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Dynamic stabilization of an Euler–Bernoulli beam equation with time delay in boundary observation
Automatica 45 (2009) 1468–1475
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Brief paper
Dynamic stabilization of an Euler–Bernoulli beam equation with time delay in boundary observationI Bao-Zhu Guo a,c,∗ , Kun-Yi Yang a,b a
Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, PR China
b
Graduate School of the Chinese Academy of Sciences, Beijing 100039, PR China
c
School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa
article
info
Article history: Received 6 May 2008 Received in revised form 17 November 2008 Accepted 1 February 2009 Available online 19 March 2009
a b s t r a c t An Euler–Bernoulli beam equation under boundary control and delayed observation is considered. An observer–predictor based scheme is developed to stabilize the equation. On a time interval where the observation is available, the state is tracked by the observer; when the observation is not available due to the delay, the state is estimated by the predictor. The estimation is then used in proportional feedback. It is shown that the state of closed-loop system decays exponentially for smooth initial values. © 2009 Elsevier Ltd. All rights reserved.
Keywords: Euler–Bernoulli beam equation Time delay Observer Feedback control Exponential stability
1. Introduction Time delays are common in practical control systems, especially in networked systems, where it takes time for the observation to be available for use in control. These delays often create stability problem or cause periodic oscillations (Gumowski & Mira, 1968). Indeed, as first noticed in Datko (1988) for one-dimensional Euler–Bernoulli beam equation, even a small amount of time delay in the stabilizing boundary output feedback schemes could destabilize the system. See also Datko (1991) for many other hyperbolic systems. According to Fiagbedzi (1988, p. 69), stabilization of distributed parameter control systems with time delay in observation and control represents a difficult mathematical challenge. This is largely still the case today. Inspired by the work in Guo and Xu (2007), we recently solved the stabilization problem for onedimensional wave equation with boundary control and delayed observation (Guo & Xu, 2008). The method we used in Guo and Xu
I This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Andrea Serrani under the direction of Editor Miroslav Krstic. ∗ Corresponding author at: Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, PR China. Tel.: +86 10 62651443; fax: +86 10 62587343. E-mail address: [email protected] (B.-Z. Guo).
0005-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2009.02.004
(2008) relies heavily on the explicit expression of analytic solution of the system. In this paper, we consider the following Euler–Bernoulli beam equation under boundary control and delayed observation:
wtt (x, t ) + wxxxx (x, t ) = 0, 0 < x < 1, t > 0, w(0, t ) = wx (0, t ) = wxx (1, t ) = 0, wxxx (1, t ) = u(t ), y(t ) = wt (1, t − τ ), t ≥ τ , w(x, 0) = w0 (x), wt (x, 0) = w1 (x),
(1)
where u is the control (or input), (w0 , w1 ) is the initial state, τ > 0 is a (known) constant time delay, and y is the delayed observation (or output). It is well known that if system (1) is time delay free (i.e., τ = 0), then the static proportional output feedback u(t ) = ky(t ), k > 0 exponentially stabilizes the system. In the presence of time delay (i.e., τ > 0), however, the closed-loop system with a static feedback is no more stable (Datko, 1988). Moreover, no dynamic stabilizing controller is robust to time delay in general for those plants like (1) with infinitely many unstable poles (Logemann, Rebarber, & Weiss, 1996). The above results lead to the search for control schemes in the presence of a given time delay. We refer to Weiss and Curtain (1997) for a general method of designing stabilizing controllers for regular systems. In this paper, we propose an observer–predictor based stabilization scheme. The state is dynamically estimated in
B.-Z. Guo, K.-Y. Yang / Automatica 45 (2009) 1468–1475
two steps: On a time interval where the observation is available, the state is tracked by the observer; when the observation is not available due to the delay, the state is predicted in the scheme. We show that with the estimated state feedback, the state of the closed-loop system decays exponentially for any smooth initial value. The idea of estimation can be explained for a finite-dimensional time-invariant system described by X˙ (t ) = AX (t ) + Bu(t ),
y(t ) = CX (t − τ ),
where τ > 0 is the time delay, A, B, C are matrices of appropriate dimensions. Let t > τ be a given time instant. For every s ∈ [0, t − τ ] the state X (s) can be tracked by a standard Luenberger observer
˙ b X (s) = Ab X (s) + Bu(s) + LCb X (s) − Ly(s + τ ), s ∈ [0, t − τ ], where L is a matrix such that A + LC is stable. To determine X (s) for s ∈ [t − τ , t ], where the observation is not available due to the delay, we design a predictor as
( ˙ b X t (s) = Ab Xt (s) + Bu(s), b Xt (t − τ ) = b X (t − τ ).
s ∈ [t − τ , t ]
We consider the system (3) in the energy state space = H × L2 (0, 1), H = HE2 (0, 1) × L2 (0, 1), HE2 (0, 1) = ∈ H 2 (0, 1)|f (0) = f 0 (0) = 0} with state variable (w(·, t ), wt (·, t ), z (·, t )), for which the inner product induced norm is defined by H {f
k(w(·, t ), wt (·, t ), z (·, t ))k2H Z 1 2 [wxx (x, t ) + wt2 (x, t ) + z 2 (x, t )]dx. = 0
The input and output spaces are the same U = Y = R. Theorem 2.1. The system (3) is well-posed: For any (w0 , w1 , z0 ) ∈ H and u ∈ L2loc (0, ∞), there exists a unique solution to (3) such that (w(·, t ), wt (·, t ), z (·, t )) ∈ C (0, ∞; H ); and for any T > 0, there exists a constant CT such that
k(w(·, T ), wt (·, T ), z (·, T ))k + 2 H
T
Z
y2 (t )dt 0
The state for t > τ is then estimated as b Xt (t ) ≈ X (t ). To comment on the proposed scheme with respect to early work, let us compute the estimate analytically. Solving the above two equations gives
˙ b X t (t ) = A + eAτ LC e−Aτ b Xt (t ) + Bu(t ) Z t − eAτ Ly(t ) − eAτ LC eA(t −τ −θ) Bu(θ )dθ .
1469
(2)
t −τ
We see that the estimate above becomes those in (17)–(18), (20)–(21) of Watanabe and Ito (1981) with D = A + eAτ LC e−Aτ , E = −eAτ L, T = I, B = B, C = C e−Aτ (see also Eq. 101 of Krstic and Smyshlyaev (2008) and Zhong (2006) on p. 40). It is noted that our scheme that views the time delay problem from a different way is closely related to the observer designs for finite-dimensional systems in Watanabe and Ito (1981) by timedomain approach (see also Klamka (1982) and Watanabe (1986)) and Zhong (2006) by frequency-domain approach, and can be easier to generalize to PDEs. Some other controllers for finitedimensional systems were presented in Curtain, Weiss, and Weiss (1996) near the end of page 1527, which are actually the same as controllers in Artstein (1982), Manitius and Olbrot (1979) and Olbrot (1978) (see also Fiagbedzi and Pearson (1986) and Zhong (2006)). The difficulty with infinite-dimensional systems is that the delay free open-loop system (τ = 0 in (1)) has infinitely many poles on the imaginary axis. We proceed as follows. In the next section, Section 2, we show the well-posedness of open-loop system. Section 3 is devoted to the design of observer and predictor. The exponential decay of closed-loop system under the estimated state feedback control is presented in Section 4. Section 5 gives some simulation results.
T
Z ≤ CT k(w0 , w1 , z0 )k2H +
u2 (t )dt .
(4)
0
Proof. It is well known that the following system
( wtt (x, t ) + wxxxx (x, t ) = 0, w(0, t ) = wx (0, t ) = wxx (1, t ) = 0, wxxx (1, t ) = u(t ), yw (t ) = wt (1, t ),
(5)
can be written as a second-order system in H :
wtt (·, t ) + Aw(·, t ) + Bu(t ) = 0, yw (t ) = B∗ wt (·, t ),
(6)
where A is a self-adjoint operator in H and B is the input operator, both are unbounded:
Af = f (4) , ∀ f ∈ D(A) = {f ∈ H 4 ∩ HE2 | f 00 (1) = f 000 (1) = 0}, B = δ(x − 1).
(7)
δ(·) denotes the Dirac distribution. It was shown in Guo and Luo (2002) that system (6)–(7) is well-posed in the sense of D. Salamon (Curtain et al., 1996): For any u ∈ L2loc (0, ∞) and (w0 , w1 ) ∈ H , there exists a unique solution (w(·, t ), wt (·, t )) ∈ C (0, ∞; H ) to (6); and for any T > 0, there exists a constant DT > 0 such that
k(w(·, T ), wt (·, T ))kH + 2
T
Z
|yw (t )|2 dt 0 T
Z ≤ DT k(w0 , w1 )k2H +
2. Well-posedness of the open-loop system
|u(t )|2 dt .
(8)
0
Introduce a new variable
Once the solution of (5) is determined, one can solve the ‘‘z’’ part in Eq. (3):
z (x, t ) = wt (1, t − τ x). Then system (1) becomes
wtt (x, t ) + wxxxx (x, t ) = 0, 0 < x < 1, t > 0, w(0, t ) = wx (0, t ) = wxx (1, t ) = 0, w (1, t ) = u(t ), xxx τ zt (x, t ) + zx (x, t ) = 0, z (0, t ) = wt (1, t ), w(x, 0) = w0 (x), wt (x, 0) = w1 (x), z ( x , 0 ) = z ( x ), 0 y(t ) = z (1, t ), where z0 is the initial value of variable z.
(3)
τ zt (x, t ) + zx (x, t ) = 0, z (0, t ) = wt (1, t ), z (x, 0) = z0 (x).
(9)
Actually, for any initial z0 ∈ L2 (0, 1), integrating along the characteristic line gives the unique (weak) solution to (9):
t t z0 x − , x≥ , τ τ z (x, t ) = wt (1, t − xτ ) , x < t . τ
(10)
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B.-Z. Guo, K.-Y. Yang / Automatica 45 (2009) 1468–1475
Therefore, for any T > 0,
there exists a unique solution to (15) such that (b w (·, s), w bs (·, s)) ∈ C (0, t − τ ; H ); and for any 0 < T ≤ t − τ , there exists a constant b C1T such that
1
Z
z 2 (x, T )dx 0
=
Z
1− Tτ 0
1 τ
z02 (x)dx +
T
Z
1
τ
w (1, t )dt , 2 t
w0 , w b1 )k2H C1T k(b k(b w (·, T ), w bs (·, T ))k2H ≤ b
T
Z
wt2 (1, t )dt ,
T ≤ τ, (11)
0
τ
T
Z
u (s)ds . 2
(16)
0
From (8), it has
y2 (t )dt 0
T +τ
Z
Z 1 z02 (x)dx, T ≤ τ , τ 1− Tτ = Z T −τ Z 1 τ wt2 (1, t )dt , z02 (x)dx + 0
y (s)ds + 2
+
T > τ,
T −τ
T
Z
T +τ
Z
τ
(12)
|u(s)|2 ds .
(17)
0
Combing (16) and (17) gives the required inequality by setting In order for (14) to be an observer for (13), we have to show its convergence. To do this, let
The importance of Theorem 2.1 lies in that for any initial value in the state space, the output belongs to L2loc (τ , ∞) as long as the input u belongs to L2loc (0, ∞). This fact is particularly important to the solvability of observer designed in the next section.
ε(x, s) = w b(x, s) − w(x, s),
3. Observer and predictor design In this section, we go back to system (1). For any given t > τ , we proceed two steps to estimate the state of (1) via the observer and predictor. Step 1: Construct an observer to estimate the state {(w(x, s), ws (x, s)), s ∈ [0, t − τ ]} from the known observation {y(s + τ )|s ∈ [0, t − τ ], t > τ }. Since the observation {y(s + τ ), s ∈ [0, t − τ ]} is known and {(w(x, s), ws (x, s)), s ∈ [0, t − τ ]} satisfies
w (x, s) + wxxxx (x, s) = 0, 0 < s < t − τ , ss w(0, s) = wx (0, s) = wxx (1, s) = 0, wxxx (1, s) = u(s), y(s + τ ) = ws (1, s),
(13)
we can construct naturally a Luenberger observer for system (13) as following:
w b (x, s) + w bxxxx (x, s) = 0, 0 < s < t − τ , ss w b(0, s) = w bx (0, s) = w bxx (1, s) = 0, w b ( 1, s) = u(s) + k1 [w bs (1, s) − y(s + τ )] , xxx w b (x , 0 ) = w b0 (x), w bs (x, 0) = w b1 (x),
(14)
where (b w0 , w b1 ) is the (arbitrarily assigned) initial state of observer. We first state the solvability and convergence of observer (14). Proposition 3.1. The system (14) is well-posed: For any t > τ , and (w0 , w1 ) ∈ H , (b w0 , w b1 ) ∈ H , u ∈ L2loc (0, ∞), there exists a unique solution to (14) such that (b w (·, s), w bs (·, s)) ∈ C (0, t − τ ; H ); and for any 0 < T ≤ t − τ , there exists a positive constant b CT such that T +τ
u2 (s)ds .
(18)
εss (x, s) + εxxxx (x, s) = 0, 0 < s < t − τ , ε(0, s) = εx (0, s) = εxx (1, s) = 0, εxxx (1, s) = k1 εs (1, s).
(19)
System (19) can be written as:
ε(x, s) ε(x, s) =B , εs (x, s) ds εs (x, s) d
(20)
where the operator B is defined as follows:
B(f , g )> = (g , −f (4) )> , D(B) = {(f , g )> ∈ (H 4 ∩ HE2 ) × HE2 | 00 f (1) = 0, f 000 (1) = k1 g (1)}.
(21)
B generates an exponential stable C0 -semigroup on H . For any (w0 , w1 ) ∈ H , (b w0 , w b1 ) ∈ H , there exists a unique solution ε to (19) which satisfies
k(ε(·, s), εs (·, s))kH ≤ Me−ωs k(b w0 − w0 , w b1 − w1 )kH , ∀ s ∈ [0, t − τ ], t > τ
(22)
for some positive constants M and ω. Step 2: Predict {(w(x, s), ws (x, s)), s ∈ (t − τ , t ]} by {(b w (x, s), w bs (x, s)), s ∈ [0, t − τ ]}. This is done by solving (1) with estimated initial value (b w (x, t − τ ), w bs (x, t − τ )) obtained from (14):
w bss (x, s, t ) + w bxxxx (x, s, t ) = 0, t − τ < s < t , w b(0, s, t ) = w bx (0, s, t ) = w bxx (1, s, t ) = 0, w bxxx (1, s, t ) = u(s), b (x , t − τ , t ) = w b(x, t − τ ), w w bs (x, t − τ , t ) = w bs (x, t − τ ).
(23)
By Theorem 2.1, for any t > τ and u ∈ L2loc (0, ∞), there exists a unique solution to (23) such that (b w (·, s, t ), w bs (·, s, t )) ∈ C (t − τ , t ; H ); and for any s ∈ (t − τ , t ], there exists a constant b C t ,s > 0 such that
0
Proof. Notice that (14) can be written as
w bss (·, s) + Aw b(·, s) + k1 BB∗ w bs (·, s) + B[u(s) − k1 y(s + τ )] = 0 for all 0 < s < t − τ , t > τ ,
0 ≤ s ≤ t − τ.
Then by (13) and (14), ε satisfies
(
k(b w (·, T ), w bs (·, T ))k2H Z ≤b CT k(w0 , w1 )k2H + k(b w0 , w b1 )k2H +
T +τ
b CT = b C1T (1 + DT ).
T > τ.
0
Collecting (8), (11) and (12) gives (4).
Z |y(s)|2 ds ≤ DT k(w0 , w1 )k2H +
(15)
where A, B are defined in (7). By Corollary 1 of Guo and Luo (2002), (15) is well-posed: For any t > τ and (b w0 , w b1 ) ∈ H , u ∈ L2loc (0, ∞), y(s +τ ) ∈ L2 (0, t −τ ) which is assured by Theorem 2.1,
k(b w (·, s, t ), w bs (·, s, t ))k2H Z ≤b Ct ,s k(b w (·, t − τ ), w bs (·, t − τ ))k2H +
t t −τ
This together with Proposition 3.1 leads to (23).
|u(s)|2 ds .
B.-Z. Guo, K.-Y. Yang / Automatica 45 (2009) 1468–1475
1471
Proposition 3.2. The system (23) is well-posed: For any t > τ , (w0 , w1 ) ∈ H , (b w0 , w b1 ) ∈ H , and u ∈ L2loc (0, ∞), there exists a unique solution to (23) such that
We consider system (30)–(32) in the state space X = H 3 . It is obvious that system (30)–(32) is equivalent to (33)–(35) for t > τ provided that u∗ ∈ L2loc (0, ∞) which will be clarified in (51).
(b w (·, s, t ), w bs (·, s, t )) ∈ C (t − τ , t ; H );
w (x, t ) + wxxxx (x, t ) = 0, t > τ , tt w(0, t ) = wx (0, t ) = wxx (1, t ) = 0, wxxx (1, t ) = k2 wt (1, t ) + k2 εs (1, t , t ), w(x, 0) = w0 (x), wt (x, 0) = w1 (x), εss (x, s) + εxxxx (x, s) = 0, 0 < s < t − τ , ε(0, s) = εx (0, s) = εxx (1, s) = 0, εxxx (1, s) = k1 εs (1, s), ε( b0 (x) − w0 (x), x, 0) = w εs ( x , 0 ) = w b1 (x) − w1 (x), εss (x, s, t ) + εxxxx (x, s, t ) = 0, t − τ < s < t , ε(0, s, t ) = εx (0, s, t ) = 0, εxx (1, s, t ) = εxxx (1, s, t ) = 0, ε( x, t − τ , t ) = ε(x, t − τ ), εs (x, t − τ , t ) = εs (x, t − τ ),
and for any s ∈ (t − τ , t ], there exists a constant Ct ,s such that
k(b w (·, s, t ), w bs (·, s, t ))k2H Z t 2 2 2 u (s)ds . w0 , w b1 )kH + ≤ Ct ,s k(w0 , w1 )kH + k(b 0
We finally get the estimated state variable by
w e(x, t ) = w b(x, t , t ),
w et (x, t ) = w bs (x, t , t ),
t > τ.
(24)
Theorem 3.1. For any t > τ , we have
k(w(·, t ) − w e(·, t ), wt (·, t ) − w et (·, t ))kH ≤ Me−ω(t −τ ) k(b w0 − w0 , w b1 − w1 )kH ,
(25)
where (b w0 , w b1 ) is the initial state of observer (14), (w0 , w1 ) is the initial state of original system (1), and M , ω are constants in (22). Proof. Let
ε(x, s, t ) = w b(x, s, t ) − w(x, s),
t − τ ≤ s ≤ t.
(26)
Then ε(x, s, t ) satisfies
εss (x, s, t ) + εxxxx (x, s, t ) = 0, t − τ < s ≤ t , ε(0, s, t ) = εx (0, s, t ) = 0, εxx (1, s, t ) = εxxx (1, s, t ) = 0, ε(x, t − τ , t ) = ε(x, t − τ ), εs (x, t − τ , t ) = εt (x, t − τ ),
(27)
This together with (22) and (24) gives (25).
(28)
Since the feedback u(t ) = k2 wt (1, t ) stabilizes exponentially the system (1), and we have the estimation w et (1, t ) of wt (1, t ), it is natural to design the estimated state feedback control law of the following (with k2 > 0): k2 w et (1, t ) = k2 w bs (1, t , t ), 0, t ∈ [0, τ ],
t > τ,
(29)
under which, the closed-loop system becomes a system of partial differential equations (30)–(32):
w (x, t ) + wxxxx (x, t ) = 0, t > τ , tt w(0, t ) = wx (0, t ) = wxx (1, t ) = 0, bs (1, t , t ), wxxx (1, t ) = k2 w w(x, 0) = w0 (x), wt (x, 0) = w1 (x), w b (x, s) + w bxxxx (x, s) = 0, 0 < s < t − τ , ss w b(0, s) = w bx (0, s) = w bxx (1, s) = 0, w b ( 1, s) = u∗ (s) + k1 [w bs (1, s) − ws (1, s)] , xxx w b(x, 0) = w b0 (x), w bs (x, 0) = w b1 (x), w bss (x, s, t ) + w bxxxx (x, s, t ) = 0, t − τ < s < t , w b(0, s, t ) = w bx (0, s, t ) = w bxx (1, s, t ) = 0, w bxxx (1, s, t ) = u∗ (s), b(x, t − τ , t ) = w b(x, t − τ ), w w bs (x, t − τ , t ) = w bs (x, t − τ ).
(35)
where ε(x, s) and ε(x, s, t ) are given by (18) and (26), respectively. System (34) is the same as system (19). Theorem 4.1. Let k1 > 0, k2 > 0 and t > τ . Then for any (w0 , w1 ) ∈ H , (b w0 , w b1 ) ∈ H , there exists a unique solution to system (33)–(35) such that (w(·, t ), wt (·, t )) ∈ C (τ , ∞; H ), (ε(·, s), εs (·, s)) ∈ C (0, t − τ ; H ), (ε(·, s, t ), εs (·, s, t )) ∈ C ([t − τ , t ] × [τ , ∞); H ) and for any t > τ , s ∈ [0, t − τ ], q ∈ (t − τ , t ], there exists a constant Ctsq > 0 such that
k(w(·, t ), wt (·, t ))kH + k(ε(·, s), εs (·, s))kH + k(ε(·, q, t ), εs (·, q, t ))kH ≤ Ctsq [k(w0 , w1 )kH + k(b w0 , w b1 )kH ].
k(w(·, t ), wt (·, t ))k2H ≤ [k(w0 , w1 )k2H + M0 kB(w0 − w b0 , w1 − w b1 )k2H ]e−2ω0 t ,
∀t≥0
(36)
for some M0 , ω0 > 0 that are independent of initial value.
4. Stabilization by the estimated state feedback
u∗ (t ) =
(34)
Moreover, for any (w0 − w b0 , w1 − w b1 ) ∈ D(B), where B is defined by (21), system (33) decays exponentially in the sense that
which is a conservative system
k(ε(·, t , t ), εs (·, t , t ))kH = k(ε(·, t − τ ), εt (·, t − τ ))kH .
(33)
(30)
(31)
Proof. For any (w0 , w1 ) ∈ H , (b w0 , w b1 ) ∈ H , since B defined by (21) generates an exponential stable C0 -semigroup on H , there is a unique solution (ε(·, s), εs (·, s)) ∈ C (0, t − τ ; H ) to (34) such that (22) holds true. Now, for any given time t > τ , write (35) as:
ε(x, s, t ) ε(x, s, t ) =A , εs (x, s, t ) ds εs (x, s, t ) d
where A is defined by
A(f , g )> = (g , −f (4) )> , D(A) = {(f , g )> ∈ (H 4 ∩ HE2 ) × HE2 | f 00 (1) = f 000 (1) = 0}. Then A is skew-adjoint in H and hence generates a conservative C0 -semigroup on H . For any (ε(·, t − τ ), εt (·, t − τ )) ∈ H that is determined by (34), there exists a unique solution to (35) such that
k(ε(·, s, t ), εs (·, s, t ))kH = k(ε(·, t − τ ), εt (·, t − τ ))kH ,
∀ s ∈ [t − τ , t ].
(37)
So, (ε(·, s, t ), εs (·, s, t )) ∈ C ([t − τ , t ] × [τ , ∞); H ). Moreover, since A is skew-adjoint with compact resolvent, the solution of (35) can be, in terms of s, represented as
X ∞ 2 φ n ( x) ε(x, s, t ) = an (t )eiωn s 2 εs (x, s, t ) iωn φn (x) n =1
(32)
+
∞ X n =1
bn (t )e−iωn s 2
φn (x) , −iωn2 φn (x)
(38)
1472
B.-Z. Guo, K.-Y. Yang / Automatica 45 (2009) 1468–1475
where {(φn (x), ±iωn2 φn (x))> }∞ n=1 is a sequence of all ω -linearly independent approximated normalized orthogonal eigenfunctions of A corresponding to eigenvalues ±iωn2 , ωn > 0 and φn satisfies
φn(4) (x) = ωn4 φn (x), 0 < x < 1, φn (0) = φn0 (0) = φn00 (1) = φn000 (1) = 0.
=
(39)
Solve (39) to obtain
00 φn (x) = −sin(ωn x) + cos(ωn x) + e−ωn x +eωn (x−1) (sin ωn + cos ωn ) +e−ωn [2 cos(ωn (1 − x)) + eωn (x−1) +e−ωn x (cos ωn − sin ωn )] +e−2ωn [sin(ωn x) + cos(ωn x)] , iωn2 φn (x) = i{sin(ωn x) − cos(ωn x) + e−ωn x +eωn (x−1) (sin ωn + cos ωn ) +e−ωn [−2 cos(ωn (1 − x)) + eωn (x−1) −ωn x +e−2ω (cos ωn − sin ωn )] −e n [sin(ωn x) + cos(ωn x)]},
+ (sin ωn + cos ωn )eωn (x−1) − 2e−ωn cos ωn (1 − x) + eωn (x−2) − (sin ωn − cos ωn )e−ωn (x+1) − e−2ωn [sin(ωn x) + cos(ωn x)] dx Z 1 i εsxx (x, t − τ ) [−sin(ωn x) − 2 ωn 0
(40)
+ cos(ωn x) + e−ωn x + (sin ωn + cos ωn )eωn (x−1) + 2e−ωn cos ωn (1 − x) + eωn (x−2) − (sin ωn − cos ωn )e−ωn (x+1) + e−2ωn [sin(ωn x) + cos(ωn x)] dx.
where (see e.g., Guo and Luo (2002))
1 π + O (n−1 ) as n → ∞. ωn = n −
(41)
2
Therefore
εxxx (1, t − τ ) −2 sin ωn + 2e−2ωn sin ωn ωn2 i + 2 εsx (1, t − τ ) 2 cos ωn + 4e−ωn ωn Z 1 1 εxxxx (x, t − τ ) + 2e−2ωn cos ωn + 2 ωn 0 × sin(ωn x) − cos(ωn x) + e−ωn x
By (40)
|ωn2 φn (1)| ≤ 12,
εs (1, t , t ) = i
∞ X
h
ωn2 an (t )e
iωn2 t
− bn (t )e
−iωn2 t
i
φn (1),
(42)
n=0
c1 ≤ `n = k(φn (x), iωn2 φn (x))> k2H ≤ c2 ,
∀n≥1
`n an (t )eiωn (t −τ ) φn (x) ε(x, t − τ ) = , εs (x, t − τ ) iωn2 φn (x) H k1 εs (1, t − τ ) = −2 sin ωn + 2e−2ωn sin ωn 2 ωn i + 2 εsx (1, t − τ ) −2 cos ωn − 4e−ωn ωn Z 1 1 −2ωn − 2e cos ωn + 2 εxxxx (x, t − τ ) ωn 0 × sin(ωn x) − cos(ωn x) + e−ωn x
Z
2 εsxx (x, t − τ )dx
1
2 εxxxx (x, t − τ )dx 0 1/2 #
1/2
0
32 + 4k1
ωn2
kB(ε(·, t − τ ), εs (·, t − τ ))kH .
(47)
By (45) and boundary condition εxxx (1, t − τ ) = k1 εs (1, t − τ ), it has
|`n bn (t )| ≤
"
1
ωn2
4k1 |εs (1, t − τ )|
+ 8|εsx (1, t − τ )| + 12
Z
1
Z
2 εsxx (x, t − τ )dx
+ 12
1
2 εxxxx (x, t − τ )dx 0 1/2 #
1/2
0
−ωn
`n bn (t )e φn (x) ε(x, t − τ ) = , εs (x, t − τ ) −iωn2 φn (x) H
4k1 |εs (1, t − τ )|
1
Z + 12 ≤
+ (sin ωn + cos ωn )e − 2e cos ωn (1 − x) + eωn (x−2) − (sin ωn − cos ωn )e−ωn (x+1) − e−2ωn [sin(ωn x) + cos(ωn x)] dx Z 1 i + 2 εsxx (x, t − τ ) −sin(ωn x) + cos(ωn x) ωn 0
−iωn2 (t −τ )
ωn2
+ 8|εsx (1, t − τ )| + 12
2
+ e−ωn x + (sin ωn + cos ωn )eωn (x−1) + 2e−ωn cos ωn (1 − x) + eωn (x−2) − (sin ωn − cos ωn )e−ωn (x+1) + e−2ωn [sin(ωn x) + cos(ωn x)] dx,
(46)
"
1
(43)
with some constants c1 , c2 > 0. Now suppose (ε(x, 0), εs (x, 0)) ∈ D(B). Then by C0 -semigroup theory, (ε(x, t − τ ), εs (x, t − τ )) ∈ D(B). This means, by (20), that εs (·, t − τ ) ∈ HE2 (0, 1), εxxxx (·, t − τ ) ∈ L2 (0, 1) for any t > τ . From (38), it follows that
ωn (x−1)
∀ n ≥ 1.
It then follows from (44) that
|`n an (t )| ≤
where
(45)
≤
32 + 4k1
ωn2
[kB(ε(·, t − τ ), εs (·, t − τ ))k]H .
(48)
Notice that in the last steps of (47) and (48), we used the facts that
(44)
Z 1 |εs (1, t − τ )| = εsx (x, t − τ )dx 0 Z 1 1/2 Z 1 1/2 2 2 ≤ εsx (x, t − τ )dx ≤ εsxx (x, t − τ )dx , 0 0 Z 1 |εsx (1, t − τ )| = εsxx (x, t − τ )dx 0 Z 1 1/2 2 ≤ εsxx (x, t − τ )dx . 0
B.-Z. Guo, K.-Y. Yang / Automatica 45 (2009) 1468–1475
By (41), collecting (42)–(48) gives
|εs (1, t , t )| ≤ C kB(ε(·, t − τ ), εs (·, t − τ ))kH
+ (49)
for some constant C > 0 independent of t. Now, by (20) and C0 semigroup theory, we have
kB(ε(·, t − τ ), εs (·, t − τ ))kH ≤ Me−ω(t −τ ) kB(ε(·, 0), εs (·, 0))kH
(50)
for any t ∈ [τ , +∞), where M , ω are given by (22). We finally get
|εs (1, t , t )| ≤ CMe ∗
In particular u ∈ be written as
−ω(t −τ )
L2loc
kB(ε(·, 0), εs (·, 0))kH .
wtt (·, t ) + Aw(·, t ) + k2 BB∗ wt (·, t ) + k2 Bεs (1, t , t ) = 0 for all t > τ ,
E (t ) =
+δ
2
k2 − 1−
(52)
2 k2 η2 4
E (t ) ≤ e
k2 η1
−
k2 δ
δ 1+δ
E (t ) + M1 εs2 (1, t , t ).
δ t − 1+δ
E (0) + M1
2
− k2 δwx (1, t )wt (1, t ) − k2 δwx (1, t )εs (1, t , t ) Z Z 1 δ 1 2 2 2 − wt (x, t ) + wxx (x, t ) dx − δ wxx (x, t )dx 2 0 0 δ η1 2 ≤ − k2 − wt2 (1, t ) + k2 wt (1, t ) 2 4 1 η2 2 1 + εs2 (1, t , t ) + k2 δ wx (1, t ) + wt2 (1, t ) η1 4 η2 1 2 δ η3 2 + k2 δ wx (1, t ) + εs (1, t , t ) − E (t ) 4 η3 1+δ Z 1 2 −δ wxx (x, t )dx 0 δ δ k2 η1 k2 δ =− E (t ) − k2 − − − wt2 (1, t ) 1+δ 2 4 η2 η η3 2 2 + k2 δ + wx (1, t ) 4 4 Z 1 k2 k2 δ 2 −δ wxx (x, t )dx + + εs2 (1, t , t ) η1 η3 0 for some constants ηi , i = 1, 2, 3. This together with the fact
4
0
t
Z
e
−(2ω− δ )t
2 [wt2 (x, t ) + wxx (x, t )]dx. 0 δ E˙ (t ) = − k2 − wt2 (1, t ) − k2 wt (1, t )εs (1, t , t )
4
(55)
δ (t −ρ) − 1+δ 2
εs (1, ρ, ρ)dρ
≤ E (0) + M1 C 2 M 2 kB(ε(·, 0), εs (·, 0))k2H 1+δ 1−e × e2ωτ δ 2ω − 1+δ
≤ E (0) +
# e
δ t − 1+δ
M1 C 2 M 2 kB(ε(·, 0), εs (·, 0))k2H · e2ωτ
1
2 wxx (x, t )dx gives ˙E (t ) ≤ − δ E (t ) − k2 − δ − k2 η1 − k2 δ wt2 (1, t ) 1+δ 2 4 η2 Z 1 k2 η2 k2 η3 2 −δ 1 − − wxx (x, t )dx
δ 1+δ
δ
(53)
0
2ω >
k δ
2 [wt2 (x, t ) + wxx (x, t )]dx
R1
> 0,
4 η2 k2 η3 − > 0, 4
"
wx2 (1, t ) ≤
(54)
≤ e− 1+δ t E (0) + M1 C 2 M 2 · kB(ε(·, 0), εs (·, 0))k2H Z t − δ (t −ρ) −2ω(ρ−τ ) e 1+δ e dρ × 0 "
0
xwx (x, t )wt (x, t )dx,
εs2 (1, t , t ).
0
where 0 < δ < 1 is a small constant. Then E (t )
−
E˙ (t ) ≤ −
0
Z
δ
k
1
Z
η3
and set η2 + η2 = M1 to get 1 3
1
Z
k2 δ
Take (51) into account and apply Gronwall’s inequality to (55) to obtain
where A, B are defined in (7). It then follows from Guo and Luo (2002) that there exists a unique solution to (52) such that (w(·, t ), wt (·, t )) ∈ C (τ , ∞; H ). In what follows, we may assume without loss of generality that the solution of (52) is a classical solution because εs (1, t , t ) can be approximated by a smooth function (see e.g., Guo and Xu (2008)). Define 1
+
Choose η1 , η2 , η3 > 0 and δ > 0 so that
(51)
(0, ∞). Furthermore, like (15), Eq. (33) can
k2
η1
1473
2ω −
δ 1+δ
The proof is complete by setting M0 = 2 δ . 2(1+δ)
# e
δ t − 1+δ
M1 C 2 M 2 (1+δ)e2ωτ 2ω(1+δ)−δ
.
, ω0 =
Remark 4.1. (1) When we take (b w0 , w b1 ) = (w0 , w1 ), (36) reduces to the usual exponential stability:
k(w(·, t ), wt (·, t ))kH ≤ k(w0 , w1 )kH e−ω0 t ,
∀ t ≥ 0.
(ii) When (w0 , w1 ) ∈ D(B), then we can take (b w0 , w b1 ) = 0; and (36) reduces to
k(w(·, t ), wt (·, t ))kH ≤ (M0 + kB−1 k)kB(w0 , w1 )kH e−ω0 t . 5. Simulation results In this section, we use finite difference method to give some numerical simulation results for the closed-loop system (33)–(35). Here we choose the space grid size N = 30, time step dt = 0.0005 and time span [0, 6]. Parameters are chosen to be τ = k1 = k2 = 1. For the initial value
w0 (x) = x2 , ε(x, 0) = x2 , , w1 (x) = 1 εs (x, 0) = 1,
∀ x ∈ [0, 1],
(1)
the latter is in H but not in the domain of B, we plot in Fig. 1 the displacement w(x, t ) and velocity wt (x, t ). It is seen that the system is still stable. This is not surprise because the asymptotical stability in Theorem 4.1 is expected for non-smooth initial value. Indeed, for a one-dimensional wave equation, it was shown in Guo and Xu (2008) that for non-smooth initial values, the closed-loop system is asymptotically stable. Obviously, the predictor–observer based scheme developed in this paper can be useful in designing stabilizing controller for any other PDEs with observation delay. But the stability proof for the latter case poses a big deal of mathematical challenge.
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B.-Z. Guo, K.-Y. Yang / Automatica 45 (2009) 1468–1475
Fig. 1. Displacement and velocity of the beam with initial value (1) and time delay τ = 1.
Fig. 2. Displacement and velocity of the beam with delay disturbance θ = 0.1, initial value (1) and time delay τ = 1.
In the end of this paper, we mention a few words about the robust to time delay for the present developed scheme. It should be pointed out that there is no positive result up to present since the problem was proposed in Datko (1988). Actually, as we mentioned earlier, it was proved in Logemann et al. (1996) that for our system, because there are infinitely many poles on the imaginary axis for delay free system, there does not exist generally a stabilizing controller that is robust to time delay. The problem was addressed earlier in Georgiou and Smith (1989) by cascading a low-pass filter to the controller. A recent study for time delay system in Hilbert space with bounded control operator by LMI approach can be found in Fridman and Orlov (2009). For our scheme, if τ = 0, it is simply an observed based scheme. From the frequency-domain point of view, this case is the same as the direct proportional output feedback control and hence is not robust to time delay by Theorem 1.1 of Logemann et al. (1996). For τ > 0, we did a numerical experiment with τ = 1 and time delay disturbance θ = 0.1. Using the same initial value (1), we get results plotted in Fig. 2. It is seen that both displacement and velocity are not convergent. We may expect at least based on numerical experiment that the scheme is not robust to time delay. The analytical proof needs further investigations. Acknowledgements The Authors would like to thank Professor Cheng-Zhong Xu of University of Lyon I, France for valuable discussions. The supports of the National Natural Science Foundation of China and the National Research Foundation of South Africa are gratefully acknowledged. References Artstein, Z. (1982). Linear systems with delayed controls: A reduction. IEEE Transactions on Automatic Control, 27, 869–879.
Curtain, R. F., Weiss, G., & Weiss, M. (1996). Coprime fasctorization for regular linear systems. Automatica, 32, 1519–1531. Datko, R. (1988). Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM Journal on Control and Optimization, 26, 697–713. Datko, R. (1991). Two questions concerning the boundary control of certain elastic systems. Journal of Differential Equations, 92, 27–44. Fiagbedzi, Y. A., & Pearson, A. E. (1986). Feedback stabilization of linear autonomous time lag systems. IEEE Transactions on Automatic Control, 31, 847–855. Fleming, W. H. (Ed.) (1988). Future directions in control theory. Philadelphia: SIAM. Fridman, E., & Orlov, Y. (2009). Exponential stability of linear distributed parameter systems with time-varying delays. Automatica, 45, 194–201. Georgiou, T. T., & Smith, M. C. (1989). W-stability of feedback systems. Systems and Control Letters, 13, 271–277. Gumowski, I., & Mira, C. (1968). Optimization in control theory and practice. Cambridge: Cambridge University Press. Guo, B. Z., & Xu, C. Z. (2008). Boundary output feedback stabilization of a onedimensional wave equation system with time delay. In Proc. 17th IFAC world congress (pp. 8755–8760). Guo, B. Z., & Xu, C. Z. (2007). The stabilization of a one-dimensional wave equation by boundary feedback with non-collocated observation. IEEE Transactions on Automatic Control, 52, 371–377. Guo, B. Z., & Luo, Y. H. (2002). Controllability and stability of a second order hyperbolic system with collocated sensor/actuator. Systems and Control Letters, 46, 45–65. Klamka, J. (1982). Observer for linear feedback control of systems with distributed delays in controls and outputs. Systems and Control Letters, 1, 326–331. Krstic, M., & Smyshlyaev, A. (2008). Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays. Systems and Control Letters, 57, 750–758. Logemann, H., Rebarber, R., & Weiss, G. (1996). Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop. SIAM Journal on Control and Optimization, 34, 572–600. Manitius, A. Z., & Olbrot, A. W. (1979). Finite spectrum assignment problem for systems with delays. IEEE Transactions on Automatic Control, 24, 541–553. Olbrot, A. W. (1978). Stabilizability, detectability, and spectrum assignment for linear autonomous systems with general time delays. IEEE Transactions on Automatic Control, 23, 887–890. Watanabe, K., & Ito, M. (1981). An observer for linear feedback control laws of multivariable systems with multiple delays in controls and outputs. Systems and Control Letters, 1, 54–59.
B.-Z. Guo, K.-Y. Yang / Automatica 45 (2009) 1468–1475 Watanabe, K. (1986). Finite spectrum assignment and observer for multivariable systems with commensurate delays. IEEE Transactions on Automatic Control, 31, 543–550. Weiss, G., & Curtain, R. F. (1997). Dynamic stabilization of regular linear systems. IEEE Transactions on Automatic Control, 42, 4–21. Zhong, Q. C. (2006). Robust control of time-delay systems. London: Springer-Verlag. Bao-Zhu Guo received his Ph.D. degree in Applied Mathematics from the Chinese University of Hong Kong in 1991. From 1993 to 2000, he had been with the Department of Applied Mathematics, Beijing Institute of Technology, China, where he was an associate professor (1993–1998) and then professor (1998–2000). Since 2000, he has been a research professor in Academy of Mathematics and System Sciences, the Chinese Academy of Sciences.
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Dr. Guo is the recipient of One Hundred Talent Program from the Chinese Academy of Sciences (1999) and the National Natural Science Foundation of China for Distinguished Young Scholars (2003). His interests include mathematical biology and infinite-dimensional system control.
Kun-Yi Yang received her B.Sc. degree in Mathematics from Hefei University of Technology, China, in 2003, the M.Sc. degree from Beijing Institute of Technology in 2006. She is currently a Ph.D. student in Academy of Mathematics and System Sciences, Academia Sinica. Her research interests focus on distributed parameter systems control.
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Brief paper
Dynamic stabilization of an Euler–Bernoulli beam equation with time delay in boundary observationI Bao-Zhu Guo a,c,∗ , Kun-Yi Yang a,b a
Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, PR China
b
Graduate School of the Chinese Academy of Sciences, Beijing 100039, PR China
c
School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa
article
info
Article history: Received 6 May 2008 Received in revised form 17 November 2008 Accepted 1 February 2009 Available online 19 March 2009
a b s t r a c t An Euler–Bernoulli beam equation under boundary control and delayed observation is considered. An observer–predictor based scheme is developed to stabilize the equation. On a time interval where the observation is available, the state is tracked by the observer; when the observation is not available due to the delay, the state is estimated by the predictor. The estimation is then used in proportional feedback. It is shown that the state of closed-loop system decays exponentially for smooth initial values. © 2009 Elsevier Ltd. All rights reserved.
Keywords: Euler–Bernoulli beam equation Time delay Observer Feedback control Exponential stability
1. Introduction Time delays are common in practical control systems, especially in networked systems, where it takes time for the observation to be available for use in control. These delays often create stability problem or cause periodic oscillations (Gumowski & Mira, 1968). Indeed, as first noticed in Datko (1988) for one-dimensional Euler–Bernoulli beam equation, even a small amount of time delay in the stabilizing boundary output feedback schemes could destabilize the system. See also Datko (1991) for many other hyperbolic systems. According to Fiagbedzi (1988, p. 69), stabilization of distributed parameter control systems with time delay in observation and control represents a difficult mathematical challenge. This is largely still the case today. Inspired by the work in Guo and Xu (2007), we recently solved the stabilization problem for onedimensional wave equation with boundary control and delayed observation (Guo & Xu, 2008). The method we used in Guo and Xu
I This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Andrea Serrani under the direction of Editor Miroslav Krstic. ∗ Corresponding author at: Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, PR China. Tel.: +86 10 62651443; fax: +86 10 62587343. E-mail address: [email protected] (B.-Z. Guo).
0005-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2009.02.004
(2008) relies heavily on the explicit expression of analytic solution of the system. In this paper, we consider the following Euler–Bernoulli beam equation under boundary control and delayed observation:
wtt (x, t ) + wxxxx (x, t ) = 0, 0 < x < 1, t > 0, w(0, t ) = wx (0, t ) = wxx (1, t ) = 0, wxxx (1, t ) = u(t ), y(t ) = wt (1, t − τ ), t ≥ τ , w(x, 0) = w0 (x), wt (x, 0) = w1 (x),
(1)
where u is the control (or input), (w0 , w1 ) is the initial state, τ > 0 is a (known) constant time delay, and y is the delayed observation (or output). It is well known that if system (1) is time delay free (i.e., τ = 0), then the static proportional output feedback u(t ) = ky(t ), k > 0 exponentially stabilizes the system. In the presence of time delay (i.e., τ > 0), however, the closed-loop system with a static feedback is no more stable (Datko, 1988). Moreover, no dynamic stabilizing controller is robust to time delay in general for those plants like (1) with infinitely many unstable poles (Logemann, Rebarber, & Weiss, 1996). The above results lead to the search for control schemes in the presence of a given time delay. We refer to Weiss and Curtain (1997) for a general method of designing stabilizing controllers for regular systems. In this paper, we propose an observer–predictor based stabilization scheme. The state is dynamically estimated in
B.-Z. Guo, K.-Y. Yang / Automatica 45 (2009) 1468–1475
two steps: On a time interval where the observation is available, the state is tracked by the observer; when the observation is not available due to the delay, the state is predicted in the scheme. We show that with the estimated state feedback, the state of the closed-loop system decays exponentially for any smooth initial value. The idea of estimation can be explained for a finite-dimensional time-invariant system described by X˙ (t ) = AX (t ) + Bu(t ),
y(t ) = CX (t − τ ),
where τ > 0 is the time delay, A, B, C are matrices of appropriate dimensions. Let t > τ be a given time instant. For every s ∈ [0, t − τ ] the state X (s) can be tracked by a standard Luenberger observer
˙ b X (s) = Ab X (s) + Bu(s) + LCb X (s) − Ly(s + τ ), s ∈ [0, t − τ ], where L is a matrix such that A + LC is stable. To determine X (s) for s ∈ [t − τ , t ], where the observation is not available due to the delay, we design a predictor as
( ˙ b X t (s) = Ab Xt (s) + Bu(s), b Xt (t − τ ) = b X (t − τ ).
s ∈ [t − τ , t ]
We consider the system (3) in the energy state space = H × L2 (0, 1), H = HE2 (0, 1) × L2 (0, 1), HE2 (0, 1) = ∈ H 2 (0, 1)|f (0) = f 0 (0) = 0} with state variable (w(·, t ), wt (·, t ), z (·, t )), for which the inner product induced norm is defined by H {f
k(w(·, t ), wt (·, t ), z (·, t ))k2H Z 1 2 [wxx (x, t ) + wt2 (x, t ) + z 2 (x, t )]dx. = 0
The input and output spaces are the same U = Y = R. Theorem 2.1. The system (3) is well-posed: For any (w0 , w1 , z0 ) ∈ H and u ∈ L2loc (0, ∞), there exists a unique solution to (3) such that (w(·, t ), wt (·, t ), z (·, t )) ∈ C (0, ∞; H ); and for any T > 0, there exists a constant CT such that
k(w(·, T ), wt (·, T ), z (·, T ))k + 2 H
T
Z
y2 (t )dt 0
The state for t > τ is then estimated as b Xt (t ) ≈ X (t ). To comment on the proposed scheme with respect to early work, let us compute the estimate analytically. Solving the above two equations gives
˙ b X t (t ) = A + eAτ LC e−Aτ b Xt (t ) + Bu(t ) Z t − eAτ Ly(t ) − eAτ LC eA(t −τ −θ) Bu(θ )dθ .
1469
(2)
t −τ
We see that the estimate above becomes those in (17)–(18), (20)–(21) of Watanabe and Ito (1981) with D = A + eAτ LC e−Aτ , E = −eAτ L, T = I, B = B, C = C e−Aτ (see also Eq. 101 of Krstic and Smyshlyaev (2008) and Zhong (2006) on p. 40). It is noted that our scheme that views the time delay problem from a different way is closely related to the observer designs for finite-dimensional systems in Watanabe and Ito (1981) by timedomain approach (see also Klamka (1982) and Watanabe (1986)) and Zhong (2006) by frequency-domain approach, and can be easier to generalize to PDEs. Some other controllers for finitedimensional systems were presented in Curtain, Weiss, and Weiss (1996) near the end of page 1527, which are actually the same as controllers in Artstein (1982), Manitius and Olbrot (1979) and Olbrot (1978) (see also Fiagbedzi and Pearson (1986) and Zhong (2006)). The difficulty with infinite-dimensional systems is that the delay free open-loop system (τ = 0 in (1)) has infinitely many poles on the imaginary axis. We proceed as follows. In the next section, Section 2, we show the well-posedness of open-loop system. Section 3 is devoted to the design of observer and predictor. The exponential decay of closed-loop system under the estimated state feedback control is presented in Section 4. Section 5 gives some simulation results.
T
Z ≤ CT k(w0 , w1 , z0 )k2H +
u2 (t )dt .
(4)
0
Proof. It is well known that the following system
( wtt (x, t ) + wxxxx (x, t ) = 0, w(0, t ) = wx (0, t ) = wxx (1, t ) = 0, wxxx (1, t ) = u(t ), yw (t ) = wt (1, t ),
(5)
can be written as a second-order system in H :
wtt (·, t ) + Aw(·, t ) + Bu(t ) = 0, yw (t ) = B∗ wt (·, t ),
(6)
where A is a self-adjoint operator in H and B is the input operator, both are unbounded:
Af = f (4) , ∀ f ∈ D(A) = {f ∈ H 4 ∩ HE2 | f 00 (1) = f 000 (1) = 0}, B = δ(x − 1).
(7)
δ(·) denotes the Dirac distribution. It was shown in Guo and Luo (2002) that system (6)–(7) is well-posed in the sense of D. Salamon (Curtain et al., 1996): For any u ∈ L2loc (0, ∞) and (w0 , w1 ) ∈ H , there exists a unique solution (w(·, t ), wt (·, t )) ∈ C (0, ∞; H ) to (6); and for any T > 0, there exists a constant DT > 0 such that
k(w(·, T ), wt (·, T ))kH + 2
T
Z
|yw (t )|2 dt 0 T
Z ≤ DT k(w0 , w1 )k2H +
2. Well-posedness of the open-loop system
|u(t )|2 dt .
(8)
0
Introduce a new variable
Once the solution of (5) is determined, one can solve the ‘‘z’’ part in Eq. (3):
z (x, t ) = wt (1, t − τ x). Then system (1) becomes
wtt (x, t ) + wxxxx (x, t ) = 0, 0 < x < 1, t > 0, w(0, t ) = wx (0, t ) = wxx (1, t ) = 0, w (1, t ) = u(t ), xxx τ zt (x, t ) + zx (x, t ) = 0, z (0, t ) = wt (1, t ), w(x, 0) = w0 (x), wt (x, 0) = w1 (x), z ( x , 0 ) = z ( x ), 0 y(t ) = z (1, t ), where z0 is the initial value of variable z.
(3)
τ zt (x, t ) + zx (x, t ) = 0, z (0, t ) = wt (1, t ), z (x, 0) = z0 (x).
(9)
Actually, for any initial z0 ∈ L2 (0, 1), integrating along the characteristic line gives the unique (weak) solution to (9):
t t z0 x − , x≥ , τ τ z (x, t ) = wt (1, t − xτ ) , x < t . τ
(10)
1470
B.-Z. Guo, K.-Y. Yang / Automatica 45 (2009) 1468–1475
Therefore, for any T > 0,
there exists a unique solution to (15) such that (b w (·, s), w bs (·, s)) ∈ C (0, t − τ ; H ); and for any 0 < T ≤ t − τ , there exists a constant b C1T such that
1
Z
z 2 (x, T )dx 0
=
Z
1− Tτ 0
1 τ
z02 (x)dx +
T
Z
1
τ
w (1, t )dt , 2 t
w0 , w b1 )k2H C1T k(b k(b w (·, T ), w bs (·, T ))k2H ≤ b
T
Z
wt2 (1, t )dt ,
T ≤ τ, (11)
0
τ
T
Z
u (s)ds . 2
(16)
0
From (8), it has
y2 (t )dt 0
T +τ
Z
Z 1 z02 (x)dx, T ≤ τ , τ 1− Tτ = Z T −τ Z 1 τ wt2 (1, t )dt , z02 (x)dx + 0
y (s)ds + 2
+
T > τ,
T −τ
T
Z
T +τ
Z
τ
(12)
|u(s)|2 ds .
(17)
0
Combing (16) and (17) gives the required inequality by setting In order for (14) to be an observer for (13), we have to show its convergence. To do this, let
The importance of Theorem 2.1 lies in that for any initial value in the state space, the output belongs to L2loc (τ , ∞) as long as the input u belongs to L2loc (0, ∞). This fact is particularly important to the solvability of observer designed in the next section.
ε(x, s) = w b(x, s) − w(x, s),
3. Observer and predictor design In this section, we go back to system (1). For any given t > τ , we proceed two steps to estimate the state of (1) via the observer and predictor. Step 1: Construct an observer to estimate the state {(w(x, s), ws (x, s)), s ∈ [0, t − τ ]} from the known observation {y(s + τ )|s ∈ [0, t − τ ], t > τ }. Since the observation {y(s + τ ), s ∈ [0, t − τ ]} is known and {(w(x, s), ws (x, s)), s ∈ [0, t − τ ]} satisfies
w (x, s) + wxxxx (x, s) = 0, 0 < s < t − τ , ss w(0, s) = wx (0, s) = wxx (1, s) = 0, wxxx (1, s) = u(s), y(s + τ ) = ws (1, s),
(13)
we can construct naturally a Luenberger observer for system (13) as following:
w b (x, s) + w bxxxx (x, s) = 0, 0 < s < t − τ , ss w b(0, s) = w bx (0, s) = w bxx (1, s) = 0, w b ( 1, s) = u(s) + k1 [w bs (1, s) − y(s + τ )] , xxx w b (x , 0 ) = w b0 (x), w bs (x, 0) = w b1 (x),
(14)
where (b w0 , w b1 ) is the (arbitrarily assigned) initial state of observer. We first state the solvability and convergence of observer (14). Proposition 3.1. The system (14) is well-posed: For any t > τ , and (w0 , w1 ) ∈ H , (b w0 , w b1 ) ∈ H , u ∈ L2loc (0, ∞), there exists a unique solution to (14) such that (b w (·, s), w bs (·, s)) ∈ C (0, t − τ ; H ); and for any 0 < T ≤ t − τ , there exists a positive constant b CT such that T +τ
u2 (s)ds .
(18)
εss (x, s) + εxxxx (x, s) = 0, 0 < s < t − τ , ε(0, s) = εx (0, s) = εxx (1, s) = 0, εxxx (1, s) = k1 εs (1, s).
(19)
System (19) can be written as:
ε(x, s) ε(x, s) =B , εs (x, s) ds εs (x, s) d
(20)
where the operator B is defined as follows:
B(f , g )> = (g , −f (4) )> , D(B) = {(f , g )> ∈ (H 4 ∩ HE2 ) × HE2 | 00 f (1) = 0, f 000 (1) = k1 g (1)}.
(21)
B generates an exponential stable C0 -semigroup on H . For any (w0 , w1 ) ∈ H , (b w0 , w b1 ) ∈ H , there exists a unique solution ε to (19) which satisfies
k(ε(·, s), εs (·, s))kH ≤ Me−ωs k(b w0 − w0 , w b1 − w1 )kH , ∀ s ∈ [0, t − τ ], t > τ
(22)
for some positive constants M and ω. Step 2: Predict {(w(x, s), ws (x, s)), s ∈ (t − τ , t ]} by {(b w (x, s), w bs (x, s)), s ∈ [0, t − τ ]}. This is done by solving (1) with estimated initial value (b w (x, t − τ ), w bs (x, t − τ )) obtained from (14):
w bss (x, s, t ) + w bxxxx (x, s, t ) = 0, t − τ < s < t , w b(0, s, t ) = w bx (0, s, t ) = w bxx (1, s, t ) = 0, w bxxx (1, s, t ) = u(s), b (x , t − τ , t ) = w b(x, t − τ ), w w bs (x, t − τ , t ) = w bs (x, t − τ ).
(23)
By Theorem 2.1, for any t > τ and u ∈ L2loc (0, ∞), there exists a unique solution to (23) such that (b w (·, s, t ), w bs (·, s, t )) ∈ C (t − τ , t ; H ); and for any s ∈ (t − τ , t ], there exists a constant b C t ,s > 0 such that
0
Proof. Notice that (14) can be written as
w bss (·, s) + Aw b(·, s) + k1 BB∗ w bs (·, s) + B[u(s) − k1 y(s + τ )] = 0 for all 0 < s < t − τ , t > τ ,
0 ≤ s ≤ t − τ.
Then by (13) and (14), ε satisfies
(
k(b w (·, T ), w bs (·, T ))k2H Z ≤b CT k(w0 , w1 )k2H + k(b w0 , w b1 )k2H +
T +τ
b CT = b C1T (1 + DT ).
T > τ.
0
Collecting (8), (11) and (12) gives (4).
Z |y(s)|2 ds ≤ DT k(w0 , w1 )k2H +
(15)
where A, B are defined in (7). By Corollary 1 of Guo and Luo (2002), (15) is well-posed: For any t > τ and (b w0 , w b1 ) ∈ H , u ∈ L2loc (0, ∞), y(s +τ ) ∈ L2 (0, t −τ ) which is assured by Theorem 2.1,
k(b w (·, s, t ), w bs (·, s, t ))k2H Z ≤b Ct ,s k(b w (·, t − τ ), w bs (·, t − τ ))k2H +
t t −τ
This together with Proposition 3.1 leads to (23).
|u(s)|2 ds .
B.-Z. Guo, K.-Y. Yang / Automatica 45 (2009) 1468–1475
1471
Proposition 3.2. The system (23) is well-posed: For any t > τ , (w0 , w1 ) ∈ H , (b w0 , w b1 ) ∈ H , and u ∈ L2loc (0, ∞), there exists a unique solution to (23) such that
We consider system (30)–(32) in the state space X = H 3 . It is obvious that system (30)–(32) is equivalent to (33)–(35) for t > τ provided that u∗ ∈ L2loc (0, ∞) which will be clarified in (51).
(b w (·, s, t ), w bs (·, s, t )) ∈ C (t − τ , t ; H );
w (x, t ) + wxxxx (x, t ) = 0, t > τ , tt w(0, t ) = wx (0, t ) = wxx (1, t ) = 0, wxxx (1, t ) = k2 wt (1, t ) + k2 εs (1, t , t ), w(x, 0) = w0 (x), wt (x, 0) = w1 (x), εss (x, s) + εxxxx (x, s) = 0, 0 < s < t − τ , ε(0, s) = εx (0, s) = εxx (1, s) = 0, εxxx (1, s) = k1 εs (1, s), ε( b0 (x) − w0 (x), x, 0) = w εs ( x , 0 ) = w b1 (x) − w1 (x), εss (x, s, t ) + εxxxx (x, s, t ) = 0, t − τ < s < t , ε(0, s, t ) = εx (0, s, t ) = 0, εxx (1, s, t ) = εxxx (1, s, t ) = 0, ε( x, t − τ , t ) = ε(x, t − τ ), εs (x, t − τ , t ) = εs (x, t − τ ),
and for any s ∈ (t − τ , t ], there exists a constant Ct ,s such that
k(b w (·, s, t ), w bs (·, s, t ))k2H Z t 2 2 2 u (s)ds . w0 , w b1 )kH + ≤ Ct ,s k(w0 , w1 )kH + k(b 0
We finally get the estimated state variable by
w e(x, t ) = w b(x, t , t ),
w et (x, t ) = w bs (x, t , t ),
t > τ.
(24)
Theorem 3.1. For any t > τ , we have
k(w(·, t ) − w e(·, t ), wt (·, t ) − w et (·, t ))kH ≤ Me−ω(t −τ ) k(b w0 − w0 , w b1 − w1 )kH ,
(25)
where (b w0 , w b1 ) is the initial state of observer (14), (w0 , w1 ) is the initial state of original system (1), and M , ω are constants in (22). Proof. Let
ε(x, s, t ) = w b(x, s, t ) − w(x, s),
t − τ ≤ s ≤ t.
(26)
Then ε(x, s, t ) satisfies
εss (x, s, t ) + εxxxx (x, s, t ) = 0, t − τ < s ≤ t , ε(0, s, t ) = εx (0, s, t ) = 0, εxx (1, s, t ) = εxxx (1, s, t ) = 0, ε(x, t − τ , t ) = ε(x, t − τ ), εs (x, t − τ , t ) = εt (x, t − τ ),
(27)
This together with (22) and (24) gives (25).
(28)
Since the feedback u(t ) = k2 wt (1, t ) stabilizes exponentially the system (1), and we have the estimation w et (1, t ) of wt (1, t ), it is natural to design the estimated state feedback control law of the following (with k2 > 0): k2 w et (1, t ) = k2 w bs (1, t , t ), 0, t ∈ [0, τ ],
t > τ,
(29)
under which, the closed-loop system becomes a system of partial differential equations (30)–(32):
w (x, t ) + wxxxx (x, t ) = 0, t > τ , tt w(0, t ) = wx (0, t ) = wxx (1, t ) = 0, bs (1, t , t ), wxxx (1, t ) = k2 w w(x, 0) = w0 (x), wt (x, 0) = w1 (x), w b (x, s) + w bxxxx (x, s) = 0, 0 < s < t − τ , ss w b(0, s) = w bx (0, s) = w bxx (1, s) = 0, w b ( 1, s) = u∗ (s) + k1 [w bs (1, s) − ws (1, s)] , xxx w b(x, 0) = w b0 (x), w bs (x, 0) = w b1 (x), w bss (x, s, t ) + w bxxxx (x, s, t ) = 0, t − τ < s < t , w b(0, s, t ) = w bx (0, s, t ) = w bxx (1, s, t ) = 0, w bxxx (1, s, t ) = u∗ (s), b(x, t − τ , t ) = w b(x, t − τ ), w w bs (x, t − τ , t ) = w bs (x, t − τ ).
(35)
where ε(x, s) and ε(x, s, t ) are given by (18) and (26), respectively. System (34) is the same as system (19). Theorem 4.1. Let k1 > 0, k2 > 0 and t > τ . Then for any (w0 , w1 ) ∈ H , (b w0 , w b1 ) ∈ H , there exists a unique solution to system (33)–(35) such that (w(·, t ), wt (·, t )) ∈ C (τ , ∞; H ), (ε(·, s), εs (·, s)) ∈ C (0, t − τ ; H ), (ε(·, s, t ), εs (·, s, t )) ∈ C ([t − τ , t ] × [τ , ∞); H ) and for any t > τ , s ∈ [0, t − τ ], q ∈ (t − τ , t ], there exists a constant Ctsq > 0 such that
k(w(·, t ), wt (·, t ))kH + k(ε(·, s), εs (·, s))kH + k(ε(·, q, t ), εs (·, q, t ))kH ≤ Ctsq [k(w0 , w1 )kH + k(b w0 , w b1 )kH ].
k(w(·, t ), wt (·, t ))k2H ≤ [k(w0 , w1 )k2H + M0 kB(w0 − w b0 , w1 − w b1 )k2H ]e−2ω0 t ,
∀t≥0
(36)
for some M0 , ω0 > 0 that are independent of initial value.
4. Stabilization by the estimated state feedback
u∗ (t ) =
(34)
Moreover, for any (w0 − w b0 , w1 − w b1 ) ∈ D(B), where B is defined by (21), system (33) decays exponentially in the sense that
which is a conservative system
k(ε(·, t , t ), εs (·, t , t ))kH = k(ε(·, t − τ ), εt (·, t − τ ))kH .
(33)
(30)
(31)
Proof. For any (w0 , w1 ) ∈ H , (b w0 , w b1 ) ∈ H , since B defined by (21) generates an exponential stable C0 -semigroup on H , there is a unique solution (ε(·, s), εs (·, s)) ∈ C (0, t − τ ; H ) to (34) such that (22) holds true. Now, for any given time t > τ , write (35) as:
ε(x, s, t ) ε(x, s, t ) =A , εs (x, s, t ) ds εs (x, s, t ) d
where A is defined by
A(f , g )> = (g , −f (4) )> , D(A) = {(f , g )> ∈ (H 4 ∩ HE2 ) × HE2 | f 00 (1) = f 000 (1) = 0}. Then A is skew-adjoint in H and hence generates a conservative C0 -semigroup on H . For any (ε(·, t − τ ), εt (·, t − τ )) ∈ H that is determined by (34), there exists a unique solution to (35) such that
k(ε(·, s, t ), εs (·, s, t ))kH = k(ε(·, t − τ ), εt (·, t − τ ))kH ,
∀ s ∈ [t − τ , t ].
(37)
So, (ε(·, s, t ), εs (·, s, t )) ∈ C ([t − τ , t ] × [τ , ∞); H ). Moreover, since A is skew-adjoint with compact resolvent, the solution of (35) can be, in terms of s, represented as
X ∞ 2 φ n ( x) ε(x, s, t ) = an (t )eiωn s 2 εs (x, s, t ) iωn φn (x) n =1
(32)
+
∞ X n =1
bn (t )e−iωn s 2
φn (x) , −iωn2 φn (x)
(38)
1472
B.-Z. Guo, K.-Y. Yang / Automatica 45 (2009) 1468–1475
where {(φn (x), ±iωn2 φn (x))> }∞ n=1 is a sequence of all ω -linearly independent approximated normalized orthogonal eigenfunctions of A corresponding to eigenvalues ±iωn2 , ωn > 0 and φn satisfies
φn(4) (x) = ωn4 φn (x), 0 < x < 1, φn (0) = φn0 (0) = φn00 (1) = φn000 (1) = 0.
=
(39)
Solve (39) to obtain
00 φn (x) = −sin(ωn x) + cos(ωn x) + e−ωn x +eωn (x−1) (sin ωn + cos ωn ) +e−ωn [2 cos(ωn (1 − x)) + eωn (x−1) +e−ωn x (cos ωn − sin ωn )] +e−2ωn [sin(ωn x) + cos(ωn x)] , iωn2 φn (x) = i{sin(ωn x) − cos(ωn x) + e−ωn x +eωn (x−1) (sin ωn + cos ωn ) +e−ωn [−2 cos(ωn (1 − x)) + eωn (x−1) −ωn x +e−2ω (cos ωn − sin ωn )] −e n [sin(ωn x) + cos(ωn x)]},
+ (sin ωn + cos ωn )eωn (x−1) − 2e−ωn cos ωn (1 − x) + eωn (x−2) − (sin ωn − cos ωn )e−ωn (x+1) − e−2ωn [sin(ωn x) + cos(ωn x)] dx Z 1 i εsxx (x, t − τ ) [−sin(ωn x) − 2 ωn 0
(40)
+ cos(ωn x) + e−ωn x + (sin ωn + cos ωn )eωn (x−1) + 2e−ωn cos ωn (1 − x) + eωn (x−2) − (sin ωn − cos ωn )e−ωn (x+1) + e−2ωn [sin(ωn x) + cos(ωn x)] dx.
where (see e.g., Guo and Luo (2002))
1 π + O (n−1 ) as n → ∞. ωn = n −
(41)
2
Therefore
εxxx (1, t − τ ) −2 sin ωn + 2e−2ωn sin ωn ωn2 i + 2 εsx (1, t − τ ) 2 cos ωn + 4e−ωn ωn Z 1 1 εxxxx (x, t − τ ) + 2e−2ωn cos ωn + 2 ωn 0 × sin(ωn x) − cos(ωn x) + e−ωn x
By (40)
|ωn2 φn (1)| ≤ 12,
εs (1, t , t ) = i
∞ X
h
ωn2 an (t )e
iωn2 t
− bn (t )e
−iωn2 t
i
φn (1),
(42)
n=0
c1 ≤ `n = k(φn (x), iωn2 φn (x))> k2H ≤ c2 ,
∀n≥1
`n an (t )eiωn (t −τ ) φn (x) ε(x, t − τ ) = , εs (x, t − τ ) iωn2 φn (x) H k1 εs (1, t − τ ) = −2 sin ωn + 2e−2ωn sin ωn 2 ωn i + 2 εsx (1, t − τ ) −2 cos ωn − 4e−ωn ωn Z 1 1 −2ωn − 2e cos ωn + 2 εxxxx (x, t − τ ) ωn 0 × sin(ωn x) − cos(ωn x) + e−ωn x
Z
2 εsxx (x, t − τ )dx
1
2 εxxxx (x, t − τ )dx 0 1/2 #
1/2
0
32 + 4k1
ωn2
kB(ε(·, t − τ ), εs (·, t − τ ))kH .
(47)
By (45) and boundary condition εxxx (1, t − τ ) = k1 εs (1, t − τ ), it has
|`n bn (t )| ≤
"
1
ωn2
4k1 |εs (1, t − τ )|
+ 8|εsx (1, t − τ )| + 12
Z
1
Z
2 εsxx (x, t − τ )dx
+ 12
1
2 εxxxx (x, t − τ )dx 0 1/2 #
1/2
0
−ωn
`n bn (t )e φn (x) ε(x, t − τ ) = , εs (x, t − τ ) −iωn2 φn (x) H
4k1 |εs (1, t − τ )|
1
Z + 12 ≤
+ (sin ωn + cos ωn )e − 2e cos ωn (1 − x) + eωn (x−2) − (sin ωn − cos ωn )e−ωn (x+1) − e−2ωn [sin(ωn x) + cos(ωn x)] dx Z 1 i + 2 εsxx (x, t − τ ) −sin(ωn x) + cos(ωn x) ωn 0
−iωn2 (t −τ )
ωn2
+ 8|εsx (1, t − τ )| + 12
2
+ e−ωn x + (sin ωn + cos ωn )eωn (x−1) + 2e−ωn cos ωn (1 − x) + eωn (x−2) − (sin ωn − cos ωn )e−ωn (x+1) + e−2ωn [sin(ωn x) + cos(ωn x)] dx,
(46)
"
1
(43)
with some constants c1 , c2 > 0. Now suppose (ε(x, 0), εs (x, 0)) ∈ D(B). Then by C0 -semigroup theory, (ε(x, t − τ ), εs (x, t − τ )) ∈ D(B). This means, by (20), that εs (·, t − τ ) ∈ HE2 (0, 1), εxxxx (·, t − τ ) ∈ L2 (0, 1) for any t > τ . From (38), it follows that
ωn (x−1)
∀ n ≥ 1.
It then follows from (44) that
|`n an (t )| ≤
where
(45)
≤
32 + 4k1
ωn2
[kB(ε(·, t − τ ), εs (·, t − τ ))k]H .
(48)
Notice that in the last steps of (47) and (48), we used the facts that
(44)
Z 1 |εs (1, t − τ )| = εsx (x, t − τ )dx 0 Z 1 1/2 Z 1 1/2 2 2 ≤ εsx (x, t − τ )dx ≤ εsxx (x, t − τ )dx , 0 0 Z 1 |εsx (1, t − τ )| = εsxx (x, t − τ )dx 0 Z 1 1/2 2 ≤ εsxx (x, t − τ )dx . 0
B.-Z. Guo, K.-Y. Yang / Automatica 45 (2009) 1468–1475
By (41), collecting (42)–(48) gives
|εs (1, t , t )| ≤ C kB(ε(·, t − τ ), εs (·, t − τ ))kH
+ (49)
for some constant C > 0 independent of t. Now, by (20) and C0 semigroup theory, we have
kB(ε(·, t − τ ), εs (·, t − τ ))kH ≤ Me−ω(t −τ ) kB(ε(·, 0), εs (·, 0))kH
(50)
for any t ∈ [τ , +∞), where M , ω are given by (22). We finally get
|εs (1, t , t )| ≤ CMe ∗
In particular u ∈ be written as
−ω(t −τ )
L2loc
kB(ε(·, 0), εs (·, 0))kH .
wtt (·, t ) + Aw(·, t ) + k2 BB∗ wt (·, t ) + k2 Bεs (1, t , t ) = 0 for all t > τ ,
E (t ) =
+δ
2
k2 − 1−
(52)
2 k2 η2 4
E (t ) ≤ e
k2 η1
−
k2 δ
δ 1+δ
E (t ) + M1 εs2 (1, t , t ).
δ t − 1+δ
E (0) + M1
2
− k2 δwx (1, t )wt (1, t ) − k2 δwx (1, t )εs (1, t , t ) Z Z 1 δ 1 2 2 2 − wt (x, t ) + wxx (x, t ) dx − δ wxx (x, t )dx 2 0 0 δ η1 2 ≤ − k2 − wt2 (1, t ) + k2 wt (1, t ) 2 4 1 η2 2 1 + εs2 (1, t , t ) + k2 δ wx (1, t ) + wt2 (1, t ) η1 4 η2 1 2 δ η3 2 + k2 δ wx (1, t ) + εs (1, t , t ) − E (t ) 4 η3 1+δ Z 1 2 −δ wxx (x, t )dx 0 δ δ k2 η1 k2 δ =− E (t ) − k2 − − − wt2 (1, t ) 1+δ 2 4 η2 η η3 2 2 + k2 δ + wx (1, t ) 4 4 Z 1 k2 k2 δ 2 −δ wxx (x, t )dx + + εs2 (1, t , t ) η1 η3 0 for some constants ηi , i = 1, 2, 3. This together with the fact
4
0
t
Z
e
−(2ω− δ )t
2 [wt2 (x, t ) + wxx (x, t )]dx. 0 δ E˙ (t ) = − k2 − wt2 (1, t ) − k2 wt (1, t )εs (1, t , t )
4
(55)
δ (t −ρ) − 1+δ 2
εs (1, ρ, ρ)dρ
≤ E (0) + M1 C 2 M 2 kB(ε(·, 0), εs (·, 0))k2H 1+δ 1−e × e2ωτ δ 2ω − 1+δ
≤ E (0) +
# e
δ t − 1+δ
M1 C 2 M 2 kB(ε(·, 0), εs (·, 0))k2H · e2ωτ
1
2 wxx (x, t )dx gives ˙E (t ) ≤ − δ E (t ) − k2 − δ − k2 η1 − k2 δ wt2 (1, t ) 1+δ 2 4 η2 Z 1 k2 η2 k2 η3 2 −δ 1 − − wxx (x, t )dx
δ 1+δ
δ
(53)
0
2ω >
k δ
2 [wt2 (x, t ) + wxx (x, t )]dx
R1
> 0,
4 η2 k2 η3 − > 0, 4
"
wx2 (1, t ) ≤
(54)
≤ e− 1+δ t E (0) + M1 C 2 M 2 · kB(ε(·, 0), εs (·, 0))k2H Z t − δ (t −ρ) −2ω(ρ−τ ) e 1+δ e dρ × 0 "
0
xwx (x, t )wt (x, t )dx,
εs2 (1, t , t ).
0
where 0 < δ < 1 is a small constant. Then E (t )
−
E˙ (t ) ≤ −
0
Z
δ
k
1
Z
η3
and set η2 + η2 = M1 to get 1 3
1
Z
k2 δ
Take (51) into account and apply Gronwall’s inequality to (55) to obtain
where A, B are defined in (7). It then follows from Guo and Luo (2002) that there exists a unique solution to (52) such that (w(·, t ), wt (·, t )) ∈ C (τ , ∞; H ). In what follows, we may assume without loss of generality that the solution of (52) is a classical solution because εs (1, t , t ) can be approximated by a smooth function (see e.g., Guo and Xu (2008)). Define 1
+
Choose η1 , η2 , η3 > 0 and δ > 0 so that
(51)
(0, ∞). Furthermore, like (15), Eq. (33) can
k2
η1
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2ω −
δ 1+δ
The proof is complete by setting M0 = 2 δ . 2(1+δ)
# e
δ t − 1+δ
M1 C 2 M 2 (1+δ)e2ωτ 2ω(1+δ)−δ
.
, ω0 =
Remark 4.1. (1) When we take (b w0 , w b1 ) = (w0 , w1 ), (36) reduces to the usual exponential stability:
k(w(·, t ), wt (·, t ))kH ≤ k(w0 , w1 )kH e−ω0 t ,
∀ t ≥ 0.
(ii) When (w0 , w1 ) ∈ D(B), then we can take (b w0 , w b1 ) = 0; and (36) reduces to
k(w(·, t ), wt (·, t ))kH ≤ (M0 + kB−1 k)kB(w0 , w1 )kH e−ω0 t . 5. Simulation results In this section, we use finite difference method to give some numerical simulation results for the closed-loop system (33)–(35). Here we choose the space grid size N = 30, time step dt = 0.0005 and time span [0, 6]. Parameters are chosen to be τ = k1 = k2 = 1. For the initial value
w0 (x) = x2 , ε(x, 0) = x2 , , w1 (x) = 1 εs (x, 0) = 1,
∀ x ∈ [0, 1],
(1)
the latter is in H but not in the domain of B, we plot in Fig. 1 the displacement w(x, t ) and velocity wt (x, t ). It is seen that the system is still stable. This is not surprise because the asymptotical stability in Theorem 4.1 is expected for non-smooth initial value. Indeed, for a one-dimensional wave equation, it was shown in Guo and Xu (2008) that for non-smooth initial values, the closed-loop system is asymptotically stable. Obviously, the predictor–observer based scheme developed in this paper can be useful in designing stabilizing controller for any other PDEs with observation delay. But the stability proof for the latter case poses a big deal of mathematical challenge.
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B.-Z. Guo, K.-Y. Yang / Automatica 45 (2009) 1468–1475
Fig. 1. Displacement and velocity of the beam with initial value (1) and time delay τ = 1.
Fig. 2. Displacement and velocity of the beam with delay disturbance θ = 0.1, initial value (1) and time delay τ = 1.
In the end of this paper, we mention a few words about the robust to time delay for the present developed scheme. It should be pointed out that there is no positive result up to present since the problem was proposed in Datko (1988). Actually, as we mentioned earlier, it was proved in Logemann et al. (1996) that for our system, because there are infinitely many poles on the imaginary axis for delay free system, there does not exist generally a stabilizing controller that is robust to time delay. The problem was addressed earlier in Georgiou and Smith (1989) by cascading a low-pass filter to the controller. A recent study for time delay system in Hilbert space with bounded control operator by LMI approach can be found in Fridman and Orlov (2009). For our scheme, if τ = 0, it is simply an observed based scheme. From the frequency-domain point of view, this case is the same as the direct proportional output feedback control and hence is not robust to time delay by Theorem 1.1 of Logemann et al. (1996). For τ > 0, we did a numerical experiment with τ = 1 and time delay disturbance θ = 0.1. Using the same initial value (1), we get results plotted in Fig. 2. It is seen that both displacement and velocity are not convergent. We may expect at least based on numerical experiment that the scheme is not robust to time delay. The analytical proof needs further investigations. Acknowledgements The Authors would like to thank Professor Cheng-Zhong Xu of University of Lyon I, France for valuable discussions. The supports of the National Natural Science Foundation of China and the National Research Foundation of South Africa are gratefully acknowledged. References Artstein, Z. (1982). Linear systems with delayed controls: A reduction. IEEE Transactions on Automatic Control, 27, 869–879.
Curtain, R. F., Weiss, G., & Weiss, M. (1996). Coprime fasctorization for regular linear systems. Automatica, 32, 1519–1531. Datko, R. (1988). Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM Journal on Control and Optimization, 26, 697–713. Datko, R. (1991). Two questions concerning the boundary control of certain elastic systems. Journal of Differential Equations, 92, 27–44. Fiagbedzi, Y. A., & Pearson, A. E. (1986). Feedback stabilization of linear autonomous time lag systems. IEEE Transactions on Automatic Control, 31, 847–855. Fleming, W. H. (Ed.) (1988). Future directions in control theory. Philadelphia: SIAM. Fridman, E., & Orlov, Y. (2009). Exponential stability of linear distributed parameter systems with time-varying delays. Automatica, 45, 194–201. Georgiou, T. T., & Smith, M. C. (1989). W-stability of feedback systems. Systems and Control Letters, 13, 271–277. Gumowski, I., & Mira, C. (1968). Optimization in control theory and practice. Cambridge: Cambridge University Press. Guo, B. Z., & Xu, C. Z. (2008). Boundary output feedback stabilization of a onedimensional wave equation system with time delay. In Proc. 17th IFAC world congress (pp. 8755–8760). Guo, B. Z., & Xu, C. Z. (2007). The stabilization of a one-dimensional wave equation by boundary feedback with non-collocated observation. IEEE Transactions on Automatic Control, 52, 371–377. Guo, B. Z., & Luo, Y. H. (2002). Controllability and stability of a second order hyperbolic system with collocated sensor/actuator. Systems and Control Letters, 46, 45–65. Klamka, J. (1982). Observer for linear feedback control of systems with distributed delays in controls and outputs. Systems and Control Letters, 1, 326–331. Krstic, M., & Smyshlyaev, A. (2008). Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays. Systems and Control Letters, 57, 750–758. Logemann, H., Rebarber, R., & Weiss, G. (1996). Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop. SIAM Journal on Control and Optimization, 34, 572–600. Manitius, A. Z., & Olbrot, A. W. (1979). Finite spectrum assignment problem for systems with delays. IEEE Transactions on Automatic Control, 24, 541–553. Olbrot, A. W. (1978). Stabilizability, detectability, and spectrum assignment for linear autonomous systems with general time delays. IEEE Transactions on Automatic Control, 23, 887–890. Watanabe, K., & Ito, M. (1981). An observer for linear feedback control laws of multivariable systems with multiple delays in controls and outputs. Systems and Control Letters, 1, 54–59.
B.-Z. Guo, K.-Y. Yang / Automatica 45 (2009) 1468–1475 Watanabe, K. (1986). Finite spectrum assignment and observer for multivariable systems with commensurate delays. IEEE Transactions on Automatic Control, 31, 543–550. Weiss, G., & Curtain, R. F. (1997). Dynamic stabilization of regular linear systems. IEEE Transactions on Automatic Control, 42, 4–21. Zhong, Q. C. (2006). Robust control of time-delay systems. London: Springer-Verlag. Bao-Zhu Guo received his Ph.D. degree in Applied Mathematics from the Chinese University of Hong Kong in 1991. From 1993 to 2000, he had been with the Department of Applied Mathematics, Beijing Institute of Technology, China, where he was an associate professor (1993–1998) and then professor (1998–2000). Since 2000, he has been a research professor in Academy of Mathematics and System Sciences, the Chinese Academy of Sciences.
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Dr. Guo is the recipient of One Hundred Talent Program from the Chinese Academy of Sciences (1999) and the National Natural Science Foundation of China for Distinguished Young Scholars (2003). His interests include mathematical biology and infinite-dimensional system control.
Kun-Yi Yang received her B.Sc. degree in Mathematics from Hefei University of Technology, China, in 2003, the M.Sc. degree from Beijing Institute of Technology in 2006. She is currently a Ph.D. student in Academy of Mathematics and System Sciences, Academia Sinica. Her research interests focus on distributed parameter systems control.