IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 6, JUNE 2003
On Standard
Control of Processes With a Single Delay Qing-Chang Zhong
Abstract—This note presents a frequency domain method to solve the control problem for processes with a single delay. For a standard norm, there exist proper stabilizing given bound on the closed-loop controllers that achieve this bound if and only if both the corresponding problem and an extended Nehari problem with a delay delay-free (or a one-block problem) are all solvable. The solvability of the extended Nehari problem (or the one-block problem) is equivalent to the nonsingularity of a delay-dependent matrix. The solvability conditions of the control problem with a delay are formulated in terms of the standard existence of solutions to two delay-independent algebraic Riccati equations and a delay-dependent nonsingularity property. All suboptimal controllers solving the three problems are, respectively, parameterized as a structure incorporating a modified Smith predictor. control, infinite-diIndex Terms—Chain scattering representation, mensional systems, [0 ]-induced norm, Nehari problem, Riccati equations, Smith predictor, time delay systems.
I. INTRODUCTION
1
The H control of processes with delay(s) has been an active research area since the mid 1980s. Early frequency response methods treated such systems within the framework of general infinite-dimensional system theory [1]–[5]. This resulted in rather complicated solutions, of which the structure is not transparent although simpler structures incorporating finite-impulse-response (FIR) blocks were mentioned in [6]–[8] for implementation, and therefore motivated some problem-oriented approaches [9]–[12] to exploit the structure of the systems. Although considerable progress has been made in this direction, most of the existing solutions still lack the transparency of the classical predictor-type controller. For example, Nagpal and Ravi [12] obtained remarkably elegant solvability conditions for the general H control problem; however, the resulting controllers are extremely complicated. The recent work of Tadmor [13] has made the solution much simpler, but still not very transparent. One of the Riccati equations of [13] depends on the delay as well as on the solution of another differential Riccati equation. A notable exception is the recent work of Meinsma and Zwart [14], who derived the solution of the two-block mixed sensitivity problem using a Smith-predictor-type controller. This approach was extended to the standard H problem [15] by reducing the four-block problem to a two-block problem. However, the solution is based on several intermediate model transformations and, as a result, the final formulas are rather involved and the Riccati equation still depends on the delay. Moreover, there is no clear relationship between the solutions of the H problems for the system with a delay and for its delay-free counterpart. It is not clear how the delay affects the achievable cost or what is the rationale behind the prediction block, especially in the four-block case; see [16] and the references therein. Recently, Mirkin [17], [18] introduced a novel approach. He treated the delay element not as a part of the generalized plant but rather as a
1
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causality constraint imposed upon the controller and then extracted the controller from the solution of the delay-free counterpart. Using this idea, the four-block problem was reduced to a one-block problem and the relationship with the delay-free counterpart becomes clear. In this approach, it is necessary to find the constraint imposed on the free parameter in the solution of the delay-free problem. This is difficult and offers few hints for the case with multiple delays. Moreover, parameterizing the transfer functions to satisfy the desired constraint is not easy. This note presents a more natural and more intuitive approach to solve the problem in the frequency domain. The four-block problem is reduced to a delay-free four-block problem and a one-block problem with a delay by inserting a unimodular matrix and its inverse. According to the chain-scattering theory [19], this transformation does not change the inherence of the problem (this idea can be generalized to the multiple-delay case or even some other cases having a constrained controller). The one-block problem with a delay is then reduced to the extended Nehari problem with a delay by inserting a unimodular FIR block. Furthermore, the extended Nehari problem is solved by inserting another FIR block and a rational matrix, which together complete the J losslessness. The solvability of the extended Nehari problem is represented in terms of the nonsingularity of a delay-dependent matrix and the solution to the extended Nehari problem is formulated in a transparent structure incorporating a modified Smith predictor (i.e., an FIR block). Using this result, a proof for a different (but equivalent) version of [17, Lemma 2] is given in this note (the original lifting-based proof is still in preparation [20] and an alternative proof can now be found in [18]). Notations
Given a matrix A, AT and A3 denote its transpose and complex conjugate transpose respectively. A03 stands for (A01 )3 when the inverse A01 exists. A rational transfer matrix G(s) = D + C (sI 0
A B A)01 B is frequently denoted as C D and its conjugate is defined 0A 0C as G (s) = [G(0s3 )]3 = B D . A truncation operator h and a completion operator h are defined as A B A h fGg =: 0 e0sh C D C : 0sh G~ (s) = G(s) 0 e
1
1
Manuscript received November 21, 2001; revised June 16, 2002 and January 3, 2003. Recommended by Associate Editor Y. Ohta. This work was supported by the Israel Science Foundation under Grant 384/99 and by the EPSRC under Grant GR/N38190/1. This note is dedicated to the memory of Prof. Yong-Feng Zhang, the former president of the Xiangtan Institute of Mechanical and Electrical Technology, Hunan, China. The author is with the Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2BT, U.K. (e-mail:
[email protected], URL: http://members.fortunecity.com/zhongqc). Digital Object Identifier 10.1109/TAC.2003.812818
and
h fGg =:
A Ce0Ah
: ^ (s) =G
B 0
0 e0sh
eAh B 0
A C
B D
0 e0sh G(s)
where h 0. This follows [17], except for a small adjustment in the notation. These two operators map any rational transfer matrix G into FIR blocks. The argument of a transfer function, s, is omitted frequently hereafter for clarity. Two operators mapping an input–output representation M M = M into the right and left chain-scattering repM M resentations are defined as
Cr (M ) =:
0018-9286/03$17.00 © 2003 IEEE
01 M22 M11 M 01 M12 0 M11 M21 21 0 1 0 1 0M21 M22 M21
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 6, JUNE 2003
and
01 M12
0M1201M11 01 M12 0 M22 M 01 M11 M22 M12 12
Cl (M ) =:
provided that M21 and M12 , respectively, are invertible. If both M21 and M12 are invertible, then Cr (M ) 1 Cl (M ) = I: Fu (1) and Fl (1) stand for the standard upper and lower linear fractional transformations (LFT). Another two less-used linear fractional transformations, called homographic transformations (HMT) [19], [21], are defined as
provided that the corresponding inverse exists. The subscript l stands for left and r for right. In the sequel, either LFT or HMT (mainly, Fl or Hr ) will be used when appropriate because Fl (M; S ) = Hr (Cr (M ); S ) when Cr (M ) exists. A signature matrix having appropriate dimensions is denoted by J = 0
0 0 I . A frequently used Hamiltonian matrix and its exponential are denoted by
H
=
and
6 =
A
0C 31 C 1
02 B 1 B 3 1
G 11 + e0sh Q
6 11 6 12 : H =e 6 21 6 22
II. PROBLEM STATEMENTS The main problem considered in this note is the standard H1 control problem of processes with a single delay. This problem is solved via reduction to a one-block problem with a delay and, furthermore, to an extended Nehari problem with a delay. Standard H1 Control Problem With a Single Delay ( h ): Given a > 0 and the general control setup for processes with a single delay as shown in Fig. 1, find a proper controller K (s) such that the closed-loop system is internally stable and
SP
Fl (P; Ke0sh ) 1 < :
(1)
When h = 0, the problem becomes the common standard H1 control problem (SP0 ). One-Block Problem With a Single Delay ( h )1 : Given a > 0 and a not necessarily stable G with G (1) = 0I I0 , find a proper (but not necessarily stable) K (s) such that
OP
Fl (G ; Ke0sh ) 1 < :
(2)
1In order to simplify the exposition, only a special case for the (1,2) and (2,1) blocks of to have a stable inverse, respectively, will be considered in this note. This is sufficient to derive the solution to the main problem.
(3)
Tadmor [22] presented a complicated state-space solution using differential game theory and Zhong [23] presented a much simpler solution in the frequency domain using J -spectral factorization. ENPh (3) is actually a stabilization problem. It can be converted to a conventional Nehari problem
esh Q1 + Q0
h
< :
1 If Q (s) 2 H1 is required (and the H1 -norm becomes L1 -norm), then it is a common Nehari problem with a delay (NPh ), for which
0A3
respectively. The rest of this note is organized as follows. The problems are stated in Section II. The main problem is reduced to a delay-free problem and an extended Nehari problem with a delay in Section III. The solutions to the problems are given in Section IV. Proof of the solution to the extended Nehari problem with a delay is given in Section V. Proof of the solution to the main problem (and the one-block problem) is given in Section VI.
G
General setup of control systems with a single delay.
Extended Nehari Problem With a Delay(ENPh ): Given a > 0 and a strictly proper, but not necessarily stable, G 11 , find a proper (but not necessarily stable) Q (s) such that G 11 + e0sh Q is stable and
Hr (M; Q) =(M11 Q + M12 )(M21 Q + M22 )01 Hl (M; Q) =0(M11 0 QM21 )01 (M12 0 QM22 )
I
Fig. 1.
1
h = k0e Q k, where 0 denotes the Hankel operator. Inspecting the transfer matrix esh Q1 , one can see that h is actually the L2 [0; h]-induced norm of G 11 , i.e., h = kG 11 kL [0; h] . Various methods [1], [25], [26] have been proposed to compute this norm. A simple representation of h is the maximal making 6 22 singular [25], i.e.,
h
= maxf : det 6 22 = 0g:
(5)
III. REDUCTION OF THE PROBLEM SPh Lemma 1. [19, Lemma 7.1]: Let 3 be any unimodular matrix, then the H1 control problem kHr (G; K0 )k1 < is solvable iff kHr (G3; K0)1k1 < is solvable. Furthermore, K0 = Hr (3; K ) or K = Hr (3 ; K0 ). This lemma provides an approach to simplify an H1 control problem by factoring out a unimodular portion from the process, as used in [19]; it also paves the way to simplify an H1 control problem by inserting a unimodular portion to the process. To the best knowledge of the author, the latter has not been exploited in the literature. A. Reducing SPh to a 1-Block Problem With a Delay (OPh ) Assume that the process in Fig. 1 is realized as
P (s) =
P11 (s) P12 (s)
A
= C1 P21 (s) P22 (s) C2
and the following standard assumptions hold:
B1
B2 0 D12 D21 0
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 6, JUNE 2003
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A1) (A; B2 ) is stabilizable and (C2 ; A) is detectable;
A0j!I C
A2)
B D
A0j!I C
and
B D
has full column rank and
full-row rank, respectively, 8 ! 2 R; 3 D12 = I and D21 D21 3 = I. A3) D12 Assumption A3) is made to simplify the exposition although, in fact, 3 D12 and D21 D21 3 is reonly the nonsingularity of the matrices D12 quired [27]. When h = 0, the problem is reduced to the common standard H1 control problem, of which the well-known results are given in [28] and [29]. The following two Hamiltonian matrices are involved:
H0
=
J0
=
A
02 B1 B13
A3
02 C13 C1
0C13C1
0A3
0B1 B13
0A
B2
0
3
0C13D12
[ D12 C1
0B1 D213
[ D21 B1
C23
0
B23 ]
3 C2 ]: (b)
Lemma 2. [28]: Assume that conditions A1)–A3) are satisfied. Then, there exists an admissible controller K (s), noted as K0 (s), such that kFl (P; K0 )k1 < (i.e., the delay-free case when h = 0 in Fig. 1) iff the following three conditions hold: i) H0 2 dom(Ric) and X = Ric(H0 ) 0; ii) J0 2 dom(Ric) and Y = Ric(J0 ) 0; iii) (XY ) < 2 . Moreover, when these conditions hold, all admissible controllers such that kFl (P; K0 )k1 < are parameterized as
K0 (s) = Hr (G (s); Q (s))
where
(6)
A 0 B 1 C 2
B 2
B 1
C 1
I
0
0
I
G (s) =
0C 2
(a)
Fig. 2. Reduction of SP to SP and a one-block delay problem. (a) SP in chain-scattering representation. (b) Decomposition of SP . Note that it is is invertible, so that C ( ) exists. This assumption does not assumed that affect the results because is not affected by this condition and the block C ( ) can be reformulated in the input–output representation as usual. A tagged right-upper corner indicates that the matrix in the block is the scattering matrix from the right-hand side to the left-hand side.
P
P
B. Reducing OPh to ENPh The transfer function involved in
Fl (G ; Ke0sh )
0L ,
=
C 1
= F Z;
02 Y C 3 C1 + B2 + 02 Y C 3 D12 F Z 1 1 0 2 3 B 2 = B2 + Y C D12 =
0
C2 + 02 D21 B13 X Z
0(B23X + D123 C1 ); L = 0(Y C23 + B1 D213 ) Z = (I 0 02 Y X )01 and Q (s) 2 H1 is a free parameter satisfying kQ (s)k1 < : F
where
1
C 2
=
and
=
G 11
G 12
G 21
G 22
A
=
0C 1 0C 2
B 1
B 2
0
I
I
0
I
12
0 I
; Kh
12 (s) is a (stable) FIR block 12 (s) = 0hfG 22 g =: 0G^ 22 (s) + e0sh G 22 (s) I
1
0
I
(8)
(9)
is a unimodular matrix. Hence, this transformation does
not change the solvability of the problem, according to Lemma 1, and OPh given in (2) is then converted to the ENPh given in (3), as shown in Fig. 3(b), where (7)
The general setup for processes with a single delay shown in Fig. 1 can be equivalently depicted in chain-scattering representation as : 1 shown in Fig. 2(a). Since G and Cr (G ) = G0 are all bistable [19, p. 214], inserting G and its inverse Cr (G ) between the process and the controller , as shown in Fig. 2(b), does not change the solvability condition of the original problem according to Lemma 1. The original four-block H1 control problem is then reduced to a delay-free problem and a one-block problem which is given in (2), where
G
= G 11 + e0sh G 12 K (I 0 G 22 Ke0sh )01 G 21
=: Kh (I + 12 Kh )01 = Hr
K
= A + LC2 +
B 1
OPh is
which is depicted in Fig. 3(a). It can be further simplified by introducing a modified Smith predictor-type controller
with
A
P
G
: with A = A 0 B 1 C 2 0 B 2 C 1 . It should be stressed that A and/or A may be unstable but A + B 2 C 1 = A 0 B 1 C 2 and A + B 1 C 2 = A 0 B 2 C 1 are always stable [19].
Q
G 12 0 ; Kh : 0 1 0 ^ 0G 21 G 22 (s) G 211
=: Hr
(10)
This means that there is a bijection from proper Q to proper Kh . Hence, the following lemma is obtained. Lemma 3: SPh is solvable iff its delay-free counterpart SP0 and the corresponding ENPh (or OPh ) are all solvable. As one of the reviewers pointed out, this transformation is not necessary. However, it might offer some useful hints for solving the problem with multiple delays. IV. MAIN RESULTS Theorem 1. (Extended Nehari Problem With a Delay): Given
( )=
G 11 s
A
0C
B
0
, there exists an admissible solution to the
extended Nehari problem with a delay iff > h . Furthermore, if the
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 6, JUNE 2003
+
+
with stable A B 2 C 1 and A B 1 C 2 , there exists an admissible solution to the one-block problem with a delay iff > h . Furthermore, if these conditions hold, then all K s satisfying (2) are parameterized as
()
I
( ) = Hr
K s
1(s)
V 01 s ; Q s
() ()
I
(14)
where and V 01 are shown in (15) and (16) at the bottom of the page, and kQ s k1 < is a free parameter. Theorem 3 (Standard H1 Problem With a Delay)2 : Assume conditions A1)–A3) hold, h is solvable iff the following conditions hold: i) H0 2 dom Ric and X Ric H0 ; ii) J0 2 dom Ric and Y Ric J0 ; iii) XY < 2 ; iv) > h . Furthermore, if these conditions hold, then all K s satisfying (1) are given in (14). Remark IV.1: The solvability conditions of the standard H control problem for systems with a single delay are those for the delay-free case plus the nonsingularity of 22 . This result is quite transparent. Remark IV.2: For delay-free systems (i.e., h ), I . 22 I is always nonsingular for any > , condition iv) disappears and V 01 s becomes G s . Hence, condition iv) can be regarded as the additional cost of the delay.
1 ()
(a)
0
(
)
SP ( ) ( )
= ( ) 0 = ( ) 0
()
1
(b) Fig. 3. Reducing the one-block problem to (b) Equivalent to .
ENP
ENP
. (a) Introducing
1 (s).
()
previous condition holds, then all Q satisfying (3) are parameterized as
0
I
( ) = Hr
Q s
11 (s) = 0h
W 01 (s) =
W 01 (s); Q(s)
11 (s)
I
A
02 B 1 B 3 1
0A3
C 31
0
02 B 3 1
0
where
0C 31 C 1
6 12 601 C 31
A
22
0C 1
02 B 3 1 3 21
6
(11)
0
0
0C 2
I
0
1(s) = 0h ( )=
= G(s)
u2 z2
=:
= Q z2
e0sh I
0
G 11
u2
I
z2 (17)
( ( ) ( )) () ()
and, hence, can be rewritten as kHr G s ; Q s k1 < , as shown in Fig. 4. It is well known [19], [14], [28] that the H1 control problem is closely related to the matrix G s JG s . Moreover, the H1 problem is solved once G s is completed to be a J -lossless transfer matrix.
()
A
02 B 1 B 3 1
0C 2
02 B 3 1
0C 31 C 1 A
V 01 s
11
2A different but equivalent version of this theorem was given in [17] as mentioned in the Introduction, however, no proof was available when this note was submitted. Hence, the contribution regarding this theorem is the proof (Section VI) rather than the theorem itself. An alternative proof can now be found in [18].
I
I
w1 u2
B 2
0
()
and
()
= 0C 1
The first part of Theorem 1 has been shown in Section II. The only thing left to be proved is to derive the parameterization of Q s under the condition > h .
(12)
and kQ s k1 < is a free parameter. Theorem 2 . (One-Block Problem With a Delay): Given
G
6
V. PROOF OF THEOREM 1
z1
(13)
B 1
=0 6
The ENPh problem (3) can be associated with the following system in chain-scattering representation:
22
0
0
()
A. Rationalization by
0603 B 1
I
A
6
+ B 2 C 1 C 1
0A3
0 02 B 3 1 6 3 21 0 C 2 6 3 22
B 2 C 31
0
(15)
1 3 B 2 0 12 0 22 C 1
6 6
603 22 B 1
I
0
0
I
(16)
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Fig. 4.
Solving the
ENP
1101
(a) .
The result of Meinsma and Zwart presented in [14] is for a two-block problem, where G s is stable, and cannot be directly used to solve the extended Nehari problem with a delay because G s is not necessarily stable and, more importantly, neither is Q . However, some ideas of Meinsma and Zwart will be borrowed. By introducing the FIR block
()
()
11 (s) = F0 (s) + F1 (s)e0sh G 11
=: 0h Fu
I
(b)
I
; 02 G 11
0
(18)
() +
2 =:
0
I
0
I
G JG
11 (s)
I
I 0 1 (s) I
becomes rational (bearing in mind that
02 B 1 B 3 1
0C 1
0
0
201 =
201 (s)
0
02 11 B 1
I
0
6
0C 31 02 6 21 B 1
0A3
02 B 3 1 6 3 21 0 02 B 3 1 6 3 11
:
( ) = Hr
201 can be further rewritten as 201 = W 01 (s) 1 J 01 1 W 0 (s) where the realization of W 01 (s) is given in (13), without the need to solve any Riccati equation but using the 6 -related identities given 3
( )=
M s
M21
M22
0
I
= G(s) :
11 (s)
I
W 01 (s)
(2 2)
= ( ( ) ( )) ( ) = Hr
Q s
+
11 (s)
I
()
; Q s
:
limjsj!1 ; Res 0; W (s) 01I (s) I0 = I0 I0 it is true I 0 that Q(1) = Hr ( 0 I ; Q (1)) = Q (1). This implies that Q(s) previously given is proper iff Q (s) is proper, yet the set of proper operators in L1 is in fact H1 [33] (see also [34, A6.26.c, A6.27]). So, if Q (s) solves ENPh , then necessarily Q(s) 1 < . The only thing left to be proved is the state-space realizations of 11 (s), which can be found as given in (12) using the well-known star k
VI. PROOF OF THEOREMS 2 AND 3 The first part of Theorems 2 and 3 have been shown in Section III. Here, only the second part, to recover the controller K s , will be shown. From (10), Kh is recovered as
()
Kh
(of which the realization is given in [30, App. A]) is stable and J -unitary. Furthermore, it is J -lossless because its ; -block M22 s can be shown to be bistable using similar arguments as in [14, p. 284: the second part of Condition 2) in Proof of Theorem 5.3]. It can now be concluded, according to [14, p. 281; Theorem 6.2], that G 11 e0sh Q Hr M s ; Q s is stable and kG 11 e0sh Q k1 < for
I
011 (s)
Since
in [31]. The following matrix:
M12
()
W s
()
product [35, Sec. 9.3].
B. Completing the J -Losslessness
M11
0
I
k
0 02 I
0
()
Qs is a unimodular
matrix). Using elementary transformations, the realization of can be obtained (see [23] and [31]) as
A
(a) Recovered structure. (b)
with any stable Q s having kQ s k1 < . This condition is also necessary, as demonstrated later applying the arguments in [32]. By construction, kG 11 e0sh Q kL < iff kQ s kL < for
the following matrix:
I 11 (s)
K (s).
Fig. 5. Recovery of the controller Simplified structure.
0 I
W 01 (s); Q(s)
3This is a notable characteristic. Due to this, and/or zeros.
G
()
may have j! -axis poles
1 G0 12
^ 22 G01 G
r
H
12
0
G 21
; Q
:
(20)
Hence, combining (20) with (8) and (19), the controller can then be recovered, as shown in Fig. 5(a), as
( )=
K s
I
r
12
H
+
(19)
=
0
I
;
r
H
1 G0 12
^ 22 G01 G
12
r
H
=
1 G0 12
I
11
0
0 I
G 21
;
W 01 ; Q
0
01 G 21 F0 + e0sh Fu (G ; 02 G 11 )G 12 G 21
r
H
1
W 01 ; Q :
1102
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According to the star product [35, Sec. 9.3], the following realization:
Fu (G ; 02 G 11 ) has
02 B 1 B 3 1
A
Fu (G ; 02 G 11 ) = 0C 31 C 1 0C 2
0A3
then
[ 0C 2
0
02 B 3 1
] 6 01
0
1 G0 12
0
I
( ) = Hr
K s
1
The definition of
V 01 s
( ) =:
REFERENCES
(21)
0
01 1 G 21 W ; Q : G 21 F0 0 F2 G0 12
I
1 G0 12
0
W 01 1 G 21 F0 0 F2 G0 12 G 21
()
offers the controller K s as given in (14), of which the structure is shown in Fig. 5(b). The realization of V 01 s can be found as given in (16) (see [30, App. B] for derivations). 3 Remark VI.1: V s is bistable because 03 22 A B 1 C 2 22 A B 1 C 2 , which is also stable [19]. I 0 V 01 is Remark VI.2: This theorem indicates that Cr P 1 I J -lossless and, hence, its lower right block is bistable.
+
()
() 6 ( +
)6
( )
VII. CONCLUDING REMARKS This note derives a complete solution to the H1 problem of rational plants with single delay. Although such a solution has been obtained by other researchers [17] (now see also [18]), the approach in this note is quite unique. This problem (SPh ) is solved by reducing the original problem to a delay-free problem and a one-block problem (OPh ) and then the OPh is further reduced to an extended Nehari problem (ENPh ), mainly using a new interpretation of Kimura’s theorem [19, Lemma 7.1]. The ENPh is solved using the techniques proposed by Meinsma and Zwart [14]. After solving the ENPh , the H1 controllers for the one-block problem and the standard H1 control problem are recovered. In summary, the contributions of this note are: 1) a unique approach to solve SPh ; 2) the solution to (a special case of) OPh ; 3) the solution to ENPh , and 4) a proof of the solution to SPh . The idea used in this note may be useful to solve the H1 control problem for processes with multiple delays or even other systems with a constrained controller. The suboptimal controller incorporates an FIR block. This FIR block can be obtained by combining two FIR blocks resulted from two steps. This might be a useful hint to solve the multidelay problem. It has been shown that the standard H1 control problem for processes having some constraints can be solved in a clear and simple way by reducing the original problem to a nonconstraint problem and a one-block problem with constraints. Once the one-block problem is solved, then the original problem is solved. Another fact that I 0 V 01 is also stable the inverse of the lower-right block of Cr P 1 I may well be useful for the robust stability analysis of time delay systems, which is a very active field (see, e.g., [36]–[38]). As shown in [30], the proposed idea can be easily used to interpret the solution to the H1 fixed-lag smoothing problem studied in [39] and [40].
( )
The author would like to thank L. Mirkin for his guidance and support during the author’s stay at Technion-Israel Institute of Technology, Israel, and greatly appreciates the help of G. Meinsma on the proof of the necessary part of Theorem 1. The author would also like to thank K. Gu, Y. Ohta, G. Weiss, K. M. Zhou, and the anonymous reviewers for their constructive suggestions, and H. Glasman–Deal for her professional writing clinic.
C 31
1(s) =: F2 (s) + Fu (G ; 02 G 11 )e0sh = 0h Fu (G ; 02 G 11 )
is an FIR block. Hence
:
B 2
H
( )=0
F2 s
C 31
02 B 3 1
Define
:
B 2
ACKNOWLEDGMENT
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[23] Q.-C. Zhong, “Frequency domain solution to delay-type Nehari problem,” Automatica, vol. 39, no. 3, pp. 499–508, 2003. [24] I. Gohberg, S. Goldberg, and M. A. Kaashoek, Classes of Linear Operators. Basel, Germany: Birkhäuser, 1993, vol. II. [25] K. Zhou and P. P. Khargonekar, “On the weighted sensitivity minimization problem for delay systems,” Syst. Control Lett., vol. 8, pp. 307–312, 1987. [26] G. Gu, J. Chen, and O. Toker, “Computation of L [0; h] induced norms,” in Proc. 35th IEEE Conf. Decision Control, Kobe, Japan, 1996, pp. 4046–4051. [27] M. Green and D. J. N. Limebeer, Linear Robust Control. Upper Saddle River, NJ: Prentice-Hall, 1995. [28] M. Green, K. Glover, D. Limebeer, and J. Doyle, “A J-spectral factorization approach to H control,” SIAM J. Control Optim., vol. 28, no. 6, pp. 1350–1371, 1990. [29] J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, “Statespace solutions to standard H and H control problems,” IEEE Trans. Automat. Contr., vol. 34, pp. 831–847, Aug. 1989. [30] Q.-C. Zhong. (2001, Nov.) On standard H control of processes with a single delay. Dept. E.E. Eng., Imperial College London, London, U.K.. [Online]http://www.ee.imperial.ac.uk/CAP/Reports/2001.html , (2001, Nov.) Frequency domain solution to the delay-type Nehari [31] problem using J-spectral factorization. Dept. E.E. Eng., Imperial College London, London, U.K.. [Online]http://www.ee.imperial.ac.uk/CAP/Reports/2001.html [32] G. Meinsma, L. Mirkin, and Q.-C. Zhong, “Control of systems with I/O delay via reduction to a one-block problem,” IEEE Trans. Automat. Contr., vol. 47, pp. 1890–1895, Nov. 2002. [33] E. G. F. Thomas, “Vector-valued integration with applications to the operator-valued H space,” J. Math. Control Inform., vol. 14, pp. 109–136, 1997. [34] R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory. New York: Springer-Verlag, 1995. [35] K. Zhou and J. C. Doyle, Essentials of Robust Control. Upper Saddle River, NJ: Prentice-Hall, 1997. [36] K. Gu and S.-I. Niculescu, “Additional dynamics in transformed timedelay systems,” IEEE Trans. Automat. Contr., vol. 45, pp. 572–575, Mar. 2000. [37] , “Further remarks on additional dynamics in various model transformations of linear delay systems,” IEEE Trans. Automat. Contr., vol. 46, pp. 497–500, Mar. 2001. [38] Q.-C. Zhong, “Robust stability analysis of simple systems controlled over communication networks,” Automatica, vol. 39, no. 7, pp. 1309–1312, July 2003. [39] L. Mirkin, “On the H fixed-lag smoothing: How to exploit the information preview,” presented at the IFAC Symp. System Structure Control, Prague, Czech Republic, 2001. [40] , “Continuous-time fixed-lag smoothing in H setting,” in Proc. 40th IEEE Conf. Decision Control, vol. 4, Orlando, FL, 2001, pp. 3512–3517.
1103
Asymptotic Stabilization via Output Feedback for Lower Triangular Systems With Output Dependent Incremental Rate Laurent Praly
Abstract—We study the global asymptotic stabilization by output feedback for systems whose dynamics are in a feedback form and where the nonlinear terms admit an incremental rate depending only on the measured output. The output feedback we consider is of the observer-controller type where the design of the controller follows from standard robust backstepping. The novelty is in the observer which is high-gain such as with a gain coming from a Riccatti equation. Index Terms—Backstepping, high-gain nonlinear observer, output nonlinear feedback, Riccatti equation.
I. INTRODUCTION We consider a nonlinear system for which we can find coordinates y1 to yn and z1 to zm such that its dynamics can be written as
y_ 1 y_ 2
= =
f1 (y1 ) + y2 f2 (y1 ; y2 ) + y3
.. .
y_ n z_1 z_2
= = =
fn (y1 ; h1 (y1 ; h2 (y1 ;
...;
yn ) + z1 + u yn ; z1 ; u) + z2 . . . ; yn ; z1 ; z2 ; u) + z3
(1)
...;
.. .
z_m
=
hm (y1 ;
...;
yn ; z1 ;
...;
zm ; u)
where y1 is the measured output in , u is the input in , the functions fi s are n + 1 times continuously differentiable and zero at the origin, the functions hi s are continuously differentiable and zero at the origin and, for all i, u, y , z , , and ', we have
jf (y1 ; y2 + 2 ; . . . ; y + ) 0 f (y1; y2 ; . . . ; y )j (y1) (j 2 j + 1 1 1 + j j) jh (y1 ; y2 + 2 ; . . . ; y + ; z1 + '1 ; . . . ; z + ' ; u) 0 h (y1; y2 ; . . . ; y ; z1 ; . . . ; z ; u)j (y1) (j 2 j + 1 1 1 + j j + j'1 j + 1 1 1 + j' j) i
i
i
n
i
i
i
i
i
n
n
i
(2)
i
i
n
i
(3)
where is a n + 1 times continuously differentiable strictly positive function. We address the problem of global asymptotic stabilization of the origin with output feedback. This problem has received a lot of attention. But until recently, the contributions were assuming that the fi s at least are linear in y2 to yn ,
Manuscript received August 29, 2002; revised January 26, 2003. Recommended by Associate Editor J. Huang. The author is with the Centre Automatique et Systèmes, École des Mines de Paris, 77305 Fontainebleau, France (e-mail:
[email protected]). Digital Object Identifier 10.1109/TAC.2003.812819 0018-9286/03$17.00 © 2003 IEEE
