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Sliding mode control to stabilization of cascaded heat PDE–ODE systems subject to boundary control matched disturbance
Automatica 52 (2015) 23–34
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Sliding mode control to stabilization of cascaded heat PDE–ODE systems subject to boundary control matched disturbance✩ Jun-Min Wang a , Jun-Jun Liu a , Beibei Ren b , Jinhao Chen b a
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
b
Department of Mechanical Engineering, Texas Tech University, Lubbock, TX 79409, USA
article
info
Article history: Received 28 January 2014 Received in revised form 30 June 2014 Accepted 15 October 2014
Keywords: Boundary control Disturbance rejection Sliding mode control Cascade systems Backstepping
abstract In this paper, we are concerned with the boundary feedback stabilization of a cascade of heat PDE–ODE system with Dirichlet/Neumann interconnection and with the external disturbance flowing the control end. In order to deal with the disturbance, the sliding model control (SMC) is integrated with the backstepping approach, where the disturbance is supposed to be bounded only. The existence and uniqueness of the solution for the closed-loop system are proved, and the monotonicity of the ‘‘reaching condition’’ is presented without differentiation of the sliding mode function, for which it may not always exist for the weak solution of the closed-loop system. Finally the numerical simulations validate the effectiveness of this method for the system with periodic and normal random disturbances respectively. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction The stabilization of partial differential equations (PDEs) subject to external disturbances has received a lot of attention in recent years as practically motivated. How to generalize the existing methods dealing with external disturbances in ordinary differential equations (ODEs) to the PDEs is quite challenging, but not impossible. The backstepping approach, which was originally used for PDEs in the ideal operational environment by Smyshlyaev and Krstic (2004, 2005), has been applied to the stabilization of wave equations with harmonic disturbances in either boundary input or observation respectively in Guo, Guo, and Shao (2011) and Guo and Guo (2013a,b,c). Due to its good performance in disturbance rejection and insensibility to uncertainties, the sliding mode control (SMC) has also been applied to some PDEs in Drakunov, Barbieri, and Silver (1996), Orlov and Utkin (1982, 1987) and Utkin (2008). Recently, the boundary SMC controllers are designed for one-dimensional heat, wave, Euler–Bernoulli, and Schrodinger equations with boundary input disturbance respectively in Cheng, Radisavljevic, and Su (2011), Guo and Jin (2013a,b),
✩ This work was supported by the National Natural Science Foundation of China (No. 61273130). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Miroslav Krstic under the direction of Editor Xiaobo Tan. E-mail addresses: [email protected] (J.-M. Wang), [email protected] (J.-J. Liu), [email protected] (B. Ren), [email protected] (J. Chen).
http://dx.doi.org/10.1016/j.automatica.2014.10.117 0005-1098/© 2014 Elsevier Ltd. All rights reserved.
Guo and Liu (2014) and Pisano, Orlov, and Usai (2011). The active disturbance rejection control (ADRC) method has been successfully applied to the attenuation of disturbance for one-dimensional anti-stable wave equation in Guo and Jin (2013a). Another powerful method in dealing with disturbances is based on Lyapunov function approach in Ge, Zhang, and He (2011), He, Ge, How, Choo, and Hong (2011) and He, Zhang, and Ge (2012). It is noted that only the PDEs are considered in the above works. To the best of our knowledge, there are few works that consider the stabilization of the PDE–ODE cascade systems subject to external disturbances. Such systems have taken place in many aspects such as electromagnetic coupling, mechanical coupling, and coupled chemical reactions. In this paper, we consider the stabilization of the heat PDE–ODE cascade system with Dirichlet and Neumann interconnection respectively:
˙ X (t ) = AX (t ) + Bu(0, t ), ut (x, t ) = uxx (x, t ), ux (0, t ) = 0, u(x, 0) = u0 (x),
t > 0, x ∈ (0, 1), t > 0, ux (1, t ) = U (t ) + d(t ), x ∈ (0, 1),
(1)
and
˙ X (t ) = AX (t ) + Bux (0, t ), ut (x, t ) = uxx (x, t ), u(0, t ) = 0, u(x, 0) = u0 (x),
t > 0, x ∈ (0, 1), t > 0 ux (1, t ) = U (t ) + d(t ), x ∈ (0, 1),
(2)
24
J.-M. Wang et al. / Automatica 52 (2015) 23–34
where K is chosen such that A + BK is Hurwitz and has real negative eigenvalues. The transformation (4) is invertible, that is x
u(x, t ) = w(x, t ) +
l(x, y)w(y, t )dy + ψ(x)X (t ), 0
where l(x, y) =
x−y
ψ(ξ )Bdξ , 0
Fig. 1. Block diagram of the coupled heat PDE–ODE system with Dirichlet interconnection (1).
where X (t ) ∈ Rn×1 and u(x, t ) ∈ R are the states of ODE and heat PDE respectively, U (t ) ∈ R is the control input to the entire system, A ∈ Rn×n , B ∈ Rn×1 , and d(t ) ∈ R is the external disturbance at the control end. The unknown disturbance d(t ) is supposed to be bounded, i.e., |d(t )| ≤ M for some M > 0 and all t ≥ 0. The objective of the paper is to design a control U (t ) which can stabilize system (1) and (2) in Rn × L2 (0, 1) with the matched disturbance d(t ) respectively. The rest of the paper is organized as follows. Section 2 is devoted to the disturbance rejection by SMC approach for system (1) with Dirichlet interconnection. The sliding mode control is designed and the existence and uniqueness of solution of the closed-loop system with Dirichlet interconnection are proved. The finite time ‘‘reaching condition’’ is presented. The Lyapunov method is used to show the closed-loop system in the sliding mode surface is exponentially stable. Section 3 is devoted to the disturbance rejection by SMC approach for system (2) with Neumann interconnection. Due to the presence of Neumann interconnection ux (0, t ) in system (2), the Riesz basis approach, instead of the Lyapunov method, is adopted to show the closed-loop system in the sliding mode surface is exponentially stable. The numerical simulations are presented in Section 4 for illustration of the effectiveness of this method. Some concluding remarks are presented in Section 5.
ψ(x) = [K 0]e
A + BK 0
0 I
x
I . 0
For [X , w] → [X , z ], we introduce X (t ) = X (t ),
(6) x
z (x, t ) = w(x, t ) −
k(x, y)w(y, t )dy,
(7)
0
where k(x, y) = −cx
I1
c (x2 − y2 )
0 ≤ y ≤ x ≤ 1,
c (x2 − y2 )
(8)
and I1 is the modified Bessel function and has the form I1 (x) =
∞
x 2n+1 .
2
n!(n + 1)! n=0 Then we obtain the target system
˙ X (t ) = (A + BK )X (t ) + Bz (0, t ), zt (x, t ) = zxx (x, t ) − cz (x, t ), zx (0, t ) = 0, 1 zx (1, t ) = U (t ) + d(t ) − qx (1, y)u(y, t )dy 0 1 − γ ′ (1)X (t ) − k(1, 1)w(1, t ) − kx (1, y)w(y, t )dy.
(9)
0
The inverse of the transformation (7) can be found as follows 2. SMC for the heat PDE–ODE system with Dirichlet interconnection
w(x, t ) = z (x, t ) +
x
p(x, y)z (y, t )dy,
(10)
0
In this section, we will integrate the backstepping approach and the sliding mode control method to design the control U (t ) such that system (1) (as depicted in Fig. 1) is stabilized.
where p(x, y) = −cx
J1
c (x2 − y2 )
c (x2 − y2 )
,
0 ≤ y ≤ x ≤ 1,
2.1. Design of sliding mode surface
and J1 is a Bessel function and has the form
We introduce a transformation [X , u] → [X , w] in the form (see Krstic, 2009):
J1 (x) =
X (t ) = X (t ),
(3)
w(x, t ) = u(x, t ) −
x
q(x, y)u(y, t )dy − γ (x)X (t ),
(4)
0
∞ (−1)n (x/2)1+2n n=0
n!(n + 1)!
Let us consider system (1) and (9) in the state space H = Rn × L2 (0, 1) with inner product given by
⟨[X , f ], [Y , g ]⟩ = X Y + ⊤
where
.
1
f (x)g (x)dx,
(11)
0
q(x, y) =
x −y
γ (σ )Bdσ ,
γ (x) = [K 0]e
0
0 I
A 0
where [X , f ], [Y , g ] ∈ H . Design the sliding mode surface Sz for system (9) as a closed-subspace of H :
x
I , 0
where I denotes an n×n identify matrix. This transformation brings system (1) into the following system:
˙ X (t ) = (A + BK )X (t ) + Bw(0, t ), wt (x, t ) = wxx (x, t ), w (0, t ) = 0, x wx (1, t ) = U (t ) + d(t ) − γ ′ (1)X (t ) 1 − q (1, y)u(y, t )dy, x
0
(5)
1
Sz = [X , f ] ∈ H
f (x)dx = 0 .
(12)
0
It is obvious that Sz is an infinite-dimensional manifold of H , on which the system (9) becomes
˙ X (t ) = (A + BK )X (t ) + Bz (0, t ), z (x, t ) = z (x, t ) − cz (x, t ), t xx 1 zx (0, t ) = z (x, t )dx = 0. 0
(13)
J.-M. Wang et al. / Automatica 52 (2015) 23–34
Theorem 1. The heat PDE–ODE system (13) on the sliding mode surface Sz is exponentially stable in Rn × L2 (0, 1), that is, there are positive constants C1 , C2 such that
|X (t )| + ∥z (t )∥ ≤ C1 e 2
2
−C 2 t
(|X0 | + ∥z0 ∥ ), 2
2
(14)
where | · | and ∥ · ∥ denote the norms in R and L (0, 1) respectively. n
2
Proof. We consider a Lyapunov function V (t ) = X (t )⊤ PX (t ) +
a 2
P (A + BK ) + (A + BK )T P = −Q , for some Q = Q T > 0, and the parameter a > 0 is the design parameter. By integrating from 0 to 1 in x at the two sides of the 1 second equation of system (13), it follows from 0 z (x, t )dx = 0 that zx (1, t ) = 0. Let λmin (Q ) be the minimum eigenvalue of matrix Q . By using Agmon’s inequality (see Krstic & Smyshlyaev, 2008), we have 5 2
Proposition 2. The original system (1) in the above sliding mode surface (15) becomes
˙ X (t ) = AX (t ) + Bu(0, t ), ut (x, t ) = uxx (x, t ), ux (0, t ) = 0, 1 x 1 q(x, y)u(y, t )dydx u ( x , t ) dx = 0 0 0
k(x, y)u(y, t )dydx γ (x)X (t )dx + + y 0 0 0 1 x q(y, τ )u(τ , t )dτ dydx k(x, y) + 0 0 0 x 1 + k(x, y)γ (y)X (t )dydx,
2
2|PB| z 2 (0, t ) λmin (Q ) − a∥zx (x, t )∥2 − ca∥z (x, t )∥2 5|PB|2 λmin (Q ) 2 |X | − ca − ∥z (x, t )∥2 ≤− 2 λmin (Q ) 6|PB|2 − a− ∥zx (x, t )∥2 . λmin (Q ) 2
which is exponentially stable in Rn × L2 (0, 1), by Theorem 1 and the equivalence between (13) and (16). 2.2. State feedback controller
S˙z (t ) =
1
zt (x, t )dx =
1
0
[zxx (x, t ) − cz (x, t )]dx 0
= zx (1, t ) − cSz (t ), and hence
|X |2 +
S˙z (t ) = U (t ) + d(t ) −
1
qx (1, y)u(y, t )dy − γ ′ (1)X (t ) 0
− k(1, 1)w(1, t ) −
1
kx (1, y)w(y, t )dy − cSz (t ). 0
Design the feedback controller: U (t ) = U0 (t ) +
Let a>
0
0
V˙ (t ) = −X ⊤ QX + 2X ⊤ PBz (0, t ) − a∥zx (x, t )∥2 − ca∥z (x, t )∥2
λmin (Q )
(16)
To facilitate the control design, we differentiate (15) with respect to t to obtain
∥z (x, t )∥2 + 3∥zx (x, t )∥2 .
Now by taking a derivative of the Lyapunov function along the solutions of system (13), we get
≤−
x
1
1
∥z ∥2 ,
where the matrix P = P T > 0 is the solution to the Lyapunov equation
z 2 (0, t ) ≤
25
1
qx (1, y)u(y, t )dy + γ ′ (1)X (t ) 0
6|PB|2
λmin (Q )
,
5|PB|2
ca >
λmin (Q )
.
+ k(1, 1)w(1, t ) +
S˙z (t ) = U0 (t ) + d(t ) − cSz (t ).
> 0.
Let U0 (t ) = −(M + η)sign(Sz (t )),
Transforming Sz by (7) and (4) into the original system (1), that is, Sz (t ) ,
1
z (x, t )dx 1
w(x, t )dx −
= 0 1
u(x, t )dx −
=
1
0
0 1
0
γ (x)X (t )dx − 0 x
k(x, y)
− 0
0 1
1
= −(M + η)sign(Sz (t ))Sz (t ) + d(t )Sz (t ) − c |Sz (t )|2 ≤ −η|Sz (t )|, η > 0.
x
k(x, y)u(y, t )dydx
q(y, τ )u(τ , t )dτ dydx 0
U (t ) =
k(x, y)γ (y)X (t )dydx
we have the following proposition.
1
qx (1, y)u(y, t )dy + γ ′ (1)X (t ) 0
0
= 0,
(20)
Note that (20) is just the well-known ‘‘reaching condition’’ for system (9). Substituting (18) into (17) results in the sliding mode controller as
0
y
(19)
Sz (t )S˙z (t )
q(x, y)u(y, t )dydx
where M > 0 is the upper bound of the external disturbance d(t ) satisfying |d(t )| ≤ M. Then
and
x
− 0
k(x, y)w(y, t )dydx
0 x
0 1
x
0
1
−
(18)
S˙z (t ) = −(M + η)sign(Sz (t )) + d(t ) − cSz (t ),
0
(17)
where U0 (t ) is an auxiliary control. Then we have
λmin (Q ) 10|PB|2 C2 = min , 2c − 2λmax (P ) aλmin (Q )
kx (1, y)w(y, t )dy, 0
Then we get V˙ (t ) ≤ −C2 V , where
Hence, we can obtain (14).
1
(15)
+ k(1, 1)w(1, t ) +
1
kx (1, y)w(y, t )dy 0
− (M + η)sign(Sz (t )).
(21)
26
J.-M. Wang et al. / Automatica 52 (2015) 23–34
Remark 3. Due to the use of sign function sign(Sz (t )), the control signal U (t ) (21) becomes discontinuous, which may excite unmodeled high-frequency system dynamics and cause the chattering phenomenon. To avoid the undesired chattering phenomenon, the sign function in U (t ) can be replaced with the following saturation function:
1x, , sat(x) = ϵ −1,
if x > ϵ
where j = 1, 2, . . . , n, m = 0, 1, 2, . . . , and
λ0j + c (1 − x) ∗ , z0j (x) = λ0j + c sinh λ0j + c ∗ z1m (x) = cos mπ x.
∗ B⊤ X0j cosh
(30)
1 Moreover {Zj0 (x), Zm (x), j = 1, 2, . . . , n, m = 0, 1, . . .} forms a
if |x| ≤ ϵ
Riesz basis for Rn × L2 (0, 1). Therefore for any [X (0), z (·, 0)] ∈ H and Sz (0) ̸= 0, there exists a tmax > 0 such that (22) admits a unique solution [X , z ] ∈ C (0, tmax ; H ) and Sz (t ) = 0 for all t > tmax .
if x < −ϵ,
where ϵ is a small positive constant. The closed-loop system of system (9) under the state feedback controller (21) is
Proof. First, we compute the eigenvalues and the corresponding eigenfunctions of A∗D . By A∗D Z = λZ , where λ ∈ σ (A∗D ) and Z = [X ∗ , z ∗ ] ∈ D(A∗D ), we have that X ∗ and z ∗ satisfy
˙ X (t ) = (A + BK )X (t ) + Bz (0, t ), zt (x, t ) = zxx (x, t ) − cz (x, t ), zx (0, t ) = 0, zx (1, t ) = −(M + η)sign(Sz (t )) + d(t ) , d˜ (t ).
(A + BK )⊤ X ∗ = λX ∗ , (z ∗ )′′ − cz ∗ = λz ∗ , ∗ ′ (z ) (0) = −B⊤ X ∗ ,
(22)
Now we are in a position to consider the existence and uniqueness of the solution to (22) and the finite time ‘‘reaching condition’’ to the sliding mode surface Sz . Write system (22) as d
(31)
(z ∗ )′ (1) = 0.
There are two cases: Case I: When X ∗ = 0, (31) becomes
(z ∗ )′′ − cz ∗ = λz ∗ , (z ∗ )′ (0) = (z ∗ )′ (1) = 0,
(32)
where Z (·, t ) = [X (t ), z (·, t )], BD = [0, δ(x − 1)], and AD is a linear operator defined in Rn × L2 (0, 1) by
∗ which has nontrivial solutions (λ1m , z1m (x)), where λ1m = −c − ∗ 2 2 m π , z1m (x) = cos mπ x, m = 0, 1, 2, . . .. Hence we get the 1 eigenvalues λ1m and the corresponding eigenfunction Zm (x) = ∗ ∗ [0, z1m (x)] of AD .
AD Z = (A + BK )X + Bf (0), f ′′ − cf ,
∗ Case II: When X ∗ ̸= 0, since (A + BK )⊤ X0j = λ0j X0j∗ , j = 1, 2, . . . , n,
dt
Z (·, t ) = AD Z (·, t ) + BD d(t ),
(23)
Z ∈ D(AD )
(24)
D(AD ) = {Z ∈ Rn × H 2 (0, 1)|f ′ (0) = f ′ (1) = 0}
∗ and, by (27), λ0j ̸= λ1m , m = 0, 1, 2, . . . , we have that z0j , given by (30), satisfies the following equation
where Z = [X , f ]. Moreover, we have A∗D , the adjoint operator of AD ,
∗ ′′ (z0j ) − cz0j∗ = λ0j z0j∗ ,
∗ ∗ ⊤ ′′ AD Z = (A + BK ) Y , g − cg , ∗ n 2 D(A∗D ) = Z ′ ∈ R ×⊤ H (0, 1), ′ g (0) + B Y = 0, g (1) = 0,
So λ0j , j = 1, 2, . . . , n is the eigenvalue of A∗D and has the corre-
(25)
(z0j∗ )′ (0) = −B⊤ X0j∗ ,
sponding eigenfunction Zj0 (x) given by (29).
where Z ∗ = [Y , g ] ∈ D(A∗D ), and the dual system of (23) is given by
1 Now we show that {Zj0 (x), Zm (x), j = 1, 2, . . . , n, m = 0, 1, . . .} forms a Riesz basis for H . Actually, by that the set of ∗ {X0j∗ , j = 1, 2, . . . , n} is a basis in Rn and the set of {z1m (x) =
cos mπ x, m ∈ N} forms an orthogonal basis in L2 (0, 1), we have ∗ {Fj0 = (X0j∗ , 0), j = 1, 2, . . . , n} ∪ {Fm1 (x) = (0, z1m (x)), m =
˙∗ X (t ) = (A + BK )⊤ X ∗ (t ), ∗ (x, t ) − cz ∗ (x, t ), zt∗ (x, t ) = zxx ⊤ ∗ ∗ zx∗ (0, t ) = − B X∗ , zx (1, t ) = 0, y(t ) = BD∗ ∗ X = z ∗ (1, t ). z (x, t )
(26)
0, 1, 2, . . .} forms a Riesz basis in H = Rn × L2 (0, 1). Moreover, by (29), we have n
∥Zj0 − Fj0 ∥2 +
j=1
Theorem 4. Suppose that |d(t )| 6 M for all t > 0, and Sz be defined by (12). Let A∗D be given by (25), let λ0j < −c , j = 1, 2, . . . , n be the simple real and negative eigenvalue of (A + BK )⊤ with the ∗ , and assume that corresponding eigenvector X0j
λ0j ∈ ̸ λ1m = −c − (mπ )2 , m = 0, 1, 2, . . . .
(27)
Then we have the eigenvalues of A∗D given by
{λ0j , λ1m = −c − (mπ )2 , j = 1, 2, . . . , n, m = 0, 1, 2, . . .},
(28)
and the corresponding eigenfunctions with respect to λ and λ given by respectively 0 j
∗ Zj0 (x) = X0j , z0j∗ (x) ,
1 ∗ Zm (x) = [0, z1m (x)],
(z0j∗ )′ (1) = 0.
1 m
(29)
n
∥Zm1 (x) − Fm1 (x)∥2
j=1
2 n B⊤ X ∗ cosh λ0j + c (1 − x) 0j < ∞, = 0 0 j =1 λj + c sinh λj + c 2
(33)
L
and from Bari’s theorem, { (x), (x), j = 1, 2, . . . , n, m = 0, 1, . . .} forms a Riesz basis for H . So A∗D generates a C0 -semigroup in H , and so does for AD . Therefore, for any Z0 = [X ∗ (0), z ∗ (·, 0)] ∈ H , assume that Zj0
1 Zm
n ∞ 0 1 Z = a Z ( x ) + bm Z m (x), 0 j j j =1
m=0
n ∞ 2 |aj |2 + |bm |2 . ∥Z 0 ∥ ≍ j =1
m=0
J.-M. Wang et al. / Automatica 52 (2015) 23–34
27
Then we get the solution of (26) given by Z (x, t ) = [X ∗ (t ), z ∗ (x, t )] n
=
aj e
λ0j t 0
Zj (x) +
j=1
∞
1 (x), bm eλm t Zm 1
m=0
where X ∗ (t ) =
n
aj e
λ0j t
∗ X0j ,
j=1
z ∗ ( x, t ) =
n
aj e
λ0j t ∗
z0j +
j=1
∞
Fig. 2. Block diagram for the coupled heat PDE–ODE system with Neumann interconnection (2).
bm eλm t cos mπ x. 1
m=0
which admits unique solution (X , u) ∈ C (0, tmax ; H ) and SU (t ) = 0 for all t > tmax . On the sliding mode surface SU (t ) = 0, the system (1) becomes
Hence y(t ) = z ∗ (1, t ) =
n
aj e
λ0j t ∗
z0j +
j =1
∞
(−1)m bm eλm t . 1
˙ X (t ) = AX (t ) + Bu(0, t ), ut (x, t ) = uxx (x, t ), u (0, t ) = 0, x 1 1 x w(x, t )dx − k(x, y)w(y, t )dydx = 0,
m=0
A direct computation shows that there exists T > 0, such that n
T
|y(t )|2 dt 6 CT1 0
|aj z0j∗ |2 + CT2
j=1
∞
|bm |2
0
m=0
6 DT ∥Z0 ∥ , 2
for some constants CT1 , CT2 , DT that depend on T only. This shows ∗
that BD∗ is admissible for eAD t and so is BD for eAD t . Therefore, for any T > 0 and [X (0), z (·, 0)] ∈ H , if Sz ∈ C [0, T ], Sz (t ) ̸= 0 for all t ∈ [0, T ), then there exists a unique solution [X , z ] ∈ C (0, T ; H ). Therefore, system (23) admits a unique solution. We may suppose without loss of generality that t ≥ 0 and Sz (0) = S0 > 0, because the proof for Sz (0) = S0 < 0 is similar. In this case, it follows from (19) that Sz (t ) = e−ct S0 −
t
e−c (t −τ ) (M + η − d(τ ))dτ
0
≤ e−ct S0 −
η c
+
η c
e−ct .
The above inequality shows that when Sz (0) > 0, there exists a unique continuous solution Sz to (19) in the maximal interval [0, tmax ), where it must have Sz (tmax ) = 0. It then follows from (20) that Sz (t ) must be decreasing in [0, tmax ) and Sz (t ) > 0 for all t ∈ [0, tmax ). Since Sz (t ) is continuous, the reaching condition (20) implies that Sz (t ) ≡ 0 for all t > tmax . The proof is complete. Returning back to the system (1) under the transformation (3)–(4), (6)–(7), and feedback control (21), we obtain the main result of this section. Theorem 5. Suppose that the disturbance d(t ) satisfies |d(t )| 6 M for all t > 0, and let SU be the sliding mode function given by SU (t ) =
1
w(x, t )dx − 0
1
0
x
k(x, y)w(y, t )dydx.
(34)
0
where w(x, t ) is given by (4). Then for any (X (0), u(·, 0)) ∈ H , SU (0) ̸= 0, there exists tmax > 0 such that the closed-loop system of (1) under the feedback control (21) is
X˙ (t ) = AX (t ) + Bu(0, t ), ut (x, t ) = uxx (x, t ), ux (0, t ) = 0 , x ux (1, t ) = qx (x, y)u(y, t )dy + γ ′ (1)X (t ) 0 1 + k ( 1 , 1 )w( 1 , t ) + kx (1, y)w(y, t )dy 0 − (M + η)sign(Sz (t )) + d(t ),
0
(36)
0
which is equivalent to (13) and hence is exponentially stable in H with the decay rate −c. It is remarked that system (22) is equivalent to system (35) under the equivalent transformation (3)–(4), and (6)–(7). 3. SMC for the heat–ODE system with Neumann interconnection In this section, we shall design the new control U (t ) for system (2) (as depicted in Fig. 2) to be stabilized. Different from the system (1), due to the presence of Neumann interconnection ux (0, t ) in system (2), the Riesz basis approach, instead of the Lyapunov method, is adopted to show the stability of the closed-loop system. 3.1. Design of sliding mode surface We introduce a transformation [X , u] → [X , w] in the form (see Krstic, 2009): X (t ) = X (t ),
w(x, t ) = u(x, t ) −
(37) x
m(x − y)u(y, t )dy − KM (x)X (t ),
(38)
0
where
m(x) = KM (x)B,
M (x) = [0 I ]e
0 I
A 0
x
I . 0
This transformation brings the system (2) into the following system:
˙ X (t ) = (A + BK )X (t ) + Bwx (0, t ), wt (x, t ) = wxx (x, t ), w(0, t ) = 0, w (1, t ) = U (t ) + d(t ) − KM ′ (1)X (t ) x 1 − m′ (1 − y)u(y, t )dy,
(39)
0
(35)
where K is chosen such that A + BK is Hurwitz. For simplicity, in this section, we have the following two hypotheses: Hypothesis. (1) Assume that A + BK has simple and real negative eigenvalue λ0j < −(9/4)π 2 , j = 1, 2, . . . , n, with
28
J.-M. Wang et al. / Automatica 52 (2015) 23–34
the corresponding eigenvector Xj0 , and assume that Xj∗ is the
eigenvector of (A + BK )⊤ with respect to λ0j , j = 1, 2, . . . , n. (2) Assume that λ0j ̸= −(m − 12 )2 π 2 , j = 1, 2, . . . , n, m ∈ N. The transformation (38) is invertible, that is u(x, t ) = w(x, t ) +
x
n(x − y)w(y, t )dy + KN (x)X (t ),
(40)
0
where n(ξ ) = KN (ξ )B,
N (ξ ) = [0 I ]e
w ∈ L2 (0, 1)
1
w(x) sin
π
0
2
1
w(x) sin 0
π 2
X˙ (t ) = (A + BK )X (t ) + Bwx (0, t ), wt (x, t ) = wxx (x, t ), 1 π w(0, t ) = w(x, t ) sin x dx = 0.
(41)
2
3.2. Stability analysis In this subsection, we consider the system (42). Define the system operator of (42) Aw : D(Aw )(⊂ Sw ) → Sw by
Y (t ) = Aw Y (t ), dt Y (0) = Y0 ,
t > 0,
(44)
where Y (t ) = [X (t ), w(·, t )].
π 2
x dx = 0
(49)
p
1
2
2
π ,
w (x) = sin m +
2
p m
1
2
π x,
(50)
m ∈ N. Moreover {wm (x), m ∈ N} forms an orthogonal basis for H. Proof. √ Since w ′′ = λw √ has the fundamental solution w(x) = c1 cosh λx + c√ λx, by w(0) = 0, we get c1 = 0 and 2 sinh 2 w(x) = π c2 sinh λx. Moreover, when λ = −π /4 and w(x) = c2 i sin 2 x , it follows from the third equality of (49) that c2 ≡ 0. Hence λ = −π 2 /4 is not an eigenvalue of (49). √ Let λ ̸= −π 2 /4. Substituting w(x) = c2 sinh λx into the third equality of (49), we get c2 cosh
√ λ 1+
√ λ π2 1 4 λ
= 0, p
which yields cosh λ = 0. Hence we get the eigenvalues {λm = −(m + 21 )2 π 2 , m ∈ N}, and the corresponding eigenfunction
wmp (x) = sin(m + 21 )π x. Moreover {wmp (x), m ∈ N} forms an orthogonal basis for H.
Theorem 8. Let Aw be given by (43) and assume that Hypothesis (1) and (2) are satisfied. Then Aw has the eigenvalues
1 Theorem 6. Let Aw be given by (43). Then A− w exists and is compact on Sw and hence σ (Aw ), the spectrum of Aw , consists of isolated eigenvalues of finitely algebraic multiplicity only.
Proof. For any given [X1 , w1 ] ∈ H , solve
Aw [X , w] = [(A + BK )X + Bw ′ (0), w ′′ ] = [X1 , w1 ].
w(x) sin
√
(43)
where Y = [X , f ], then (42) can be written as an evolution equation in H :
d
λ =− m+ p m
(42)
′ ′′ Aw Y = [(A + BK )nX + Bf2 (0), f ], Y ∈ D(Aw ), D(Aw ) = {Y ∈ R × H (0, 1) ∩ Sw |f (0) = 0},
1
0
x dx = 0 .
w(0) = p
On the sliding mode surface Sw , system (39) becomes
0
w ′′ (x) = λw(x),
has the eigenpairs (λm , wm (x)) given by
x dx = 0 .
Sw = Rn × H
= [X , w] ∈ H
(48)
2
Lemma 7. The eigenvalue problem
Design the sliding mode surface Sw for system (39):
′ w′′ (x) = λw(x), (A + BK )X + Bw (0) = λX , π 1 w(x) sin x dx = 0. w(0) =
A + BK ξ I 0 . 0
0 I
We also consider systems (2) and (39) in the state space H = Rn × L2 (0, 1) with inner product given by (11). Let H =
Now we are in a position to consider the eigenvalue problem of Aw . By Aw Y = λY , where Y = [X , w], we have
0
2003, p. 85). Therefore, σ (Aw ) consists of isolated eigenvalues of finite algebraic multiplicity only. The proof is complete.
(45)
{λ , j = 1, 2, . . . , n} ∪ λ = − m + 0 j
p m
1 2
2
π , m ∈ N , (51) 2
p
and the corresponding eigenfunctions with respect to λ0j and λm respectively Wj0 = [Xj0 , 0],
p Wmp (x) = [Xm0 , wm (x)], p m
(52) p m
where Xm0 = (m + )π [λ − (A + BK )]−1 B and w (x) = sin(m +
We get
(46)
2
0
with the solution
X = (A + BK )−1 (X1 − Bw ′ (0)), x w(x) = w ′ (0)x + (x − ξ )w1 (ξ )dξ , 0 π x π2 1 w′ (0) = − (x − ξ )w1 (ξ )dξ sin x dx. 4
0
0
)π x is given by (50). Moreover, {Wj0 , Wmp (x), j = 1, 2, . . . , n, m ∈ N} forms a Riesz basis for Sw . 1 2
′ ′′ (A + BK )X + Bw (0) = X1 , w (x) = w1 (x), π 1 x dx = 0, w(x) sin w(0) =
1 2
Proof. When w(x) = 0, (48) becomes (A + BK )X = λX . Since λ0j , j = 1, 2, . . . , n is the simple and real negative eigenvalue of A + BK with the corresponding eigenvector Xj0 , we have that Aw
has the eigenvalues λ0j , j = 1, 2, . . . , n and the corresponding (47)
2
1 Hence, we get the unique [X , w] ∈ D(Aw ). A− w exists and is compact on Sw by the Sobolev embedding theorem (Adams & Fournier,
eigenvector Wj0 = [Xj0 , 0]. When w(x) ̸= 0, by Lemma 7, the eigenvalue problem (49) has p p the eigenpairs (λm , wm (x)), m ∈ N, given by (50). Moreover, λ0j ̸=
λpm , m ∈ N, j = 1, 2, . . . , n, and (wmp )′ (0) = (m + 21 )π , for each (λpm , wmp (x)), the algebraic equation (A + BK )X +(m + 21 )π B = λpm X p has the unique solution Xm0 = (m + 12 )π[λm − (A + BK )]−1 B. So
J.-M. Wang et al. / Automatica 52 (2015) 23–34 p
λm , m ∈ N, is the eigenvalue of Aw and it has the corresponding p p eigenfunction Wm (x) = [Xm0 , wm (x)]. p 0 Now we show that {Wj , Wm (x), j = 1, 2, . . . , n, m ∈ N} forms p
a Riesz basis for Sw . Since {Xj0 , j = 1, 2, . . . , n} and {wm (x), m ∈ N} form the orthogonal bases for Rn and H respectively, we have that {Fj0 , Fmp (x), j = 1, 2, . . . , n, m ∈ N} forms an orthogonal basis for
Sw , where
Fj0
Xj0
= [
, 0] and
p Fm
29
3.3. State feedback controller Transforming Sw (41) by (38) into the original system (2), that is, Sw (t ) ,
1
w(x, t ) sin
π
(x) = [0, w (x)]. It follows from
1
u(x, t ) sin
=
(52) that
π
∥Wj0 − Fj0 ∥2 +
j =1
∞
∥Wmp (x) − Fmp (x)∥2 =
m=0
∞
∥Xm0 ∥2Rn ,
(53)
1
m+
−2
2
π
∥
Xm0 2Rn
∥
π 2
x dydx
π 2
x X (t )dx.
˙ X (t ) = AX (t ) + Bux (0, t ), u t (x, t ) = uxx (x, t ), u(0, t ) = 0, 1 π x dx u(x, t ) sin 2
⊤ 1 1 Υ = I − p (A + BK ) I − p (A + BK ) . λm λm
p
It follows from (51) that when m → ∞, λm → −∞. So there is a positive number N such that for m > N, we have
−1 1 =I +O , I − p (A + BK ) λm |λpm | 1
m > N,
0 1 x π = m ( x − y ) u ( y , t ) sin x dydx 2 0 0 π 1 x X (t )dx, + KM (x) sin
and as m → ∞,
which is exponentially stable by Theorem 9, and the equivalence between (42) and (55). To facilitate the control design, we differentiate (54) with respect to t to obtain
∥B∥2Rn 2 m + 21 π 2
1+O
1
p m
−
∥ +
∞
|λ |
∥
Wmp
(x) −
Fmp
By using Bari’s theorem, { , forms a Riesz basis for Sw .
p Wm
wxx (x, t ) sin
2
∞
= wx (1, t ) − ∥
Xm0 2Rn
∥
< ∞.
(x), j = 1, 2, . . . , n, m ∈ N}
π2 4
2
(1) Aw generates a C0 -semigroup eAw t on H . (2) The spectrum-determined growth condition holds true for eAw t , that is, ω(Aw ) = s(Aw ), where
x dx
π 2
x dx
Sw (t ),
and hence S˙w (t ) = U (t ) + d(t ) −
1
m′ (1 − y)u(y, t )dy 0
− KM (1)X (t ) −
π2
Sw (t ). 4 Design the feedback controller: ′
Theorem 9. Let Aw be given by (43) and assume that Hypothesis (1) and (2) are satisfied. Then we have the following assertions:
U (t ) =
1
m′ (1 − y)u(y, t )dy + KM ′ (1)X (t ) + U0 (t ),
S˙w (t ) = −(π 2 /4)Sw (t ) + U0 (t ) + d(t ).
t
is the growth bound of eAw t , and
Let
s(Aw ) = sup{Reλ|λ ∈ σ (Aw )}
U0 (t ) = −(M + η)sign(Sw (t )),
is the spectral bound of Aw . (3) The C0 -semigroup eAw t is exponentially stable in the sense
∥eAw t ∥ ≤ Mw e−(9/4)π t . 2
p Wm
Proof. By Theorem 8, we have that the eigenfunctions { , (x), j = 1, 2, . . . , n, m ∈ N} of Aw forms an orthogonal basis Wj0
for Sw . As a direct conclusion, by Theorem 6, Aw generates a C0 semigroup eAw t on H and the spectrum-determined growth condition ω(Aw ) = s(Aw ) holds true for eAw t . Moreover, by Hypothesis (1), we have sup{−(9/4)π 2 , λ0j , j = 1, 2, . . . , n} = −(9/4)π 2 , so there is a positive constant Mw > 0 such that
∥eAw t ∥ ≤ Mw e−(9/4)π
2t
∀ t ≥ 0.
(56)
0
where U0 is an auxiliary control. Then
∥eAw t ∥
t →∞
1
=
m=0
Wj0
ω(Aw ) = lim
π
0
(x)∥ =
m=0
j =1
wt (x, t ) sin
.
Therefore, this together with (53) yields Fj0 2
1
0
(55)
2
0
S˙w (t ) =
∥Xm0 ∥2Rn =
(54)
The original system (2) becomes
where
∥
KM (x) sin 0
−2
= |λpm |−2 B⊤ Υ −1 B,
Wj0
1
−
= B⊤ ([λpm − (A + BK )]−1 )⊤ [λpm − (A + BK )]−1 B
n
m(x − y)u(y, t ) sin
0
0
m=0
= ∥[λpm − (A + BK )]−1 B∥2Rn
x
−
where ∥ · ∥Rn denotes the norm of a vector in Rn and
1
x dx
2
0 n
x dx
2
0
p m
(57)
then S˙w (t ) = −
π2 4
Sw (t ) − (M + η)sign(Sw (t )) + d(t ).
(58)
Therefore, Sw (t )S˙w (t )
=−
π2 4
2 Sw (t ) − (M + η)sign(Sw (t ))Sw (t ) + d(t )Sw (t )
≤ −η|Sw (t )|,
η > 0.
(59)
Note that (59) is just the well-known ‘‘reaching condition’’ for system (39). By combining (56) with (57), the sliding mode
30
J.-M. Wang et al. / Automatica 52 (2015) 23–34
Proof. By A∗N Z = λZ , where λ ∈ σ (A∗N ) and Z = [X ∗ , z ∗ ] ∈ D(A∗N ), we have that X ∗ and z ∗ satisfy
controller is derived as U (t ) =
1
m′ (1 − y)u(y, t )dy + KM ′ (1)X (t )
0
− (M + η)sign(Sw (t )).
(60)
The closed-loop system of system (39) under the state feedback controller (60) is
There are two cases:
(z ∗ )′′ = λz ∗ ,
(61)
d W (·, t ) = AN W (·, t ) + BN d(t ), (62) dt where W (t ) = [X (t ), w(·, t )] and BN = [0, δ(x − 1)]. Define the system operator of (61) by: for W = [X , f ],
AN W = [(A + BK )X + Bf ′ (0), f ′′ ], W ∈ D(AN ) D(AN ) = {W ∈ Rn × H 2 (0, 1)|f (0) = f ′ (1) = 0}
p
λ = λ and X = Xj , the equation 0 j
(z ∗ )′′ = λ0j z ∗ ,
λ ,λ = − m −
2
(64)
has the solution zj0 (x) given by (68). Hence, we get the
(65)
(x) = Xj ,
zj0
( x) ,
2
[Xj∗ , 0] and Fmp (x) = [0, zmp (x)]. It then follows from (67) and (68) that n
∥Zj0 (x) − Fj0 ∥2 +
∞ m=0
2 n B⊤ X ∗ cosh λ0j (1 − x) j < ∞. = 0 j =1 cosh λj p
By using Bari’s Theorem again, {Zj0 , Zm (x), j = 1, 2, . . . , n, m ∈ N} forms a Riesz basis for H . So A∗N generates a C0 -semigroup ∗
eAN t on H , and so does for AN . Moreover, for any Z0∗ (·, 0) = [X ∗ (0), z ∗ (·, 0)] ∈ H , we assume that
π , j = 1, 2, . . . , n, m = 1, 2, . . . , 2
Z0∗ (x, 0) =
n
aj Zj0 (x) +
j =1
p Zm
(x) = [0,
∥Zmp (x) − Fmp (x)∥2
L2
p
∗
p
a Riesz basis for R × L (0, 1). It is known that {Fj0 , Fm (x), j = n
j=1
and the eigenfunctions corresponding to λ0j and λm are respectively
(z ∗ )′ (1) = 0
1, 2, . . . , n, m ∈ N}, forms an orthogonal basis for H with Fj0 =
(66)
Zj0
z ∗ (0) = B⊤ Xj∗ ,
p
Theorem 10. Suppose that |d(t )| 6 M for all t > 0, and Sw be defined by (41). Let A∗N be given by (64) and assume that Hypothesis (1) and (2) are satisfied. Then the eigenvalues of A∗N are
2
∗
∗
Now we show that {Zj0 , Zm (x), j = 1, 2, . . . , n, m ∈ N} forms
˙∗ X (t ) = (A + BK )⊤ X ∗ (t ), ∗ zt∗ (x, t ) = zxx (x, t ), ∗ ⊤ ∗ ∗ z (0, t ) = B X ,∗ zx (1, t ) = 0, y(t ) = BN∗ ∗ X = z ∗ (1, t ). z ( x, t )
1
p zm
(x)],
(67)
∥Z0∗ ∥2 ≍
n
∞
p bm Zm (x),
m=1
|aj |2 +
j =1
∞
|bm |2 .
m=1
Then we get the solution of (65): Z (x, t ) = [X ∗ (t ), z ∗ (x, t )]
where
λ0j (1 − x) zj0 (x) = , cosh λ0j 1 p zm (x) = sin m − π x, B⊤ Xj∗ cosh
=
n
aj e
λ0j t 0
Zj (x) +
j =1
(68)
2
p Zj0 Zm 2
∞
p
p bm eλm t Zm (x),
m=1
where X ∗ (t ) =
n
aj e
λ0j t
Xj∗ ,
, (x), j = 1, 2, . . . , n, m ∈ N} forms a Riesz basis for R × L (0, 1). Then for any W (·, 0) = [X (0), w(·, 0)] ∈ H , Sw (0) ̸= 0, there exists tmax > 0 such that (61) admits a unique solution W (·, t ) = [X (t ), w(·, t )] ∈ C (t0 , tmax ; H ) and Sw (t ) = 0 for all
z ∗ (x, t ) =
t > tmax .
Hence, it follows from (65) and (68) that
and { n
p
eigenpairs (λ0j , Zj0 (x)), j = 1, 2, . . . , n, given by (66) and (67).
We claim that AN generates a C0 -semigroup eAN t on H . To this ∗ purpose, it suffices to show that A∗N generates a C0 -semigroup eAN t on H . The dual system of (62) is hence given by
p
(2), we have that λ0j ̸= λm , j = 1, 2, . . . , n, m ∈ N. Then for
(63)
∗ ∗ [(A + BK )⊤ Y , g ′′ ], W ∗ ∈ D(A∗N ) AN W = W ∗ ∈ Rn × H 2 (0, 1), D(A∗N ) = g (0) = B⊤ Y , g ′ (1) = 0.
p m
p
which has solutions (λm , zm (x)), m ∈ N, where λm , zm (x) are p given by (66), (68) respectively, and {zm (x), m ∈ N} forms 2 an orthogonal basis in L (0, 1). Hence, we get the eigenpairs (λpm , Zmp (x)), m ∈ N, given by (66) and (67). (2) When X ∗ ̸= 0, by Hypothesis (1), (A+BK )⊤ has the eigenvalues λ0j , j = 1, 2, . . . , and the related eigenvector Xj∗ . By Hypothesis
which has the adjoint operator A∗N : for W ∗ = [Y , g ]
0 j
z ∗ (0) = (z ∗ )′ (1) = 0 p
The next result confirms the existence and uniqueness of the solution to (61) and the finite time ‘‘reaching condition’’ to the sliding mode surface Sw . Write system (61) as
(69)
(1) When X ∗ = 0, the eigenvalue problem (69) becomes
˙ X (t ) = (A + BK )X (t ) + Bwx (0, t ), wt (x, t ) = wxx (x, t ), w(0, t ) = 0, wx (1, t ) = −(M + η)sign(Sw (t )) + d(t ) , d˜ (t ).
(A + BK )⊤ X ∗ = λX ∗ , (z ∗ )′′ = λz ∗ , ∗ ⊤ ∗ ′ z (0) = B X , (z ) (1) = 0.
j =1 n j =1
aj e
λ0j t 0
zj (x) +
∞
p
p bm eλm t zm (x).
m=1
J.-M. Wang et al. / Automatica 52 (2015) 23–34
(a) The ODE state X (t ).
31
(b) The heat PDE state u(x, t ).
(c) The external disturbance d(t ). Fig. 3. Open loop response with Dirichlet interconnection (1) in the presence of the sinusoidal boundary matched disturbance.
zj0 (1) = B⊤ Xj∗ / cosh
1
p zm (1) = sin m −
Theorem 11. Suppose that |d(t )| 6 M for all t > 0, and let SU be the sliding mode function given by
λ0j ,
2
π = (−1)m+1 , SU (t ) =
and
u(x, t ) sin 0
y(t ) = z (1, t ) = ∗
n
aj e
λ0j t 0
zj (1) +
j =1
∞
(−1)
m+1
p
bm e
λm t
T
|y(t )|2 dt 6 CT1
0
.
n
|aj |2 + CT2
∞
|bm |2
m=1
for some constants CT1 , CT2 , DT that depend on T only. This shows A∗ t
that BN∗ is admissible for e N and so is BN for eAN t . Therefore, for any T > 0 and W (·, 0) = [X (0), w(·, 0)] ∈ H , if Sw ∈ C [0, T ], Sw (t ) ̸= 0 for all t ∈ [0, T ), then there exists a unique solution W (·, t ) = [X (t ), w(·, t )] ∈ C (0, T ; H ). Therefore, system (62) admits a unique solution. We may suppose without loss of generality that t ≥ 0 and Sw (0) = S0 > 0, since the proof for Sw (0) = S0 < 0 is similar. In this case, it follows from (58) that t
S0 −
2 e−(π /4)(t −τ ) (M + η − d(τ ))dτ
0
≤e
−(π 2 /4)t
2
4η 2 S0 − 2 1 − e−(π /4)t . π
The above inequality shows that when Sw (0) > 0, there exists a unique continuous solution Sw to (58) in the maximal interval [0, tmax ), where it must have Sw (tmax ) = 0. It then follows from (59) that Sw (t ) must be decreasing in [0, tmax ) and Sw (t ) > 0 for all t ∈ [0, tmax ). Since Sw (t ) is continuous, the reaching condition (59) implies that Sw (t ) ≡ 0 for all t > tmax . Returning back to the system (2) under the transformation (37) and feedback control (60), we obtain the main result of this section from Theorem 10.
π
0 1
KM (x) sin
−
x dx
m(x − y)u(y, t ) sin
2
x dydx
π
0
6 DT ∥Z ∗ (·, 0)∥2
Sw (t ) = e
0
π
x
−
m=1
j =1
−(π 2 /4)t
1
Through a direct computation, there exists T > 0, such that
1
2
x X (t )dx.
(70)
Then for any [X (0), u(·, 0)] ∈ H , SU (0) ̸= 0, there exists tmax > 0 such that the closed-loop system of (2) under the feedback control (60) is
˙ X (t ) = AX (t ) + Bux (0, t ), ut (x, t ) = uxx (x, t ), u(0, t ) = 0, 1 ( 1 , t ) = m′ (1 − y)u(y, t )dy + KM ′ (1)X (t ) u x 0 − (M + η)sign(Sw (t )) + d(t ),
(71)
which admits a unique solution [X , u] ∈ C (0, tmax ; H ) and SU (t ) = 0 for all t > tmax . On the sliding mode surface SU (t ) = 0, the system (2) becomes
˙ X (t ) = AX (t ) + Bux (0, t ), u t (x, t ) = uxx (x, t ), u(0, t ) = 0, 1 π x dx u(x, t ) sin 2
0 1 x π = m ( x − y ) u ( y , t ) sin x dydx 2 0 0 π 1 + KM (x) sin x X (t )dx, 0
(72)
2
which is equivalent to (42) and hence is exponentially stable in H with the decay rate −π 2 /4.
32
J.-M. Wang et al. / Automatica 52 (2015) 23–34
(a) The ODE state X (t ).
(b) The heat PDE state u(x, t ).
(c) The input signal U (t ) using the sign function.
(d) The input signal U (t ) using the saturation function.
Fig. 4. Closed-loop response with Dirichlet interconnection (1) in the presence of the sinusoidal boundary matched disturbance.
(a) The ODE state X (t ).
(c) The external disturbance d(t ).
(b) The heat PDE state u(x, t ).
(d) The input signal U (t ) using the sign function.
(e) The input signal U (t ) using the saturation function. Fig. 5. Closed-loop response with Neumann interconnection (2) in the presence of the normal distribution random boundary matched disturbance.
J.-M. Wang et al. / Automatica 52 (2015) 23–34
It is remarked that system (61) is equivalent to system (71) under the equivalent transformation (38) and (40). 4. Simulation studies In this section, we will show the effectiveness of the proposed controller for the heat PDE–ODE cascade system with both Dirichlet interconnection (1) and Neumann interconnection (2). Figs. 3 and 4 show the open loop and closed loop results respectively for the Dirichlet interconnection case with A = 2, B = 1, d(t ) = cos(5t ). The initial conditions are chosen as X (0) = 1, u(x, 0) = x (0 < x ≤ 1) and ux (0, 0) = 0. It can be seen that without control both the states of ODE and heat PDE are not stabilized in Fig. 3. After the proposed sliding mode controller (21) is applied with the design parameters are chosen as K = −50, c = 0.1, M = 1, η = 25, both the ODE and heat PDE are stabilized to zero when they reach the sliding surface Sz after some time as shown in Fig. 4. According to Slotine and Li (1991), starting from any initial condition, the state trajectory reaches the sliding mode surface Sz (t ) = 0 in a finite time smaller than Sz (0)/η, i.e., treach ≤ tmax = Sz (0)/η = 3.3 s. It is noted that the chattering in the control signal U (t ) as Fig. 4(c), caused by the sign function, could be mitigated by replacing the sign function with the saturation function as shown in Fig. 4(d). The closed loop simulation results with the Neumann interconnection are shown in Fig. 5 with the normal distribution random disturbance considered. Other parameters and initial conditions are chosen as A = 2, B = 1, X (0) = 1, u(x, 0) = x (0 < x ≤ 1) and u(0, 0) = 0. It is shown that the good control performance is achieved by adopting the proposed sliding mode controller (60) with K = −50, M = 1, η = 25 in the presence of an external normal distribution random disturbance. For the Neumann interconnection case, the reach time is calculated as treach ≤ tmax = Sw (0)/η = 1.27 s. 5. Conclusion In this paper, the sliding mode control was designed to achieve the stabilization of the heat PDE–ODE cascade systems subject to boundary control matched disturbances. Both Dirichlet boundary interconnection and Neumann boundary interconnection were considered. The existence and uniqueness of the solution for the closed-loop via SMC have been proved. The effectiveness of the proposed method was validated with numerical simulations. Acknowledgments The authors would like to thank the anonymous reviewers for providing valuable comments and suggestions to improve the paper. References Adams, R. A., & Fournier, J. F. (2003). Pure and applied mathematics (amsterdam): vol. 140. Sobolev spaces (2nd ed.). Amsterdam: Elsevier/Academic Press. Cheng, M. B., Radisavljevic, V., & Su, W. C. (2011). Sliding mode boundary control of a parabolic pde system with parameter variations and boundary uncertainties. Automatica, 47(2), 381–387. Drakunov, S., Barbieri, E., & Silver, D. A. (1996). Sliding mode control of a heat equation with application to arc welding. In Proc. IEEE int. conf. control appl (pp. 668–672). Ge, S. S., Zhang, S., & He, W. (2011). Vibration control of an Euler–Bernoulli beam under unknown spatiotemporally varying disturbance. International Journal of Control, 84, 947–960. Guo, W., & Guo, B. Z. (2013a). Adaptive output feedback stabilization for one-dimensional wave equation with corrupted observation by harmonic disturbance. SIAM Journal on Control and Optimization, 51, 1679–1706.
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Guo, W., & Guo, B. Z. (2013b). Parameter estimation and non-collocated adaptive stabilization for a wave equation subject to general boundary harmonic disturbance. IEEE Transactions on Automatic Control, 58, 1631–1643. Guo, W., & Guo, B. Z. (2013c). Stabilization and regulator design for a onedimensional unstable wave equation with input harmonic disturbance. International Journal of Robust and Nonlinear Control, 23, 514–533. Guo, W., Guo, B. Z., & Shao, Z. C. (2011). Parameter estimation and stabilization for a wave equation with boundary output harmonic disturbance and noncollocated control. International Journal of Robust and Nonlinear Control, 21, 1297–1321. Guo, B., & Jin, F. (2013a). Sliding mode and active disturbance rejection control to the stabilization of anti-stable one-dimensional wave equation subject to boundary input disturbance. IEEE Transactions on Automatic Control, 58, 1269–1274. Guo, B., & Jin, F. (2013b). The active disturbance rejection and sliding mode control approach to the stabilization of Euler–Bernoulli beam equation with boundary input disturbance. Automatica, 49, 2911–2918. Guo, B., & Liu, J. (2014). 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, 24, 2194–2212. He, W., Ge, S. S., How, B. V. E., Choo, Y. S., & Hong, K. S. (2011). Robust adaptive boundary control of a flexible marine riser with vessel dynamics. Automatica, 47, 722–732. He, W., Zhang, S., & Ge, S. S. (2012). Boundary output- feedback stabilization of a Timoshenko beam using disturbance observer. IEEE Transactions on Industrial Electronics, 99, 1–9. Krstic, M. (2009). Compensating actuator and sensor dynamics governed by diffusion PDEs. Systems & Control Letters, 58(5), 372–377. Krstic, M., & Smyshlyaev, A. (2008). Boundary control of PDEs. a course on backstepping designs. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). Orlov, Y. V., & Utkin, V. I. (1982). Use of sliding modes in distributed system control problems. Automation and Remote Control, 43(9), 1127–1135. Orlov, Y. V., & Utkin, V. I. (1987). Sliding mode control in infinite-dimensional systmes. Automatica, 23(6), 753–757. Pisano, A., Orlov, Y., & Usai, E. (2011). Tracking control of the uncertain heat and wave equation via power-fractional and sliding-mode techniques. SIAM Journal on Control and Optimization, 49(2), 363–382. Slotine, J.-J. E., & Li, W. (1991). Applied nonlinear control. Englewood Cliffs, NJ: Prentice-Hall. Smyshlyaev, A., & Krstic, M. (2004). Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations. IEEE Transactions on Automatic Control, 49(12), 2185–2202. Smyshlyaev, A., & Krstic, M. (2005). Backstepping observers for a class of parabolic PDEs. Systems & Control Letters, 54, 613–625. Utkin, V. I. (2008). Sliding mode control: Mathematical tools, design and applications. In Ser. Lecture notes in math.: vol. 1932. Nonlinear and optimal control theory (pp. 289–347).
Jun-Min Wang received his B.S. degree in Mathematics from Shanxi University in 1995, and M.S. degree in Applied Mathematics from Bering Institute of Technology in 1998, respectively, and Ph.D. degree in Applied Mathematics from the University of Hong Kong in 2004. From 1998 to 2000, he worked as a teaching assistant at the Beijing Institute of Technology. He was a postdoctoral fellow at the University of the Witwatersrand, South Africa from 2004 to 2006. From 2006 to 2009 he was an associate professor at the Beijing Institute of Technology. Since 2009, he has been with Beijing Institute of Technology as a full professor in Applied Mathematics. His research interests focus on the control theory of distributed parameter systems. Dr. Wang is the recipient for New Century Excellent Talents Program from Ministry of Higher Education of China (2007).
Jun-Jun Liu received his B.S. degree in Mathematics at Changzhi University, China, in 2008, M.S. degree from Xinjiang University in 2011. He is currently a Ph.D. candidate at Beijing Institute of Technology. His research interests focus on distributed parameter systems control.
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Beibei Ren received the B. Eng. degree in the Mechanical Electronic Engineering and the M. Eng. degree in Automation from Xidian University, Xi’an, China, in 2001 and in 2004, respectively, and the Ph.D. degree in the Electrical and Computer Engineering from the National University of Singapore, Singapore, in 2010. From 2010 to 2013, she was a postdoctoral scholar in the Department of Mechanical Aerospace Engineering, University of California, San Diego, CA, USA. Since 2013, she has been with the Department of Mechanical Engineering, Texas Tech University, Lubbock, TX, USA, as an Assistant Professor. Her current research interests include adaptive control, robust control, distributed parameter systems, extremum seeking and their applications.
Jinhao Chen received his B. Eng. degrees from School of Information Science and Technology (SIST), Sun YatSen University (SYSU) in 2012. He is currently a Ph.D. student in the Department of Mechanical Engineering, Texas Tech University, Lubbock, TX, USA. His research interests include robust control, neural network, machine learning and their applications.
Contents lists available at ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Sliding mode control to stabilization of cascaded heat PDE–ODE systems subject to boundary control matched disturbance✩ Jun-Min Wang a , Jun-Jun Liu a , Beibei Ren b , Jinhao Chen b a
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
b
Department of Mechanical Engineering, Texas Tech University, Lubbock, TX 79409, USA
article
info
Article history: Received 28 January 2014 Received in revised form 30 June 2014 Accepted 15 October 2014
Keywords: Boundary control Disturbance rejection Sliding mode control Cascade systems Backstepping
abstract In this paper, we are concerned with the boundary feedback stabilization of a cascade of heat PDE–ODE system with Dirichlet/Neumann interconnection and with the external disturbance flowing the control end. In order to deal with the disturbance, the sliding model control (SMC) is integrated with the backstepping approach, where the disturbance is supposed to be bounded only. The existence and uniqueness of the solution for the closed-loop system are proved, and the monotonicity of the ‘‘reaching condition’’ is presented without differentiation of the sliding mode function, for which it may not always exist for the weak solution of the closed-loop system. Finally the numerical simulations validate the effectiveness of this method for the system with periodic and normal random disturbances respectively. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction The stabilization of partial differential equations (PDEs) subject to external disturbances has received a lot of attention in recent years as practically motivated. How to generalize the existing methods dealing with external disturbances in ordinary differential equations (ODEs) to the PDEs is quite challenging, but not impossible. The backstepping approach, which was originally used for PDEs in the ideal operational environment by Smyshlyaev and Krstic (2004, 2005), has been applied to the stabilization of wave equations with harmonic disturbances in either boundary input or observation respectively in Guo, Guo, and Shao (2011) and Guo and Guo (2013a,b,c). Due to its good performance in disturbance rejection and insensibility to uncertainties, the sliding mode control (SMC) has also been applied to some PDEs in Drakunov, Barbieri, and Silver (1996), Orlov and Utkin (1982, 1987) and Utkin (2008). Recently, the boundary SMC controllers are designed for one-dimensional heat, wave, Euler–Bernoulli, and Schrodinger equations with boundary input disturbance respectively in Cheng, Radisavljevic, and Su (2011), Guo and Jin (2013a,b),
✩ This work was supported by the National Natural Science Foundation of China (No. 61273130). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Miroslav Krstic under the direction of Editor Xiaobo Tan. E-mail addresses: [email protected] (J.-M. Wang), [email protected] (J.-J. Liu), [email protected] (B. Ren), [email protected] (J. Chen).
http://dx.doi.org/10.1016/j.automatica.2014.10.117 0005-1098/© 2014 Elsevier Ltd. All rights reserved.
Guo and Liu (2014) and Pisano, Orlov, and Usai (2011). The active disturbance rejection control (ADRC) method has been successfully applied to the attenuation of disturbance for one-dimensional anti-stable wave equation in Guo and Jin (2013a). Another powerful method in dealing with disturbances is based on Lyapunov function approach in Ge, Zhang, and He (2011), He, Ge, How, Choo, and Hong (2011) and He, Zhang, and Ge (2012). It is noted that only the PDEs are considered in the above works. To the best of our knowledge, there are few works that consider the stabilization of the PDE–ODE cascade systems subject to external disturbances. Such systems have taken place in many aspects such as electromagnetic coupling, mechanical coupling, and coupled chemical reactions. In this paper, we consider the stabilization of the heat PDE–ODE cascade system with Dirichlet and Neumann interconnection respectively:
˙ X (t ) = AX (t ) + Bu(0, t ), ut (x, t ) = uxx (x, t ), ux (0, t ) = 0, u(x, 0) = u0 (x),
t > 0, x ∈ (0, 1), t > 0, ux (1, t ) = U (t ) + d(t ), x ∈ (0, 1),
(1)
and
˙ X (t ) = AX (t ) + Bux (0, t ), ut (x, t ) = uxx (x, t ), u(0, t ) = 0, u(x, 0) = u0 (x),
t > 0, x ∈ (0, 1), t > 0 ux (1, t ) = U (t ) + d(t ), x ∈ (0, 1),
(2)
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J.-M. Wang et al. / Automatica 52 (2015) 23–34
where K is chosen such that A + BK is Hurwitz and has real negative eigenvalues. The transformation (4) is invertible, that is x
u(x, t ) = w(x, t ) +
l(x, y)w(y, t )dy + ψ(x)X (t ), 0
where l(x, y) =
x−y
ψ(ξ )Bdξ , 0
Fig. 1. Block diagram of the coupled heat PDE–ODE system with Dirichlet interconnection (1).
where X (t ) ∈ Rn×1 and u(x, t ) ∈ R are the states of ODE and heat PDE respectively, U (t ) ∈ R is the control input to the entire system, A ∈ Rn×n , B ∈ Rn×1 , and d(t ) ∈ R is the external disturbance at the control end. The unknown disturbance d(t ) is supposed to be bounded, i.e., |d(t )| ≤ M for some M > 0 and all t ≥ 0. The objective of the paper is to design a control U (t ) which can stabilize system (1) and (2) in Rn × L2 (0, 1) with the matched disturbance d(t ) respectively. The rest of the paper is organized as follows. Section 2 is devoted to the disturbance rejection by SMC approach for system (1) with Dirichlet interconnection. The sliding mode control is designed and the existence and uniqueness of solution of the closed-loop system with Dirichlet interconnection are proved. The finite time ‘‘reaching condition’’ is presented. The Lyapunov method is used to show the closed-loop system in the sliding mode surface is exponentially stable. Section 3 is devoted to the disturbance rejection by SMC approach for system (2) with Neumann interconnection. Due to the presence of Neumann interconnection ux (0, t ) in system (2), the Riesz basis approach, instead of the Lyapunov method, is adopted to show the closed-loop system in the sliding mode surface is exponentially stable. The numerical simulations are presented in Section 4 for illustration of the effectiveness of this method. Some concluding remarks are presented in Section 5.
ψ(x) = [K 0]e
A + BK 0
0 I
x
I . 0
For [X , w] → [X , z ], we introduce X (t ) = X (t ),
(6) x
z (x, t ) = w(x, t ) −
k(x, y)w(y, t )dy,
(7)
0
where k(x, y) = −cx
I1
c (x2 − y2 )
0 ≤ y ≤ x ≤ 1,
c (x2 − y2 )
(8)
and I1 is the modified Bessel function and has the form I1 (x) =
∞
x 2n+1 .
2
n!(n + 1)! n=0 Then we obtain the target system
˙ X (t ) = (A + BK )X (t ) + Bz (0, t ), zt (x, t ) = zxx (x, t ) − cz (x, t ), zx (0, t ) = 0, 1 zx (1, t ) = U (t ) + d(t ) − qx (1, y)u(y, t )dy 0 1 − γ ′ (1)X (t ) − k(1, 1)w(1, t ) − kx (1, y)w(y, t )dy.
(9)
0
The inverse of the transformation (7) can be found as follows 2. SMC for the heat PDE–ODE system with Dirichlet interconnection
w(x, t ) = z (x, t ) +
x
p(x, y)z (y, t )dy,
(10)
0
In this section, we will integrate the backstepping approach and the sliding mode control method to design the control U (t ) such that system (1) (as depicted in Fig. 1) is stabilized.
where p(x, y) = −cx
J1
c (x2 − y2 )
c (x2 − y2 )
,
0 ≤ y ≤ x ≤ 1,
2.1. Design of sliding mode surface
and J1 is a Bessel function and has the form
We introduce a transformation [X , u] → [X , w] in the form (see Krstic, 2009):
J1 (x) =
X (t ) = X (t ),
(3)
w(x, t ) = u(x, t ) −
x
q(x, y)u(y, t )dy − γ (x)X (t ),
(4)
0
∞ (−1)n (x/2)1+2n n=0
n!(n + 1)!
Let us consider system (1) and (9) in the state space H = Rn × L2 (0, 1) with inner product given by
⟨[X , f ], [Y , g ]⟩ = X Y + ⊤
where
.
1
f (x)g (x)dx,
(11)
0
q(x, y) =
x −y
γ (σ )Bdσ ,
γ (x) = [K 0]e
0
0 I
A 0
where [X , f ], [Y , g ] ∈ H . Design the sliding mode surface Sz for system (9) as a closed-subspace of H :
x
I , 0
where I denotes an n×n identify matrix. This transformation brings system (1) into the following system:
˙ X (t ) = (A + BK )X (t ) + Bw(0, t ), wt (x, t ) = wxx (x, t ), w (0, t ) = 0, x wx (1, t ) = U (t ) + d(t ) − γ ′ (1)X (t ) 1 − q (1, y)u(y, t )dy, x
0
(5)
1
Sz = [X , f ] ∈ H
f (x)dx = 0 .
(12)
0
It is obvious that Sz is an infinite-dimensional manifold of H , on which the system (9) becomes
˙ X (t ) = (A + BK )X (t ) + Bz (0, t ), z (x, t ) = z (x, t ) − cz (x, t ), t xx 1 zx (0, t ) = z (x, t )dx = 0. 0
(13)
J.-M. Wang et al. / Automatica 52 (2015) 23–34
Theorem 1. The heat PDE–ODE system (13) on the sliding mode surface Sz is exponentially stable in Rn × L2 (0, 1), that is, there are positive constants C1 , C2 such that
|X (t )| + ∥z (t )∥ ≤ C1 e 2
2
−C 2 t
(|X0 | + ∥z0 ∥ ), 2
2
(14)
where | · | and ∥ · ∥ denote the norms in R and L (0, 1) respectively. n
2
Proof. We consider a Lyapunov function V (t ) = X (t )⊤ PX (t ) +
a 2
P (A + BK ) + (A + BK )T P = −Q , for some Q = Q T > 0, and the parameter a > 0 is the design parameter. By integrating from 0 to 1 in x at the two sides of the 1 second equation of system (13), it follows from 0 z (x, t )dx = 0 that zx (1, t ) = 0. Let λmin (Q ) be the minimum eigenvalue of matrix Q . By using Agmon’s inequality (see Krstic & Smyshlyaev, 2008), we have 5 2
Proposition 2. The original system (1) in the above sliding mode surface (15) becomes
˙ X (t ) = AX (t ) + Bu(0, t ), ut (x, t ) = uxx (x, t ), ux (0, t ) = 0, 1 x 1 q(x, y)u(y, t )dydx u ( x , t ) dx = 0 0 0
k(x, y)u(y, t )dydx γ (x)X (t )dx + + y 0 0 0 1 x q(y, τ )u(τ , t )dτ dydx k(x, y) + 0 0 0 x 1 + k(x, y)γ (y)X (t )dydx,
2
2|PB| z 2 (0, t ) λmin (Q ) − a∥zx (x, t )∥2 − ca∥z (x, t )∥2 5|PB|2 λmin (Q ) 2 |X | − ca − ∥z (x, t )∥2 ≤− 2 λmin (Q ) 6|PB|2 − a− ∥zx (x, t )∥2 . λmin (Q ) 2
which is exponentially stable in Rn × L2 (0, 1), by Theorem 1 and the equivalence between (13) and (16). 2.2. State feedback controller
S˙z (t ) =
1
zt (x, t )dx =
1
0
[zxx (x, t ) − cz (x, t )]dx 0
= zx (1, t ) − cSz (t ), and hence
|X |2 +
S˙z (t ) = U (t ) + d(t ) −
1
qx (1, y)u(y, t )dy − γ ′ (1)X (t ) 0
− k(1, 1)w(1, t ) −
1
kx (1, y)w(y, t )dy − cSz (t ). 0
Design the feedback controller: U (t ) = U0 (t ) +
Let a>
0
0
V˙ (t ) = −X ⊤ QX + 2X ⊤ PBz (0, t ) − a∥zx (x, t )∥2 − ca∥z (x, t )∥2
λmin (Q )
(16)
To facilitate the control design, we differentiate (15) with respect to t to obtain
∥z (x, t )∥2 + 3∥zx (x, t )∥2 .
Now by taking a derivative of the Lyapunov function along the solutions of system (13), we get
≤−
x
1
1
∥z ∥2 ,
where the matrix P = P T > 0 is the solution to the Lyapunov equation
z 2 (0, t ) ≤
25
1
qx (1, y)u(y, t )dy + γ ′ (1)X (t ) 0
6|PB|2
λmin (Q )
,
5|PB|2
ca >
λmin (Q )
.
+ k(1, 1)w(1, t ) +
S˙z (t ) = U0 (t ) + d(t ) − cSz (t ).
> 0.
Let U0 (t ) = −(M + η)sign(Sz (t )),
Transforming Sz by (7) and (4) into the original system (1), that is, Sz (t ) ,
1
z (x, t )dx 1
w(x, t )dx −
= 0 1
u(x, t )dx −
=
1
0
0 1
0
γ (x)X (t )dx − 0 x
k(x, y)
− 0
0 1
1
= −(M + η)sign(Sz (t ))Sz (t ) + d(t )Sz (t ) − c |Sz (t )|2 ≤ −η|Sz (t )|, η > 0.
x
k(x, y)u(y, t )dydx
q(y, τ )u(τ , t )dτ dydx 0
U (t ) =
k(x, y)γ (y)X (t )dydx
we have the following proposition.
1
qx (1, y)u(y, t )dy + γ ′ (1)X (t ) 0
0
= 0,
(20)
Note that (20) is just the well-known ‘‘reaching condition’’ for system (9). Substituting (18) into (17) results in the sliding mode controller as
0
y
(19)
Sz (t )S˙z (t )
q(x, y)u(y, t )dydx
where M > 0 is the upper bound of the external disturbance d(t ) satisfying |d(t )| ≤ M. Then
and
x
− 0
k(x, y)w(y, t )dydx
0 x
0 1
x
0
1
−
(18)
S˙z (t ) = −(M + η)sign(Sz (t )) + d(t ) − cSz (t ),
0
(17)
where U0 (t ) is an auxiliary control. Then we have
λmin (Q ) 10|PB|2 C2 = min , 2c − 2λmax (P ) aλmin (Q )
kx (1, y)w(y, t )dy, 0
Then we get V˙ (t ) ≤ −C2 V , where
Hence, we can obtain (14).
1
(15)
+ k(1, 1)w(1, t ) +
1
kx (1, y)w(y, t )dy 0
− (M + η)sign(Sz (t )).
(21)
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J.-M. Wang et al. / Automatica 52 (2015) 23–34
Remark 3. Due to the use of sign function sign(Sz (t )), the control signal U (t ) (21) becomes discontinuous, which may excite unmodeled high-frequency system dynamics and cause the chattering phenomenon. To avoid the undesired chattering phenomenon, the sign function in U (t ) can be replaced with the following saturation function:
1x, , sat(x) = ϵ −1,
if x > ϵ
where j = 1, 2, . . . , n, m = 0, 1, 2, . . . , and
λ0j + c (1 − x) ∗ , z0j (x) = λ0j + c sinh λ0j + c ∗ z1m (x) = cos mπ x.
∗ B⊤ X0j cosh
(30)
1 Moreover {Zj0 (x), Zm (x), j = 1, 2, . . . , n, m = 0, 1, . . .} forms a
if |x| ≤ ϵ
Riesz basis for Rn × L2 (0, 1). Therefore for any [X (0), z (·, 0)] ∈ H and Sz (0) ̸= 0, there exists a tmax > 0 such that (22) admits a unique solution [X , z ] ∈ C (0, tmax ; H ) and Sz (t ) = 0 for all t > tmax .
if x < −ϵ,
where ϵ is a small positive constant. The closed-loop system of system (9) under the state feedback controller (21) is
Proof. First, we compute the eigenvalues and the corresponding eigenfunctions of A∗D . By A∗D Z = λZ , where λ ∈ σ (A∗D ) and Z = [X ∗ , z ∗ ] ∈ D(A∗D ), we have that X ∗ and z ∗ satisfy
˙ X (t ) = (A + BK )X (t ) + Bz (0, t ), zt (x, t ) = zxx (x, t ) − cz (x, t ), zx (0, t ) = 0, zx (1, t ) = −(M + η)sign(Sz (t )) + d(t ) , d˜ (t ).
(A + BK )⊤ X ∗ = λX ∗ , (z ∗ )′′ − cz ∗ = λz ∗ , ∗ ′ (z ) (0) = −B⊤ X ∗ ,
(22)
Now we are in a position to consider the existence and uniqueness of the solution to (22) and the finite time ‘‘reaching condition’’ to the sliding mode surface Sz . Write system (22) as d
(31)
(z ∗ )′ (1) = 0.
There are two cases: Case I: When X ∗ = 0, (31) becomes
(z ∗ )′′ − cz ∗ = λz ∗ , (z ∗ )′ (0) = (z ∗ )′ (1) = 0,
(32)
where Z (·, t ) = [X (t ), z (·, t )], BD = [0, δ(x − 1)], and AD is a linear operator defined in Rn × L2 (0, 1) by
∗ which has nontrivial solutions (λ1m , z1m (x)), where λ1m = −c − ∗ 2 2 m π , z1m (x) = cos mπ x, m = 0, 1, 2, . . .. Hence we get the 1 eigenvalues λ1m and the corresponding eigenfunction Zm (x) = ∗ ∗ [0, z1m (x)] of AD .
AD Z = (A + BK )X + Bf (0), f ′′ − cf ,
∗ Case II: When X ∗ ̸= 0, since (A + BK )⊤ X0j = λ0j X0j∗ , j = 1, 2, . . . , n,
dt
Z (·, t ) = AD Z (·, t ) + BD d(t ),
(23)
Z ∈ D(AD )
(24)
D(AD ) = {Z ∈ Rn × H 2 (0, 1)|f ′ (0) = f ′ (1) = 0}
∗ and, by (27), λ0j ̸= λ1m , m = 0, 1, 2, . . . , we have that z0j , given by (30), satisfies the following equation
where Z = [X , f ]. Moreover, we have A∗D , the adjoint operator of AD ,
∗ ′′ (z0j ) − cz0j∗ = λ0j z0j∗ ,
∗ ∗ ⊤ ′′ AD Z = (A + BK ) Y , g − cg , ∗ n 2 D(A∗D ) = Z ′ ∈ R ×⊤ H (0, 1), ′ g (0) + B Y = 0, g (1) = 0,
So λ0j , j = 1, 2, . . . , n is the eigenvalue of A∗D and has the corre-
(25)
(z0j∗ )′ (0) = −B⊤ X0j∗ ,
sponding eigenfunction Zj0 (x) given by (29).
where Z ∗ = [Y , g ] ∈ D(A∗D ), and the dual system of (23) is given by
1 Now we show that {Zj0 (x), Zm (x), j = 1, 2, . . . , n, m = 0, 1, . . .} forms a Riesz basis for H . Actually, by that the set of ∗ {X0j∗ , j = 1, 2, . . . , n} is a basis in Rn and the set of {z1m (x) =
cos mπ x, m ∈ N} forms an orthogonal basis in L2 (0, 1), we have ∗ {Fj0 = (X0j∗ , 0), j = 1, 2, . . . , n} ∪ {Fm1 (x) = (0, z1m (x)), m =
˙∗ X (t ) = (A + BK )⊤ X ∗ (t ), ∗ (x, t ) − cz ∗ (x, t ), zt∗ (x, t ) = zxx ⊤ ∗ ∗ zx∗ (0, t ) = − B X∗ , zx (1, t ) = 0, y(t ) = BD∗ ∗ X = z ∗ (1, t ). z (x, t )
(26)
0, 1, 2, . . .} forms a Riesz basis in H = Rn × L2 (0, 1). Moreover, by (29), we have n
∥Zj0 − Fj0 ∥2 +
j=1
Theorem 4. Suppose that |d(t )| 6 M for all t > 0, and Sz be defined by (12). Let A∗D be given by (25), let λ0j < −c , j = 1, 2, . . . , n be the simple real and negative eigenvalue of (A + BK )⊤ with the ∗ , and assume that corresponding eigenvector X0j
λ0j ∈ ̸ λ1m = −c − (mπ )2 , m = 0, 1, 2, . . . .
(27)
Then we have the eigenvalues of A∗D given by
{λ0j , λ1m = −c − (mπ )2 , j = 1, 2, . . . , n, m = 0, 1, 2, . . .},
(28)
and the corresponding eigenfunctions with respect to λ and λ given by respectively 0 j
∗ Zj0 (x) = X0j , z0j∗ (x) ,
1 ∗ Zm (x) = [0, z1m (x)],
(z0j∗ )′ (1) = 0.
1 m
(29)
n
∥Zm1 (x) − Fm1 (x)∥2
j=1
2 n B⊤ X ∗ cosh λ0j + c (1 − x) 0j < ∞, = 0 0 j =1 λj + c sinh λj + c 2
(33)
L
and from Bari’s theorem, { (x), (x), j = 1, 2, . . . , n, m = 0, 1, . . .} forms a Riesz basis for H . So A∗D generates a C0 -semigroup in H , and so does for AD . Therefore, for any Z0 = [X ∗ (0), z ∗ (·, 0)] ∈ H , assume that Zj0
1 Zm
n ∞ 0 1 Z = a Z ( x ) + bm Z m (x), 0 j j j =1
m=0
n ∞ 2 |aj |2 + |bm |2 . ∥Z 0 ∥ ≍ j =1
m=0
J.-M. Wang et al. / Automatica 52 (2015) 23–34
27
Then we get the solution of (26) given by Z (x, t ) = [X ∗ (t ), z ∗ (x, t )] n
=
aj e
λ0j t 0
Zj (x) +
j=1
∞
1 (x), bm eλm t Zm 1
m=0
where X ∗ (t ) =
n
aj e
λ0j t
∗ X0j ,
j=1
z ∗ ( x, t ) =
n
aj e
λ0j t ∗
z0j +
j=1
∞
Fig. 2. Block diagram for the coupled heat PDE–ODE system with Neumann interconnection (2).
bm eλm t cos mπ x. 1
m=0
which admits unique solution (X , u) ∈ C (0, tmax ; H ) and SU (t ) = 0 for all t > tmax . On the sliding mode surface SU (t ) = 0, the system (1) becomes
Hence y(t ) = z ∗ (1, t ) =
n
aj e
λ0j t ∗
z0j +
j =1
∞
(−1)m bm eλm t . 1
˙ X (t ) = AX (t ) + Bu(0, t ), ut (x, t ) = uxx (x, t ), u (0, t ) = 0, x 1 1 x w(x, t )dx − k(x, y)w(y, t )dydx = 0,
m=0
A direct computation shows that there exists T > 0, such that n
T
|y(t )|2 dt 6 CT1 0
|aj z0j∗ |2 + CT2
j=1
∞
|bm |2
0
m=0
6 DT ∥Z0 ∥ , 2
for some constants CT1 , CT2 , DT that depend on T only. This shows ∗
that BD∗ is admissible for eAD t and so is BD for eAD t . Therefore, for any T > 0 and [X (0), z (·, 0)] ∈ H , if Sz ∈ C [0, T ], Sz (t ) ̸= 0 for all t ∈ [0, T ), then there exists a unique solution [X , z ] ∈ C (0, T ; H ). Therefore, system (23) admits a unique solution. We may suppose without loss of generality that t ≥ 0 and Sz (0) = S0 > 0, because the proof for Sz (0) = S0 < 0 is similar. In this case, it follows from (19) that Sz (t ) = e−ct S0 −
t
e−c (t −τ ) (M + η − d(τ ))dτ
0
≤ e−ct S0 −
η c
+
η c
e−ct .
The above inequality shows that when Sz (0) > 0, there exists a unique continuous solution Sz to (19) in the maximal interval [0, tmax ), where it must have Sz (tmax ) = 0. It then follows from (20) that Sz (t ) must be decreasing in [0, tmax ) and Sz (t ) > 0 for all t ∈ [0, tmax ). Since Sz (t ) is continuous, the reaching condition (20) implies that Sz (t ) ≡ 0 for all t > tmax . The proof is complete. Returning back to the system (1) under the transformation (3)–(4), (6)–(7), and feedback control (21), we obtain the main result of this section. Theorem 5. Suppose that the disturbance d(t ) satisfies |d(t )| 6 M for all t > 0, and let SU be the sliding mode function given by SU (t ) =
1
w(x, t )dx − 0
1
0
x
k(x, y)w(y, t )dydx.
(34)
0
where w(x, t ) is given by (4). Then for any (X (0), u(·, 0)) ∈ H , SU (0) ̸= 0, there exists tmax > 0 such that the closed-loop system of (1) under the feedback control (21) is
X˙ (t ) = AX (t ) + Bu(0, t ), ut (x, t ) = uxx (x, t ), ux (0, t ) = 0 , x ux (1, t ) = qx (x, y)u(y, t )dy + γ ′ (1)X (t ) 0 1 + k ( 1 , 1 )w( 1 , t ) + kx (1, y)w(y, t )dy 0 − (M + η)sign(Sz (t )) + d(t ),
0
(36)
0
which is equivalent to (13) and hence is exponentially stable in H with the decay rate −c. It is remarked that system (22) is equivalent to system (35) under the equivalent transformation (3)–(4), and (6)–(7). 3. SMC for the heat–ODE system with Neumann interconnection In this section, we shall design the new control U (t ) for system (2) (as depicted in Fig. 2) to be stabilized. Different from the system (1), due to the presence of Neumann interconnection ux (0, t ) in system (2), the Riesz basis approach, instead of the Lyapunov method, is adopted to show the stability of the closed-loop system. 3.1. Design of sliding mode surface We introduce a transformation [X , u] → [X , w] in the form (see Krstic, 2009): X (t ) = X (t ),
w(x, t ) = u(x, t ) −
(37) x
m(x − y)u(y, t )dy − KM (x)X (t ),
(38)
0
where
m(x) = KM (x)B,
M (x) = [0 I ]e
0 I
A 0
x
I . 0
This transformation brings the system (2) into the following system:
˙ X (t ) = (A + BK )X (t ) + Bwx (0, t ), wt (x, t ) = wxx (x, t ), w(0, t ) = 0, w (1, t ) = U (t ) + d(t ) − KM ′ (1)X (t ) x 1 − m′ (1 − y)u(y, t )dy,
(39)
0
(35)
where K is chosen such that A + BK is Hurwitz. For simplicity, in this section, we have the following two hypotheses: Hypothesis. (1) Assume that A + BK has simple and real negative eigenvalue λ0j < −(9/4)π 2 , j = 1, 2, . . . , n, with
28
J.-M. Wang et al. / Automatica 52 (2015) 23–34
the corresponding eigenvector Xj0 , and assume that Xj∗ is the
eigenvector of (A + BK )⊤ with respect to λ0j , j = 1, 2, . . . , n. (2) Assume that λ0j ̸= −(m − 12 )2 π 2 , j = 1, 2, . . . , n, m ∈ N. The transformation (38) is invertible, that is u(x, t ) = w(x, t ) +
x
n(x − y)w(y, t )dy + KN (x)X (t ),
(40)
0
where n(ξ ) = KN (ξ )B,
N (ξ ) = [0 I ]e
w ∈ L2 (0, 1)
1
w(x) sin
π
0
2
1
w(x) sin 0
π 2
X˙ (t ) = (A + BK )X (t ) + Bwx (0, t ), wt (x, t ) = wxx (x, t ), 1 π w(0, t ) = w(x, t ) sin x dx = 0.
(41)
2
3.2. Stability analysis In this subsection, we consider the system (42). Define the system operator of (42) Aw : D(Aw )(⊂ Sw ) → Sw by
Y (t ) = Aw Y (t ), dt Y (0) = Y0 ,
t > 0,
(44)
where Y (t ) = [X (t ), w(·, t )].
π 2
x dx = 0
(49)
p
1
2
2
π ,
w (x) = sin m +
2
p m
1
2
π x,
(50)
m ∈ N. Moreover {wm (x), m ∈ N} forms an orthogonal basis for H. Proof. √ Since w ′′ = λw √ has the fundamental solution w(x) = c1 cosh λx + c√ λx, by w(0) = 0, we get c1 = 0 and 2 sinh 2 w(x) = π c2 sinh λx. Moreover, when λ = −π /4 and w(x) = c2 i sin 2 x , it follows from the third equality of (49) that c2 ≡ 0. Hence λ = −π 2 /4 is not an eigenvalue of (49). √ Let λ ̸= −π 2 /4. Substituting w(x) = c2 sinh λx into the third equality of (49), we get c2 cosh
√ λ 1+
√ λ π2 1 4 λ
= 0, p
which yields cosh λ = 0. Hence we get the eigenvalues {λm = −(m + 21 )2 π 2 , m ∈ N}, and the corresponding eigenfunction
wmp (x) = sin(m + 21 )π x. Moreover {wmp (x), m ∈ N} forms an orthogonal basis for H.
Theorem 8. Let Aw be given by (43) and assume that Hypothesis (1) and (2) are satisfied. Then Aw has the eigenvalues
1 Theorem 6. Let Aw be given by (43). Then A− w exists and is compact on Sw and hence σ (Aw ), the spectrum of Aw , consists of isolated eigenvalues of finitely algebraic multiplicity only.
Proof. For any given [X1 , w1 ] ∈ H , solve
Aw [X , w] = [(A + BK )X + Bw ′ (0), w ′′ ] = [X1 , w1 ].
w(x) sin
√
(43)
where Y = [X , f ], then (42) can be written as an evolution equation in H :
d
λ =− m+ p m
(42)
′ ′′ Aw Y = [(A + BK )nX + Bf2 (0), f ], Y ∈ D(Aw ), D(Aw ) = {Y ∈ R × H (0, 1) ∩ Sw |f (0) = 0},
1
0
x dx = 0 .
w(0) = p
On the sliding mode surface Sw , system (39) becomes
0
w ′′ (x) = λw(x),
has the eigenpairs (λm , wm (x)) given by
x dx = 0 .
Sw = Rn × H
= [X , w] ∈ H
(48)
2
Lemma 7. The eigenvalue problem
Design the sliding mode surface Sw for system (39):
′ w′′ (x) = λw(x), (A + BK )X + Bw (0) = λX , π 1 w(x) sin x dx = 0. w(0) =
A + BK ξ I 0 . 0
0 I
We also consider systems (2) and (39) in the state space H = Rn × L2 (0, 1) with inner product given by (11). Let H =
Now we are in a position to consider the eigenvalue problem of Aw . By Aw Y = λY , where Y = [X , w], we have
0
2003, p. 85). Therefore, σ (Aw ) consists of isolated eigenvalues of finite algebraic multiplicity only. The proof is complete.
(45)
{λ , j = 1, 2, . . . , n} ∪ λ = − m + 0 j
p m
1 2
2
π , m ∈ N , (51) 2
p
and the corresponding eigenfunctions with respect to λ0j and λm respectively Wj0 = [Xj0 , 0],
p Wmp (x) = [Xm0 , wm (x)], p m
(52) p m
where Xm0 = (m + )π [λ − (A + BK )]−1 B and w (x) = sin(m +
We get
(46)
2
0
with the solution
X = (A + BK )−1 (X1 − Bw ′ (0)), x w(x) = w ′ (0)x + (x − ξ )w1 (ξ )dξ , 0 π x π2 1 w′ (0) = − (x − ξ )w1 (ξ )dξ sin x dx. 4
0
0
)π x is given by (50). Moreover, {Wj0 , Wmp (x), j = 1, 2, . . . , n, m ∈ N} forms a Riesz basis for Sw . 1 2
′ ′′ (A + BK )X + Bw (0) = X1 , w (x) = w1 (x), π 1 x dx = 0, w(x) sin w(0) =
1 2
Proof. When w(x) = 0, (48) becomes (A + BK )X = λX . Since λ0j , j = 1, 2, . . . , n is the simple and real negative eigenvalue of A + BK with the corresponding eigenvector Xj0 , we have that Aw
has the eigenvalues λ0j , j = 1, 2, . . . , n and the corresponding (47)
2
1 Hence, we get the unique [X , w] ∈ D(Aw ). A− w exists and is compact on Sw by the Sobolev embedding theorem (Adams & Fournier,
eigenvector Wj0 = [Xj0 , 0]. When w(x) ̸= 0, by Lemma 7, the eigenvalue problem (49) has p p the eigenpairs (λm , wm (x)), m ∈ N, given by (50). Moreover, λ0j ̸=
λpm , m ∈ N, j = 1, 2, . . . , n, and (wmp )′ (0) = (m + 21 )π , for each (λpm , wmp (x)), the algebraic equation (A + BK )X +(m + 21 )π B = λpm X p has the unique solution Xm0 = (m + 12 )π[λm − (A + BK )]−1 B. So
J.-M. Wang et al. / Automatica 52 (2015) 23–34 p
λm , m ∈ N, is the eigenvalue of Aw and it has the corresponding p p eigenfunction Wm (x) = [Xm0 , wm (x)]. p 0 Now we show that {Wj , Wm (x), j = 1, 2, . . . , n, m ∈ N} forms p
a Riesz basis for Sw . Since {Xj0 , j = 1, 2, . . . , n} and {wm (x), m ∈ N} form the orthogonal bases for Rn and H respectively, we have that {Fj0 , Fmp (x), j = 1, 2, . . . , n, m ∈ N} forms an orthogonal basis for
Sw , where
Fj0
Xj0
= [
, 0] and
p Fm
29
3.3. State feedback controller Transforming Sw (41) by (38) into the original system (2), that is, Sw (t ) ,
1
w(x, t ) sin
π
(x) = [0, w (x)]. It follows from
1
u(x, t ) sin
=
(52) that
π
∥Wj0 − Fj0 ∥2 +
j =1
∞
∥Wmp (x) − Fmp (x)∥2 =
m=0
∞
∥Xm0 ∥2Rn ,
(53)
1
m+
−2
2
π
∥
Xm0 2Rn
∥
π 2
x dydx
π 2
x X (t )dx.
˙ X (t ) = AX (t ) + Bux (0, t ), u t (x, t ) = uxx (x, t ), u(0, t ) = 0, 1 π x dx u(x, t ) sin 2
⊤ 1 1 Υ = I − p (A + BK ) I − p (A + BK ) . λm λm
p
It follows from (51) that when m → ∞, λm → −∞. So there is a positive number N such that for m > N, we have
−1 1 =I +O , I − p (A + BK ) λm |λpm | 1
m > N,
0 1 x π = m ( x − y ) u ( y , t ) sin x dydx 2 0 0 π 1 x X (t )dx, + KM (x) sin
and as m → ∞,
which is exponentially stable by Theorem 9, and the equivalence between (42) and (55). To facilitate the control design, we differentiate (54) with respect to t to obtain
∥B∥2Rn 2 m + 21 π 2
1+O
1
p m
−
∥ +
∞
|λ |
∥
Wmp
(x) −
Fmp
By using Bari’s theorem, { , forms a Riesz basis for Sw .
p Wm
wxx (x, t ) sin
2
∞
= wx (1, t ) − ∥
Xm0 2Rn
∥
< ∞.
(x), j = 1, 2, . . . , n, m ∈ N}
π2 4
2
(1) Aw generates a C0 -semigroup eAw t on H . (2) The spectrum-determined growth condition holds true for eAw t , that is, ω(Aw ) = s(Aw ), where
x dx
π 2
x dx
Sw (t ),
and hence S˙w (t ) = U (t ) + d(t ) −
1
m′ (1 − y)u(y, t )dy 0
− KM (1)X (t ) −
π2
Sw (t ). 4 Design the feedback controller: ′
Theorem 9. Let Aw be given by (43) and assume that Hypothesis (1) and (2) are satisfied. Then we have the following assertions:
U (t ) =
1
m′ (1 − y)u(y, t )dy + KM ′ (1)X (t ) + U0 (t ),
S˙w (t ) = −(π 2 /4)Sw (t ) + U0 (t ) + d(t ).
t
is the growth bound of eAw t , and
Let
s(Aw ) = sup{Reλ|λ ∈ σ (Aw )}
U0 (t ) = −(M + η)sign(Sw (t )),
is the spectral bound of Aw . (3) The C0 -semigroup eAw t is exponentially stable in the sense
∥eAw t ∥ ≤ Mw e−(9/4)π t . 2
p Wm
Proof. By Theorem 8, we have that the eigenfunctions { , (x), j = 1, 2, . . . , n, m ∈ N} of Aw forms an orthogonal basis Wj0
for Sw . As a direct conclusion, by Theorem 6, Aw generates a C0 semigroup eAw t on H and the spectrum-determined growth condition ω(Aw ) = s(Aw ) holds true for eAw t . Moreover, by Hypothesis (1), we have sup{−(9/4)π 2 , λ0j , j = 1, 2, . . . , n} = −(9/4)π 2 , so there is a positive constant Mw > 0 such that
∥eAw t ∥ ≤ Mw e−(9/4)π
2t
∀ t ≥ 0.
(56)
0
where U0 is an auxiliary control. Then
∥eAw t ∥
t →∞
1
=
m=0
Wj0
ω(Aw ) = lim
π
0
(x)∥ =
m=0
j =1
wt (x, t ) sin
.
Therefore, this together with (53) yields Fj0 2
1
0
(55)
2
0
S˙w (t ) =
∥Xm0 ∥2Rn =
(54)
The original system (2) becomes
where
∥
KM (x) sin 0
−2
= |λpm |−2 B⊤ Υ −1 B,
Wj0
1
−
= B⊤ ([λpm − (A + BK )]−1 )⊤ [λpm − (A + BK )]−1 B
n
m(x − y)u(y, t ) sin
0
0
m=0
= ∥[λpm − (A + BK )]−1 B∥2Rn
x
−
where ∥ · ∥Rn denotes the norm of a vector in Rn and
1
x dx
2
0 n
x dx
2
0
p m
(57)
then S˙w (t ) = −
π2 4
Sw (t ) − (M + η)sign(Sw (t )) + d(t ).
(58)
Therefore, Sw (t )S˙w (t )
=−
π2 4
2 Sw (t ) − (M + η)sign(Sw (t ))Sw (t ) + d(t )Sw (t )
≤ −η|Sw (t )|,
η > 0.
(59)
Note that (59) is just the well-known ‘‘reaching condition’’ for system (39). By combining (56) with (57), the sliding mode
30
J.-M. Wang et al. / Automatica 52 (2015) 23–34
Proof. By A∗N Z = λZ , where λ ∈ σ (A∗N ) and Z = [X ∗ , z ∗ ] ∈ D(A∗N ), we have that X ∗ and z ∗ satisfy
controller is derived as U (t ) =
1
m′ (1 − y)u(y, t )dy + KM ′ (1)X (t )
0
− (M + η)sign(Sw (t )).
(60)
The closed-loop system of system (39) under the state feedback controller (60) is
There are two cases:
(z ∗ )′′ = λz ∗ ,
(61)
d W (·, t ) = AN W (·, t ) + BN d(t ), (62) dt where W (t ) = [X (t ), w(·, t )] and BN = [0, δ(x − 1)]. Define the system operator of (61) by: for W = [X , f ],
AN W = [(A + BK )X + Bf ′ (0), f ′′ ], W ∈ D(AN ) D(AN ) = {W ∈ Rn × H 2 (0, 1)|f (0) = f ′ (1) = 0}
p
λ = λ and X = Xj , the equation 0 j
(z ∗ )′′ = λ0j z ∗ ,
λ ,λ = − m −
2
(64)
has the solution zj0 (x) given by (68). Hence, we get the
(65)
(x) = Xj ,
zj0
( x) ,
2
[Xj∗ , 0] and Fmp (x) = [0, zmp (x)]. It then follows from (67) and (68) that n
∥Zj0 (x) − Fj0 ∥2 +
∞ m=0
2 n B⊤ X ∗ cosh λ0j (1 − x) j < ∞. = 0 j =1 cosh λj p
By using Bari’s Theorem again, {Zj0 , Zm (x), j = 1, 2, . . . , n, m ∈ N} forms a Riesz basis for H . So A∗N generates a C0 -semigroup ∗
eAN t on H , and so does for AN . Moreover, for any Z0∗ (·, 0) = [X ∗ (0), z ∗ (·, 0)] ∈ H , we assume that
π , j = 1, 2, . . . , n, m = 1, 2, . . . , 2
Z0∗ (x, 0) =
n
aj Zj0 (x) +
j =1
p Zm
(x) = [0,
∥Zmp (x) − Fmp (x)∥2
L2
p
∗
p
a Riesz basis for R × L (0, 1). It is known that {Fj0 , Fm (x), j = n
j=1
and the eigenfunctions corresponding to λ0j and λm are respectively
(z ∗ )′ (1) = 0
1, 2, . . . , n, m ∈ N}, forms an orthogonal basis for H with Fj0 =
(66)
Zj0
z ∗ (0) = B⊤ Xj∗ ,
p
Theorem 10. Suppose that |d(t )| 6 M for all t > 0, and Sw be defined by (41). Let A∗N be given by (64) and assume that Hypothesis (1) and (2) are satisfied. Then the eigenvalues of A∗N are
2
∗
∗
Now we show that {Zj0 , Zm (x), j = 1, 2, . . . , n, m ∈ N} forms
˙∗ X (t ) = (A + BK )⊤ X ∗ (t ), ∗ zt∗ (x, t ) = zxx (x, t ), ∗ ⊤ ∗ ∗ z (0, t ) = B X ,∗ zx (1, t ) = 0, y(t ) = BN∗ ∗ X = z ∗ (1, t ). z ( x, t )
1
p zm
(x)],
(67)
∥Z0∗ ∥2 ≍
n
∞
p bm Zm (x),
m=1
|aj |2 +
j =1
∞
|bm |2 .
m=1
Then we get the solution of (65): Z (x, t ) = [X ∗ (t ), z ∗ (x, t )]
where
λ0j (1 − x) zj0 (x) = , cosh λ0j 1 p zm (x) = sin m − π x, B⊤ Xj∗ cosh
=
n
aj e
λ0j t 0
Zj (x) +
j =1
(68)
2
p Zj0 Zm 2
∞
p
p bm eλm t Zm (x),
m=1
where X ∗ (t ) =
n
aj e
λ0j t
Xj∗ ,
, (x), j = 1, 2, . . . , n, m ∈ N} forms a Riesz basis for R × L (0, 1). Then for any W (·, 0) = [X (0), w(·, 0)] ∈ H , Sw (0) ̸= 0, there exists tmax > 0 such that (61) admits a unique solution W (·, t ) = [X (t ), w(·, t )] ∈ C (t0 , tmax ; H ) and Sw (t ) = 0 for all
z ∗ (x, t ) =
t > tmax .
Hence, it follows from (65) and (68) that
and { n
p
eigenpairs (λ0j , Zj0 (x)), j = 1, 2, . . . , n, given by (66) and (67).
We claim that AN generates a C0 -semigroup eAN t on H . To this ∗ purpose, it suffices to show that A∗N generates a C0 -semigroup eAN t on H . The dual system of (62) is hence given by
p
(2), we have that λ0j ̸= λm , j = 1, 2, . . . , n, m ∈ N. Then for
(63)
∗ ∗ [(A + BK )⊤ Y , g ′′ ], W ∗ ∈ D(A∗N ) AN W = W ∗ ∈ Rn × H 2 (0, 1), D(A∗N ) = g (0) = B⊤ Y , g ′ (1) = 0.
p m
p
which has solutions (λm , zm (x)), m ∈ N, where λm , zm (x) are p given by (66), (68) respectively, and {zm (x), m ∈ N} forms 2 an orthogonal basis in L (0, 1). Hence, we get the eigenpairs (λpm , Zmp (x)), m ∈ N, given by (66) and (67). (2) When X ∗ ̸= 0, by Hypothesis (1), (A+BK )⊤ has the eigenvalues λ0j , j = 1, 2, . . . , and the related eigenvector Xj∗ . By Hypothesis
which has the adjoint operator A∗N : for W ∗ = [Y , g ]
0 j
z ∗ (0) = (z ∗ )′ (1) = 0 p
The next result confirms the existence and uniqueness of the solution to (61) and the finite time ‘‘reaching condition’’ to the sliding mode surface Sw . Write system (61) as
(69)
(1) When X ∗ = 0, the eigenvalue problem (69) becomes
˙ X (t ) = (A + BK )X (t ) + Bwx (0, t ), wt (x, t ) = wxx (x, t ), w(0, t ) = 0, wx (1, t ) = −(M + η)sign(Sw (t )) + d(t ) , d˜ (t ).
(A + BK )⊤ X ∗ = λX ∗ , (z ∗ )′′ = λz ∗ , ∗ ⊤ ∗ ′ z (0) = B X , (z ) (1) = 0.
j =1 n j =1
aj e
λ0j t 0
zj (x) +
∞
p
p bm eλm t zm (x).
m=1
J.-M. Wang et al. / Automatica 52 (2015) 23–34
(a) The ODE state X (t ).
31
(b) The heat PDE state u(x, t ).
(c) The external disturbance d(t ). Fig. 3. Open loop response with Dirichlet interconnection (1) in the presence of the sinusoidal boundary matched disturbance.
zj0 (1) = B⊤ Xj∗ / cosh
1
p zm (1) = sin m −
Theorem 11. Suppose that |d(t )| 6 M for all t > 0, and let SU be the sliding mode function given by
λ0j ,
2
π = (−1)m+1 , SU (t ) =
and
u(x, t ) sin 0
y(t ) = z (1, t ) = ∗
n
aj e
λ0j t 0
zj (1) +
j =1
∞
(−1)
m+1
p
bm e
λm t
T
|y(t )|2 dt 6 CT1
0
.
n
|aj |2 + CT2
∞
|bm |2
m=1
for some constants CT1 , CT2 , DT that depend on T only. This shows A∗ t
that BN∗ is admissible for e N and so is BN for eAN t . Therefore, for any T > 0 and W (·, 0) = [X (0), w(·, 0)] ∈ H , if Sw ∈ C [0, T ], Sw (t ) ̸= 0 for all t ∈ [0, T ), then there exists a unique solution W (·, t ) = [X (t ), w(·, t )] ∈ C (0, T ; H ). Therefore, system (62) admits a unique solution. We may suppose without loss of generality that t ≥ 0 and Sw (0) = S0 > 0, since the proof for Sw (0) = S0 < 0 is similar. In this case, it follows from (58) that t
S0 −
2 e−(π /4)(t −τ ) (M + η − d(τ ))dτ
0
≤e
−(π 2 /4)t
2
4η 2 S0 − 2 1 − e−(π /4)t . π
The above inequality shows that when Sw (0) > 0, there exists a unique continuous solution Sw to (58) in the maximal interval [0, tmax ), where it must have Sw (tmax ) = 0. It then follows from (59) that Sw (t ) must be decreasing in [0, tmax ) and Sw (t ) > 0 for all t ∈ [0, tmax ). Since Sw (t ) is continuous, the reaching condition (59) implies that Sw (t ) ≡ 0 for all t > tmax . Returning back to the system (2) under the transformation (37) and feedback control (60), we obtain the main result of this section from Theorem 10.
π
0 1
KM (x) sin
−
x dx
m(x − y)u(y, t ) sin
2
x dydx
π
0
6 DT ∥Z ∗ (·, 0)∥2
Sw (t ) = e
0
π
x
−
m=1
j =1
−(π 2 /4)t
1
Through a direct computation, there exists T > 0, such that
1
2
x X (t )dx.
(70)
Then for any [X (0), u(·, 0)] ∈ H , SU (0) ̸= 0, there exists tmax > 0 such that the closed-loop system of (2) under the feedback control (60) is
˙ X (t ) = AX (t ) + Bux (0, t ), ut (x, t ) = uxx (x, t ), u(0, t ) = 0, 1 ( 1 , t ) = m′ (1 − y)u(y, t )dy + KM ′ (1)X (t ) u x 0 − (M + η)sign(Sw (t )) + d(t ),
(71)
which admits a unique solution [X , u] ∈ C (0, tmax ; H ) and SU (t ) = 0 for all t > tmax . On the sliding mode surface SU (t ) = 0, the system (2) becomes
˙ X (t ) = AX (t ) + Bux (0, t ), u t (x, t ) = uxx (x, t ), u(0, t ) = 0, 1 π x dx u(x, t ) sin 2
0 1 x π = m ( x − y ) u ( y , t ) sin x dydx 2 0 0 π 1 + KM (x) sin x X (t )dx, 0
(72)
2
which is equivalent to (42) and hence is exponentially stable in H with the decay rate −π 2 /4.
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J.-M. Wang et al. / Automatica 52 (2015) 23–34
(a) The ODE state X (t ).
(b) The heat PDE state u(x, t ).
(c) The input signal U (t ) using the sign function.
(d) The input signal U (t ) using the saturation function.
Fig. 4. Closed-loop response with Dirichlet interconnection (1) in the presence of the sinusoidal boundary matched disturbance.
(a) The ODE state X (t ).
(c) The external disturbance d(t ).
(b) The heat PDE state u(x, t ).
(d) The input signal U (t ) using the sign function.
(e) The input signal U (t ) using the saturation function. Fig. 5. Closed-loop response with Neumann interconnection (2) in the presence of the normal distribution random boundary matched disturbance.
J.-M. Wang et al. / Automatica 52 (2015) 23–34
It is remarked that system (61) is equivalent to system (71) under the equivalent transformation (38) and (40). 4. Simulation studies In this section, we will show the effectiveness of the proposed controller for the heat PDE–ODE cascade system with both Dirichlet interconnection (1) and Neumann interconnection (2). Figs. 3 and 4 show the open loop and closed loop results respectively for the Dirichlet interconnection case with A = 2, B = 1, d(t ) = cos(5t ). The initial conditions are chosen as X (0) = 1, u(x, 0) = x (0 < x ≤ 1) and ux (0, 0) = 0. It can be seen that without control both the states of ODE and heat PDE are not stabilized in Fig. 3. After the proposed sliding mode controller (21) is applied with the design parameters are chosen as K = −50, c = 0.1, M = 1, η = 25, both the ODE and heat PDE are stabilized to zero when they reach the sliding surface Sz after some time as shown in Fig. 4. According to Slotine and Li (1991), starting from any initial condition, the state trajectory reaches the sliding mode surface Sz (t ) = 0 in a finite time smaller than Sz (0)/η, i.e., treach ≤ tmax = Sz (0)/η = 3.3 s. It is noted that the chattering in the control signal U (t ) as Fig. 4(c), caused by the sign function, could be mitigated by replacing the sign function with the saturation function as shown in Fig. 4(d). The closed loop simulation results with the Neumann interconnection are shown in Fig. 5 with the normal distribution random disturbance considered. Other parameters and initial conditions are chosen as A = 2, B = 1, X (0) = 1, u(x, 0) = x (0 < x ≤ 1) and u(0, 0) = 0. It is shown that the good control performance is achieved by adopting the proposed sliding mode controller (60) with K = −50, M = 1, η = 25 in the presence of an external normal distribution random disturbance. For the Neumann interconnection case, the reach time is calculated as treach ≤ tmax = Sw (0)/η = 1.27 s. 5. Conclusion In this paper, the sliding mode control was designed to achieve the stabilization of the heat PDE–ODE cascade systems subject to boundary control matched disturbances. Both Dirichlet boundary interconnection and Neumann boundary interconnection were considered. The existence and uniqueness of the solution for the closed-loop via SMC have been proved. The effectiveness of the proposed method was validated with numerical simulations. Acknowledgments The authors would like to thank the anonymous reviewers for providing valuable comments and suggestions to improve the paper. References Adams, R. A., & Fournier, J. F. (2003). Pure and applied mathematics (amsterdam): vol. 140. Sobolev spaces (2nd ed.). Amsterdam: Elsevier/Academic Press. Cheng, M. B., Radisavljevic, V., & Su, W. C. (2011). Sliding mode boundary control of a parabolic pde system with parameter variations and boundary uncertainties. Automatica, 47(2), 381–387. Drakunov, S., Barbieri, E., & Silver, D. A. (1996). Sliding mode control of a heat equation with application to arc welding. In Proc. IEEE int. conf. control appl (pp. 668–672). Ge, S. S., Zhang, S., & He, W. (2011). Vibration control of an Euler–Bernoulli beam under unknown spatiotemporally varying disturbance. International Journal of Control, 84, 947–960. Guo, W., & Guo, B. Z. (2013a). Adaptive output feedback stabilization for one-dimensional wave equation with corrupted observation by harmonic disturbance. SIAM Journal on Control and Optimization, 51, 1679–1706.
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Guo, W., & Guo, B. Z. (2013b). Parameter estimation and non-collocated adaptive stabilization for a wave equation subject to general boundary harmonic disturbance. IEEE Transactions on Automatic Control, 58, 1631–1643. Guo, W., & Guo, B. Z. (2013c). Stabilization and regulator design for a onedimensional unstable wave equation with input harmonic disturbance. International Journal of Robust and Nonlinear Control, 23, 514–533. Guo, W., Guo, B. Z., & Shao, Z. C. (2011). Parameter estimation and stabilization for a wave equation with boundary output harmonic disturbance and noncollocated control. International Journal of Robust and Nonlinear Control, 21, 1297–1321. Guo, B., & Jin, F. (2013a). Sliding mode and active disturbance rejection control to the stabilization of anti-stable one-dimensional wave equation subject to boundary input disturbance. IEEE Transactions on Automatic Control, 58, 1269–1274. Guo, B., & Jin, F. (2013b). The active disturbance rejection and sliding mode control approach to the stabilization of Euler–Bernoulli beam equation with boundary input disturbance. Automatica, 49, 2911–2918. Guo, B., & Liu, J. (2014). 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, 24, 2194–2212. He, W., Ge, S. S., How, B. V. E., Choo, Y. S., & Hong, K. S. (2011). Robust adaptive boundary control of a flexible marine riser with vessel dynamics. Automatica, 47, 722–732. He, W., Zhang, S., & Ge, S. S. (2012). Boundary output- feedback stabilization of a Timoshenko beam using disturbance observer. IEEE Transactions on Industrial Electronics, 99, 1–9. Krstic, M. (2009). Compensating actuator and sensor dynamics governed by diffusion PDEs. Systems & Control Letters, 58(5), 372–377. Krstic, M., & Smyshlyaev, A. (2008). Boundary control of PDEs. a course on backstepping designs. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). Orlov, Y. V., & Utkin, V. I. (1982). Use of sliding modes in distributed system control problems. Automation and Remote Control, 43(9), 1127–1135. Orlov, Y. V., & Utkin, V. I. (1987). Sliding mode control in infinite-dimensional systmes. Automatica, 23(6), 753–757. Pisano, A., Orlov, Y., & Usai, E. (2011). Tracking control of the uncertain heat and wave equation via power-fractional and sliding-mode techniques. SIAM Journal on Control and Optimization, 49(2), 363–382. Slotine, J.-J. E., & Li, W. (1991). Applied nonlinear control. Englewood Cliffs, NJ: Prentice-Hall. Smyshlyaev, A., & Krstic, M. (2004). Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations. IEEE Transactions on Automatic Control, 49(12), 2185–2202. Smyshlyaev, A., & Krstic, M. (2005). Backstepping observers for a class of parabolic PDEs. Systems & Control Letters, 54, 613–625. Utkin, V. I. (2008). Sliding mode control: Mathematical tools, design and applications. In Ser. Lecture notes in math.: vol. 1932. Nonlinear and optimal control theory (pp. 289–347).
Jun-Min Wang received his B.S. degree in Mathematics from Shanxi University in 1995, and M.S. degree in Applied Mathematics from Bering Institute of Technology in 1998, respectively, and Ph.D. degree in Applied Mathematics from the University of Hong Kong in 2004. From 1998 to 2000, he worked as a teaching assistant at the Beijing Institute of Technology. He was a postdoctoral fellow at the University of the Witwatersrand, South Africa from 2004 to 2006. From 2006 to 2009 he was an associate professor at the Beijing Institute of Technology. Since 2009, he has been with Beijing Institute of Technology as a full professor in Applied Mathematics. His research interests focus on the control theory of distributed parameter systems. Dr. Wang is the recipient for New Century Excellent Talents Program from Ministry of Higher Education of China (2007).
Jun-Jun Liu received his B.S. degree in Mathematics at Changzhi University, China, in 2008, M.S. degree from Xinjiang University in 2011. He is currently a Ph.D. candidate at Beijing Institute of Technology. His research interests focus on distributed parameter systems control.
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J.-M. Wang et al. / Automatica 52 (2015) 23–34
Beibei Ren received the B. Eng. degree in the Mechanical Electronic Engineering and the M. Eng. degree in Automation from Xidian University, Xi’an, China, in 2001 and in 2004, respectively, and the Ph.D. degree in the Electrical and Computer Engineering from the National University of Singapore, Singapore, in 2010. From 2010 to 2013, she was a postdoctoral scholar in the Department of Mechanical Aerospace Engineering, University of California, San Diego, CA, USA. Since 2013, she has been with the Department of Mechanical Engineering, Texas Tech University, Lubbock, TX, USA, as an Assistant Professor. Her current research interests include adaptive control, robust control, distributed parameter systems, extremum seeking and their applications.
Jinhao Chen received his B. Eng. degrees from School of Information Science and Technology (SIST), Sun YatSen University (SYSU) in 2012. He is currently a Ph.D. student in the Department of Mechanical Engineering, Texas Tech University, Lubbock, TX, USA. His research interests include robust control, neural network, machine learning and their applications.