Stabilization of linear strict-feedback systems with delayed integrators
Automatica 46 (2010) 1902–1910
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Brief paper
Stabilization of linear strict-feedback systems with delayed integrators✩ Nikolaos Bekiaris-Liberis ∗ , Miroslav Krstic Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, USA
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Article history: Received 10 December 2009 Received in revised form 29 May 2010 Accepted 8 July 2010 Available online 21 August 2010 Keywords: Delay systems Predictor Strict-feedback systems
abstract The problem of compensation of input delays for unstable linear systems was solved in the late 1970s. Systems with simultaneous input and state delay have remained a challenge, although exponential stabilization has been solved for systems that are not exponentially unstable, such as chains of delayed integrators and systems in the ‘feedforward’ form. We consider a general system in strict-feedback form with delayed integrators, which is an example of a particularly challenging class of exponentially unstable systems with simultaneous input and state delays, and design a predictor feedback controller for this class of systems. Exponential stability is proven with the aid of a Lyapunov–Krasovskii functional that we construct using the PDE backstepping approach. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Stabilization of linear systems with input delays continues to be an active area of research. Various control schemes for systems with input delay have been developed, with the starting point for many of them being the Smith predictor (Smith, 1959). The most important extensions of the Smith predictor have been designs based on the finite spectrum assignment framework (Artstein, 1982; Fiagbedzi & Pearson, 1986; Jankovic, 2009a, 2010; Kwon & Pearson, 1980; Manitius & Olbrot, 1979; Mondie & Michiels, 2003; Olbrot, 1978; Richard, 2003; Zhong, 2006). In addition to these designs, adaptive versions of predictorbased linear controllers are proposed in Evesque, Annaswamy, Niculescu, and Dowling (2003a), Liu and Krstic (2001), Niculescu and Annaswamy (2003b), Zhou, Wang, and Wen (2008), whereas adaptive controllers for unknown delay have been developed recently in Bekiaris-Liberis and Krstic (2010), Bresch-Pietri and Krstic (2009a,b) and Yildiray, Annaswamy, Kolmanovsky, and Yanakiev (2010). Moreover, various control designs for nonlinear systems exist (Jankovic, 2001, 2003, 2009b; Karafyllis, 2006; Krstic, 2008a, 2010; Mazenc & Bliman, 2004; Mazenc, Mondie, & Francisco, 2004). Despite the fact that numerous papers deal with linear systems with input delay, the problem of controller design for systems with simultaneous input and state delay has been tackled in only a
✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Faryar Jabbari under the direction of Editor Roberto Tempo. ∗ Corresponding author. Tel.: +1 858 822 1936; fax: +1 858 822 3107. E-mail addresses: [email protected], [email protected] (N. Bekiaris-Liberis), [email protected] (M. Krstic).
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few Refs. Fiagbedzi and Pearson (1986), Jankovic (2009a), Jankovic (2010), Loiseau (2000), Manitius and Olbrot (1979) and Watanabe, Nobuyama, Kitamori, and Ito (1992). In this work we consider a specially chosen possibly open-loop unstable linear system, with the special form of a strict-feedback system with delayed integrators. Specifically, we consider the following n-dimensional linear system X˙¯ 1 (t ) = a¯ 11 X¯ 1 (t ) + b1 X¯ 2 (t − D1 )
X˙¯ 2 (t ) = a¯ 21 X¯ 1 (t ) + a¯ 22 X¯ 2 (t ) + b2 X¯ 3 (t − D2 )
.. . X˙¯ n (t ) = a¯ n1 X¯ 1 (t ) + · · · + a¯ nn X¯ n (t ) + bn U¯ (t − Dn ), where X¯ i (t ), a¯ ij , U¯ (t ) ∈ R, bi �= 0, and Di ∈ R+ .
(1) (2)
(3)
For this system we develop a predictor-based controller (Section 2). To achieve this we use tools from the boundary control of first order linear hyperbolic PDEs (Krstic & Smyshlyaev, 2008a), together with the classical backstepping procedure (Krstic, Kanellakopoulos, & Kokotovic, 1995). Specifically, an infinite dimensional backstepping transformation is used, together with a control law, to convert the system to an exponentially stable (in a certain sense) target system. Using the boundness of the backstepping transformation and its inverse, we then prove exponential stability of the closed-loop system using a suitably weighted Lyapunov–Krasovskii functional (Section 3). The effectiveness of the proposed controller is illustrated by a simulation example of a second order unstable system (Section 4). 2. Controller design We start by redefining the states of system (1)–(3) such that the coefficients in front of the delayed terms are unity. That is, we
N. Bekiaris-Liberis, M. Krstic / Automatica 46 (2010) 1902–1910
define X1 (t ) = X¯ 1 (t )
(4)
X3 (t ) = b1 b2 X¯ 3 (t )
(6)
Xn (t ) = b1 b2 . . . bn−1 X¯ n (t ).
(7)
X2 (t ) = b1 X¯ 2 (t )
P2 (t ) = X2 (t ) +
.. .
U (t ) = b1 b2 . . . bn U¯ (t ) aij =
�
a¯ ij , bj . . . bi−1 a¯ ij ,
�
+ an2 P2 θ −
(8)
�
if i = j . if i > j
(9)
X˙ 1 (t ) = a11 X1 (t ) + X2 (t − D1 )
(10)
X˙ 2 (t ) = a21 X1 (t ) + a22 X2 (t ) + X3 (t − D2 )
(11)
.. . X˙ n (t ) = an1 X1 (t ) + · · · + ann Xn (t ) + U (t − Dn ).
(12)
We state here our controller and in Section 3 we analyze the stability properties of the closed-loop system. The controller for the system (10)–(12) is given by
= −an1 P1 t −
Dk
k=1
�
αi (x, t ) = −ai1 P1 t − � − aii Pi t − � �
k=1 n
�
k=i n
− ci Pi t − +
Dk + x
Dk + x
� k=i
�
Dk + x
∂αi−1 (Di−1 + x, t ) , ∂x �
α1 (x, t ) = −(a11 + c1 )P1 t + x − � � n � x ∈ 0, Dk ,
− αi−1 (Di−1 + x, t ) � � n � x ∈ 0, Dk k=1
Dk
�
�
t
t−
n �
k=1
Dk
�
P2 (θ ) = X2 (0) +
Dk
k=1
�
� · · · + ann Pn (θ ) + U (θ ) dθ,
(18)
(a11 P1 (σ ) + P2 (σ ))dσ
(19)
θ −
�
n �
k=1
Dk
θ
−
n �
k=2
Dk
(a21 P1 (σ − D1 )
+ a22 P2 (σ ) + P3 (σ ))dσ
.. .
Pn (θ ) = Xn (0) +
�
�
θ −D n
n −1 �
�
Dk
k=2
an1 P1
�
(D
�
σ−
(20)
n −1 �
Dk
k=1
�
�
· · · + ann Pn (σ ) + U (σ ) dσ , + x ,t )
�n
k=i
(21)
Dk , 0]. Note here
∂α
(x� ,t )
3. Stability analysis
(14)
k=i
n 1�
Ω (t ) =
2 i =1
+
,
(a11 P1 (θ ) + P2 (θ ))dθ
(22)
where
(15)
and the ci , i = 1, 2 . . . n are arbitrary positive constants. In the �n above control scheme we use the Pi (t ) signals, the k=i Dk seconds ahead predictors of the Xi (t ) state � (this fact becomes clear later on). n That is, it holds that Pi (t ) = Xi (t + k=i Dk ). These signals are given by
�
Dk
�
θ−
n−1 �
Ω (t ) ≤ κ Ω (0)e−λt ,
k=1
P1 (t ) = X1 (t ) +
an1 P1
�
Theorem 1. System (10)–(12) with the controller (13) is exponentially stable in the sense that there exist constants κ and λ such that
�
�
t −D n
�
(17)
We first state a theorem describing our main stability result and then we prove it using a series of technical lemmas.
− ···
n
t
∂α
(13)
�
(a21 P1 (θ − D1 )
that the notation i−1 ∂ix−1 corresponds to i−∂1x� |x� =x+Di−1 which includes the time derivatives of the signals P1 (t ), . . . , Pi−1 (t ). These derivatives are obtained from (10)–(12) and (16)–(18).
where n �
k=2
Dk
where θ is defined in each Pi (θ ) as θ ∈ [−
− · · · − ann Pn (t )
∂αn−1 (Dn−1 + Dn , t ) , ∂x �
n �
k=2
+ an2 P2 σ −
− cn (Pn (t ) − αn−1 (Dn−1 + Dn , t )) +
t−
with initial conditions
U (t ) = u(Dn , t )
n−1 �
t
n−1 �
P1 (θ ) = X1 (0) +
In the new variables, system (1)–(3) is transformed to
= αn (Dn , t ) �
�
Pn (t ) = Xn (t ) +
Moreover, for notational consistency we define
1903
+ a22 P2 (θ ) + P3 (θ ))dθ
(5)
.. .
�
(16)
and
� �
t t −Di−1 t
t −D n
1 2
�
Xi2 (t ) + t t −D n
U (θ )dθ =
2 i=2
t t −Di−1
Xi2 (θ )dθ
U 2 (θ )dθ,
(23)
�
(24)
Xi2 (θ )dθ = 2
n � 1�
�
Di−1 0
Dn
0
ξi2 (x, t )dx = �ξi (t )�2
u2 (x, t )dx = �u(t )�2
ξi (x, t ) = Xi (t + x − Di−1 ), u(x, t ) = U (t + x − Dn ),
x ∈ [0, Di−1 ]
x ∈ [0, Dn ].
We first give and prove the following lemmas.
(25) (26) (27)
N. Bekiaris-Liberis, M. Krstic / Automatica 46 (2010) 1902–1910
1904
Lemma 1. The signals Pi (t ) defined in (16)–(18) are, respectively the �n D seconds ahead predictors of the Xi (t ) states. Moreover an k k=i equivalent representation for (16)–(18) is given by p1
p2
� �
n �
Dk , t
k=1
n �
Dk , t
k=2
� �
= X1 (t ) +
= X2 (t ) +
k=1
0
Dk
(a11 p1 (y, t )
�
n �
k=2
0
Dk
(28)
(a21 p1 (y, t )
+ a22 p2 (y, t ) + p3 (y − D2 , t ))dy
pn (Dn , t ) = Xn (t ) +
pi (x, t ) = Pi
n �
+ p2 (y − D1 , t ))dy
.. .
where
�
�
Dn 0
(an1 p1 (y, t ) + · · ·
+ ann pn (y, t ) + u(y, t ))dy,
�
t +x−
n �
Dk
k =i
�
,
x∈
�
0,
(29)
(30)
n �
�
Dk .
k=i
By taking into account (44) we have that pi (x, t ) = Xi (t + x),
� n �
Dk , t
k=i
�
= Pi (t ), �
p1 (x, t ) = X1 (t ) +
X˙ 1 (t ) = a11 X1 (t ) + ξ2 (0, t )
(32)
p2 (x, t ) = X2 (t ) +
ξ2 (D1 , t ) = X2 (t ) X˙ 2 (t ) = a21 X1 (t ) + a22 X2 (t ) + ξ3 (0, t )
(34)
(33) (35)
ξ3t (x, t ) = ξ3x (x, t )
(36)
ξ3 (D2 , t ) = X3 (t ) .. . X˙ n (t ) = an1 X1 (t ) + · · · + ann Xn (t ) + u(0, t ) ut (x, t ) = ux (x, t ) u(Dn , t ) = U (t ).
(37)
p1x (x, t ) = a11 p1 (x, t ) + p2 (x − D1 , t )
(41)
(38) (39) (40)
Consider the following ODEs in x (to become clear that these are ODEs in x, consider the time t to act as a parameter rather as a running variable), p2x (x, t ) = a21 p1 (x, t ) + a22 p2 (x, t ) + p3 (x − D2 , t )
(42)
.. .
�n
(43)
where, for each pi (x, t ), x varies in [0, k=i Dk ]. The initial conditions for the above system of ODEs are given by
and
Dk , ∀i.
(47)
∀i,
(48)
.. .
x 0
�
(a11 p1 (y, t ) + p2 (y − D1 , t ))dy x
0
(a21 p1 (y, t ) + a22 p2 (y, t )
+ p3 (y − D2 , t ))dy
pn (x, t ) = Xn (t ) +
�
0
x
(49)
(50)
(an1 p1 (y, t ) + · · ·
+ ann pn (y, t ) + u(y, t ))dy. (51) �n By setting in each pi (x, t ), x = k=i Dk and using (31) we get (28)–(30).
�
It is important here to observe that the total delay from the input �n to each state Xi (t ) is D . This explains the fact that our k k=i predictor �n intervals are different for each state and specifically must be k=i Dk seconds for each state Xi (t ). Our controller design is based on a recursive procedure that transforms system (10)–(12) to a target system which is exponentially stable with the controller (13). Then, using the invertibility of this transformation, we prove exponential stability of the original system. We now state this transformation, along with its inverse. Lemma 2. The state transformation defined by
pnx (x, t ) = an1 p1 (x, t ) + · · · + ann pn (x, t ) + u(x, t ),
pi (0, t ) = Xi (t ),
k=i
�
we get (31). By integrating from 0 to x (41)–(43) we get
Proof. Consider the equivalent representation of system (10)–(12) using transport PDEs for the delayed states and control
ξ2t (x, t ) = ξ2x (x, t )
0,
n �
To see that (46) holds it is sufficient to prove that (47) is the unique solution of the ODEs in x given by (41)–(43) with the initial conditions (44)–(45). Since then, the pi (x, t ) are functions of only one variable, namely x + t, and consequently (46) holds. Thus, it remains to prove that (47) is the unique solution of the initial value problem (41)–(45). Toward this end, by taking into account (27) we point out that (47) satisfy the initial value problem (41)–(45). Then, assuming that Xi (t + θi ), i = 2, . . . , n are continuous for all θi ∈ [−Di−1 , 0], using Theorem 2.1 from Hale and Verduyn Lunel (1993) we can conclude that (47) is the unique solution of the ODEs in x given by (41)–(43) with the initial conditions (44)–(45). Thus, (46) holds. From relation (47) becomes clear that the pi (x, t ) are the x seconds ahead predictors of the states. By defining pi
(31)
x∈
�
∀i,
pi (θi , t ) = Xi (t + θi ),
θi ∈ [−Di−1 , 0], i = 2, . . . , n.
(44)
(45)
Since system (41)–(43) is driven by the input u(x, t ), which satisfies a transport PDE, the same holds for all the pi (x, t ) (see for example Krstic (2008a, 2010)). Thus,
∂ pi (x, t ) ∂ pi (x, t ) = , ∂t ∂x
x∈
�
0,
n � k=i
�
Dk , ∀i.
(46)
Z1 (t ) = X1 (t )
(52)
Zi+1 (t ) = Xi+1 (t ) − αi (Di , t ),
i = 1, 2, . . . , n − 1,
(53)
along with the transformation of the actuator state
w(x, t ) = u(x, t ) − αn (x, t ),
x ∈ [0, Dn ],
(54)
where the αi (x, t ) are defined as in (14)–(15), transforms the system (10)–(12) to the target system with the control law given by (13). The target system is given by Z˙1 (t ) = −c1 Z1 (t ) + Z2 (t − D1 ) Z˙2 (t ) = −c2 Z2 (t ) + Z3 (t − D2 )
.. . Z˙n (t ) = −cn Zn (t ) + W (t − Dn ),
(55) (56)
(57)
N. Bekiaris-Liberis, M. Krstic / Automatica 46 (2010) 1902–1910
where
1905
If we now define
W (θ) = 0,
θ ≥ 0.
(58)
Proof. Before we start our recursive procedure we rewrite the target system using transport PDEs as Z˙1 (t ) = −c1 Z1 (t ) + ζ2 (0, t )
(59)
ζ2 (D1 , t ) = Z2 (t ) Z˙2 (t ) = −c2 Z2 (t ) + ζ3 (0, t )
(61)
ζ3 (D2 , t ) = Z3 (t ) .. . Z˙n (t ) = −cn Zn (t ) + w(0, t ) wt (x, t ) = wx (x, t ) w(Dn , t ) = 0.
(64)
ζ2t (x, t ) = ζ2x (x, t )
(60) (62)
ζ3t (x, t ) = ζ3x (x, t )
(63)
(65) (66)
Z˙3 (t ) = a31 X1 (t ) + a32 X2 (t ) + a33 X3 (t ) + ζ4 (0, t )
+ α3 (0, t ) −
x ∈ [0, Di−1 ].
(68)
ζ2 (x, t ) = ξ2 (x, t ) − α1 (x, t ),
(69)
X˙ 1 (t ) = a11 X1 (t ) + ζ2 (0, t ) + α1 (0, t ).
(70)
By choosing α1 (x, t ) = −(a11 + c1 )p1 (x, t ) (note the equivalent representation of α1 (x, t ) using (31)) and by using (44) we get Z˙1 (t ) = −c1 Z1 (t ) + ζ2 (0, t ).
(71)
Z2 (t ) = X2 (t ) − α1 (D1 , t ).
(72)
ζ3 (x, t ) = ξ3 (x, t ) − α2 (x, t ),
(73)
From (69) with x = D1 and (61) it follows that
Z˙i−1 (t ) = −ci−1 Zi−1 (t ) + ζi (0, t ),
(80)
ζi+1 (x, t ) = ξi+1 (x, t ) − αi (x, t ).
(81)
and define ζi+1 (x, t ) as
Then from (53) with x = Di−1 we have that
Z˙i (t ) = ai1 X1 (t ) + · · · + aii Xi (t ) + ζi+1 (0, t )
+ αi (0, t ) −
∂αi−1 (Di−1 , t ) . ∂x
(82)
αi (x, t ) = −ai1 p1 (x, t ) − · · · − aii pi (x, t ) ∂αi−1 (Di−1 + x, t ) − ci (pi (x, t ) − αi−1 (Di−1 + x, t )) + , ∂x
(83)
we get (84)
Step n. In the last step we choose the controller U (t ). Since Z˙n (t ) = an1 X1 (t ) + · · · + ann Xn (t ) + u(0, t )
∂αn−1 (Dn−1 , t ) . ∂x Then using (14) for i = n we have that −
(85)
Z˙n (t ) = −cn Zn (t ) + w(0, t ),
(86)
wt (x, t ) = wx (x, t ) w(Dn , t ) = 0.
(87)
w(x, 0) = w0 (x),
(89)
and using (13)
By setting now
and using (69), (33) and (35) we have
∂α1 (D1 , t ) + ζ3 (0, t ) + α2 (0, t ) − , ∂x
(74)
∂α (x,t )
where 1 ∂ x1 corresponds to 1∂ x |x=D1 and we use the fact that α1t (x, t ) = α1x (x, t ) (which is a consequence of relation (15)).
and by defining a new variable W (·) as
w0 (x) = W (x − D), we get that
Step 2. By choosing
α2 (x, t ) = −a21 p1 (x, t ) − a22 p2 (x, t ) − c2 (p2 (x, t ) ∂α1 (x + D1 , t ) − α1 (D1 + x, t )) + ∂x = −a21 p1 (x, t ) − a22 p2 (x, t ) − c2 (p2 (x, t ) + (a11 + c1 )p1 (D1 + x, t )) − (a11 + c1 )(a11 p1 (x + D1 , t ) + p2 (x, t )),
w(x, t ) =
�
x ∈ [0, Dn ],
W (t + x − D), 0,
� −D ≤ t + x − D ≤ 0 . t +x−D≥0
(90)
(91)
Defining θ = t + x − D one gets (58). Note here that based on (54), w0 (x) is given by
w0 (x) = u(x, 0) − αn (x, 0), (75)
we get from (74) (with the help of (44)) that
By setting now x = D2 in (73) and using (64) we get
(88)
Assuming an initial condition for (87) as
Z˙2 (t ) = ζ2x (x, t )|x=D1 = a21 X1 (t ) + a22 X2 (t )
Z3 (t ) = X3 (t ) − α2 (D2 , t ) .
(79)
Step i. Assume now that
Z˙i (t ) = −ci Zi (t ) + ζi+1 (0, t ).
then using (32) we get
Z˙2 (t ) = −c2 Z2 (t ) + ζ3 (0, t ).
∂α2 (D2 , t ) . ∂x
Hence, with
Step 1. Following the backstepping procedure we first stabilize X1 (t ) with the virtual input α1 (D1 , t ). We define
∂α (D ,t )
(78)
then with the help of (36) we get
(67)
Note that
ζi (x, t ) = Zi (t + x − Di−1 ),
ζ4 (x, t ) = ξ4 (x, t ) − α3 (x, t ),
(76)
(77)
x ∈ [0, Dn ]. �
(92)
We now define the inverse transformation of (52)–(54). Lemma 3. The inverse transformation of (52)–(54) is defined as X1 ( t ) = Z 1 ( t )
Xi+1 (t ) = Zi+1 (t ) + βi (Di , t ), u(x, t ) = w(x, t ) + βn (x, t ),
(93) i = 1, 2, . . . , n − 1.
x ∈ [0, Dn ],
(94) (95)
N. Bekiaris-Liberis, M. Krstic / Automatica 46 (2010) 1902–1910
1906
where the βi (x, t ) are now given by
β1 (x, t ) = −(a11 + c1 )�1 (x, t ),
x∈
�
0,
n �
Dk
k=1
�
βi (x, t ) = −ai1 �1 (x, t ) − ai2 (�2 (x, t ) + β1 (D1 + x, t )) − · · · − aii (�i (x, t ) + βi−1 (Di−1 + x, t )) ∂βi−1 (Di−1 + x, t ) − ci �i (x, t ) + , ∂x � � n � x ∈ 0, Dk , ∀i = 2, . . . , n,
(96)
derivative of the above function along the solutions of the Z (t ) system and by exploiting the fact that ζi (x, t ) and w(x, t ) satisfy transport PDEs (based on (60), (63) and (66)), it follows that V˙ (t ) = −
(97)
−
and the �i (x, t ) (the predictors of the transformed states) are given by the following relations
�1 (x, t ) = Z1 (t ) + �2 (x, t ) = Z2 (t ) + .. . � n ( x, t ) = Z n ( t ) +
� �
x 0 x 0
x 0
2
t −D n
(−cn �n (y, t ) + w(y, t ))dy, �n
k=i
(100)
Dk ].
Ξ (0)e
m − M1 t 1
(101)
,
� �
t t −Di−1 t
t −D n
Zi2
n � 1�
(θ )dθ =
W 2 (θ )dθ =
2 i =2
t t −Di−1
∀i,
(103)
�
�
2
(104)
w 2 (x, t )dx = �w(t )�2 .
(105)
0
0
2 i
ζ (x, t )dx = �ζi (t )�
n 1�
2 i=1
+
ki Zi2 (t ) +
λn+1 2
�
Dn 0
�
λ n +1 2
Dn
0
Di−1 0
ζi2 (x, t )dx
w 2 (x, t )dx,
(107)
λ2 =
(108) ci2−1
1 2c1
n 1�
2 i =2
λi
�
0
Di−1
(109)
,
V˙ (t ) ≤ −
−
n 1�
ci ki Zi2 (t ) −
2 i=1
�
λn+1 2
Defining
Dn 0
�
ci ki 2
,
it follows that V˙ (t ) ≤ −
m1 M1
m2 = min
Ξ (t ) ≤
�
n 1�
2 i=2
λi
�
Di−1 0
ζi2 (x, t )dx
w2 (x, t )dx.
(110)
i = 1, 2, . . . , n.
�
λi+1 , 2 (1 + Di )
(111)
i = 1, 2, . . . , n,
(112)
V (t ).
(113)
ki λi+1 2
,
2
�
,
i = 1, 2, . . . , n,
(114)
m1 m V (0) − M M1 (1 + Dmax ) t − 1t e 1 ≤ Ξ (0)e M1 . � m2 m2
(115)
We give now the following lemma which we prove in the Appendix.
(1 + x)ζi2 (x, t )dx
(1 + x)w 2 (x, t )dx.
i = 3, . . . , n + 1
,
and after some manipulations that incorporate completion of squares we get
then
Proof. We consider the following Lyapunov-like function V (t ) =
2 i=2
�
λi
i = 2, . . . , n
λi−1 (1 + Di−2 )
If we now define
Di−1
Dn
λi = 4
(1 + Di−1 ),
ci
m1 = min
Zi2 (θ )dθ (102)
and
w 2 (0, t ) −
M1 = max {ki , λi+1 } ,
W 2 (θ )dθ
Dmax = max {Di } ,
λi
k1 = 2
Zi2 (t ) + t
2
n 1�
λi ζi2 (0, t ) −
ki = 2
where
+
λn+1
λi (1 + Di−1 )Zi2 (t )
2 i=2
(99)
M1 (1 + Dmax )
2 i=1 � 1
2 i =2
n 1�
(−c2 �2 (y, t ) + �3 (y − D2 , t ))dy
Lemma 4. The target system is exponentially stable in the sense that there exist constants M1 , m1 and m2 such that
n 1�
−
n 1�
ki Zi (t )ζi+1 (0, t )
where we used integration by parts in the above integrals. By choosing the weights as
We now prove the stability of the transformed system.
Ξ (t ) =
i=1
(98)
Proof. Applying similar arguments as in Lemma 2 we prove that the inverse transformation of (52)–(54) and (14)–(15) is given by (93)–(97). �
m2
n−1 �
(−c1 �1 (y, t ) + �2 (y − D1 , t ))dy
where in each �i (x, t ), x varies in [0,
Ξ (t ) ≤
i=1
ci ki Zi2 (t ) +
+ kn Zn (t )w(0, t ) +
k=i
�
n �
Lemma 5. There exist constants Gi such that (106)
Note that the above functional can be considered as a Control Lyapunov Functional in the sense of Karafyllis and Jiang (in press). This fact reinforces the strength of the present result: a Control Lyapunov Functional is actually constructed. By taking the time
2
pi (x, t ) ≤ Gi
+
�
|X (t )|2 +
�
Dn 0
2
n � � i =2
Di−1 0
�
u (y, t )dy ,
ξi2 (y, t )dy �
∀x ∈ 0,
n � k=i
�
Dk ,
(116)
N. Bekiaris-Liberis, M. Krstic / Automatica 46 (2010) 1902–1910
3
where
|X (t )|2 =
n � i =1
Xi2 (t ),
2
(117)
1
and the bound (116) is independent of x.
0
Lemma 6. There exists a constant M such that
Ξ (t ) ≤ M Ω (t ).
(118)
–1 –2
Proof. From (52)–(54) it follows that Zi2 (t ) ≤ 2(Xi2 (t ) + αi2−1 (Di , t )), 2 i
2 i
i = 2, . . . , n
2 i−1
(119)
ζ (x, t ) ≤ 2(ξ (x, t ) + α (x, t )), x ∈ [0, Di−1 ], i = 2, . . . , n 2
2
2 n
w (x, t ) ≤ 2(u (x, t ) + α (x, t )),
x ∈ [0, Dn ].
αi2 (x, t ) ≤ bi
i �
p2k (x, t ),
k=1
x∈
�
0,
–3
(120)
n �
Dk
k=i
�
(122)
for some constants bi . By employing the bound of Lemma 5, the lemma is proven. �
�
2
� (x, t ) ≤ Fi |Z (t )| + +
�
Dn 0
n � � i =2
Di−1
Ω (t ) ≤
0
�
w2 (y, t )dy ,
ζ (y, t )dy x∈
�
0,
n �
�
Dk .
k=i
Ω (t ) ≤
(t ) ≤ 2( (t ) + β 2 i
(Di , t )),
i = 2, . . . , n
2 i−1
ξ (x, t ) ≤ 2(ζ (x, t ) + β (x, t )), x ∈ [0, Di−1 ], i = 2, . . . , n 2
2
2 n
u (x, t ) ≤ 2(w (x, t ) + β (x, t )),
(125) (126)
x ∈ [0, Dn ].
(127)
By observing that βi (x, t ) are linearly dependent on �1 (x, t ), . . . , �i (x, t ) we conclude that there exist constants di such that 2 i
β (x, t ) ≤ di
i � k=1
2 k
� (x, t ),
x∈
�
0,
n �
�
Dk .
k=i
Using Lemma 7 the lemma is proven.
(128)
�
Proof of Theorem 1. Combining Lemmas 6 and 8 we have that M Ω (t ) ≤ Ξ (t ) ≤ M Ω (t ).
(130)
,
MM1 (1 + Dmax ) Mm2
Ω (0)e
m
− M1 t 1
.
(131)
MM1 (1 + Dmax ) Mm2
m1 M1
. �
(132) (133)
We illustrate here our controller with a second order example with parameters a11 = a21 = a22 = 0.2, D1 = 0.4, D2 = 0.8 and c1 = c2 = 2. The initial conditions for the controller are given by (19)–(21) and for the system are X1 (0) = X2 (0) = 1 and X2 (θ ) = 1, θ ∈ [−D1 , 0]. This system is unstable (to see this one can use Olgac and Sipahi (2002)). In the present case the controller will have the form U (t ) = u(D2 , t )
Proof. Using relations (93)–(95) we get
2 i
M
4. Simulations
(124)
2 i −1
Ξ (t )
(123)
Lemma 8. There exists a constant M such that
Zi2
6
Thus Theorem 1 is proven with
λ=
Proof. Immediately note that the relation for the �i (x, t ) is similar to the relation for pi (x, t ). Note here that in this case the derivation of the explicit bound is easier due to the special form of the �i (x, t ) in (98)–(100). �
M Ω (t ) ≤ Ξ (t ).
4
and by Lemma 4 we get
κ=
2 i
2
Hence,
Lemma 7. There exist constants Fi such that 2 i
0
Fig. 1. System’s response for the simulation example.
(121)
Moreover, from relations (14)–(15) one can see that the αi (x, t ) are linear functions of the predictors p1 (x, t ), . . . , pi (x, t ), hence it holds that
Xi2
1907
(129)
= α2 (D2 , t ) = −a21 p1 (D2 , t ) − a22 p2 (D2 , t ) − c2 (p2 (D2 , t ) + (a11 + c1 )p1 (D1 + D2 , t )) − (a11 + c1 )(a11 p1 (D2 + D1 , t ) + p2 (D2 , t )) = −a21 P1 (t − D1 ) − a22 P2 (t ) − c2 (P2 (t ) + (a11 + c1 )P1 (t )) − (a11 + c1 )(a11 P1 (t ) + P2 (t )), (134)
where P1 (t ) and P2 (t ) are calculated using the integral representation (16)–(18). Note also that these integrals are computed using the trapezoidal rule. Fig. 1 shows that the predictor controller exponentially stabilizes the system. The control signal reaches first X2 (t ), since the delay from the input to X2 (t ) is 0.4, which is smaller than the total delay from the input to X1 (t ). After 1.2 s, which is the total delay from the input to X1 (t ), the controller starts stabilizing X1 (t ). Then both X1 (t ) and X2 (t ) converge exponentially to zero (see Fig. 2).
N. Bekiaris-Liberis, M. Krstic / Automatica 46 (2010) 1902–1910
1908
Appendix Here we give the proof of Lemma 5. Proof of Lemma 5. By solving (41)–(43), and by taking into account that this ODE in x system is in strict-feedback form, we get p1 (x, t ) =
p2 (x, t ) = Fig. 2. Control effort for the simulation example.
5. Conclusions We present a backstepping design for an exponentially unstable system with simultaneous input and state delay. Our design is predictor-based since it uses the predicted values of the states on given intervals. Using the boundness of the backstepping transformation and its inverse, we prove exponential stability of the closed-loop system using a properly weighted Lyapunov–Krasovskii functional. A backstepping-like design for linear systems with only state delay is the one considered in Jankovic (2009a). The major difference with the design in Jankovic (2009a) and the one considered here, is that in Jankovic (2009a) delays are not allowed in the virtual inputs (which is the difficult case considered here). The present procedure can be modified to incorporate state delays that are in other positions other than the virtual inputs. In the case of a system with only input delay (irrespective of the form of the system, i.e., either if the system is a chain of integrators with input delay, e.g. Mazenc, Mondie, and Niculescu (2003), or a system in feedforward form, e.g. Jankovic (2010), etc.) the resulting control law is the predictor-based/finite spectrum assignment controller from Artstein (1982), Fiagbedzi and Pearson (1986), Krstic and Smyshlyaev (2008a) and Manitius and Olbrot (1979) with the gain K being designed using the classical backstepping procedure for linear systems from Krstic et al. (1995). In the case of a system with simultaneous input and state delays a backstepping-like design comparable with the one considered here is the one in Jankovic (2010) for the special case of a chain of delayed integrators and input delay. In this special case the resulting control law from the present work turns out to be the same with the one in Jankovic (2010). The present results can be also applied in the case where there are delays in other states too, and not just in the virtual inputs. Thus, the class of systems such that the present method can be applied is not limited. Considering the problem where the delays or the coefficients aij are unknown, is a completely different and very challenging problem. Following the infinite dimensional backstepping technique, this problem has been solved for the case where there is only unknown input delay (Bresch-Pietri & Krstic, 2009a) and extended to the case of unknown input delay and plant parameters in Bresch-Pietri and Krstic (2009b). In BekiarisLiberis and Krstic (2010) a problem with unknown input and state delays is solved for a class of linear feedforward systems. Application of the design methods from Bekiaris-Liberis and Krstic (2010), Bresch-Pietri and Krstic (2009a,b) in the present case seems promising and can be pursued as a forthcoming research topic.
�
x−D1
v11 (x − y − D1 )p2 (y, t )dy � � n � + v11 (x)X1 (t ), x ∈ 0, Dk −D 1
2 � �
x−Di
v2i (x − y − Di )pi+1 (y, t )dy � � 2 n � � + v2i (x)Xi (t ), x ∈ 0, Dk i=1
−Di
i=1
.. . pn (x, t ) =
(135)
k=1
n−1 � � i=1
x −D i
−Di
n �
+
i=1
(136)
k=2
vni (x − y − Di )pi+1 (y, t )dy �
vni (x)Xi (t ) +
x ∈ [0, Dn ] ,
x 0
vnn (x − y)u(y, t )dy, (137)
where
eA 0 x
v11 (x) v21 (x) = .. .
0
v22 (x) .. .
vn1 (x)
a11
a21 A0 = .. . an1
0 0
.. . ...
vn2 (x)
0 a22
0 0
.. .
.. . ...
an2
... ... .. . ...
0 0
... ... .. .
.. . vnn (x)
...
0 0
(138)
.. . .
(139)
ann
By applying Young’s and Cauchy–Schwarz’s inequalities to Eqs. (135)–(137) we get p21 (x, t ) ≤ A1 p22
(x, t ) ≤ A2
�
�
X12
X22
�
.. . p2n
X12 (t ) +
�
(x, t ) ≤ An
�
+
(t ) +
n � i=1 x
0
Xi2
x−D1
−D 1
(t ) +
(t ) +
2
2 � � i=1
n −1 � � i =1
�
�
p22 (y, t )dy
x −D i
−Di
x−Di
−D i
(140) p2i+1
(y, t )dy
u (y, t )dy ,
2 Ai = 2i max � sup � vi1 (x), . . . , x∈ 0 ,
n �
k=i
Dk
(141)
p2i+1 (y, t )dy (142)
where, in each of the above bounds, x ∈ [0, Also
�
�
�n
k=i
Dk ], respectively.
sup � vii2 (x),
x∈ 0,
n �
k=i
Dk
N. Bekiaris-Liberis, M. Krstic / Automatica 46 (2010) 1902–1910
�
sup �
x∈ 0 ,
n �
k=i
Dk
sup � �
x∈ 0 ,
n �
k=i
Dk
�
x−D1
−D 1
�
x−Di
−D i
vi1 (x − y − D1 )2 dy, . . . ,
p23 (x, t ) ≤ A3
vii (x − y − Di )2 dy .
(143)
+ A3 A2 e
If we take now into account that pi (x − Di−1 , t ) = Xi (t + x − Di−1 ) = ξi (x, t ) we can rewrite (140)–(142) as p21
(x, t ) ≤ A1
�
X12
p22 (x, t ) ≤ A2
2
�
k=1 2
i=1
.. . (x, t ) ≤ An +
(t ) + �ξ2 (t )� +
��
+
p2n
�
2
�
n �
k =1 n−1
�� i=1
2
�
Xi2 (t ) + x −D i 0
i=1
0
x−D1
p22
0
�
(y, t )dy
(144)
�ξi+1 (t )�2 �
p2i+1 (y, t )dy
n−1 �
Xi2 (t ) + x −D i
�
i=1
p2i+1
(145)
�ξi+1 (t )�2
(y, t )dy +
�
x 0
�
2
u (y, t )dy ,
(146)
�n
where in each of the above relations, x ∈ [0, k=i Dk ], respectively. From the above equations, recursively, we can take the upper bound of the lemma. To see this, we start from relation (144) and observe that the boundness of p21 (x, t ) depends only on the boundness of X1 (t ) and ξ2 (x, t ) (thats is, p21 (x, t ) remains �n �n bounded for all x ∈ [0, k=1 Dk ]), if for all x ∈ [0, k=2 Dk ], p22 (x, t ) is upper bounded. We proceed now by proving that the boundness of p22 (x, t ) depends only on the boundness of X1 (t ), X2 (t ), ξ2 (x, t ) and ξ3 (x, t ) (thats is, p22 (x, t ) remains bounded �n �n for all x ∈ [0, k=2 Dk ]), if for all x ∈ [0, k=3 Dk ], p23 (x, t ) is upper bounded. From relation (145) (and by noting
� x −D
�x
that 0 1 p22 (y, t )dy ≤ 0 p22 (y, t )dy for all x for which this �n equation holds, i.e., ∀x ∈ [0, k=2 Dk ]) by using the comparison A2 x | A2 | principle �n and by exploiting the fact that e ≤ e [0, k=2 Dk ], we get that
�
x
0
p22
(y, t )dy ≤ A2 e 2
+
� i=1
|A2 |
2
�ξi+1 (t )�
n �
k=2
Dk
�� � x 0
�
n �
Dk
k=2
y−D2 0
p23
�
2 � k=1
�n
k=2 Dk
, ∀x ∈
Xi2 (t )
�
(r , t )drdy .
(147)
Plugging the above bound in to relation (145) we get a bound of p22 (x, t ) that depends on p23 (x, t ). Moreover, using the relation p23
(x, t ) ≤ A3
�
3 � k =1
��
Xi2 (t ) +
3
+
i=1
x −D i 0
3 � i=1
p2i+1
�ξi+1 (t )�2 �
(y, t )dy ,
and the previous bound, we get
+ A3 A2 e
(148)
+ A3
�
�
k=1
| A2 |
|A2 |
n �
n �
k=2
k=2
x−D3 0
3 �
Dk
p24
1909
Xi2 (t ) + Dk
n �
Dk
k=2
� x� 0
y 0
(x, t ),
2 � i =1
�
p23
x∈
�ξi+1 (t )�2
2 � k=1
Xi2
(t ) +
� 2 � i =1
(r , t )drdy + A3 �
0,
n � k=3
�
�
� �ξi+1 (t )� 2
x
0
p23 (x, t )dy
Dk .
(149)
Note that the delayed terms in the integral for p23 (x, t ) can be �n removed since now x ∈ [0, k=3 Dk ] which is the domain of definition for p3 (x, t ) (and of course this integral is larger than the delayed one). By changing the order of integration in the double integral of the previous relation, we can rewrite
� x� 0
y 0
p23 (r , t )drdy =
�x
�
x 0
(x − y)p23 (y, t )dy. �n
(150)
�x
2 By observing that 0 (x − y)p23 (y, t )dy ≤ k=3 Dk 0 p3 (y, t )dy, �n ∀ k=3 Dk , and applying again the comparison principle for � xx ∈2 p ( y , t )dy we can bound p23 (x, t ) from p24 (x, t ) and consequently 0 3 2 also p2 (x, t ). Repeating this process until p2n (x, t ) (the boundness of which depends only on the boundness of �u(x, t )�2 ), we derive the bound of the lemma. �
References Artstein, Z. (1982). Linear systems with delayed controls: a reduction. IEEE Transactions on Automatic Control, 27, 869–879. Bekiaris-Liberis, N., & Krstic, M. (2010). Delay-adaptive feedback for linear feedforward systems. Systems and Control Letters, 59, 277–283. Bresch-Pietri, D., & Krstic, Miroslav (2009a). Delay-adaptive full-state predictor feedback for systems with unknown long actuator delay. American Control Conference,. Bresch-Pietri, D., & Krstic, Miroslav (2009b). Adaptive trajectory tracking despite unknown input delay and plant parameters. Automatica, 45, 2074–2081. Evesque, S., Annaswamy, A. M., Niculescu, S., & Dowling, A. P. (2003). Adaptive control of a class of time-delay systems. ASME Transactions on Dynamics, Systems, Measurement, and Control, 125, 186–193. Fiagbedzi, Y. A., & Pearson, A. E. (1986). Feedback stabilization of linear autonomous time lag systems. IEEE Transactions on Automatic Control, 31, 847–855. Hale, J. K., & Verduyn Lunel, S. M. (1993). Introduction to functional differential equations. New York: Springer-Verlag. Jankovic, M. (2001). Control Lyapunov–Razumikhin functions and robust stabilization of time delay systems. IEEE Transactions on Automatic Control, 46, 1048–1060. Jankovic, M. (2003). Control of nonlinear systems with time delay. IEEE Conference on Decision and Control,. Jankovic, M. (2009a). Forwarding, backstepping, and finite spectrum assignment for time delay systems. Automatica, 45(1), 2–9. Jankovic, M. (2009b). Cross-term forwarding for systems with time delay. IEEE Transactions on Automatic Control, 54(3), 498–511. Jankovic, M. (2010). Recursive predictor design for state and output feedback controllers for linear time delay systems. Automatica, 46(3), 510–517. Karafyllis, I. (2006). Finite-time global stabilization by means of time-varying distributed delay feedback. SIAM Journal on Control and Optimization, 45(1), 320–342. Karafyllis, I., & Jiang, Z. P. (2008). Necessary and sufficient Lyapunov-like conditions for robust nonlinear stabilization. ESAIM Control, Optimization and Calculus of Variations, in press, (doi:10.1051/cocv/2009029), available at: http://www.esaim-cocv.org/.. Krstic, M. (2008a). On compensating long actuator delays in nonlinear control. IEEE Transactions on Automatic Control, 53, 1684–1688. Krstic, M. (2010). Input delay compensation for forward complete and feedforward nonlinear systems. IEEE Transactions on Automatic Control, 55, 287–303. Krstic, M., Kanellakopoulos, I., & Kokotovic, P. V. (1995). Nonlinear and adaptive control design. Wiley. Krstic, M., & Smyshlyaev, A. (2008a). Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays. Systems and Control Letters, 57, 750–758. Kwon, W. H., & Pearson, A. E. (1980). Feedback stabilization of linear systems with delayed control. IEEE Transactions on Automatic Control, 25, 266–269.
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Nikolaos Bekiaris-Liberis received his B.S. degree in Electrical and Computer Engineering from the National Technical University of Athens in 2007. He is now working toward the Ph.D. degree in the Department of Mechanical and Aerospace Engineering at University of California, San Diego. His research interests include control of delay systems, control of distributed parameter systems and nonlinear control.
Miroslav Krstic is the Daniel L. Alspach Professor and the founding Director of the Cymer Center for Control Systems and Dynamics (CCSD) at UC San Diego. He received his Ph.D. in 1994 from UC Santa Barbara and was Assistant Professor at University of Maryland until 1997. He is a coauthor of eight books: Nonlinear and Adaptive Control Design (Wiley, 1995), Stabilization of Nonlinear Uncertain Systems (Springer, 1998), Flow Control by Feedback (Springer, 2002), Real-time Optimization by Extremum Seeking Control (Wiley, 2003), Control of Turbulent and Magnetohydrodynamic Channel Flows (Birkhauser, 2007), Boundary Control of PDEs: A Course on Backstepping Designs (SIAM, 2008), Delay Compensation for Nonlinear, Adaptive, and PDE Systems (Birkhauser, 2009), and Adaptive Control of Parabolic PDEs (Princeton, 2010). Krstic is a Fellow of IEEE and IFAC and has received the Axelby and Schuck paper prizes, NSF Career, ONR Young Investigator, and PECASE award. He has held the appointment of Springer Distinguished Visiting Professor of Mechanical Engineering at UC Berkeley.
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Automatica journal homepage: www.elsevier.com/locate/automatica
Brief paper
Stabilization of linear strict-feedback systems with delayed integrators✩ Nikolaos Bekiaris-Liberis ∗ , Miroslav Krstic Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, USA
article
info
Article history: Received 10 December 2009 Received in revised form 29 May 2010 Accepted 8 July 2010 Available online 21 August 2010 Keywords: Delay systems Predictor Strict-feedback systems
abstract The problem of compensation of input delays for unstable linear systems was solved in the late 1970s. Systems with simultaneous input and state delay have remained a challenge, although exponential stabilization has been solved for systems that are not exponentially unstable, such as chains of delayed integrators and systems in the ‘feedforward’ form. We consider a general system in strict-feedback form with delayed integrators, which is an example of a particularly challenging class of exponentially unstable systems with simultaneous input and state delays, and design a predictor feedback controller for this class of systems. Exponential stability is proven with the aid of a Lyapunov–Krasovskii functional that we construct using the PDE backstepping approach. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Stabilization of linear systems with input delays continues to be an active area of research. Various control schemes for systems with input delay have been developed, with the starting point for many of them being the Smith predictor (Smith, 1959). The most important extensions of the Smith predictor have been designs based on the finite spectrum assignment framework (Artstein, 1982; Fiagbedzi & Pearson, 1986; Jankovic, 2009a, 2010; Kwon & Pearson, 1980; Manitius & Olbrot, 1979; Mondie & Michiels, 2003; Olbrot, 1978; Richard, 2003; Zhong, 2006). In addition to these designs, adaptive versions of predictorbased linear controllers are proposed in Evesque, Annaswamy, Niculescu, and Dowling (2003a), Liu and Krstic (2001), Niculescu and Annaswamy (2003b), Zhou, Wang, and Wen (2008), whereas adaptive controllers for unknown delay have been developed recently in Bekiaris-Liberis and Krstic (2010), Bresch-Pietri and Krstic (2009a,b) and Yildiray, Annaswamy, Kolmanovsky, and Yanakiev (2010). Moreover, various control designs for nonlinear systems exist (Jankovic, 2001, 2003, 2009b; Karafyllis, 2006; Krstic, 2008a, 2010; Mazenc & Bliman, 2004; Mazenc, Mondie, & Francisco, 2004). Despite the fact that numerous papers deal with linear systems with input delay, the problem of controller design for systems with simultaneous input and state delay has been tackled in only a
✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Faryar Jabbari under the direction of Editor Roberto Tempo. ∗ Corresponding author. Tel.: +1 858 822 1936; fax: +1 858 822 3107. E-mail addresses: [email protected], [email protected] (N. Bekiaris-Liberis), [email protected] (M. Krstic).
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few Refs. Fiagbedzi and Pearson (1986), Jankovic (2009a), Jankovic (2010), Loiseau (2000), Manitius and Olbrot (1979) and Watanabe, Nobuyama, Kitamori, and Ito (1992). In this work we consider a specially chosen possibly open-loop unstable linear system, with the special form of a strict-feedback system with delayed integrators. Specifically, we consider the following n-dimensional linear system X˙¯ 1 (t ) = a¯ 11 X¯ 1 (t ) + b1 X¯ 2 (t − D1 )
X˙¯ 2 (t ) = a¯ 21 X¯ 1 (t ) + a¯ 22 X¯ 2 (t ) + b2 X¯ 3 (t − D2 )
.. . X˙¯ n (t ) = a¯ n1 X¯ 1 (t ) + · · · + a¯ nn X¯ n (t ) + bn U¯ (t − Dn ), where X¯ i (t ), a¯ ij , U¯ (t ) ∈ R, bi �= 0, and Di ∈ R+ .
(1) (2)
(3)
For this system we develop a predictor-based controller (Section 2). To achieve this we use tools from the boundary control of first order linear hyperbolic PDEs (Krstic & Smyshlyaev, 2008a), together with the classical backstepping procedure (Krstic, Kanellakopoulos, & Kokotovic, 1995). Specifically, an infinite dimensional backstepping transformation is used, together with a control law, to convert the system to an exponentially stable (in a certain sense) target system. Using the boundness of the backstepping transformation and its inverse, we then prove exponential stability of the closed-loop system using a suitably weighted Lyapunov–Krasovskii functional (Section 3). The effectiveness of the proposed controller is illustrated by a simulation example of a second order unstable system (Section 4). 2. Controller design We start by redefining the states of system (1)–(3) such that the coefficients in front of the delayed terms are unity. That is, we
N. Bekiaris-Liberis, M. Krstic / Automatica 46 (2010) 1902–1910
define X1 (t ) = X¯ 1 (t )
(4)
X3 (t ) = b1 b2 X¯ 3 (t )
(6)
Xn (t ) = b1 b2 . . . bn−1 X¯ n (t ).
(7)
X2 (t ) = b1 X¯ 2 (t )
P2 (t ) = X2 (t ) +
.. .
U (t ) = b1 b2 . . . bn U¯ (t ) aij =
�
a¯ ij , bj . . . bi−1 a¯ ij ,
�
+ an2 P2 θ −
(8)
�
if i = j . if i > j
(9)
X˙ 1 (t ) = a11 X1 (t ) + X2 (t − D1 )
(10)
X˙ 2 (t ) = a21 X1 (t ) + a22 X2 (t ) + X3 (t − D2 )
(11)
.. . X˙ n (t ) = an1 X1 (t ) + · · · + ann Xn (t ) + U (t − Dn ).
(12)
We state here our controller and in Section 3 we analyze the stability properties of the closed-loop system. The controller for the system (10)–(12) is given by
= −an1 P1 t −
Dk
k=1
�
αi (x, t ) = −ai1 P1 t − � − aii Pi t − � �
k=1 n
�
k=i n
− ci Pi t − +
Dk + x
Dk + x
� k=i
�
Dk + x
∂αi−1 (Di−1 + x, t ) , ∂x �
α1 (x, t ) = −(a11 + c1 )P1 t + x − � � n � x ∈ 0, Dk ,
− αi−1 (Di−1 + x, t ) � � n � x ∈ 0, Dk k=1
Dk
�
�
t
t−
n �
k=1
Dk
�
P2 (θ ) = X2 (0) +
Dk
k=1
�
� · · · + ann Pn (θ ) + U (θ ) dθ,
(18)
(a11 P1 (σ ) + P2 (σ ))dσ
(19)
θ −
�
n �
k=1
Dk
θ
−
n �
k=2
Dk
(a21 P1 (σ − D1 )
+ a22 P2 (σ ) + P3 (σ ))dσ
.. .
Pn (θ ) = Xn (0) +
�
�
θ −D n
n −1 �
�
Dk
k=2
an1 P1
�
(D
�
σ−
(20)
n −1 �
Dk
k=1
�
�
· · · + ann Pn (σ ) + U (σ ) dσ , + x ,t )
�n
k=i
(21)
Dk , 0]. Note here
∂α
(x� ,t )
3. Stability analysis
(14)
k=i
n 1�
Ω (t ) =
2 i =1
+
,
(a11 P1 (θ ) + P2 (θ ))dθ
(22)
where
(15)
and the ci , i = 1, 2 . . . n are arbitrary positive constants. In the �n above control scheme we use the Pi (t ) signals, the k=i Dk seconds ahead predictors of the Xi (t ) state � (this fact becomes clear later on). n That is, it holds that Pi (t ) = Xi (t + k=i Dk ). These signals are given by
�
Dk
�
θ−
n−1 �
Ω (t ) ≤ κ Ω (0)e−λt ,
k=1
P1 (t ) = X1 (t ) +
an1 P1
�
Theorem 1. System (10)–(12) with the controller (13) is exponentially stable in the sense that there exist constants κ and λ such that
�
�
t −D n
�
(17)
We first state a theorem describing our main stability result and then we prove it using a series of technical lemmas.
− ···
n
t
∂α
(13)
�
(a21 P1 (θ − D1 )
that the notation i−1 ∂ix−1 corresponds to i−∂1x� |x� =x+Di−1 which includes the time derivatives of the signals P1 (t ), . . . , Pi−1 (t ). These derivatives are obtained from (10)–(12) and (16)–(18).
where n �
k=2
Dk
where θ is defined in each Pi (θ ) as θ ∈ [−
− · · · − ann Pn (t )
∂αn−1 (Dn−1 + Dn , t ) , ∂x �
n �
k=2
+ an2 P2 σ −
− cn (Pn (t ) − αn−1 (Dn−1 + Dn , t )) +
t−
with initial conditions
U (t ) = u(Dn , t )
n−1 �
t
n−1 �
P1 (θ ) = X1 (0) +
In the new variables, system (1)–(3) is transformed to
= αn (Dn , t ) �
�
Pn (t ) = Xn (t ) +
Moreover, for notational consistency we define
1903
+ a22 P2 (θ ) + P3 (θ ))dθ
(5)
.. .
�
(16)
and
� �
t t −Di−1 t
t −D n
1 2
�
Xi2 (t ) + t t −D n
U (θ )dθ =
2 i=2
t t −Di−1
Xi2 (θ )dθ
U 2 (θ )dθ,
(23)
�
(24)
Xi2 (θ )dθ = 2
n � 1�
�
Di−1 0
Dn
0
ξi2 (x, t )dx = �ξi (t )�2
u2 (x, t )dx = �u(t )�2
ξi (x, t ) = Xi (t + x − Di−1 ), u(x, t ) = U (t + x − Dn ),
x ∈ [0, Di−1 ]
x ∈ [0, Dn ].
We first give and prove the following lemmas.
(25) (26) (27)
N. Bekiaris-Liberis, M. Krstic / Automatica 46 (2010) 1902–1910
1904
Lemma 1. The signals Pi (t ) defined in (16)–(18) are, respectively the �n D seconds ahead predictors of the Xi (t ) states. Moreover an k k=i equivalent representation for (16)–(18) is given by p1
p2
� �
n �
Dk , t
k=1
n �
Dk , t
k=2
� �
= X1 (t ) +
= X2 (t ) +
k=1
0
Dk
(a11 p1 (y, t )
�
n �
k=2
0
Dk
(28)
(a21 p1 (y, t )
+ a22 p2 (y, t ) + p3 (y − D2 , t ))dy
pn (Dn , t ) = Xn (t ) +
pi (x, t ) = Pi
n �
+ p2 (y − D1 , t ))dy
.. .
where
�
�
Dn 0
(an1 p1 (y, t ) + · · ·
+ ann pn (y, t ) + u(y, t ))dy,
�
t +x−
n �
Dk
k =i
�
,
x∈
�
0,
(29)
(30)
n �
�
Dk .
k=i
By taking into account (44) we have that pi (x, t ) = Xi (t + x),
� n �
Dk , t
k=i
�
= Pi (t ), �
p1 (x, t ) = X1 (t ) +
X˙ 1 (t ) = a11 X1 (t ) + ξ2 (0, t )
(32)
p2 (x, t ) = X2 (t ) +
ξ2 (D1 , t ) = X2 (t ) X˙ 2 (t ) = a21 X1 (t ) + a22 X2 (t ) + ξ3 (0, t )
(34)
(33) (35)
ξ3t (x, t ) = ξ3x (x, t )
(36)
ξ3 (D2 , t ) = X3 (t ) .. . X˙ n (t ) = an1 X1 (t ) + · · · + ann Xn (t ) + u(0, t ) ut (x, t ) = ux (x, t ) u(Dn , t ) = U (t ).
(37)
p1x (x, t ) = a11 p1 (x, t ) + p2 (x − D1 , t )
(41)
(38) (39) (40)
Consider the following ODEs in x (to become clear that these are ODEs in x, consider the time t to act as a parameter rather as a running variable), p2x (x, t ) = a21 p1 (x, t ) + a22 p2 (x, t ) + p3 (x − D2 , t )
(42)
.. .
�n
(43)
where, for each pi (x, t ), x varies in [0, k=i Dk ]. The initial conditions for the above system of ODEs are given by
and
Dk , ∀i.
(47)
∀i,
(48)
.. .
x 0
�
(a11 p1 (y, t ) + p2 (y − D1 , t ))dy x
0
(a21 p1 (y, t ) + a22 p2 (y, t )
+ p3 (y − D2 , t ))dy
pn (x, t ) = Xn (t ) +
�
0
x
(49)
(50)
(an1 p1 (y, t ) + · · ·
+ ann pn (y, t ) + u(y, t ))dy. (51) �n By setting in each pi (x, t ), x = k=i Dk and using (31) we get (28)–(30).
�
It is important here to observe that the total delay from the input �n to each state Xi (t ) is D . This explains the fact that our k k=i predictor �n intervals are different for each state and specifically must be k=i Dk seconds for each state Xi (t ). Our controller design is based on a recursive procedure that transforms system (10)–(12) to a target system which is exponentially stable with the controller (13). Then, using the invertibility of this transformation, we prove exponential stability of the original system. We now state this transformation, along with its inverse. Lemma 2. The state transformation defined by
pnx (x, t ) = an1 p1 (x, t ) + · · · + ann pn (x, t ) + u(x, t ),
pi (0, t ) = Xi (t ),
k=i
�
we get (31). By integrating from 0 to x (41)–(43) we get
Proof. Consider the equivalent representation of system (10)–(12) using transport PDEs for the delayed states and control
ξ2t (x, t ) = ξ2x (x, t )
0,
n �
To see that (46) holds it is sufficient to prove that (47) is the unique solution of the ODEs in x given by (41)–(43) with the initial conditions (44)–(45). Since then, the pi (x, t ) are functions of only one variable, namely x + t, and consequently (46) holds. Thus, it remains to prove that (47) is the unique solution of the initial value problem (41)–(45). Toward this end, by taking into account (27) we point out that (47) satisfy the initial value problem (41)–(45). Then, assuming that Xi (t + θi ), i = 2, . . . , n are continuous for all θi ∈ [−Di−1 , 0], using Theorem 2.1 from Hale and Verduyn Lunel (1993) we can conclude that (47) is the unique solution of the ODEs in x given by (41)–(43) with the initial conditions (44)–(45). Thus, (46) holds. From relation (47) becomes clear that the pi (x, t ) are the x seconds ahead predictors of the states. By defining pi
(31)
x∈
�
∀i,
pi (θi , t ) = Xi (t + θi ),
θi ∈ [−Di−1 , 0], i = 2, . . . , n.
(44)
(45)
Since system (41)–(43) is driven by the input u(x, t ), which satisfies a transport PDE, the same holds for all the pi (x, t ) (see for example Krstic (2008a, 2010)). Thus,
∂ pi (x, t ) ∂ pi (x, t ) = , ∂t ∂x
x∈
�
0,
n � k=i
�
Dk , ∀i.
(46)
Z1 (t ) = X1 (t )
(52)
Zi+1 (t ) = Xi+1 (t ) − αi (Di , t ),
i = 1, 2, . . . , n − 1,
(53)
along with the transformation of the actuator state
w(x, t ) = u(x, t ) − αn (x, t ),
x ∈ [0, Dn ],
(54)
where the αi (x, t ) are defined as in (14)–(15), transforms the system (10)–(12) to the target system with the control law given by (13). The target system is given by Z˙1 (t ) = −c1 Z1 (t ) + Z2 (t − D1 ) Z˙2 (t ) = −c2 Z2 (t ) + Z3 (t − D2 )
.. . Z˙n (t ) = −cn Zn (t ) + W (t − Dn ),
(55) (56)
(57)
N. Bekiaris-Liberis, M. Krstic / Automatica 46 (2010) 1902–1910
where
1905
If we now define
W (θ) = 0,
θ ≥ 0.
(58)
Proof. Before we start our recursive procedure we rewrite the target system using transport PDEs as Z˙1 (t ) = −c1 Z1 (t ) + ζ2 (0, t )
(59)
ζ2 (D1 , t ) = Z2 (t ) Z˙2 (t ) = −c2 Z2 (t ) + ζ3 (0, t )
(61)
ζ3 (D2 , t ) = Z3 (t ) .. . Z˙n (t ) = −cn Zn (t ) + w(0, t ) wt (x, t ) = wx (x, t ) w(Dn , t ) = 0.
(64)
ζ2t (x, t ) = ζ2x (x, t )
(60) (62)
ζ3t (x, t ) = ζ3x (x, t )
(63)
(65) (66)
Z˙3 (t ) = a31 X1 (t ) + a32 X2 (t ) + a33 X3 (t ) + ζ4 (0, t )
+ α3 (0, t ) −
x ∈ [0, Di−1 ].
(68)
ζ2 (x, t ) = ξ2 (x, t ) − α1 (x, t ),
(69)
X˙ 1 (t ) = a11 X1 (t ) + ζ2 (0, t ) + α1 (0, t ).
(70)
By choosing α1 (x, t ) = −(a11 + c1 )p1 (x, t ) (note the equivalent representation of α1 (x, t ) using (31)) and by using (44) we get Z˙1 (t ) = −c1 Z1 (t ) + ζ2 (0, t ).
(71)
Z2 (t ) = X2 (t ) − α1 (D1 , t ).
(72)
ζ3 (x, t ) = ξ3 (x, t ) − α2 (x, t ),
(73)
From (69) with x = D1 and (61) it follows that
Z˙i−1 (t ) = −ci−1 Zi−1 (t ) + ζi (0, t ),
(80)
ζi+1 (x, t ) = ξi+1 (x, t ) − αi (x, t ).
(81)
and define ζi+1 (x, t ) as
Then from (53) with x = Di−1 we have that
Z˙i (t ) = ai1 X1 (t ) + · · · + aii Xi (t ) + ζi+1 (0, t )
+ αi (0, t ) −
∂αi−1 (Di−1 , t ) . ∂x
(82)
αi (x, t ) = −ai1 p1 (x, t ) − · · · − aii pi (x, t ) ∂αi−1 (Di−1 + x, t ) − ci (pi (x, t ) − αi−1 (Di−1 + x, t )) + , ∂x
(83)
we get (84)
Step n. In the last step we choose the controller U (t ). Since Z˙n (t ) = an1 X1 (t ) + · · · + ann Xn (t ) + u(0, t )
∂αn−1 (Dn−1 , t ) . ∂x Then using (14) for i = n we have that −
(85)
Z˙n (t ) = −cn Zn (t ) + w(0, t ),
(86)
wt (x, t ) = wx (x, t ) w(Dn , t ) = 0.
(87)
w(x, 0) = w0 (x),
(89)
and using (13)
By setting now
and using (69), (33) and (35) we have
∂α1 (D1 , t ) + ζ3 (0, t ) + α2 (0, t ) − , ∂x
(74)
∂α (x,t )
where 1 ∂ x1 corresponds to 1∂ x |x=D1 and we use the fact that α1t (x, t ) = α1x (x, t ) (which is a consequence of relation (15)).
and by defining a new variable W (·) as
w0 (x) = W (x − D), we get that
Step 2. By choosing
α2 (x, t ) = −a21 p1 (x, t ) − a22 p2 (x, t ) − c2 (p2 (x, t ) ∂α1 (x + D1 , t ) − α1 (D1 + x, t )) + ∂x = −a21 p1 (x, t ) − a22 p2 (x, t ) − c2 (p2 (x, t ) + (a11 + c1 )p1 (D1 + x, t )) − (a11 + c1 )(a11 p1 (x + D1 , t ) + p2 (x, t )),
w(x, t ) =
�
x ∈ [0, Dn ],
W (t + x − D), 0,
� −D ≤ t + x − D ≤ 0 . t +x−D≥0
(90)
(91)
Defining θ = t + x − D one gets (58). Note here that based on (54), w0 (x) is given by
w0 (x) = u(x, 0) − αn (x, 0), (75)
we get from (74) (with the help of (44)) that
By setting now x = D2 in (73) and using (64) we get
(88)
Assuming an initial condition for (87) as
Z˙2 (t ) = ζ2x (x, t )|x=D1 = a21 X1 (t ) + a22 X2 (t )
Z3 (t ) = X3 (t ) − α2 (D2 , t ) .
(79)
Step i. Assume now that
Z˙i (t ) = −ci Zi (t ) + ζi+1 (0, t ).
then using (32) we get
Z˙2 (t ) = −c2 Z2 (t ) + ζ3 (0, t ).
∂α2 (D2 , t ) . ∂x
Hence, with
Step 1. Following the backstepping procedure we first stabilize X1 (t ) with the virtual input α1 (D1 , t ). We define
∂α (D ,t )
(78)
then with the help of (36) we get
(67)
Note that
ζi (x, t ) = Zi (t + x − Di−1 ),
ζ4 (x, t ) = ξ4 (x, t ) − α3 (x, t ),
(76)
(77)
x ∈ [0, Dn ]. �
(92)
We now define the inverse transformation of (52)–(54). Lemma 3. The inverse transformation of (52)–(54) is defined as X1 ( t ) = Z 1 ( t )
Xi+1 (t ) = Zi+1 (t ) + βi (Di , t ), u(x, t ) = w(x, t ) + βn (x, t ),
(93) i = 1, 2, . . . , n − 1.
x ∈ [0, Dn ],
(94) (95)
N. Bekiaris-Liberis, M. Krstic / Automatica 46 (2010) 1902–1910
1906
where the βi (x, t ) are now given by
β1 (x, t ) = −(a11 + c1 )�1 (x, t ),
x∈
�
0,
n �
Dk
k=1
�
βi (x, t ) = −ai1 �1 (x, t ) − ai2 (�2 (x, t ) + β1 (D1 + x, t )) − · · · − aii (�i (x, t ) + βi−1 (Di−1 + x, t )) ∂βi−1 (Di−1 + x, t ) − ci �i (x, t ) + , ∂x � � n � x ∈ 0, Dk , ∀i = 2, . . . , n,
(96)
derivative of the above function along the solutions of the Z (t ) system and by exploiting the fact that ζi (x, t ) and w(x, t ) satisfy transport PDEs (based on (60), (63) and (66)), it follows that V˙ (t ) = −
(97)
−
and the �i (x, t ) (the predictors of the transformed states) are given by the following relations
�1 (x, t ) = Z1 (t ) + �2 (x, t ) = Z2 (t ) + .. . � n ( x, t ) = Z n ( t ) +
� �
x 0 x 0
x 0
2
t −D n
(−cn �n (y, t ) + w(y, t ))dy, �n
k=i
(100)
Dk ].
Ξ (0)e
m − M1 t 1
(101)
,
� �
t t −Di−1 t
t −D n
Zi2
n � 1�
(θ )dθ =
W 2 (θ )dθ =
2 i =2
t t −Di−1
∀i,
(103)
�
�
2
(104)
w 2 (x, t )dx = �w(t )�2 .
(105)
0
0
2 i
ζ (x, t )dx = �ζi (t )�
n 1�
2 i=1
+
ki Zi2 (t ) +
λn+1 2
�
Dn 0
�
λ n +1 2
Dn
0
Di−1 0
ζi2 (x, t )dx
w 2 (x, t )dx,
(107)
λ2 =
(108) ci2−1
1 2c1
n 1�
2 i =2
λi
�
0
Di−1
(109)
,
V˙ (t ) ≤ −
−
n 1�
ci ki Zi2 (t ) −
2 i=1
�
λn+1 2
Defining
Dn 0
�
ci ki 2
,
it follows that V˙ (t ) ≤ −
m1 M1
m2 = min
Ξ (t ) ≤
�
n 1�
2 i=2
λi
�
Di−1 0
ζi2 (x, t )dx
w2 (x, t )dx.
(110)
i = 1, 2, . . . , n.
�
λi+1 , 2 (1 + Di )
(111)
i = 1, 2, . . . , n,
(112)
V (t ).
(113)
ki λi+1 2
,
2
�
,
i = 1, 2, . . . , n,
(114)
m1 m V (0) − M M1 (1 + Dmax ) t − 1t e 1 ≤ Ξ (0)e M1 . � m2 m2
(115)
We give now the following lemma which we prove in the Appendix.
(1 + x)ζi2 (x, t )dx
(1 + x)w 2 (x, t )dx.
i = 3, . . . , n + 1
,
and after some manipulations that incorporate completion of squares we get
then
Proof. We consider the following Lyapunov-like function V (t ) =
2 i=2
�
λi
i = 2, . . . , n
λi−1 (1 + Di−2 )
If we now define
Di−1
Dn
λi = 4
(1 + Di−1 ),
ci
m1 = min
Zi2 (θ )dθ (102)
and
w 2 (0, t ) −
M1 = max {ki , λi+1 } ,
W 2 (θ )dθ
Dmax = max {Di } ,
λi
k1 = 2
Zi2 (t ) + t
2
n 1�
λi ζi2 (0, t ) −
ki = 2
where
+
λn+1
λi (1 + Di−1 )Zi2 (t )
2 i=2
(99)
M1 (1 + Dmax )
2 i=1 � 1
2 i =2
n 1�
(−c2 �2 (y, t ) + �3 (y − D2 , t ))dy
Lemma 4. The target system is exponentially stable in the sense that there exist constants M1 , m1 and m2 such that
n 1�
−
n 1�
ki Zi (t )ζi+1 (0, t )
where we used integration by parts in the above integrals. By choosing the weights as
We now prove the stability of the transformed system.
Ξ (t ) =
i=1
(98)
Proof. Applying similar arguments as in Lemma 2 we prove that the inverse transformation of (52)–(54) and (14)–(15) is given by (93)–(97). �
m2
n−1 �
(−c1 �1 (y, t ) + �2 (y − D1 , t ))dy
where in each �i (x, t ), x varies in [0,
Ξ (t ) ≤
i=1
ci ki Zi2 (t ) +
+ kn Zn (t )w(0, t ) +
k=i
�
n �
Lemma 5. There exist constants Gi such that (106)
Note that the above functional can be considered as a Control Lyapunov Functional in the sense of Karafyllis and Jiang (in press). This fact reinforces the strength of the present result: a Control Lyapunov Functional is actually constructed. By taking the time
2
pi (x, t ) ≤ Gi
+
�
|X (t )|2 +
�
Dn 0
2
n � � i =2
Di−1 0
�
u (y, t )dy ,
ξi2 (y, t )dy �
∀x ∈ 0,
n � k=i
�
Dk ,
(116)
N. Bekiaris-Liberis, M. Krstic / Automatica 46 (2010) 1902–1910
3
where
|X (t )|2 =
n � i =1
Xi2 (t ),
2
(117)
1
and the bound (116) is independent of x.
0
Lemma 6. There exists a constant M such that
Ξ (t ) ≤ M Ω (t ).
(118)
–1 –2
Proof. From (52)–(54) it follows that Zi2 (t ) ≤ 2(Xi2 (t ) + αi2−1 (Di , t )), 2 i
2 i
i = 2, . . . , n
2 i−1
(119)
ζ (x, t ) ≤ 2(ξ (x, t ) + α (x, t )), x ∈ [0, Di−1 ], i = 2, . . . , n 2
2
2 n
w (x, t ) ≤ 2(u (x, t ) + α (x, t )),
x ∈ [0, Dn ].
αi2 (x, t ) ≤ bi
i �
p2k (x, t ),
k=1
x∈
�
0,
–3
(120)
n �
Dk
k=i
�
(122)
for some constants bi . By employing the bound of Lemma 5, the lemma is proven. �
�
2
� (x, t ) ≤ Fi |Z (t )| + +
�
Dn 0
n � � i =2
Di−1
Ω (t ) ≤
0
�
w2 (y, t )dy ,
ζ (y, t )dy x∈
�
0,
n �
�
Dk .
k=i
Ω (t ) ≤
(t ) ≤ 2( (t ) + β 2 i
(Di , t )),
i = 2, . . . , n
2 i−1
ξ (x, t ) ≤ 2(ζ (x, t ) + β (x, t )), x ∈ [0, Di−1 ], i = 2, . . . , n 2
2
2 n
u (x, t ) ≤ 2(w (x, t ) + β (x, t )),
(125) (126)
x ∈ [0, Dn ].
(127)
By observing that βi (x, t ) are linearly dependent on �1 (x, t ), . . . , �i (x, t ) we conclude that there exist constants di such that 2 i
β (x, t ) ≤ di
i � k=1
2 k
� (x, t ),
x∈
�
0,
n �
�
Dk .
k=i
Using Lemma 7 the lemma is proven.
(128)
�
Proof of Theorem 1. Combining Lemmas 6 and 8 we have that M Ω (t ) ≤ Ξ (t ) ≤ M Ω (t ).
(130)
,
MM1 (1 + Dmax ) Mm2
Ω (0)e
m
− M1 t 1
.
(131)
MM1 (1 + Dmax ) Mm2
m1 M1
. �
(132) (133)
We illustrate here our controller with a second order example with parameters a11 = a21 = a22 = 0.2, D1 = 0.4, D2 = 0.8 and c1 = c2 = 2. The initial conditions for the controller are given by (19)–(21) and for the system are X1 (0) = X2 (0) = 1 and X2 (θ ) = 1, θ ∈ [−D1 , 0]. This system is unstable (to see this one can use Olgac and Sipahi (2002)). In the present case the controller will have the form U (t ) = u(D2 , t )
Proof. Using relations (93)–(95) we get
2 i
M
4. Simulations
(124)
2 i −1
Ξ (t )
(123)
Lemma 8. There exists a constant M such that
Zi2
6
Thus Theorem 1 is proven with
λ=
Proof. Immediately note that the relation for the �i (x, t ) is similar to the relation for pi (x, t ). Note here that in this case the derivation of the explicit bound is easier due to the special form of the �i (x, t ) in (98)–(100). �
M Ω (t ) ≤ Ξ (t ).
4
and by Lemma 4 we get
κ=
2 i
2
Hence,
Lemma 7. There exist constants Fi such that 2 i
0
Fig. 1. System’s response for the simulation example.
(121)
Moreover, from relations (14)–(15) one can see that the αi (x, t ) are linear functions of the predictors p1 (x, t ), . . . , pi (x, t ), hence it holds that
Xi2
1907
(129)
= α2 (D2 , t ) = −a21 p1 (D2 , t ) − a22 p2 (D2 , t ) − c2 (p2 (D2 , t ) + (a11 + c1 )p1 (D1 + D2 , t )) − (a11 + c1 )(a11 p1 (D2 + D1 , t ) + p2 (D2 , t )) = −a21 P1 (t − D1 ) − a22 P2 (t ) − c2 (P2 (t ) + (a11 + c1 )P1 (t )) − (a11 + c1 )(a11 P1 (t ) + P2 (t )), (134)
where P1 (t ) and P2 (t ) are calculated using the integral representation (16)–(18). Note also that these integrals are computed using the trapezoidal rule. Fig. 1 shows that the predictor controller exponentially stabilizes the system. The control signal reaches first X2 (t ), since the delay from the input to X2 (t ) is 0.4, which is smaller than the total delay from the input to X1 (t ). After 1.2 s, which is the total delay from the input to X1 (t ), the controller starts stabilizing X1 (t ). Then both X1 (t ) and X2 (t ) converge exponentially to zero (see Fig. 2).
N. Bekiaris-Liberis, M. Krstic / Automatica 46 (2010) 1902–1910
1908
Appendix Here we give the proof of Lemma 5. Proof of Lemma 5. By solving (41)–(43), and by taking into account that this ODE in x system is in strict-feedback form, we get p1 (x, t ) =
p2 (x, t ) = Fig. 2. Control effort for the simulation example.
5. Conclusions We present a backstepping design for an exponentially unstable system with simultaneous input and state delay. Our design is predictor-based since it uses the predicted values of the states on given intervals. Using the boundness of the backstepping transformation and its inverse, we prove exponential stability of the closed-loop system using a properly weighted Lyapunov–Krasovskii functional. A backstepping-like design for linear systems with only state delay is the one considered in Jankovic (2009a). The major difference with the design in Jankovic (2009a) and the one considered here, is that in Jankovic (2009a) delays are not allowed in the virtual inputs (which is the difficult case considered here). The present procedure can be modified to incorporate state delays that are in other positions other than the virtual inputs. In the case of a system with only input delay (irrespective of the form of the system, i.e., either if the system is a chain of integrators with input delay, e.g. Mazenc, Mondie, and Niculescu (2003), or a system in feedforward form, e.g. Jankovic (2010), etc.) the resulting control law is the predictor-based/finite spectrum assignment controller from Artstein (1982), Fiagbedzi and Pearson (1986), Krstic and Smyshlyaev (2008a) and Manitius and Olbrot (1979) with the gain K being designed using the classical backstepping procedure for linear systems from Krstic et al. (1995). In the case of a system with simultaneous input and state delays a backstepping-like design comparable with the one considered here is the one in Jankovic (2010) for the special case of a chain of delayed integrators and input delay. In this special case the resulting control law from the present work turns out to be the same with the one in Jankovic (2010). The present results can be also applied in the case where there are delays in other states too, and not just in the virtual inputs. Thus, the class of systems such that the present method can be applied is not limited. Considering the problem where the delays or the coefficients aij are unknown, is a completely different and very challenging problem. Following the infinite dimensional backstepping technique, this problem has been solved for the case where there is only unknown input delay (Bresch-Pietri & Krstic, 2009a) and extended to the case of unknown input delay and plant parameters in Bresch-Pietri and Krstic (2009b). In BekiarisLiberis and Krstic (2010) a problem with unknown input and state delays is solved for a class of linear feedforward systems. Application of the design methods from Bekiaris-Liberis and Krstic (2010), Bresch-Pietri and Krstic (2009a,b) in the present case seems promising and can be pursued as a forthcoming research topic.
�
x−D1
v11 (x − y − D1 )p2 (y, t )dy � � n � + v11 (x)X1 (t ), x ∈ 0, Dk −D 1
2 � �
x−Di
v2i (x − y − Di )pi+1 (y, t )dy � � 2 n � � + v2i (x)Xi (t ), x ∈ 0, Dk i=1
−Di
i=1
.. . pn (x, t ) =
(135)
k=1
n−1 � � i=1
x −D i
−Di
n �
+
i=1
(136)
k=2
vni (x − y − Di )pi+1 (y, t )dy �
vni (x)Xi (t ) +
x ∈ [0, Dn ] ,
x 0
vnn (x − y)u(y, t )dy, (137)
where
eA 0 x
v11 (x) v21 (x) = .. .
0
v22 (x) .. .
vn1 (x)
a11
a21 A0 = .. . an1
0 0
.. . ...
vn2 (x)
0 a22
0 0
.. .
.. . ...
an2
... ... .. . ...
0 0
... ... .. .
.. . vnn (x)
...
0 0
(138)
.. . .
(139)
ann
By applying Young’s and Cauchy–Schwarz’s inequalities to Eqs. (135)–(137) we get p21 (x, t ) ≤ A1 p22
(x, t ) ≤ A2
�
�
X12
X22
�
.. . p2n
X12 (t ) +
�
(x, t ) ≤ An
�
+
(t ) +
n � i=1 x
0
Xi2
x−D1
−D 1
(t ) +
(t ) +
2
2 � � i=1
n −1 � � i =1
�
�
p22 (y, t )dy
x −D i
−Di
x−Di
−D i
(140) p2i+1
(y, t )dy
u (y, t )dy ,
2 Ai = 2i max � sup � vi1 (x), . . . , x∈ 0 ,
n �
k=i
Dk
(141)
p2i+1 (y, t )dy (142)
where, in each of the above bounds, x ∈ [0, Also
�
�
�n
k=i
Dk ], respectively.
sup � vii2 (x),
x∈ 0,
n �
k=i
Dk
N. Bekiaris-Liberis, M. Krstic / Automatica 46 (2010) 1902–1910
�
sup �
x∈ 0 ,
n �
k=i
Dk
sup � �
x∈ 0 ,
n �
k=i
Dk
�
x−D1
−D 1
�
x−Di
−D i
vi1 (x − y − D1 )2 dy, . . . ,
p23 (x, t ) ≤ A3
vii (x − y − Di )2 dy .
(143)
+ A3 A2 e
If we take now into account that pi (x − Di−1 , t ) = Xi (t + x − Di−1 ) = ξi (x, t ) we can rewrite (140)–(142) as p21
(x, t ) ≤ A1
�
X12
p22 (x, t ) ≤ A2
2
�
k=1 2
i=1
.. . (x, t ) ≤ An +
(t ) + �ξ2 (t )� +
��
+
p2n
�
2
�
n �
k =1 n−1
�� i=1
2
�
Xi2 (t ) + x −D i 0
i=1
0
x−D1
p22
0
�
(y, t )dy
(144)
�ξi+1 (t )�2 �
p2i+1 (y, t )dy
n−1 �
Xi2 (t ) + x −D i
�
i=1
p2i+1
(145)
�ξi+1 (t )�2
(y, t )dy +
�
x 0
�
2
u (y, t )dy ,
(146)
�n
where in each of the above relations, x ∈ [0, k=i Dk ], respectively. From the above equations, recursively, we can take the upper bound of the lemma. To see this, we start from relation (144) and observe that the boundness of p21 (x, t ) depends only on the boundness of X1 (t ) and ξ2 (x, t ) (thats is, p21 (x, t ) remains �n �n bounded for all x ∈ [0, k=1 Dk ]), if for all x ∈ [0, k=2 Dk ], p22 (x, t ) is upper bounded. We proceed now by proving that the boundness of p22 (x, t ) depends only on the boundness of X1 (t ), X2 (t ), ξ2 (x, t ) and ξ3 (x, t ) (thats is, p22 (x, t ) remains bounded �n �n for all x ∈ [0, k=2 Dk ]), if for all x ∈ [0, k=3 Dk ], p23 (x, t ) is upper bounded. From relation (145) (and by noting
� x −D
�x
that 0 1 p22 (y, t )dy ≤ 0 p22 (y, t )dy for all x for which this �n equation holds, i.e., ∀x ∈ [0, k=2 Dk ]) by using the comparison A2 x | A2 | principle �n and by exploiting the fact that e ≤ e [0, k=2 Dk ], we get that
�
x
0
p22
(y, t )dy ≤ A2 e 2
+
� i=1
|A2 |
2
�ξi+1 (t )�
n �
k=2
Dk
�� � x 0
�
n �
Dk
k=2
y−D2 0
p23
�
2 � k=1
�n
k=2 Dk
, ∀x ∈
Xi2 (t )
�
(r , t )drdy .
(147)
Plugging the above bound in to relation (145) we get a bound of p22 (x, t ) that depends on p23 (x, t ). Moreover, using the relation p23
(x, t ) ≤ A3
�
3 � k =1
��
Xi2 (t ) +
3
+
i=1
x −D i 0
3 � i=1
p2i+1
�ξi+1 (t )�2 �
(y, t )dy ,
and the previous bound, we get
+ A3 A2 e
(148)
+ A3
�
�
k=1
| A2 |
|A2 |
n �
n �
k=2
k=2
x−D3 0
3 �
Dk
p24
1909
Xi2 (t ) + Dk
n �
Dk
k=2
� x� 0
y 0
(x, t ),
2 � i =1
�
p23
x∈
�ξi+1 (t )�2
2 � k=1
Xi2
(t ) +
� 2 � i =1
(r , t )drdy + A3 �
0,
n � k=3
�
�
� �ξi+1 (t )� 2
x
0
p23 (x, t )dy
Dk .
(149)
Note that the delayed terms in the integral for p23 (x, t ) can be �n removed since now x ∈ [0, k=3 Dk ] which is the domain of definition for p3 (x, t ) (and of course this integral is larger than the delayed one). By changing the order of integration in the double integral of the previous relation, we can rewrite
� x� 0
y 0
p23 (r , t )drdy =
�x
�
x 0
(x − y)p23 (y, t )dy. �n
(150)
�x
2 By observing that 0 (x − y)p23 (y, t )dy ≤ k=3 Dk 0 p3 (y, t )dy, �n ∀ k=3 Dk , and applying again the comparison principle for � xx ∈2 p ( y , t )dy we can bound p23 (x, t ) from p24 (x, t ) and consequently 0 3 2 also p2 (x, t ). Repeating this process until p2n (x, t ) (the boundness of which depends only on the boundness of �u(x, t )�2 ), we derive the bound of the lemma. �
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Nikolaos Bekiaris-Liberis received his B.S. degree in Electrical and Computer Engineering from the National Technical University of Athens in 2007. He is now working toward the Ph.D. degree in the Department of Mechanical and Aerospace Engineering at University of California, San Diego. His research interests include control of delay systems, control of distributed parameter systems and nonlinear control.
Miroslav Krstic is the Daniel L. Alspach Professor and the founding Director of the Cymer Center for Control Systems and Dynamics (CCSD) at UC San Diego. He received his Ph.D. in 1994 from UC Santa Barbara and was Assistant Professor at University of Maryland until 1997. He is a coauthor of eight books: Nonlinear and Adaptive Control Design (Wiley, 1995), Stabilization of Nonlinear Uncertain Systems (Springer, 1998), Flow Control by Feedback (Springer, 2002), Real-time Optimization by Extremum Seeking Control (Wiley, 2003), Control of Turbulent and Magnetohydrodynamic Channel Flows (Birkhauser, 2007), Boundary Control of PDEs: A Course on Backstepping Designs (SIAM, 2008), Delay Compensation for Nonlinear, Adaptive, and PDE Systems (Birkhauser, 2009), and Adaptive Control of Parabolic PDEs (Princeton, 2010). Krstic is a Fellow of IEEE and IFAC and has received the Axelby and Schuck paper prizes, NSF Career, ONR Young Investigator, and PECASE award. He has held the appointment of Springer Distinguished Visiting Professor of Mechanical Engineering at UC Berkeley.
