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Stabilization for the multi-dimensional heat equation with disturbance on the controller
Automatica (
)
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Technical communique
Stabilization for the multi-dimensional heat equation with disturbance on the controller✩ Guojie Zheng, Jun Li College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, PR China
article
info
Article history: Received 29 March 2016 Received in revised form 15 January 2017 Accepted 7 March 2017 Available online xxxx
abstract In this paper, we consider stabilization for a multi-dimensional heat equation with internal control matched disturbance. The active disturbance rejection control approach is adopted in this work. First of all, we estimate the disturbance in terms of the output. Then we cancel the disturbance by its approximation. In the end, we design some control strategies to stabilize the anti-stable system. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Heat equations Feedback Active disturbance rejection control
1. Introduction
Christofides, 1998). It is well known that the Dirichlet boundary value problem
In this work, we consider the stabilization of the following multi-dimensional heat equation:
yt (x, t ) − △y(x, t ) − ay(x, t ) = χω (U (x, t ) + d(x, t )), (x, t ) ∈ Ω × (0, +∞), y(x, t ) = 0, (x, t ) ∈ ∂ Ω × (0, +∞),
(1)
where Ω is a bounded domain in Rd (d ≥ 1) with Lipschitz boundary, ω ⊂ Ω is an nonempty subdomain of Ω , χω is the characteristic function of ω, a > 0 is a fixed number, U (x, t ) is the control function, and d(x, t ) can be regarded as an uncertainty disturbance on the control, which comes from outside of the system. Usually, the disturbance on the control cannot be avoided when control plans are carried out in the system. In this paper, we suppose that
χω d(x, t ) ∈ L∞ (0, ∞; L2 (ω)), χω dt (x, t ) ∈ L∞ (0, ∞; L2 (ω)).
(2)
The heat equation is one of the most important second order partial differential equations, and this controlled system plays an important role in industry application for temperature control (see
✩ This work was partially supported by the Natural Science Foundation of Henan Province (No. 162300410176). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Xiaobo Tan under the direction of Editor André L. Tits. E-mail addresses: [email protected] (G. Zheng), [email protected] (J. Li).
http://dx.doi.org/10.1016/j.automatica.2017.04.011 0005-1098/© 2017 Elsevier Ltd. All rights reserved.
yt (x, t ) − △y(x, t ) − ay(x, t ) = 0, (x, t ) ∈ Ω × (0, +∞), (3) y(x, t ) = 0, (x, t ) ∈ ∂ Ω × (0, +∞).
is unstable if a is bigger than the first eigenvalue of −△. In the case without disturbance, system (1) can be exponentially stabilized with the feedback control U (x, t ) = −ky(x, t ),
(x, t ) ∈ Ω × (0, +∞),
(4)
where k is a positive number big enough (see Barbu, 2013; Barbu, Lefter, & Tessitore, 2002 and Barbu & Wang, 2003). However, control (4) fails to deal with the disturbance on the control. To the best of our knowledge, this problem for PDEs was first studied in Guo and Jin (2013) for the stability of the one-dimensional antistable wave equation. The main idea in Guo and Jin (2013) is that the disturbance can be estimated in terms of the outputs, and then it can be canceled. This strategy is also called the active disturbance rejection control (ADRC) strategy, which was first proposed by Han in Han (2009). The results of the ADRC for general nonlinear lumped parameter systems are available recently in Guo and Zhao (2013). Another important paper we should mention is Krstic and Smyshlyaev (2008). In Krstic and Smyshlyaev (2008), the authors introduced the adaptive design to handle the parabolic PDEs with disturbance and anti-damping. For other related works on this subject, we refer to Freidovich and Khalil (2008), Guo and Guo (2013b), Han (2009), Krstic (2010), Krstic, Guo, Balogh, and Smyshlyaev (2008), Vazqueza and Krstic (2008) and Smyshlyaev and Krstic (2007a,b), and the references therein. It should be pointed out that many control methods aforementioned have also been applied to
2
G. Zheng, J. Li / Automatica (
deal with uncertainties in PDEs. However, there is few work on stabilization for multi-dimensional PDEs. In our work, we mainly deal with the stabilization of multi-dimensional anti-stable heat equations with a disturbance on the control. The paper is organized as follows. In Section 2, we will present the main result and its proof. The numerical simulation will be given in Section 3. 2. The main result In this work, we study the stability of system (1) when (2) holds. The objective of our work is to design a continuous control U (x, t ), which is based on the output, to stabilize system (1) with disturbance d(x, t ) on the control. First of all, we introduce some notations. Let {λi }∞ i=1 , 0 < λ1 < λ2 ≤ · · · ≤ λn . . . , be the eigenvalues of −△ with the Dirichlet boundary condition on Ω , and {ei }∞ i=1 be the corresponding eigenfunctions satisfying that ∥ei (x)∥L2 (Ω ) = 1, i = 1, 2, 3 . . . , which constitutes an orthonormal basis for L2 (Ω ). Then, the disturbance function χω d(x, t ) can be written as
χω d(x, t ) =
∞
di (t )ei (x),
(5)
i =1
where di (t ) = Ω ei (x)χω d(x, t )dx, i = 1, 2, 3 . . . , are the Fourier coefficients of χω d(x, t ). In addition to (2), we suppose that the following condition holds.
• Condition (C): The Fourier series (5) is uniformly convergent in L2 (Ω ) for any t ∈ [0, +∞). Namely, for any δ > 0, there exists a positive number N = N (δ), which is independent of t, such that
)
–
where ε > 0 is the small tuning parameter, then the errors Y˜iε (t ) = Yi (t ) − Yˆiε (t ),
d˜ iε (t ) = di (t ) − dˆ iε (t )
(10)
satisfy
1 Y˜i′ε (t ) = d˜ iε (t ) − Y˜iε (t ), ε d˜ ′ (t ) = − 1 Y˜iε (t ) + d′ (t ). i iε 4µε 2
(11)
Lemma 2.1. Suppose that (2) holds, and µ > 1. Then, for any given T > 0, it follows that
|Y˜iε (t )| + |d˜ iε (t )| → 0,
(12)
as ε → 0, uniformly for t ∈ [T , +∞), and i = 1, 2, 3 . . . . Remark 2.2. While the main idea is derived from Theorem 2.1 of Guo and Zhao (2011) for the case of ODEs, we would rather give the proof here in detail for the sake of completeness. Proof. Let Z˜iε (t ) = 1ε Y˜iε (t ), i = 1, 2, 3 . . . , in (11),
Φiε (t ) =
A=
Z˜iε (t ) , d˜ iε (t )
0 , di (t ) ′
−1 −
Di (t ) =
1
.
1
0
4µ
Then, we can rewrite (11) as
di (t )ei (x) 2 i>N
< δ,
for any t ∈ [0, +∞).
(6)
L (Ω )
Φi′ε (t ) =
1
ε
AΦiε (t ) + Di (t ).
(13)
The eigenvalues of A are Remark 2.1. (1) In Guo and Guo (2013a,b), and Guo, Guo, and Shao (2011), the authors discussed the case that the disturbance function was taken as harmonic disturbance, in which case Condition (C) holds naturally. (2) If the disturbance function d(x, t ) = f (t )G(x), where G(x) ∈ L2 (Ω ) and f (t ) is a bounded function from [0, +∞) to R, then Condition (C) also holds. In this paper, we suppose the output measurement to be Yi (t ) =
Ω
ei (x)y(x, t )dx,
i = 1, 2, 3 . . . .
Ω
(7)
1−
1−
1
µ
,
µ2 = −
1
ei (x)χω U (x, t )dx + di (t ).
(8)
We design a state observer to estimate Yi (t ) and di (t ) as in Guo and Zhao (2011):
′ ˆ Yiε (t ) = ei (x)χω U (x, t )dx + diε (t ) Ω 1 +(a − λi )Yi (t ) + Yi (t ) − Yˆiε (t ) , ε dˆ ′ (t ) = 1 ˆ Y ( t ) − Y ( t ) i i ε iε 4µε 2
1+
1−
1
µ
.
1
(14)
It follows that
R2
t 1 ε A(t −τ ) D (τ )dτ + e i
≤ Le ε µ1 t ∥Φiε (0)∥R2 + Ω
2
∥e ε At ∥ ≤ Le ε µ1 t . The solution of (13) can be written as t 1 1 At ε Φiε (t ) = e Φiε (0) + e ε A(t −τ ) Di (τ )dτ .
1
1
It is easy to check that µ2 < µ1 < 0, when µ > 1. Thus, there exists a constant L > 0, which is independent of ε , such that
1 ∥Φiε (t )∥R2 ≤ e ε At Φiε (0)
ei (x)yt (x, t )dx
= −λi Yi (t ) + aYi (t ) +
2
0
By direct computation, Yi′ (t ) =
µ1 = −
1
(9)
1
≤ Le ε µ1 t ∥Φiε (0)∥R2 +
0
t
Le ε µ1 (t −τ ) ∥Di (τ )∥R2 dτ
0
Lε
µ1
R2
1
∥d′i (t )∥L∞ (0,∞) .
(15)
By (2) and (5), we obtain that ∥d′i (t )∥L∞ (0,∞) are uniformly bounded for i = 1, 2, 3 . . . . Thus, for any T > 0,
1 Y˜iε (t ) + |d˜ iε (t )| → 0, ε
as ε → 0,
(16)
uniformly for t ∈ [T , +∞), and i = 1, 2, 3 . . . . This shows (12) holds.
G. Zheng, J. Li / Automatica (
By Lemma 2.1, we can regard dˆ iε (t ) as an approximation of the Fourier coefficient di (t ) of the disturbance χω d(x, t ). Now, we define dˆ ε (x, t ) =
N (ε)
dˆ iε (t )ei (x),
(17)
i=1
√
where N√(ε) = [1/ ε], which stands for the biggest integer less than 1/ ε . Lemma 2.2. Suppose Condition (C) holds, and let dˆ ε (x, t ) be defined by (17). Then, for any T > 0, lim ∥χω dˆ ε (x, t ) − χω d(x, t )∥L2 (Ω ) = 0,
(18)
ε→0
uniformly for t ∈ [T , +∞). Proof. By (5), we have ∞ di (t )χω ei (x). χω d(x, t ) = χω χω d(x, t ) =
(19)
i =1
Thus,
1
1
1
≤ L √ e ε µ1 t ∥Φiε (0)∥R2 ε
max
i=1,...,N (ε)
|d˜ iε (t )| → 0,
∥d′i (t )∥L∞ (0,∞)
(21)
for any φ(x) ∈ D(Aω ),
(22)
2
Ω
|∇x φ(x)| dx + k
|φ(x)| dx ≥ (λω − δ) 2
ω
∥y(x, t )∥2L2 (Ω ) =− |∇x y(x, t )|2 dx + a |y(x, t )|2 dx Ω Ω − k |y(x, t )|2 dx − y(x, t ) d(x, t ) − dε (x, t ) dx.
2 dt
ω
ω
For (26), there exists a positive number δ > 0, such that a + 3δ < λω . It follows from Lemma 2.3 that there exists a positive number kδ such that ∀ φ(x) ∈ H01 (Ω ), and ∀ k > kδ ,
|∇x φ(x)| dx + k 2
ω
|φ(x)| dx ≥ (λω − δ) 2
Ω
|φ(x)|2 dx.
(28)
This, together with Young’s inequality, shows that 1 d
∥y(x, t )∥2L2 (Ω ) ≤ −(λω − δ) |y(x, t )|2 dx + a |y(x, t )|2 dx Ω Ω +δ |y(x, t )|2 dx + C (δ)∥d(x, t ) − dε (x, t )∥2L2 (ω) Ω ≤ −δ |y(x, t )|2 dx + C (δ)∥d(x, t ) − dε (x, t )∥2L2 (ω) .
2 dt
Lemma 2.3. For any δ > 0 there exists kδ > 0 such that ∀ φ(x) ∈ H01 (Ω ), and ∀ k > kδ , 2
1 d
Ω
where D(Aω ) = (Ω \ ω) ∩ H (Ω \ ω). The following lemma is taken from Barbu et al. (2002).
(27)
Proof. By direct calculation,
µ1 √ L ε ′ + ∥d (t )∥L∞ (0,∞) , (20) µ1 i
as ε → 0, ∀ t ∈ [T , +∞).
H01
(25)
(26)
lim lim ∥y(x, t )∥L2 (Ω ) = 0.
Let λω be the first eigenvalue of the operator Aω = − △x on Ω \ ω with Dirichlet boundary conditions. Namely, Aω φ = − △x φ,
yt (x, t ) − △y(x, t ) − ay(x, t ) = χω (U (x, t ) + d(x, t )), y (x, t ) = 0, ei (x)y(x, t )dx, Yi (t ) = Ω ′ ˆ ei (x)χω U (x, t )dx + diε (t ) Yiε (t ) = Ω 1 Yi (t ) − Yˆiε (t ) , +(a − λi )Yi (t ) + ε 1 ′ ˆiε (t ) , dˆ iε (t ) = Y ( t ) − Y i 4µε 2 N (ε) ˆ d ( x , t ) = dˆ iε (t )ei (x), ε i=1 U (x, t ) = −ky(x, t ) − dˆ ε (x, t ), where i = 1, 2, 3 . . . , N (ε).
ε→0 t →∞
, it follows that ε
In summary, we can complete the proof of this lemma.
(24)
Then, there exists a positive number kδ > 0 such that for any k > kδ > 0 in (24) such that
for i = 1, 2, . . . , N (ε). Thus, N (ε)
U (x, t ) = −ky(x, t ) − dˆ ε (x, t ),
where k is a positive number which will be given according to the number a. From (1), (7), (9), (17) and (24), we obtain the overall closed-loop system as follows.
λ1 < a < λω .
√1
Lε
By this lemma, we can design the control for system (1) as
Theorem 2.1. Suppose that (2) and condition (C) hold. Let µ > 1, and
On the one hand, it follows from Condition (C) that for any t ∈ [0, +∞), ∥ ∞ i=N (ε)+1 di (t )ei (x)∥L2 (Ω ) → 0 as ε → 0. On the other
N (ε)|d˜ iε (t )| ≤ N (ε) Le ε µ1 t ∥Φiε (0)∥R2 +
3
The main results of this paper are stated as follows.
L2 (Ω )
hand, according to (15) and N (ε) = [ ε ] ≤
–
Remark 2.3. From the views of physics, Yi (t ) in (7), i = 1, 2, . . . , N (ε), can be regarded as the low frequency components of y(x, t ). This kind of observer can be regarded as a reduced order observer.
∥χω dˆ ε (x, t ) − χω d(x, t )∥L2 (Ω ) N (ε) ˆ diε (t ) − di (t ) χω ei (x) ≤ 2 i=1 L (Ω ) ∞ + χω di (t )ei (x) i=N (ε)+1 N (ε) ∞ di (t )ei (x) ≤ |d˜ iε (t )| ∥ei (x)∥L2 (Ω ) + i=1 i=N (ε)+1 L2 (Ω ) ∞ . d (t )ei (x) ≤ N (ε) max |d˜ iε (t )| + i=N (ε)+1 i i=1,...,N (ε)
√1
)
|φ(x)| dx. 2
Ω
(23)
Ω
Combining this with Lemma 2.2, we complete the proof of this theorem.
4
G. Zheng, J. Li / Automatica (
)
–
Fig. 3. The state of initial data.
Fig. 1. The state of the heat equation without control in 1-d case.
Fig. 4. The state of solution of (25) in t = 1. Fig. 2. The state of the heat equation with control in 1-d case.
Remark 2.4. For any positive number a > λ1 , we can easily find an open subset ω ⊂ Ω such that the first eigenvalue λω > a. 3. Numerical simulation In this section, we will carry out some numerical simulation to illustrate the theoretical results. In the case of spatial dimension d = 1, we take Ω = (0, 1) and ω = (0.5, 1) in system (1). The first eigenvalue of −△ is π 2 with eigenfunction sin(π x). Let a = 10. Then a > π 2 . We set the initial data y(x, 0) = 100(x − x2 ). Let the disturbance function d(x, t ) = sin t sin x. In Fig. 1, we can see that system (1) is unstable without control. In order to stabilize the system, we take ε = 0.1, µ = 2 and k = 50 in (24). Fig. 2 shows that the control we designed yields great performance in this case. Next, we show the experimental result in the case of spatial dimension d = 2. Taking Ω = (0, 1) × (0, 1), and ω = (0.5, 1) × (0, 1) in system (1). Then the first eigenvalue of −△ is 2π 2 with eigenfunction sin(π x1 )· sin(π x2 ). Take a = 20. Then, a > 2π 2 . Letting the initial data y(x1 , x2 , 0) = 10 sin(π x1 ) sin(2π x2 ), the disturbance function d(x1 , x2 , t ) = sin t sin x1 sin x2 , we can compare the state of solution of (1) at t = 0 and t = 1 by taking ε = 0.1, µ = 2 and k = 50. Combining Figs. 3 and 4, we conclude that the closed-loop system (25) with the control designed in (24) can be stabilized in the space L2 (Ω ). (Related numerical results could be found at http://guojiezheng.blog, 0000).
4. Conclusions In this paper, we develop a systematic way to deal with an unstable multi-dimensional heat equation with disturbance flowing into the control channel by the active disturbance rejection control approach. By observing the low frequency components of the state, a feedback control strategy is designed to stabilize system (1). An analogous discussion can be performed on other multidimensional PDEs. References Barbu, V. (2013). Note on the internal stabilization of stochastic parabolic equations with linearly multiplicative gaussian noise. ESAIM: Control, Optimisation and Calculus of Variations, 19(4), 1055–1063. Barbu, V., Lefter, C., & Tessitore, G. (2002). A note on the stabilizability of stochastic heat eqautions with multiplicative noise. Comptes Rendus de l’Academie des Sciences, Serie I, 334(4), 311–316. Barbu, V., & Wang, G. (2003). Internal stabilization of semilinear parabolic systems. Journal of Mathematical Analysis and Applications, 285(2), 387–407. Christofides, P. D. (1998). Robust control of parabolic PDE systems. Chemical Engineering Science, 53(16), 2949–2965. Freidovich, L. B., & Khalil, H. K. (2008). Performance recovery of feedbacklinearization based designs. IEEE Transactions on Automatic Control, 53(10), 2324–2334. Guo, W., & 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., & Guo, B. Z. (2013b). Stabilization and regulator design for a one dimensional unstable wave equation with input harmonic disturbance. International Journal of Robust and Nonlinear Control, 23(5), 514–533.
G. Zheng, J. Li / Automatica ( 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(11), 1297–1321. Guo, B. Z., & Jin, F. F. (2013). Sliding mode and active disturbance rejection control to stabilization of one-dimensional anti-stable wave equations subject to disturbance in boundary input. IEEE Transactions on Automatic Control, 58(5), 1269–1274. Guo, B. Z., & Zhao, Z. L. (2011). On the convergence of an extended state observer for nonlinear systems with uncertainty. System Control Letters, 60(6), 420–430. Guo, B. Z., & Zhao, Z. L. (2013). On convergence of the nonlinear active disturbance rejection control for MIMO Systems. SIAM Journal on Control and Optimization, 51(2), 1727–1757. Han, J. Q. (2009). From PID to active disturbance rejection control. IEEE Transactions on Industrial Electronics, 56(3), 900–906. http://guojiezheng.blog.163.com/blog/static/18420145620170584826482/? newFollowBlog.
)
–
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Krstic, M. (2010). Adaptive control of an anti-stable wave PDE. Dynamics of Continuous, Discrete and Impulsive Systems, Series A. Mathematical Analysis, 17(6), 853–882. Krstic, M., Guo, B. Z., Balogh, A., & Smyshlyaev, A. (2008). Output-feedback stabilization of an unstable wave equation. Automatica, 44(1), 63–74. Krstic, M., & Smyshlyaev, A. (2008). Adaptive boundary control for unstable parabolic PDEs-Part I: Lyapunov design. IEEE Transactions on Automatic Control, 53(7), 1575–1591. Smyshlyaev, A., & Krstic, M. (2007a). Adaptive boundary control for unstable parabolic PDEs-Part II: Estimation-based designs. Automatica, 43(9), 1543–1556. Smyshlyaev, A., & Krstic, M. (2007b). Adaptive boundary control for unstable parabolic PDEs-Part III: Output-feedback examples with swapping identifiers. Automatica, 43(9), 1557–1564. Vazqueza, R., & Krstic, M. (2008). Control of 1-D parabolic PDEs with Volterra nonlinearities, Part I: Design. Automatica, 44(11), 2778–2790.
)
–
Contents lists available at ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Technical communique
Stabilization for the multi-dimensional heat equation with disturbance on the controller✩ Guojie Zheng, Jun Li College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, PR China
article
info
Article history: Received 29 March 2016 Received in revised form 15 January 2017 Accepted 7 March 2017 Available online xxxx
abstract In this paper, we consider stabilization for a multi-dimensional heat equation with internal control matched disturbance. The active disturbance rejection control approach is adopted in this work. First of all, we estimate the disturbance in terms of the output. Then we cancel the disturbance by its approximation. In the end, we design some control strategies to stabilize the anti-stable system. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Heat equations Feedback Active disturbance rejection control
1. Introduction
Christofides, 1998). It is well known that the Dirichlet boundary value problem
In this work, we consider the stabilization of the following multi-dimensional heat equation:
yt (x, t ) − △y(x, t ) − ay(x, t ) = χω (U (x, t ) + d(x, t )), (x, t ) ∈ Ω × (0, +∞), y(x, t ) = 0, (x, t ) ∈ ∂ Ω × (0, +∞),
(1)
where Ω is a bounded domain in Rd (d ≥ 1) with Lipschitz boundary, ω ⊂ Ω is an nonempty subdomain of Ω , χω is the characteristic function of ω, a > 0 is a fixed number, U (x, t ) is the control function, and d(x, t ) can be regarded as an uncertainty disturbance on the control, which comes from outside of the system. Usually, the disturbance on the control cannot be avoided when control plans are carried out in the system. In this paper, we suppose that
χω d(x, t ) ∈ L∞ (0, ∞; L2 (ω)), χω dt (x, t ) ∈ L∞ (0, ∞; L2 (ω)).
(2)
The heat equation is one of the most important second order partial differential equations, and this controlled system plays an important role in industry application for temperature control (see
✩ This work was partially supported by the Natural Science Foundation of Henan Province (No. 162300410176). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Xiaobo Tan under the direction of Editor André L. Tits. E-mail addresses: [email protected] (G. Zheng), [email protected] (J. Li).
http://dx.doi.org/10.1016/j.automatica.2017.04.011 0005-1098/© 2017 Elsevier Ltd. All rights reserved.
yt (x, t ) − △y(x, t ) − ay(x, t ) = 0, (x, t ) ∈ Ω × (0, +∞), (3) y(x, t ) = 0, (x, t ) ∈ ∂ Ω × (0, +∞).
is unstable if a is bigger than the first eigenvalue of −△. In the case without disturbance, system (1) can be exponentially stabilized with the feedback control U (x, t ) = −ky(x, t ),
(x, t ) ∈ Ω × (0, +∞),
(4)
where k is a positive number big enough (see Barbu, 2013; Barbu, Lefter, & Tessitore, 2002 and Barbu & Wang, 2003). However, control (4) fails to deal with the disturbance on the control. To the best of our knowledge, this problem for PDEs was first studied in Guo and Jin (2013) for the stability of the one-dimensional antistable wave equation. The main idea in Guo and Jin (2013) is that the disturbance can be estimated in terms of the outputs, and then it can be canceled. This strategy is also called the active disturbance rejection control (ADRC) strategy, which was first proposed by Han in Han (2009). The results of the ADRC for general nonlinear lumped parameter systems are available recently in Guo and Zhao (2013). Another important paper we should mention is Krstic and Smyshlyaev (2008). In Krstic and Smyshlyaev (2008), the authors introduced the adaptive design to handle the parabolic PDEs with disturbance and anti-damping. For other related works on this subject, we refer to Freidovich and Khalil (2008), Guo and Guo (2013b), Han (2009), Krstic (2010), Krstic, Guo, Balogh, and Smyshlyaev (2008), Vazqueza and Krstic (2008) and Smyshlyaev and Krstic (2007a,b), and the references therein. It should be pointed out that many control methods aforementioned have also been applied to
2
G. Zheng, J. Li / Automatica (
deal with uncertainties in PDEs. However, there is few work on stabilization for multi-dimensional PDEs. In our work, we mainly deal with the stabilization of multi-dimensional anti-stable heat equations with a disturbance on the control. The paper is organized as follows. In Section 2, we will present the main result and its proof. The numerical simulation will be given in Section 3. 2. The main result In this work, we study the stability of system (1) when (2) holds. The objective of our work is to design a continuous control U (x, t ), which is based on the output, to stabilize system (1) with disturbance d(x, t ) on the control. First of all, we introduce some notations. Let {λi }∞ i=1 , 0 < λ1 < λ2 ≤ · · · ≤ λn . . . , be the eigenvalues of −△ with the Dirichlet boundary condition on Ω , and {ei }∞ i=1 be the corresponding eigenfunctions satisfying that ∥ei (x)∥L2 (Ω ) = 1, i = 1, 2, 3 . . . , which constitutes an orthonormal basis for L2 (Ω ). Then, the disturbance function χω d(x, t ) can be written as
χω d(x, t ) =
∞
di (t )ei (x),
(5)
i =1
where di (t ) = Ω ei (x)χω d(x, t )dx, i = 1, 2, 3 . . . , are the Fourier coefficients of χω d(x, t ). In addition to (2), we suppose that the following condition holds.
• Condition (C): The Fourier series (5) is uniformly convergent in L2 (Ω ) for any t ∈ [0, +∞). Namely, for any δ > 0, there exists a positive number N = N (δ), which is independent of t, such that
)
–
where ε > 0 is the small tuning parameter, then the errors Y˜iε (t ) = Yi (t ) − Yˆiε (t ),
d˜ iε (t ) = di (t ) − dˆ iε (t )
(10)
satisfy
1 Y˜i′ε (t ) = d˜ iε (t ) − Y˜iε (t ), ε d˜ ′ (t ) = − 1 Y˜iε (t ) + d′ (t ). i iε 4µε 2
(11)
Lemma 2.1. Suppose that (2) holds, and µ > 1. Then, for any given T > 0, it follows that
|Y˜iε (t )| + |d˜ iε (t )| → 0,
(12)
as ε → 0, uniformly for t ∈ [T , +∞), and i = 1, 2, 3 . . . . Remark 2.2. While the main idea is derived from Theorem 2.1 of Guo and Zhao (2011) for the case of ODEs, we would rather give the proof here in detail for the sake of completeness. Proof. Let Z˜iε (t ) = 1ε Y˜iε (t ), i = 1, 2, 3 . . . , in (11),
Φiε (t ) =
A=
Z˜iε (t ) , d˜ iε (t )
0 , di (t ) ′
−1 −
Di (t ) =
1
.
1
0
4µ
Then, we can rewrite (11) as
di (t )ei (x) 2 i>N
< δ,
for any t ∈ [0, +∞).
(6)
L (Ω )
Φi′ε (t ) =
1
ε
AΦiε (t ) + Di (t ).
(13)
The eigenvalues of A are Remark 2.1. (1) In Guo and Guo (2013a,b), and Guo, Guo, and Shao (2011), the authors discussed the case that the disturbance function was taken as harmonic disturbance, in which case Condition (C) holds naturally. (2) If the disturbance function d(x, t ) = f (t )G(x), where G(x) ∈ L2 (Ω ) and f (t ) is a bounded function from [0, +∞) to R, then Condition (C) also holds. In this paper, we suppose the output measurement to be Yi (t ) =
Ω
ei (x)y(x, t )dx,
i = 1, 2, 3 . . . .
Ω
(7)
1−
1−
1
µ
,
µ2 = −
1
ei (x)χω U (x, t )dx + di (t ).
(8)
We design a state observer to estimate Yi (t ) and di (t ) as in Guo and Zhao (2011):
′ ˆ Yiε (t ) = ei (x)χω U (x, t )dx + diε (t ) Ω 1 +(a − λi )Yi (t ) + Yi (t ) − Yˆiε (t ) , ε dˆ ′ (t ) = 1 ˆ Y ( t ) − Y ( t ) i i ε iε 4µε 2
1+
1−
1
µ
.
1
(14)
It follows that
R2
t 1 ε A(t −τ ) D (τ )dτ + e i
≤ Le ε µ1 t ∥Φiε (0)∥R2 + Ω
2
∥e ε At ∥ ≤ Le ε µ1 t . The solution of (13) can be written as t 1 1 At ε Φiε (t ) = e Φiε (0) + e ε A(t −τ ) Di (τ )dτ .
1
1
It is easy to check that µ2 < µ1 < 0, when µ > 1. Thus, there exists a constant L > 0, which is independent of ε , such that
1 ∥Φiε (t )∥R2 ≤ e ε At Φiε (0)
ei (x)yt (x, t )dx
= −λi Yi (t ) + aYi (t ) +
2
0
By direct computation, Yi′ (t ) =
µ1 = −
1
(9)
1
≤ Le ε µ1 t ∥Φiε (0)∥R2 +
0
t
Le ε µ1 (t −τ ) ∥Di (τ )∥R2 dτ
0
Lε
µ1
R2
1
∥d′i (t )∥L∞ (0,∞) .
(15)
By (2) and (5), we obtain that ∥d′i (t )∥L∞ (0,∞) are uniformly bounded for i = 1, 2, 3 . . . . Thus, for any T > 0,
1 Y˜iε (t ) + |d˜ iε (t )| → 0, ε
as ε → 0,
(16)
uniformly for t ∈ [T , +∞), and i = 1, 2, 3 . . . . This shows (12) holds.
G. Zheng, J. Li / Automatica (
By Lemma 2.1, we can regard dˆ iε (t ) as an approximation of the Fourier coefficient di (t ) of the disturbance χω d(x, t ). Now, we define dˆ ε (x, t ) =
N (ε)
dˆ iε (t )ei (x),
(17)
i=1
√
where N√(ε) = [1/ ε], which stands for the biggest integer less than 1/ ε . Lemma 2.2. Suppose Condition (C) holds, and let dˆ ε (x, t ) be defined by (17). Then, for any T > 0, lim ∥χω dˆ ε (x, t ) − χω d(x, t )∥L2 (Ω ) = 0,
(18)
ε→0
uniformly for t ∈ [T , +∞). Proof. By (5), we have ∞ di (t )χω ei (x). χω d(x, t ) = χω χω d(x, t ) =
(19)
i =1
Thus,
1
1
1
≤ L √ e ε µ1 t ∥Φiε (0)∥R2 ε
max
i=1,...,N (ε)
|d˜ iε (t )| → 0,
∥d′i (t )∥L∞ (0,∞)
(21)
for any φ(x) ∈ D(Aω ),
(22)
2
Ω
|∇x φ(x)| dx + k
|φ(x)| dx ≥ (λω − δ) 2
ω
∥y(x, t )∥2L2 (Ω ) =− |∇x y(x, t )|2 dx + a |y(x, t )|2 dx Ω Ω − k |y(x, t )|2 dx − y(x, t ) d(x, t ) − dε (x, t ) dx.
2 dt
ω
ω
For (26), there exists a positive number δ > 0, such that a + 3δ < λω . It follows from Lemma 2.3 that there exists a positive number kδ such that ∀ φ(x) ∈ H01 (Ω ), and ∀ k > kδ ,
|∇x φ(x)| dx + k 2
ω
|φ(x)| dx ≥ (λω − δ) 2
Ω
|φ(x)|2 dx.
(28)
This, together with Young’s inequality, shows that 1 d
∥y(x, t )∥2L2 (Ω ) ≤ −(λω − δ) |y(x, t )|2 dx + a |y(x, t )|2 dx Ω Ω +δ |y(x, t )|2 dx + C (δ)∥d(x, t ) − dε (x, t )∥2L2 (ω) Ω ≤ −δ |y(x, t )|2 dx + C (δ)∥d(x, t ) − dε (x, t )∥2L2 (ω) .
2 dt
Lemma 2.3. For any δ > 0 there exists kδ > 0 such that ∀ φ(x) ∈ H01 (Ω ), and ∀ k > kδ , 2
1 d
Ω
where D(Aω ) = (Ω \ ω) ∩ H (Ω \ ω). The following lemma is taken from Barbu et al. (2002).
(27)
Proof. By direct calculation,
µ1 √ L ε ′ + ∥d (t )∥L∞ (0,∞) , (20) µ1 i
as ε → 0, ∀ t ∈ [T , +∞).
H01
(25)
(26)
lim lim ∥y(x, t )∥L2 (Ω ) = 0.
Let λω be the first eigenvalue of the operator Aω = − △x on Ω \ ω with Dirichlet boundary conditions. Namely, Aω φ = − △x φ,
yt (x, t ) − △y(x, t ) − ay(x, t ) = χω (U (x, t ) + d(x, t )), y (x, t ) = 0, ei (x)y(x, t )dx, Yi (t ) = Ω ′ ˆ ei (x)χω U (x, t )dx + diε (t ) Yiε (t ) = Ω 1 Yi (t ) − Yˆiε (t ) , +(a − λi )Yi (t ) + ε 1 ′ ˆiε (t ) , dˆ iε (t ) = Y ( t ) − Y i 4µε 2 N (ε) ˆ d ( x , t ) = dˆ iε (t )ei (x), ε i=1 U (x, t ) = −ky(x, t ) − dˆ ε (x, t ), where i = 1, 2, 3 . . . , N (ε).
ε→0 t →∞
, it follows that ε
In summary, we can complete the proof of this lemma.
(24)
Then, there exists a positive number kδ > 0 such that for any k > kδ > 0 in (24) such that
for i = 1, 2, . . . , N (ε). Thus, N (ε)
U (x, t ) = −ky(x, t ) − dˆ ε (x, t ),
where k is a positive number which will be given according to the number a. From (1), (7), (9), (17) and (24), we obtain the overall closed-loop system as follows.
λ1 < a < λω .
√1
Lε
By this lemma, we can design the control for system (1) as
Theorem 2.1. Suppose that (2) and condition (C) hold. Let µ > 1, and
On the one hand, it follows from Condition (C) that for any t ∈ [0, +∞), ∥ ∞ i=N (ε)+1 di (t )ei (x)∥L2 (Ω ) → 0 as ε → 0. On the other
N (ε)|d˜ iε (t )| ≤ N (ε) Le ε µ1 t ∥Φiε (0)∥R2 +
3
The main results of this paper are stated as follows.
L2 (Ω )
hand, according to (15) and N (ε) = [ ε ] ≤
–
Remark 2.3. From the views of physics, Yi (t ) in (7), i = 1, 2, . . . , N (ε), can be regarded as the low frequency components of y(x, t ). This kind of observer can be regarded as a reduced order observer.
∥χω dˆ ε (x, t ) − χω d(x, t )∥L2 (Ω ) N (ε) ˆ diε (t ) − di (t ) χω ei (x) ≤ 2 i=1 L (Ω ) ∞ + χω di (t )ei (x) i=N (ε)+1 N (ε) ∞ di (t )ei (x) ≤ |d˜ iε (t )| ∥ei (x)∥L2 (Ω ) + i=1 i=N (ε)+1 L2 (Ω ) ∞ . d (t )ei (x) ≤ N (ε) max |d˜ iε (t )| + i=N (ε)+1 i i=1,...,N (ε)
√1
)
|φ(x)| dx. 2
Ω
(23)
Ω
Combining this with Lemma 2.2, we complete the proof of this theorem.
4
G. Zheng, J. Li / Automatica (
)
–
Fig. 3. The state of initial data.
Fig. 1. The state of the heat equation without control in 1-d case.
Fig. 4. The state of solution of (25) in t = 1. Fig. 2. The state of the heat equation with control in 1-d case.
Remark 2.4. For any positive number a > λ1 , we can easily find an open subset ω ⊂ Ω such that the first eigenvalue λω > a. 3. Numerical simulation In this section, we will carry out some numerical simulation to illustrate the theoretical results. In the case of spatial dimension d = 1, we take Ω = (0, 1) and ω = (0.5, 1) in system (1). The first eigenvalue of −△ is π 2 with eigenfunction sin(π x). Let a = 10. Then a > π 2 . We set the initial data y(x, 0) = 100(x − x2 ). Let the disturbance function d(x, t ) = sin t sin x. In Fig. 1, we can see that system (1) is unstable without control. In order to stabilize the system, we take ε = 0.1, µ = 2 and k = 50 in (24). Fig. 2 shows that the control we designed yields great performance in this case. Next, we show the experimental result in the case of spatial dimension d = 2. Taking Ω = (0, 1) × (0, 1), and ω = (0.5, 1) × (0, 1) in system (1). Then the first eigenvalue of −△ is 2π 2 with eigenfunction sin(π x1 )· sin(π x2 ). Take a = 20. Then, a > 2π 2 . Letting the initial data y(x1 , x2 , 0) = 10 sin(π x1 ) sin(2π x2 ), the disturbance function d(x1 , x2 , t ) = sin t sin x1 sin x2 , we can compare the state of solution of (1) at t = 0 and t = 1 by taking ε = 0.1, µ = 2 and k = 50. Combining Figs. 3 and 4, we conclude that the closed-loop system (25) with the control designed in (24) can be stabilized in the space L2 (Ω ). (Related numerical results could be found at http://guojiezheng.blog, 0000).
4. Conclusions In this paper, we develop a systematic way to deal with an unstable multi-dimensional heat equation with disturbance flowing into the control channel by the active disturbance rejection control approach. By observing the low frequency components of the state, a feedback control strategy is designed to stabilize system (1). An analogous discussion can be performed on other multidimensional PDEs. References Barbu, V. (2013). Note on the internal stabilization of stochastic parabolic equations with linearly multiplicative gaussian noise. ESAIM: Control, Optimisation and Calculus of Variations, 19(4), 1055–1063. Barbu, V., Lefter, C., & Tessitore, G. (2002). A note on the stabilizability of stochastic heat eqautions with multiplicative noise. Comptes Rendus de l’Academie des Sciences, Serie I, 334(4), 311–316. Barbu, V., & Wang, G. (2003). Internal stabilization of semilinear parabolic systems. Journal of Mathematical Analysis and Applications, 285(2), 387–407. Christofides, P. D. (1998). Robust control of parabolic PDE systems. Chemical Engineering Science, 53(16), 2949–2965. Freidovich, L. B., & Khalil, H. K. (2008). Performance recovery of feedbacklinearization based designs. IEEE Transactions on Automatic Control, 53(10), 2324–2334. Guo, W., & 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., & Guo, B. Z. (2013b). Stabilization and regulator design for a one dimensional unstable wave equation with input harmonic disturbance. International Journal of Robust and Nonlinear Control, 23(5), 514–533.
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