E{ E {e(0)}} ˆ xe '(0)=O.
0
R
0
R
We have already shown that, for each k∈ [0,N],
E{ E {e(k) r '(k)1
R
d, c
{m(k)=i}}}=O
k
in section 4. Suppose
E{ E {e(k) ˆ x '(k)1
R
e
{m(k)=i}}}=O.
k
Then, since ωd(k) is zero mean, not correlated with ˆ xe (k) and rd, c (k) and independent of
m(k), we have
E{ E {e(k + 1) ˆ x ’ +
R
e (k
1) 1
}
{m k+1 =i}
k+1
}
(
)
= ∑ p ⎡⎡A + M C ⎤ k E E {e k ˆ
⎡
+
⎤
d, ij ⎢⎣ d,i
i i ⎦ ( )
⎣
{ ( ) x '
R
e (k ) {m
1 k =i} ⎣
⎦
k
}
( ) } Ad,i MiC2d,i ’(k)
i∈ J( k)
+ ⎡G
+ M H ⎤ k E E {ω k ˆ
⎡
+
⎤
⎣ d,i
i
d,i ⎦ ( ) {
x
R
d ( )
'
e (k ) {m
1 k =i} ⎣
⎦
k
}
( ) } Ad,i MiC2d,i ’(k)
+ ⎡A
+
C
⎤
⎣
⎦(k)E{ E {e(k) ud, c*
i 2d,i
'(k)
R
{m
1 k =i}
k
}}B2d,i’(k)
d,i
M
( )
+ ⎡G
+
⎤
ω
⎣ d,i MiHd,i ⎦(k)E{ E {
u
R
d (k ) d, c* '(k ) 1
} B
’ k
{m k =i}
k
}
( )
2d,i ( )
+ ⎡A
+ M C
⎤
⎣
⎦(k)E{ E {e(k) rd,
d,i
i 2d,i
c '(k )
R
{m
1 (k)=i}
k
}}B3d,i’(k)
+ ⎡G
+
H ⎤
ω
⎣
d,i ⎦ (k )E{ E {
r
R
d (k ) d, c '(k ) 1
} B
’ k
{m k =i}
k
} 3d,i ( )
d,i
Mi
( )
− ⎡A
+
⎤
⎣ d,i MiC2d,i ⎦(k)E{ E {e(k)y'(k)
R
{m
1 k =i}
k
}
( ) } Mi ’(k)
− G
+
ω
⎤
⎡
⎤
⎣ d,i MiHd,i ⎦(k)E{ E {
R
d (k ) y'(k ) 1
} M ’ k
{m k =i}
⎥
k
}
( )
i ( )⎦
= ∑ p ⎡−⎡
+
⎤
⎢⎣
{
(k)1
} M ’ k
{m k =i} }
i ( )
d,
A
ij
⎣ d,i MiC2d,i ⎦(k)E E {e(k)y'
R
( )
k
i∈ J( k)
− G
+
ω
⎤
⎡
⎤
⎣ d,i MiHd,i ⎦(k)E{ E {
R
d (k ) y'(k ) {m
1 (k)=i}
⎥
k
}}Mi’(k)⎦
128
Discrete Time Systems
where ud, c* (k)=F2,i(k) ˆ xe (k) , i=1, ···,M. Notice that
y(k)= C2d,m(k)(k)x(k)+Hd,m(k)(k)ωd(k)= C2d,m(k)(k)(e(k)+ ˆ xe (k))+Hd,m(k)(k)ωd(k).
Then, by induction on k, we obtain
E{ E {e(k)y’(k)1
E {e(k)e'(k)1
E {e(k) ˆ x '(k)1
R
{m(k)=i}}}= E{
{m(k)=i}}}C2d,i'(k)+E{
e
{m(k)=i}}}C2d,i'(k)
k
Rk
Rk
+ E{ E {e(k)ω
R
d'(k)1{m(k)=i}}}Hd,i'(k)
k
=Yi(k)C2d,i'(k)
We also obtain
E{ E {ω
R
d(k)y’(k)1{m(k)=i}}}
k
= E{ E {ω
E {ω
x '(k)1
R
d(k)e'(k)1{m(k)=i}}}C2d,i'(k)+E{
d(k) ˆ e
{m(k)=i}}}C2d,i'(k)
k
Rk
+ E{ E {ω
R
d(k)ωd'(k)1{m(k)=i}}}Hd,i'(k)
k
= E{ωd(k)ωd'(k)}P{m(k)=i}Hd,i'(k)= πi(k)Hd,i'(k).
Then considering the assumption A4 Gd,i(k)Hd,i'(k) = O, i=1, ···,M, and
Mi(k)(Hd,iHd,i’πi(k)+ C2d,iYi(k)C2d,i’)= - Ad,iYi(k)C2d,i’
by (11), we finally obtain
E{ E
{e(k+1) ˆ x '(k+1) 1
R
e
{m(k+1)=i}}}
k+1
= ∑ pd, ij [-[Ad,i+MiC2d,i](k)Yi(k)C2d,i'(k)-[Gd,i+MiHd,i](k)πi(k)Hd,i'(k)]Mi’(k)
i∈ J( k)
= ∑ pd, ij [-Ad,iYi(k)C2d,i'(k)-Mi(k)(Hd,iHd,i’πi(k)+ C2d,iYi(k)C2d,i’)]Mi’(k)
i∈ J( k)
= ∑ pd, ij [-Ad,iYi(k)C2d,i'(k)+ Ad,iYi(k)C2d,i’]Mi’(k)
i∈ J( k)
=0
which concludes the proof. (Q.E.D.)
7. References
E. K. Boukas. (2006). Stochastic Switching Systems: Analysis and Design, Birkhauser, 0-8176-
3782-6, Boston,
A. Cohen. & U. Shaked. (1997). Linear Discrete-Time H∞-Optimal Tracking with Preview.
IEEE Trans. Automat. Contr., 42, 2, 270-276
O. L. V. Costa. & E. F. Tuesta. (2003). Finite Horizon Quadratic Optimal Control and a
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O. L. V. Costa.; M. D. Fragoso. & R. P. Marques. (2005). Discrete-Time Markov Jump Linear
Systems, Springer, 1-85233-761-3, London
V. Dragan. & T. Morozan. (2004). The linear quadratic optimization problems for a class of
linear stochastic systems with multiplicative white noise and Markovian jumping.
IEEE Trans. Automat. Contr., 49, 5, 665-675
M. D. Fragoso. (1989). Discrete-Time Jump LQG Problem. Int. J. Systems Science, 20, 12, 2539-
2545
M. D. Fragoso.; J. B. R. do Val . & D. L. Pinto Junior. (1995). Jump Linear H∞ Control: the
discrete-time case. Control-Theory and Advanced Technology, 10, 4, 1459-1474
E. Gershon.; D. J. N. Limebeer.; U. Shaked. & I. Yaesh. (2004). Stochastic H∞ Tracking with
Preview for State-Multiplicative Systems. IEEE Trans. Automat. Contr., 49, 11, 2061-
2068
E. Gershon.; U. Shaked. & I. Yaesh. (2004). H∞ tracking of linear continuous-time systems
with stochastic uncertainties and preview. Int. J. Robust and Nonlinear Control, 14, 7,
607-626
E. Gershon.; U. Shaked. & I. Yaesh. (2005). H∞ Control and Estimation of State-Multiplicative
Linear Systems, LNCIS 318, Springer, 1-85233-997-7, London
J.-W. Lee. & P. P. Khargonekar. (2008). Optimal output regulation for discrete-time switched
and Markovian jump linear systems, SIAM J. Control Optim., 47, 1, 40-72
M. Mariton. (1990). Jump Linear Systems in Automatic Control, Marcel Dekker, 0-8247-8200-3,
New York
G. Nakura. (2008a). Noncausal Optimal Tracking for Linear Switched Systems. In: Hybrid
Systems: Computation and Control: 11th International Workshop, HSCC 2008, St. Louis,
MO, USA, April, 2008, Proceedings, LNCS 4981, M. Egerstedt. & B. Mishra. (eds.),
pp.372-385, Springer, 3-540-78928-6, Berlin, Heidelberg.
G. Nakura. (2008b). H∞ Tracking with Preview for Linear Systems with Impulsive Effects -
State Feedback and Full Information Cases-. Proceedings of the 17th IFAC World
Congress, TuA08.4 (CD-ROM), Seoul, Korea
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Impulsive Effects. Proceedings of the 17th IFAC World Congress, TuA08.5 (CD-ROM),
Seoul, Korea
G. Nakura. (2008d). Stochastic Optimal Tracking with Preview for Linear Continuous-Time
Markovian Jump Systems. Proceedings of SICE Annual Conference 2008, 2A09-2 (CD-
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G. Nakura. (2010). Stochastic Optimal Tracking with Preview by State Feedback for Linear
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8
The Design of a Discrete Time Model Following
Control System for Nonlinear Descriptor System
Shigenori Okubo1 and Shujing Wu2
1Yamagata University
2Shanghai University of Engineering Science
1Japan
2P. R. China
1. Introduction
This paper studies the design of a model following control system (MFCS) for nonlinear
descriptor system in discrete time. In previous studies, a method of nonlinear model
following control system with disturbances was proposed by Okubo,S. and also a nonlinear
model following control system with unstable zero of the linear part, a nonlinear model
following control system with containing inputs in nonlinear parts, and a nonlinear model
following control system using stable zero assignment. In this paper, the method of MFCS
will be extended to descriptor system in discrete time, and the effectiveness of the method
will be verified by numerical simulation.
2. Expressions of the problem
The controlled object is described below, which is a nonlinear descriptor system in discrete
time.
(
Ex k + 1) = A (
x k) +
(
Bu k) + B f ( (
v k)) + (
d k) (1)
f
(
v k) = C (
x k) (2)
f
y( k) = Cx( k) + 0
d ( k) (3)
The reference model is given below, which is assumed controllable and observable.
x ( k + 1) = A x ( k) + B r ( k)
m
m m
m m
(4)
y ( k) = C x ( k)
m
m m
(5)
, where
(
x k)
n
∈ R , (
d k)
n
∈ R , (
u k)∈ R , y( k)∈ R , y ( k)∈ R ,
f
∈
∈
m
0
d ( k) R , f ( (
v k)) R ,
(
v k)
f
∈ R , r ( k)
m
∈ R , x ( k)
m
n
∈ R ,
m
m
y( k) is the available states output vector, (
v k) is the
measurement output vector, (
u k) is the control input vector, x( k) is the internal state vector
132
Discrete Time Systems
whose elements are available, d( k), 0
d ( k) are bounded disturbances, y ( k)
m
is the model
output.
The basic assumptions are as follows:
1. Assume that ( C, A, )
B is controllable and observable, i.e.
⎡ zE − A⎤
ran [
k zE − A, B] = n, rank
=
⎢
⎥ n .
⎣ C ⎦
2. In order to guarantee the existence and uniqueness of the solution and have exponential
function mode but an impulse one for (1), the following conditions are assumed.
zE − A ≡/ 0, rankE = deg zE − A = r ≤ n
3. Zeros of [
] 1
C zE A −
−
B are stable.
In this system, the nonlinear function f ( (
v k)) is available and satisfies the following
constraint.
f ( (
v k)) ≤ +
(
v k) γ
α β
,
where α ≥ 0,β ≥ 0,0 ≤ γ < 1, ⋅ is Euclidean norm, disturbances d( k), 0
d ( k) are bounded and
satisfy
( ) ( ) 0
d
D z d k = (6)
( ) 0( ) 0
d
D z d k = . (7)
Here,
( )
d
D z is a scalar characteristic polynomial of disturbances. Output error is given as
(
e k) = y( k) − y ( k)
m
. (8)
The aim of the control system design is to obtain a control law which makes the output error
zero and keeps the internal states be bounded.
3. Design of a nonlinear model following control system
Let z be the shift operator, Eq.(1) can be rewritten as follows.
C[ zE A] 1
−
−
B = N( z) / (
D z)
C[ zE A] 1
−
−
B = N ( z) / (
D z) ,
f
f
where (
D z) = zE − A , ∂ ( N( z)) = σ and ∂ ( N ( z)) = σ .
i
r
i
i
r
f
fi
Then the representations of input-output equation is given as
(
D z) y( k) = N( z) (
u k) + N ( z) f ( (
v k)) + (
w k)
f
.
(9)
Here (
w k) = Cadj[ zE − A] (
d k) + (
D z) 0
d ( k) , ( C , A , B )
m
m
m is controllable and observable. Hence,
C [ zI A ] 1
−
−
B = N ( z) / D ( z)
m
m
m
m
m
.
Then, we have
The Design of a Discrete Time Model Following Control System for Nonlinear Descriptor System 133
D ( z) y ( k) = N ( z) r ( k) , (10)
m