Active disturbance rejection control and sliding ... - Semantic Scholar
IMA Journal of Mathematical Control and Information (2015) 32, 97–117 doi:10.1093/imamci/dnt034 Advance Access publication on October 10, 2013
Active disturbance rejection control and sliding mode control of one-dimensional unstable heat equation with boundary uncertainties
[Received on 18 December 2012; revised on 8 May 2013; accepted on 15 September 2013] In this paper, we are concerned with the boundary stabilization of a one-dimensional unstable heat equation with the external disturbance flowing into the control end. The active disturbance rejection control (ADRC) and the sliding mode control (SMC) are adopted in investigation. By the ADRC approach, the disturbance is estimated through an external observer and cancelled online by the approximated one in the closed-loop. It is shown that the external disturbance can be attenuated in the sense that the resulting closed-loop system under the extended state feedback tends to any arbitrary given vicinity of zero as the time goes to infinity. In the second part, we use the SMC to reject the disturbance with the assumption in which the disturbance is supposed to be bounded. The reaching condition, and the existence and uniqueness of the solution for all states in the state space via SMC are established. Simulation examples are presented for both control strategies. Keywords: heat equation; active disturbance rejection control; sliding mode control; boundary control; disturbance rejection.
1. Introduction In the past two decades, the boundary control of systems described by partial differential equations (PDEs) has become an important research in the area of distributed parameter systems control. Many contributions have been made, such as, Chen et al. (1987), Guo & Xu (2007), Krstic & Smyshlyaev (2008), Luo et al. (1999), and the references therein. Traditionally, the system is controlled in the ideal operational environment with exact mathematical model and no internal/external disturbances, for instance, the stabilization for Schrödinger, wave and flexible beam equations by Machtyngier (1994), Luo et al. (1999) and Chen et al. (1987), respectively. Generally speaking, there are two different types of control methods, collocated and non-collocated control designs, which are used to stabilize the PDE systems without any disturbance. The collocated control design is based on the passive principle that makes the closed-loop system dissipative and hence the system is stable at least in the sense of Lyapunov (see Chen et al., 1987). On the other hand, due to the backstepping method introduced into the PDE systems in the last few years by Krstic & Smyshlyaev (2008) (see also Guo & Xu, 2007), the non-collocated method is systematically applied to stabilize some unstable or even anti-stable wave and heat equations (see Smyshlyaev & Krstic, 2004; Krstic & Smyshlyaev, 2008; Krstic et al., 2008). It is known that if there is no uncertainty in the system, the control or the environment, feedback control is largely unnecessary (see Brockett, 2001). So, when the external disturbances enter the system from the boundary/internal of the spatial domain, the new approach is needed to deal with the uncertainties. There are three powerful methods in dealing with the uncertainties: One is the traditionally adaptive c The authors 2013. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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Jun-Jun Liu∗ and Jun-Min Wang School of Mathematics, Beijing Institute of Technology, Beijing 100081, P.R. China ∗ Corresponding author: Email: [email protected]
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J.-J. LIU AND J.-M. WANG
where u is the state and U(t) is the control input. The system represents an unstable distributed parameter system in the sense that for q > 0 large enough, there are finitely many eigenvalues of the system (with no control and disturbance) located on the right-half complex plane. The main aim of this paper is to apply both the ADRC and SMC approaches to attenuate and reject the disturbance in the stabilization of (1.1) for the unknown disturbance d supposed to have bounded ˙ M ) and to be uniformly bounded measurable (|d| M ), respectively. derivative (|d| The rest of the paper is organized as follows. In Section 2, we use the ADRC approach to attenuate the disturbance by designing a high gain estimator to estimate the disturbance. After cancelling the disturbance by the approximated one, we design the state feedback controller. The closed-loop system is shown to attend any arbitrary given vicinity of zero as the time goes to infinity and the gain tuning parameter tends to zero. The SMC for disturbance rejection is presented in Section 3. The SMC is designed and the existence and uniqueness of solution of the closed-loop system are proved. Simulation results are given in Section 4 and some concluding remarks are presented in Section 5. 2. The ADRC approach ˙ is bounded measurable. Firstly, In this section, we design the ADRC control for (1.1) and suppose |d| we introduce a transformation (see Krstic & Smyshlyaev, 2008, pp. 76–77):
x
w(x, t) = u(x, t) + (c0 + q)
eq(x−y) u(y, t) dy, 0
c0 > 0.
(2.1)
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control method in dealing with the systems with the unknown parameters (see Guo et al., 2011; Guo & Guo, 2013 or Krstic, 2010). The second is the active disturbance rejection control (ADRC) method (initially proposed by Han in 1990s) in dealing with the disturbance, which is an unconventional and effective control design for lumped parameter systems in the absence of proper models and in the presence of model uncertainty. The challenge convergence problem for the ADRC method is settled by Guo & Zhao (2011) recently and the ADRC method has been successfully applied to the attenuation of disturbance for a one-dimensional anti-stable wave equation by Guo & Jin (2013). The third is the sliding mode control (SMC) method to reject the disturbance (see Breger et al., 1980; Drakunov et al., 1996; Cheng et al., 2011 or Orlov & Utkin, 1987). In Orlov & Utkin (1987), based on the semigroup theory, the SMC is used to deal with a class of abstract infinite-dimensional systems where the control and disturbance are bounded. The boundary stabilization for a one-dimensional heat equation with boundary disturbance is studied in Drakunov et al. (1996), where the SMC is designed for the first-order PDEs obtained through an integral transformation on the heat equation (which is second order in spatial variable). The sliding mode boundary stabilizer is also designed for a one-dimensional unstable heat, wave and Schrödinger equation by Cheng et al. (2011), Guo & Jin (2013) and Guo & Liu (2013), respectively. The perturbed heat equation with diffusivity and homogeneous Neumann-type boundary condition was studied by Pisano & Orlov (2012), where the initial condition of the heat equation is assumed to belong to H 4 and the proposed infinite-dimensional treatment of the system retains robustness features against non-vanishing matched disturbances. In this paper, we are concerned with the following unstable heat equation in L2 (0, 1): ⎧ ⎪ x ∈ (0, 1), t > 0, ⎨ut (x, t) = uxx (x, t), (1.1) q > 0, t 0, ux (0, t) = −qu(0, t), ⎪ ⎩ ux (1, t) = U(t) + d(t), t 0,
ACTIVE DISTURBANCE REJECTION CONTROL AND SLIDING MODE CONTROL
It is known that (2.1) has an invertible transformation: x e−c0 (x−y) w(y, t) dy. u(x, t) = w(x, t) − (c0 + q)
99
(2.2)
0
x ∈ (0, 1), t > 0, t 0, (2.3) t 0.
0
It is seen that the unstable factor −qu(0, t) in (1.1) becomes the dissipative term c0 w(0, t) in (2.3) under the transformation (2.1), both at the end x = 0. In what follows, we consider the stabilization of system (2.3) until the final step to go back system (1.1) under the inverse transformation (2.2). By introducing a new controller U0 (t) so that 1 U(t) = U0 (t) − (c0 + q)w(1, t) + c0 (c0 + q) e−c0 (1−x) w(x, t) dx, (2.4) 0
(2.3) becomes
⎧ ⎪ ⎨wt (x, t) = wxx (x, t), wx (0, t) = c0 w(0, t), ⎪ ⎩ wx (1, t) = U0 (t) + d(t).
x ∈ (0, 1), t > 0, t 0,
(2.5)
Now we write (2.5) into the operator form in L2 (0, 1). Define the operator A : D(A )(⊂ L2 (0, 1)) → L2 (0, 1) as follows: A f (x) = f (x), (2.6) D(A ) = {f ∈ H 2 (0, 1) | f (0) = c0 f (0), f (1) = 0}. A direct computation gives A ∗ f (x) = f (x), D(A ∗ ) = {f ∈ H 2 (0, 1) | f (0) = c0 f (0), f (1) = 0} with
⎧ x ⎪ ∗ −1 ⎪ f =C + (x − τ )f (τ ) dτ ∀f ∈ L2 (0, 1), ⎨A 1 1 1 1 ⎪ ⎪ f (x) dx − xf (x) dx. ⎩C = − c0 0 0
(2.7)
(2.8)
It is seen that A is self-adjoint in L2 (0, 1) and (2.5) can be written as an evolution equation in L2 (0, 1) as d w(·, t) = A w(·, t) + B(U0 (t) + d(t)), dt where B = δ(x − 1). We have the following lemma directly.
(2.9)
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Then (2.1) transforms (1.1) into the following equivalent system: ⎧ ⎪ wt (x, t) = wxx (x, t), ⎪ ⎪ ⎪ ⎪ w ⎨ x (0, t) = c0 w(0, t), wx (1, t) = U(t) + d(t) + (c0 + q)w(1, t) − c0 (c0 + q) ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎩ e−c0 (1−x) w(x, t) dx,
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Lemma 2.1 Let A be defined by (2.6). Then A is dissipative and generates an exponential stable C0 -semigroup eA t on L2 (0, 1). Definition 2.1 (see Tucsnak & Weiss, 2000, p. 126) Let B ∈ L(U, X−1 ) and τ 0. We define Φτ ∈ L(L2 ([0, ∞); U), X−1 ) by τ Φτ u = Tt−τ Bu(σ ) dσ . (2.10) 0
Lemma 2.2 Let A and B be defined in (2.9). Then B is admissible to the semigroup eA t . Proof. By (2.8), we have B ∗ A ∗ −1 f = C
∀f ∈ L2 (0, 1).
Hence, B ∗ A ∗ −1 is bounded from L2 (0, 1) to C. Consider the dual system of (2.9): ⎧ ⎨ d w∗ (·, t) = A ∗ w∗ (·, t), dt ⎩ y(t) = B ∗ w∗ (·, t), that is,
⎧ ∗ wt (x, t) = w∗xx (x, t), ⎪ ⎪ ⎪ ⎨w∗ (0, t) = c w∗ (0, t), 0 x ∗ ⎪ (1, t) = 0, w ⎪ x ⎪ ⎩ y(t) = w∗ (1, t).
x ∈ (0, 1), t > 0, t 0, t 0,
(2.11)
A direct computation gives that the eigen-pairs {μn , gn (x)} of A ∗ are μn = −2c0 − (nπ )2 + O(n−1 ) as n → ∞. gn (x) = cos nπ x + O(n−1 ) Since A is self-adjoint, {gn (x)} forms an orthogonal basis for L2 (0, 1). So, the solution w∗ (x, t) of (2.11) can be written as ∞ w∗ (x, t) = bn eμn t gn (x). n=0
Hence, y(t) =
∞ n=0
bn eμn t gn (1).
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The operator B ∈ L(U; X−1 ) is called an admissible control operator for T if for some τ > 0, Ran Φτ ⊂ X . Note that if B is admissible, then in (2.10) (with t = τ ) we integrate in X−1 , but the integral is in X , a dense subspace of X−1 . The operator B (as in the above definition) is called bounded if B ∈ L(U, X ) (and unbounded otherwise). Obviously, every bounded B is admissible for T.
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ACTIVE DISTURBANCE REJECTION CONTROL AND SLIDING MODE CONTROL
A direct computation shows that, there exists a T > 0 such that
T
|y(t)|2 dt C0T
0
∞
|bn gn (1)|2 C1T
n=0
∞
|bn |2 C2T w∗ (·, 0) 2
n=0
for some constants CiT , i = 0, 1, 2 that depend on T only. This together with boundedness of B ∗ A ∗ −1 shows that B is admissible to the semigroup generated by A (see Weiss, 1989a,b or Tucsnak & Weiss, 2000). y1 (t) =
1
(2x3 − 3x2 )w(x, t) dx,
0
y2 (t) =
1
(12x − 6)w(x, t) dx.
(2.12)
0
Since B is admissible to the C0 -semigroup eA t , the solution of (2.5) is understood in the sense of d w(·, t), f = w(·, t), A ∗ f + f (1)(U0 (t) + d(t)) dt
∀f ∈ D(A ∗ ).
(2.13)
Let f (x) = 2x3 − 3x2 ∈ D(A ∗ ) in (2.13) to obtain y˙ 1 (t) = −U0 (t) − d(t) + y2 (t).
(2.14)
This implies that for any initial value w(·, 0) ∈ L2 (0, 1), the (weak) solution of (2.5) must satisfy (2.14). Now, we design the high gain estimators for y1 and d as follows: ⎧ 1 ⎪ ˆ ⎨y˙ˆ (t) = −(U0 (t) + d(t)) + y2 (t) − (ˆy(t) − y1 (t)), ε (2.15) 1 ⎪ ˙ˆ ⎩d(t) = 2 (ˆy(t) − y1 (t)), ε where ε > 0 is the tuning small parameter and dˆ is regarded as an approximation of d. Let y˜ (t) = yˆ (t) − y1 (t), be the errors. Then y˜ and d˜ satisfy ⎛ 1
− d y˜ (t) ⎜ ε = ⎝ ˜ 1 dt d(t) ε2
−1 0
˜ = d(t) ˆ − d(t) d(t)
(2.16)
⎞
y˜ (t) 0 ˙ ⎟ y˜ (t) ˙ + Bd(t), d(t) = A ˜ + ⎠ ˜ −1 d(t) d(t)
(2.17)
where
⎛ 1 ⎞
− −1 0 ⎜ ⎟ A=⎝ ε , B = , ⎠ 1 −1 0 ε2 and A is stable because the eigenvalues of A are √ √ 3 3 1 1 j, λ2 = − − j. λ1 = − + 2ε 2ε 2ε 2ε
(2.18)
(2.19)
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Let
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The state feedback controller to (2.5) is then designed as follows: ˆ U0 (t) = −d(t).
(2.20)
Proposition 2.1 Suppose that d˙ is uniformly bounded measurable. Then for any initial value w(·, 0) ∈ L2 (0, 1), the closed-loop system (2.21) admits an unique solution w(x, t) ∈ C((0, ∞); L2 (0, 1)). Moreover, the solution of system (2.21) tends to any arbitrary given vicinity of zero as t → ∞, ε → 0. ˜ defined in (2.16), we can write the equivalent system of (2.21) Proof. Using the error variables (˜y, d) as follows: ⎧ wt (x, t) = wxx (x, t), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ (0, t) = c w(0, t), t 0, w x 0 ⎪ ⎪ ⎪ ⎨w (1, t) = −d(t), ˜ t 0, x (2.22) 1 ⎪ ˜ ⎪ t 0, y˙˜ (t) = − y˜ (t) + d(t), ⎪ ⎪ ε ⎪ ⎪ ⎪ ⎪ ˙˜ = 1 y˜ (t) − d(t), ⎩d(t) ˙ t 0. ε2 ˜ is an external model for the ‘w part’ of the system (see Medvedev & It is seen from (2.22) that (˜y, d) Hillerström, 1995). So, we can solve this ODE separately to be
t y˜ (t) ˜ (0) At y ˙ ds, = e + eA(t−s) Bd(s) (2.23) ˜ ˜ d(t) d(0) 0 where A and B are defined by (2.18), and ⎛
⎞ λ1 ε 2 λ 1 λ 2 λ2 t λ2 λ1 t λ2 t λ1 t e − e (e − e ) ⎜ λ2 − λ1 ⎟ λ2 − λ1 λ2 − λ1 ⎟, eAt = ⎜ ⎝ ⎠ λ2 λ1 1 λ1 t λ2 t λ1 t λ2 t (e − e ) − e + e ε2 (λ2 − λ1 ) λ2 − λ1 λ2 − λ1 ⎞ ⎛ 2 ε λ 1 λ 2 λ2 t (e − eλ1 t ) ⎟ ⎜ λ 2 − λ1 At ⎟. ⎜ e B=−⎝ ⎠ λ2 λ 1 − eλ1 t + eλ2 t λ2 − λ1 λ2 − λ1
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It is clearly seen from (2.20) that the controller U0 (t) is just used to cancel the disturbance d because A generates an exponential stable C0 -semigroup. This estimation/cancellation strategy (2.20) is obviously an economic strategy. Under the feedback (2.20), the closed-loop system of (2.5) becomes ⎧ wt (x, t) = wxx (x, t), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ t 0, wx (0, t) = c0 w(0, t), ⎪ ⎪ ⎪ ⎨w (1, t) = −d(t) ˆ + d(t), t 0, x (2.21) 1 ⎪ ⎪y˙ˆ (t) = y2 (t) − (ˆy(t) − y1 (t)), ⎪ ⎪ ε ⎪ ⎪ ⎪ 1 ⎪ ˙ˆ ⎩d(t) = 2 (ˆy(t) − y1 (t)). ε
ACTIVE DISTURBANCE REJECTION CONTROL AND SLIDING MODE CONTROL
103
It is seen that in (2.23), the first term above can be arbitrarily small as t → 0 by the exponential stability of eAt , and the second term can also be arbitrarily small as ε → 0 due to boundedness of d˙ and the ˜ of (2.23) satisfies expression of eAt B. As a result, the solution (˜y, d) ˜ (˜y(t), d(t)) → 0 as t → ∞, ε → 0.
(2.24)
Now, we consider the ‘w part’ of the system (2.22) which is rewritten as x ∈ (0, 1), t > 0, t 0, t 0.
(2.25)
System (2.25) can be rewritten as an evolution equation in L2 (0, 1) as d ˜ w(·, t) = A w(·, t) − B d(t), dt
(2.26)
where A and B are the same as that in (2.9). Since A generates an exponential stable C0 -semigroup on L2 (0, 1) and B is admissible to eA t , for any initial value w(·, 0) ∈ L2 (0, 1), there exists a unique solution w ∈ C((0, ∞); L2 (0, 1)), which can be written as t At ˜ ds. eA (t−s) B d(s) (2.27) w(·, t) = e w(·, 0) − 0
˜ By (2.24), for any given ε0 > 0, there exist t0 > 0 and ε1 > 0 such that |d(t)| < ε0 for all t > t0 and 0 < ε < ε1 . We rewrite solution of (2.27) as w(·, t) = eA t w(·, 0) − eA (t−t0 )
t0
˜ ds − eA (t0 −s) B d(s)
t
˜ ds. eA (t−s) B d(s)
(2.28)
∀d˜ ∈ L∞ (0, ∞)
(2.29)
t0
0
The admissibility of B implies that t 2 eA (t−s) B d(s) ˜ ds Ct d
˜ 22 ˜ 2 Lloc (0,t) t Ct d L∞ (0,t) 0
˜ Since eA t is exponentially stable, it follows from Propofor some constant Ct that is independent of d. sition 2.5 in Weiss (1989a,b) that t t eA (t−s) B d(s) ˜ ˜ ˜ ds = eA (t−s) B(0 ♦ d)(s) ds L d L∞ (0,∞) Lε0 , t t0
0
(2.30)
0
˜ and where L is a constant that is independent of d, d1 (t), (d1 ♦ d2 )(t) = τ d2 (t − τ ),
0 t < τ, t τ,
(2.31)
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⎧ ⎪ ⎨wt (x, t) = wxx (x, t), wx (0, t) = c0 w(0, t), ⎪ ⎩ ˜ wx (1, t) = −d(t),
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J.-J. LIU AND J.-M. WANG
where the left-hand side of (2.31) denotes the τ -concatenation of d1 and d2 (see Weiss, 1989a,b). Suppose that eA t L0 e−ωt for some L0 , ω > 0. By (2.28–2.30), we have ˜ L∞ (0,t0 ) + Lε0 .
w(·, t) L0 e−ωt w(·, 0) + L0 Ct0 e−ω(t−t0 ) d
(2.32)
As t → ∞, the first two terms of (2.32) tend to zero. The result is then proved by the arbitrariness of ε0 .
Theorem 2.1 Suppose that d˙ is also uniformly bounded measurable. Then for any initial value u(·, 0) ∈ L2 (0, 1), the closed-loop system of (1.1) following: ⎧ ⎪ ut (x, t) = uxx (x, t), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ t 0, ⎨ux (0, t) = −qu(0, t), ˆ (1, t) = − d(t) − (c + q)u(1, t) − q(c + q) u 0 0 ⎪ x ⎪ ⎪ ⎪ 1 q(1−x) ⎪ ⎩ e u(x, t) dx + d(t), t 0,
(2.33)
0
admits a unique solution u ∈ C((0, ∞); L2 (0, 1)), and the solution of system (2.33) tends to any arbitrary given vicinity of zero as t → ∞, ε → 0 in L2 (0, 1), where the feedback control is ˆ − (c0 + q)u(1, t) − q(c0 + q) U(t) = −d(t)
1
eq(1−x) u(x, t) dx,
t 0,
(2.34)
0
and dˆ satisfies
with
⎧ 1 ⎪ ⎨y˙ˆ (t) = y2 (t) − (ˆy(t) − y1 (t)), ε 1 ⎪ ˙ˆ ⎩d(t) = 2 (ˆy(t) − y1 (t)) ε
(2.35)
⎧ 1 x ⎪ 3 2 q(x−y) ⎪ y (t) = (2x − 3x ) u(x, t) + (c + q) e u(y, t) dy dx, ⎪ 0 ⎨ 1 0
0
1 x ⎪ ⎪ ⎪ ⎩y2 (t) = (12x − 6) u(x, t) + (c0 + q) eq(x−y) u(y, t) dy dx. 0
(2.36)
0
3. SMC approach In this section, we design the SMC control for system (1.1) and suppose that d is uniformly bounded measurable, that is, |d(t)| M for some M > 0 and all t > 0. As given by Smyshlyaev & Krstic (2004),
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Returning back to system (1.1) by the inverse transformation (2.2), feedback control (2.4) and (2.20), and new variable (2.12), we have proved, from Proposition 2.1 the main result of this section.
ACTIVE DISTURBANCE REJECTION CONTROL AND SLIDING MODE CONTROL
105
we introduce another transformation:
x
w(x, t) = u(x, t) −
k(x, y)u(y, t) dy,
(3.1)
0
where the gain kernel k satisfies the following PDE: ⎧ k (x, y) − kyy (x, y) = ck(x, y), ⎪ ⎪ ⎨ xx ky (x, 0) + qk(x, 0) = 0, ⎪ ⎪ ⎩k(x, x) = − c x − q. 2
c > 0,
Lemma 3.1 The problem (3.2) admits a unique solution which is twice continuously differentiable in 0 y x. Proof. We introduce new variables ξ = x + y,
η=x−y
and define the function
G(ξ , η) = k(x, y) = k
(3.3)
ξ +η ξ −η , 2 2
.
(3.4)
0ηξ 2
(3.5)
It follows from (3.2) that G(ξ , η) satisfies the following PDE: c Gξ η (ξ , η) = G(ξ , η), 4 with the boundary conditions: ⎧ ⎪ Gξ (ξ , ξ ) − Gη (ξ , ξ ) + qG(ξ , ξ ) = 0, ⎪ ⎪ ⎪ ⎪ ⎨ ξc G(ξ , 0) = − − q, 4 ⎪ ⎪ ⎪ ⎪ c ⎪ ⎩Gξ (ξ , 0) = − . 4
0 ξ 2, 0 ξ 2,
(3.6)
By integrating (3.5), first with respect to η between 0 and η, and then with respect to ξ between η and ξ , we have an integral equation for G(ξ , η): G(ξ , η) = −qeqη −
c 2
η
eq(η−τ ) dτ +
0
c c − (ξ − η) + 4 4
ξ
c 2
η
eq(η−τ )
0
0
τ
G(τ , s) ds dτ 0
η
G(τ , s) ds dτ . η
(3.7)
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(3.2)
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J.-J. LIU AND J.-M. WANG
We now use the method of successive approximations to show that this equation has a unique continuous solution. Set ⎧ c η q(η−τ ) c ⎪ dτ − (ξ − η), ⎨G0 (ξ , η) = −qeqη − 0 e 2 4 (3.8) η q(η−τ ) τ c c ξ η ⎪ ⎩Gn (ξ , η) = e G (τ , s) ds dτ + G (τ , s) ds dτ , n 1. n−1 n−1 0 2 0 4 η 0 Then we have the estimate for G0 (ξ , η):
Suppose that |Gn (ξ , η)| M0n+1
(ξ + η)n . n!
Then, we have cM0n+1 |Gn+1 (ξ , η)| 4n!
2
η
τ
eη q
0
(τ + s) ds dτ + n
0
η
ξ
0
η
(τ + s) ds dτ n
cM0n+1 q (ξ + η)n+1 (2e + 2) 4n! n+1
M0n+2
(ξ + η)n+1 . (n + 1)!
(3.9)
By mathematical induction, (3.9) is true for all n 0. It then follows from the Weierstrass M-test that the series ∞ G(ξ , η) = Gn (ξ , η) n=0
converges absolutely and uniformly in 0 η ξ 2. Furthermore, by Theorem 4.17 of Liu (2009, p. 156), we deduce that G(ξ , η) =
∞
Gn (ξ , η) = −qeqη −
n=0
+
c 2
0
η
c eq(η−τ ) dτ − (ξ − η) 4
∞ ∞ c ξ η c η q(η−τ ) τ e Gn−1 (τ , s) ds dτ + Gn−1 (τ , s) ds dτ 2 0 4 η 0 0 n=1 n=1 c 2
η
c eq(η−τ ) dτ − (ξ − η) 4 0 τ η ∞ ∞ c c ξ η q(η−τ ) + e Gn−1 (τ , s) ds dτ + Gn−1 (τ , s) ds dτ 2 0 4 η 0 n=1 0 n=1
= −qeqη −
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c c |G0 (ξ , η)| qeq + eq η + |ξ − η| (2c + q)eq = M0 . 2 4
ACTIVE DISTURBANCE REJECTION CONTROL AND SLIDING MODE CONTROL
c − 2
η
c eq(η−τ ) dτ − (ξ − η) 4 0 τ η c η η c q(η−τ ) e G(τ , s) ds dτ + G(τ , s) ds dτ + 2 0 4 ξ 0 0
= −qe
qη
107
(3.10)
has zero solution only. Let Ω0 = {G | G(ξ , η) is continuous in 0 η ξ 2}. Define the mapping F : Ω0 → Ω0 by c ξ η c η q(η−τ ) τ e G(τ , s) ds dτ + G(τ , s) ds dτ (F G)(ξ , η) = 2 0 4 η 0 0
∀G ∈ Ω0 .
Then F is a compact operator on Ω0 . By (3.9), the spectral radius of F is zero. So, 0 is the unique spectrum of F . Therefore, (3.11) has zero solution only. The proof is complete. Remark 3.1 By the proof of Lemma 3.1, it is found that k(x, y) = G(ξ , η) = lim
N→∞
N
Gn (ξ , η),
ξ = x + y, η = x − y,
(3.12)
n=0
is uniformly convergent in x ∈ [0, 1], y ∈ [0, x], where Gn (ξ , η) is given by (3.8). This can be used to approximate the kernel function k numerically. The transformation (3.1) brings (1.1) into the following system: ⎧ ⎪ ⎪ ⎪wt (x, t) = wxx (x, t) − cw(x, t), ⎨ wx (0, t) = 0, 1 ⎪ ⎪ ⎪ kx (1, y)u(y, t) dy, ⎩wx (1, t) = U(t) + d(t) − k(1, 1)u(1, t) −
x ∈ (0, 1), t > 0, t 0,
(3.13)
t 0.
0
Now, we design the sliding mode surface W 2 S = f ∈ L (0, 1)
1
f (x) dx = 0
,
(3.14)
0
which is a closed subspace of L2 (0, 1). The corresponding sliding mode function for (3.13) is 1 w(x, t) dx. SW (t) = 0
(3.15)
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which shows that G(ξ , η) is a continuous solution of (3.7). Hence, G(ξ , η) is twice continuously differ˜ exp(M0 (ξ + η)). entiable in 0 η ξ 2, and |G(ξ , η)| M Now, we claim that this solution is unique. To this purpose, it suffices to show that the equation c ξ η c η q(η−τ ) τ e G(τ , s) ds dτ + G(τ , s) ds dτ (3.11) G(ξ , η) = 2 0 4 η 0 0
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J.-J. LIU AND J.-M. WANG
On the sliding mode surface SW (t) ≡ 0, (3.13) becomes ⎧ ⎨wt (x, t) = wxx (x, t) − cw(x, t), 1 ⎩wx (0, t) = w(x, t) dx = 0,
x ∈ (0, 1), t > 0, t 0.
(3.16)
0
Proposition 3.1 System (3.16) associates with a C0 -semigroup of contractions on S W , and is exponentially stable in S W with the decay rate −c. Proof. Define the operator A : D(A)(⊂ S W ) → S W as follows: Af (x) D(A)
= f (x) − cf (x) ∀f ∈ D(A), = {f ∈ S W ∩ H 2 (0, 1) | f ∈ S W , f (0) = 0}.
It is easy to show that for any f ∈ D(A), f ∈ S W if and only if f (1) = 0. So, we can write A as Af (x) = f (x) − cf (x) ∀f ∈ D(A), D(A) = {f ∈ S W ∩ H 2 (0, 1) | f (0) = f (1) = 0}. For any f ∈ D(A), we have
1
Re Af , f = − 0
|f (x)|2 dx − c
1
|f (x)|2 dx 0.
(3.17)
0
Hence A + cI is dissipative and so is for A. For any g ∈ S W solve Af = g, this is, f (x) − cf (x) = g(x), f (0) = f (1) = 0, to obtain the unique solution f as x √ ⎧ √ √ √ 1 cx − cx ⎪ ⎪ f (x) = c0 (e + e (e c(x−s) − e− c(x−s) )g(s) ds, )+ √ ⎪ ⎨ 2 c 0 1 √ ⎪ √ 1 ⎪ ⎪ ⎩c0 = − √ √ √ (e c(1−s) − e− c(1−s) ) ds. c − c 2 c(e + e ) 0 So, A−1 exists and is bounded on S W . By the Lumer–Phillips theorem (see Theorem 4.3 by Pazy, 1983, p. 14), A generates a C0 -semigroup of contractions on S W , and so does for A + cI. Therefore, the semigroup generated by A is exponentially stable with the decay rate −c.
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The Proposition 3.1 shows that system (3.16) decays exponentially in L2 (0, 1) as t → ∞ with the decay rate −c.
ACTIVE DISTURBANCE REJECTION CONTROL AND SLIDING MODE CONTROL
109
To motivate the control design, we differentiate SW (t) formally with respect to t to obtain
S˙ W (t) =
1
1
wt (x, t) dx =
0
[wxx (x, t) − cw(x, t)] dx
0
1
= wx (1, t) − c
w(x, t) dx = wx (1, t) − cSW (t),
(3.18)
0
and hence
1
kx (1, y)u(y, t) dy − cSW (t).
0
This suggests us to design the feedback controller: U(t) = k(1, 1)u(1, t) +
1
kx (1, y)u(y, t) dy + U0 (t),
(3.19)
0
where U0 is a new control. Then S˙ W (t) = U0 (t) + d(t) − cSw (t). For SW (t) = | 0, let U0 (t) = −(M + η)
SW (t) , |SW (t)|
where η > 0. Then we have SW (t) S˙ W (t) = −(M + η) + d(t) − cSw (t) |SW (t)|
for SW (t) = | 0.
(3.20)
Therefore, d |SW (t)|2 = 2SW (t)S˙ W (t) = −2(M + η)|SW (t)| + 2d(t)SW (t) − 2c|SW (t)|2 dt −2η|SW (t)|,
(3.21)
which is just the well-known finite time ‘reaching condition’. The sliding mode controller is U(t) = k(1, 1)u(1, t) + 0
1
kx (1, y)u(y, t) dy − (M + η)
SW (t) |SW (t)|
for SW (t) = | 0.
Under the control (3.22), the closed-loop of the target system (3.13) becomes ⎧ ⎪ ⎪ ⎪wt (x, t) = wxx (x, t) − cw(x, t), ⎪ ⎨w (0, t) = 0, x ⎪ ⎪ SW (t) ⎪ ⎪ ˜ | 0, + d(t) = d(t), SW (t) = ⎩wx (1, t) = −(M + η) |SW (t)|
(3.22)
(3.23)
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S˙ W (t) = U(t) + d(t) − k(1, 1)u(1, t) −
110 where
J.-J. LIU AND J.-M. WANG
˜ = −(M + η) SW (t) + d(t). d(t) |SW (t)|
(3.24)
The next result confirms the existence and uniqueness of the solution to (3.23) and the finite time ‘reaching condition’ to the sliding mode surface S W .
Proof. We rewrite system (3.23) as d ˜ w(·, t) = A0 w(·, t) + B0 d(t), dt where A0 is given by
B0 = δ(x − 1),
A0 f (x) = f (x) − cf (x), D(A0 ) = {f ∈ H 2 (0, 1) | f (0) = 0, f (1) = 0}.
A straightforward calculation shows that ⎧ ⎨A∗0 g(x) = g (x) − cg(x), ⎩D(∗ ) = {g ∈ H 2 (0, 1) | g (0) = g (1) = 0}. 0
(3.25)
(3.26)
(3.27)
It is easy to see that A∗0 = A0 is self-adjoint and dissipative. Hence, A0 generates a C0 -semigroup on L2 (0, 1). On the other hand, the dual system of (3.25) is ⎧ ⎪ w∗ (x, t) = w∗xx (x, t) − cw∗ (x, t), ⎪ ⎪ t ⎨ (3.28) w∗x (0, t) = w∗x (1, t) = 0, ⎪ ⎪ ⎪ ⎩ y(t) = B∗0 w∗ = w∗ (1, t) and the eigen-pairs (λn , ϕn ) of A∗0 are λn = −c − (nπ )2 , ϕn (x) = cos nπ x,
n = 0, 1, 2, . . . ,
2 ∗ 2 where {ϕn }∞ n=0 forms an orthogonal basis for L (0, 1). Therefore, for any w (·, 0) ∈ L (0, 1), we have
w∗ (x, 0) =
∞
an ϕn (x),
an = 2 w∗ (x, 0), ϕn (x)
n=0
and ∗
w∗ (x, t) = eA0 t w∗ (x, 0) =
∞ n=0
an eλn t ϕn (x).
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Proposition 3.2 Suppose that d is measurable and |d(t)| M for all t 0, and let SW be defined by | 0, there exists a tmax > 0 such that (3.23) admits a unique (3.15). Then for any w(·, 0) ∈ L2 (0, 1), SW (0) = solution w ∈ C((0, tmax ); L2 (0, 1)) and SW (t) = 0 for all t tmax .
ACTIVE DISTURBANCE REJECTION CONTROL AND SLIDING MODE CONTROL
111
So, y(t) =
∞
eλn t an ϕn (1).
n=0
A direct computation shows that, for any T > 0, there are positive constants CT , DT > 0 dependent on T only, such that T ∞ |y(t)|2 dt CT |an ϕn (1)|2 DT w∗ (·, 0) 2 . (3.29) n=0 A∗0 t
This shows that B∗0 is admissible for e and so is B0 for eA0 t (see Weiss, 1989a,b). Therefore, for | 0 for all t ∈ [0, T), then there exists a unique any T > 0 and w(·, 0) ∈ L2 (0, 1), if SW ∈ C[0, T], SW (t) = solution w ∈ C((0, T); L2 (0, 1)) to (3.23). Remark 3.2 We need to remark that the discontinuous term S(t)/|S(t)| is actually the unit controller and more details for the unit controllers of infinite-dimensional systems can be found in the monograph by Orlov (2009). Now we go back the original system (1.1). Under the feedback control (3.22), we get the closed-loop system for (1.1): ⎧ ut (x, t) = uxx (x, t), ⎪ ⎪ ⎪ ⎪ ⎪ ux (0, t) = −qu(0, t), ⎪ ⎪ ⎨
1 S(t) ux (1, t) = k(1, 1)u(1, t) + + d(t) kx (1, y)u(y, t) dy − (M + η) ⎪ |S(t)| ⎪ 0 ⎪ 1 ⎪ ⎪ ⎪ ⎪ ˜ ⎩ = k(1, 1)u(1, t) + kx (1, y)u(y, t) dy + d(t),
(3.30)
0
where
1
S(t) = SW (t) =
u(x, t) dx −
0
1
x
k(x, y)u(y, t) dy dx. 0
(3.31)
0
It is noted that system (3.30) is equivalent to (3.23) under the equivalent transformation (3.1). So, by (3.1), we can define a bounded invertible operator F : L2 (0, 1) → L2 (0, 1) by x w(x) = (F u)(x) = u(x) − k(x, y)u(y) dy ∀u ∈ L2 (0, 1), (3.32) 0
and the solution u(x, t) of (3.30) can be given by u(x, t) = (F −1 w)(x, t)
(3.33)
with w(x, 0) = (F u)(x, 0). Theorem 3.1 Suppose that d is bounded measurable and |d(t)| M . Let S(t) be given by (3.31). | 0, there exists a t0 > 0 such that (3.30) admits a unique solution Then for any u(·, 0) ∈ L2 (0, 1), S(0) = u ∈ C((0, t0 ); L2 (0, 1)) and S(t) = 0 for all t t0 . Moreover, S(t) is continuous, monotone in [0, t0 ]. On
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0
112
J.-J. LIU AND J.-M. WANG
the sliding surface S(t) = 0, the system (1.1) becomes ⎧ ⎪ ut (x, t) = uxx (x, t), ⎪ ⎪ ⎪ ⎨u (0, t) = −qu(0, t), x 1 x ⎪ 1 ⎪ ⎪ ⎪ u(x, t) dx − k(x, y)u(y, t) dy dx = 0, ⎩ 0
0
(3.34)
0
Proof. We only need to prove that S(t) is continuous, monotone in [0, t0 ]. Suppose, without any loss of generality, that S(0) > 0 since the proof for S(0) < 0 is similar. Since (3.30) is equivalent to (3.23) and B0 is admissible for eA0 t , the solution to the system (3.23) can be written as t ˜ ds. (3.35) eA0 (t−s) B0 d(s) w(x, t) = eA0 t w(x, 0) + 0
So, by (3.32) and (3.33), the solution to the system (3.30) can be written as u(x, t) = [F −1 eA0 t F u](x, 0) +
t
˜ ds. F −1 eA0 (t−s) B0 d(s)
(3.36)
0
Moreover, for any T > 0, there exists a constant CT > 0 such that (see Weiss, 1989a,b) ˜ L2 (0,T) ]
u(·, t) CT [ u(·, 0) + d
∀t ∈ [0, T].
(3.37)
Now for d˜ ∈ L2 (0, T) defined by (3.24), since H01 (0, T) is dense in L2 (0, T), take d˜ n ∈ H01 (0, T) such that ˜ L2 (0,T) = 0. lim d˜ n − d
n→∞
Let un (·, t) be the solution of (3.25) corresponding to d˜ n and the initial value un (·, 0) ∈ D(A0 ) (note that D(A0 ) is dense in L2 (0, 1)), where lim un (·, 0) − u(·, 0) = 0.
n→∞
It follows from (3.37) that lim un (·, t) − u(·, t) = 0
n→∞
uniformly in t ∈ [0, T]. Since un (·, t) and wn (·, t) are the classical solutions of (3.30) and (3.23), respectively, by Proposition 4.2.1 in Weiss (1989a,b, p. 120), it follows from (3.20) that
1
un (x, t) dx −
0 −ct
=e
0
0 1
1
x
k(x, y)un (y, t) dy dx
0
un (x, 0) r dx − 0
1
0
x
t k(x, y)un (y, 0) dy dx + e−c(t−τ ) d˜ n (τ ) dτ . 0
(3.38)
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which is exponentially stable in L2 (0, 1) with the decay rate −c.
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ACTIVE DISTURBANCE REJECTION CONTROL AND SLIDING MODE CONTROL
Passing to the limit as n → ∞ in the above equality yields
1
1
u(x, t) dx −
0
= e−ct
k(x, y)u(y, t) dy dx 0
1
x
0
1
u(x, 0) rdx −
0
0
x
t ˜ ) dτ . k(x, y)u(y, 0) dy dx + e−c(t−τ ) d(τ
0
(3.39)
0
S(t) =
1
0 −ct
1
u(x, t) dx −
=e
0
= e−ct e−ct
1
1
1
0
1
u(x, 0) dx −
0
0
u(x, 0) dx −
0
x
k(x, y)u(y, t) rdy dx 0
1
1
0
t ˜ ) dτ k(x, y)u(y, 0) dy dx + e−c(t−τ ) d(τ
x
t k(x, y)u(y, 0) dy dx − e−c(t−τ ) [(M + η) − d(τ )] dτ
0
0
u(x, 0) dx −
x
0
0
0
0
x
η η k(x, y)u(y, 0) dy dx − + e−ct . c c
(3.40)
Hence, S(t) is decreasing in t. Since S(0) > 0, there exists a t0 > 0 such that S(t) > 0 for t ∈ [0, t0 ) and S(t) = 0 for all t t0 . The proof is complete.
Fig. 1. The displacement with d(t) = 2 sin t by SMC.
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Since T is arbitrary, we see that S(t) defined by (3.31) with d˜ defined by (3.24) is continuous in the whole half line [0, ∞) for any initial value in L2 (0, 1). Furthermore, by (3.39) and S(0) > 0, we have
114
J.-J. LIU AND J.-M. WANG 10
5
0
−10
−15
0
0.5
1
1.5
2
Fig. 2. The controller by SMC.
Fig. 3. The displacement with d(t) = 2 sin t by ADRC.
4. Numerical simulations In this section, the finite difference method is applied to compute the displacements numerically for both SMC and ADRC to illustrate the effect of the controllers. Figure 1 shows the displacement of system (3.30). Here the steps of space and time are taken as 0.1 and 0.0001, respectively. We choose q = 1, c = 10, M0 = 4, η = 1, u(x, 0) = 10x3 and d = 2 sin t. The kernel function is approximated by
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−5
ACTIVE DISTURBANCE REJECTION CONTROL AND SLIDING MODE CONTROL
115
20
15
10
5
0
−5
0
0.5
1
1.5
2
Fig. 5. The controller by ADRC.
(3.12) with N = 20. Figure 2 shows the controller by SMC. Owing to discontinuity, the control vibrates rapidly after some time. Figure 3 shows the spatiotemporal profile of the solution of the system (2.33) with the same space and time sizes used in SMC. Other parameters are q = c0 = 1, ε = 0.01, u(x, 0) = x, d(t) = cos t. It is seen that in both cases, the displacements are obviously convergent. Moreover, Fig. 4 shows that convergence of dˆ to the disturbance d. Figure 5 shows the controller by ADRC. It is much better than that by SMC.
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ˆ (red dotted line) and disturbance d(t) = cos t (blue solid line) by ADRC. Fig. 4. The d(t)
116
J.-J. LIU AND J.-M. WANG
5. Concluding remarks
Acknowledgements The authors thank the anonymous referees’ comments and suggestions for improving the paper. Funding This work was supported by the National Natural Science Foundation of China. References Breger, A. M., Butkovskii, A. G., Kubyshkin, V. A. & Utkin, V. I. (1980) Sliding modes for control of distributed parameter entities subjected to a mobile multicycle signal. Automat. Remote Control, 41, 346–355. Brockett, R. W. (2001) New issues in the mathematics of control. Mathematics Unlimited–2001 and Beyond (B. Engquist and W. Schmid, eds). Berlin: Springer, pp. 189–219. Chen, G., Delfour, M. C., Krall, A. M. & Payre, G. (1987) Modeling, stabilization and control of serially connected beam. SIAM J. Control Optim., 25, 526–546. 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, 381–387. Drakunov, S., Barbieeri, E. & Silver, D. A. (1996) Sliding mode control of a heat equation with application to arc welding. Proceedings of the 1996 IEEE International Conference On Control Applications Dearborn, MI, September 15–18, 668–672. Guo, B. Z. & Jin, F. F. (2013) Sliding mode and active disturbance rejection control to the stabilization of antistable one-dimensional wave equation subject to boundary input disturbance. IEEE Trans. Automat. Control, 58, 1269–1274. Guo, B. Z. & Liu, J. J. (2013) Sliding mode control and active disturbance rejection control to the stabilization of one-dimensional Schrodinger equation subject to boundary control matched disturbance. Int. J. Robust Nonlinear Control, in press. Guo, B. Z. & Xu, C. Z. (2007) The stabilization of a one-dimensional wave equation by boundary feedback with noncollocated observation. IEEE Trans. Automat. Control, 52, 371–377. Guo, B. Z. & Zhao, Z. L. (2011) On the convergence of extended state observer for nonlinear systems with uncertainty. Systems Control Lett., 60, 420–430. Guo, W. & Guo, B. Z. (2013) Stabilization and regulator design for a one-dimensional unstable wave equation with input harmonic disturbance. Internat. J. Robust 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 non-collocated control. Internat. J. Robust Nonlinear Control, 21, 1297–1321.
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In this paper, we apply two different approaches of the ADRC and the sliding mode control (SMC) to stabilize a one-dimensional unstable heat equation subject to boundary control matched disturbance. We first apply the ADRC to cancel the disturbance which is supposed to have a bounded derivative. The ADRC is an online estimation/cancellation control strategy, and a high gain estimator is designed to estimate the disturbance. The well-posedness of the closed-loop system is presented, and it is shown that the closed-loop system can reach any arbitrary given vicinity of zero as time goes to infinity and high gain tuning parameter goes to zero. Secondly, we apply the SMC approach to reject the disturbance which is supposed to be bounded. The sliding mode surface is found to be a closed subspace of L2 (0, 1). On the sliding mode surface, the system is shown to be exponentially stable with arbitrary prescribed decay rate.
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Krstic, M. (2010) Adaptive control of an anti-stable wave PDE. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 17, 853–882. Krstic, M., Guo, B. Z., Balogh, A. & Smyshlyaev, A. (2008) Output-feedback stabilization of an unstable wave equation. Automatica, 44, 63–74. Krstic, M. & Smyshlyaev, A. (2008) Boundary Control of PDEs: A Course on Backstepping Designs. Philadelphia, PA: SIAM. Liu, W. J. (2009) Elementary Feedback Stabilization of the Linear Reaction–Convection–Diffusion Equation and the Wave Equation. New York: Springer. Luo, Z. H., Guo, B. Z. & Morgul, O. (1999) Stability and Stabilization of Infinite Dimensional Systems with Applications. London: Springer. Machtyngier, E. (1994) Exact controllability for the Schrödinger equation. SIAM J. Control Optim., 32, 24–34. Medvedev, A. & Hillerström, G. (1995) An external model control system. Control-Theory Adv. Technol., 10, 1643–1665. Orlov, Y. V. (2009) Discontinuous Systems Lyapunov Analysis and Robust Synthesis under Uncertainty Conditions. London: Springer. Orlov, Y. V. & Utkin, V. I. (1987) Sliding mode control in infinite-dimensional systems. Automatica, 23, 753–757. Pazy, A. (1983) Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer. Pisano, A. & Orlov, Y. (2012) Boundary second-order sliding-mode control of an uncertain heat process with unbounded matched perturbation. Automatica., 48, 1768–1775. Smyshlyaev, A. & Krstic, M. (2004) Closed-form boundary state feedbacks for a class of 1-D partial integrodifferential equations. IEEE Trans. Automat. Control, 49, 2185–2202. Tucsnak, M. & Weiss, G. (2000) Observation and Control for Operator Semigroups. Basel: Birkhäuser. Weiss, G. (1989a) Admissibility of unbounded control operators. SIAM J. Control Optim., 27, 527–545. Weiss, G. (1989b) Admissible observation operators for linear semigroups. Israel J. Math., 65, 17–43.
Active disturbance rejection control and sliding mode control of one-dimensional unstable heat equation with boundary uncertainties
[Received on 18 December 2012; revised on 8 May 2013; accepted on 15 September 2013] In this paper, we are concerned with the boundary stabilization of a one-dimensional unstable heat equation with the external disturbance flowing into the control end. The active disturbance rejection control (ADRC) and the sliding mode control (SMC) are adopted in investigation. By the ADRC approach, the disturbance is estimated through an external observer and cancelled online by the approximated one in the closed-loop. It is shown that the external disturbance can be attenuated in the sense that the resulting closed-loop system under the extended state feedback tends to any arbitrary given vicinity of zero as the time goes to infinity. In the second part, we use the SMC to reject the disturbance with the assumption in which the disturbance is supposed to be bounded. The reaching condition, and the existence and uniqueness of the solution for all states in the state space via SMC are established. Simulation examples are presented for both control strategies. Keywords: heat equation; active disturbance rejection control; sliding mode control; boundary control; disturbance rejection.
1. Introduction In the past two decades, the boundary control of systems described by partial differential equations (PDEs) has become an important research in the area of distributed parameter systems control. Many contributions have been made, such as, Chen et al. (1987), Guo & Xu (2007), Krstic & Smyshlyaev (2008), Luo et al. (1999), and the references therein. Traditionally, the system is controlled in the ideal operational environment with exact mathematical model and no internal/external disturbances, for instance, the stabilization for Schrödinger, wave and flexible beam equations by Machtyngier (1994), Luo et al. (1999) and Chen et al. (1987), respectively. Generally speaking, there are two different types of control methods, collocated and non-collocated control designs, which are used to stabilize the PDE systems without any disturbance. The collocated control design is based on the passive principle that makes the closed-loop system dissipative and hence the system is stable at least in the sense of Lyapunov (see Chen et al., 1987). On the other hand, due to the backstepping method introduced into the PDE systems in the last few years by Krstic & Smyshlyaev (2008) (see also Guo & Xu, 2007), the non-collocated method is systematically applied to stabilize some unstable or even anti-stable wave and heat equations (see Smyshlyaev & Krstic, 2004; Krstic & Smyshlyaev, 2008; Krstic et al., 2008). It is known that if there is no uncertainty in the system, the control or the environment, feedback control is largely unnecessary (see Brockett, 2001). So, when the external disturbances enter the system from the boundary/internal of the spatial domain, the new approach is needed to deal with the uncertainties. There are three powerful methods in dealing with the uncertainties: One is the traditionally adaptive c The authors 2013. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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Jun-Jun Liu∗ and Jun-Min Wang School of Mathematics, Beijing Institute of Technology, Beijing 100081, P.R. China ∗ Corresponding author: Email: [email protected]
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J.-J. LIU AND J.-M. WANG
where u is the state and U(t) is the control input. The system represents an unstable distributed parameter system in the sense that for q > 0 large enough, there are finitely many eigenvalues of the system (with no control and disturbance) located on the right-half complex plane. The main aim of this paper is to apply both the ADRC and SMC approaches to attenuate and reject the disturbance in the stabilization of (1.1) for the unknown disturbance d supposed to have bounded ˙ M ) and to be uniformly bounded measurable (|d| M ), respectively. derivative (|d| The rest of the paper is organized as follows. In Section 2, we use the ADRC approach to attenuate the disturbance by designing a high gain estimator to estimate the disturbance. After cancelling the disturbance by the approximated one, we design the state feedback controller. The closed-loop system is shown to attend any arbitrary given vicinity of zero as the time goes to infinity and the gain tuning parameter tends to zero. The SMC for disturbance rejection is presented in Section 3. The SMC is designed and the existence and uniqueness of solution of the closed-loop system are proved. Simulation results are given in Section 4 and some concluding remarks are presented in Section 5. 2. The ADRC approach ˙ is bounded measurable. Firstly, In this section, we design the ADRC control for (1.1) and suppose |d| we introduce a transformation (see Krstic & Smyshlyaev, 2008, pp. 76–77):
x
w(x, t) = u(x, t) + (c0 + q)
eq(x−y) u(y, t) dy, 0
c0 > 0.
(2.1)
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control method in dealing with the systems with the unknown parameters (see Guo et al., 2011; Guo & Guo, 2013 or Krstic, 2010). The second is the active disturbance rejection control (ADRC) method (initially proposed by Han in 1990s) in dealing with the disturbance, which is an unconventional and effective control design for lumped parameter systems in the absence of proper models and in the presence of model uncertainty. The challenge convergence problem for the ADRC method is settled by Guo & Zhao (2011) recently and the ADRC method has been successfully applied to the attenuation of disturbance for a one-dimensional anti-stable wave equation by Guo & Jin (2013). The third is the sliding mode control (SMC) method to reject the disturbance (see Breger et al., 1980; Drakunov et al., 1996; Cheng et al., 2011 or Orlov & Utkin, 1987). In Orlov & Utkin (1987), based on the semigroup theory, the SMC is used to deal with a class of abstract infinite-dimensional systems where the control and disturbance are bounded. The boundary stabilization for a one-dimensional heat equation with boundary disturbance is studied in Drakunov et al. (1996), where the SMC is designed for the first-order PDEs obtained through an integral transformation on the heat equation (which is second order in spatial variable). The sliding mode boundary stabilizer is also designed for a one-dimensional unstable heat, wave and Schrödinger equation by Cheng et al. (2011), Guo & Jin (2013) and Guo & Liu (2013), respectively. The perturbed heat equation with diffusivity and homogeneous Neumann-type boundary condition was studied by Pisano & Orlov (2012), where the initial condition of the heat equation is assumed to belong to H 4 and the proposed infinite-dimensional treatment of the system retains robustness features against non-vanishing matched disturbances. In this paper, we are concerned with the following unstable heat equation in L2 (0, 1): ⎧ ⎪ x ∈ (0, 1), t > 0, ⎨ut (x, t) = uxx (x, t), (1.1) q > 0, t 0, ux (0, t) = −qu(0, t), ⎪ ⎩ ux (1, t) = U(t) + d(t), t 0,
ACTIVE DISTURBANCE REJECTION CONTROL AND SLIDING MODE CONTROL
It is known that (2.1) has an invertible transformation: x e−c0 (x−y) w(y, t) dy. u(x, t) = w(x, t) − (c0 + q)
99
(2.2)
0
x ∈ (0, 1), t > 0, t 0, (2.3) t 0.
0
It is seen that the unstable factor −qu(0, t) in (1.1) becomes the dissipative term c0 w(0, t) in (2.3) under the transformation (2.1), both at the end x = 0. In what follows, we consider the stabilization of system (2.3) until the final step to go back system (1.1) under the inverse transformation (2.2). By introducing a new controller U0 (t) so that 1 U(t) = U0 (t) − (c0 + q)w(1, t) + c0 (c0 + q) e−c0 (1−x) w(x, t) dx, (2.4) 0
(2.3) becomes
⎧ ⎪ ⎨wt (x, t) = wxx (x, t), wx (0, t) = c0 w(0, t), ⎪ ⎩ wx (1, t) = U0 (t) + d(t).
x ∈ (0, 1), t > 0, t 0,
(2.5)
Now we write (2.5) into the operator form in L2 (0, 1). Define the operator A : D(A )(⊂ L2 (0, 1)) → L2 (0, 1) as follows: A f (x) = f (x), (2.6) D(A ) = {f ∈ H 2 (0, 1) | f (0) = c0 f (0), f (1) = 0}. A direct computation gives A ∗ f (x) = f (x), D(A ∗ ) = {f ∈ H 2 (0, 1) | f (0) = c0 f (0), f (1) = 0} with
⎧ x ⎪ ∗ −1 ⎪ f =C + (x − τ )f (τ ) dτ ∀f ∈ L2 (0, 1), ⎨A 1 1 1 1 ⎪ ⎪ f (x) dx − xf (x) dx. ⎩C = − c0 0 0
(2.7)
(2.8)
It is seen that A is self-adjoint in L2 (0, 1) and (2.5) can be written as an evolution equation in L2 (0, 1) as d w(·, t) = A w(·, t) + B(U0 (t) + d(t)), dt where B = δ(x − 1). We have the following lemma directly.
(2.9)
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Then (2.1) transforms (1.1) into the following equivalent system: ⎧ ⎪ wt (x, t) = wxx (x, t), ⎪ ⎪ ⎪ ⎪ w ⎨ x (0, t) = c0 w(0, t), wx (1, t) = U(t) + d(t) + (c0 + q)w(1, t) − c0 (c0 + q) ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎩ e−c0 (1−x) w(x, t) dx,
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Lemma 2.1 Let A be defined by (2.6). Then A is dissipative and generates an exponential stable C0 -semigroup eA t on L2 (0, 1). Definition 2.1 (see Tucsnak & Weiss, 2000, p. 126) Let B ∈ L(U, X−1 ) and τ 0. We define Φτ ∈ L(L2 ([0, ∞); U), X−1 ) by τ Φτ u = Tt−τ Bu(σ ) dσ . (2.10) 0
Lemma 2.2 Let A and B be defined in (2.9). Then B is admissible to the semigroup eA t . Proof. By (2.8), we have B ∗ A ∗ −1 f = C
∀f ∈ L2 (0, 1).
Hence, B ∗ A ∗ −1 is bounded from L2 (0, 1) to C. Consider the dual system of (2.9): ⎧ ⎨ d w∗ (·, t) = A ∗ w∗ (·, t), dt ⎩ y(t) = B ∗ w∗ (·, t), that is,
⎧ ∗ wt (x, t) = w∗xx (x, t), ⎪ ⎪ ⎪ ⎨w∗ (0, t) = c w∗ (0, t), 0 x ∗ ⎪ (1, t) = 0, w ⎪ x ⎪ ⎩ y(t) = w∗ (1, t).
x ∈ (0, 1), t > 0, t 0, t 0,
(2.11)
A direct computation gives that the eigen-pairs {μn , gn (x)} of A ∗ are μn = −2c0 − (nπ )2 + O(n−1 ) as n → ∞. gn (x) = cos nπ x + O(n−1 ) Since A is self-adjoint, {gn (x)} forms an orthogonal basis for L2 (0, 1). So, the solution w∗ (x, t) of (2.11) can be written as ∞ w∗ (x, t) = bn eμn t gn (x). n=0
Hence, y(t) =
∞ n=0
bn eμn t gn (1).
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The operator B ∈ L(U; X−1 ) is called an admissible control operator for T if for some τ > 0, Ran Φτ ⊂ X . Note that if B is admissible, then in (2.10) (with t = τ ) we integrate in X−1 , but the integral is in X , a dense subspace of X−1 . The operator B (as in the above definition) is called bounded if B ∈ L(U, X ) (and unbounded otherwise). Obviously, every bounded B is admissible for T.
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ACTIVE DISTURBANCE REJECTION CONTROL AND SLIDING MODE CONTROL
A direct computation shows that, there exists a T > 0 such that
T
|y(t)|2 dt C0T
0
∞
|bn gn (1)|2 C1T
n=0
∞
|bn |2 C2T w∗ (·, 0) 2
n=0
for some constants CiT , i = 0, 1, 2 that depend on T only. This together with boundedness of B ∗ A ∗ −1 shows that B is admissible to the semigroup generated by A (see Weiss, 1989a,b or Tucsnak & Weiss, 2000). y1 (t) =
1
(2x3 − 3x2 )w(x, t) dx,
0
y2 (t) =
1
(12x − 6)w(x, t) dx.
(2.12)
0
Since B is admissible to the C0 -semigroup eA t , the solution of (2.5) is understood in the sense of d w(·, t), f = w(·, t), A ∗ f + f (1)(U0 (t) + d(t)) dt
∀f ∈ D(A ∗ ).
(2.13)
Let f (x) = 2x3 − 3x2 ∈ D(A ∗ ) in (2.13) to obtain y˙ 1 (t) = −U0 (t) − d(t) + y2 (t).
(2.14)
This implies that for any initial value w(·, 0) ∈ L2 (0, 1), the (weak) solution of (2.5) must satisfy (2.14). Now, we design the high gain estimators for y1 and d as follows: ⎧ 1 ⎪ ˆ ⎨y˙ˆ (t) = −(U0 (t) + d(t)) + y2 (t) − (ˆy(t) − y1 (t)), ε (2.15) 1 ⎪ ˙ˆ ⎩d(t) = 2 (ˆy(t) − y1 (t)), ε where ε > 0 is the tuning small parameter and dˆ is regarded as an approximation of d. Let y˜ (t) = yˆ (t) − y1 (t), be the errors. Then y˜ and d˜ satisfy ⎛ 1
− d y˜ (t) ⎜ ε = ⎝ ˜ 1 dt d(t) ε2
−1 0
˜ = d(t) ˆ − d(t) d(t)
(2.16)
⎞
y˜ (t) 0 ˙ ⎟ y˜ (t) ˙ + Bd(t), d(t) = A ˜ + ⎠ ˜ −1 d(t) d(t)
(2.17)
where
⎛ 1 ⎞
− −1 0 ⎜ ⎟ A=⎝ ε , B = , ⎠ 1 −1 0 ε2 and A is stable because the eigenvalues of A are √ √ 3 3 1 1 j, λ2 = − − j. λ1 = − + 2ε 2ε 2ε 2ε
(2.18)
(2.19)
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Let
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The state feedback controller to (2.5) is then designed as follows: ˆ U0 (t) = −d(t).
(2.20)
Proposition 2.1 Suppose that d˙ is uniformly bounded measurable. Then for any initial value w(·, 0) ∈ L2 (0, 1), the closed-loop system (2.21) admits an unique solution w(x, t) ∈ C((0, ∞); L2 (0, 1)). Moreover, the solution of system (2.21) tends to any arbitrary given vicinity of zero as t → ∞, ε → 0. ˜ defined in (2.16), we can write the equivalent system of (2.21) Proof. Using the error variables (˜y, d) as follows: ⎧ wt (x, t) = wxx (x, t), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ (0, t) = c w(0, t), t 0, w x 0 ⎪ ⎪ ⎪ ⎨w (1, t) = −d(t), ˜ t 0, x (2.22) 1 ⎪ ˜ ⎪ t 0, y˙˜ (t) = − y˜ (t) + d(t), ⎪ ⎪ ε ⎪ ⎪ ⎪ ⎪ ˙˜ = 1 y˜ (t) − d(t), ⎩d(t) ˙ t 0. ε2 ˜ is an external model for the ‘w part’ of the system (see Medvedev & It is seen from (2.22) that (˜y, d) Hillerström, 1995). So, we can solve this ODE separately to be
t y˜ (t) ˜ (0) At y ˙ ds, = e + eA(t−s) Bd(s) (2.23) ˜ ˜ d(t) d(0) 0 where A and B are defined by (2.18), and ⎛
⎞ λ1 ε 2 λ 1 λ 2 λ2 t λ2 λ1 t λ2 t λ1 t e − e (e − e ) ⎜ λ2 − λ1 ⎟ λ2 − λ1 λ2 − λ1 ⎟, eAt = ⎜ ⎝ ⎠ λ2 λ1 1 λ1 t λ2 t λ1 t λ2 t (e − e ) − e + e ε2 (λ2 − λ1 ) λ2 − λ1 λ2 − λ1 ⎞ ⎛ 2 ε λ 1 λ 2 λ2 t (e − eλ1 t ) ⎟ ⎜ λ 2 − λ1 At ⎟. ⎜ e B=−⎝ ⎠ λ2 λ 1 − eλ1 t + eλ2 t λ2 − λ1 λ2 − λ1
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It is clearly seen from (2.20) that the controller U0 (t) is just used to cancel the disturbance d because A generates an exponential stable C0 -semigroup. This estimation/cancellation strategy (2.20) is obviously an economic strategy. Under the feedback (2.20), the closed-loop system of (2.5) becomes ⎧ wt (x, t) = wxx (x, t), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ t 0, wx (0, t) = c0 w(0, t), ⎪ ⎪ ⎪ ⎨w (1, t) = −d(t) ˆ + d(t), t 0, x (2.21) 1 ⎪ ⎪y˙ˆ (t) = y2 (t) − (ˆy(t) − y1 (t)), ⎪ ⎪ ε ⎪ ⎪ ⎪ 1 ⎪ ˙ˆ ⎩d(t) = 2 (ˆy(t) − y1 (t)). ε
ACTIVE DISTURBANCE REJECTION CONTROL AND SLIDING MODE CONTROL
103
It is seen that in (2.23), the first term above can be arbitrarily small as t → 0 by the exponential stability of eAt , and the second term can also be arbitrarily small as ε → 0 due to boundedness of d˙ and the ˜ of (2.23) satisfies expression of eAt B. As a result, the solution (˜y, d) ˜ (˜y(t), d(t)) → 0 as t → ∞, ε → 0.
(2.24)
Now, we consider the ‘w part’ of the system (2.22) which is rewritten as x ∈ (0, 1), t > 0, t 0, t 0.
(2.25)
System (2.25) can be rewritten as an evolution equation in L2 (0, 1) as d ˜ w(·, t) = A w(·, t) − B d(t), dt
(2.26)
where A and B are the same as that in (2.9). Since A generates an exponential stable C0 -semigroup on L2 (0, 1) and B is admissible to eA t , for any initial value w(·, 0) ∈ L2 (0, 1), there exists a unique solution w ∈ C((0, ∞); L2 (0, 1)), which can be written as t At ˜ ds. eA (t−s) B d(s) (2.27) w(·, t) = e w(·, 0) − 0
˜ By (2.24), for any given ε0 > 0, there exist t0 > 0 and ε1 > 0 such that |d(t)| < ε0 for all t > t0 and 0 < ε < ε1 . We rewrite solution of (2.27) as w(·, t) = eA t w(·, 0) − eA (t−t0 )
t0
˜ ds − eA (t0 −s) B d(s)
t
˜ ds. eA (t−s) B d(s)
(2.28)
∀d˜ ∈ L∞ (0, ∞)
(2.29)
t0
0
The admissibility of B implies that t 2 eA (t−s) B d(s) ˜ ds Ct d
˜ 22 ˜ 2 Lloc (0,t) t Ct d L∞ (0,t) 0
˜ Since eA t is exponentially stable, it follows from Propofor some constant Ct that is independent of d. sition 2.5 in Weiss (1989a,b) that t t eA (t−s) B d(s) ˜ ˜ ˜ ds = eA (t−s) B(0 ♦ d)(s) ds L d L∞ (0,∞) Lε0 , t t0
0
(2.30)
0
˜ and where L is a constant that is independent of d, d1 (t), (d1 ♦ d2 )(t) = τ d2 (t − τ ),
0 t < τ, t τ,
(2.31)
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⎧ ⎪ ⎨wt (x, t) = wxx (x, t), wx (0, t) = c0 w(0, t), ⎪ ⎩ ˜ wx (1, t) = −d(t),
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J.-J. LIU AND J.-M. WANG
where the left-hand side of (2.31) denotes the τ -concatenation of d1 and d2 (see Weiss, 1989a,b). Suppose that eA t L0 e−ωt for some L0 , ω > 0. By (2.28–2.30), we have ˜ L∞ (0,t0 ) + Lε0 .
w(·, t) L0 e−ωt w(·, 0) + L0 Ct0 e−ω(t−t0 ) d
(2.32)
As t → ∞, the first two terms of (2.32) tend to zero. The result is then proved by the arbitrariness of ε0 .
Theorem 2.1 Suppose that d˙ is also uniformly bounded measurable. Then for any initial value u(·, 0) ∈ L2 (0, 1), the closed-loop system of (1.1) following: ⎧ ⎪ ut (x, t) = uxx (x, t), x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ t 0, ⎨ux (0, t) = −qu(0, t), ˆ (1, t) = − d(t) − (c + q)u(1, t) − q(c + q) u 0 0 ⎪ x ⎪ ⎪ ⎪ 1 q(1−x) ⎪ ⎩ e u(x, t) dx + d(t), t 0,
(2.33)
0
admits a unique solution u ∈ C((0, ∞); L2 (0, 1)), and the solution of system (2.33) tends to any arbitrary given vicinity of zero as t → ∞, ε → 0 in L2 (0, 1), where the feedback control is ˆ − (c0 + q)u(1, t) − q(c0 + q) U(t) = −d(t)
1
eq(1−x) u(x, t) dx,
t 0,
(2.34)
0
and dˆ satisfies
with
⎧ 1 ⎪ ⎨y˙ˆ (t) = y2 (t) − (ˆy(t) − y1 (t)), ε 1 ⎪ ˙ˆ ⎩d(t) = 2 (ˆy(t) − y1 (t)) ε
(2.35)
⎧ 1 x ⎪ 3 2 q(x−y) ⎪ y (t) = (2x − 3x ) u(x, t) + (c + q) e u(y, t) dy dx, ⎪ 0 ⎨ 1 0
0
1 x ⎪ ⎪ ⎪ ⎩y2 (t) = (12x − 6) u(x, t) + (c0 + q) eq(x−y) u(y, t) dy dx. 0
(2.36)
0
3. SMC approach In this section, we design the SMC control for system (1.1) and suppose that d is uniformly bounded measurable, that is, |d(t)| M for some M > 0 and all t > 0. As given by Smyshlyaev & Krstic (2004),
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Returning back to system (1.1) by the inverse transformation (2.2), feedback control (2.4) and (2.20), and new variable (2.12), we have proved, from Proposition 2.1 the main result of this section.
ACTIVE DISTURBANCE REJECTION CONTROL AND SLIDING MODE CONTROL
105
we introduce another transformation:
x
w(x, t) = u(x, t) −
k(x, y)u(y, t) dy,
(3.1)
0
where the gain kernel k satisfies the following PDE: ⎧ k (x, y) − kyy (x, y) = ck(x, y), ⎪ ⎪ ⎨ xx ky (x, 0) + qk(x, 0) = 0, ⎪ ⎪ ⎩k(x, x) = − c x − q. 2
c > 0,
Lemma 3.1 The problem (3.2) admits a unique solution which is twice continuously differentiable in 0 y x. Proof. We introduce new variables ξ = x + y,
η=x−y
and define the function
G(ξ , η) = k(x, y) = k
(3.3)
ξ +η ξ −η , 2 2
.
(3.4)
0ηξ 2
(3.5)
It follows from (3.2) that G(ξ , η) satisfies the following PDE: c Gξ η (ξ , η) = G(ξ , η), 4 with the boundary conditions: ⎧ ⎪ Gξ (ξ , ξ ) − Gη (ξ , ξ ) + qG(ξ , ξ ) = 0, ⎪ ⎪ ⎪ ⎪ ⎨ ξc G(ξ , 0) = − − q, 4 ⎪ ⎪ ⎪ ⎪ c ⎪ ⎩Gξ (ξ , 0) = − . 4
0 ξ 2, 0 ξ 2,
(3.6)
By integrating (3.5), first with respect to η between 0 and η, and then with respect to ξ between η and ξ , we have an integral equation for G(ξ , η): G(ξ , η) = −qeqη −
c 2
η
eq(η−τ ) dτ +
0
c c − (ξ − η) + 4 4
ξ
c 2
η
eq(η−τ )
0
0
τ
G(τ , s) ds dτ 0
η
G(τ , s) ds dτ . η
(3.7)
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(3.2)
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J.-J. LIU AND J.-M. WANG
We now use the method of successive approximations to show that this equation has a unique continuous solution. Set ⎧ c η q(η−τ ) c ⎪ dτ − (ξ − η), ⎨G0 (ξ , η) = −qeqη − 0 e 2 4 (3.8) η q(η−τ ) τ c c ξ η ⎪ ⎩Gn (ξ , η) = e G (τ , s) ds dτ + G (τ , s) ds dτ , n 1. n−1 n−1 0 2 0 4 η 0 Then we have the estimate for G0 (ξ , η):
Suppose that |Gn (ξ , η)| M0n+1
(ξ + η)n . n!
Then, we have cM0n+1 |Gn+1 (ξ , η)| 4n!
2
η
τ
eη q
0
(τ + s) ds dτ + n
0
η
ξ
0
η
(τ + s) ds dτ n
cM0n+1 q (ξ + η)n+1 (2e + 2) 4n! n+1
M0n+2
(ξ + η)n+1 . (n + 1)!
(3.9)
By mathematical induction, (3.9) is true for all n 0. It then follows from the Weierstrass M-test that the series ∞ G(ξ , η) = Gn (ξ , η) n=0
converges absolutely and uniformly in 0 η ξ 2. Furthermore, by Theorem 4.17 of Liu (2009, p. 156), we deduce that G(ξ , η) =
∞
Gn (ξ , η) = −qeqη −
n=0
+
c 2
0
η
c eq(η−τ ) dτ − (ξ − η) 4
∞ ∞ c ξ η c η q(η−τ ) τ e Gn−1 (τ , s) ds dτ + Gn−1 (τ , s) ds dτ 2 0 4 η 0 0 n=1 n=1 c 2
η
c eq(η−τ ) dτ − (ξ − η) 4 0 τ η ∞ ∞ c c ξ η q(η−τ ) + e Gn−1 (τ , s) ds dτ + Gn−1 (τ , s) ds dτ 2 0 4 η 0 n=1 0 n=1
= −qeqη −
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c c |G0 (ξ , η)| qeq + eq η + |ξ − η| (2c + q)eq = M0 . 2 4
ACTIVE DISTURBANCE REJECTION CONTROL AND SLIDING MODE CONTROL
c − 2
η
c eq(η−τ ) dτ − (ξ − η) 4 0 τ η c η η c q(η−τ ) e G(τ , s) ds dτ + G(τ , s) ds dτ + 2 0 4 ξ 0 0
= −qe
qη
107
(3.10)
has zero solution only. Let Ω0 = {G | G(ξ , η) is continuous in 0 η ξ 2}. Define the mapping F : Ω0 → Ω0 by c ξ η c η q(η−τ ) τ e G(τ , s) ds dτ + G(τ , s) ds dτ (F G)(ξ , η) = 2 0 4 η 0 0
∀G ∈ Ω0 .
Then F is a compact operator on Ω0 . By (3.9), the spectral radius of F is zero. So, 0 is the unique spectrum of F . Therefore, (3.11) has zero solution only. The proof is complete. Remark 3.1 By the proof of Lemma 3.1, it is found that k(x, y) = G(ξ , η) = lim
N→∞
N
Gn (ξ , η),
ξ = x + y, η = x − y,
(3.12)
n=0
is uniformly convergent in x ∈ [0, 1], y ∈ [0, x], where Gn (ξ , η) is given by (3.8). This can be used to approximate the kernel function k numerically. The transformation (3.1) brings (1.1) into the following system: ⎧ ⎪ ⎪ ⎪wt (x, t) = wxx (x, t) − cw(x, t), ⎨ wx (0, t) = 0, 1 ⎪ ⎪ ⎪ kx (1, y)u(y, t) dy, ⎩wx (1, t) = U(t) + d(t) − k(1, 1)u(1, t) −
x ∈ (0, 1), t > 0, t 0,
(3.13)
t 0.
0
Now, we design the sliding mode surface W 2 S = f ∈ L (0, 1)
1
f (x) dx = 0
,
(3.14)
0
which is a closed subspace of L2 (0, 1). The corresponding sliding mode function for (3.13) is 1 w(x, t) dx. SW (t) = 0
(3.15)
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which shows that G(ξ , η) is a continuous solution of (3.7). Hence, G(ξ , η) is twice continuously differ˜ exp(M0 (ξ + η)). entiable in 0 η ξ 2, and |G(ξ , η)| M Now, we claim that this solution is unique. To this purpose, it suffices to show that the equation c ξ η c η q(η−τ ) τ e G(τ , s) ds dτ + G(τ , s) ds dτ (3.11) G(ξ , η) = 2 0 4 η 0 0
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J.-J. LIU AND J.-M. WANG
On the sliding mode surface SW (t) ≡ 0, (3.13) becomes ⎧ ⎨wt (x, t) = wxx (x, t) − cw(x, t), 1 ⎩wx (0, t) = w(x, t) dx = 0,
x ∈ (0, 1), t > 0, t 0.
(3.16)
0
Proposition 3.1 System (3.16) associates with a C0 -semigroup of contractions on S W , and is exponentially stable in S W with the decay rate −c. Proof. Define the operator A : D(A)(⊂ S W ) → S W as follows: Af (x) D(A)
= f (x) − cf (x) ∀f ∈ D(A), = {f ∈ S W ∩ H 2 (0, 1) | f ∈ S W , f (0) = 0}.
It is easy to show that for any f ∈ D(A), f ∈ S W if and only if f (1) = 0. So, we can write A as Af (x) = f (x) − cf (x) ∀f ∈ D(A), D(A) = {f ∈ S W ∩ H 2 (0, 1) | f (0) = f (1) = 0}. For any f ∈ D(A), we have
1
Re Af , f = − 0
|f (x)|2 dx − c
1
|f (x)|2 dx 0.
(3.17)
0
Hence A + cI is dissipative and so is for A. For any g ∈ S W solve Af = g, this is, f (x) − cf (x) = g(x), f (0) = f (1) = 0, to obtain the unique solution f as x √ ⎧ √ √ √ 1 cx − cx ⎪ ⎪ f (x) = c0 (e + e (e c(x−s) − e− c(x−s) )g(s) ds, )+ √ ⎪ ⎨ 2 c 0 1 √ ⎪ √ 1 ⎪ ⎪ ⎩c0 = − √ √ √ (e c(1−s) − e− c(1−s) ) ds. c − c 2 c(e + e ) 0 So, A−1 exists and is bounded on S W . By the Lumer–Phillips theorem (see Theorem 4.3 by Pazy, 1983, p. 14), A generates a C0 -semigroup of contractions on S W , and so does for A + cI. Therefore, the semigroup generated by A is exponentially stable with the decay rate −c.
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The Proposition 3.1 shows that system (3.16) decays exponentially in L2 (0, 1) as t → ∞ with the decay rate −c.
ACTIVE DISTURBANCE REJECTION CONTROL AND SLIDING MODE CONTROL
109
To motivate the control design, we differentiate SW (t) formally with respect to t to obtain
S˙ W (t) =
1
1
wt (x, t) dx =
0
[wxx (x, t) − cw(x, t)] dx
0
1
= wx (1, t) − c
w(x, t) dx = wx (1, t) − cSW (t),
(3.18)
0
and hence
1
kx (1, y)u(y, t) dy − cSW (t).
0
This suggests us to design the feedback controller: U(t) = k(1, 1)u(1, t) +
1
kx (1, y)u(y, t) dy + U0 (t),
(3.19)
0
where U0 is a new control. Then S˙ W (t) = U0 (t) + d(t) − cSw (t). For SW (t) = | 0, let U0 (t) = −(M + η)
SW (t) , |SW (t)|
where η > 0. Then we have SW (t) S˙ W (t) = −(M + η) + d(t) − cSw (t) |SW (t)|
for SW (t) = | 0.
(3.20)
Therefore, d |SW (t)|2 = 2SW (t)S˙ W (t) = −2(M + η)|SW (t)| + 2d(t)SW (t) − 2c|SW (t)|2 dt −2η|SW (t)|,
(3.21)
which is just the well-known finite time ‘reaching condition’. The sliding mode controller is U(t) = k(1, 1)u(1, t) + 0
1
kx (1, y)u(y, t) dy − (M + η)
SW (t) |SW (t)|
for SW (t) = | 0.
Under the control (3.22), the closed-loop of the target system (3.13) becomes ⎧ ⎪ ⎪ ⎪wt (x, t) = wxx (x, t) − cw(x, t), ⎪ ⎨w (0, t) = 0, x ⎪ ⎪ SW (t) ⎪ ⎪ ˜ | 0, + d(t) = d(t), SW (t) = ⎩wx (1, t) = −(M + η) |SW (t)|
(3.22)
(3.23)
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S˙ W (t) = U(t) + d(t) − k(1, 1)u(1, t) −
110 where
J.-J. LIU AND J.-M. WANG
˜ = −(M + η) SW (t) + d(t). d(t) |SW (t)|
(3.24)
The next result confirms the existence and uniqueness of the solution to (3.23) and the finite time ‘reaching condition’ to the sliding mode surface S W .
Proof. We rewrite system (3.23) as d ˜ w(·, t) = A0 w(·, t) + B0 d(t), dt where A0 is given by
B0 = δ(x − 1),
A0 f (x) = f (x) − cf (x), D(A0 ) = {f ∈ H 2 (0, 1) | f (0) = 0, f (1) = 0}.
A straightforward calculation shows that ⎧ ⎨A∗0 g(x) = g (x) − cg(x), ⎩D(∗ ) = {g ∈ H 2 (0, 1) | g (0) = g (1) = 0}. 0
(3.25)
(3.26)
(3.27)
It is easy to see that A∗0 = A0 is self-adjoint and dissipative. Hence, A0 generates a C0 -semigroup on L2 (0, 1). On the other hand, the dual system of (3.25) is ⎧ ⎪ w∗ (x, t) = w∗xx (x, t) − cw∗ (x, t), ⎪ ⎪ t ⎨ (3.28) w∗x (0, t) = w∗x (1, t) = 0, ⎪ ⎪ ⎪ ⎩ y(t) = B∗0 w∗ = w∗ (1, t) and the eigen-pairs (λn , ϕn ) of A∗0 are λn = −c − (nπ )2 , ϕn (x) = cos nπ x,
n = 0, 1, 2, . . . ,
2 ∗ 2 where {ϕn }∞ n=0 forms an orthogonal basis for L (0, 1). Therefore, for any w (·, 0) ∈ L (0, 1), we have
w∗ (x, 0) =
∞
an ϕn (x),
an = 2 w∗ (x, 0), ϕn (x)
n=0
and ∗
w∗ (x, t) = eA0 t w∗ (x, 0) =
∞ n=0
an eλn t ϕn (x).
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Proposition 3.2 Suppose that d is measurable and |d(t)| M for all t 0, and let SW be defined by | 0, there exists a tmax > 0 such that (3.23) admits a unique (3.15). Then for any w(·, 0) ∈ L2 (0, 1), SW (0) = solution w ∈ C((0, tmax ); L2 (0, 1)) and SW (t) = 0 for all t tmax .
ACTIVE DISTURBANCE REJECTION CONTROL AND SLIDING MODE CONTROL
111
So, y(t) =
∞
eλn t an ϕn (1).
n=0
A direct computation shows that, for any T > 0, there are positive constants CT , DT > 0 dependent on T only, such that T ∞ |y(t)|2 dt CT |an ϕn (1)|2 DT w∗ (·, 0) 2 . (3.29) n=0 A∗0 t
This shows that B∗0 is admissible for e and so is B0 for eA0 t (see Weiss, 1989a,b). Therefore, for | 0 for all t ∈ [0, T), then there exists a unique any T > 0 and w(·, 0) ∈ L2 (0, 1), if SW ∈ C[0, T], SW (t) = solution w ∈ C((0, T); L2 (0, 1)) to (3.23). Remark 3.2 We need to remark that the discontinuous term S(t)/|S(t)| is actually the unit controller and more details for the unit controllers of infinite-dimensional systems can be found in the monograph by Orlov (2009). Now we go back the original system (1.1). Under the feedback control (3.22), we get the closed-loop system for (1.1): ⎧ ut (x, t) = uxx (x, t), ⎪ ⎪ ⎪ ⎪ ⎪ ux (0, t) = −qu(0, t), ⎪ ⎪ ⎨
1 S(t) ux (1, t) = k(1, 1)u(1, t) + + d(t) kx (1, y)u(y, t) dy − (M + η) ⎪ |S(t)| ⎪ 0 ⎪ 1 ⎪ ⎪ ⎪ ⎪ ˜ ⎩ = k(1, 1)u(1, t) + kx (1, y)u(y, t) dy + d(t),
(3.30)
0
where
1
S(t) = SW (t) =
u(x, t) dx −
0
1
x
k(x, y)u(y, t) dy dx. 0
(3.31)
0
It is noted that system (3.30) is equivalent to (3.23) under the equivalent transformation (3.1). So, by (3.1), we can define a bounded invertible operator F : L2 (0, 1) → L2 (0, 1) by x w(x) = (F u)(x) = u(x) − k(x, y)u(y) dy ∀u ∈ L2 (0, 1), (3.32) 0
and the solution u(x, t) of (3.30) can be given by u(x, t) = (F −1 w)(x, t)
(3.33)
with w(x, 0) = (F u)(x, 0). Theorem 3.1 Suppose that d is bounded measurable and |d(t)| M . Let S(t) be given by (3.31). | 0, there exists a t0 > 0 such that (3.30) admits a unique solution Then for any u(·, 0) ∈ L2 (0, 1), S(0) = u ∈ C((0, t0 ); L2 (0, 1)) and S(t) = 0 for all t t0 . Moreover, S(t) is continuous, monotone in [0, t0 ]. On
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0
112
J.-J. LIU AND J.-M. WANG
the sliding surface S(t) = 0, the system (1.1) becomes ⎧ ⎪ ut (x, t) = uxx (x, t), ⎪ ⎪ ⎪ ⎨u (0, t) = −qu(0, t), x 1 x ⎪ 1 ⎪ ⎪ ⎪ u(x, t) dx − k(x, y)u(y, t) dy dx = 0, ⎩ 0
0
(3.34)
0
Proof. We only need to prove that S(t) is continuous, monotone in [0, t0 ]. Suppose, without any loss of generality, that S(0) > 0 since the proof for S(0) < 0 is similar. Since (3.30) is equivalent to (3.23) and B0 is admissible for eA0 t , the solution to the system (3.23) can be written as t ˜ ds. (3.35) eA0 (t−s) B0 d(s) w(x, t) = eA0 t w(x, 0) + 0
So, by (3.32) and (3.33), the solution to the system (3.30) can be written as u(x, t) = [F −1 eA0 t F u](x, 0) +
t
˜ ds. F −1 eA0 (t−s) B0 d(s)
(3.36)
0
Moreover, for any T > 0, there exists a constant CT > 0 such that (see Weiss, 1989a,b) ˜ L2 (0,T) ]
u(·, t) CT [ u(·, 0) + d
∀t ∈ [0, T].
(3.37)
Now for d˜ ∈ L2 (0, T) defined by (3.24), since H01 (0, T) is dense in L2 (0, T), take d˜ n ∈ H01 (0, T) such that ˜ L2 (0,T) = 0. lim d˜ n − d
n→∞
Let un (·, t) be the solution of (3.25) corresponding to d˜ n and the initial value un (·, 0) ∈ D(A0 ) (note that D(A0 ) is dense in L2 (0, 1)), where lim un (·, 0) − u(·, 0) = 0.
n→∞
It follows from (3.37) that lim un (·, t) − u(·, t) = 0
n→∞
uniformly in t ∈ [0, T]. Since un (·, t) and wn (·, t) are the classical solutions of (3.30) and (3.23), respectively, by Proposition 4.2.1 in Weiss (1989a,b, p. 120), it follows from (3.20) that
1
un (x, t) dx −
0 −ct
=e
0
0 1
1
x
k(x, y)un (y, t) dy dx
0
un (x, 0) r dx − 0
1
0
x
t k(x, y)un (y, 0) dy dx + e−c(t−τ ) d˜ n (τ ) dτ . 0
(3.38)
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which is exponentially stable in L2 (0, 1) with the decay rate −c.
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ACTIVE DISTURBANCE REJECTION CONTROL AND SLIDING MODE CONTROL
Passing to the limit as n → ∞ in the above equality yields
1
1
u(x, t) dx −
0
= e−ct
k(x, y)u(y, t) dy dx 0
1
x
0
1
u(x, 0) rdx −
0
0
x
t ˜ ) dτ . k(x, y)u(y, 0) dy dx + e−c(t−τ ) d(τ
0
(3.39)
0
S(t) =
1
0 −ct
1
u(x, t) dx −
=e
0
= e−ct e−ct
1
1
1
0
1
u(x, 0) dx −
0
0
u(x, 0) dx −
0
x
k(x, y)u(y, t) rdy dx 0
1
1
0
t ˜ ) dτ k(x, y)u(y, 0) dy dx + e−c(t−τ ) d(τ
x
t k(x, y)u(y, 0) dy dx − e−c(t−τ ) [(M + η) − d(τ )] dτ
0
0
u(x, 0) dx −
x
0
0
0
0
x
η η k(x, y)u(y, 0) dy dx − + e−ct . c c
(3.40)
Hence, S(t) is decreasing in t. Since S(0) > 0, there exists a t0 > 0 such that S(t) > 0 for t ∈ [0, t0 ) and S(t) = 0 for all t t0 . The proof is complete.
Fig. 1. The displacement with d(t) = 2 sin t by SMC.
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Since T is arbitrary, we see that S(t) defined by (3.31) with d˜ defined by (3.24) is continuous in the whole half line [0, ∞) for any initial value in L2 (0, 1). Furthermore, by (3.39) and S(0) > 0, we have
114
J.-J. LIU AND J.-M. WANG 10
5
0
−10
−15
0
0.5
1
1.5
2
Fig. 2. The controller by SMC.
Fig. 3. The displacement with d(t) = 2 sin t by ADRC.
4. Numerical simulations In this section, the finite difference method is applied to compute the displacements numerically for both SMC and ADRC to illustrate the effect of the controllers. Figure 1 shows the displacement of system (3.30). Here the steps of space and time are taken as 0.1 and 0.0001, respectively. We choose q = 1, c = 10, M0 = 4, η = 1, u(x, 0) = 10x3 and d = 2 sin t. The kernel function is approximated by
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−5
ACTIVE DISTURBANCE REJECTION CONTROL AND SLIDING MODE CONTROL
115
20
15
10
5
0
−5
0
0.5
1
1.5
2
Fig. 5. The controller by ADRC.
(3.12) with N = 20. Figure 2 shows the controller by SMC. Owing to discontinuity, the control vibrates rapidly after some time. Figure 3 shows the spatiotemporal profile of the solution of the system (2.33) with the same space and time sizes used in SMC. Other parameters are q = c0 = 1, ε = 0.01, u(x, 0) = x, d(t) = cos t. It is seen that in both cases, the displacements are obviously convergent. Moreover, Fig. 4 shows that convergence of dˆ to the disturbance d. Figure 5 shows the controller by ADRC. It is much better than that by SMC.
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ˆ (red dotted line) and disturbance d(t) = cos t (blue solid line) by ADRC. Fig. 4. The d(t)
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5. Concluding remarks
Acknowledgements The authors thank the anonymous referees’ comments and suggestions for improving the paper. Funding This work was supported by the National Natural Science Foundation of China. References Breger, A. M., Butkovskii, A. G., Kubyshkin, V. A. & Utkin, V. I. (1980) Sliding modes for control of distributed parameter entities subjected to a mobile multicycle signal. Automat. Remote Control, 41, 346–355. Brockett, R. W. (2001) New issues in the mathematics of control. Mathematics Unlimited–2001 and Beyond (B. Engquist and W. Schmid, eds). Berlin: Springer, pp. 189–219. Chen, G., Delfour, M. C., Krall, A. M. & Payre, G. (1987) Modeling, stabilization and control of serially connected beam. SIAM J. Control Optim., 25, 526–546. 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, 381–387. Drakunov, S., Barbieeri, E. & Silver, D. A. (1996) Sliding mode control of a heat equation with application to arc welding. Proceedings of the 1996 IEEE International Conference On Control Applications Dearborn, MI, September 15–18, 668–672. Guo, B. Z. & Jin, F. F. (2013) Sliding mode and active disturbance rejection control to the stabilization of antistable one-dimensional wave equation subject to boundary input disturbance. IEEE Trans. Automat. Control, 58, 1269–1274. Guo, B. Z. & Liu, J. J. (2013) Sliding mode control and active disturbance rejection control to the stabilization of one-dimensional Schrodinger equation subject to boundary control matched disturbance. Int. J. Robust Nonlinear Control, in press. Guo, B. Z. & Xu, C. Z. (2007) The stabilization of a one-dimensional wave equation by boundary feedback with noncollocated observation. IEEE Trans. Automat. Control, 52, 371–377. Guo, B. Z. & Zhao, Z. L. (2011) On the convergence of extended state observer for nonlinear systems with uncertainty. Systems Control Lett., 60, 420–430. Guo, W. & Guo, B. Z. (2013) Stabilization and regulator design for a one-dimensional unstable wave equation with input harmonic disturbance. Internat. J. Robust 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 non-collocated control. Internat. J. Robust Nonlinear Control, 21, 1297–1321.
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In this paper, we apply two different approaches of the ADRC and the sliding mode control (SMC) to stabilize a one-dimensional unstable heat equation subject to boundary control matched disturbance. We first apply the ADRC to cancel the disturbance which is supposed to have a bounded derivative. The ADRC is an online estimation/cancellation control strategy, and a high gain estimator is designed to estimate the disturbance. The well-posedness of the closed-loop system is presented, and it is shown that the closed-loop system can reach any arbitrary given vicinity of zero as time goes to infinity and high gain tuning parameter goes to zero. Secondly, we apply the SMC approach to reject the disturbance which is supposed to be bounded. The sliding mode surface is found to be a closed subspace of L2 (0, 1). On the sliding mode surface, the system is shown to be exponentially stable with arbitrary prescribed decay rate.
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Krstic, M. (2010) Adaptive control of an anti-stable wave PDE. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 17, 853–882. Krstic, M., Guo, B. Z., Balogh, A. & Smyshlyaev, A. (2008) Output-feedback stabilization of an unstable wave equation. Automatica, 44, 63–74. Krstic, M. & Smyshlyaev, A. (2008) Boundary Control of PDEs: A Course on Backstepping Designs. Philadelphia, PA: SIAM. Liu, W. J. (2009) Elementary Feedback Stabilization of the Linear Reaction–Convection–Diffusion Equation and the Wave Equation. New York: Springer. Luo, Z. H., Guo, B. Z. & Morgul, O. (1999) Stability and Stabilization of Infinite Dimensional Systems with Applications. London: Springer. Machtyngier, E. (1994) Exact controllability for the Schrödinger equation. SIAM J. Control Optim., 32, 24–34. Medvedev, A. & Hillerström, G. (1995) An external model control system. Control-Theory Adv. Technol., 10, 1643–1665. Orlov, Y. V. (2009) Discontinuous Systems Lyapunov Analysis and Robust Synthesis under Uncertainty Conditions. London: Springer. Orlov, Y. V. & Utkin, V. I. (1987) Sliding mode control in infinite-dimensional systems. Automatica, 23, 753–757. Pazy, A. (1983) Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer. Pisano, A. & Orlov, Y. (2012) Boundary second-order sliding-mode control of an uncertain heat process with unbounded matched perturbation. Automatica., 48, 1768–1775. Smyshlyaev, A. & Krstic, M. (2004) Closed-form boundary state feedbacks for a class of 1-D partial integrodifferential equations. IEEE Trans. Automat. Control, 49, 2185–2202. Tucsnak, M. & Weiss, G. (2000) Observation and Control for Operator Semigroups. Basel: Birkhäuser. Weiss, G. (1989a) Admissibility of unbounded control operators. SIAM J. Control Optim., 27, 527–545. Weiss, G. (1989b) Admissible observation operators for linear semigroups. Israel J. Math., 65, 17–43.
