306
Discrete Time Systems
exist a real scalar κ ∈]0, 2[ such that for θ ∈]0, 1], κ( κ − 2) = − θ. Thus, replacing block (4, 4) of
(52) by κ( κ − 2)I p, the optimization variables W and Wd by K F and Kd F , respectively, using the definitions given by (10)–(11) and pre- and post-multiplying the resulting LMI by TH (on
the left) and by T T
H (on the right), with
⎡
⎤
T
−1
0
TH = ⎣
G 0
⎦
(53)
I
with T given by (48) and G ∈ R p× p, it is possible to obtain ˜ΨH i < 0, with ˜ΨH i given by
⎡ F− T ˜ Pi F−1 − F− T − F−1
F− T ( Ai + BiK)
F− T ( Adi + BiKd)
⎢
⎢
⎢
˜
A
˜
i
Adi
˜
Ψ
⎢
β F − T ˜
H
Q
i = ⎢
⎢
i F −1 − F − T ˜
Pi F −1
0
⎢
−F− T ˜ Q
⎣
i F −1
⎤
0
FBw
( CT +
)
⎥
i
KTDT
i
GT
0
⎥
⎥
˜
⎥
Ci
⎥
( CdT +
)
⎥ (54)
i
KT
d DT
i
GT
0
⎥
⎥
˜
⎥
Cdi
⎥
G κ 2 − 2 κ GT
GDw ⎦
− μI
Observe that, assuming G = − 1 κI p, block (4, 4) of (54) can be rewritten as
G κ 2 − 2 κ GT =
− 1 κI p κ 2 − 2 κ −1 κI p
= 1 − 2 κ I p
= I p − 1 κI p − 1 κI p
= I p + G + GT
(55)
assuring the feasibility of ΨH i < 0 given in (36) with Pi = F − T ˜ Pi F −1, Qi = F − T ˜
Qi F −1,
i ∈ I[1, N], and
⎡
⎤
F−1 0
⎢
⎢ 0 0 ⎥
⎥
X
⎢
⎥
H = ⎢ 0
0
⎣
⎥
0
G ⎦
0
0
completing the proof.
Uncertain Discrete-Time Systems with Delayed State:
Robust Stabilization with Performance Specification via LMI Formulations
307
Theorem 4 provides a solution to Problem 4. This kind of solution can be efficiently achieved
by means of, for example, interior point algorithms. Note that all matrices of the system can
be affected by polytopic uncertainties which states a difference w.r.t. most of the proposals
found in the literature. Another remark concerns the technique used to obtain the synthesis
condition: differently from the usual approach for delay free systems, here it is not enough
to replace matrices in the analysis conditions with their respective closed-loop versions and
to make a linearizing change of variables. This makes clear that the H∞ control of systems
with delayed state is more complex than with delay free systems. Also, note that the design
of state feedback gains K and Kd can be done minimizing the guaranteed H∞ cost, γ = √ μ,
of the uncertain closed-loop system. In this case, it is enough to solve the following convex
optimization problem:
⎧
⎪
⎪
μ
⎨
min
˜
S
Pi > 0; ˜
Qi > 0; 0 < θ ≤ 1;
H :
(56)
∞
⎪
⎪
W; W
⎩
d; F
such that
(52) is feasible
Example 3 (H∞ Design). A physically motivated problem is considered in this example. It consists of
a fifth order state space model of an industrial electric heater investigated in Chu (1995). This furnace is
divided into five zones, each of them with a thermocouple and a electric heater as indicated in Figure 2.
The state variables are the temperatures in each zone (x 1, . . . , x 5 ), measured by thermocouples, and
the control inputs are the electrical power signals (u 1, . . . , u 5 ) applied to each electric heater. The
u 1
u 2
u 3
u 4
u 5
x 1
x 2
x 3
x 4
x 5
Fig. 2. Schematic diagram of the industrial electric heater.
temperature of each zone of the process must be regulated around its respective nominal operational
conditions (see Chu (1995) for details). The dynamics of this system is slow and can be subject to several
load disturbances. Also, a time-varying delay can be expected, since the velocity of the displacement of
the mass across the furnace may vary. A discrete-time with delayed state model for this system has been
obtained as given by (1) with dk = d = 15 , where
⎡
⎤
0.97421 0.15116 0.19667 −0.05870 0.07144
⎢
⎢ −0.01455 0.88914 0.26953 0.11866 −0.22047 ⎥
⎥
A = A
⎢
⎥
0 = ⎢ 0.06376 0.12056 1.00049 −0.03491 −0.02766
(57)
⎣
⎥
−0.05084 0.09254 0.28774 0.82569 0.02570 ⎦
0.01723 0.01939 0.29285 0.03544
0.87111
308
Discrete Time Systems
⎡
⎤
−0.01000 −0.08837 −0.06989 0.18874 0.20505
⎢
⎢ 0.02363 0.03384 0.05282 −0.09906 −0.00191 ⎥
⎥
A
⎢
⎥
d = Ad 0 = ⎢ −0.04468 −0.00798 0.05618
0.00157
0.03593
,
(58)
⎣
⎥
−0.04082 0.01153 −0.07116 0.16472 0.00083 ⎦
−0.02537 0.03878 −0.04683 0.05665 −0.03130
⎡
⎤
0.53706 −0.11185 0.09978 0.04652
0.25867
⎢
⎢ −0.51718 0.73519 0.57518 0.40668 −0.12472 ⎥
⎥
B = B
⎢
⎥
0 = ⎢ 0.29469
0.31528 1.16420 −0.29922 0.23883
,
(59)
⎣
⎥
−0.20191 0.19739 0.41686 0.66551 0.11366 ⎦
−0.11835 0.16287 0.20378 0.23261 0.36525
and C = D = I5 , Cd = 0 , Dw = 0 , Bw = 0.1I with A 0 , Ad 0 , and B 0 being the nominal matrices of this system. Note that, this nominal system has unstable modes. The design of a stabilizing state
feedback gain for this system has been considered in Chu (1995) by using optimal control theory,
designed by an augmented delay-free system with order equal to 85 and a time-invariant delay d = 15 ,
by means of a Riccati equation.
Here, robust H∞ state feedback gains are calculated to stabilize this system subject to uncertain
parameters given by | ρ| ≤ 0.4 , | η| ≤ 0.4 and | σ| ≤ 0.08 that affect the matrices of the system as follows:
A( ρ) = A(1 + ρ),
Ad( θ) = Ad(1 + θ),
B( σ) = B(1 + σ)
(60)
This set of uncertainties defines a polytope with 8 vertices, obtained by combination of the upper and
lower bounds of uncertain parameters. Also, it is supposed in this example that the system has a
time-varying delay given by 10 ≤ dk ≤ 20 .
In these conditions, an H∞ guaranteed cost γ = 6.37 can be obtained by applying Theorem 4 that
yields the robust state feedback gains presented in the sequel.
⎡
⎤
−2.2587 −1.0130 −0.0558 0.4113 0.9312
⎢
⎢ −2.0369 −2.1037 0.0822 1.5032 0.0380 ⎥
⎥
K = ⎢
⎢ 0.9410 0.5645 −0.7523 −0.8688 0.3801 ⎥
(61)
⎣
⎥
−0.5796 −0.2559 0.0454 −1.0495 0.4072 ⎦
−0.0801 0.4106 −0.4369 0.5415 −2.4452
⎡
⎤
−0.0625 0.2592 0.0545 −0.2603 −0.5890
⎢
⎢ −0.1865 0.1056 −0.0508 0.1911 −0.4114 ⎥
⎥
K
⎢
⎥
d = ⎢ 0.1108 −0.0460 −0.0483 −0.0612 0.1551
(62)
⎣
⎥
0.0309
0.0709
0.1404 −0.3511 −0.1736 ⎦
0.0516 −0.1016 0.1324 −0.0870 0.1158
5. Extensions
In this section some extensions to the conditions presented in sections 3 and 4 are presented.
5.1 Quadratic stability approach
The quadratic stability approach is the source of many results of control theory presented in
the literature. In such approach, the Lyapunov matrices are taken constant and independent of
the uncertain parameter. As a consequence, their achieved results may be very conservative,
specially when applied to uncertain time-invariant systems. See, for instance, the works
of Leite & Peres (2003), de Oliveira et al. (2002) and Leite et al. (2004). Perhaps the main
Uncertain Discrete-Time Systems with Delayed State:
Robust Stabilization with Performance Specification via LMI Formulations
309
advantages of the quadratic stability approach are the simple formulation — with low
numerical complexity — and the possibility to deal with time-varying systems. In this case, all
equations given in Section 2 can be reformulated by using time-dependency on the uncertain
parameter, i.e., by using α = αk. In special, the uncertain open-loop system (1) can be
described by
Ω
xk+1 = A( αk) xk + Ad( αk) xk− d + B( αk) uk + Bw( αk) wk, v( α
k
k ) :
(63)
zk = C( αk) xk + Cd( αk) xk− d + D( α
k
k ) uk + Dw ( αk ) wk,
with αk ∈ Υ v
N
Υ v = αk : ∑ αki = 1, αki ≥ 0, i ∈ I[1, N]
(64)
i=1
which allows to define the polytope P given in (2) with αk replacing α. Still considering control
law (6), the resulting closed-loop system is given by
˜
Ω
xk+1 = ˜
A( αk) xk + ˜
Ad( αk) xk− d + Bw( αk) wk,
v( α
k
k ) :
(65)
zk = ˜
C( αk) xk + ˜
Cd( αk) xk− d + D
k
w ( αk) wk,
with ˜
Ω( αk) ∈ ˜P given in (8) with αk replacing α.
The convex conditions presented can be simplified to match with quadratic stability
formulation. This can be done in the analysis cases by imposing Pi = P > 0 and Qi = Q > 0
in (20) and (36) and, in the synthesis cases, by imposing ˜
Pi = ˜ P > 0, ˜
Qi = ˜
Q > 0, i = 1, . . . , N.
This procedure allows to establish the following Corollary.
Corollary 1 (Quadratic stability). The following statements are equivalent and sufficient for the
quadratic stability of system ˜
Ω v( αk) given in (65):
i) There exist symmetric matrices 0 < P ∈ R n× n, 0 < Q ∈ R n× n, matrices F ∈ R n× n, G ∈ R n× n and H ∈ R n× n ∈ R n× n, dk ∈ I[ d, ¯
d] with ¯
d and d belonging to N∗ , such that
⎡
⎤
P + FT + F
GT − FAi
HT − FAdi
Ψ
⎣
⎦
qi =
βQ − P − ATGT − GA
HT − GT A
< 0,
(66)
i
i
− ATi
di
−( Q + HAdi + AT HT)
di
is verified for i = 1, . . . , N.
ii) There exist symmetric matrices 0 < P ∈ R n× n, 0 < Q ∈ R n× n, dk ∈ I[ d, ¯
d] with ¯
d and d
belonging to N∗ , such that
Φ
PAi + βQ − P
ATPA
i
di
i =
ATi
< 0
(67)
AT PA
di
di − Q
is verified for i = 1, . . . , N.
Proof. Condition (66) can be obtained from (20) by imposing Pi = P > 0 and Qi = Q > 0.
This leads to a Lyapunov-Krasovskii function given by
k−1
1− d
k−1
V( xk) = xkPxk +
∑ xjQxj + ∑
∑ xjQxj
j= k− d( k)
=2− ¯ d j= k+ −1
310
Discrete Time Systems
which is sufficient for the quadratic stability of ˜
Ω v( αk). This condition is not necessary for
the quadratic stability because this function is also not necessary, even for the stability of the
precisely known system. The equivalence between (66) and (67) can be stated as follows: i) ⇒
ii) if (66) is verified, then (67) can be recovered by Φ i = T TΨ
qi
qi T qi with
T qi = Ai Adi
I2 n
i) ⇐ ii) On the other hand, if (67) is verified, then it is possible by its Schur’s complement to
obtain
⎡
⎤
− P PAi PAdi
Φ