functions) and the one used, in general, for the stability of switched systems with time-varying
delay. The latter gives nonconvex conditions for switched systems where each operation mode
is subject to a norm-bounded uncertainty and constant delay.
The problem of robust filtering for discrete-time uncertain systems with delayed state is
considered in some papers. Delayed state systems with norm-bounded uncertainties are
studied by Yu & Gao (2001), Chen et al. (2004) and Xu et al. (2007) and with polytopic
uncertainties by Du et al. (2007). The results of Gao et al. (2004) were improved by Liu et al.
(2006), but the approach is based on QS and the design conditions are nonconvex depending
directly on the Lyapunov-Krasovskii matrices.
The problem of output feedback has attracted attention for discrete-time systems with delay in
the state and the works of Gao et al. (2004), He et al. (2008) and Liu et al. (2006) can be cited as
examples of on going research. In special, He et al. (2008) present results for precisely known
systems with time-varying delay including both static output feedback (SOF) and dynamic
output feedback (DOF). However, the conditions are presented as an interactive method that
relax some matrix inequalities.
The main objective of this chapter is to study the robust analysis and synthesis of discrete-time
systems with state delay. This chapter is organized as follows. In Section 2 some notations
and statements are presented, together the problems that are studied and solved in the next
sections. In sections 3 and 4 solutions are presented for, respectively, robust stability analysis
and robust design, based in a L-K function presented in section 2. In Section 5 some additional
results are given by the application of the techniques developed in previous sections are
presented, such as: extensions for switched systems, to treat actuator failure and to make
design with pole location. In the last section it is presented the final comments.
2. Preliminaries and problem statement
In this chapter the uncertain discrete time system with time-varying delay in the state vector
is given by
Ω( α)
x
+ B( α) u
:
k+1 = A( α) xk + Ad( α) xk− dk
k + Bw ( α) wk,
(1)
zk = C( α) xk + Cd( α) xk− d + D( α) u
k
k + Dw ( α) wk,
where k is the k-th sample-time, matrices A( α), Ad( α), B( α), Bw, C( α), Cd( α), D( α) and Dw( α) are time-invariant, uncertain and with adequate dimensions defined in function of the signals
xk = x( k) ∈ R n, the state vector at sample-time k, uk = u( k) ∈ R m, representing the control vector with m control signals, wk = w( k) ∈ R , the exogenous input vector with
input
Uncertain Discrete-Time Systems with Delayed State:
Robust Stabilization with Performance Specification via LMI Formulations
297
signals, and zk = z( k) ∈ R p, the output vector with p weight output signals. These matrices
can be described by a polytope P with known vertices
N
P = Ω( α) ∈ R n+ p×2 n+ m+ : Ω( α) = ∑ αiΩ i, α ∈ Υ ,
(2)
i=1
where
N
Υ = α : ∑ αi = 1, αi ≥ 0, i ∈ I[1, N]
(3)
i=1
and
Ω i = Ai Adi Bi Bwi , i ∈ I[1, N].
(4)
Ci Cdi Di Dwi
The delay, denoted by dk, is supposed to be time-varying and given by:
dk ∈ I d, ¯
d , ( d, ¯
d) ∈ N2∗
(5)
with d, ¯
d representing the minimum and maximum values of dk, respectively. Thus, any
system Ω( α) ∈ P can be written as a convex combination of the N vertices Ω i, i ∈ I[1, N], of
P.
The following control law is considered in this chapter:
uk = Kxk + Kdxk− d
(6)
k
with [ K| Kd] ∈ R m×2 n. By replacing (6) in (1)-(4), the resulting uncertain closed-loop system is
given by
˜
Ω( α)
x
+ B
:
k+1 = ˜
A( α) xk + ˜
Ad( α) xk− dk
w ( α) wk
(7)
zk = ˜
C( α) xk + ˜
Cd( α) xk− d + D
k
w ( α) wk
with ˜
Ω( α) ∈ ˜P,
N
˜
P =
˜
Ω( α) ∈ R n+ p×2 n+ : ˜Ω( α) = ∑ α ˜
iΩ i, α ∈ Υ
(8)
i=1
where
˜
˜
˜
Ω
Ai Adi Bwi
i =
˜
, i ∈ I[1, N].
(9)
C
˜
i Cdi Dwi
and matrices ˜
Ai, ˜
Adi, ˜
Ci e ˜
Cdi are defined by
˜
Ai = Ai + BiK,
˜
Adi = Adi + BiKd,
(10)
˜
Ci = Ci + DiK,
˜
Cdi = Cdi + DiKd
(11)
Note that, control law (6) requires that both xk and xk− d are available at each sample-time.
k
Eventually, this can be achieved in physical systems by employing, for instance, a
time-stamped in the measurements or in the estimated states Srinivasagupta et al. (2004). In
case of dk is not known, it is sufficient to assume Kd = 0.
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Discrete Time Systems
2.1 Stability conditions
Since the stability of system ˜
Ω( α) given in (7) plays a central rule in this work, it is addressed
in the sequence. Note that, without loss of generality, it is possible to consider the stability of
the system (7) with wk = 0, ∀ k ∈ N.
Consider the sequence composed by ¯
d + 1 null vectors
ˆ
φ ¯ = {
d
0, . . . , 0}
( ¯ d+1) terms
In this chapter null initial conditions are always assumed, that is,
xk = φ 0, k = ˆ φ ¯ d, k ∈ I[− ¯ d, 0]
(12)
If φt, k = ˆ φ ¯ d, then an equilibrium solution for system (7) with wk = 0, ∀ k ∈ N, is achieved because xk+1 = xk = 0, ∀ k > t and α ∈ ˜Ω.
Definition 1 (Uniform asymptotic stability). For a given α ∈ Υ , the trivial solution of (7) with
wk = 0 , ∀ k ∈ N is said uniformly asymptotically stable if for any κ ∈ R+ such that for all initial conditions xk ∈ φ ¯ d ∈ Φ κ, k ∈ I[− ¯
d, 0] , it is verified
0, k
¯
d
lim φ ¯ d
= 0, ∀ j ∈ I[1, ¯ d+ 1]
t→∞ t, j, k
This allows the following definition:
Definition 2 (Robust stability). System (7) subject to (3), (5) and (8) is said robustly stable if its
respective trivial solution is uniformly asymptotically stable ∀ α ∈ Υ .
The main objective in this work is to formulate convex optimization problems, expressed as
LMIs, allowing an efficient numerical solution to a set of stability and performance problems.
2.2 Problems
Two sets of problems are investigated in this chapter. The first set concerns stability issues
related to uncertain discrete time with time varying delay in the state vector as presented in
the sequence.
Problem 1 (Robust stability analysis). Determine if system (7) subject to (3), (5) and (8) is robustly
stable.
Problem 2 (Robust control design). Determine a pair of static feedback gains, K and Kd, such that
(1)-(5) controlled by (6) is robustly stable.
The other set of problems is related to the performance of the class of systems considered in
this chapter. In this proposal, the H∞ index is used to quantify the performance of the system
as stated in the following problems:
Problem 3 (H∞ guaranteed cost). Given the uncertain system ˜
Ω( α) ∈ ˜P , determine an estimation
for γ > 0 such that for all wk ∈ 2 there exist zk ∈ 2 satisfying
zk 2 < γ wk 2
(13)
for all α ∈ Υ . In this case, γ is called an H∞ guaranteed cost for (7).
Uncertain Discrete-Time Systems with Delayed State:
Robust Stabilization with Performance Specification via LMI Formulations
299
Problem 4 (Robust H∞ control design). Given the uncertain system Ω( α) ∈ ˜
P , (1), and a scalar
γ > 0 , determine robust state feedback gains K and Kd, such that the uncertain closed-loop system
˜
Ω( α) ∈ ˜P , (7), is robustly stable and, additionally, satisfies (13) for all wk and zk belonging to 2 .
It is worth to say that, in cases where time-delay depends on a physical parameter (such as
velocity of a transport belt, the position of a steam valve, etc.) it may be possible to determine
the delay value at each sample-time. As a special case, consider the regenerative chatter in
metal cutting. In this process a cylindrical workpiece has an angular velocity while a machine
tool (lathe) translates along the axis of this workpiece. For details, see (Gu et al., 2003, pp. 2). In
this case the delay depends on the angular velocity and can be recovered at each sample-time
k. However, the study of a physical application is not the objective in this chapter.
The following parameter dependent L-K function is used in this paper to investigate problems
1-4:
3
V( α, k) = ∑ Vv( α, k) > 0
(14)
v=1
with
V 1( α, k) = xkP( α) xk,
(15)
k−1
V 2( α, k) = ∑ xjQ( α) xj,
(16)
j= k− dk
1− d
k−1
V 3( α, k) = ∑
∑ xjQ( α) xj,
(17)
=2− ¯ d j= k+ −1
The dependency of matrices P( α) and Q( α) on the uncertain parameter α is a key issue on
reducing the conservatism of the resulting conditions. Here, a linear relation on α is assumed.
Thus, consider the following structure for these matrices:
N
N
P( α) = ∑ αiPi; Q( α) = ∑ αiQi
(18)
i=1
i=1
with α ∈ Υ.
Note that, more general structures such as P( α) and Q( α) depending
homogeneously on α — see Oliveira & Peres (2005) — may result in less conservative
conditions, but at the expense of a higher numerical complexity of the resulting conditions.
To be a L-K function, the candidate (14) must be positive definite and satisfy
Δ V( α, k) = V( α, k + 1) − V( α, k) < 0
(19)
T
for all xT xT
=
k
k− d
0 and α ∈ Υ.
k
The following result is used in this work to obtain less conservative results and to decouple
the matrices of the system from the L-K matrices P( α) and Q( α).
Lemma 1 (Finsler’s Lemma). Let ϕ ∈ R n, M( α) = M( α) T ∈ R n× n and G( α) ∈ R m× n such that rank(G( α)) < n. Then, the following statements are equivalents:
i) ϕT M( α) ϕ < 0, ∀ ϕ : G( α) ϕ = 0, ϕ = 0
ii) G( α)⊥ T M( α)G( α)⊥ < 0 ,
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Discrete Time Systems
iii) ∃ μ( α) ∈ R+ : M( α) − μ( α)G( α) T G( α) < 0
iv) ∃ X ( α) ∈ R n× m : M( α) + X ( α)G( α) + G( α) T X ( α) T < 0
In the case of parameter independent matrices, the proof of this theorem can be found in
de Oliveira & Skelton (2001). The proof for the case depending on α follows similar steps.
3. Robust stability analysis and H∞ guaranteed cost
In this section it is presented the conditions for stability analysis and calculation of H∞
guaranteed cost for system (7). The objective here is to present sufficient convex conditions
for solving problems 1 and 3.
3.1 Robust stability analysis
Theorem 1. If there exist symmetric matrices 0 < Pi ∈ R n× n, 0 < Qi ∈ R n× n, a matrix X ∈
R3 n× n, dk ∈ I[ d, ¯ d] with ¯ d and d belonging to N∗ , such that
Ψ i = Q i + X B i + B T X T <
i
0;
i = 1, . . . , N
(20)
with
⎡
⎤
Pi
0
0
Q
⎣
⎦
i =
βQi − Pi 0
(21)
− Qi
β = ¯ d− d + 1
(22)
and
B i = I − Ai − Adi
(23)
is verified ∀ α admissible, then system (7) subject to (5) is robustly stable. Besides, (14)-(17) is a
Lyapunov-Krasovskii function assuring the robust stability of the considered system.
Proof. The positivity of the function (14) is assured with the hypothesis of Pi = PT > 0,
i
Qi = QT > 0. For the equation (14) be a Lyapunov-Krasovskii function, besides its positivity,
i
it is necessary to verify (19) ∀ α ∈ Ω. From hereafter, the α dependency is omitted in the
expressions Vv( k), v = 1, . . . , 3, To calculate (19), consider
Δ V 1( k) = xTk+1 P(