) = 0, i.e., (27) S(α,β )T = {( f , g)T ∈ H | g(1) = 0}. For the system (1) − (4), the sliding surface is Z 1 n K vu (1, ξ ) f (ξ )dξ S(u,v)T = ( f , g)T ∈ H | g(1) − 0 Z 1 o K vv (1, ξ )g(ξ )dξ = 0 . (28) − 0
(13)
Secondly, a sliding mode boundary control is designed as Lβ α (x, ξ )α(ξ ,t)dξ
Z 1
U(t) =
0
Lβ β (x, ξ )β (ξ ,t)dξ ,
(14)
ε1 (x)Lxαβ − ε2 (ξ )Lξ = ε20 (ξ )Lαβ + c1 (x)Lβ β
(16)
= ε10 (ξ )Lβ α − c2 (x)Lαα
(17)
= −ε20 (ξ )Lβ β
(18)
αβ
βα ε2 (x)Lxβ α − ε1 (ξ )Lξ ββ ε2 (x)Lxβ β + ε2 (ξ )Lξ
− c2 (x)L
αβ
with boundary conditions c1 (x) ε2 (0) αβ L (x, 0), Lαβ (x, x) = Lαα (x, 0) = , qε1 (0) ε1 (x) + ε2 (x) (19) qε1 (0) β α c2 (x) , Lβ β (x, 0) = L (x, 0). Lβ α (x, x) = − ε1 (x) + ε2 (x) ε2 (0) (20) The transformation (5) − (6) brings the system (1) − (4) into the following system : αt (x,t) = − ε1 (x)αx (x,t) (21) βt (x,t) =ε2 (x)βx (x,t) (22) α(0,t) =qβ (0,t) (23) β (1,t) =U(t) + d(t) Z 1 Z 1 vu K vu (1, η)Lαα (η, ξ )dη α(ξ ,t) K (1, ξ ) + − ξ 0 Z 1 K vv (1, η)Lβ α (η, ξ )dη dξ + ξ Z 1 Z 1 K vu (1, η)Lαβ (η, ξ )dη β (ξ ,t) K vv (1, ξ ) + − ξ 0 Z 1 vv ββ K (1, η)L (η, ξ )dη dξ . (24) +
K vv (1, ξ )v(ξ ,t)dξ
0
−K
where the continuous kernel functions are uniquely determined by the following system of coupled PDEs: (15)
Z 1
0
0
ε1 (x)Lxαα + ε1 (ξ )Lξαα = −ε10 (ξ )Lαα + c1 (x)Lβ α
K vu (1, ξ )u(ξ ,t)dξ + Z t S (u,v)T (τ) 0
|S(u,v)T (τ)|
dτ for S(u,v)T (t) 6= 0.
(29)
The following main result is then proved in Tang and Krstic (2014). Theorem 1. Suppose that d and d˙ are bounded measurable in time, then for any initial data (u(·, 0), v(·, 0))T ∈ H, there exists Tmax ≥ 0, depending on initial data, such that the system (1) − (4) with controller (29) admits a unique solution (u(·,t), v(·,t))T ∈ C([0, Tmax ]; H)
(30)
and S(u,v)T (t) =v(1,t) − −
Z 1
Z 1
K vu (1, ξ )u(ξ ,t)dξ
0
K vv (1, ξ )v(ξ ,t)dξ = 0
(31)
0
for all t ≥ Tmax . Moreover, S(u,v)T (t) is continuous and monotone in [0, Tmax ]. On the sliding mode surface S(u,v)T (t) = 0, the system (1) − (4) becomes exponentially stable. 4. ACTIVE DISTURBANCE REJECTION CONTROL Inspired by the controller (29) from SMC design, we implement a to-be-designed controller U0 (t) in U(t): Z 1
U(t) =
K vu (1, ξ )u(ξ ,t)dξ +
Z 1 0
0
K vv (1, ξ )v(ξ ,t)dξ +U0 (t), (32)
that is,
ξ
Z 1
U(t) =
α(ξ ,t) (K vu (1, ξ )
0
Z 1
+ Z 1
K vv (1, η)Lβ α (η, ξ )dη dξ ξ Z 1 Z 1 K vu (1, η)Lαβ (η, ξ )dη β (ξ ,t) K vv (1, ξ ) + + ξ 0 Z 1 vv ββ K (1, η)L (η, ξ )dη dξ +U0 (t). (33) + +
3. RESULTS FROM SLIDING MODE CONTROL DESIGN Consider the systems (1) − (4) and (21) − (24) in the state 2 Hilbert space H = L2 (0, 1) with an induced norm from the following inner product 11386
K vu (1, η)Lαα (η, ξ )dη
ξ
ξ
19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014
Then the system (21) − (24) becomes αt (x,t) = − ε1 (x)αx (x,t) (34) βt (x,t) =ε2 (x)βx (x,t) (35) α(0,t) =qβ (0,t) (36) β (1,t) =U0 (t) + d(t), (37) which can be written into the following operator form: d α α + B(U0 (t) + d(t)). (38) =A β β dt Here the operator A : D(A )(⊂ H) → H is defined as follows: T A ( f , g)T = −ε1 (x) f 0 , ε2 (x)g0 , ∀( f , g)T ∈ D(A ), (39) 2 D(A ) = {( f , g)T ∈ H 1 (0, 1) | f (0) = qg(0), g(1) = 0}, (40) and 0 B= , (41) 4q2 δ (x − 1) where δ (·) denotes the Dirac distribution. In Tang and Krstic (2014), it has been proved that A generates a C0 -semigroup eA t of contractions in H and B is admissible for eA t .
The following state feedback controller to (34) − (37) is designed: U0 (t) = −dˆε (t), (53) and then the closed-loop system becomes αt (x,t) = −ε1 (x)αx (x,t) (54) βt (x,t) = ε2 (x)βx (x,t) (55) α(0,t) = qβ (0,t) (56) β (1,t) = −dˆε (t) + d(t) (57)
The adjoint operator of A is T φ ψ ∗ T 0 0 A (φ , ψ) = ε1 (x) φ + , −ε2 (x) ψ + , x−2 x+1 ∀(φ , ψ)T ∈ D(A ∗ ), (42) 2 ∗ T 1 D(A ) = {(φ , ψ) ∈ H (0, 1) | φ (0) = qψ(0), φ (1) = 0}. (43) Choose 0 φ ∈ D(A ∗ ) (44) = x ψ and let Z 1 2 2q (1 + x) φ α = , y1 (t) = β (x,t)xdx, (45) ψ β ε2 (x) 0 Z 1 α φ , A∗ y2 (t) = = −2q2 β (x,t)(1 + 2x)dx, β ψ 0 (46) then we can get y˙1 (t) = 4q2 (U0 (t) + d(t)) + y2 (t). (47) Design the following extended state observer for y1 (t) and d(t): 1 yˆ˙ε (t) = 4q2 (U0 (t) + dˆε (t)) + y2 (t) + (y1 (t) − yˆε (t)) ε 1 (48) d˙ˆε (t) = 2 2 (y1 (t) − yˆε (t)), 4q ε where ε > 0 is the small tuning parameter, then the errors y˜ε = y1 − yˆε , d˜ε = d − dˆε (49) satisfy d y˜ε (t) y˜ε (t) ˙ =A ˜ + Bd(t), (50) dε (t) dt d˜ε (t) where 1 2 − 4q 0 ε A= (51) , B = 1 . 1 − 2 2 0 4q ε The eigenvalues of A are √ 3 1 j. (52) λ =− ± 2ε 2ε
Proof. The system (54) − (59) is equivalent to the following system: αt (x,t) = −ε1 (x)αx (x,t) (61) βt (x,t) = ε2 (x)βx (x,t) (62) α(0,t) = qβ (0,t) (63) ˜ β (1,t) = dε (t) (64) 1 2 (65) y˙˜ε (t) = − y˜ε (t) + 4q d˜ε (t) ε 1 ˙ d˙˜ε (t) = − 2 2 y˜ε (t) + d(t). (66) 4q ε T Firstly, the y˜ε (t), d˜ε (t) -subsystem (65) − (66) can be solved separately: Zt y˜ε (t) y˜ε (0) At ˙ =e (67) + eA(t−τ) Bd(τ)dτ. d˜ε (t) d˜ε (0) 0
1 y˙ˆε (t) = y2 (t) + (y1 (t) − yˆε (t)) ε 1 ˙ ˆ dε (t) = 2 2 (y1 (t) − yˆε (t)). 4q ε Lemma 2. Suppose that d and d˙ are uniformly bounded surable, then for any initial data (α(·, 0), β (·, 0))T ∈ H, exists a unique solution
(58) (59) meathere
(α(·,t), β (·,t))T ∈ C([0, ∞); H) (60) to the closed-loop system (54) − (59). Moreover, the solution tends to any arbitrary vicinity of zero as t → ∞, ε → 0.
From (52), it can be derived that y˜ε (t), d˜ε (t) → 0 as t → ∞, ε → 0.
(68)
Secondly, the (α, β )T -subsystem (61) − (64) can be written as d α α + B d˜ε (t). (69) =A β dt β Since from Tang and Krstic (2014), it can be proved that for the system d γ γ(·, 0) γ ∈ H, (70) , =A δ (·, 0) δ dt δ there exists a unique solution (γ(·,t), δ (·,t))T ∈ C([0, ∞); H),
(71)
and
T T
(γ(·,t), δ (·,t)) 6 e−a/2t (γ(·, 0), δ (·, 0)) ) , H
H
(72)
where a=
1 min {ε1 (x), ε2 (x)} > 0, 2 x∈[0,1]
(73)
then it can be derived that the C0 -semigroup eA t generated by A is exponentially stable. Moreover, since B is admissible for eA t , there exists a unique solution to the system (69):
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19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014
α(·,t) β (·,t)
= eA t
α(·, 0) β (·, 0)
Z t
+
0
eA (t−τ) B d˜ε (τ)dτ. (74)
From (68), for any given ε0 > 0, there exist t0 > 0 and ε1 > 0 such that |d˜ε (t)| < ε0 for all t > t0 , 0 < ε < ε1 . Thus, Zt α(·,t) α(·, 0) =eA t + eA (t−τ) B d˜ε (τ)dτ β (·,t) β (·, 0) t0 + eA (t−t0 )
Z t0 0
eA (t0 −τ) B d˜ε (τ)dτ.
(75)
With admissibility of B, it can be derived that
Z t
2
0 A (t −τ)
0
˜ε (τ)dτ e B d
0
H
2
2 6 Ct0 d˜ε L2 (0,t ) 6 Ct0 t02 d˜ε L∞ (0,t ) , ∀d˜ε ∈ L∞ (0, ∞), loc
0
where Ct0 is a constant that is independent of d˜ε . With exponential stability of eA t , it can be derived that
Z t
Z t
eA (t−τ) B d˜ε (τ)dτ = eA (t−τ) B(0 d˜ε )(τ)dτ
0
t t0 0 H H
(77) 6N d˜ε L∞ (0,∞) 6 Nε0 , where N is a constant independent of d˜ε , and d1 d2 denotes the s
s-concatenation of d1 and d2 . Since keA t k 6 e−a/2t , then from (75), (76) and (77),
α(·,t)
6e−a/2t α(·, 0)
β (·,t)
β (·, 0) H
H +Ct e−a/2(t−t0 ) d˜ε ∞ + Nε0 . (78) L (0,t0 )
With arbitrariness of ε0 , the proof is completed. By equivalence between the transformations (5) − (6) and (13) − (14), we summarize our closed-loop construction in the following main theorem. Theorem 3. Suppose that d and d˙ are uniformly bounded measurable, then for any initial data (u(·, 0), v(·, 0))T ∈ H, there exists a unique solution
Consider the system (1) − (4) with ε1 (x) = 0.1, ε2 (x) = 0.2, c1 (x) = 0.03, c2 (x) = 0.04, q = 1/4 and disturbance d(t) = 10 sint. Set initial data u(x, 0) = 25 (1 − x), v(x, 0) = 10(1 − x). Take the time length, steps of time and space as 50, 0.01 and 0.01, then open-loop response and and closed-loop response with ADRC (choosing ε = 0.001) of the (u, v)T −system are shown in Fig. 1 and Fig. 3, respectively. The control kernel functions K vu (1, ξ ), K vv (1, ξ ) are depicted in Fig. 2. Also, for the closed-loop (u, v)T −system with SMC (choosing K = 20), a figure very similar to Fig. 3 is obtained, which is not shown here due to the page limit. As can be seen from these figures, the designed ADRC, as well as SMC, has stabilized the (u, v)T −system to very small vicinities of zero. Remark 4. From the above simulation results, it’s worth noting that the first order SMC designed in Tang and Krstic (2014) exhibits a non-chattering character and could achieve satisfactory results as well as ADRC. However, if we set the time step to be bigger (0.05), a figure very similar to Fig. 3 is obtained for the closed-loop (u, v)T −system with ADRC, while Fig. 4 is obtained for that with SMC. Thus, considering SMC’s higher requirement to integrators than ADRC, ADRC is a better choice than SMC. 5.2 Example 2
(79)
Consider the system (1) − (4), where system parameters, disturbance and initial data are chosen to be the same as those in Example 1 except that c1 (x) = 0.3, c2 (x) = 0.4.
(80) (81) (82) (83)
Take the time length, steps of time and space as 50, 0.01 and 0.01, then open-loop response and and closed-loop response with ADRC (choosing ε = 0.001) of the system are shown in Fig. 5 and Fig. 7, respectively. The control kernel functions K vu (1, ξ ), K vv (1, ξ ) are depicted in Fig. 6. If choosing K = 20, a figure very similar as Fig. 7 is obtained for the closed-loop system with SMC. Thus, although the open-loop system blows up, ADRC, as well as SMC, has still successfully stabilized them.
to the following closed-loop system: ut (x,t) = −ε1 (x)ux (x,t) + c1 (x)v(x,t) vt (x,t) = ε2 (x)vx (x,t) + c2 (x)u(x,t) u(0,t) = qv(0,t) v(1,t) = U(t) + d(t) 1 y˙ˆε (t) = y2 (t) + (y1 (t) − yˆε (t)) ε 1 ˙ ˆ dε (t) = 2 2 (y1 (t) − yˆε (t)), 4q ε
0
Moreover, the solution tends to any arbitrary vicinity of zero as t → ∞, ε → 0.
5.1 Example 1
(76)
(u(·,t), v(·,t))T ∈ C([0, ∞); H)
Z x
5. NUMERICAL SIMULATIONS
0
0
Z 1 2 2q (1 + x)
K vu (x, ξ )u(ξ ,t)dξ x v(x,t) − ε2 (x) 0 0 Z x K vv (x, ξ )v(ξ ,t)dξ dx, (87) − 0 Z x Z 1 K vu (x, ξ )u(ξ ,t)dξ y2 (t) = − 2q2 (1 + 2x) v(x,t) − 0 0 Z x vv K (x, ξ )v(ξ ,t)dξ dx. (88) −
y1 (t) =
(84) (85)
where the control is Z 1
U(t) = 0
K vu (1, ξ )u(ξ ,t)dξ +
Z 1 0
(86) and
REFERENCES
K vv (1, ξ )v(ξ ,t)dξ − dˆε (t),
Aamo, O.M. (2013). Disturbance Rejection in 2 × 2 Linear Hyperbolic Systems. IEEE Transactions on Automatic Control, 58(5), 1095–1106.
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19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014
(a)
(a)
(b) (b)
Fig. 1. Simulation results for open-loop (u, v)T −systems (time step=0.01, space step=0.01) −0.1315
Fig. 3. Simulation results for closed-loop (u, v)T −systems with ADRC (time step=0.01, space step=0.01)
−0.015
−0.132 −0.1325
−0.02
Kvv(x,ξ)
Kvu(x,ξ)
−0.133 −0.1335 −0.134 −0.1345
−0.025
−0.03
−0.135 −0.1355
−0.035
−0.136 −0.1365
0
0.2
0.4
0.6
x
0.8
1
−0.04
0
0.2
0.4
0.6
0.8
1
x
Fig. 2. Control kernel functions K vu (1, ξ ) (left) and K vv (1, ξ ) (right) Bastin, G. and Coron, J.M. (2010). Further results on boundary feedback stabilisation of 2 × 2 hyperbolic systems over a bounded interval. Proceedings of the 8th IFAC Symposium on Nonlinear Control Systems, 1081–1085. Curr´o, C., Fusco, D., and Manganaro, N. (2011). A reduction procedure for generalized Riemann problems with application to nonlinear transmission lines. Journal of Physics A: Mathematical and Theoretical, 44(33), 335205. Goatin, P. (2006). The Aw–Rascle vehicular traffic flow model with phase transitions. Mathematical and Computer Modelling, 44, 287–303. Gugat, M. and Dick, M. (2011). Gas flow in fan-shaped networks: classical solutions and feedback stabilization. SIAM Journal of Control and Optimizaiton, 49(5), 2101–2117. Gugat, M. and Leugering, G. (2003). Global boundary controllability of the de St. Venant equations between steady states. Annales de l’Institut Henri Poincare, 20(1), 1–11. Guo, B.Z. and Jin, F.F. (2013a). The active disturbance rejection and sliding mode control approach to the stabilization
(a)
(b)
Fig. 4. Simulation results for closed-loop (u, v)T −systems with SMC (time step=0.05, space step=0.01)
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19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014
Guo, W. and Guo, B.Z. (2013a). Parameter Estimation and Non-Collocated Adaptive Stabilization for a Wave Equation Subject to General Boundary Harmonic Disturbance. IEEE Transactions on Automatic Control, 58(7), 1631–1643. Guo, W. and Guo, B.Z. (2013b). Stabilization and regulator design for a one-dimensional unstable wave equation with
(a)
(a)
(b)
Fig. 5. Simulation results for open-loop (u, v)T −systems (time step=0.01, space step=0.01) −0.95
−0.1
(b)
−0.2 −0.3
−1.1
−0.4
−1.15
−0.5
Kvv(x,ξ)
Kvu(x,ξ)
−1 −1.05
−1.2 −1.25
Fig. 7. Simulation results for closed-loop (u, v)T −systems with ADRC (time step=0.01, space step=0.01)
−0.6 −0.7 −0.8
−1.3
−0.9 −1.35
−1 −1.4
0
0.2
0.4
0.6
x
0.8
1
−1.1
0
0.2
0.4
0.6
0.8
1
x
Fig. 6. Control kernel functions K vu (1, ξ ) (left) and K vv (1, ξ ) (right) of the Euler-Bernoulli beam equation with boundary input disturbance. Automatica, 49, 2911–2918. Guo, B.Z. and Jin, F.F. (2013b). Sliding Mode and Active Disturbance Rejection Control to Stabilization of OneDimensional Anti-Stable Wave Equations Subject to Disturbance in Boundary Input. IEEE Transactions on Automatic Control, 58(5), 1269–1274. Guo, B.Z. and Liu, J.J. (2013). Sliding mode control and active disturbance rejection control to the stabilization of one-dimensional Schrodinger ¨ equation subject to boundary control matched disturbance. International Journal of Robust and Nonlinear Control.
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