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On designing filters for uncertain sampled-data nonlinear systems
Systems & Control Letters 41 (2000) 305–316
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On designing ÿlters for uncertain sampled-data nonlinear systems Sing Kiong Nguanga; ∗ , Peng Shib a Department
of Electrical and Electronic Engineering, Systems and Control Group, University of Auckland, Private Bag 92019, Auckland, New Zealand b Land Operations Division, Defence Science & Technology Organisation, P.O. Box 1500, Salibury, SA 5108, Australia Received 9 November 1998; received in revised form 16 May 2000; accepted 22 June 2000
Abstract This paper is concerned with the problem of nonlinear H∞ ÿltering for sampled-data systems with nonlinear time-varying parameter uncertainty. The aim is to design a digital ÿlter such that the ratio between the energy of the estimation errors and the energy of the exogenous inputs is minimised or guaranteed to be less or equal to a prescribed value for all admissible uncertainties. A nonlinear bounded real lemma for sampled-data systems with nonlinear time-varying parameter uncertainty is provided. Based on this nonlinear bounded real lemma, the robust H∞ ÿltering problem is solved in terms c 2000 Elsevier Science B.V. All rights reserved. of both continuous and discrete Hamilton–Jacobi equations. Keywords: Nonlinear systems; Sampled-data; Filter
1. Introduction Recently, a number of various techniques have been proposed to handle linear sampled-data systems. The techniques are: (1) lifting technique [1,2,11,12,4], consists of transforming the original sampled-data system into an equivalent LTI discrete-time system with inÿnite-dimensional input–output signal space. The L2 -induced norm of the sampled-data system is shown to be ¡ 1 if and only if the H∞ norm of this equivalent discrete system is ¡ 1, (2) descriptor system technique [3], 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, and (3) technique based on linear systems with jumps [6 –10], 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. Recently in [5], the nonlinear H∞ ÿltering theory [4] has been used to solve the ÿltering problem for nonlinear sampled-data systems. We have obtained some sucient conditions for the existence of a digital ÿlter that guarantee the L2 gain from an exogenous input to an estimation error is less than or equal to a prescribed value. These conditions are expressed in terms of both continuous and discrete Hamilton–Jacobi ∗ Corresponding author. Tel.: 0064-3737-599#8285; fax: 0064-9-3737-461. E-mail addresses: [email protected] (S.K. Nguang), [email protected] (P. Shi).
c 2000 Elsevier Science B.V. All rights reserved. 0167-6911/00/$ - see front matter PII: S 0 1 6 7 - 6 9 1 1 ( 0 0 ) 0 0 0 6 4 - 5
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equations. It has been shown that this result can be viewed as an extension of [6 –10] which treat linear systems. In the present work, we consider nonlinear sampled-data systems subject to uncertainties that are described by an integral functional constraint and input disturbance. The problem addressed here is to design a nonlinear digital ÿlter, such that the ratio between the energy of the estimation errors and the energy of the erogenous inputs is minimised or guaranteed to be less or equal to a prescribed value for all admissible uncertainties. We ÿrst provide a nonlinear bounded real lemma for sampled-data systems with nonlinear time-varying parameter uncertainty. Then based on this nonlinear bounded real lemma for sampled-data systems with nonlinear time-varying parameter uncertainty, a complete solution to the robust nonlinear H∞ ÿltering problem is provided in terms of both continuous and discrete Hamilton–Jacobi equations. This result can be viewed as a generalisation of robust linear H∞ ÿltering results in [6] to a class of nonlinear sampled-data systems with uncertainties. 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. L2 [0; T ] stands for the space of square integrable vector functions over [0; T ]; l2 (0; T ) is the space of square summable vector sequences over (0; T ). f( − ) and f( + ) denote for the left limit and right limit of a function f(), respectively. The gradient of a scalar-valued function f() along the vector ∈ Rn is a column vector denoted by f(), and the gradient of a column vector-valued function 1 () 2 () .. . m ()
is a matrix deÿned by @ 1 () @ 1 () @ 1 () ··· @1 @ @n 2 @ () @ () @ 2 () 2 2 ··· @1 @2 @n D () = . . .. . .. .. .. . @ m () @ m () @ m () ··· @1 @2 @n
:
2. Nonlinear bounded real lemma Consider the nonlinear time-varying system with ÿnite discrete jumps: x(t) ˙ = A(x(t); t) + A(x(t); t) + B(x(t); t)w(t); 1 :
t ∈ [0; T ]; t 6= ih; x(0) = 0;
x(ih) = Ad (x(ih− ); ih) + Ad (x(ih− ); ih) + Bd (x(ih− ); ih)v(ih); z(t) = L(x(t); t);
t ∈ [0; T ]; t 6= ih;
(2.1)
zd (ih) = Ld (x(ih− ); ih); where x(t) ∈ Rn is the state, w(t) ∈ Rp and v(ih) ∈ Rq are the continuous and discrete exogenous inputs which belong to L2 [0; T ] and l2 (0; T ), respectively, z(t) ∈ Rr and zd (ih) ∈ Rs are the continuous and discrete outputs, 0 ¡ h ∈ R is the sampling period, i is a positive integer, A(x(t); t); B(x(t); t); Ad (x(ih− ); ih); Bd (x(ih− ); ih), Ld (x(ih− ); ih) and L(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 L(x(t); t) being piecewise continuous. A(x; t) and Ad (x(ih− ); ih) represent the uncertainties in the system.
S.K. Nguang, P. Shi / Systems & Control Letters 41 (2000) 305–316
307
Assumption 2.1. A(x(t); t) = H (x(t); t)F(x(t); t)E(x(t); t); Ad (x(ih− ); ih) = Hd (x(ih− ); ih)Fd (x(ih− ); ih)Ed (x(ih− ); ih);
(2.2)
where H (x(t); t); Hd (x(ih− ); ih); Ed (x(ih− ); ih) and E(x(t); t) are known matrix functions which characterize the structure of the uncertainties with E(0; t) = 0 and Ed (0; ih) = 0. Furthermore, the following inequalities hold: kE(x(t); t)k2 ¿kF(x(t); t)E(x(t); t)k2
(2.3)
kEd (x(ih− ); ih)k2 ¿Fd (x(ih− ); ih)E(x(ih− ); ih)k2 :
(2.4)
and
Deÿnition 2.1. Suppose is a given positive real number. A system 1 of the form (2.1) is said to have L2 gain less than or equal to , if for all admissible uncertainties # "Z Z T k k T X X t t 2 t t z (t)z(t) dt + zd (ih)zd (ih)6 w (t)w(t) dt + v (ih)v(ih) ; (2.5) 0
i=1
0
i=1
where k is the largest integer such that k ¡ T . Note that the left-hand side of (2.5) can be viewd as a ‘mixed L2 =l2 ’ norm for the output signals z and zd , and the right-hand side of (2.5) as a ‘mixed L2 =l2 ’ norm of the input signals which comprise of w and v. In the sequel, without loss of generality, we assume = 1. The following is a nonlinear bounded real lemma for sampled-data systems with nonlinear time-varying parameter uncertainty. Theorem 2.1. Consider the system 1 of the form (2:1) satisfying Assumption 2:1. Given a positive constant . Suppose there exist positive-deÿnite functions (x(t); t) and Q(x(t); t) with (0; t) = 0 and Q(0; t) = 0 such that for all x ∈ Rn the following conditions hold: (C1) HJE(x(t); t) , (x(t); ˙ t) + xt (x(t); t)A(x(t); t) t 1 1 1 + xt (x(t); t) B(x(t); t) H (x(t); t) B(x(t); t) H (x(t); t) x (x(t); t) 4 +Lt (x(t); t)L(x(t); t) + 2 E t (x(t); t)E(x(t); t) + Q(x(t); t) = 0:
(2.6)
(C2) DHJE(x (ih− ); va∗ (ih); ih) , (Ad (x(ih− ); ih) + [Bd (x(ih− ); ih) Hd (x(ih− ); ih)]va∗ (ih); ih+ ) − (x(ih− ); ih) − vat∗ (ih)va∗ (ih) + Ltd (x(ih− ); ih)Ld (x(ih− ); ih) + 2 Edt (x(ih− ); ih)Ed (x(ih− ); ih) + Q(x(ih− ); ih) = 0
(2.7)
where va∗ (ih) satisfying v (Ad (x(ih− ); ih) + [Bd (x(ih− ); ih) Hd (x(ih− ); ih)]va∗ (ih); ih+ ) − 2va∗ (ih) = 0
(2.8)
Dv v (Ad (x(ih− ); ih) + [Bd (x(ih− ); ih) + Hd (x(ih− ); ih)]va∗ (ih); ih+ ) − 2I ¡ 0:
(2.9)
and
Then the system 1 is having L2 -gain less than or equal to 1.
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Remark 2.1. Conditions (2.8) and (2.9) imply that va∗ (ih) is the maximiser for DHJE(x(ih− ); va (ih); ih), i.e., DHJE(x(ih− ); va (ih); ih)6DHJE(x(ih− ); va∗ (ih); ih) where
v(ih)
(2.10)
: va (ih) = 1 Fd (x(ih− ); ih)Ed (x(ih− ); ih) Proof of Theorem 2.1. For ∈ (ih; ih + h), Z d {(x; t)} dt = (x( ); ) − (x(ih+ ); ih+ ): ih+ dt First, let us consider and denote the left-hand side of (2.12) as Z d {(x; t)} dt (x(t); t) = dt + ih Z [(x(t); ˙ t) + xt (x(t); t){A(x(t); t) + A(x(t); t) + B(x(t); t)w(t)}] dt: = ih+
(2.11)
(2.12)
(2.13)
Next adding and subtracting (wt (t)w(t) − z t (t)z(t)) to and from (2.13), it follows that Z [(x(t); ˙ t) + xt (x(t); t){A(x(t); t) + A(x(t); t) + B(x(t); t)w(t)} (x(t); t) = ih+
+ wt (t)w(t) − z t (t)z(t) − wt (t)w(t) + z t (t)z(t)] dt:
(2.14)
Note that z(t) = L(x(t); t). Using Assumption 2.1 and completing the square, one has Z 1 1 (x(t); ˙ t) + xt (x(t); t)A(x(t); t) + xt (x(t); t) B(x(t); t) H (x(t); t) (x(t); t) = 4 ih+ t t 1 t 1 H (x(t); t)x (x(t); t) − F(x(t); t)E(x(t); t) × B(x(t); t) H (x(t); t) x (x(t); t) − 2 1 t × H (x(t); t)x (x(t); t) − F(x(t); t)E(x(t); t) + 2 E t (x(t); t)F t (x(t); t)F(x(t); t)E(x(t); t) 2 t 1 t 1 t t B (x(t); t)x (x(t); t)) − w(t) B (x(t); t)x (x(t); t)) − w(t) +L (x(t); t)L(x(t); t) − 2 2 + wt (t)w(t) − z t (t)z(t)] dt;
(2.15)
where ¿ 0. Employing (2.6), we have Z [wt (t)w(t) − z t (t)z(t) − Q(x(t); t) − 2 (kE(x(t); t)k2 − kF(x(t); t)E(x(t); t)k2 )] dt: (x(t); t)6 ih+
(2.16)
Following from (2.12) and (2.16), we obtain Z [wt (t)w(t) − z t (t)z(t) − Q(x(t); t) − 2 (kE(x(t); t)k2 − kF(x(t); t)E(x(t); t)k2 )] dt ih+
¿(x( ); ) − (x(ih+ ); ih+ ):
(2.17)
Now let us consider at the sampling instant +
+ + − − (x(t); t)|ih ih− = (x(ih ); ih ) − (x(ih ); ih )
= (x(ih); ih+ ) − (x(ih− ); ih):
(2.18)
S.K. Nguang, P. Shi / Systems & Control Letters 41 (2000) 305–316
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Next adding and subtracting the term zdt (ih)zd (ih) − vat (ih)va (ih) to and from (2.18), we obtain d (x(ih); ih) = (x(ih); ih+ ) − (x(ih− ); ih) + zdt (ih)zd (ih) − vat (ih)va (ih) − zdt (ih)zd (ih) + vat (ih)va (ih);
(2.19)
where va (ih) is given in (2.11). Knowing that zd (ih) = Ld (x(ih− ); ih) and using Assumption 2.1, we have 1 d (x(ih); ih) = Ad (x(ih− ); ih) + Bd (x(ih− ); ih) Hd va (ih); ih+ − (x(ih− ); ih) + Ltd (x(ih− ); ih)Ld (x(ih− ); ih) − vat (ih)va (ih) − zdt (ih)zd (ih) + vat (ih)v(ih):
(2.20)
Using (2.7) and the maximiser va∗ (ih) given in (2.8) and (2.9), we get d (x(ih); ih)6vat (ih)va (ih) − zdt (ih)zd (ih) − Q(x(ih− ); ih) − 2 kEdt (x(ih− ); ih)k2 :
(2.21)
Utilising (2.11), we obtain d (x(ih); ih) 6 vt (ih)v(ih) − zdt (ih)zd (ih) − Q(x(ih− ); ih)
− 2 (kEdt (x(ih− ); ih)k2 − kFd (x(ih− ); ih)Ed (x(ih− ); ih)k2 ):
(2.22)
By combining (2.16) and (2.22) over all possible t on [0; T ], one has Z T k X [wt (t)w(t) − z t (t)z(t)] dt + vt (ih)v(ih) (x(T ); T ) − (0; 0) 6 0
−
k X
zdt (ih)zd (ih) −
Z 0
i=1
− 2 − 2
Z
T
0 k X
i=1 T
Q(x(t); t) dt −
k X
Q(x(ih− ); ih)
i=1
{kEdt (x(ih− ); ih)k2 − kFd (x(ih−1 ); ih)Ed (x(ih− ); ih)k2 } dt {kEdt (x(ih− ); ih)k2 − kFd (x(ih−1 ); ih)Ed (x(ih− ); ih)k2 }:
(2.23)
i=1
Using Assumption 2.1 and knowing that (0; t) = 0 and Q(x(t); t) ¿ 0, ∀x(t) 6= 0, we obtain Z T Z T k k X X zdt (ih)zd (ih) + z t (t)z(t) dt 6 wt (t)w(t) dt + vt (ih)v(ih) − (x(T ); T ) i=1
0
0
Z 6
i=1 T
wt (t)w(t) dt +
0
k X
vt (ih)v(ih):
(2.24)
i=1
This shows that the system 1 has the L2 -gain less than or equal to 1. To prove the stability, we rearrange (2.23) to Z T Z T k X Q(x(t); t) dt + Q(x(ih− ); ih) 6 [wt (t)w(t) − z t (t)z(t)] dt (x(T ); T ) − (0; 0) + 0
0
i=1
+
k X
vt (ih)v(ih) −
i=1
Since (0; 0) = 0, we obtain Z Z T k X Q(x(t); t) dt + Q(x(ih− ); ih)6 0
i=1
0
T
wt (t)w(t) dt +
k X i=1
vt (ih)v(ih):
k X
zdt (ih)zd (ih): (2.25)
i=1
(2.26)
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S.K. Nguang, P. Shi / Systems & Control Letters 41 (2000) 305–316
This means that "Z "Z # k T X − Q(x(t); t) dt + Q(x(ih ); ih) 6 lim lim T 7→∞
0
T 7→∞
i=1
0
T
t
w (t)w(t) dt +
k X
# t
v (ih)v(ih) ¡ ∞:
(2.27)
i=1
Since Q(x(t); t) is a positive-deÿnite function, we conclude that the system 1 is bounded input and bounded state stable. Remark 2.2. In the case of a linear system without uncertain, i.e. linearise (2.1) at the origin with A(x(t); t)= 0 and Ad (x(ih− ); ih) = 0: x(t) ˙ = A1 (t)x(t) + B(0; t)w(t);
t 6= ih; x(0) = 0;
−
1l :
x(ih) = Ad1 (ih)x(ih ) + Bd (0; ih)v(ih);
∀ih ∈ (0; T );
z(t) = L1 (t)x(t);
(2.28)
zd (ih) = Ld1 (ih)x(ih−1 ); where A1 (t) = Dx A(0; t), L1 (t) = Dx L(0; t), Ad1 (ih) = Dx(ih− ) Ad (0; ih) and Ld1 (ih) = Dx(ih− ) Ld (0; ih). It is easy to show that for (x(t); t) = 12 xt (t)P(t)x(t), the Hamilton–Jacobi equation (2.6) reduces to the following Riccati inequality: ˙ + At1 (t)P(t) + P(t)A1 (t) + P(t)B(0; t)Bt (0; t)P(t) + Lt1 (t)L1 (t) ¡ 0; t 6= ih; P(T ) = 0 P(t) (2.29) and the discrete Hamilton–Jacobi equation (2.7) boils down to the following discrete Riccati inequality: Atd1 (ih)P(ih+ )Ad1 (ih) + Atd1 (ih)P(ih+ )Bd (0; ih)[I − Bd (0; ih)P(ih+ )Bd (0; ih)]−1 ×Bdt (0; ih)P(ih+ )Ad1 (ih) + Ltd1 (ih)Ld1 (ih) − P(ih) ¡ 0
(2.30)
with I − Bd (0; ih)P(ih+ )Bd (0; ih) ¿ 0:
(2.31)
In [10,6], the authors also showed that (2.29) and (2.30) are the necessary conditions for the system 1l to be BIBOS with L2 -gain less than or equal to 1. 3. Robust nonlinear H∞ ÿltering Consider the following class of uncertain nonlinear sampled-data systems: ) x(t) ˙ = A(x(t); t) + A(x(t); t) + B(x(t); t)w(t); ∀t ∈ [0; T ]; t 6= ih; x(0) = 0; z(t) = L(x(t); t); : zd (ih−1 ) = Ld (x(ih−1 ); ih);
(3.1)
y(ih) = C(x(ih−1 ); ih) + C(x(ih−1 ); ih) + D(x(ih−1 ); ih)v(ih); where x(t) ∈ Rn is the state, w(t) ∈ Rp and v(ih) ∈ Rq are the continuous and discrete inputs which belong to L2 [0; T ] and ‘2 (0; T ), respectively, z(t) ∈ Rr and zd (ih) ∈ Rs are the continuous and discrete regulated outputs, 0 ¡ h ∈ R 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), Ld (x(ih− ); ih) and L(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 L(x(t); t) being piecewise continuous. A(x; t) and C(x(ih− ); ih) represent the uncertainties in the system. Assumption 3.1. A(x(t); t) = H (x(t); t)F(x(t); t)E(x(t); t); C(x(ih− ); ih) = Hd (x(ih− ); ih)Fd (x(ih− ); ih)Ed (x(ih− ); ih); R(x(ih− ); ih) = [D(x(ih− ); ih) Hd (x(ih− 1); ih)][D(x(ih− ); ih) Hd (x(ih− ); ih)]t ¿ 0;
(3.2)
S.K. Nguang, P. Shi / Systems & Control Letters 41 (2000) 305–316
311
where H (x(t); t); Hd (x(ih− ); ih); Ed (x(ih− ); ih) and E(x(t); t) are known matrix functions which characterise the structure of the uncertainties with E(0; t) = 0 and Ed (0; ih) = 0. Further, the following inequalities hold: kE(x(t); t)k2 ¿kF(x(t); t)E(x(t); t)k2
(3.3)
kEd (x(ih− ); ih)k2 ¿kFd (x(ih− ); ih)E(x(ih− ); ih)k2 :
(3.4)
and
Robust nonlinear H∞ ÿltering problem: Given a scalar ¿ 0, design a nonlinear digital ÿlter of the form ˙ = a((t); t); (t)
t ∈ [0; T ]; t 6= ih;
(ih) = (ih− ) + b((ih− ); ih)[y(ih) − C(x(ih− ); ih)];
K:
z(t) ˆ = l((t); t)
(3.5)
∀t ∈ [0; T ]; t 6= ih;
zˆd (ih) = ld ((ih− ); ih); ˆ ∈ Rp and zˆd (ih) ∈ Rq are the estimate of z(t) and zd (ih), respecwhere (t) ∈ Rn is the state of the ÿlter, z(t) − tively, a((t); t); b((ih ); ih); l((t); t) and ld ((ih− ); ih) are matrix functions with appropriate dimensions, a(0; t) = 0; l(0; t) = 0 and ld (0; ih) = 0. The objective is such that the L2 gain from the disturbances w(t) and v(t) to the estimation errors z(t) − z(t) ˆ and zd (ih) − zˆd (ih) for the augmented system with K is less than or equal to , i.e. for all T ¿0, all w(t) ∈ L2 [0; T ], all v(ih) ∈ ‘2 (0; T ) and all admissible uncertainties Z
T 0
(z(t) − z(t)) ˆ t (z(t) − z(t)) ˆ dt +
(zd (ih) − zˆd (ih))t (zd (ih) − zˆd (ih))
i=1
Z 6
k X
T
wt (t)w(t) dt +
0
k X
vt (ih)v(ih);
(3.6)
i=1
where k is the largest integer such that k ¡ T . Remark 3.1. The performance requirement given in (3.6) is deÿned directly in terms of the continuous-time signals which means the intersample behaviour is incorporated in the design. In the sequel, without loss of generality, we assume = 1. Theorem 3.1. Consider the system satisfying Assumption 2:1. Given a scaling parameter ¿ 0; suppose there exist nonnegative scalar functions (x; ˜ t) and Q(x(t); ˜ t); and a matrix function b((ih− ); ih) such that 2n for all x(t) ˜ ∈ R the following conditions hold: (C1) t x(t); x(t); ˜ t)A( ˜ t) + 14 xt˜(x(t); ˜ t)B( ˜ t)B (x(t); ˜ t)x˜(x(t); ˜ t) HJ , ( ˙ x(t); ˜ t) + xt˜(x(t); t
t
x(t); x(t); ˜ t)L( ˜ t) + 2 E (x(t); ˜ t)E( ˜ t) + Q(x(t); ˜ t) = 0; + L (x(t); where
x(t) x(t) ˜ = ; (t)
x(t); A( ˜ t) =
B((t); t) = [B((t); t)
(3.7)
A(x(t); t) ; A((t); t) + 12 B((t); t)Bt ((t); t)x (; )
1 H ((t); t)];
x(t); L( ˜ t) = L(x(t); t) − L((t); t);
E(x(t); t) E(x(t); ˜ t) = 0
312
and
S.K. Nguang, P. Shi / Systems & Control Letters 41 (2000) 305–316
B((t); t) x(t); B( ˜ t) = : 0
(C2) ˜ − ); ih) + B d (x(ih ˜ − ); ih)va∗ (ih); ih+ ) − (x(ih ˜ − ); ih) − va∗t (ih)va∗ (ih) (A d (x(ih t
t
˜ − ); ih)Ld (x(ih ˜ − ); ih) + Q(x(ih ˜ − ); ih) + 2 E d (x(ih ˜ − ); ih)E d (x(ih ˜ − ); ih) = 0; + Ld (x(ih
(3.8)
where va∗ (ih) satisfying ˜ − ); ih) + B d (x(ih ˜ − ); ih)va∗ (ih); ih+ ) − 2va∗ (ih) = 0; va (A d (x(ih
(3.9)
˜ − ); ih) + B d (x(ih ˜ − ); ih)va∗ (ih); ih+ ) − 2I ¡ 0; Dv v (A d (x(ih
(3.10)
x(ih− ) ; (ih− ) + b((ih− ); ih)[C(x(ih− ); ih) − C((ih− ); ih)] Ed (x(ih− ); ih) ˜ − ); ih) = Ld (x(ih− ); ih) − Ld ((ih− ); ih); E d (x(ih ˜ − ); ih) = Ld (x(ih 0
˜ − ); ih) = A d (x(ih
and ˜ − ); ih) = B d (x(ih
0 : b((ih− ); ih)[D(x(ih− ); ih) 1 Hd (x(ih− ); ih)]
If this is the case; then there exists a ÿlter of the form ˙ = A((t); t) + 1 B((t); t)Bt ((t); t)x (; ); (t) 2
t ∈ [0; T ]; t 6= ih;
(ih) = (ih− ) + b((ih− ); ih)[y(ih) − [C(x(ih− ); ih)]; z(t) ˆ = L((t); t)
t ∈ [0; T ]; t 6= ih;
zˆd (ih) = Ld ((ih− ); ih)
(3.11)
which will render (3:6) for all x˜ ∈ R2n . Proof. The augmented system with (3.11) as follows:
t 1 ˙ (t) = A((t); t) + 2 B((t); t)B ((t); t)x (; ); x(t) ˙ = A(x(t); t) + A(x(t); t) + B(x(t); t)w(t);
z(t) = L(x(t); t);
∀t ∈ [0T ]; t 6= ih; x(0) = 0;
z(t) ˆ = L((t); t); (ih) = (ih− ) + b((ih− ); ih)[C(x(ih− ); ih) + C(x(ih− ); ih) + D(x(ih− ); ih)v(ih)
as :
− C((ih− ); ih)]; zd (ih− ) = Ld (x(ih− ); ih); zˆd (ih) = Ld ((ih− ); ih): Deÿning
x(t) x˜ = ; (t)
(3.12)
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313
the system as can be recast into the following form: x(t); x(t); x(t); x(t) ˜˙ = A( ˜ t) + A( ˜ t) + B( ˜ t)w(t);
t 6= ih; x(0) ˜ = 0;
˜ − ); ih) + A d (x(ih ˜ − ); ih) + B d (x(ih ˜ − ); ih)v(ih); x(ih) ˜ = A d (x(ih " # x(t); L( ˜ t) z(t) = z(t) − z(t) ˆ = ; E(x(t); # " t) ˜ − ); ih) Ld (x(ih ; zd (ih) = zd (ih) − zˆd (ih) = Ed ((ih− ); ih− ) where x(t); A( ˜ t) =
A(x(t); t) ; 0
˜ − ); ih) = A d (x(ih
∀ih ∈ (0; T ); (3.13)
0 ; C(x(ih− ); ih
x(t); x(t); x(t); ˜ − ); ih); B d (x(ih ˜ − ); ih); L( ˜ t), and Ld (x(ih ˜ − ); ih) are deÿned in Theorem 3.1. A( ˜ t); B( ˜ t); A d (x(ih x(t); Before applying Theorem 2.1, we need to show that A( ˜ t) and A d (x(ih ˜ − ); ih) satisfying Assumption − x(t); ˜ ); ih) can be rewritten as 2.1. Using Assumption 3.1, A( ˜ t) and A d (x(ih H (x(t); t) x(t); A( ˜ t) = F(x(t); t)E(x(t); t) (3.14) 0 and ˜ − ); ih) = A d (x(ih
0 Fd (x(ih− ); ih)Ed (x(ih− ); ih): Hd (x(ih− ); ih)
(3.15)
By Assumption 3.1, clearly one can see that Assumption 2.1 is satisÿed. Now, applying Theorem 2.1, we obtain Theorem 3.1 Remark 3.2. In the linear case, the optimal b(0; t) = S(ih)C t (t). If we choose (x(t); (t)) = xt (t)P(t)x(t) + (x(t) − (t))t (Q(t) − P(t))(x(t) − (t)) and linearise the Hamilton–Jacobi inequality (3.7), we will have LP − LQ LQ x(t) ¡ 0; t 6= ih; (3.16) [x(t) (t)]t LP − LQ LQ − LP (t) where t
˙ + At1 (t)P(t) + P(t)A1 (t) + P(t)B(0; t)B (0; t)P(t) + 2 E1t (t)E1 (t); LP = P(t)
(3.17)
˙ + A 1 (t)Q(t) + Q(t)A t1 (t) + Q(t)L1 (t)t L1 (t)Q(t) + B(0; t)B t (0; t): LQ = Q(t)
(3.18)
Condition (3.16) implies that LP ¡ 0 and LQ ¡ 0. Similarly, the discrete Hamilton–Jacobi equation (3.8) boils down to LPd − LQd x(ih) t LQd ¡ 0; (3.19) [x(ih) (ih)] LPd − LQd LQd − LPd (ih) where
(3.20)
(3.21)
LPd (ih) = P(ih) + Edt (ih− )Ed (ih− ) − P(ih− ); LQd (ih) = [Q−1 (ih− ) − Ltd1 (ih− )Ld1 (ih− ) + C1t (ih− )R−1 (0; ih)C1 (ih− )]−1 − Q(ih):
Eq. (3.18) implies that LPd ¡ 0 and LQd ¡ 0. A1 (t) = Dx A(0; t); L1 (t) = Dx L1 (0; t); Ld1 (ih) = Dx(ih− ) Ld (0; ih); t)B t (0; t)P t) = [B(0; t) (1= )H (0; t)]; E1t (t) = Dx(t) E(0; t); Edt (ih− ) = Dx(ih− ) Ed1 (0; ih); A 1 (t) = A1 (t) + B(0; B(0; 1 and C1t (ih− ) = Dx(ih− ) C1 (0; ih). In [6,7], these conditions have been shown to be the necessary and sucient conditions for the existence of a estimator such that inequality (3.6) is satisÿed.
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4. An example Consider the problem of robust nonlinear H∞ ÿltering of the uncertain nonlinear sampled-data system x˙1 (t) = −10x1 (t) − 20x2 (t) − sin(x1 (t)) + 0:1f(t)[x1 (t) + x2 (t)] + 0:1w; x˙2 (t) = x1 (t) + 0:1w; z(t) = x1 (t); zd (ih) =x1 (ih− ); 1 0 2x1 (ih− ) + 0:1fd (ih− )x1 (ih− ) + v(ih− ); y(ih) = 0 1 2x2 (ih− ) + 0:1fd (ih− )x2 (ih− )
(4.1)
where f(t) and fd (ih− ) are uncertain time-varying parameters satisfying |f(t)|61 and |fd (ih− )|61, respectively. The sampling period h is 0.1 and we are required to design a digital ÿlter which guarantees (3.6). The above system is of the form of the system , with −10x1 (t) − 20x2 (t) − sin(x1 (t)) + 0:1f(t)[x1 (t) + x2 (t)] ; A(x(t); t) = x1 (t) 0:1 H (x(t); t) = ; 0 2x1 (ih− ) ; E(x(t); t) = x1 (t) + x2 (t); L(x(t); t) = x1 (t); Ld (x(ih− ); ih) = x1 (ih− ); C(x(ih− ; ih) = 2x2 (ih− ) 0:1 0 − ; Hd (x(ih ); ih) = 0 0:1 1 0 0:1 D(x(ih− ); ih) = and B(x(t); t) = : Ed (x(ih− ); ih) = 1; 0 1 0:1 We now apply the approach proposed in Section 3 to solve the associated robust nonlinear H∞ ÿltering problem. The storage function is chosen to be of the form (x(t); (t)) = xt (t)P(t)x(t) + (x(t) − (t))t (Q(t) − P(t))(x(t) − (t)): After some algebra manipulations a stationary robust nonlinear H∞ ÿlter is found to be −9:9870 −19:8650 sin(1 (t)) ; t ∈ [0; T ]; t 6= ih; xi(t) ˙ = (t) − 0 1:0078 0:1325 0:2243 −0:1078 21 (ih− ) − y(ih) − ; (ih) = (ih ) + −0:1078 0:0648 22 (ih− ) z(t) ˆ = 1 (t);
t ∈ [0; T ];
(4.2)
(4.3)
t 6= ih;
−
zˆd (ih) = 1 (ih ): In comparison, we also have a robust linear H∞ ÿlter corresponding to the linearised system of (4.1). A linear ÿlter is found to be −9:9870 −19:8650 ˙ (t) = (t); t ∈ [0; T ]; t 6= ih; 1:0078 0:1325 0:2243 −0:1078 21 (ih− ) y(ih) − ; (ih) = (ih− ) + (4.4) −0:1078 0:0648 22 (ih− ) z(t) ˆ = 1 (t);
t ∈ [0; T ]; t 6= ih; −
zˆd (ih) = 1 (ih ): Simulation results for the ratio RT Pk (z(t) − z(t)) ˆ t (z(t) − z(t)) ˆ dt + i=1 (zd (ih) − zˆd (ih))t zd (ih) − zˆd (ih)) 0 RT Pk wt (t)w(t) dt + i=1 vt (ih)v(ih) 0
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315
Fig. 1. Ratio of the estimator error energy to the disturbance energy for both robust nonlinear H∞ ÿlter and robust linear H∞ ÿlter.
Fig. 2. The disturbance input, w(t).
obtained by using the robust nonlinear H∞ ÿlter (4.3) and that by using the robust linear H∞ ÿlter (4.4) for the linearised system of (4.1) are depicted in Fig. 1. It can be seen that the robust nonlinear H∞ ÿlter (4.3) achieves much more preferable noise attenuation than the robust linear H∞ ÿlter (4.4). Fig. 2 shows the input disturbance signal w(t) which was used during the simulation.
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5. Conclusion In this paper, we have provided a nonlinear bounded real lemma for sampled-data systems with nonlinear time-varying parameter uncertainty. Based on this nonlinear bounded real lemma, the sucient conditions for the existence of a digital ÿlter that guarantees the L2 gain from an exogenous input to an estimation error is less than or equal to a prescribed value for all admissible uncertainties have been derived. These conditions are expressed in terms of both continuous and discrete Hamilton–Jacobi equations. This result can also be viewed as an extension of [7,8] which treat linear sampled-data systems with uncertainties. References [1] B. Bamieh, J.B. Pearson, A general framework for linear periodic systems with application to H∞ sampled-data control, IEEE Trans. Automat. Control 37 (1992) 418–435. [2] B. Bamieh, J. Pearson, B. Francis, A. Tannenbaum, A lifting technique for linear periodic systems with applications to sampled-data control, Systems Control Lett. 17 (1991) 78–88. [3] S. Hara, P.T. Kabamba, Worst case analysis and design of sampled-data control systems, Proceedings of 29th IEEE Conference on Decision Control, Honolulu, HI, 1990, pp. 202–203. [4] S.K. Nguang, M. Fu, Robust nonlinear H∞ ÿltering, Automatica 32 (1996) 1195–1199. [5] S.K. Nguang, P. Shi, Nonlinear H∞ ÿltering of sampled-data systems, Automatica 36 (2000) 303–310. [6] P. Shi, Issues in robust ÿltering and control of sampled-data systems, Technical Report, The University of Newcastle, 1994; Ph.D. Thesis. [7] P. Shi, Robust ÿltering for uncertain systems with sampled measurements, Int. J. Systems Sci. 27 (12) (1996) 1403–1415. [8] P. Shi, Filtering on sampled-data systems with parametric uncertainty, IEEE Trans. Automat. Control 43 (1998) 1022–1027. [9] N. Sivashankar, P. Khargonekar, Characterization of the L2 -induced norm for linear systems with jumps with application to sampled-data systems, SIAM J. Control Optim. 32 (1994) 1128–1150. [10] W. Sun, K.M. Nagpal, P.P. Khargonekar, H∞ control and ÿltering for sampled-data systems, IEEE Trans. Automat. Control 38 (1993) 1162–1175. [11] H.T. Toivonen, Sampled-data control of continuous-time system with an H∞ optimality criterion, Automatica 28 (1992) 45–54. [12] Y. Yamamoto, On the state space and frequency domain characterization of H∞ norm of sampled-data systems, Systems Control Lett. 21 (1991) 163–172.
www.elsevier.com/locate/sysconle
On designing ÿlters for uncertain sampled-data nonlinear systems Sing Kiong Nguanga; ∗ , Peng Shib a Department
of Electrical and Electronic Engineering, Systems and Control Group, University of Auckland, Private Bag 92019, Auckland, New Zealand b Land Operations Division, Defence Science & Technology Organisation, P.O. Box 1500, Salibury, SA 5108, Australia Received 9 November 1998; received in revised form 16 May 2000; accepted 22 June 2000
Abstract This paper is concerned with the problem of nonlinear H∞ ÿltering for sampled-data systems with nonlinear time-varying parameter uncertainty. The aim is to design a digital ÿlter such that the ratio between the energy of the estimation errors and the energy of the exogenous inputs is minimised or guaranteed to be less or equal to a prescribed value for all admissible uncertainties. A nonlinear bounded real lemma for sampled-data systems with nonlinear time-varying parameter uncertainty is provided. Based on this nonlinear bounded real lemma, the robust H∞ ÿltering problem is solved in terms c 2000 Elsevier Science B.V. All rights reserved. of both continuous and discrete Hamilton–Jacobi equations. Keywords: Nonlinear systems; Sampled-data; Filter
1. Introduction Recently, a number of various techniques have been proposed to handle linear sampled-data systems. The techniques are: (1) lifting technique [1,2,11,12,4], consists of transforming the original sampled-data system into an equivalent LTI discrete-time system with inÿnite-dimensional input–output signal space. The L2 -induced norm of the sampled-data system is shown to be ¡ 1 if and only if the H∞ norm of this equivalent discrete system is ¡ 1, (2) descriptor system technique [3], 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, and (3) technique based on linear systems with jumps [6 –10], 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. Recently in [5], the nonlinear H∞ ÿltering theory [4] has been used to solve the ÿltering problem for nonlinear sampled-data systems. We have obtained some sucient conditions for the existence of a digital ÿlter that guarantee the L2 gain from an exogenous input to an estimation error is less than or equal to a prescribed value. These conditions are expressed in terms of both continuous and discrete Hamilton–Jacobi ∗ Corresponding author. Tel.: 0064-3737-599#8285; fax: 0064-9-3737-461. E-mail addresses: [email protected] (S.K. Nguang), [email protected] (P. Shi).
c 2000 Elsevier Science B.V. All rights reserved. 0167-6911/00/$ - see front matter PII: S 0 1 6 7 - 6 9 1 1 ( 0 0 ) 0 0 0 6 4 - 5
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equations. It has been shown that this result can be viewed as an extension of [6 –10] which treat linear systems. In the present work, we consider nonlinear sampled-data systems subject to uncertainties that are described by an integral functional constraint and input disturbance. The problem addressed here is to design a nonlinear digital ÿlter, such that the ratio between the energy of the estimation errors and the energy of the erogenous inputs is minimised or guaranteed to be less or equal to a prescribed value for all admissible uncertainties. We ÿrst provide a nonlinear bounded real lemma for sampled-data systems with nonlinear time-varying parameter uncertainty. Then based on this nonlinear bounded real lemma for sampled-data systems with nonlinear time-varying parameter uncertainty, a complete solution to the robust nonlinear H∞ ÿltering problem is provided in terms of both continuous and discrete Hamilton–Jacobi equations. This result can be viewed as a generalisation of robust linear H∞ ÿltering results in [6] to a class of nonlinear sampled-data systems with uncertainties. 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. L2 [0; T ] stands for the space of square integrable vector functions over [0; T ]; l2 (0; T ) is the space of square summable vector sequences over (0; T ). f( − ) and f( + ) denote for the left limit and right limit of a function f(), respectively. The gradient of a scalar-valued function f() along the vector ∈ Rn is a column vector denoted by f(), and the gradient of a column vector-valued function 1 () 2 () .. . m ()
is a matrix deÿned by @ 1 () @ 1 () @ 1 () ··· @1 @ @n 2 @ () @ () @ 2 () 2 2 ··· @1 @2 @n D () = . . .. . .. .. .. . @ m () @ m () @ m () ··· @1 @2 @n
:
2. Nonlinear bounded real lemma Consider the nonlinear time-varying system with ÿnite discrete jumps: x(t) ˙ = A(x(t); t) + A(x(t); t) + B(x(t); t)w(t); 1 :
t ∈ [0; T ]; t 6= ih; x(0) = 0;
x(ih) = Ad (x(ih− ); ih) + Ad (x(ih− ); ih) + Bd (x(ih− ); ih)v(ih); z(t) = L(x(t); t);
t ∈ [0; T ]; t 6= ih;
(2.1)
zd (ih) = Ld (x(ih− ); ih); where x(t) ∈ Rn is the state, w(t) ∈ Rp and v(ih) ∈ Rq are the continuous and discrete exogenous inputs which belong to L2 [0; T ] and l2 (0; T ), respectively, z(t) ∈ Rr and zd (ih) ∈ Rs are the continuous and discrete outputs, 0 ¡ h ∈ R is the sampling period, i is a positive integer, A(x(t); t); B(x(t); t); Ad (x(ih− ); ih); Bd (x(ih− ); ih), Ld (x(ih− ); ih) and L(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 L(x(t); t) being piecewise continuous. A(x; t) and Ad (x(ih− ); ih) represent the uncertainties in the system.
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307
Assumption 2.1. A(x(t); t) = H (x(t); t)F(x(t); t)E(x(t); t); Ad (x(ih− ); ih) = Hd (x(ih− ); ih)Fd (x(ih− ); ih)Ed (x(ih− ); ih);
(2.2)
where H (x(t); t); Hd (x(ih− ); ih); Ed (x(ih− ); ih) and E(x(t); t) are known matrix functions which characterize the structure of the uncertainties with E(0; t) = 0 and Ed (0; ih) = 0. Furthermore, the following inequalities hold: kE(x(t); t)k2 ¿kF(x(t); t)E(x(t); t)k2
(2.3)
kEd (x(ih− ); ih)k2 ¿Fd (x(ih− ); ih)E(x(ih− ); ih)k2 :
(2.4)
and
Deÿnition 2.1. Suppose is a given positive real number. A system 1 of the form (2.1) is said to have L2 gain less than or equal to , if for all admissible uncertainties # "Z Z T k k T X X t t 2 t t z (t)z(t) dt + zd (ih)zd (ih)6 w (t)w(t) dt + v (ih)v(ih) ; (2.5) 0
i=1
0
i=1
where k is the largest integer such that k ¡ T . Note that the left-hand side of (2.5) can be viewd as a ‘mixed L2 =l2 ’ norm for the output signals z and zd , and the right-hand side of (2.5) as a ‘mixed L2 =l2 ’ norm of the input signals which comprise of w and v. In the sequel, without loss of generality, we assume = 1. The following is a nonlinear bounded real lemma for sampled-data systems with nonlinear time-varying parameter uncertainty. Theorem 2.1. Consider the system 1 of the form (2:1) satisfying Assumption 2:1. Given a positive constant . Suppose there exist positive-deÿnite functions (x(t); t) and Q(x(t); t) with (0; t) = 0 and Q(0; t) = 0 such that for all x ∈ Rn the following conditions hold: (C1) HJE(x(t); t) , (x(t); ˙ t) + xt (x(t); t)A(x(t); t) t 1 1 1 + xt (x(t); t) B(x(t); t) H (x(t); t) B(x(t); t) H (x(t); t) x (x(t); t) 4 +Lt (x(t); t)L(x(t); t) + 2 E t (x(t); t)E(x(t); t) + Q(x(t); t) = 0:
(2.6)
(C2) DHJE(x (ih− ); va∗ (ih); ih) , (Ad (x(ih− ); ih) + [Bd (x(ih− ); ih) Hd (x(ih− ); ih)]va∗ (ih); ih+ ) − (x(ih− ); ih) − vat∗ (ih)va∗ (ih) + Ltd (x(ih− ); ih)Ld (x(ih− ); ih) + 2 Edt (x(ih− ); ih)Ed (x(ih− ); ih) + Q(x(ih− ); ih) = 0
(2.7)
where va∗ (ih) satisfying v (Ad (x(ih− ); ih) + [Bd (x(ih− ); ih) Hd (x(ih− ); ih)]va∗ (ih); ih+ ) − 2va∗ (ih) = 0
(2.8)
Dv v (Ad (x(ih− ); ih) + [Bd (x(ih− ); ih) + Hd (x(ih− ); ih)]va∗ (ih); ih+ ) − 2I ¡ 0:
(2.9)
and
Then the system 1 is having L2 -gain less than or equal to 1.
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Remark 2.1. Conditions (2.8) and (2.9) imply that va∗ (ih) is the maximiser for DHJE(x(ih− ); va (ih); ih), i.e., DHJE(x(ih− ); va (ih); ih)6DHJE(x(ih− ); va∗ (ih); ih) where
v(ih)
(2.10)
: va (ih) = 1 Fd (x(ih− ); ih)Ed (x(ih− ); ih) Proof of Theorem 2.1. For ∈ (ih; ih + h), Z d {(x; t)} dt = (x( ); ) − (x(ih+ ); ih+ ): ih+ dt First, let us consider and denote the left-hand side of (2.12) as Z d {(x; t)} dt (x(t); t) = dt + ih Z [(x(t); ˙ t) + xt (x(t); t){A(x(t); t) + A(x(t); t) + B(x(t); t)w(t)}] dt: = ih+
(2.11)
(2.12)
(2.13)
Next adding and subtracting (wt (t)w(t) − z t (t)z(t)) to and from (2.13), it follows that Z [(x(t); ˙ t) + xt (x(t); t){A(x(t); t) + A(x(t); t) + B(x(t); t)w(t)} (x(t); t) = ih+
+ wt (t)w(t) − z t (t)z(t) − wt (t)w(t) + z t (t)z(t)] dt:
(2.14)
Note that z(t) = L(x(t); t). Using Assumption 2.1 and completing the square, one has Z 1 1 (x(t); ˙ t) + xt (x(t); t)A(x(t); t) + xt (x(t); t) B(x(t); t) H (x(t); t) (x(t); t) = 4 ih+ t t 1 t 1 H (x(t); t)x (x(t); t) − F(x(t); t)E(x(t); t) × B(x(t); t) H (x(t); t) x (x(t); t) − 2 1 t × H (x(t); t)x (x(t); t) − F(x(t); t)E(x(t); t) + 2 E t (x(t); t)F t (x(t); t)F(x(t); t)E(x(t); t) 2 t 1 t 1 t t B (x(t); t)x (x(t); t)) − w(t) B (x(t); t)x (x(t); t)) − w(t) +L (x(t); t)L(x(t); t) − 2 2 + wt (t)w(t) − z t (t)z(t)] dt;
(2.15)
where ¿ 0. Employing (2.6), we have Z [wt (t)w(t) − z t (t)z(t) − Q(x(t); t) − 2 (kE(x(t); t)k2 − kF(x(t); t)E(x(t); t)k2 )] dt: (x(t); t)6 ih+
(2.16)
Following from (2.12) and (2.16), we obtain Z [wt (t)w(t) − z t (t)z(t) − Q(x(t); t) − 2 (kE(x(t); t)k2 − kF(x(t); t)E(x(t); t)k2 )] dt ih+
¿(x( ); ) − (x(ih+ ); ih+ ):
(2.17)
Now let us consider at the sampling instant +
+ + − − (x(t); t)|ih ih− = (x(ih ); ih ) − (x(ih ); ih )
= (x(ih); ih+ ) − (x(ih− ); ih):
(2.18)
S.K. Nguang, P. Shi / Systems & Control Letters 41 (2000) 305–316
309
Next adding and subtracting the term zdt (ih)zd (ih) − vat (ih)va (ih) to and from (2.18), we obtain d (x(ih); ih) = (x(ih); ih+ ) − (x(ih− ); ih) + zdt (ih)zd (ih) − vat (ih)va (ih) − zdt (ih)zd (ih) + vat (ih)va (ih);
(2.19)
where va (ih) is given in (2.11). Knowing that zd (ih) = Ld (x(ih− ); ih) and using Assumption 2.1, we have 1 d (x(ih); ih) = Ad (x(ih− ); ih) + Bd (x(ih− ); ih) Hd va (ih); ih+ − (x(ih− ); ih) + Ltd (x(ih− ); ih)Ld (x(ih− ); ih) − vat (ih)va (ih) − zdt (ih)zd (ih) + vat (ih)v(ih):
(2.20)
Using (2.7) and the maximiser va∗ (ih) given in (2.8) and (2.9), we get d (x(ih); ih)6vat (ih)va (ih) − zdt (ih)zd (ih) − Q(x(ih− ); ih) − 2 kEdt (x(ih− ); ih)k2 :
(2.21)
Utilising (2.11), we obtain d (x(ih); ih) 6 vt (ih)v(ih) − zdt (ih)zd (ih) − Q(x(ih− ); ih)
− 2 (kEdt (x(ih− ); ih)k2 − kFd (x(ih− ); ih)Ed (x(ih− ); ih)k2 ):
(2.22)
By combining (2.16) and (2.22) over all possible t on [0; T ], one has Z T k X [wt (t)w(t) − z t (t)z(t)] dt + vt (ih)v(ih) (x(T ); T ) − (0; 0) 6 0
−
k X
zdt (ih)zd (ih) −
Z 0
i=1
− 2 − 2
Z
T
0 k X
i=1 T
Q(x(t); t) dt −
k X
Q(x(ih− ); ih)
i=1
{kEdt (x(ih− ); ih)k2 − kFd (x(ih−1 ); ih)Ed (x(ih− ); ih)k2 } dt {kEdt (x(ih− ); ih)k2 − kFd (x(ih−1 ); ih)Ed (x(ih− ); ih)k2 }:
(2.23)
i=1
Using Assumption 2.1 and knowing that (0; t) = 0 and Q(x(t); t) ¿ 0, ∀x(t) 6= 0, we obtain Z T Z T k k X X zdt (ih)zd (ih) + z t (t)z(t) dt 6 wt (t)w(t) dt + vt (ih)v(ih) − (x(T ); T ) i=1
0
0
Z 6
i=1 T
wt (t)w(t) dt +
0
k X
vt (ih)v(ih):
(2.24)
i=1
This shows that the system 1 has the L2 -gain less than or equal to 1. To prove the stability, we rearrange (2.23) to Z T Z T k X Q(x(t); t) dt + Q(x(ih− ); ih) 6 [wt (t)w(t) − z t (t)z(t)] dt (x(T ); T ) − (0; 0) + 0
0
i=1
+
k X
vt (ih)v(ih) −
i=1
Since (0; 0) = 0, we obtain Z Z T k X Q(x(t); t) dt + Q(x(ih− ); ih)6 0
i=1
0
T
wt (t)w(t) dt +
k X i=1
vt (ih)v(ih):
k X
zdt (ih)zd (ih): (2.25)
i=1
(2.26)
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This means that "Z "Z # k T X − Q(x(t); t) dt + Q(x(ih ); ih) 6 lim lim T 7→∞
0
T 7→∞
i=1
0
T
t
w (t)w(t) dt +
k X
# t
v (ih)v(ih) ¡ ∞:
(2.27)
i=1
Since Q(x(t); t) is a positive-deÿnite function, we conclude that the system 1 is bounded input and bounded state stable. Remark 2.2. In the case of a linear system without uncertain, i.e. linearise (2.1) at the origin with A(x(t); t)= 0 and Ad (x(ih− ); ih) = 0: x(t) ˙ = A1 (t)x(t) + B(0; t)w(t);
t 6= ih; x(0) = 0;
−
1l :
x(ih) = Ad1 (ih)x(ih ) + Bd (0; ih)v(ih);
∀ih ∈ (0; T );
z(t) = L1 (t)x(t);
(2.28)
zd (ih) = Ld1 (ih)x(ih−1 ); where A1 (t) = Dx A(0; t), L1 (t) = Dx L(0; t), Ad1 (ih) = Dx(ih− ) Ad (0; ih) and Ld1 (ih) = Dx(ih− ) Ld (0; ih). It is easy to show that for (x(t); t) = 12 xt (t)P(t)x(t), the Hamilton–Jacobi equation (2.6) reduces to the following Riccati inequality: ˙ + At1 (t)P(t) + P(t)A1 (t) + P(t)B(0; t)Bt (0; t)P(t) + Lt1 (t)L1 (t) ¡ 0; t 6= ih; P(T ) = 0 P(t) (2.29) and the discrete Hamilton–Jacobi equation (2.7) boils down to the following discrete Riccati inequality: Atd1 (ih)P(ih+ )Ad1 (ih) + Atd1 (ih)P(ih+ )Bd (0; ih)[I − Bd (0; ih)P(ih+ )Bd (0; ih)]−1 ×Bdt (0; ih)P(ih+ )Ad1 (ih) + Ltd1 (ih)Ld1 (ih) − P(ih) ¡ 0
(2.30)
with I − Bd (0; ih)P(ih+ )Bd (0; ih) ¿ 0:
(2.31)
In [10,6], the authors also showed that (2.29) and (2.30) are the necessary conditions for the system 1l to be BIBOS with L2 -gain less than or equal to 1. 3. Robust nonlinear H∞ ÿltering Consider the following class of uncertain nonlinear sampled-data systems: ) x(t) ˙ = A(x(t); t) + A(x(t); t) + B(x(t); t)w(t); ∀t ∈ [0; T ]; t 6= ih; x(0) = 0; z(t) = L(x(t); t); : zd (ih−1 ) = Ld (x(ih−1 ); ih);
(3.1)
y(ih) = C(x(ih−1 ); ih) + C(x(ih−1 ); ih) + D(x(ih−1 ); ih)v(ih); where x(t) ∈ Rn is the state, w(t) ∈ Rp and v(ih) ∈ Rq are the continuous and discrete inputs which belong to L2 [0; T ] and ‘2 (0; T ), respectively, z(t) ∈ Rr and zd (ih) ∈ Rs are the continuous and discrete regulated outputs, 0 ¡ h ∈ R 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), Ld (x(ih− ); ih) and L(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 L(x(t); t) being piecewise continuous. A(x; t) and C(x(ih− ); ih) represent the uncertainties in the system. Assumption 3.1. A(x(t); t) = H (x(t); t)F(x(t); t)E(x(t); t); C(x(ih− ); ih) = Hd (x(ih− ); ih)Fd (x(ih− ); ih)Ed (x(ih− ); ih); R(x(ih− ); ih) = [D(x(ih− ); ih) Hd (x(ih− 1); ih)][D(x(ih− ); ih) Hd (x(ih− ); ih)]t ¿ 0;
(3.2)
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where H (x(t); t); Hd (x(ih− ); ih); Ed (x(ih− ); ih) and E(x(t); t) are known matrix functions which characterise the structure of the uncertainties with E(0; t) = 0 and Ed (0; ih) = 0. Further, the following inequalities hold: kE(x(t); t)k2 ¿kF(x(t); t)E(x(t); t)k2
(3.3)
kEd (x(ih− ); ih)k2 ¿kFd (x(ih− ); ih)E(x(ih− ); ih)k2 :
(3.4)
and
Robust nonlinear H∞ ÿltering problem: Given a scalar ¿ 0, design a nonlinear digital ÿlter of the form ˙ = a((t); t); (t)
t ∈ [0; T ]; t 6= ih;
(ih) = (ih− ) + b((ih− ); ih)[y(ih) − C(x(ih− ); ih)];
K:
z(t) ˆ = l((t); t)
(3.5)
∀t ∈ [0; T ]; t 6= ih;
zˆd (ih) = ld ((ih− ); ih); ˆ ∈ Rp and zˆd (ih) ∈ Rq are the estimate of z(t) and zd (ih), respecwhere (t) ∈ Rn is the state of the ÿlter, z(t) − tively, a((t); t); b((ih ); ih); l((t); t) and ld ((ih− ); ih) are matrix functions with appropriate dimensions, a(0; t) = 0; l(0; t) = 0 and ld (0; ih) = 0. The objective is such that the L2 gain from the disturbances w(t) and v(t) to the estimation errors z(t) − z(t) ˆ and zd (ih) − zˆd (ih) for the augmented system with K is less than or equal to , i.e. for all T ¿0, all w(t) ∈ L2 [0; T ], all v(ih) ∈ ‘2 (0; T ) and all admissible uncertainties Z
T 0
(z(t) − z(t)) ˆ t (z(t) − z(t)) ˆ dt +
(zd (ih) − zˆd (ih))t (zd (ih) − zˆd (ih))
i=1
Z 6
k X
T
wt (t)w(t) dt +
0
k X
vt (ih)v(ih);
(3.6)
i=1
where k is the largest integer such that k ¡ T . Remark 3.1. The performance requirement given in (3.6) is deÿned directly in terms of the continuous-time signals which means the intersample behaviour is incorporated in the design. In the sequel, without loss of generality, we assume = 1. Theorem 3.1. Consider the system satisfying Assumption 2:1. Given a scaling parameter ¿ 0; suppose there exist nonnegative scalar functions (x; ˜ t) and Q(x(t); ˜ t); and a matrix function b((ih− ); ih) such that 2n for all x(t) ˜ ∈ R the following conditions hold: (C1) t x(t); x(t); ˜ t)A( ˜ t) + 14 xt˜(x(t); ˜ t)B( ˜ t)B (x(t); ˜ t)x˜(x(t); ˜ t) HJ , ( ˙ x(t); ˜ t) + xt˜(x(t); t
t
x(t); x(t); ˜ t)L( ˜ t) + 2 E (x(t); ˜ t)E( ˜ t) + Q(x(t); ˜ t) = 0; + L (x(t); where
x(t) x(t) ˜ = ; (t)
x(t); A( ˜ t) =
B((t); t) = [B((t); t)
(3.7)
A(x(t); t) ; A((t); t) + 12 B((t); t)Bt ((t); t)x (; )
1 H ((t); t)];
x(t); L( ˜ t) = L(x(t); t) − L((t); t);
E(x(t); t) E(x(t); ˜ t) = 0
312
and
S.K. Nguang, P. Shi / Systems & Control Letters 41 (2000) 305–316
B((t); t) x(t); B( ˜ t) = : 0
(C2) ˜ − ); ih) + B d (x(ih ˜ − ); ih)va∗ (ih); ih+ ) − (x(ih ˜ − ); ih) − va∗t (ih)va∗ (ih) (A d (x(ih t
t
˜ − ); ih)Ld (x(ih ˜ − ); ih) + Q(x(ih ˜ − ); ih) + 2 E d (x(ih ˜ − ); ih)E d (x(ih ˜ − ); ih) = 0; + Ld (x(ih
(3.8)
where va∗ (ih) satisfying ˜ − ); ih) + B d (x(ih ˜ − ); ih)va∗ (ih); ih+ ) − 2va∗ (ih) = 0; va (A d (x(ih
(3.9)
˜ − ); ih) + B d (x(ih ˜ − ); ih)va∗ (ih); ih+ ) − 2I ¡ 0; Dv v (A d (x(ih
(3.10)
x(ih− ) ; (ih− ) + b((ih− ); ih)[C(x(ih− ); ih) − C((ih− ); ih)] Ed (x(ih− ); ih) ˜ − ); ih) = Ld (x(ih− ); ih) − Ld ((ih− ); ih); E d (x(ih ˜ − ); ih) = Ld (x(ih 0
˜ − ); ih) = A d (x(ih
and ˜ − ); ih) = B d (x(ih
0 : b((ih− ); ih)[D(x(ih− ); ih) 1 Hd (x(ih− ); ih)]
If this is the case; then there exists a ÿlter of the form ˙ = A((t); t) + 1 B((t); t)Bt ((t); t)x (; ); (t) 2
t ∈ [0; T ]; t 6= ih;
(ih) = (ih− ) + b((ih− ); ih)[y(ih) − [C(x(ih− ); ih)]; z(t) ˆ = L((t); t)
t ∈ [0; T ]; t 6= ih;
zˆd (ih) = Ld ((ih− ); ih)
(3.11)
which will render (3:6) for all x˜ ∈ R2n . Proof. The augmented system with (3.11) as follows:
t 1 ˙ (t) = A((t); t) + 2 B((t); t)B ((t); t)x (; ); x(t) ˙ = A(x(t); t) + A(x(t); t) + B(x(t); t)w(t);
z(t) = L(x(t); t);
∀t ∈ [0T ]; t 6= ih; x(0) = 0;
z(t) ˆ = L((t); t); (ih) = (ih− ) + b((ih− ); ih)[C(x(ih− ); ih) + C(x(ih− ); ih) + D(x(ih− ); ih)v(ih)
as :
− C((ih− ); ih)]; zd (ih− ) = Ld (x(ih− ); ih); zˆd (ih) = Ld ((ih− ); ih): Deÿning
x(t) x˜ = ; (t)
(3.12)
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the system as can be recast into the following form: x(t); x(t); x(t); x(t) ˜˙ = A( ˜ t) + A( ˜ t) + B( ˜ t)w(t);
t 6= ih; x(0) ˜ = 0;
˜ − ); ih) + A d (x(ih ˜ − ); ih) + B d (x(ih ˜ − ); ih)v(ih); x(ih) ˜ = A d (x(ih " # x(t); L( ˜ t) z(t) = z(t) − z(t) ˆ = ; E(x(t); # " t) ˜ − ); ih) Ld (x(ih ; zd (ih) = zd (ih) − zˆd (ih) = Ed ((ih− ); ih− ) where x(t); A( ˜ t) =
A(x(t); t) ; 0
˜ − ); ih) = A d (x(ih
∀ih ∈ (0; T ); (3.13)
0 ; C(x(ih− ); ih
x(t); x(t); x(t); ˜ − ); ih); B d (x(ih ˜ − ); ih); L( ˜ t), and Ld (x(ih ˜ − ); ih) are deÿned in Theorem 3.1. A( ˜ t); B( ˜ t); A d (x(ih x(t); Before applying Theorem 2.1, we need to show that A( ˜ t) and A d (x(ih ˜ − ); ih) satisfying Assumption − x(t); ˜ ); ih) can be rewritten as 2.1. Using Assumption 3.1, A( ˜ t) and A d (x(ih H (x(t); t) x(t); A( ˜ t) = F(x(t); t)E(x(t); t) (3.14) 0 and ˜ − ); ih) = A d (x(ih
0 Fd (x(ih− ); ih)Ed (x(ih− ); ih): Hd (x(ih− ); ih)
(3.15)
By Assumption 3.1, clearly one can see that Assumption 2.1 is satisÿed. Now, applying Theorem 2.1, we obtain Theorem 3.1 Remark 3.2. In the linear case, the optimal b(0; t) = S(ih)C t (t). If we choose (x(t); (t)) = xt (t)P(t)x(t) + (x(t) − (t))t (Q(t) − P(t))(x(t) − (t)) and linearise the Hamilton–Jacobi inequality (3.7), we will have LP − LQ LQ x(t) ¡ 0; t 6= ih; (3.16) [x(t) (t)]t LP − LQ LQ − LP (t) where t
˙ + At1 (t)P(t) + P(t)A1 (t) + P(t)B(0; t)B (0; t)P(t) + 2 E1t (t)E1 (t); LP = P(t)
(3.17)
˙ + A 1 (t)Q(t) + Q(t)A t1 (t) + Q(t)L1 (t)t L1 (t)Q(t) + B(0; t)B t (0; t): LQ = Q(t)
(3.18)
Condition (3.16) implies that LP ¡ 0 and LQ ¡ 0. Similarly, the discrete Hamilton–Jacobi equation (3.8) boils down to LPd − LQd x(ih) t LQd ¡ 0; (3.19) [x(ih) (ih)] LPd − LQd LQd − LPd (ih) where
(3.20)
(3.21)
LPd (ih) = P(ih) + Edt (ih− )Ed (ih− ) − P(ih− ); LQd (ih) = [Q−1 (ih− ) − Ltd1 (ih− )Ld1 (ih− ) + C1t (ih− )R−1 (0; ih)C1 (ih− )]−1 − Q(ih):
Eq. (3.18) implies that LPd ¡ 0 and LQd ¡ 0. A1 (t) = Dx A(0; t); L1 (t) = Dx L1 (0; t); Ld1 (ih) = Dx(ih− ) Ld (0; ih); t)B t (0; t)P t) = [B(0; t) (1= )H (0; t)]; E1t (t) = Dx(t) E(0; t); Edt (ih− ) = Dx(ih− ) Ed1 (0; ih); A 1 (t) = A1 (t) + B(0; B(0; 1 and C1t (ih− ) = Dx(ih− ) C1 (0; ih). In [6,7], these conditions have been shown to be the necessary and sucient conditions for the existence of a estimator such that inequality (3.6) is satisÿed.
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4. An example Consider the problem of robust nonlinear H∞ ÿltering of the uncertain nonlinear sampled-data system x˙1 (t) = −10x1 (t) − 20x2 (t) − sin(x1 (t)) + 0:1f(t)[x1 (t) + x2 (t)] + 0:1w; x˙2 (t) = x1 (t) + 0:1w; z(t) = x1 (t); zd (ih) =x1 (ih− ); 1 0 2x1 (ih− ) + 0:1fd (ih− )x1 (ih− ) + v(ih− ); y(ih) = 0 1 2x2 (ih− ) + 0:1fd (ih− )x2 (ih− )
(4.1)
where f(t) and fd (ih− ) are uncertain time-varying parameters satisfying |f(t)|61 and |fd (ih− )|61, respectively. The sampling period h is 0.1 and we are required to design a digital ÿlter which guarantees (3.6). The above system is of the form of the system , with −10x1 (t) − 20x2 (t) − sin(x1 (t)) + 0:1f(t)[x1 (t) + x2 (t)] ; A(x(t); t) = x1 (t) 0:1 H (x(t); t) = ; 0 2x1 (ih− ) ; E(x(t); t) = x1 (t) + x2 (t); L(x(t); t) = x1 (t); Ld (x(ih− ); ih) = x1 (ih− ); C(x(ih− ; ih) = 2x2 (ih− ) 0:1 0 − ; Hd (x(ih ); ih) = 0 0:1 1 0 0:1 D(x(ih− ); ih) = and B(x(t); t) = : Ed (x(ih− ); ih) = 1; 0 1 0:1 We now apply the approach proposed in Section 3 to solve the associated robust nonlinear H∞ ÿltering problem. The storage function is chosen to be of the form (x(t); (t)) = xt (t)P(t)x(t) + (x(t) − (t))t (Q(t) − P(t))(x(t) − (t)): After some algebra manipulations a stationary robust nonlinear H∞ ÿlter is found to be −9:9870 −19:8650 sin(1 (t)) ; t ∈ [0; T ]; t 6= ih; xi(t) ˙ = (t) − 0 1:0078 0:1325 0:2243 −0:1078 21 (ih− ) − y(ih) − ; (ih) = (ih ) + −0:1078 0:0648 22 (ih− ) z(t) ˆ = 1 (t);
t ∈ [0; T ];
(4.2)
(4.3)
t 6= ih;
−
zˆd (ih) = 1 (ih ): In comparison, we also have a robust linear H∞ ÿlter corresponding to the linearised system of (4.1). A linear ÿlter is found to be −9:9870 −19:8650 ˙ (t) = (t); t ∈ [0; T ]; t 6= ih; 1:0078 0:1325 0:2243 −0:1078 21 (ih− ) y(ih) − ; (ih) = (ih− ) + (4.4) −0:1078 0:0648 22 (ih− ) z(t) ˆ = 1 (t);
t ∈ [0; T ]; t 6= ih; −
zˆd (ih) = 1 (ih ): Simulation results for the ratio RT Pk (z(t) − z(t)) ˆ t (z(t) − z(t)) ˆ dt + i=1 (zd (ih) − zˆd (ih))t zd (ih) − zˆd (ih)) 0 RT Pk wt (t)w(t) dt + i=1 vt (ih)v(ih) 0
S.K. Nguang, P. Shi / Systems & Control Letters 41 (2000) 305–316
315
Fig. 1. Ratio of the estimator error energy to the disturbance energy for both robust nonlinear H∞ ÿlter and robust linear H∞ ÿlter.
Fig. 2. The disturbance input, w(t).
obtained by using the robust nonlinear H∞ ÿlter (4.3) and that by using the robust linear H∞ ÿlter (4.4) for the linearised system of (4.1) are depicted in Fig. 1. It can be seen that the robust nonlinear H∞ ÿlter (4.3) achieves much more preferable noise attenuation than the robust linear H∞ ÿlter (4.4). Fig. 2 shows the input disturbance signal w(t) which was used during the simulation.
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5. Conclusion In this paper, we have provided a nonlinear bounded real lemma for sampled-data systems with nonlinear time-varying parameter uncertainty. Based on this nonlinear bounded real lemma, the sucient conditions for the existence of a digital ÿlter that guarantees the L2 gain from an exogenous input to an estimation error is less than or equal to a prescribed value for all admissible uncertainties have been derived. These conditions are expressed in terms of both continuous and discrete Hamilton–Jacobi equations. This result can also be viewed as an extension of [7,8] which treat linear sampled-data systems with uncertainties. References [1] B. Bamieh, J.B. Pearson, A general framework for linear periodic systems with application to H∞ sampled-data control, IEEE Trans. Automat. Control 37 (1992) 418–435. [2] B. Bamieh, J. Pearson, B. Francis, A. Tannenbaum, A lifting technique for linear periodic systems with applications to sampled-data control, Systems Control Lett. 17 (1991) 78–88. [3] S. Hara, P.T. Kabamba, Worst case analysis and design of sampled-data control systems, Proceedings of 29th IEEE Conference on Decision Control, Honolulu, HI, 1990, pp. 202–203. [4] S.K. Nguang, M. Fu, Robust nonlinear H∞ ÿltering, Automatica 32 (1996) 1195–1199. [5] S.K. Nguang, P. Shi, Nonlinear H∞ ÿltering of sampled-data systems, Automatica 36 (2000) 303–310. [6] P. Shi, Issues in robust ÿltering and control of sampled-data systems, Technical Report, The University of Newcastle, 1994; Ph.D. Thesis. [7] P. Shi, Robust ÿltering for uncertain systems with sampled measurements, Int. J. Systems Sci. 27 (12) (1996) 1403–1415. [8] P. Shi, Filtering on sampled-data systems with parametric uncertainty, IEEE Trans. Automat. Control 43 (1998) 1022–1027. [9] N. Sivashankar, P. Khargonekar, Characterization of the L2 -induced norm for linear systems with jumps with application to sampled-data systems, SIAM J. Control Optim. 32 (1994) 1128–1150. [10] W. Sun, K.M. Nagpal, P.P. Khargonekar, H∞ control and ÿltering for sampled-data systems, IEEE Trans. Automat. Control 38 (1993) 1162–1175. [11] H.T. Toivonen, Sampled-data control of continuous-time system with an H∞ optimality criterion, Automatica 28 (1992) 45–54. [12] Y. Yamamoto, On the state space and frequency domain characterization of H∞ norm of sampled-data systems, Systems Control Lett. 21 (1991) 163–172.