CL−1 =
I p 0 . Using such matrix L in a similarity
transformation applied to (1) it yields
ˆ xk+1 = ˆ
A( α) ˆ xk + ˆ
Ad( α) ˆ xk− d + ˆ B( α) u
k
k,
(83)
Uncertain Discrete-Time Systems with Delayed State:
Robust Stabilization with Performance Specification via LMI Formulations
315
0
−0.05
−0.1
Kσ x
k − k 0.15
−0.2
−0.25
0
100
200
300
400
500
600
700
800
k
0
−0.05
−0.1
Kdσ x
k
k −
−0.15
dk
−0.2
−0.25
0
100
200
300
400
500
600
700
800
k
Fig. 5. Control signal uk = Kσ x
x
, with K x
x
shown in the top
k
k + Kd, σ
σ
k
k− dk
k
k and Kd, σk k− dk
and bottom parts, respectively.
where ˆ
A( α) = L ˜
A( α) L−1, ˆ
Ad( α) = L ˜
Ad( α) L−1 and ˆ B( α) = L ˜ B( α), ˆ xk = Lxk and the output
signal is given by yk = I P 0 ˆ xk. Thus, the objective here is to find robust static feedback
gains K ∈ R p× and K d ∈ R p× such that (83) is robustly stabilizable by the control law
uk = K yk + K dyk− d
(84)
k
These gains can be determined by using the conditions of theorems 3, 4, 6 with the following
structures
F = F11
o
0
F21
,
W = W K 0 , Wd = W K 0
d
o
F22
o
with F 11
o
∈ R p× p, F21
o
∈ R( n− p)× p, F22
o
∈ R( n− p)×( n− p), W K ∈ R p× n, W K ∈ R p× n which d
yields
K = K 0
and
Kd = K d 0
Note that,
similarly to the decentralized case, no constraint is taken over the
Lyapunov-Krasovskii function matrices leading to less conservative conditions, in general.
5.6 Input delay
Another relevant issue in Control Theory is the study of stability and stabilization of input
delay systems, which is quite frequent in many real systems Yu & Gao (2001), Chen et al.
316
Discrete Time Systems
(2004). In this case, consider the controlled system given by
xk+1 = A( α) xk + B( α) uk− d
(85)
k
with A( α) and B( α) belonging to polytope (2), Adi = 0 and α ∈ Υ. In Zhang et al. (2007) this system is detailed investigated and the problem is converted into an optimization problem
in Krein space with an stochastic model associated. Here, the delayed input control signal is
considered as
uk− d = K
(86)
k
d xk− dk
The closed-loop-system is given by
xk+1 = ˜
A( α) xk + ˜
Ad( α) xk− d
(87)
k
with ˜
A( α) = A( α), ˜
Ad( α) = B( α) Kd. Thus, with known Kd, closed-loop system (87) is
equivalent to (7) with null exogenous signal wk. This leads to simple analysis stability
conditions obtained from Theorem 1 replacing ˜
Ai by Ai and ˜
Adi by BiKd, i = 1, . . . , N.
Besides, similar replacements can be used with conditions presented in theorems 2 and 5 and
in Corollary 1. The possibility to address both controller fragility and input delay is a side
result of this proposal. In the former it is required that no uncertainty affects the input matrix,
i.e., B( α) = B, ∀ α ∈ Υ, while the latter can be used to investigate the bounds of stability
of a closed-loop system with a delay due to, for example, digital processing or information
propagation.
In case of the design of Kd it is possible to take similar steps with conditions of theorems 3, 4
and 6. In this case, it is sufficient to impose, Adi = 0, i = 1, . . . , N and W = 0 that yield K = 0.
Finally, observe that static delayed output feedback control can be additionally addressed here
by considering what is pointed out in Subsection 5.5.
5.7 Performance by delay-free model specification
Some well developed techniques related to model-following control (or internal model
control) can be applied in the context of delayed state systems. The major advantage of
such techniques for delayed systems concerns with the design with performance specification
based on zero-pole location. See, for example, the works of Mao & Chu (2009) and Silva
et al. (2009). Generally, the model-following control design is related to an input-output
closed-loop model, specified from its poles, zeros and static gain, from which the controller
is calculated. As the proposal presented in this chapter is based on state feedback control,
it does not match entirely with the requirements for following-model, because doing state
feedback only the poles can be redesigned, but not the zeros and the static gain. To develop a
complete following model approach an usual way is to deal with output feedback, that yields
a non-convex formulation. One way to match all the requirements of following model by
using state feedback and maintaining the convexity of the formulation, is to use the technique
presented by Coutinho et al. (2009) where the model to be matched is separated into two
parts: One of them is used to coupe the static gain and zeros of the closed loop system with
the prescribed model and the other part is matched by state feedback control. Consider the
block diagram presented in Figure 6. In this figure, Ω( α) is the system to controlled with
signal uk. This system is subject to input wk which is required to be reject at the output yk.
Please, see equation (1). Ω m stands for a specified delay-free model with realization given by
Am Bm . The model receives the same exogenous input of the system to be controlled, w
C
k,
m Dm
and has an output signal ymk at the instant k.
Uncertain Discrete-Time Systems with Delayed State:
Robust Stabilization with Performance Specification via LMI Formulations
317
wk
yk
Ω( α) = A( α) Ad( α) B( α) Bw( α)
C( α) Cd( α) D( α) Dw( α)
uk
xk
+
−
ek
K
xk− dk
+
+
Kd
ymk
Ω m = Am Bm
Cm Dm
Fig. 6. Following model inspired problem.
The objective here is to design robust state feedback gains K and Kd to implement the control
law (6) such that the H∞ guaranteed cost between the input wk and the output ek = yk − ymk is
minimized. In other words, it is desired that the disturbance rejection of the uncertain system
with time-varying delay in the state have a behavior as close as possible to the behavior of
the specified delay-free model Ω m. The dashed line in Figure 6 identifies the enlarged system
required to have its H∞ guaranteed cost minimized.
Taking the closed-loop system (7) and the specified model of perturbation rejection given by
xmk+1 = Amxmk + Bmwk
(88)
ymk = Cmxmk + Dmwk
(89)
where xmk ∈ R nm is the model state vector at the k-th sample-time, ymk ∈ R p is the output
of the model at the same sample-time and wk ∈ R is the same perturbation affecting the
controlled system, the difference ek = ymk − zk is obtained as
⎡
⎤
xmk
e
⎣
⎦
k =
Cm −( C( α) + D( α) K) −( Cd( α) + D( α) Kd)
xk
xk− dk
+ Dm − Dw( α) wk (90)
Thus, by using (1) with (88)-(89) and (90) it is possible to construct an augmented system
composed by the state of the system and those from model yielding the following system
ˆ
Ω( α)
ˆ x
+ ˆ B
:
k+1 = ˆ
A( α) ˆ xk + ˆ
Ad( α) ˆ xk− dk
w ( α) wk
(91)
ek = ˆ
C( α) ˆ xk + ˆ
Cd( α) ˆ xk− d + ˆ D
k
w ( α) wk
318
Discrete Time Systems
T
with ˆ xk = xT xT
∈ R nm+ n, ˆΩ( α) ∈ ˆP,
mk
k
N
ˆ
P =
ˆ
Ω( α) ∈ R n+ nm+ p×2( n+ nm)+ : ˆΩ( α) = ∑ α ˆ
iΩ i, α ∈ Υ
(92)
i=1
where
ˆ
ˆ
ˆ
ˆ
Ω
Ai Adi Bwi
i =
ˆ
C
ˆ
ˆ
i Cdi Dwi
⎡
⎤
Am
0
0
0
Bm
= ⎣ 0
A
⎦
i + BiK
0
Adi + BiKd
Bwi
, i ∈ I[1, N].
(93)
Cm − ( Ci + DiK) 0 − ( Cdi + DiKd) Dm − Dwi
Therefore, matrices in (93) — ˆ
Ai, ˆ
Adi, ˆ Bwi, ˆ
Ci, ˆ
Cdi, ˆ
Dwi — can be used to replace their respective
in (38) and (23). As a consequence, LMI (36) becomes with 3( n + nm) + 2( p + ) rows. Since
the main interest in this section is to design K and Kd that minimize the H∞ guaranteed
cost between ek and wk, only the design condition is presented in the sequel. To achieve
such condition, similar steps of those taken in the proof of Theorem 4 are taken. The main
differences are related to i) the size and structure of the matrices and ii) the manipulations
done to keep the convexity of the formulation.
˜
P
˜
P
Theorem 7. If there exist symmetric matrices 0 < ˜
P
11 i
12 i
i =
˜
∈ R n+ nm× n+ nm, 0 < ˜ Q
P
i =
22 i
˜
Q
˜
11 i Q 12 i
˜
∈ R n+ nm× n+ nm, matrices F =
F11 F12 ∈ R n+ nm× n+ nm, Λ ∈ R n× nm is a given
Q 22 i
F22Λ F22
matrix, W ∈ R p× n, Wd ∈ R p× n, a scalar variable θ ∈]0, 1] and for a given μ = γ 2 such that
⎡ ˜ P
˜
11 i − F11 − F T P 12 i − F12 − Λ T F T
⎢
11
22
Am F11
Am F12
⎢
˜
P 22 i − F22 − F T
( Ai F22 + BiW)Λ Ai F22 + BiW
⎢
22
⎢
β ˜
Q
⎢
11 i − ˜
P 11 i
β ˜
Q 12 i − ˜ P 12 i
¯
Ψ
⎢
β ˜
Q 22 i − ˜ P 22 i
i = ⎢
⎢
⎢
⎢
⎣
⎤
0
0
0
Bm
( A
⎥
di F22 + BiWd)Λ Adi F22 + BiWd
0
Bwi
⎥
0
0
F T CT
+ F T
)
0
⎥
11 m − Λ T ( W T DT
i
22 CT
i
⎥
0
0
F T
+ F T
)
⎥
12 CT
m − ( W T DT
0
i
22 CT
i