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Nonlinear H∞ filtering of sampled-data systems
Automatica 36 (2000) 303}310
Brief Paper
Nonlinear H "ltering of sampled-data systemsq = Sing Kiong Nguang!,*, Peng Shi" !Systems and Control Group, Department of Electrical and Electronic Engineering, The University of Auckland, Private Bag 92019, Auckland, New Zealand "Center for Industrial and Applied Mathematics, School of Mathematics, The University of South Australia, The Levels, SA 5095 Australia Received 8 June 1998; revised 1 December 1998; received in "nal form 20 May 1999
Abstract This paper considers the problem of H "ltering for nonlinear sampled-data systems. Su$cient conditions are obtained for the = existence of "lters that guarantee the L gain from an exogenous input to an estimation error is less than or equal to a prescribed 2 value. These conditions are expressed in terms of both continuous and discrete Hamilton}Jacobi equations. ( 1999 Elsevier Science Ltd. All rights reserved. Keywords: Sampled-data systems; H control; Filtering problem; Discrete systems; Estimators =
1. Introduction Past years have seen a numerous industrial systems that are continuous time processes but controlled by a digital controller. In general, there are two classic approaches which have been adopted in engineering practice for the design of a discrete controller. The "rst approach is, "rst designing a continuous controller such that the closed-loop system has satisfactory performances, and then discretising the continuous controller into a discrete controller using one of the discretisation methods. This technique has the advantage that the synthesis is done in continuous time, but it may have a serious degradation in performances when the sampling is not fast enough. The second approach is, discretising the process "rst, then designing a discrete controller. The main bene"t of the approach is that the synthesis procedure is again simpli"ed. However, the discrete models obtained by discretisation fail to represent the intersample behaviour of the systems. The main drawback of
q This paper was presented at the 14th IFAC World Congress Beijing, P.R. China, July 1999. This paper was recommended for publication in revised form by Associate Editor Kenko Uchida under the direction of Editor T. Bas7 ar. * Corresponding author. Tel.: 00-64-9-3737599d8285; fax: 00-64-93737461. E-mail addresses: [email protected] (S.K. Nguang), [email protected] (P. Shi)
these approaches is neither of them o!ers an adequate framework for analysis of the sampled-data system which requires the consideration of both sampled and intersample behaviours. Recently, a number of di!erent approaches have been proposed to incorporate intersample behaviour. These approaches include lifting technique (Bamieh, Pearson, Francis & Tannenbaum, 1991; Toivonen, 1992; Bamieh & Pearson, 1992; Yamamoto, 1991); descriptor system technique (Hara & Kabamba, 1990), and technique based on linear systems with jumps (Shi, 1994, 1996, 1998; Sun, Nagpal & Khargonekar, 1993; Sivashankar & Khargonekar, 1994). The lifting technique consists of transforming the original sampled-data system into an equivalent LTI discrete-time system with in"nite-dimensional input}output signal space. Then L -induced 2 norm of the sampled-data system is shown to be less than one if and only if the H of this equivalent discrete = system is less than one. In the descriptor system approach, on the other hand, the system is represented by a hybrid state-space model, from which the descriptor system is formulated. The solution of the H sampled= data problem is then characterised by the solution of certain associated Hamilton}Jacobi equation. In contrast with these procedures, the theory of linear systems with jumps allows a direct characterisation of the problem in the similar terms to those of standard LTI H control problems, and leads to a pair of Riccati = equations. Despite the procedural di!erences in all these
0005-1098/00/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 1 4 1 - 7
304
S.K. Nguang, P. Shi / Automatica 36 (2000) 303}310
approaches, the results obtained are mathematically equivalent. At the same time, the problem of nonlinear H control = problem has been studied by a number of researchers; see for instance (Ball & Helton, 1989; Basar & Olsder, 1982; van der Schaft, 1992; Isidori & Astol", 1992; Isidori, 1991). There are two commonly used approaches for providing solutions to nonlinear H control problems. = One is based on the dissipativity theory and theory of di!erential games (Basar, 1991; Ball & Helton, 1989). Another is based on the nonlinear version of classical Bounded Real Lemma as developed by Willems (1972), and Hill and Moylan (1980); see, e.g., van der Schaft (1991); Isidori (1991) and Isidori and Astol" (1992). Both of these approaches convert the problem of nonlinear H control to the solvability of the so-called Hamil= ton}Jacobi equation (HJE). A nice feature of these results is that they are parallel to the linear H results. Further = research along the line of the dissipativity theory and theory of di!erential games has been attempted (see, e.g., Ball, Helton & Walker, 1993; Isidori and Astol", 1992; Isidori, 1991) where results on disturbance attenuation for nonlinear systems via state feedback and/or output feedback have been provided. In Nguang and Fu (1994,1996); Berman and Shaked (1996) solutions to the nonlinear H "ltering problem have been obtained. = These continuous nonlinear H results have further been = extended to discrete nonlinear systems (Lin & Byrnes, 1995, 1996). In Suzuki, Isidori and Tarn (1995), su$cient conditions for the existence of a solution to the problem of H control nonlinear systems with sampled-data have = been developed. Their su$cient conditions are expressed in terms of both Hamilton}Jacobi equations and Riccati di!erential equation with jump. In the present work, we extend the nonlinear H "lter= ing theory to a nonlinear H estimator for nonlinear = sampled-data systems. This problem can be stated as follows: given a dynamic system with exogenous input and sampled output measurements at discrete instants of time, design a "lter to estimate an unmeasured output such that the L norm of the mapping from the 2 exogenous input to the estimation error is minimised or no larger than some prescribed level. The performance requirements are de"ned directly in terms of the continuous-time signals which means the intersample behaviour is incorporated in the design. Based on a nonlinear version of the classical bounded real lemma theorem for sampled-data systems, su$cient conditions for the existence of a "lter that guarantees the L gain from an 2 exogenous input to an estimation error is less than or equal to a prescribed value are derived. These conditions are expressed in terms of both continuous and discrete Hamilton}Jacobi equations. Since "ltering results are closely related to output feedback results, from Suzuki et al. (1995) it is easy to see that their "ltering condition is expressed in terms of Riccati di!erential equation with
jump. However, our su$cient conditions are expressed in terms of both continuous and discrete Hamilton}Jacobi equations, which are much more general than the "ltering condition given in Suzuki et al. (1995). Another di!erence between our results and their results Suzuki et al. (1995) is the derivation of results. Our results are based on classical bounded real lemma theorem for sampleddata systems, their results Suzuki et al. (1995) are based on dissipativity theory and theory of di!erential games. Our results can also be viewed as an extension of Sun et al. (1993) and Shi (1994), which treat the linear systems. We end this introduction with some remarks on the notations. The notations used in this paper are fairly standard. The superscript t denotes matrix transposition. L [0, ¹] stands for the space of square integrable vector 2 functions over [0, ¹], l (0, ¹) is the space of square 2 summable vector sequences over (0, ¹). f (h~) and f (h`) denote for the left limit and right limit of a function f (h), respectively. The gradient of a scalar-valued function f (m) along the vector m3Rn is a column vector denoted by + f (m), and the gradient of a column vector-valued funcm tion
CD / (m) 1 / (m) 2 F
(1.1)
/ (m) m is a matrix de"ned by
C
L/ (m) 1 Lm 1 L/ (m) 2 D /(m)" Lm 1 m F
L/ (m) 1 Lm 2 L/ (m) 2 Lm 2 F
2 2 }
D
L/ (m) 1 Lm n L/ (m) 2 Lm . n F
(1.2)
L/ (m) L/ (m) L/ (m) m m m 2 Lm Lm Lm n 2 1
2. De5nition and preliminary results
Consider the nonlinear time-varying system with "nite discrete jumps: x5 (t)"A(x(t), t)#B(x(t), t)w(t), t3[0, ¹], tOih; x(0)"0, & : x(ih)"A (x(ih~), ih)#B (x(ih~), ih)v(ih), 1 $ $ z(t)"¸(x(t), t), t3[0, ¹], tOih,
(2.1)
z (ih)"¸ (x(ih~), ih), $ $ where x(t)3Rn is the state, w(t)3Rp and v(ih)3Rq are the continuous and discrete exogenous inputs which belong to L [0, ¹] and l (0, ¹), respectively, z(t)3Rr and 2 2 z (ih)3Rs are the continuous and discrete outputs, $ 0(h3R is the sampling period, i is a positive integer,
S.K. Nguang, P. Shi / Automatica 36 (2000) 303}310
A(x(t), t), B(x(t), t), A (x(ih~), ih), B (x(ih~), ih) ¸ (x(ih~), $ $ $ ih) and ¸(x(t), t) are known real time-varying bounded nonlinear matrix functions of appropriate dimensions with A(x(t), t), B(x(t), t) and ¸(x(t), t) being piecewise continuous.
305
Remark 1. Conditions (2.5) and (2.6) imply that vH(ih) is the maximiser for DHJE(x(ih~), v(ih), ih), i.e., DHJE(x(ih~), v(ih), ih)4DHJE(x(ih~), vH(ih), ih).
(2.7)
Proof of Theorem 2.1. For q3(ih, ih#h), De5nition 1. Suppose c is a given positive real number. A system & of the form (2.1) is said to have L gain less 1 2 than or equal to c if
P
CP
T T k z5(t)z(t) dt# + z5 (ih)z (ih)4c2 w5(t)w(t) dt $ $ 0 0 i/1 k # + v5(ih)v(ih) , i/1 where k is the largest integer such that k(¹.
D
P
q d Me(x, t)N dt"e(x(q), q)!e(x(ih`), ih`). dt ih`
First, let us consider the left-hand side of (2.8):
P P
q d Me(x, t)N dt ` dt ih q [e5 (x(t), t)#+5 e(x(t), t)MA(x(t), t) " x ih`
H(x(t), t)" (2.2)
Note that the left-hand side of (2.2) can be viewed as a &mixed ¸ /l ' norm for the output signals z and z , and 2 2 $ the right-hand side of (2.2) as a &mixed ¸ /l ' norm of the 2 2 input signals which comprise of w and v. In the sequel, without loss of generality, we assume c"1. Theorem 1. Consider system & of the form (2.1). Suppose 1 there exist positive-dexnite functions e(x(t), t) and Q(x(t), t) with e(0, t)"0 and Q(0, t)"0 which satisfy (C1) HJE(x(t), t)Oe5 (x(t), t)#+5 e(x(t), t)A(x(t), t) x #1+5 e(x(t), t)B(x(t), t)B5(x(t), t)+ e(x(t), t) 4 x x #¸5(x(t), t)¸(x(t), t)#Q(x(t), t)"0, (2.3)
#B(x(t), t)w(t)N] dt.
P
H(x(t), t)"
where
q
[e5 (x(t), t)#+5 e(x(t), t)MA(x(t), t) x ih`
#B(x(t), t)w(t)N#w5(t)w(t)!z5(t)z(t) ! w5(t)w(t)#z5(t)z(t)] dt.
(2.10)
Completing the square and knowing that z(t)"¸(x(t), t), one has
P
H(x(t), t)"
q
ih
[e5 (x(t), t)#+5 e(x(t), t)A(x(t), t) x
`
#1+5 e(x(t), t)B(x(t), t)B5(x(t), t)+ e(x(t), t) 4 x x #¸5(x(t), t)¸(x(t), t)!(1B5(x(t), t)+ e(x(t), t)) 2 x
DHJE(x(ih~), vH(ih), ih)Oe(A (x(ih~), ih) $ #B (x(ih~), ih)vH(ih), ih`) $ !e(x(ih~), ih)! v5H(ih)vH(ih) #¸5 (x(ih~), ih)¸ (x(ih~), ih) $ $ #Q(x(ih~), ih)"0, (2.4)
(2.9)
Next adding and subtracting (w5(t)w(t)!z5(t)z(t)) to and from (2.9), it follows that
(C2)
+ e(A (x(ih~), ih)#B (x(ih~), ih)vH(ih), ih`) v $ $ !2vH(ih)"0
(2.8)
!w(t))5(1B5(x(t), t)+ e(x(t), t))!w(t)) 2 x #w5(t)w(t)!z5(t)z(t)] dt.
(2.11)
Employing (2.3), we have
P
H(x(t), t)4
q
[w5(t)w(t)!z5(t)z(t)!Q(x(t), t)] dt. (2.12) ih`
From (2.8) (2.5)
and D + e(A (x(ih~), ih)#B (x(ih~), ih)vH(ih), ih`)!2I(0. v v $ $ (2.6) Then the system & is having L -gain less than or equal 1 2 to 1.
P
q
[w5(t)w(t)!z5(t)z(t)!Q(x(t), t)] dt
ih` 5e(x(q), q)!e(x(ih`), ih`).
(2.13)
Now, let us consider at the sampling instant e(x(t), t)Dih`~"e(x(ih`), ih`)!e(x(ih~), ih~) ih "e(x(ih), ih`)!e(x(ih~), ih).
(2.14)
306
S.K. Nguang, P. Shi / Automatica 36 (2000) 303}310
Next adding and subtracting the term z5 (ih)z (ih)! $ $ v5(ih)v(ih) to and from (2.14), we obtain H (x(ih), ih)"e(x(ih), ih`)!e(x(ih~), ih)#z5 (ih)z (ih) $ $ $ !v5(ih)v(ih)!z5 (ih)z (ih)#v5(ih)v(ih). $ $ (2.15) Knowing that z (ih)"¸ (x(ih~), ih), we have $ $ H (x(ih), ih)"e(A (x(ih~), ih)#B (x(ih~), ih)v(ih), ih`) $ $ $ !e(x(ih~), ih)#¸5 (x(ih~), ih)¸ (x(ih~), ih) $ $ !v5(ih)v(ih)!z5 (ih)z (ih)#v5(ih)v(ih). $ $ (2.16) Using (2.4) and the maximiser vH(ih) given in (2.5) and (2.6), we get H (x(ih), ih)4v5(ih)v(ih)!z5 (ih)z (ih)!Q(x(ih~), ih). $ $ $ (2.17) By combining (2.12) and (2.17) over all possible ih on [0, ¹], one has e(x(¹), ¹)!e(0, 0)
P
T k [w5(t)w(t)!z5(t)z(t)] dt# + v5(ih)v(ih) 0 i/1 T k ! + z5 (ih)z (ih)! Q(x(t), t) dt $ $ 0 i/1 k ! + Q(x(ih~), ih). (2.18) i/1 Knowing that e(0, t)"0 and Q(x(t), t)'0, ∀x(t)O0, we obtain 4
P
P
T k + z5 (ih)z (ih)# z5(t)z(t) dt $ $ 0 i/1 T k 4 w5(t)w(t) dt# + v5(ih)v(ih)!e(x(¹), ¹) 0 i/1 T k 4 w5(t)w(t) dt# + v5(ih)v(ih). (2.19) 0 i/1 This shows that the system & has the L -gain less 1 2 than or equal to 1. To prove the stability, we rearrange (2.18) to
P P
P
T
k Q(x(t), t) dt# + Q(x(ih~), ih) 0 i/1 T k 4 [w5(t)w(t)!z5(t)z(t)] dt# + v5(ih)v(ih) 0 i/1 k (2.20) ! + z5 (ih)z (ih) $ $ i/1
e(x(¹), ¹)!e(0, 0)#
P
Since e(0, 0)"0, we obtain
P
T k Q(x(t), t) dt# + Q(x(ih~), ih) 0 i/1
P
4
T k w5(t)w(t) dt# + v5(ih)v(ih). 0 i/1
(2.21)
This means that lim TC=
CP
D
T k Q(x(t), t) dt# + Q(x(ih~), ih) 0 i/1
4 lim TC=
CP
D
T k w5(t)w(t) dt# + v5(ih)v(ih) (R. 0 i/1
(2.22)
Since Q(x(t), t) is a positive-de"nite function, we conclude that system & is bounded input and bounded state 1 stable. h Remark 2. In the case of a linear system, i.e., linearise (2.1) at the origin: x5 (t)"A (t)x(t)#B(0, t)w(t), tOih; x(0)"0, 1 x(ih)"A 1(ih)x(ih~)#B (0, ih)v(ih), ∀ih3(0, ¹), d $ & l: 1 z(t)"¸ (t)x(t), (2.23) 1 z (ih)"¸ 1(ih)x(ih~), $ d where A (t)"D A(0, t), ¸ (t)"D ¸(0, t), 1 x 1 x A 1(ih)"D ~ A (0, ih) x(ih ) $ d and ¸ 1(ih)"D ~ ¸ (0, ih), d x(ih ) $ it is easy to show that for e(x(t), t)"1x5(t)P(t)x(t), the 2 Hamilton}Jacobi equation (2.3) reduces to the following Riccati inequality: PQ (t)#A5 (t)P(t)#P(t)A (t)#P(t)B(0, t)B5(0, t)P(t) 1 1 #¸5 (t)¸ (t)(0, tOih, 1 1
P(¹)"0
(2.24)
and the discrete Hamilton}Jacobi equation (2.4) boils down to the following discrete Riccati inequality: A5 1(ih)P(ih`)A 1(ih)#A5 1(ih)P(ih`)B (0, ih) d d d $ [I!B (0, ih)P(ih`)B (0, ih)]~1 B5 (0, ih)P(ih`)A 1(ih) $ d $ $ #¸5 1(ih)¸ 1(ih)!P(ih)(0 d d
(2.25)
with I!B (0, ih)P(ih`)B (0, ih)'0. $ $
(2.26)
S.K. Nguang, P. Shi / Automatica 36 (2000) 303}310
In Sun et al. (1993) and Shi (1994), the authors also showed that (2.24) and (2.25) are the necessary conditions for the system & l to be BIBOS with L -gain less than or 1 2 equal to 1.
Consider the following class of nonlinear sampled-data systems:
H
x5 (t)"A(x(t))#B(x(t))w(t),
∀t3[0, ¹], tOih, x(0)"0,
& : 2 z (ih~)"¸ (x(ih~), ih), $ $ y(ih)"C(x(ih~))#D(x(ih~))v(ih),
(3.1)
Problem formulation: Given a scalar c'0, design a nonlinear "lter of the form mQ (t)"a(m(t)), t3[0, ¹], tOih, (3.2)
C D x(t) m(t)
,
C
AI (x8 (t), t)"
A(x(t), t)
D
A(m(t), t))
,
¸I (x8 (t), t)"¸(x(t), t)!¸(m(t), t) and BI t(x8 (t), t)"[B5(x(t), t) 0]. (C2) e(AI (x8 (ih~), ih)#BI (x8 (ih~), ih)vH(ih), ih`) $ $ !e(x8 (ih~), ih)!vtH(ih)vH(ih)
+ e(AI (x8 (ih~), ih)#BI (x8 (ih~), ih)v(ih), ih`)!2vH(ih)"0, v $ $ (3.6) D + e(AI (x8 (ih~), ih)#BI (x8 (ih~), ih)v(ih), ih`)!2I(0, v v $ $ (3.7)
C
such that
x(ih~)m(ih~)
"
D
b(m(ih~), ih)[C(x(ih~), ih)!C(m(ih~), ih)
P
T (z(t)!z( (t))5(z(t)!z( (t)) dt 0
P
for all x8 (t)3R2n, where
AI (x8 (ih~), ih) $
z( (ih)"¸ (m(ih~), ih) $ $
k # + (z (ih)!z( (ih))5(z (ih)!z( (ih)) $ $ $ $ i/1 T k 4 w5(t)w(t) dt# + v5(ih)v(ih) 0 i/1 holds.
(3.4)
where
R(x(ih~), ih)"D(x(ih~), ih)D5(x(ih~), ih)'0.
z( (t)"l(m(t)), ∀t3[0, ¹], tOih,
#¸I 5(x8 (t), t)¸I (x8 (t), t)#Q(x8 (t), t)"0
#¸I 5 (x8 (ih~), ih)¸I (x8 (ih~), ih)#Q(x(ih~), ih)"0, (3.5) $ $
Assumption 1.
m(ih)"m(ih~)#b(m(ih~), ih)[y(ih)!C(m(ih~))],
#1+58 e(x8 (t), t)BI (x8 (t), t)BI 5(x8 (t), t)+ 8 e(x8 (t), t) 4 x x
x8 (t)"
where x(t)3Rn is the state, w(t)3Rp and v(ih)3Rq are the continuous and discrete inputs which belong to L [0, ¹] and l (0, ¹), respectively, z(t)3Rr and 2 2 z (ih)3Rs are the continuous and discrete regulated out$ puts, 0(h3R is the sampling period, i is a positive integer, A(x(t), t), B(x(t), t), C(x(ih~), ih), D(x(ih~), ih), ¸ (x(ih~), ih) and ¸(x(t), t) are known real time-varying $ bounded nonlinear matrix functions of appropriate dimensions with A(x(t), t), B(x(t), t) and ¸(x(t), t) being piecewise continuous.
K:
loop system of & with K satisxes (3.3) if there exist 2 e(x8 , t)'0, Q(x8 , t)'0 and b(m(ih~), ih) such that the following conditions hold: (C1) HJOe5(x8 (t), t)#+58 e(x8 (t), t)AI (x8 (t), t) x
3. Nonlinear H 5ltering =
z(t)"¸(x(t)),
307
,
¸I (x8 (ih~), ih)"¸ (x(ih~), ih)!¸ (m(ih~), ih) and BI 5 (x8 (ih~), $ $ $ $ ih)"[0 b(m(ih~), ih)D(x(ih~), ih)]. If this is the case, then a suitable xlter of the form K is given by mQ (t)"A(m(t), t), (3.3)
Theorem 2. Consider system & satisfying Assumption 3.1. 2 Then, there exists a nonlinear xlter K such that the closed-
t3[0, ¹], tOih, m(0)"0,
m(ih)"m(ih~)#b(m(ih~), ih)[y(ih)!C(m(ih~), ih)], z( (t)"¸(m, t), t3[0, ¹], tOih,
(3.8)
z( (ih)"¸ (m(ih~), ih). $ $ The "lter given above is a nonlinear counterpart of the "lter given in Sun et al. (1993) and Shi (1994).
308
S.K. Nguang, P. Shi / Automatica 36 (2000) 303}310
Remark 3. In general, the optimal choice of b(m(ih~), ih) is still an open problem as pointed out in lin1 for the discrete nonlinear case.
(1993) and Shi (1994), the authors showed that these are the necessary and su$cient conditions for the existence of an estimator such that the inequality (3.3) is satis"ed.
Proof of Theorem 3.1. Rewrite the augmented system & with (3.8) as follows: 2 x5 (t)"A(x(t), t)#B(x(t), t)w(t), mQ (t)"A(m(t), t), z(t)"¸(x(t), t),
H
∀t3[0, ¹], tOih, x(0)"0,
& : z( (t)"¸(m(t), t), as m(ih)"m(ih~)#b(m(ih~), ih)[C(x(ih~), ih)#D(x(ih~), ih)v(ih)!C(m(ih~), ih)], z (ih~)"¸ (x(ih~), ih), $ $ z( (ih)"¸ (m(ih~), ih). $ $
(3.9)
De"ning
C D
x8 "
x(t) m(t)
4. A numerical example ,
Consider the following nonlinear system:
the system & can be recast into the following form: !4 5x8 (t)"AI (x8 (t), t)#BI (x8 (t), t)w(t), tOih, x8 (0)"0, x8 (ih)"AI (x8 (ih~), ih)#BI (x8 (ih~), ih)v(ih), ∀ih3(0, ¹), $ $ z8 (t)"z(t)!z( (t)"¸I (x8 (t), t), z8 (ih)"z (ih)!z( (ih)"¸I (x8 (ih~), ih), $ $ $ $
(3.10)
where AI (x8 (t), t), BI (x8 (t), t), AI (x8 (ih~), ih), BI (x8 (ih~), ih), $ $ ¸I (x8 (t), t), and ¸I (x8 (ih~), ih) are de"ned in Theorem 3.1. $ Applying Theorem 2.1, we obtain Theorem 3.1. h Remark 4. For the linear case, it can be shown that for e(x8 (t), t)"(x(t)!m(t))5P(t)(x(t)!m(t)), the Hamilton-Jacobi equation (3.4) reduces to the following Riccati inequality: A (t)5P(t)#P(t)A (t)#P(t)B(0, t)B5(0, t)P(t) 1 1 #¸5 (t)¸ (t)(0, tOih; P(¹)"0 1 1
(3.11)
and similarly, the discrete Hamilton}Jacobi equation (3.5) boils down to the discrete Riccati inequality [P~1(ih~)!¸5 1(ih)¸ 1(ih) d d #C5 (ih)R~1(0, ih)C (ih)]~1!P(ih)'0, 1 1
(3.12)
where [P~1(ih~)!¸5 1(ih)¸ 1(ih)#C5 (ih)R~1(0, ih)C (ih)]'0, d 1 1 d (3.13) A (t)"D A(0 , t) , ¸ (t)"D ¸(0 , t),¸ 1(ih)"D ~ 1 x 1 x d x(ih ) ¸ (0, ih) and C5 (ih)"D ~ C(0, ih). And in Sun et al. 1 x(ih ) $
C D C C D C D
D CD
x5 (t) !8x (t)!15x (t)#sin(x (t)) 1 1 " 1 2 1 # w(t), x5 (t) x (t) 1 2 1
x (ih~) 1 0 1 # v(ih), x (ih~) 0 1 2 z(t)"x (t), 1 z (ih)"x (ih~). (4.1) $ 1 We now apply the approach proposed in Theorem 3.1 to solve the associated nonlinear H estimation problem. = The prescribed level of noise attenuation is assumed to be c"1, and the storage function is chosen to be of the form y(ih)"
e(t, x8 )"(x!m)5P(t)(x!m)
(4.2)
After a lengthly algebraic manipulation, a stationary solution has been found, i.e.,
C
P"
6.2857
D
8.5714
8.5714 17.1429
.
(4.3)
Consequently, the nonlinear estimator is given by x5( (t) !8x( (t)!15x( (t)#sin(x( (t)) 1 " 1 2 1 , tOih, 5x( (t) x( (t) 2 1 x( (ih) x( (ih) 1 " 1 x( (ih) x( (ih) 2 2 0.5000 !0.2500 x( (ih~) # y(ih)! 1 , (4.4) !0.2500 0.1833 x( (ih~) 2 z( (t)"x( (t), tOih, 1 z( (ih)"x( (ih~). d 1
C D C C D C D C
D
DA
C
DB
S.K. Nguang, P. Shi / Automatica 36 (2000) 303}310
309
Fig. 1. Ratio of the estimator error energy to the disturbance energy for both nonlinear H "lter and linear H "lter. = =
In comparison, we also have a linear H "lter corre= sponding to the linearised system (4.1). It can be shown that this estimator is given by x(5 (t) !7x( (t)!15x( (t) 1 " 1 2 , tOih, x(5 (t) x( (t) 2 1 x( (ih) x( (ih) 1 " 1 x( (ih) x( (ih) 2 2 0.5715 !0.2500 x( (ih~) # y(ih)! 1 !0.2500 0.1714 x( (ih~) 2 z( (t)"x( (t), tOih, 1 z( (ih)"x( (ih~). d 1 Fig. 1 shows simulation results for the ratio
C D C C D C D C
D
DA
C
DB
conditions are obtained for the existence of "lters that guarantee the L gain from an exogenous input to an 2 estimation error is less than or equal to a prescribed value. The underlying problem can be solved if certain continuous and discrete Hamilton}Jacobi equations can be solved. This result can be viewed as an extension of Sun et al. (1993) and Shi (1994) which treat linear systems. References
,
(4.5)
Ball, J. A., & Helton, J. W. (1989). H control for nonlinear plants: = Connection with di!erential games. Proceedings of the 28th IEEE Conference Decision Control (pp. 956}962). Ball, J. A., Helton, J. W., & Walker, M. L. (1993). H control for = nonlinear systems with output feedback. IEEE Transactions on Automatic Control, 38, 546}559.
:T(z(t)!z( (t))5(z(t)!z( (t))#+k (z (ih)!z( (ih))5(z (ih)!z( (ih)) dt i/1 $ $ $ $ 0 :Tw5(t)w(t) dt#+k v5(ih)v(ih) i/1 0 obtained by using the nonlinear H estimator (4.4) = and that by using the linear H estimator (4.5) for the = linearised system (4.1). It can be observed that the nonlinear H estimator (4.4) achieves much more preferable = noise attenuation than the linear H estimator (4.5). = 5. Conclusion In this paper the problem of nonlinear H "ltering for = sampled-data systems has been considered. Su$cient
(4.6)
Bamieh, B., & Pearson, J. B. (1992). A general framework for linear periodic systems with application to H sampled-data = control. IEEE Transactions on Automatic Control, 37, 418}435. Bamieh, B., Pearson, J., Francis, B., & Tannenbaum, A. (1991). A lifting technique for linear periodic systems with applications to sampleddata control. Systems and Control Letters, 17, 78}88. Basar, T. (1991). Optimum performance levels for minimax "lters, predictors and smoothers. Systems and Control Letters, 16, 309}317. Basar, T., & Olsder, G. J. (1982). Dynamic Noncooperative Game Theory. New York: Academic Press.
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Hara, S., & Kabamba, P. T. (1990). Worst case analysis and design of sampled-data control systems. Proceedings of 29th IEEE Conference Decision Control, Honolulu, HI (pp. 202}203). Hill, D. J., & Moylan, P. J. (1980). Dissipative dynamical systems: Basic input}output and state properties. Journal of Franklin Institute, 309, 327}357. Isidori, A. (1991). Feedback control of nonlinear systems. Proceedings of the First European Control Conference (pp. 1001}1012). Isidori, A., & Astol", A. (1992). Disturbance attenuation and H = control via measurement feedback in nonlinear systems. IEEE Transactions on Automatic Control, 37, 1283}1293. Lin, W., & Byrnes, C. I. (1995). Discrete-time nonlinear H control with = measurement feedback. Automatica, 31, 419}434. Lin, W., & Byrnes, C. I. (1996). H control of discrete time nonlinear = systems. IEEE Transactions on Automatic Control, 41, 494}510. Nguang, S. K., & Fu, M. (1994). H "ltering for known and uncertain = nonlinear systems. Proceedings of IFAC, Symposium Robust Control Design (pp. 347}352). Nguang, S. K., & Fu, M. (1996). Robust nonlinear H "ltering. = Automatica, 32, 1195}1199. Shi, P. (1994). Issues in robust xltering and control of sampled-data systems. Technical Report. The University of Newcastle (Ph.D Thesis). Shi, P. (1996). Robust "ltering for uncertain systems with sampled measurements. International Journal of Systems Science, 27(12), 1403}1415. Shi, P. (1998). Filtering on sampled-data systems with parametric uncertainty IEEE Transactions on Automatic Control, to appear. Sivashankar, N., & Khargonekar, P. (1994). Characterization of the L -induced norm for linear systems with jumps with application to 2 sampled-data systems. SIAM Journal on Control and Optimisation, 32, 1128}1150. Sun, W., Nagpal, K. M., & Khargonekar, P. P. (1993). H control and = "ltering for sampled-data systems. IEEE Transactions on Automatic Control, 38, 1162}1175. Suzuki, S., Isidori, A., & Tarn, T. J. (1995). S control of nonlinear = systems with sampled measurements. Journal of Mathematical Systems, Estimation, and Control, 5, 1}12. Toivonen, H. T. (1992). Sampled-data control of continuous-time system with an H optimality criterion. Automatica, 28, 45}54. =
van der Schaft, A. J. (1991). A state-space approach to nonlinear H control. Systems and Control Letters, 16, 1}8. = van der Schaft, A. J. (1992). L -gain analysis of nonlinear systems and 2 nonlinear state feedback H control. IEEE Transactions on Auto= matic Control, 37, 770}784. Willems, J. C. (1972). Dissipative dynamical systems Part I: General theory. Archives of Rational Mechanical Analysis, 45, 321}351. Yamamoto, Y. (1991). On the state space and frequency domain characterization of H norm of sampled-data systems. Systems and Control = Letters, 21, 163}172.
Sing Kiong Nguang was born in Sibu, Malaysia. He received the B.E. (Hon. I) and Ph.D. degrees from the Department of Electrical and Computer Engineering, the University of Newcastle, Australia in 1992 and 1995, respectively. Since 1995, he has been with the Department of Electrical and Electronic Engineering, the University of Auckland, New Zealand. His research interests include robust nonlinear control and its applications. Peng Shi received the BS degree in mathematics from Harbin Institute of Technology in 1982, the ME degree in modern control theory and applications from Harbin University of Engineering and Heilongjiang Institute of Applied Mathematics in 1985, the Ph.D. degree in electrical engineering from the University of Newcastle, Australia in 1994. He also has a doctor degree in mathematics from the University of South Australia (UniSA) in 1998. He was a lecturer in applied mathematics at Heilongjiang University, People's Republic of China, from 1985 to 1989. He held a postdoctoral research associate position at Centre for Industrial and Applicable Mathematics in UniSA from 1995 to 1997. Since 1998, he has been a lecturer of mathematics in UniSA. His research interests include robust control and "ltering, sampled-data systems, hybrid systems, markovian jump systems, and singularly perturbed systems.
Brief Paper
Nonlinear H "ltering of sampled-data systemsq = Sing Kiong Nguang!,*, Peng Shi" !Systems and Control Group, Department of Electrical and Electronic Engineering, The University of Auckland, Private Bag 92019, Auckland, New Zealand "Center for Industrial and Applied Mathematics, School of Mathematics, The University of South Australia, The Levels, SA 5095 Australia Received 8 June 1998; revised 1 December 1998; received in "nal form 20 May 1999
Abstract This paper considers the problem of H "ltering for nonlinear sampled-data systems. Su$cient conditions are obtained for the = existence of "lters that guarantee the L gain from an exogenous input to an estimation error is less than or equal to a prescribed 2 value. These conditions are expressed in terms of both continuous and discrete Hamilton}Jacobi equations. ( 1999 Elsevier Science Ltd. All rights reserved. Keywords: Sampled-data systems; H control; Filtering problem; Discrete systems; Estimators =
1. Introduction Past years have seen a numerous industrial systems that are continuous time processes but controlled by a digital controller. In general, there are two classic approaches which have been adopted in engineering practice for the design of a discrete controller. The "rst approach is, "rst designing a continuous controller such that the closed-loop system has satisfactory performances, and then discretising the continuous controller into a discrete controller using one of the discretisation methods. This technique has the advantage that the synthesis is done in continuous time, but it may have a serious degradation in performances when the sampling is not fast enough. The second approach is, discretising the process "rst, then designing a discrete controller. The main bene"t of the approach is that the synthesis procedure is again simpli"ed. However, the discrete models obtained by discretisation fail to represent the intersample behaviour of the systems. The main drawback of
q This paper was presented at the 14th IFAC World Congress Beijing, P.R. China, July 1999. This paper was recommended for publication in revised form by Associate Editor Kenko Uchida under the direction of Editor T. Bas7 ar. * Corresponding author. Tel.: 00-64-9-3737599d8285; fax: 00-64-93737461. E-mail addresses: [email protected] (S.K. Nguang), [email protected] (P. Shi)
these approaches is neither of them o!ers an adequate framework for analysis of the sampled-data system which requires the consideration of both sampled and intersample behaviours. Recently, a number of di!erent approaches have been proposed to incorporate intersample behaviour. These approaches include lifting technique (Bamieh, Pearson, Francis & Tannenbaum, 1991; Toivonen, 1992; Bamieh & Pearson, 1992; Yamamoto, 1991); descriptor system technique (Hara & Kabamba, 1990), and technique based on linear systems with jumps (Shi, 1994, 1996, 1998; Sun, Nagpal & Khargonekar, 1993; Sivashankar & Khargonekar, 1994). The lifting technique consists of transforming the original sampled-data system into an equivalent LTI discrete-time system with in"nite-dimensional input}output signal space. Then L -induced 2 norm of the sampled-data system is shown to be less than one if and only if the H of this equivalent discrete = system is less than one. In the descriptor system approach, on the other hand, the system is represented by a hybrid state-space model, from which the descriptor system is formulated. The solution of the H sampled= data problem is then characterised by the solution of certain associated Hamilton}Jacobi equation. In contrast with these procedures, the theory of linear systems with jumps allows a direct characterisation of the problem in the similar terms to those of standard LTI H control problems, and leads to a pair of Riccati = equations. Despite the procedural di!erences in all these
0005-1098/00/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 1 4 1 - 7
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S.K. Nguang, P. Shi / Automatica 36 (2000) 303}310
approaches, the results obtained are mathematically equivalent. At the same time, the problem of nonlinear H control = problem has been studied by a number of researchers; see for instance (Ball & Helton, 1989; Basar & Olsder, 1982; van der Schaft, 1992; Isidori & Astol", 1992; Isidori, 1991). There are two commonly used approaches for providing solutions to nonlinear H control problems. = One is based on the dissipativity theory and theory of di!erential games (Basar, 1991; Ball & Helton, 1989). Another is based on the nonlinear version of classical Bounded Real Lemma as developed by Willems (1972), and Hill and Moylan (1980); see, e.g., van der Schaft (1991); Isidori (1991) and Isidori and Astol" (1992). Both of these approaches convert the problem of nonlinear H control to the solvability of the so-called Hamil= ton}Jacobi equation (HJE). A nice feature of these results is that they are parallel to the linear H results. Further = research along the line of the dissipativity theory and theory of di!erential games has been attempted (see, e.g., Ball, Helton & Walker, 1993; Isidori and Astol", 1992; Isidori, 1991) where results on disturbance attenuation for nonlinear systems via state feedback and/or output feedback have been provided. In Nguang and Fu (1994,1996); Berman and Shaked (1996) solutions to the nonlinear H "ltering problem have been obtained. = These continuous nonlinear H results have further been = extended to discrete nonlinear systems (Lin & Byrnes, 1995, 1996). In Suzuki, Isidori and Tarn (1995), su$cient conditions for the existence of a solution to the problem of H control nonlinear systems with sampled-data have = been developed. Their su$cient conditions are expressed in terms of both Hamilton}Jacobi equations and Riccati di!erential equation with jump. In the present work, we extend the nonlinear H "lter= ing theory to a nonlinear H estimator for nonlinear = sampled-data systems. This problem can be stated as follows: given a dynamic system with exogenous input and sampled output measurements at discrete instants of time, design a "lter to estimate an unmeasured output such that the L norm of the mapping from the 2 exogenous input to the estimation error is minimised or no larger than some prescribed level. The performance requirements are de"ned directly in terms of the continuous-time signals which means the intersample behaviour is incorporated in the design. Based on a nonlinear version of the classical bounded real lemma theorem for sampled-data systems, su$cient conditions for the existence of a "lter that guarantees the L gain from an 2 exogenous input to an estimation error is less than or equal to a prescribed value are derived. These conditions are expressed in terms of both continuous and discrete Hamilton}Jacobi equations. Since "ltering results are closely related to output feedback results, from Suzuki et al. (1995) it is easy to see that their "ltering condition is expressed in terms of Riccati di!erential equation with
jump. However, our su$cient conditions are expressed in terms of both continuous and discrete Hamilton}Jacobi equations, which are much more general than the "ltering condition given in Suzuki et al. (1995). Another di!erence between our results and their results Suzuki et al. (1995) is the derivation of results. Our results are based on classical bounded real lemma theorem for sampleddata systems, their results Suzuki et al. (1995) are based on dissipativity theory and theory of di!erential games. Our results can also be viewed as an extension of Sun et al. (1993) and Shi (1994), which treat the linear systems. We end this introduction with some remarks on the notations. The notations used in this paper are fairly standard. The superscript t denotes matrix transposition. L [0, ¹] stands for the space of square integrable vector 2 functions over [0, ¹], l (0, ¹) is the space of square 2 summable vector sequences over (0, ¹). f (h~) and f (h`) denote for the left limit and right limit of a function f (h), respectively. The gradient of a scalar-valued function f (m) along the vector m3Rn is a column vector denoted by + f (m), and the gradient of a column vector-valued funcm tion
CD / (m) 1 / (m) 2 F
(1.1)
/ (m) m is a matrix de"ned by
C
L/ (m) 1 Lm 1 L/ (m) 2 D /(m)" Lm 1 m F
L/ (m) 1 Lm 2 L/ (m) 2 Lm 2 F
2 2 }
D
L/ (m) 1 Lm n L/ (m) 2 Lm . n F
(1.2)
L/ (m) L/ (m) L/ (m) m m m 2 Lm Lm Lm n 2 1
2. De5nition and preliminary results
Consider the nonlinear time-varying system with "nite discrete jumps: x5 (t)"A(x(t), t)#B(x(t), t)w(t), t3[0, ¹], tOih; x(0)"0, & : x(ih)"A (x(ih~), ih)#B (x(ih~), ih)v(ih), 1 $ $ z(t)"¸(x(t), t), t3[0, ¹], tOih,
(2.1)
z (ih)"¸ (x(ih~), ih), $ $ where x(t)3Rn is the state, w(t)3Rp and v(ih)3Rq are the continuous and discrete exogenous inputs which belong to L [0, ¹] and l (0, ¹), respectively, z(t)3Rr and 2 2 z (ih)3Rs are the continuous and discrete outputs, $ 0(h3R is the sampling period, i is a positive integer,
S.K. Nguang, P. Shi / Automatica 36 (2000) 303}310
A(x(t), t), B(x(t), t), A (x(ih~), ih), B (x(ih~), ih) ¸ (x(ih~), $ $ $ ih) and ¸(x(t), t) are known real time-varying bounded nonlinear matrix functions of appropriate dimensions with A(x(t), t), B(x(t), t) and ¸(x(t), t) being piecewise continuous.
305
Remark 1. Conditions (2.5) and (2.6) imply that vH(ih) is the maximiser for DHJE(x(ih~), v(ih), ih), i.e., DHJE(x(ih~), v(ih), ih)4DHJE(x(ih~), vH(ih), ih).
(2.7)
Proof of Theorem 2.1. For q3(ih, ih#h), De5nition 1. Suppose c is a given positive real number. A system & of the form (2.1) is said to have L gain less 1 2 than or equal to c if
P
CP
T T k z5(t)z(t) dt# + z5 (ih)z (ih)4c2 w5(t)w(t) dt $ $ 0 0 i/1 k # + v5(ih)v(ih) , i/1 where k is the largest integer such that k(¹.
D
P
q d Me(x, t)N dt"e(x(q), q)!e(x(ih`), ih`). dt ih`
First, let us consider the left-hand side of (2.8):
P P
q d Me(x, t)N dt ` dt ih q [e5 (x(t), t)#+5 e(x(t), t)MA(x(t), t) " x ih`
H(x(t), t)" (2.2)
Note that the left-hand side of (2.2) can be viewed as a &mixed ¸ /l ' norm for the output signals z and z , and 2 2 $ the right-hand side of (2.2) as a &mixed ¸ /l ' norm of the 2 2 input signals which comprise of w and v. In the sequel, without loss of generality, we assume c"1. Theorem 1. Consider system & of the form (2.1). Suppose 1 there exist positive-dexnite functions e(x(t), t) and Q(x(t), t) with e(0, t)"0 and Q(0, t)"0 which satisfy (C1) HJE(x(t), t)Oe5 (x(t), t)#+5 e(x(t), t)A(x(t), t) x #1+5 e(x(t), t)B(x(t), t)B5(x(t), t)+ e(x(t), t) 4 x x #¸5(x(t), t)¸(x(t), t)#Q(x(t), t)"0, (2.3)
#B(x(t), t)w(t)N] dt.
P
H(x(t), t)"
where
q
[e5 (x(t), t)#+5 e(x(t), t)MA(x(t), t) x ih`
#B(x(t), t)w(t)N#w5(t)w(t)!z5(t)z(t) ! w5(t)w(t)#z5(t)z(t)] dt.
(2.10)
Completing the square and knowing that z(t)"¸(x(t), t), one has
P
H(x(t), t)"
q
ih
[e5 (x(t), t)#+5 e(x(t), t)A(x(t), t) x
`
#1+5 e(x(t), t)B(x(t), t)B5(x(t), t)+ e(x(t), t) 4 x x #¸5(x(t), t)¸(x(t), t)!(1B5(x(t), t)+ e(x(t), t)) 2 x
DHJE(x(ih~), vH(ih), ih)Oe(A (x(ih~), ih) $ #B (x(ih~), ih)vH(ih), ih`) $ !e(x(ih~), ih)! v5H(ih)vH(ih) #¸5 (x(ih~), ih)¸ (x(ih~), ih) $ $ #Q(x(ih~), ih)"0, (2.4)
(2.9)
Next adding and subtracting (w5(t)w(t)!z5(t)z(t)) to and from (2.9), it follows that
(C2)
+ e(A (x(ih~), ih)#B (x(ih~), ih)vH(ih), ih`) v $ $ !2vH(ih)"0
(2.8)
!w(t))5(1B5(x(t), t)+ e(x(t), t))!w(t)) 2 x #w5(t)w(t)!z5(t)z(t)] dt.
(2.11)
Employing (2.3), we have
P
H(x(t), t)4
q
[w5(t)w(t)!z5(t)z(t)!Q(x(t), t)] dt. (2.12) ih`
From (2.8) (2.5)
and D + e(A (x(ih~), ih)#B (x(ih~), ih)vH(ih), ih`)!2I(0. v v $ $ (2.6) Then the system & is having L -gain less than or equal 1 2 to 1.
P
q
[w5(t)w(t)!z5(t)z(t)!Q(x(t), t)] dt
ih` 5e(x(q), q)!e(x(ih`), ih`).
(2.13)
Now, let us consider at the sampling instant e(x(t), t)Dih`~"e(x(ih`), ih`)!e(x(ih~), ih~) ih "e(x(ih), ih`)!e(x(ih~), ih).
(2.14)
306
S.K. Nguang, P. Shi / Automatica 36 (2000) 303}310
Next adding and subtracting the term z5 (ih)z (ih)! $ $ v5(ih)v(ih) to and from (2.14), we obtain H (x(ih), ih)"e(x(ih), ih`)!e(x(ih~), ih)#z5 (ih)z (ih) $ $ $ !v5(ih)v(ih)!z5 (ih)z (ih)#v5(ih)v(ih). $ $ (2.15) Knowing that z (ih)"¸ (x(ih~), ih), we have $ $ H (x(ih), ih)"e(A (x(ih~), ih)#B (x(ih~), ih)v(ih), ih`) $ $ $ !e(x(ih~), ih)#¸5 (x(ih~), ih)¸ (x(ih~), ih) $ $ !v5(ih)v(ih)!z5 (ih)z (ih)#v5(ih)v(ih). $ $ (2.16) Using (2.4) and the maximiser vH(ih) given in (2.5) and (2.6), we get H (x(ih), ih)4v5(ih)v(ih)!z5 (ih)z (ih)!Q(x(ih~), ih). $ $ $ (2.17) By combining (2.12) and (2.17) over all possible ih on [0, ¹], one has e(x(¹), ¹)!e(0, 0)
P
T k [w5(t)w(t)!z5(t)z(t)] dt# + v5(ih)v(ih) 0 i/1 T k ! + z5 (ih)z (ih)! Q(x(t), t) dt $ $ 0 i/1 k ! + Q(x(ih~), ih). (2.18) i/1 Knowing that e(0, t)"0 and Q(x(t), t)'0, ∀x(t)O0, we obtain 4
P
P
T k + z5 (ih)z (ih)# z5(t)z(t) dt $ $ 0 i/1 T k 4 w5(t)w(t) dt# + v5(ih)v(ih)!e(x(¹), ¹) 0 i/1 T k 4 w5(t)w(t) dt# + v5(ih)v(ih). (2.19) 0 i/1 This shows that the system & has the L -gain less 1 2 than or equal to 1. To prove the stability, we rearrange (2.18) to
P P
P
T
k Q(x(t), t) dt# + Q(x(ih~), ih) 0 i/1 T k 4 [w5(t)w(t)!z5(t)z(t)] dt# + v5(ih)v(ih) 0 i/1 k (2.20) ! + z5 (ih)z (ih) $ $ i/1
e(x(¹), ¹)!e(0, 0)#
P
Since e(0, 0)"0, we obtain
P
T k Q(x(t), t) dt# + Q(x(ih~), ih) 0 i/1
P
4
T k w5(t)w(t) dt# + v5(ih)v(ih). 0 i/1
(2.21)
This means that lim TC=
CP
D
T k Q(x(t), t) dt# + Q(x(ih~), ih) 0 i/1
4 lim TC=
CP
D
T k w5(t)w(t) dt# + v5(ih)v(ih) (R. 0 i/1
(2.22)
Since Q(x(t), t) is a positive-de"nite function, we conclude that system & is bounded input and bounded state 1 stable. h Remark 2. In the case of a linear system, i.e., linearise (2.1) at the origin: x5 (t)"A (t)x(t)#B(0, t)w(t), tOih; x(0)"0, 1 x(ih)"A 1(ih)x(ih~)#B (0, ih)v(ih), ∀ih3(0, ¹), d $ & l: 1 z(t)"¸ (t)x(t), (2.23) 1 z (ih)"¸ 1(ih)x(ih~), $ d where A (t)"D A(0, t), ¸ (t)"D ¸(0, t), 1 x 1 x A 1(ih)"D ~ A (0, ih) x(ih ) $ d and ¸ 1(ih)"D ~ ¸ (0, ih), d x(ih ) $ it is easy to show that for e(x(t), t)"1x5(t)P(t)x(t), the 2 Hamilton}Jacobi equation (2.3) reduces to the following Riccati inequality: PQ (t)#A5 (t)P(t)#P(t)A (t)#P(t)B(0, t)B5(0, t)P(t) 1 1 #¸5 (t)¸ (t)(0, tOih, 1 1
P(¹)"0
(2.24)
and the discrete Hamilton}Jacobi equation (2.4) boils down to the following discrete Riccati inequality: A5 1(ih)P(ih`)A 1(ih)#A5 1(ih)P(ih`)B (0, ih) d d d $ [I!B (0, ih)P(ih`)B (0, ih)]~1 B5 (0, ih)P(ih`)A 1(ih) $ d $ $ #¸5 1(ih)¸ 1(ih)!P(ih)(0 d d
(2.25)
with I!B (0, ih)P(ih`)B (0, ih)'0. $ $
(2.26)
S.K. Nguang, P. Shi / Automatica 36 (2000) 303}310
In Sun et al. (1993) and Shi (1994), the authors also showed that (2.24) and (2.25) are the necessary conditions for the system & l to be BIBOS with L -gain less than or 1 2 equal to 1.
Consider the following class of nonlinear sampled-data systems:
H
x5 (t)"A(x(t))#B(x(t))w(t),
∀t3[0, ¹], tOih, x(0)"0,
& : 2 z (ih~)"¸ (x(ih~), ih), $ $ y(ih)"C(x(ih~))#D(x(ih~))v(ih),
(3.1)
Problem formulation: Given a scalar c'0, design a nonlinear "lter of the form mQ (t)"a(m(t)), t3[0, ¹], tOih, (3.2)
C D x(t) m(t)
,
C
AI (x8 (t), t)"
A(x(t), t)
D
A(m(t), t))
,
¸I (x8 (t), t)"¸(x(t), t)!¸(m(t), t) and BI t(x8 (t), t)"[B5(x(t), t) 0]. (C2) e(AI (x8 (ih~), ih)#BI (x8 (ih~), ih)vH(ih), ih`) $ $ !e(x8 (ih~), ih)!vtH(ih)vH(ih)
+ e(AI (x8 (ih~), ih)#BI (x8 (ih~), ih)v(ih), ih`)!2vH(ih)"0, v $ $ (3.6) D + e(AI (x8 (ih~), ih)#BI (x8 (ih~), ih)v(ih), ih`)!2I(0, v v $ $ (3.7)
C
such that
x(ih~)m(ih~)
"
D
b(m(ih~), ih)[C(x(ih~), ih)!C(m(ih~), ih)
P
T (z(t)!z( (t))5(z(t)!z( (t)) dt 0
P
for all x8 (t)3R2n, where
AI (x8 (ih~), ih) $
z( (ih)"¸ (m(ih~), ih) $ $
k # + (z (ih)!z( (ih))5(z (ih)!z( (ih)) $ $ $ $ i/1 T k 4 w5(t)w(t) dt# + v5(ih)v(ih) 0 i/1 holds.
(3.4)
where
R(x(ih~), ih)"D(x(ih~), ih)D5(x(ih~), ih)'0.
z( (t)"l(m(t)), ∀t3[0, ¹], tOih,
#¸I 5(x8 (t), t)¸I (x8 (t), t)#Q(x8 (t), t)"0
#¸I 5 (x8 (ih~), ih)¸I (x8 (ih~), ih)#Q(x(ih~), ih)"0, (3.5) $ $
Assumption 1.
m(ih)"m(ih~)#b(m(ih~), ih)[y(ih)!C(m(ih~))],
#1+58 e(x8 (t), t)BI (x8 (t), t)BI 5(x8 (t), t)+ 8 e(x8 (t), t) 4 x x
x8 (t)"
where x(t)3Rn is the state, w(t)3Rp and v(ih)3Rq are the continuous and discrete inputs which belong to L [0, ¹] and l (0, ¹), respectively, z(t)3Rr and 2 2 z (ih)3Rs are the continuous and discrete regulated out$ puts, 0(h3R is the sampling period, i is a positive integer, A(x(t), t), B(x(t), t), C(x(ih~), ih), D(x(ih~), ih), ¸ (x(ih~), ih) and ¸(x(t), t) are known real time-varying $ bounded nonlinear matrix functions of appropriate dimensions with A(x(t), t), B(x(t), t) and ¸(x(t), t) being piecewise continuous.
K:
loop system of & with K satisxes (3.3) if there exist 2 e(x8 , t)'0, Q(x8 , t)'0 and b(m(ih~), ih) such that the following conditions hold: (C1) HJOe5(x8 (t), t)#+58 e(x8 (t), t)AI (x8 (t), t) x
3. Nonlinear H 5ltering =
z(t)"¸(x(t)),
307
,
¸I (x8 (ih~), ih)"¸ (x(ih~), ih)!¸ (m(ih~), ih) and BI 5 (x8 (ih~), $ $ $ $ ih)"[0 b(m(ih~), ih)D(x(ih~), ih)]. If this is the case, then a suitable xlter of the form K is given by mQ (t)"A(m(t), t), (3.3)
Theorem 2. Consider system & satisfying Assumption 3.1. 2 Then, there exists a nonlinear xlter K such that the closed-
t3[0, ¹], tOih, m(0)"0,
m(ih)"m(ih~)#b(m(ih~), ih)[y(ih)!C(m(ih~), ih)], z( (t)"¸(m, t), t3[0, ¹], tOih,
(3.8)
z( (ih)"¸ (m(ih~), ih). $ $ The "lter given above is a nonlinear counterpart of the "lter given in Sun et al. (1993) and Shi (1994).
308
S.K. Nguang, P. Shi / Automatica 36 (2000) 303}310
Remark 3. In general, the optimal choice of b(m(ih~), ih) is still an open problem as pointed out in lin1 for the discrete nonlinear case.
(1993) and Shi (1994), the authors showed that these are the necessary and su$cient conditions for the existence of an estimator such that the inequality (3.3) is satis"ed.
Proof of Theorem 3.1. Rewrite the augmented system & with (3.8) as follows: 2 x5 (t)"A(x(t), t)#B(x(t), t)w(t), mQ (t)"A(m(t), t), z(t)"¸(x(t), t),
H
∀t3[0, ¹], tOih, x(0)"0,
& : z( (t)"¸(m(t), t), as m(ih)"m(ih~)#b(m(ih~), ih)[C(x(ih~), ih)#D(x(ih~), ih)v(ih)!C(m(ih~), ih)], z (ih~)"¸ (x(ih~), ih), $ $ z( (ih)"¸ (m(ih~), ih). $ $
(3.9)
De"ning
C D
x8 "
x(t) m(t)
4. A numerical example ,
Consider the following nonlinear system:
the system & can be recast into the following form: !4 5x8 (t)"AI (x8 (t), t)#BI (x8 (t), t)w(t), tOih, x8 (0)"0, x8 (ih)"AI (x8 (ih~), ih)#BI (x8 (ih~), ih)v(ih), ∀ih3(0, ¹), $ $ z8 (t)"z(t)!z( (t)"¸I (x8 (t), t), z8 (ih)"z (ih)!z( (ih)"¸I (x8 (ih~), ih), $ $ $ $
(3.10)
where AI (x8 (t), t), BI (x8 (t), t), AI (x8 (ih~), ih), BI (x8 (ih~), ih), $ $ ¸I (x8 (t), t), and ¸I (x8 (ih~), ih) are de"ned in Theorem 3.1. $ Applying Theorem 2.1, we obtain Theorem 3.1. h Remark 4. For the linear case, it can be shown that for e(x8 (t), t)"(x(t)!m(t))5P(t)(x(t)!m(t)), the Hamilton-Jacobi equation (3.4) reduces to the following Riccati inequality: A (t)5P(t)#P(t)A (t)#P(t)B(0, t)B5(0, t)P(t) 1 1 #¸5 (t)¸ (t)(0, tOih; P(¹)"0 1 1
(3.11)
and similarly, the discrete Hamilton}Jacobi equation (3.5) boils down to the discrete Riccati inequality [P~1(ih~)!¸5 1(ih)¸ 1(ih) d d #C5 (ih)R~1(0, ih)C (ih)]~1!P(ih)'0, 1 1
(3.12)
where [P~1(ih~)!¸5 1(ih)¸ 1(ih)#C5 (ih)R~1(0, ih)C (ih)]'0, d 1 1 d (3.13) A (t)"D A(0 , t) , ¸ (t)"D ¸(0 , t),¸ 1(ih)"D ~ 1 x 1 x d x(ih ) ¸ (0, ih) and C5 (ih)"D ~ C(0, ih). And in Sun et al. 1 x(ih ) $
C D C C D C D
D CD
x5 (t) !8x (t)!15x (t)#sin(x (t)) 1 1 " 1 2 1 # w(t), x5 (t) x (t) 1 2 1
x (ih~) 1 0 1 # v(ih), x (ih~) 0 1 2 z(t)"x (t), 1 z (ih)"x (ih~). (4.1) $ 1 We now apply the approach proposed in Theorem 3.1 to solve the associated nonlinear H estimation problem. = The prescribed level of noise attenuation is assumed to be c"1, and the storage function is chosen to be of the form y(ih)"
e(t, x8 )"(x!m)5P(t)(x!m)
(4.2)
After a lengthly algebraic manipulation, a stationary solution has been found, i.e.,
C
P"
6.2857
D
8.5714
8.5714 17.1429
.
(4.3)
Consequently, the nonlinear estimator is given by x5( (t) !8x( (t)!15x( (t)#sin(x( (t)) 1 " 1 2 1 , tOih, 5x( (t) x( (t) 2 1 x( (ih) x( (ih) 1 " 1 x( (ih) x( (ih) 2 2 0.5000 !0.2500 x( (ih~) # y(ih)! 1 , (4.4) !0.2500 0.1833 x( (ih~) 2 z( (t)"x( (t), tOih, 1 z( (ih)"x( (ih~). d 1
C D C C D C D C
D
DA
C
DB
S.K. Nguang, P. Shi / Automatica 36 (2000) 303}310
309
Fig. 1. Ratio of the estimator error energy to the disturbance energy for both nonlinear H "lter and linear H "lter. = =
In comparison, we also have a linear H "lter corre= sponding to the linearised system (4.1). It can be shown that this estimator is given by x(5 (t) !7x( (t)!15x( (t) 1 " 1 2 , tOih, x(5 (t) x( (t) 2 1 x( (ih) x( (ih) 1 " 1 x( (ih) x( (ih) 2 2 0.5715 !0.2500 x( (ih~) # y(ih)! 1 !0.2500 0.1714 x( (ih~) 2 z( (t)"x( (t), tOih, 1 z( (ih)"x( (ih~). d 1 Fig. 1 shows simulation results for the ratio
C D C C D C D C
D
DA
C
DB
conditions are obtained for the existence of "lters that guarantee the L gain from an exogenous input to an 2 estimation error is less than or equal to a prescribed value. The underlying problem can be solved if certain continuous and discrete Hamilton}Jacobi equations can be solved. This result can be viewed as an extension of Sun et al. (1993) and Shi (1994) which treat linear systems. References
,
(4.5)
Ball, J. A., & Helton, J. W. (1989). H control for nonlinear plants: = Connection with di!erential games. Proceedings of the 28th IEEE Conference Decision Control (pp. 956}962). Ball, J. A., Helton, J. W., & Walker, M. L. (1993). H control for = nonlinear systems with output feedback. IEEE Transactions on Automatic Control, 38, 546}559.
:T(z(t)!z( (t))5(z(t)!z( (t))#+k (z (ih)!z( (ih))5(z (ih)!z( (ih)) dt i/1 $ $ $ $ 0 :Tw5(t)w(t) dt#+k v5(ih)v(ih) i/1 0 obtained by using the nonlinear H estimator (4.4) = and that by using the linear H estimator (4.5) for the = linearised system (4.1). It can be observed that the nonlinear H estimator (4.4) achieves much more preferable = noise attenuation than the linear H estimator (4.5). = 5. Conclusion In this paper the problem of nonlinear H "ltering for = sampled-data systems has been considered. Su$cient
(4.6)
Bamieh, B., & Pearson, J. B. (1992). A general framework for linear periodic systems with application to H sampled-data = control. IEEE Transactions on Automatic Control, 37, 418}435. Bamieh, B., Pearson, J., Francis, B., & Tannenbaum, A. (1991). A lifting technique for linear periodic systems with applications to sampleddata control. Systems and Control Letters, 17, 78}88. Basar, T. (1991). Optimum performance levels for minimax "lters, predictors and smoothers. Systems and Control Letters, 16, 309}317. Basar, T., & Olsder, G. J. (1982). Dynamic Noncooperative Game Theory. New York: Academic Press.
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S.K. Nguang, P. Shi / Automatica 36 (2000) 303}310
Hara, S., & Kabamba, P. T. (1990). Worst case analysis and design of sampled-data control systems. Proceedings of 29th IEEE Conference Decision Control, Honolulu, HI (pp. 202}203). Hill, D. J., & Moylan, P. J. (1980). Dissipative dynamical systems: Basic input}output and state properties. Journal of Franklin Institute, 309, 327}357. Isidori, A. (1991). Feedback control of nonlinear systems. Proceedings of the First European Control Conference (pp. 1001}1012). Isidori, A., & Astol", A. (1992). Disturbance attenuation and H = control via measurement feedback in nonlinear systems. IEEE Transactions on Automatic Control, 37, 1283}1293. Lin, W., & Byrnes, C. I. (1995). Discrete-time nonlinear H control with = measurement feedback. Automatica, 31, 419}434. Lin, W., & Byrnes, C. I. (1996). H control of discrete time nonlinear = systems. IEEE Transactions on Automatic Control, 41, 494}510. Nguang, S. K., & Fu, M. (1994). H "ltering for known and uncertain = nonlinear systems. Proceedings of IFAC, Symposium Robust Control Design (pp. 347}352). Nguang, S. K., & Fu, M. (1996). Robust nonlinear H "ltering. = Automatica, 32, 1195}1199. Shi, P. (1994). Issues in robust xltering and control of sampled-data systems. Technical Report. The University of Newcastle (Ph.D Thesis). Shi, P. (1996). Robust "ltering for uncertain systems with sampled measurements. International Journal of Systems Science, 27(12), 1403}1415. Shi, P. (1998). Filtering on sampled-data systems with parametric uncertainty IEEE Transactions on Automatic Control, to appear. Sivashankar, N., & Khargonekar, P. (1994). Characterization of the L -induced norm for linear systems with jumps with application to 2 sampled-data systems. SIAM Journal on Control and Optimisation, 32, 1128}1150. Sun, W., Nagpal, K. M., & Khargonekar, P. P. (1993). H control and = "ltering for sampled-data systems. IEEE Transactions on Automatic Control, 38, 1162}1175. Suzuki, S., Isidori, A., & Tarn, T. J. (1995). S control of nonlinear = systems with sampled measurements. Journal of Mathematical Systems, Estimation, and Control, 5, 1}12. Toivonen, H. T. (1992). Sampled-data control of continuous-time system with an H optimality criterion. Automatica, 28, 45}54. =
van der Schaft, A. J. (1991). A state-space approach to nonlinear H control. Systems and Control Letters, 16, 1}8. = van der Schaft, A. J. (1992). L -gain analysis of nonlinear systems and 2 nonlinear state feedback H control. IEEE Transactions on Auto= matic Control, 37, 770}784. Willems, J. C. (1972). Dissipative dynamical systems Part I: General theory. Archives of Rational Mechanical Analysis, 45, 321}351. Yamamoto, Y. (1991). On the state space and frequency domain characterization of H norm of sampled-data systems. Systems and Control = Letters, 21, 163}172.
Sing Kiong Nguang was born in Sibu, Malaysia. He received the B.E. (Hon. I) and Ph.D. degrees from the Department of Electrical and Computer Engineering, the University of Newcastle, Australia in 1992 and 1995, respectively. Since 1995, he has been with the Department of Electrical and Electronic Engineering, the University of Auckland, New Zealand. His research interests include robust nonlinear control and its applications. Peng Shi received the BS degree in mathematics from Harbin Institute of Technology in 1982, the ME degree in modern control theory and applications from Harbin University of Engineering and Heilongjiang Institute of Applied Mathematics in 1985, the Ph.D. degree in electrical engineering from the University of Newcastle, Australia in 1994. He also has a doctor degree in mathematics from the University of South Australia (UniSA) in 1998. He was a lecturer in applied mathematics at Heilongjiang University, People's Republic of China, from 1985 to 1989. He held a postdoctoral research associate position at Centre for Industrial and Applicable Mathematics in UniSA from 1995 to 1997. Since 1998, he has been a lecturer of mathematics in UniSA. His research interests include robust control and "ltering, sampled-data systems, hybrid systems, markovian jump systems, and singularly perturbed systems.