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Robust 1 Filtering for Uncertain Two-Dimensional Continuous Systems with Time-varying Delays C. El-KASRI, A. HMAMED, E.H. TISSIR, F. TADEO
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Abstract This paper deals with the problem of delay-dependent robust H1 filtering for uncertain two-dimensional (2-D) continuous systems described by Roesser state space model with time-varying delays, with the uncertain parameters assumed to be of polytopic type. A sufficient condition for H1 noise attenuation is derived in terms of linear matrix inequalities, so a robust H1 filter can be obtained by solving a convex optimization problem. Finally, some examples are provided to illustrate the effectiveness of the proposed methodology. Key words 2-D Continuous Systems, Uncertainty, Delayed States, tering, Linear Matrix Inequality (LMI).
H1 Fil-
1 Introduction The noise filtering problem is known to play an important role in signal processing and other applications [12], [21], [22], [23]. In the last few years, delaydependent and delay-independent designs for filters have been proposed: see, for example, [10], [11], [15], [19], [20] and [24]. One of the most popular ways to deal with the filtering problem is the celebrated Kalman filtering approach, which provides an optimal estimation of the state variables in the sense that the covariance of the estimation error is minimized [2]. This approach requires precise information on the external This work is funded by AECI AP/034911/11 and MiCInn DPI2010-21589-c05 C. El-KASRI, A. HMAMED and E.H. TISSIR are with LESSI, Department of Physics, Faculty of Sciences Dhar El Mehraz, BP 1796, Fes-Atlas, Morocco. chakir_elkasri, hmamed_abdelaziz,
[email protected] F. TADEO is with the Department of Systems Engineering and Automatic Control, University of Valladolid, 47005 Valladolid, Spain.
[email protected] Address(es) of author(s) should be given
noises characteristics, which is not always possible in practical applications. To overcome this limitation, an alternative methodology called H1 filtering has been introduced [12], [34], which guarantees a prescribed noise attenuation level over the entire frequency range: an H1 filter is robust when this attenuation level is guaranteed even in the presence of uncertainty in the internal model of the process, which is very appropriate in practical situations. To handle problems with modeling uncertainties, several robust H1 filtering methods for one-dimensional (1-D) systems have been proposed in the literature: for example, [16], [17] and [32]. H1 filtering is here studied for the two-dimensional (2-D) case in the presence of time-varying delays in the states: this is motivated by the fact that there are many examples of practical 2-D systems with inherent delays [28], [33]. Although much work has been reported on H1 problems for 1-D state-delayed systems (see, e.g., [15], [29], and references therein), there are few research results on uncertain 2-D state-delayed systems: we can cite [30], [35], and the mixed H2 =H1 filter design problem proposed in [5], [9] that uses a parameter-dependent Lyapunov function approach similar to the one that will be used here. These works are based on the delay-independent approach; However, delay-dependent results are generally less conservative especially when it is known beforehand that the delays involved are small. Therefore, it is natural to try to derive delay-dependent conditions for 2-D systems with delayed states, as done here.
Thus, in this paper, based on the structured polynomially parameterdependent method, we study the problem of robust H1 filtering for polytopic 2-D continuous systems with state-varying delays described by the Roesser model. The reported results are based on homogeneous polynomially parameterdependent matrices of arbitrary degree. It is shown that as the degree of the polynomial increases, the number of linear matrix inequalities and free variables also increases and the tests become less conservative. The purpose of the problem under investigation is then to design a 2-D filter such that, for all admissible uncertainties, the filtering error dynamics are asymptotically stable, and a prescribed H1 -norm performance level is achieved, within specific delay ranges in both horizontal and vertical directions. Effective methods to solve the robust H1 filtering problem by using a parameter-dependent Lyapunov function [6], [7], [31] will be derived. Different from the quadratic stability framework [29], the use of a parameter-dependent Lyapunov function allows different Lyapunov matrices to be set for different parts of the entire polytope domain, and produces less conservative designs. Notation: Throughout this paper, for real symmetric matrices X and Y , the notation X Y (respectively, X > Y ) means that the matrix X Y is positive semi-definite (respectively, positive definite). I is the identity matrix with appropriate dimension. The superscript T represents the transpose of a matrix; diag f:::g denotes a block-diagonal matrix, her(S ) stands for S + S T . The Euclidean vector norm is denoted by k:k. The l2 norm of a 2-D signal
w(t1 ; t2 ) is given by
kw(t1 ; t2 )k2 =
1Z 1
Z 0
0
w(t1 ; t2 )T w(t1 ; t2 )dt1 dt2 ;
and the symmetric term in a symmetric matrix is denoted by , e.g.
XY Z
=
X Y Y T Z : Matrices, if not explicitly stated, are assumed to have compatible
dimensions.
2 PROBLEM FORMULATION Consider a 2-D continuous system with delays described by the following Roesser’s state-space model:
( ) : x_ (t1 ; t2 ) = A x(t1 ; t2 ) + Ad x(t1 y(t1 ; t2 ) = C1 x(t1 ; t2 ) + C1d x(t1 z (t1 ; t2 ) = C x(t1 ; t2 ) h h x (t ; t ) i _ ( with ( 1 2 ) = xv (t1 ; t2 ) 1 1 2 h h i x (t1 1 (t); t2 ) , where 2 ( )) = xv (t1 ; t2 2 (t))
x t ;t
; x t ; t2 ) =
1 (t); t2 2 (t)) + B !(t1 ; t2 ) (1) 1 (t); t2 2 (t)) + D1 !(t1 ; t2 )(2) (3) @ h @t1 x (t1 ; t2 ) and x(t 1 @ v @t2 x (t1 ; t2 )
1 (t); t2
xh (t1 ; t2 ) 2 i=1 4 C1di D1i 5 > > : ; C 0 C 0 Ci 0
where
is the unit simplex: (
= ( 1 ; 2 ; :::; N ) :
N X i=1
)
i = 1; i 0 :
In this paper we consider the following 2-D continuous filter, in order to estimate z (t1 ; t2 ): #
"
@ x^h (t1 ;t2 ) x ^h (t1 ; t2 ) @t 1 (f ) : @ x^h (t1 ;t2 ) = Af x^v (t1 ; t2 ) + Bf y(t1 ; t2 ) @t2 x ^h (t1 ; t2 ) z^(t1 ; t2 ) = Cf x^v (t ; t ) 1 2 x^h (0; t ) = 0; x^v (t ; 0) = 0; 8t ; t > 0 2
1
1
(7) (8)
2
where x ^h (t1 ; t2 ) 2 0; Qh > 0; Zh > 0; Pv > 0; Qv > 0 and be found. The variation V_ (t1 ; t2 ) is given by
Zv > 0 are matrices to
(t1 ; t2 ) @V2 (t1 ; t2 ) : V_ (t1 ; t2 ) , @V1@t + @t 1
Moreover, we define
1Z 1
Z
0
0
V_ (t1 ; t2 )dt1 dt2 ,
Z 0
2
1 @V (t ; t ) 1 1 2 dt
@t1
Z 1
+
Then, along the solution of the nominal process
@V11 (t1 ; t2 ) + @V21 (t1 ; t2 ) @t1 @t2
0
1 @V (t ; t ) 2 1 2 dt2 :
@t2
0 , we have that
= 2x(t1 ; t2 )T P (A + Ad )x(t1 ; t2 ) 2x(t1 ; t2 )T PAd (1 ; 2 ) = 2x(t1 ; t2 )T P (A + Ad )x(t1 ; t2 ) + 2x(t1 ; t2 )T (Y PAd ) (1 ; 2 ) +2x(t1 1 (t); t2 2 (t))T W (1 ; 2 ) f2x(t1 ; t2 )T Y (1 ; 2 ) +2x(t1 1 (t); t2 2 (t))T W (1 ; 2 )g = 2x(t1 ; t2 )T (PA + Y )x(t1 ; t2 ) +2x(t1 ; t2 )T (PAd Y + W T )x(t1 1 (t); t2 2 (t)) 2x(t1 1 (t); t2 2 (t))T Wx(t1 1 (t); t2 2 (t)) 2x(t1 ; t2 )T Y (1 ; 2 ) 2x(t1 1 (t); t2 2 (t))T W (1 ; 2 ) or
# " # R t1 R t2 t1 @xh (1 ;t2 ) d d @xh (1 ;t2 ) d 1 1 1 2 @ @ t (t) 1 2 (t) t1 1 (t) t2 2 (t) 1 1 1 R t1 R t2 (1 ; 2 ) = R t2 @xv (t1 ;2 ) d d @xv (t1 ;2 ) d = 1 2 1 2 @2 1 (t) t1 1 (t) t2 2 (t) @2 t2 2 (t) # " Z t1 Z t2 h @x (1 ;t2 ) (t) @1 = d1 d2 (t1 ;2 ) 1 (t) 2 (t) t1 1 (t) t2 2 (t) @xv@ 2 "R
@V11 (t1 ; t2 ) + @V21 (t1 ; t2 ) : @t1 @t2
=
Z t2 Z t1 1 T 1 (t) 2 (t) t1 1 (t) t2 2 (t) f2x(t1 ; t2 ) (PA + Y )x(t1 ; t2 )
+2x(t1 ; t2 )T (PAd 2x(t1 1 (t); t2
"
Y W T )x(t1 1 (t); t2 2 (t)) 2 (t))T Wx(t1 1 (t); t2 2 (t)) #
2x(t1 ; t2 )T (t)Y
@xh (1 ;t2 ) @1 @xv (t1 ;2 ) @2
2x(t1
2 (t))T (t)W
"
1 (t); t2
(t)1 (t) 1 2
Z t1
Z t2
2x(t1 ; t2 )T Y 2x(t1
1 (t); t2
@xh (1 ;t2 ) @1 @xv (t1 ;2 ) @2
#)
d1 d2
f2x(t1 ; t2 )T (PA + Y )x(t1 ; t2 )
t1 1 (t) t2 2 (t) T +2x(t1 ; t2 ) (PAd Y W T )x(t1 2x(t1 1 (t); t2 2 (t))T Wx(t1 "
@xh (1 ;t2 ) @1 @xv (t1 ;2 ) @2
1 (t); t2 2 (t)) 1 (t); t2 2 (t))
#
"
2 (t))T W
@xh (1 ;t2 ) @1 @xv (t1 ;2 ) @2
#)
d1 d2
(17)
@V12 (t1 ; t2 ) + @V22 (t1 ; t2 ) @t1 @t2
= _1 (t)
Z t1
t1 1 (t)
x_ h (1 ; t2 )T Zh x_ h (1 ; t2 )d1 +
Z
0
x_ h (t1 + 1 ; t2 )T Zh x_ h (t1 + 1 ; t2 )]d 1 + _2 (t) Z
+
0
2 (t)
[x_ v (t1 ; t2 )T Zv x_ v (t1 ; t2 )
[x_ h (t1 ; t2 )T Zh x_ h (t1 ; t2 )
1 (t) Z t2
t2 2 (t)
x_ v (t1 ; 2 )T Zv x_ v (t1 ; 2 )d2
x_ v (t1 ; t2 + 2 )T Zv x_ v (t1 ; t2 + 2 )]d 2
= + =
Z t1
[x_ h (t1 ; t2 )T Zh x_ (t1 ; t2 )
x_ h (1 ; t2 )T (1 _1 (t))Zh x_ h (1 ; t2 )]d1
[x_ v (t1 ; t2 )T Zv x_ v (t1 ; t2 )
x_ v (t1 ; 2 )T (1 _2 (t))Zv x_ v (t1 ; 2 )]d2
t1 1 (t) Z t2 t2 2 (t)
Z t2 Z t1 1 h T h 1 (t) 2 (t) t1 1 (t) t2 2 (t) fx_ (t1 ; t2 ) 1 (t)Zh x_ (t1 ; t2 )
+x_ v (t1 ; t2 )T 2 (t)Zv x_ v (t1 ; t2 ) x_ h (1 ; t2 )T 1 (t)(1 _1 (t))Zh x_ h (1 ; t2 ) x_ v (t1 ; 2 )T 2 (t)(1 _2 (t))Zv x_ v (t1 ; 2 )gd1 d2 Z t2 Z t1 1 T = 1 (t) 2 (t) t1 1 (t) t2 2 (t) fx_ (t1 ; t2 ) (t)Z x_ (t1 ; t2 ) " #T " #) @xh (1 ;t2 ) @xh (1 ;t2 ) @ @ 1 1 (t)(I _ (t))Z @xv (t ; ) d1 d2 @xv (t ; ) 1
2
@2
(t)1 (t) 1 2 "
1
@xh (1 ;t2 ) @1 @xv (t1 ;2 ) @2
(t)1 (t) 1 2
Z t2
Z t1
@2
fx_ (t1 ; t2 )T (t)Z x_ (t1 ; t2 )
t1 1 (t) t2 2 (t) #T "
(t)(I )Z
Z t1
Z t2
2
@xh (1 ;t2 ) @1 @xv (t1 ;2 ) @2
#)
d1 d2
f[Ax(t1 ; t2 )
t1 1 (t) t2 2 (t) +Ad x(t1 1 (t); t2 2 (t))]T Z [Ax(t1 ; t2 ) + Ad x(t1 #T " #) " @xh (1 ;t2 ) @xh (1 ;t2 ) @1 1 (I )Z @xv@ d1 d2 @xv (t1 ;2 ) (t1 ;2 ) @2 @2
1 (t); t2 2 (t))] (18)
@V12 (t1 ; t2 ) + @V22 (t1 ; t2 ) @t1 @t2 = xh (t1 ; t2 )T Qh xh (t1 ; t2 ) xh (t1 1 (t); t2 )T (1 _1 (t))Qh xh (t1 1 (t); t2 ) +xv (t1 ; t2 )T Qv xv (t1 ; t2 ) xv (t1 ; t2 2 (t))T (1 _2 (t))Qv xv (t1 ; t2 2 (t)) = x(t1 ; t2 )T Qx(t1 ; t2 ) x(t1 1 (t); t2 2 (t))T (1 _ (t))Qx(t1 1 (t); t2 2 (t)) Z t1 Z t2 1 T = 1 (t) 2 (t) t (t) t (t) [x(t1 ; t2 ) Qx(t1 ; t2 ) 1
1
2
2
x(t1 1 (t); t2 2 (t))T (1 _ (t))Qx(t1 1 (t); t2 2 (t))]d1 d2
Z t1 Z t2 1 T 1 (t) 2 (t) t1 1 (t) t2 2 (t) [x(t1 ; t2 ) Qx(t1 ; t2 )
x(t1 1 (t); t2 2 (t))T (1 )Qx(t1 1 (t); t2 2 (t))]d1 d2
It then follows from (17)-(19), that Z t1 Z t2 1 T _ V (t1 ; t2 ) (t) (t) 1 2 t1 1 (t) t2 2 (t)
d1 d2
(19)
where
"
"
= x(t1 ; t2 )T
and
x(t1 1 (t); t2
2 (t))T
2
T T 6 her (A P + Y ) + A ZA + Q T T 6 = 4 Ad P Y + W + ATd ZA Y T
@xh (1 ;t2 ) @1 @xv (t1 ;2 ) @2
#T #T
3
7 her(W ) + ATd ZAd 7 5 (I )Q W T (I )Z
(20) If < 0, then V_ (t1 ; t2 ) < 0 for any 6= 0. Now, applying the Schur complement equivalence to (15) gives that Z > 0 and < 0. So (0 ) is asymptotically stable if LMI (15) holds, which completes the proof. 3.2 Performance analysis of robust
H1 filtering
In this section, we consider the robust H1 performance analysis problem using the parameter-dependent approach, in the light of which, the filter design problem is dealt with in the next section. Theorem 2 Given a scalar > 0 and < I , the time-delay 2-D continuous system (e ) is asymptotically stable with disturbance attenuation level for any time-varying delay (t) satisfying 0 < (t) < if there exist matrices P = diag (Ph ; Pv ) > 0, Z = diag (Zh ; Zv ) > 0, Q = diag(Qh ; Qv ) > 0 and W , Y such that the following LMI holds: 2
her(P A~ + Y ) + Q 6 A~T P Y T + W d 6 6 Y T 6 6 Z A~ 4 B~ T P C~
her(W ) )Q (I )Z W T 0 Z Z A~d 0 B~ T Z
(I
0 0
0
0
Proof 2 To simplify the proof we denote Y := Y and W := W
x~h (t1 1 (t); t2 ) = x~h (t1 ; t2 ) x~v (t1 ; t2 2 (t)) = x~v (t1 ; t2 )
P
:=
2I
I
0
P ; Z
:=
3 7 7 7 7 0, and < 1, the 2-D continuous system with time-varying delays in (31) is asymptotically stable for any delay (t) satisfying 0 < (t) < if there exist matrices P = diag(Ph ; Pv ) > 0, Z v > 0, Qv > 0, W 22 , Y1 , Q = diag(0; Qv ), Z = 0 Zv , W = 0 W22 , and Y = 0 Y1 ; such that the following LMI holds: 2
her(P A + Y ) + Q 3 T P Y T + W T A her ( W ) (1 ) Q 5 < 0: 22 v 1 22 4 d Y1T W22T (1 )Zv Z A Z Ad 0 Zv
(32)
3.4 Solution using parameter-dependent polynomials To solve the parameter-dependent LMI conditions of Theorems 1 to 3 the polynomially parameter-dependent method is used; this method includes results in the quadratic framework and the linearly parameter-dependent framework as particular cases, for zeroth and first degree polynomials, respectively. Now, before presenting the Theorem 3 using homogeneous parameterdependent polynomials, some definitions and preliminaries from [14] are recalled: For the matrices P1 , we take a homogeneous polynomially dependent Lyapunov function given by
P1 (g) =
JX (g)
k1 k2 ::: kNN P1Kj (g) ; 1
2
j =1 k1 k2 :::kN = Kj (g);
(33)
Similar definitions for the matrices Q11 > 0, Q22 > 0, Z1 > 0, Q12 , Y11 , Y12 , Y21 , Y22 , W11 , W12 , W21 and W22 are used. To facilitate the presentation, we denote ji (j + 1) in [14] by h; using this notation we now present the Theorem 4. Theorem 4 If there exists a solution P1Kj (g) = diag (P1hKj (g) ; P1v ) > 0, P2 = diag (P2h ; P2v ) > 0, Q11Kj (g) = diag (Q11hKj (g) ; Q11vKj (g) ) > 0, Q22Kj (g) = diag(Q22hKj (g) ; Q22vKj (g) ) > 0, Z1Kj (g) = diag(Z1hKj (g) ; Z1vKj (g) ) >
0 and Q12K (g) = diag (Q12hK (g) ; Q12vK (g) ), Y11K (g) , Y12K (g) , Y21K (g) , Y22K (g) , W11K (g) , W12K (g) , W21K (g) , W22K (g) , Kj (g) 2 K(g), j = 1; :::; J(g), such that the following LMIs hold for all Kl (g + 1) 2 K(g + l), l = 1; :::; J(g + l): 2 3 11 6 22 77 X 6 21 6 31 32 33 77 < 0 (34) 6 4 0 44 5 i2ll (g+l) 41 42 51 52 53 54 55 j
j
j
j
j
j
j
j
j
j
j
where
3 her(P1 ( + ) Ai + hSb C1i 7 6 +Y11 ) + Q11 ( + ) ( + ) 7 6 7 6 hP2 A + hS C1 + hS T i b i 11 = 6 a 7 her ( Y T 22 7 6 +Y21 + Y12 ( + ) ( + ) 5 4 ( + ) + h S ) + Q a 22 ( + ) +QT12 ( + ) 3 2 T T (Adi P1 ( + ) + hCdT SbT Y11 ( + ) h 2 i M1 7 h I +W11 ( + ) ) 21 = 6 5 ; 55 = 4 0 hI M2 Y12T ( + ) + W21 ( + ) M1 = hATdi P2 + hCdTi SbT Y12T ( + ) + W12 ( + ) ; T Bi Z1 ( + ) + 1 hDiT SbT 1 hBiT P2 + 2 hDiT SbT T M2 = W22 ( + ) Y22 ( + ) ; 54 = 0 0 3 2 her(W11 ( + ) ) 7 6 i h +(I )Q11 ( + ) 7 6 0 0 ; 22 = 6 W21 7 52 = T + W12 hCi hSd C1i 0 her(W22 ( + ) ) 5 4 ( + ) ( + ) +( I ) Q +(I )Q12 ( + ) 22 ( + ) " # " T W W21T ( + ) # Y11 ( + ) Y12 ( + ) 11 ( + ) 31 = Y21 Y22 ( + ) ; 32 = W12T ( + ) W22T ( + ) ( + ) Z1 ( + ) Ai + 1 hSb C1i 1 hSa (I )Z1 ( + ) 1 h(I )P2 ; 41 = 33 = hP2 2 (I ) hP2 1 hP2 Ai + 2 hSb C1i 2 hSa 1 (I ) Z1 ( + ) 1 hP2 Z1 ( + ) Adi + 1 hSb C1di 0 ; 44 = ; 42 = 1 hP2 T2 hP2 T 1 hP2 Adi + 2ThSbTC1diT 0 T T Bi P1 ( + ) + hDi Sb hBi P2 + hDi Sb 0 Bi Z1 ( + ) + 1 hDiT SbT ; 53 = 51 = 0 0 hCi hSd C1i hSc 2
Ki g l
Ki g l
1
Ki g l
1
Ki g l
Ki g l
1
Ki g l
Ki g l
1
Ki g l
1
Ki g l
Ki g l
1
Ki g l
Ki g l
1
1
Ki g l
1
1
1
1
1
Ki g l
1
1
Ki g l
Ki g l
Ki g l
Ki g l
1
Ki g l
1
Ki g l
1
1
Ki g l
1
Ki g l
1
Ki g l
Ki g l
1
Ki g l
1
Ki g l
Ki g l
Ki g l
1
Ki g l
1
1
Ki g l
1
Ki g l
Ki g l
1
Ki g l
1
1
1
Ki g l
1
1
Ki g l
1
Ki g l
Ki g l
Ki g l
1
1
1
1
Ki g l
1
then the homogeneous polynomially parameter-dependent matrices given by (33) assure (29) for all 2 . Moreover, if the LMI of (34) is fulfilled ^ is for a given degree g, then the LMI corresponding to any degree g > g also satisfied. Proof 4 The proof is parallel to that of Theorem 3 in [14], using the result in Theorem 3, so it is omitted. Remark 2 Note that, when , 1 and 2 , are fixed, (34) is linear in the variables for the solutions of Theorem 4. To find the optimal values of ,
Table 1 Upper bounds on the allowable time delay for different
2 2
0 2.4601
0.1 2.3752
0.3 2.1843
0.4 2.0749
2
0.6 1.8126
0.9 1.1418
1 and 2 an optimization problem can be solved, using, for example, the
Matlab command Fminsearch. Thus, an optimal H1 filter is obtained by solving the following convex optimization problem. minimize subject to
P1K (g) > 0; P2 > 0; Q11K (g) > 0; Q22K (g) > 0; Z1K (g) > 0 and (34) with = 2 j
j
j
j
Remark 3 It should be noted that the proposed approach could be extended to other related problems, such as Marchesini-Fornasini models, or even multidimensional systems of more than two dimensions (see [1] and [27]).
4 NUMERICAL EXAMPLES Example 1 Consider the dynamical processes in gas absorption, water stream heating and air drying, which can be described by the following Darboux equation with time delays [8]:
@ 2 s(x;t) = a @s(x;t) + a @s(x;t) + a S (x; t) + a s(x; t 1 2 @x 0 3 @x@t @t
2 (t)) + bf (x; t);
(35) where s(x; t) is an unknown function at x(space)2 [0; xf ] and t(time)2 [0; 1], a0 ; a1 ; a2 ; a3 ; and b are real coefficients, 2 (t) is the time-varying delay, and f (x; t) is the input function. Define
r(x; t) = @s(@tx; t) a2 s(x; t)
(36)
Then, equation (35) can be converted into a Roesser model of the form (31) with i h i h i h A = a11 a0 +a2a1 a2 ; Ad = a03 ; B = 0b :
For the specific parameters a0 = 0:2, a1 = 3, a2 = 1, a3 = 0:4 and b = 0 the delay-independent results in [4] and [20] do not provide a feasible solution. On the contrary, using Corollary 1 a feasible solution can be found for a time-varying delay with upper bounds shown in Table 1 for different values of 2 .
Example 2 Consider the 2-D time-varying delay system in the form of (1)-(3) with h
i
h
i
h i
A = 02 00:9 ; Ad = 11 01 ; B = 10 ; C = 2 1 ; C1 = 0 1 ; C1d = 1 0 : The objective is to calculate the minimum H1 performance opt
when the delay is bounded. For this example, the existing method on robust H1 filtering for uncertain 2-D continuous systems with delays [20] does not provide a feasible solution. On the other hand, by applying Theorem 4 in this paper we can find feasible solutions with guaranteed performance: Fig. 1 shows the 1 = 2 value plots of the error dynamics H1 filter; for 1 = 2 = 0:1, good H1 performance bounds are obtained with fixed parameters = 1 = 2 = 1, with the performance further improved when the parameters are optimized (in this case, using the Matlab command Fminsearch gives = 2:0843; 1 = 2:4425 and 2 = 0:1472). The role of the scalar parameters ; 1 ; and 2 in the LMI conditions is to provide extra degrees and to reduce the conservativeness of the LMI tests. Some feasible solutions are listed in Table 2 for different values of 1 and 2 , illustrating clearly that the proposed delay-dependent approach is less conservative than the previous approach in [20], that is always infeasible for this specific example. Moreover, using the proposed approach, upper bounds on the time delay and the minimum noise attenuation values can be easily obtained from Theorem 4: they are listed in Table 2 for both s fixed and searched. As an example of a numerical solution, for = 2:0843; 1 = 2:4425; 2 = 0:1472; = 1:4762, 1 = 2 = 1, 1 = 2 = 0:3, the filter gains obtained are
Af =
6:4808 6:1976 2:1208 12:5568
5:7803 ; Bf = 11 :8212
; Cf = 0:0530 0:0769 :
For the purpose of simulation, the responses x ~h1 (t1 ; t2 ), x ~v1 (t1 ; t2 ) of the error system are shown in Figs. 2 and 3, respectively. Fig. 4 gives the response of the error z~(t1 ; t2 ), starting from the boundary conditions f (; t2 ) = 0:4 and g (t1 ; ) = 0:6, with delays fixed to be 1 = 2 = 1. It is clear that effectively, the 2-D system is asymptotically stable and converges towards zero.
Example 3 Consider the 2-D time-delay system in the form of (1)-(3) with
B
0:05 0:97 :68 1:44 ; A = ; A2 = 01:94 d 1 0 : 01 0:22 1 = ; C = 2 1 ; C1 = 0 1 ; C1d = 1 2 : 0
A1 =
1 1
1 1
;
Table 2 Minimum
Table 3 Minimum
H1 performance min for different values of 1 = 2 and 1 = 2 =1 = 2:0843 1 = 1 1 = 2:4425 2 = 1 2 = 0:1472 1 1
0 1:1575 1:1072 0:3 1:2454 1:1611 0.1 0:9 1:4521 1:279 0 1:5146 1:2056 0:3 2:2313 1:4762 1 0:8 infeasible 21:0835 0 3:6731 2:0389 0:3 7:5098 2:5348 2 0:6 infeasible 2:9869 0 3:376 2:9538 0:3 24:5 4:2971 3 0:6 infeasible 7:19 0 35:32 8:3255 0:3 infeasible infeasible 4 0:6 infeasible infeasible 0 infeasible 18:6871 0:3 infeasible infeasible 4.5 0:6 infeasible infeasible H1 performance min for different values of 1 = 2 and 1 = 2
min 1 1 g=0 g=1 g=2 3:62141 3:6220 0 4:3118 3:9562 3:9566 0:3 4:1178 0.1 4:6518 4:6532 0:6 4:6895 8:4404 8:4681 0:9 9:0179 0 8:2624 4:0566 3:9662 0:3 8:5850 4:3890 4:2291 0.2 0:6 10:7976 5:6474 5:5250 0 infeas 4:8858 4:6220 0:3 infeas 5:9323 5:5410 0.3 0:6 infeas 31:8347 15:0570 0 infeas 7:1805 6:0547 0:3 infeas 56:1311 11:7152 0.4 0:4 infeas infeas infeas 0 infeas infeas 14:0397 0.5
By Theorem 4 in this paper we can find feasible solutions with guaranteed performance, whereas the previous result in [10] does not provide a feasible solution for any g. Table 3 gives some H1 performance bounds. It can be seen that the best results are obtained for g = 2 ( = 1:5973; 1 = 0:0272; and 2 = 0:1259), when the result in Theorem 4 correspond to a quadratic framework formulation, compared with g = 1 (linearly parameter-dependent polynomial), and g = 0.
Fig. 1 The achieved performance opt for different values of the tuning parameter delays smaller than 1 = 0:1 (Example 2))
Fig. 2 Response of x ~h 1 (t1 ; t2 ) with the proposed filter (Example 2).
1 , and
Fig. 3 Response of x ~v1 (t1 ; t2 ) with the proposed filter (Example 2).
Fig. 4 Filtering error response of z~(t1 ; t2 ) with the proposed filter (Example 2).
5 CONCLUSION Based on homogeneous polynomially parameter-dependent matrices, we have studied the delay-dependent H1 filtering problem for 2-D continuous systems with state delay, described by Roesser’s state space model with polytopic uncertainty. By increasing the degree of some polynomials used in the solution, the designed H1 filters reduce the conservativeness inherent to robust filters. Our design method uses slacks variables that provide flexibility in the optimization. Examples are given to illustrate the application of the proposed methodology. It must be pointed out that further work is being done using finite frequency (FF) filtering ([13], [26]), as existence conditions of robust FF H1 filters for two-dimensional systems can be obtained in terms of an optimization problem (see [3], [25], [36]).
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