z( k) =
, Pd ( k) = ⎢
⎥ , PA Q = ⎢
⎥ ,
=
⎢
⎥
. (44)
s
s
P s
B
⎢
⎥
⎣ 2
z ( k)⎦
⎣ s
d 2( k)⎦
⎣ 0
I ⎦
⎣ s
B 2 ⎦
Let C Q = [ C 1 C 2 ],
v
v
v
then
(
v k) = C
+
1 1
z ( k) C 2 2
z ( k)
v
v
. (45)
From Eq.(43) and Eq.(45), we have
(
v k) + C
=
−
2 B 2 f ( (
v k)) C 1 z( k) C 2 d 2( k)
v
s
v
v
s
. (46)
From Eq.(46), we have
∂
f
∂ ( (
v k))
( (
v k) + C B f v k = I + C B
.
T
v 2 s 2 ( ( ))
v 2 s 2
v
∂ ( k)
T
v
∂ ( k)
Existing condition of (
v k) is
f
∂ ( (
v k))
I + C
≠
v 2 s
B 2
0 . (47)
T
v
∂ ( k)
From Eq.(44), we have
zI − A
0
s 1
P zE −
= α
−
=
= α
−
s
A Q
PQ zE
s
A
I zI
s
A 1 . (48)
0
− I
Here, α
α
PQ and I are fixed. So, from Eq.(39),
s
A 1 is a stable system matrix.
Consider a quadratic Lyapunov function candidate
V( k)
T
= 1
z ( k) s
P 1
z ( k) . (49)
138
Discrete Time Systems
The difference of V( k) along the trajectories of system Eq.(42) is given by
V
Δ ( k) = V( k + 1) − V( k)
T
T
=
+
+ −
(50)
1
z ( k 1) s
P 1
z ( k 1)
1
z ( k) s
P 1
z ( k)
= [
T
T
+
+
+
+
−
s
A 1 1
z ( k)
s
B 1 f ( (
v k))
s
d 1( k ]) s
P [ s
A 1 1
z ( k)
s
B 1 f ( (
v k))
s
d 1( k ]) 1
z ( k) s
P 1
z ( k)
T
−
= −
s
A 1 s
P s
A 1
s
P
s
Q , (51)
where s
Q and s
P are symmetric positive definite matrices defined by Eq.(51). If s
A 1 is a
stable matrix, we can get a unique s
P from Eq.(51) when s
Q is given. As 1( )
s
d k is bounded
and 0 ≤ γ < 1 , V
Δ ( k) satisfies
V
Δ ( k)
T
≤ − z ( k) Q z ( k) + X z ( k) f ( (
v k))
1
s 1
1
1
(52)
2
+ X
+ μ
+
+
2
1
z ( k)
2 f ( (
v k))
X 3 f ( (
v k))
X 4
From Eq.(40), we have
≤
1
z ( k)
M (
z k) . (53)
Here, M is positive constant. From Eq.(52), Eq.(53), we have
2
1
V
Δ ( k)
+γ
≤ −μ
+
+
1
(
z k)
X 5 1
z ( k)
X 6
2
≤ −μ
(
z k) + X
c
(54)
2
≤ − μ
+
c 1
1
z ( k)
X
≤ −μ V( k) + X,
m
where 0 < μ = λ
μ ≥
< μ < μ <
μ
μ μ
1
min (
), 2 0
s
Q
and 0
min( 1,1)
m
c
. Also, 1, 2 , X ( i = 1 ∼ 6)
i
and
X are positive constants. As a result of Eq.(54), V( k) is bounded:
V( k) ≤ V(0) + X / μ m . (55)
Hence, 1
z ( k) is bounded. From Eq.(43), 2
z ( k) is also bounded. Therefore, (
z k) is bounded.
The above result is summarized as Theorem1.
[Theorem1]
In the nonlinear system
Ex( k + 1) = Ax( k) +
(
Bu k) + B f ( (
v k)) + (
d k)
f
(
v k) = C x( k)
f
(56)
y( k) = C (
x k) + 0
d ( k),
where ( )
n
∈
, ( )∈
, ( )∈
, ( )
f
∈
, ( )
n
x k
R u k
R y k
R v k
R
d k ∈ R ,
f
∈
∈
0
d ( k) R , f ( (
v k)) R , d( k) and
0
d ( k) are assumed to be bounded. All the internal states are bounded and the output error
(
e k) = y( k) − y ( k)
m
asymptotically converges to zero in the design of the model following
control system for a nonlinear descriptor system in discrete time, if the following conditions
are held:
The Design of a Discrete Time Model Following Control System for Nonlinear Descriptor System 139
1. Both the controlled object and the reference model are controllable and observable.
2.
N ≠ 0.
r
3. Zeros of [
] 1
C zE A −
−
B are stable.
4.
f ( (
v k)) ≤ +
(
v k) γ
α β
,(α ≥ 0,β ≥ 0,0 ≤ γ < 1) .
5. Existing condition of (
v k) is
f
∂ ( (
v k))
I + C
≠ .
v 2 s
B 2
0
T
v
∂ ( k)
6.
zE − A ≡/ 0 and rankE = deg zE − A = r ≤ n .
5. Numerical simulation
An example is given as follows:
⎡1 0 1⎤
⎡ 0
1
0 ⎤
⎡0 0⎤
⎡1⎤
⎡0⎤
⎢0 1 1⎥ ( xk 1) ⎢ 0
0
1 ⎥ (
x k) ⎢1 0⎥ (
u k) ⎢0⎥ f ( (
v k)) ⎢1⎥
+
=
+
+
+
(
d k)
⎢
⎥
⎢
⎥
⎢
⎥
⎢ ⎥
⎢ ⎥
⎢1 0 1⎥
⎢0.2
0
−
⎣
⎦
⎣
.5 0.6⎥
⎢
⎦
⎣0 1⎥
⎢
⎦
⎣1⎥
⎢
⎦
⎣1⎥⎦
(
v k) = [1 1 1] (
x k) ,
⎡ 0 0.1 0 ⎤
⎡1⎤
y( k) =
⎥ (
x k) +
⎢
(57)
0.1 0 0.1
⎢1⎥
⎣
⎦
⎣ ⎦
3
3 v ( k) + 4 (
v k) + 1
f ( (
v k)) =
.
4
1 + v ( k)
Reference model is given by
⎡ 0
1 ⎤
⎡0⎤
x ( k + 1) =
x ( k) +
⎢
⎥
⎢ ⎥ r ( k)
m
0.12
−
⎣
0.7 m
⎦
⎣1 m
⎦
y ( k) =
m
[1 0] x ( k)
m
(58)
r ( k) = sin( kπ /16) .
m
In this example, disturbances d( k) and 0
d ( k) are ramp and step disturbances respectively.
Then d( k) and 0
d ( k) are given as
d( k) = 0.05( k − 85),(85 ≤ k ≤ 100)
(59)
=
≤ ≤
0
d ( k) 1.2,(20 k 50)
We show a result of simulation in Fig. 1. It can be concluded that the output signal follows
the reference even if disturbances exit in the system.
6. Conclusion
In the responses (Fig. 1) of the discrete time model following control system for nonlinear
descriptor system, the output signal follows the references even though disturbances exit in
the system. The effectiveness of this method has thus been verified. The future topic is that
the case of nonlinear system for γ ≥ 1 will be proved and analysed.
140
Discrete Time Systems
Fig. 1. Responses of the system for nonlinear descriptor system in discrete time
7. References
Wu,S.; Okubo,S.; Wang,D. (2008). Design of a Model Following Control System for
Nonlinear Descriptor System in Discrete Time, Kybernetika, vol.44,no.4,pp.546-556.
Byrnes,C.I; Isidori,A. (1991). Asymptotic stabilization of minimum phase nonlinear system,
IEEE Transactions on Automatic Control, vol.36,no.10,pp.1122-1137.
Casti,J.L. (1985). Nonlinear Systems Theory, Academic Press, London.
Furuta,K. (1989). Digital Control (in Japanese), Corona Publishing Company, Tokyo.
Ishidori,A. (1995). Nonlinear Control Systems, Third edition, Springer-Verlag, New York.
Khalil,H.K. (1992). Nonlinear Systems, MacMillan Publishing Company, New York.
Mita,T. (1984). Digital Control Theory (in Japanese), Shokodo Company, Tokyo.
Mori,Y. (2001). Control Engineering (in Japanese), Corona Publishing Company, Tokyo.
Okubo,S. (1985). A design of nonlinear model following control system with disturbances
(in Japanese), Transactions on Instrument and Control Engineers, vol.21,no.8,pp.792-
799.
Okubo,S. (1986). A nonlinear model following control system with containing inputs in
nonlinear parts (in Japanese), Transactions on Instrument and Control Engineers,
vol.22,no.6,pp.714-716.
Okubo,S. (1988). Nonlinear model following control system with unstable zero points of the
linear part (in Japanese), Transactions on Instrument and Control Engineers,
vol.24,no.9,pp.920-926.
Okubo,S. (1992). Nonlinear model following control system using stable zero assignment (in
Japanese), Transactions on Instrument and Control Engineers, vol.28, no.8, pp.939-946.
Takahashi,Y. (1985). Digital Control (in Japanese), Iwanami Shoten,Tokyo.
Zhang,Y; Okubo,S. (1997). A design of discrete time nonlinear model following control
system with disturbances (in Japanese), Transactions on The Institute of Electrical
Engineers of Japan, vol.117-C,no.8,pp.1113-1118.
9
Output Feedback Control of Discrete-time
LTI Systems: Scaling LMI Approaches
Jun Xu
National University of Singapore
Singapore
1. Introduction
Most physical systems have only limited states to be measured and fed back for system
controls.
Although sometimes, a reduced-order observer can be designed to meet the
requirements of full-state feedback, it does introduce extra dynamics, which increases the
complexity of the design. This naturally motivates the employment of output feedback, which
only use measurable output in its feedback design. From implementation point of view, static
feedback is more cost effective, more reliable and easier to implement than dynamic feedback
(Khalil, 2002; Kučera & Souza, 1995; Syrmos et al., 1997). Moreover, many other problems are
reducible to some variation of it. Simply stated, the static output feedback problem is to find
a static output feedback so that the closed-loop system has some desirable characteristics, or
determine the nonexistence of such a feedback (Syrmos et al., 1997). This problem, however,
still marked as one important open question even for LTI systems in control engineering.
Although this problem is also known NP-hard (Syrmos et al., 1997), the curious fact to
note here is that these early negative results have not prevented researchers from studying
output feedback problems. In fact, there are a lot of existing works addressing this problem
using different approaches, say, for example, Riccati equation approach, rank-constrained
conditions, approach based on structural properties, bilinear matrix inequality (BMI)
approaches and min-max optimization techniques (e.g., Bara & Boutayeb (2005; 2006); Benton
(Jr.); Gadewadikar et al. (2006); Geromel, de Oliveira & Hsu (1998); Geromel et al. (1996);
Ghaoui et al. (2001); Henrion et al. (2005); Kučera & Souza (1995); Syrmos et al. (1997) and the
references therein). Nevertheless, the LMI approaches for this problem remain popular (Bara
& Boutayeb, 2005; 2006; Cao & Sun, 1998; Geromel, de Oliveira & Hsu, 1998; Geromel et al.,
1996; Prempain & Postlethwaite, 2001; Yu, 2004; Zečević & Šiljak, 2004) due to simplicity and
efficiency.
Motivated by the recent work (Bara & Boutayeb, 2005; 2006; Geromel et al., 1996; Xu & Xie,
2005a;b; 2006), this paper proposes several scaling linear matrix inequality (LMI) approaches
to static output feedback control of discrete-time linear time invariant (LTI) plants. Based on
whether a similarity matrix transformation is applied, we divide these approaches into two
parts. Some approaches with similarity transformation are concerned with the dimension
and rank of system input and output. Several different methods with respect to the system
state dimension, output dimension and input dimension are given based on whether the
distribution matrix of input B or the distribution matrix of output C is full-rank. The other
142
Discrete Time Systems