Chapter 6
Discrete-time delay systems
In this chapter, we consider stability, performance analysis and control of linear
(probably, uncertain) discrete-time systems. Some of the presented ideas may also
be useful in the nonlinear case. Differently from continuous-time TDSs, the discretetime ones can be reduced to non-delay systems by state/input/output augmentation. However, such an augmentation may lead to complicated systems if the delay is uncertain, variable or not small. Therefore, also in the discrete-time case,
Lyapunov-Krasovskii and Lyapunov-Razumikhin methods and the small gain theorem approach lead to efficient conditions for analysis and control.
6.1 Stability and performance analysis of discrete-time TDSs
Consider the linear discrete-time system
x(k + 1) = Ax(k) + A1 x(k − h),
k ∈ Z+ ,
h ∈ N,
x(k) ∈ Rn
(6.1)
with constant matrices and a constant delay h. As in the continuous-time case, x(k)
can no longer be regarded as the state of this system. It should be augmented by its
finite history
xaug (k) = col{x(k), x(k − 1), ..., x(k − h)},
(6.2)
resulting in the following augmented system without delay
xaug (k + 1) = Aaug xaug (k),
where

A
 In

Aaug =  .
 ..
k ∈ Z+ ,
0
0
..
.
...
...
..
.
xaug (k) ∈ R(h+1)n ,
(6.3)

A1
0 

..  .
. 
0 ... In 0
233
234
6 Discrete-time delay systems
The initial condition for (6.1) should be given as
col{x(0), x(−1), ..., x(−h)} = col{φ (0), φ (−1), ..., φ (−h)}.
(6.4)
Unlike the continuous case, the discrete-time system with (bounded) delay is finitedimensional.
The stability notion for the discrete-time system (6.1) is defined similarly to the
one in the continuous-time: the system is stable if x(k) (k = 0, 1, 2, ...) is small for
small enough initial condition (6.4). The system is asymptotically stable if it is stable
and limk→∞ x(k) = 0.
6.1.1 Analysis of discrete-time delay systems via augmentation
The stability analysis of (6.1) can be reduced to the analysis of the higher-order
system without delay (6.3). If we apply the Lyapunov method to (6.3), then a necessary and sufficient condition for the asymptotic stability of (6.1) is the existence of
a Lyapunov function V (xaug (k)) = xTaug (k)Paug xaug (k) with Paug > 0 such that along
(6.3) the following holds (see [142]):
Vaug(xaug (k + 1)) − Vaug(xaug (k)) ≤ −α |xaug (k)|2 , α > 0.
Differently from the continuous-time case, the general Lyapunov function Vaug that
corresponds to the necessary and sufficient stability condition can be easily found
by the augmentation of the discrete-time LTI system. However, such augmentation
suffers from the curse of dimensionality if h is large.
Application of the augmentation becomes complicated if the delay is unknown
or/and time-varying. Thus, for the linear system with a time-varying delay
x(k + 1) = Ax(k) + A1 x(k − τk ),
k ∈ Z+ ,
τk ∈ N,
0 ≤ τk ≤ h,
(6.5)
the augmentation (6.2) leads to a system with a time-varying uncertain matrix
Aaug (k). For all uncertain τk ≤ h there are h + 1 possibilities of (h + 1)n × (h + 1)n
matrices




A A1 0 ... 0
A + A1 0 ... 0
 In 0 0 ... 0 
 In
0 ... 0



(1) = 
,
A
A(0) =  .

 .. . . . . . . ..  , ...,
. . . . .. 
.
 ..
. . .. 
. . .

A
 In

A(h) =  .
 ..
0
0
0
..
.
. . . In 0
... 0 A1
... 0 0 

. . .. .. 
.. . 
0 ... In 0 0
0 . . . In 0 0
6.1 Stability and performance analysis of discrete-time TDSs
235
for Aaug (k), k ∈ Z+ .
The stability of (6.5) is reduced, thus, to the stability of the LTI switched system
with an arbitrary switching (Hetel et al. [117], 2008)
xaug (k + 1) = A( j) xaug (k),
k ∈ Z+ ,
xaug (k) ∈ R(h+1)n .
(6.6)
Remark 6.1 Note that, as in the continuous-time, we are interested in the stability
that is uniform in the initial time k0 . Therefore, for simplicity, in this chapter we
consider k0 = 0 for (6.5).
6.1.2 Transfer function of discrete-time TDS
Consider the discrete-time system
x(k + 1) = ∑ri=0 Ai x(k − hi ) + ∑ri=0 Bi u(k − hi),
y(k) = ∑ri=0 Ci x(k − hi ) + ∑ri=0 Di u(k − hi)
k ∈ Z+ ,
hi ∈ Z+ ,
(6.7)
with the constant matrices and the delays
0 = h0 < h1 < · · · < hr = h.
Taking Z-transform in (6.7) we arrive at Y (z) = G(z)U(z), where the transfer function matrix G is given by
−1 r
G(z) = ∑ri=0 Di z−hi + ∑ri=0 Ci z−hi zI − ∑ri=0 Ai z−hi
∑i=0 Bi z−hi .
Each entry of G(z) is a quotient of polynomials.
6.1.3 LMI stability conditions: the direct Lyapunov method
To derive simple sufficient stability conditions, discrete-time counterparts of the
Lyapunov-Razumikhin functions or the Lyapunov-Krasovskii functionals can be applied.
6.1.3.1 Delay-independent conditions in the case of time-varying delays
As in the continuous-time situation, this case is treated by adopting the LyapunovRazumikhin approach (see Zhang and Chen [250], 1998). In the Lyapunov-Razumikhin
approach, the Lyapunov functions V : R × Rn → R+ are used and the condition
V (k + 1, x(k + 1)) − V(k, x(k)) ≤ −α |x(k)|2
∀k ∈ Z+
236
6 Discrete-time delay systems
for some α > 0 along the system should be satisfied under the Razumikhin condition:
∃ρ > 1 : V (k + i, x(k + i)) ≤ ρ V (k, x(k)), −h ≤ i ≤ −1.
Proposition 6.1 [73] Consider the system (6.5) with time-varying delay. Given a
tuning scalar q ∈ (0, 1) this system is asymptotically stable if there exists an n × n
matrix P > 0 that satisfies the following LMI:
T
AT PA1
∆ A PA − (1 − q)P
< 0.
(6.8)
Γind =
∗
AT1 PA1 − qP
Proof. Choosing the Lyapunov-Razumikhin function V (x(k)) = x(k)T Px(k) and assuming that for some ρ > 1 the Razumikhin condition
x(k − τk )T Px(k − τk ) ≤ ρ x(k)T Px(k),
−h ≤ i ≤ −1, k ≥ 0
holds, we find, by using the S-procedure with q > 0, that
V (x(k + 1)) − V(x(k)) = [xT (k)AT + xT (k − τk )AT1 ]P[Ax(k) + A1x(k − τk )]
T
−x(k)T Px(k) ≤ [xT (k)AT + xT (k − τk )AT1 ]P[Ax(k) + A1 x(k − τk )] −
x(k) Px(k)
x(k)
+q[ρ xT (k)Px(k) − xT (k − τk )Px(k − τk )] ≤ [xT (k) xT (k − τk )]Γ̄ind
,
x(k − τk )
where
Γ̄ind
AT PA − (1 − qρ )P AT PA1
=
.
∗
AT1 PA1 − qP
Note that Γ̄ind < 0 is feasible with ρ = 1 + ε for some small enough ε > 0 if Γind <
0. From Γind < 0 it follows that q < 1. Hence, under the Razumikhin condition,
V (x(k + 1)) − V (x(k)) ≤ −α |x(k)|2 for some α > 0, which implies the asymptotic
stability of (6.5). ⊓
⊔
Remark 6.2 The feasibility of the delay-independent conditions (6.8) yields the
Lyapunov inequalities AT PA− P < 0 and AT1 PA1 − P < 0, i.e. that both, A and A1 are
Schur matrices (with all the eigenvalues inside of the unit circle). This is different
from the continuous-time case, where the delay-independent stability implies that A
only is a Hurwitz matrix.
Note that the LMI (6.8) yields for all 0 6= x(k) ∈ Rn
h
i
x(k)
[xT (k) ± xT (k)]Γind
= xT (k) (A ± A1)T P(A ± A1) − P x(k) < 0.
±x(k)
Thus, the Lyapunov inequality (A ± A1 )T P(A ± A1 ) − P < 0 holds, implying that
A ± A1 are Schur matrices. The latter is similar to the delay-independent stability in
the continuous-time, where A ± A1 are Hurwitz matrices.
Example 6.1. We consider (6.5), where
6.1 Stability and performance analysis of discrete-time TDSs
0 0.5
A=
,
0.5 0.2
237
−0.4 0
and A1 =
.
0 0
By (6.8), it is verified that the system is delay-independently stable. This is achieved
by choosing q = 0.5.
6.1.3.2 The Lyapunov-Krasovskii method and delay-dependent LMIs
Denote (similar to the continuous-time case)
∆
xk ( j) = x(k + j),
j = −h, ..., −1, 0.
Lemma 6.1 If there exist positive numbers α , β and a functional V : Rn × ... × Rn →
|
{z
}
h+1 times
R+ such that for all k = 0, 1, 2...
and
0 ≤ V (xk ) ≤ β {max j∈[−h,0] |x(k + j)|2 },
(6.9)
V (xk+1 ) − V (xk ) ≤ −α |x(k)|2
(6.10)
for x(k) satisfying (6.5), then (6.5) is asymptotically stable.
Proof. From (6.10) it follows that
k
∑ (V (x j+1) − V (x j )) = V (xk+1 ) − V (x0 ) ≤ −α
j=0
k
∑ |x( j)|2 .
j=0
Therefore, for x(k) satisfying (6.5), (6.4) we have, due to (6.9),
|x(k)|2 ≤
k
1
β
max |φ ( j)|2
∑ |x( j)|2 ≤ α V (x0 ) ≤ α j∈[−h,0]
j=0
∀k ∈ Z+ .
(6.11)
The inequality (6.11) implies that |x(k)|2 is small for small enough max j∈[−h,0] |φ ( j)|2 .
Moreover,
∞
1
∑ |x( j)|2 ≤ α V (x0 ) < ∞.
j=0
Hence, |x( j)|2 → 0 for j → ∞. ⊓
⊔
The Lyapunov-based delay-dependent analysis of discrete-time delay systems
employs the following form of Jensen’s inequality: for all ζ ( j) ∈ Rn ( j = 0, . . . N −
1) and n × n matrix R > 0
!
!
N−1
N−1
1 N−1 T
T
(6.12)
∑ ζ ( j)Rζ ( j) ≥ N ∑ ζ ( j) R ∑ ζ ( j) .
j=0
j=0
j=0
238
6 Discrete-time delay systems
For (6.5), the discrete-time counterpart of the Lyapunov-Krasovkii functional (3.101)
has the form
V (xk ) = VP (k) + VS (k) + VR(k),
VP (k) = xT (k)Px(k),
k−1
T
VR (k) = h ∑−1
m=−h ∑ j=k+m ȳ ( j)Rȳ( j),
P > 0, R ≥ 0, S ≥ 0,
T
VS (k) = ∑k−1
j=k−h x ( j)Sx( j),
ȳ( j) = x( j + 1) − x( j).
(6.13)
The technique for the derivation of LMI conditions is similar to the continuoustime case (see e.g. Chen et al.[24], 2004; Fridman & Shaked [73], 2005). We have
VP (k + 1) − VP(k) = 2xT (k)Pȳ(k) + ȳT (k)Pȳ(k),
T
VS (k + 1) − VS(k) = ∑kj=k+1−h xT ( j)Sx( j) − ∑k−1
j=k−h x ( j)Sx( j)
= xT (k)Sx(k) − xT (k − h)Sx(k − h),
−1
k−1
k
T
T
VR (k + 1) − VR(k) = h ∑−1
m=−h ∑ j=k+m+1 ȳ ( j)Rȳ( j) − h ∑m=−h ∑ j=k+m ȳ ( j)Rȳ( j)
k−1
2
T
T
= h ȳ (k)Rȳ(k) − h ∑ j=k−h ȳ ( j)Rȳ( j).
We employ the representation
k−τ −1
k−1
T
T
T
k
−h ∑k−1
j=k−h ȳ ( j)Rȳ( j) = −h ∑ j=k−h ȳ ( j)Rȳ( j) − h ∑ j=k−τk ȳ ( j)Rȳ( j)
and apply further Jensen’s inequality twice:
k−τk −1
k−τk −1 T
k−τk −1 T
ȳ ( j)Rȳ( j) ≥ h−hτk ∑ j=k−h
ȳ ( j) R ∑ j=k−h
ȳ( j)
h ∑ j=k−h
= h−hτk [x(k − τk ) − x(k − h)]T R[x(k − τk ) − x(k − h)],
k−1
T
ȳ
(
j)
R
ȳ(
j)
∑
∑k−1
j=k−τk
j=k−τk
= τhk [x(k) − x(k − τk )]T R[x(k) − x(k − τk )].
T
h ∑k−1
j=k−τk ȳ ( j)Rȳ( j) ≥
h
τk
By using the reciprocally convex approach (see Lemma 3.4), we obtain
h
T
T
−h ∑k−1
j=k−h ȳ ( j)Rȳ( j) ≤ − h−τk [x(k − τk ) − x(k − h)] R[x(k − τk ) − x(k − h)]
− τh [x(k) − x(k − τk )]T R[x(k) − x(k − τk )]
k T x(k)−x(k − τk )
x(k)−x(k − τk )
R S12
≤−
∗ R
x(k − τk )−x(k − h)
x(k − τk ) − x(k − h)
(6.14)
for some S12 ∈ Rn×n , where the latter inequality holds if the following
R S12
≥0
(6.15)
∗ R
is feasible.
Then, by the descriptor method, where the right-hand side of the expression
0 = 2[xT (k)P2T + ȳT (k)P3T ][(A − I)x(k) + A1x(k − τk ) − ȳ(k)]
6.1 Stability and performance analysis of discrete-time TDSs
239
with some P2 , P3 ∈ Rn×n is added to V (xk+1 ) − V (xk ), and setting
ηdis (k) = col{x(k), ȳ(k), x(k − h), x(k − τk )},
we obtain that
T
V (xk+1 ) − V (xk ) ≤ ηdis
(k)Φdis ηdis (k) ≤ −α |x(k)|2 ,
is satisfied for some α > 0, and, thus, (6.5) is asymptotically stable if the following
LMI


S12
R − S12 + P2T A1
Φ11 Φ12

 ∗ Φ22
0
P3T A1
<0
(6.16)
Φdis = 
T

 ∗ ∗ −(S + R)
R − S12
T
∗ ∗
∗
−2R + S12 + S12
is feasible, where
Φ11 = (AT − I)P2 + P2T (A − I) + S − R,
Φ12 = P − P2T + (AT − I)P3 , Φ22 = −P3 − P3T + P + h2R.
(6.17)
We have thus proved the following:
Theorem 6.1 Given 0 ≤ h. If there exist n × n matrices P > 0, S > 0, R > 0, S12 ,
P2 , P3 that satisfy the LMIs (6.15) and (6.16), then the system (6.5) is asymptotically
stable for all time-varying delays 0 ≤ τk ≤ h.
For a constant delay τ ≤ h, we obtain the following by modifying the above and
the similar arguments for the continuous-time case (cf. (3.100)):
Corollary 6.1 Given 0 ≤ h. If there exist n × n matrices P > 0, S > 0, R > 0, P2 , P3
that satisfy the LMI


(A − I)T P2 + P2T (A − I) + S − R P − P2T + (A − I)T P3 P2T A1 + R

∗
−P3 − P3T + P + h2R P3T A1  < 0, (6.18)
∗
∗
−S − R
then the system (6.5) is asymptotically stable for all constant delays 0 ≤ τ ≤ h.
Moreover, if (6.18) is feasible with R = 0, then (6.5) is delay-independently stable
for all constant delays τ .
Note that as in the continuous-time case, the descriptor-based LMIs (6.16) and
(6.18) are affine in the system matrices and can be easily applied to uncertain systems with polytopic type uncertainties.
Example 6.2. We consider the system
0.8 0
−0.1
0
x(k + 1) =
x(k) +
x(k − τk )
0 0.97
−0.1 −0.1
(6.19)
240
6 Discrete-time delay systems
0.9 0
borrowed from [73]. Here A− A1 =
is unstable with an eigenvalue 1.07 >
0.1 1.07
1. Thus, the delay-independent conditions cannot be feasible. For a constant τk ≡ h,
by using the augmentation, it is found that the system is asymptotically stable for
all h ≤ 18 (since the eigenvalues of the resulting Aaug are inside the unit circle).
For the constant delay, by verifying the feasibility of (6.18), we find that the system
is asymptotically stable for all 0 ≤ τ ≤ 16. For a time-varying τk , by verifying the
feasibility of (6.16), we find that the system is asymptotically stable for a slightly
smaller interval 0 ≤ τk ≤ 15. Note that (6.16) with S12 = 0 leads to the smaller
interval 0 ≤ τk ≤ 12.
Problem 6.1. Prove Jensen’s inequality (6.12).
Problem 6.2. As in the continuous-time case, delay-dependent conditions can be
rewritten without using the descriptor method. By modifying the derivation of Theorem 6.1 derive the corresponding LMIs. Note that
VP (k + 1) − VP(k) = [Ax(k) + A1 x(k − τk )]T P[Ax(k) + A1x(k − τk )] − xT (k)Px(k).
Apply Schur complement to the first term in the right-hand side of the latter equation.
Problem 6.3. Consider (6.5) with an interval delay h ≤ τk ≤ h1 , h > 0. The discretetime counterpart of the Lyapunov functional (3.127) has a form
−1
k−1
T
T
V (xk ) = xT (k)Px(k) + ∑k−1
j=k−h x ( j)Sx( j) + h ∑m=−h ∑ j=k+m ȳ ( j)Rȳ( j)
k−1
−h−1
k−h−1 T
T
(6.20)
+ ∑ j=k−h1 x ( j)S1 x( j) + (h1 − h) ∑m=−h1 ∑ j=k+m ȳ ( j)R1 ȳ( j),
ȳ( j) = x( j + 1) − x( j), P > 0, S > 0, R > 0, S1 > 0, R1 > 0.
Derive LMI conditions for the asymptotic stability of (6.5) with h ≤ τk ≤ h1 via
V (xk ) of (6.20). Use the reciprocally convex approach.
6.1.4 l2 -gain analysis via the Krasovskii approach
The direct Lyapunov-Krasovskii method is applicable not only to stability, but also
to the performance analysis. We will consider below the l2 -gain and the input-tostate stability analysis.
Consider the perturbed linear system
x(k + 1) = Ax(k) + A1 x(k − τk ) + B1w(k),
k ∈ Z+ ,
0 ≤ τk ≤ h,
(6.21)
where x(k) ∈ Rn is the state vector, and w(k) ∈ Rnw is a disturbance. Let
z(k) = C0 x(k) + C1 x(k − τk )
(6.22)
be the controlled output. Given γ > 0, define the following performance index
6.1 Stability and performance analysis of discrete-time TDSs
241
∞
J=
∑ [zT (k)z(k) − γ 2 wT (k)w(k)].
(6.23)
k=0
The system (6.21), (6.22) has an induced l2 -gain less than γ if J < 0 for all the
solutions of (6.21) that start at zero and for 0 6= w ∈ l2 [0, ∞).
The following inequality along (6.21)
V (xk+1 ) − V (xk ) + zT (k)z(k) − γ 2 wT (k)w(k) ≤ −α [|x(k)|2 + |w(k)|2 ]
for some α > 0 yields J < 0. This can be seen from summation in the latter inequality in k from 0 till ∞. Similarly to the continuous-time case, we arrive at the LMIs
that guarantee l2 -gain to be less than γ :


| P2T B1 C0T

| P3T B1 0 


 Φdis | 0 0 


R S12
T

|
0
C
≥ 0,
(6.24)
<
0,
1 

∗ R
 − − − −


 ∗ | −γ 2 I 0 
∗ | ∗ −I
where Φdis is defined in (6.16). We note that if the LMIs (6.24) are feasible, then
Φdis < 0 and, thus, (6.21) with w ≡ 0 is asymptotically stable.
Proposition 6.2 Given γ > 0 and h > 0, let there exist n × n matrices P > 0, P2 , P3 ,
R > 0, S > 0, S12 such that the LMIs (6.24) hold, where Φdis is given by (6.16). Then
the system (6.21), (6.22) is internally stable and has an l2 -gain less than γ for all
0 ≤ τk ≤ h.
6.1.5 Exponential and input-to-state stability
Consider the linear perturbed system (6.21) with the initial condition (6.4). The
system (6.21) is said to be ISS if there exist constants c ≥ 1, λ ∈ (0, 1) and γ > 0
such that for any initial condition φ and for any disturbance w the solution satisfies
|x(k)|2 ≤ cλ k max{|φ (0)|2 , ..., |φ (−h)|2 } + γ max{|w(0)|2 , ..., |w(k)|2 }
∀k ∈ Z+ .
(6.25)
The unperturbed system
that
satisfies
(6.25)
with
γ
=
0
is
called
exponentially
stable
√
with the decay rate λ .
The following discrete-time counterpart of Lemma 4.1 is useful for the exponential and input-to-state stability analysis:
Lemma 6.2 Let V : Z+ → R+ and w(k) ∈ Rnw . If there exist scalars λ ∈ (0, 1) and
b > 0 such that
242
6 Discrete-time delay systems
∆
W (k) = V (k + 1) − λ V(k) − b|w(k)|2 ≤ 0, k ∈ Z+
(6.26)
b
max{|w(0)|2 , ..., |w(k)|2 }.
1− λ
(6.27)
then
V (k) ≤ λ kV (0) +
Proof. From (6.26) we have
V (k) ≤ λ V (k − 1) + b|w(k − 1)|2
≤ λ 2V (k − 2) + λ b|w(k − 2)|2 + b|w(k − 1)|2 ≤ ...
Finally we arrive at
k−1
V (k) ≤ λ kV (0) + b ∑ λ k− j−1 |w( j)|2 ,
(6.28)
j=0
that implies (6.27). ⊓
⊔
Remark 6.3 Note that the right-hand side of (6.28) is the solution of the equation
V (k + 1) − λ V(k) − b|w(k)|2 = 0,
k ∈ Z+
Therefore, similarly to Lemma 4.1, the above Lemma follows from the discrete-time
comparison principle (see e.g. Lakshmikantham et al. [144], 1989): the solution of
the discrete-time inequality
v(k + 1) ≤ f (k, v(k)),
v(k0 ) ≤ v0
for f Lipschitz in v satisfies v(k) ≤ u(k), k ≥ k0 , where u(k) is the solution of the
discrete-time equation
u(k + 1) = f (k, u(k)), u(k0 ) = u0 .
We will derive LMIs for ISS of the system (6.21) by using Lemma 6.2 via the
following Lyapunov functional V :
V (xk ) = VP (k) + VS (k) + VR(k),
(6.29)
where
VP (k) = xT (k)Px(k), P > 0,
k− j−1 xT ( j)Sx( j),
VS (k) = ∑k−1
S > 0,
j=k−h λ
−1
k− j−1 T
VR (k) = h ∑m=−h ∑k−1
ȳ
(
j)R
ȳ( j),
λ
j=k+m
ȳ(k) = x(k + 1) − x(k),
R > 0.
For λ = 1 the above functional coincides with (6.13). The λ -dependent terms for
the exponential stability were considered e.g. in (Sun et al. [216], 2008).
We have
6.1 Stability and performance analysis of discrete-time TDSs
243
VP (k + 1) − λ VP(k) = VP (k + 1) − VP(k) + (1 − λ )VP(k)
= 2xT (k)Pȳ(k) + ȳT (k)Pȳ(k) + xT (k)(1 − λ )Px(k),
and
k− j xT ( j)Sx( j)
VS (k + 1) − λ VS(k) = ∑kj=k+1−h λ k− j xT ( j)Sx( j) − ∑k−1
j=k−h λ
T
h
T
= x (k)Sx(k) − λ x (k − h)Sx(k − h)
Similarly,
T
VR (k + 1) − λ VR(k) ≤ h2 ȳT (k)Rȳ(k) − hλ h ∑k−1
j=k−h ȳ ( j)Rȳ( j).
Then
W (k) ≤2xT (k)Pȳ(k) + ȳT (k)Pȳ(k) + xT (k)[S + (1 − λ )P]x(k) + h2ȳT (k)Rȳ(k)
T
2
−λ h xT (k − h)Sx(k − h) − λ hh ∑k−1
j=k−h ȳ ( j)Rȳ( j) − b|w(k)| .
(6.30)
Denote ηdis (k) = col{x(k), ȳ(k), x(k − h), x(k − τk ), w(k)}. Taking into account
(6.14) and (6.15) and applying the descriptor method, where the right-hand side of
the following expression
0 = 2[xT (k)P2T + ȳT (k)P3T ][(A − I)x(k) + A1x(k − τk ) + B1w(k) − ȳ(k)]
T (k)Φ η (k) ≤
with some P2 , P3 ∈ Rn×n is added to W (k), we obtain that W (k) ≤ ηdis
ISS dis
0, if the following LMI


Φ11 Φ12
λ h S12
λ h (R − S12) + P2T A1 P2T B1
 ∗ Φ22
0
P3T A1
P3T B1 


h
h
T

(6.31)
ΦISS =  ∗ ∗ −λ (S + R)
λ (R − S12)
0 
<0
T
 ∗ ∗
) 0 
∗
−2λ h(R − S12 − S12
∗ ∗
∗
∗
−bI
is feasible, where
Φ11 = (AT − I)P2 + P2T (A − I) + S + (1 − λ )P− λ h R,
Φ12 = P − P2T + (AT − I)P3 , Φ22 = −P3 − P3T + P + h2R.
(6.32)
We have thus proved the following result:
Proposition 6.3 Given λ ∈ (0, 1) and h > 0, let there exist n × n matrices P > 0,
S > 0, R > 0, S12 , P2 , P3 and a scalar b > 0 that satisfy the LMIs (6.15) and (6.31)
with notations given in (6.32). Then the solutions of (6.21) satisfy
xT (k)Px(k) ≤ V (xk ) ≤ λ kV (x0 ) +
b
max{|w(0)|2 , ..., |w(k)|2 },
1−λ
k ∈ Z+
for all delays 0 ≤ τk ≤ h, i.e. (6.21) is ISS. Moreover, given ∆ > 0, the ellipsoid
244
6 Discrete-time delay systems
X∞ = {x ∈ Rn : xT Px ≤
b
∆ 2}
1−λ
is exponentially attractive for all initial functions and for all bounded w with
supk∈Z+ |w(k)|2 ≤ ∆ 2 .
Problem 6.4. (i) Consider the linear system (6.5) with an interval time-varying delay h0 ≤ τk ≤ h1 , h0 ≥ 0. Derive LMIs for the exponential stability of this system.
(ii) Applying the conditions of (i), find a decay rate of the system (6.19), where
3 ≤ τk ≤ 5.
6.1.5.1 The discrete-time Halanay inequality
As in the continuous-time, the exponential stability of discrete-time delay systems
can be analyzed by using the discrete-time Halanay’s inequality (see e.g. [213]):
Lemma 6.3 (Halanay’s inequality) Let
V (−h),V (−h + 1), . . .,V (k), . . .
be a sequence of nonnegative numbers. Assume that for some positive constants
δ1 < δ0 the following inequality holds:
V (k + 1) − V(k) ≤ −2δ0V (k) + 2δ1 max{V (k), . . . ,V (k − h)},
Then
V (k) ≤ λ0k max{V (0), . . . ,V (−h)},
k ∈ Z+ ,
k ∈ Z+ . (6.33)
(6.34)
where λ0 ∈ (0, 1) is a unique positive solution of the equation
λ − 1 + h ln λ + δ0 − δ1 = 0.
Problem 6.5. By using the discrete-time Halanay’s inequality derive sufficient conditions for the exponential stability of (6.5). Apply the conditions to the system of
Example 6.1 and find the resulting λ0 for h = 3.
248
6 Discrete-time delay systems
6.2 Control of discrete-time delay systems
6.2.1 Infinite horizon LQR for LTI discrete-time delay systems
Consider the linear discrete-time system
x(k + 1) = Ax(k) + A1 x(k − h) + Bu(k),
k ∈ Z+ , h ∈ Z+ , x(k) ∈ Rn , u(k) ∈ Rnu ,
col{x(0), x(−1), ..., x(−h)} = col{φ (0), φ (−1), ..., φ (−h)}.
(6.47)
with constant matrices and constant delay h. The augmentation of the state vector
xaug (k) = col{x(k), x(k − 1), ..., x(k − h)},
results in the following augmented system without delay
xaug (k + 1) = Aaug xaug (k) + Baugu(k),
where

A
 In

Aaug =  .
 ..
0
0
..
.
...
...
..
.

A1
0 

..  ,
. 
0 ... In 0
The initial condition for (6.47) is given as
k ∈ Z+ ,
xaug (k) ∈ R(h+1)n ,
(6.48)
 
B
0 
 
and Baug =  .  .
 .. 
0
col{x(0), x(−1), ..., x(−h)} = col{φ (0), φ (−1), ..., φ (−h)}.
Consider e.g. the following infinite-horizon LQR problem: find a control law
u that asymptotically stabilizes (6.47) and minimizes the following quadratic cost
function:
∞
J=
∑ [xT (k)Qx(k) + uT (k)Ru(k)], Q ≥ 0, R > 0.
(6.49)
k=0
This problem is reduced to a LQR problem for the augmented system (6.48) without
delay with J of (6.49). It is well-known [142], that if (Aaug , Baug ) is stabilizable
and (Aaug , Qaug ) is detectable, where Qaug = diag{Q, 0}, then there exists a unique
optimal control law given by
u(k) = Kxaug (k),
K = −(BTaug PBaug + R)−1BTaug PAaug,
(6.50)
where P ∈ R(h+1)n×(h+1)n is the unique solution P ≥ 0 to the discrete-time algebraic
Riccati equation [142]
P = ATaug [P − PBaug(BTaug PBaug + R)−1BTaug P]Aaug + Qaug
(6.51)
6.2 Control of discrete-time delay systems
249
such that Aaug + Baug K is a Schur matrix. This results in the optimal feedback with
the distributed delay for the original system
u(k) = Kcol{x(k), x(k − 1), ..., x(k − h)}.
Note that also in the continuous-time case, the LQR problem for systems with state
delay has an optimal feedback with the distributed delay (see Section 5.1.4).
Consider next the LQR problem with a delayed control input
x(k + 1) = Ax(k) + Bu(k − h), k ∈ Z+ ,
col{x(0), u(−1), ..., u(−h)} = col{φ (0), 0, ..., 0},
T
T
Jh = ∑∞
k=0 [x (k)Qx(k) + u (k − h)Ru(k − h)], Q ≥ 0, R > 0,
(6.52)
where x(k) ∈ Rn , u(k) ∈ Rnu . Here, one can also use the augmentation
xaug (k) = col{x(k), u(k − 1), ..., u(k − h)}.
Taking into account that
∞
∑ uT (k − h)Ru(k − h) =
k=0
∞
∑ uT (k)Ru(k),
k=0
the following non-singular (with R > 0) LQR for the non-delay system is obtained:


 
A 0 ...
B
0
 0 0 ...

 Inu 
0


 

 
0
xaug (k + 1) =  0 Inu ...
 xaug (k) +  0  u(k),
 ..

 .. 
.
..
.. 
.
. 
.
0
|
... Inu 0
{z
}
0
| {z }
Baug
Aaug
T
T
Jaug = ∑∞
k=0 [xaug (k)Qaug xaug (k) + u (k)Ru(k)], Qaug = diag{Q, 0} ≥ 0, R > 0.
Note that Aaug is a singular matrix. If (A, B) (for h = 0) is stabilizable, then
(Aaug , Baug ) is stabilizable. This follows from Hautus criterion [142]: the matrix

sIn − A 0
 0
sInu


[sI − Aaug Baug ] = 
 0 −Inu

 0
0
0
0

−B 0
0 Inu 


0 0 0


.. ..
.
. 0 0
0 −Inu sInu 0
0
0
..
.
0
0
has full rank for all complex
s with |s| ≥ 1, because p
[sIn − A B] has full rank for
√
|s| ≥ 1. Similarly, if (A,
Q)
is
detectable,
then
(A
,
Qaug ) is detectable.
aug
√
Therefore, if (A, B, Q) is stabilizable-detectable (for h = 0), then there exists a
unique optimal control given by (6.50) where P ≥ 0 is the unique solution P ≥ 0
250
6 Discrete-time delay systems
to the discrete algebraic Riccati equation (6.51) such that Aaug + BaugK is a Schur
matrix. The optimal control
u(k) = Kcol{x(k), u(k − 1), ..., u(k − h)}
has the distributed input delay (similar to the predictor-based design of the continuoustime system).
6.2.2 The predictor-based design and the reduction approach
To derive a reduced-order solution for the LQR problem (6.52) with the input delay
(the one based on the n × n matrix Riccati equation), we follow the arguments of the
predictor-based design (see Section 5.1). Denote v(k) = u(k − h). Then, the cost Jh
can be presented as
h−1
Jh =
∑ φ T (0)(AT ) j QA j φ (0) + J¯h,
j=0
where
J¯h =
∞
∑ [xT (k)Qx(k) + vT (k)Rv(k)].
k=h
Therefore, minimization of Jh subject to (6.52) is reduced to the minimization of J¯h
along the non-delayed system
x(k + 1) = Ax(k) + Bv(k), k ≥ h, x(h) = Ah φ (0).
√
If (A, B) is stabilizable and (A, Q) is detectable, then the unique optimal control in
the latter problem is given by
v(k) = u(k − h) = Kx(k),
K = −(BT PB + R)−1BT PA,
where P ∈ Rn×n is the unique non-negative solution to the discrete algebraic Riccati
equation
P = AT [P − PB(BT PB + R)−1BT PA]A + Q.
Moreover, the optimal feedback stabilizes the system leading to a Schur matrix A +
BK.
We then also have u(k) = v(k + h) = Kx(k + h), where
x(k + h) = Ah x(k) +
k+h−1
∑
j=k
Ak+h− j−1Bu( j − h).
Changing the indices in the latter sum, we arrive at
6.2 Control of discrete-time delay systems
251
u(k) = K[Ah x(k) +
−1
∑
A− j−1 Bu(k + j)].
j=−h
The optimal value Jh∗ is given by
Jh∗ =
h−1
∑ φ T (0)(A j )T QA j φ (0) + φ T (0)(Ah )T PAhφ (0).
j=0
The reduction approach has been extended to the discrete-time linear uncertain
input-delayed systems (see Gonzalez et al. [89], 2012). Consider the following system
x(k + 1) = Ax(k) + Bu(k − τk ),
x(k) ∈ Rn , u(k) ∈ Rnu , k = 0, 1, . . .
(6.53)
with an uncertain non-small delay τk = h + ηk . Here h is a known constant nominal
value and |ηk | ≤ µ < h is a delay uncertainty. The change of state variable
z(k) = Ah x(k) +
−1
∑
A− j−1 Bu(k + j)
(6.54)
j=−h
leads (6.53) to
z(k + 1) = Az(k) + Bu(k) + AhB[u(k − τk ) − u(k − h)].
(6.55)
Let there exist a gain K such that the matrix A + BK is Schur. Then the feedback
u(k) = Kz(k) stabilizes (6.55) if the following system is stable
z(k + 1) = (A + BK)z(k) + AhBK[z(k − τk ) − z(k − h)].
(6.56)
As in the continuous-time case, the stability of (6.56) can be analyzed by using
the direct Lyapunov-Krasovskii approach for systems with non-small delays. Thus,
the following Lyapunov functional can be employed [73]
µ −1
T
V (k) = zT (k)Pz(k) + ∑ j=−µ ∑k−1
s=k+ j−h ζ (s)R1 ζ (s),
ζ (s) = z(s + 1) − z(s), P > 0, R1 > 0 k ∈ Z+ .
(6.57)
The stability of (6.56) can also be analyzed via the input-output approach.
Problem 6.7. Consider the LQR problem with input and output delay
x(k + 1) = Ax(k) + A1x(k − h0 ) + Bu(k − h), k ∈ Z+ ,
col{x(0), ..., x(−h0 ), u(−1), ..., u(−h)} = col{φ (0), ..., φ (−h0 ), 0, ..., 0}, (6.58)
T
T
Jh = ∑∞
k=0 [x (k)Qx(k) + u (k − h)Ru(k − h)], Q ≥ 0, R > 0,
where h0 , h ∈ N, x(k) ∈ Rn , u(k) ∈ Rnu . By using an augmentation in x and the
predictor-based arguments, find the solution to the LQR problem and the conditions
that guarantee its existence and uniqueness.
252
6 Discrete-time delay systems
Problem 6.8. Derive LMI conditions for the asymptotic stability of (6.56) via V of
(6.57).
6.2.3 LMI-based design: time-varying delays
LMI-based design problems can be solved similarly to the continuous-time case. We
consider below the guaranteed cost control problem for the system with uncertain
time-varying delay τk ∈ [0, h]
x(k + 1) = Ax(k) + Bu(k − τk ), k ≥ 0,
x(0) = φ (0), u(k) = 0, k < 0,
(6.59)
where x(k) ∈ Rn , u(k) ∈ Rnu , A and B are constant matrices of the appropriate dimensions. We also consider the following cost function:
∞
J=
∑ zT (k)z(k)
(6.60)
j=0
where the objective vector z(k) ∈ Rnz is defined by
z(k) = Lx(k) + Du(k − τk )
(6.61)
for matrices L and D of the appropriate dimensions.
For systems with delays and/or system matrices that are uncertain, the optimal
control and cost (as in the LQR problem) do not exist. Instead, given initial condition
x(0), a control law can be found which leads to guaranteed (as small as possible) cost
δ for J, namely, J ≤ δ for all uncertainties. A state-feedback control u(k) = Kx(k)
is sought below that, for a given x(0), leads to a minimum guaranteed cost δ for J,
namely, J(x0 ) ≤ δ for all uncertain delays τk . The closed-loop system has a form
x(k + 1) = Ax(k) + BKx(k − τk ),
x(0) = φ (0), x(k) = 0, k < 0.
z(k) = Lx(k) + DKx(k − τk ),
k ∈ Z+ ,
(6.62)
For simplicity, assume that k − τk ≥ 0 for all k ∈ Z+ and that x(k) = x(0), k < 0.
This assumption allows to avoid additional solution bounds on the first delay interval
(cf. Sections 5.4.2 and 6.3.1).
Choosing V (xk ) of (6.13) we have along (6.62)
V (xk+1 ) − V (xk ) + zT (k)z(k) ≤ ξ̄ T (k)Γ ξ̄ (k) < −α |x(k)|2 , α > 0
where ξ̄ (k) = col{x(k), y(k), x(k − h), x(k − τk ), z(k)} if
(6.63)
6.2 Control of discrete-time delay systems

Φdis

Γ =

−
∗
253

| LT
|
0 

T
| K DT 
 < 0,
− − 
| −I
R S12
≥ 0.
∗ R
(6.64)
Here Φdis is given by (6.16) with A1 = BK. We take the sum of the two sides of
(6.63), from 0 to N, and obtain that
N
∑ zT (k)z(k) ≤ V (x0 ) − V(xN+1 ) ≤ V (x0)
k=0
and thus (under the constant initial condition)
J ≤ V (x0 ) = xT (0)(P + hS)x(0).
(6.65)
We choose, as in the continuous-time case, P3 = ε P2 and denote
P̄2 = P2−1 , P̄ = P̄2T PP̄2 , R̄ = P̄2T RP̄2 , S̄ = P̄2T SP̄2, S̄12 = P̄2T S12 P̄2 , Y = K P̄2 .
Multiplying the first inequality (6.64) by diag{P̄2, P̄2 , P̄2 , P̄2 , I} and its transpose,
from the right and the left, respectively, and the second one by diag{P̄2, P̄2 } and its
transpose, the following is obtained:


S̄12
BY + R̄ − S̄12
LT
Θ P̄ − P̄2 + ε P̄2T (A − I)T
 ∗ P̄ − ε P̄2 − ε P̄T + h2R̄
ε BY
0 
0
2


T
∗
∗
−(
S̄
+
R̄)
R̄
−
S̄
0 
12
 < 0,

T Y T DT 
∗
(6.66)
∗
∗
−2R̄ + S̄12 + S̄12
∗
∗
∗
∗
−I
R̄ S̄12
≥ 0,
∗ R̄
where
Θ = (A − I)P̄2 + P̄2T (A − I)T + S̄ − R̄.
Further, given δ > 0, V (0) =
xT (0)(P + hS)x(0) <
(6.67)
δ if
−δ xT (0)(P + hS)
<0
∗
−(P + hS)
or similarly to (5.98) if
−δ
x(0)T
< 0.
∗ −P̄2 − P̄2T + P̄ + hS̄
Indeed, since P + hS > 0 and P̄ + hS̄ = P̄2T (P + hS)P̄2, we have
[(P + hS)−1 − P̄2T ](P + hS)[(P + hS)−1 − P̄2] ≥ 0,
(6.68)
254
6 Discrete-time delay systems
which implies that
(P + hS)−1 ≥ P̄2 + P̄2T − (P̄ + hS̄)
or
P + hS ≤ (P̄2 + P̄2T − P̄ − hS̄)−1 .
Hence, by Schur complement, if (6.68) holds, it follows that
V (x0 ) = xT0 (P + hS)x0 ≤ xT0 (P̄2 + P̄2T − (P̄ + hS̄))−1 x0 < δ .
Proposition 6.5 Given initial condition x(0) ∈ Rn , constant h > 0 and a tuning
parameter ε , let there exist n × n matrices P̄ > 0, R̄ > 0, P̄2 , S̄ > 0, S̄12 , an nu × n
matrix Y and δ > 0 such that the LMIs (6.66) and (6.68) with notation (6.67) are
feasible. Then for all delays τk ∈ [0, h] the closed-loop system (6.62) is exponentially
stable and achieves a guaranteed cost δ for J, namely, J ≤ δ . The state-feedback
gain is given by K = Y P̄2−1 .
To minimize the guaranteed cost the following optimization problem can be
solved: minimize δ > 0 subject to (6.66) and (6.68).
Remark 6.4 As in the continuous-time case, LMIs for the analysis and design of
Propositions 6.2 and 6.5 are affine in the system matrices. Therefore, in the case
of polytopic type time-varying uncertainty in these matrices one has to solve these
LMIs simultaneously for all the vertices, applying the same decision matrices.
Example 6.3. Consider the system (6.59)-(6.61) with
 
10
√
1.06 0.1
0
A=
, B=
, L = 0.10 1 ,
0 0.74
1
00
1
x(0) =
.
0

0
√
D = 0.1 0 ,
1
Applying the conditions of Proposition 6.5 with h = 3 and ε = 1 we obtain that the
feedback u(k) = Kx(k) with K = −[0.4030 0.1260] leads the closed-loop system
to a cost bound of δ = 0.1488.
Problem 6.9. (i) Formulate the discrete-time counterpart of the H∞ -filtering problem considered in the previous chapter. Derive an LMI solution for the H∞ filter in
the case of an interval measurement delay h0 ≤ τk ≤ h1 , h0 ≥ 0 (without augmentation of the state).
(ii) Apply the conditions of (i) to the system
0.8 0
0.1
x(k + 1) =
x(k) +
w(k),
0.1 0.97
−1
y(k) = [1 1]x(k − τk ) + 0.1w(k)
with
6.2 Control of discrete-time delay systems
255
∞
J=
∑ [eT (k)[1 0]T [1 0]e(k) − γ 2wT (k)w(k)].
k=0
Choose h0 = 2, h1 = 5.
(iii) Find an improved value of the l2 -gain for the error system of (ii) (with
the gain Ko found in (ii)) by analyzing the augmented error system with the state
eaug (k) = col{e(k), e(k − 1), e(k − 2)}. The latter augmentation results in the system with the small measurement delay τk − 2 ∈ [0, 3].