CHAPTER 3. STABILITY
Using the Schwartz inequality we have
τ
2
τ
2
e−α 0( τ−s) |u( s) |ds
=
e−( α 0 − δ 1 )( τ−s)
( τ −s)
2
e− δ 12
|u( s) |ds
0
0
τ
τ
≤
e−(2 α 0 −δ 1)( τ−s) ds
e−δ 1( τ−s) |u( s) | 2 ds
0
0
1
τ
≤
e−δ 1( τ−s) |u( s) | 2 ds
2 α 0 − δ 1 0
i.e.,
1
1
t
τ
2
Qt 2 δ ≤ √
e−δ( t−τ)
e−δ 1( τ−s) |u( s) | 2 dsdτ
(3.3.19)
2 α 0 − δ 1
0
0
Interchanging the sequence of integration, (3.3.19) becomes
1
1
t
t
2
Qt 2 δ ≤ √
e−δt+ δ 1 s|u( s) | 2
e−( δ 1 −δ) τ dτ ds
2 α 0 − δ 1
0
s
1
1
t
2
=
√
e−δt+ δ 1 s|u( s) | 2 e−( δ 1 −δ) s − e−( δ 1 −δ) t ds 2 α 0 − δ 1
0
δ 1 − δ
1
1
t e−δ( t−s) − e−δ 1( t−s)
2
=
√
|u( s) | 2 ds
2 α 0 − δ 1
0
δ 1 − δ
1
1
t
2
≤
e−δ( t−s) |u( s) | 2 ds
(2 α 0 − δ 1)( δ 1 − δ)
0
for any δ < δ 1 < 2 α 0. Because
t 2 δ ≤ t, the proof of (ii) follows.
The proof of (iii) follows directly by noting that
yt 2 δ ≤ ( C x) t 2 δ + ( Du) t 2 δ ≤ xt 2 δ sup C ( t) + ut 2 δ sup D( t) t
t
✷
A useful extension of Lemma 3.3.3, applicable to the case where A( t) is
not necessarily stable and δ = δ 0 > 0 is a given fixed constant, is given by
the following Lemma that makes use of the following definition.
Definition 3.3.3 The pair ( C( t) ,A( t)) in (3.3.15) is uniformly completely
observable (UCO) if there exist constants β 1 , β 2 , ν > 0 such that for all
t 0 ≥ 0 ,
β 2 I ≥ N( t 0 , t 0 + ν) ≥ β 1 I
3.3. INPUT/OUTPUT STABILITY
91
where N ( t 0 , t 0 + ν) = t 0+ ν Φ ( τ, t
t
0) C( τ ) C ( τ )Φ( τ, t 0) dτ is the so-called
0
observability grammian [1, 201] and Φ( t, τ ) is the state transition matrix
associated with A( t) .
Lemma 3.3.4 Consider a linear time-varying system of the same form as
(3.3.15) where ( C( t) , A( t)) is UCO, and the elements of A, B, C, and D are
bounded continuous functions of time. For any given finite constant δ 0 > 0 ,
we have
(i)
|x( t) | ≤
λ 1
√
( c
+ c
) +
2 α
1 ut 2 δ
2 yt 2 δ
t
1 −δ 0
0
0
(ii)
x( t) 2 δ ≤
λ 1
√
( c
+ c
) +
0
1 ut 2 δ
2 yt 2 δ
1
( δ
0
0
1 −δ 0)(2 α 1 −δ 1)
(iii)
yt 2 δ ≤ x
sup
sup
0
t 2 δ 0
t C ( t) + ut 2 δ 0
t D( t)
where c 1 , c 2 ≥ 0 are some finite constants; δ 1 , α 1 satisfy δ 0 < δ 1 < 2 α 1 , and t is an exponentially decaying to zero term because x 0 = 0 .
Proof Because ( C, A) is uniformly completely observable, there exists a matrix
K( t) with bounded elements such that the state transition matrix Φ c( t, τ) of Ac( t) ∆
=
A( t) − K( t) C ( t) satisfies
Φ c( t, τ ) ≤ λ 1 e−α 1( t−τ)
for some constants α 1 , δ 1 , λ 1 that satisfy α 1 > δ 1 > δ 0 , λ
2
2
1 > 0. Let us now rewrite
(3.3.15), by using what is called “output injection,” as
˙ x = ( A − KC ) x + Bu + KC x
Because C x = y − Du, we have
˙ x = Ac( t) x + ¯
Bu + Ky
where ¯
B = B − KD. Following exactly the same procedure as in the proof of
Lemma 3.3.3, we obtain
λ
|x( t) | ≤
1
√
( c
+ c
) +
2 α
1 ut 2 δ 0
2 yt 2 δ 0
t
1 − δ 0
where c
¯
1 = sup t B( t) , c 2 = sup t K( t)
and t is an exponentially decaying to
zero term due to x(0) = x 0. Similarly,
λ
x
1
t 2 δ ≤
( c
+ c
) +
0
1 ut 2 δ
2 yt 2 δ
t
( δ
0
0
1 − δ 0)(2 α 1 − δ 1)
92
CHAPTER 3. STABILITY
by following exactly the same steps as in the proof of Lemma 3.3.3. The proof of
(iii) follows directly from the expression of y.
✷
Instead of the interval [0 , t), the L 2 δ norm can be defined over any arbi-
trary interval of time as follows:
1
t
2
x
∆
t,t
=
e−δ( t−τ) x ( τ ) x( τ ) dτ
1
2 δ
t 1
for any t 1 ≥ 0 and t ≥ t 1. This definition allow us to use the properties of
the L 2 δ norm over certain intervals of time that are of interest. We develop
some of these properties for the LTI, SISO system
˙ x = Ax + Bu,
x(0) = x 0
y = C x + Du
(3.3.20)
whose transfer function is given by
y = [ C ( sI − A) − 1 B + D] u = H( s) u
(3.3.21)
Lemma 3.3.5 Consider the LTI system (3.3.20), where A is a stable matrix
and u ∈ L 2 e. Let α 0 , λ 0 be the positive constants that satisfy eA( t−τ) ≤
λ 0 e−α 0( t−τ) . Then for any constant δ ∈ [0 , δ 1) where 0 < δ 1 < 2 α 0 is arbitrary, for any finite t 1 ≥ 0 and t ≥ t 1 we have
(a) |x( t) | ≤ λ
(i)
0 e−α 0( t−t 1) |x( t 1) | + c 1 ut,t 1 2 δ
(b)
x
( t−t
t,t
1) |x( t
1
2 δ ≤ c 0 e− δ 2
1) | + c 2 ut,t 1 2 δ
(ii)
y
( t−t
t,t
1) |x( t
1
2 δ ≤ c 3 e− δ 2
1) | + H ( s) ∞δ ut,t 1 2 δ
(iii) Furthermore if D = 0 , i.e., H( s) is strictly proper, then
|y( t) | ≤ c 4 e−α 0( t−t 1) |x( t 1) | + H( s) 2 δ ut,t 1 2 δ
where
λ
B λ
c
0
0
1 =
B c 0 , c 0 = √
,
c
2 α
2 =
0 − δ
( δ 1 − δ)(2 α 0 − δ 1)
c 3 = C
c 0 ,
c 4 = C
λ 0
3.3. INPUT/OUTPUT STABILITY
93
Proof Define v( τ ) as
0
if τ < t
v( τ ) =
1
u( τ ) if τ ≥ t 1
From (3.3.20) we have
x( t) = eA( t−t 1) x( t 1) + ¯ x( t)
∀t ≥ t 1
(3.3.22)
where
t
¯
x( t) =
eA( t−τ) Bu( τ ) dτ
∀t ≥ t 1
t 1
We can now rewrite ¯
x( t) as
t
¯
x( t) =
eA( t−τ) Bv( τ ) dτ
∀t ≥ 0
(3.3.23)
0
Similarly
y( t) = C eA( t−t 1) x( t 1) + ¯ y( t) ∀t ≥ t 1
(3.3.24)
t
¯
y( t) =
C eA( t−τ) Bv( τ ) dτ + Dv( t) ∀t ≥ 0
(3.3.25)
0
It is clear that ¯
x in (3.3.23) and ¯
y in (3.3.25) are the solutions of the system
˙¯ x = A¯ x + Bv,
¯
x(0) = 0
¯
y = C ¯
x + Dv
(3.3.26)
whose transfer function is C ( sI − A) − 1 B + D = H( s).
Because A is a stable matrix, there exists constants λ 0 , α 0 > 0 such that
eA( t−τ) ≤ λ 0 e−α 0( t−τ)
which also implies that H( s) is analytic in Re[ s] ≥ −α 0.
Let us now apply the results of Lemma 3.3.3 to (3.3.26). We have
B λ
|¯
x( t) | ≤
0
√
v
2 α
t 2 δ = c 1 vt 2 δ
0 − δ
B λ
¯
x
0
t 2 δ
≤
vt 2 δ = c 2 vt 2 δ
( δ 1 − δ)(2 α 0 − δ 1)
for some δ 1 > 0, δ > 0 such that 0 < δ < δ 1 < 2 α 0. Because vt 2 δ = ut,t 1 2 δ and
¯
xt,t 1 2 δ ≤ ¯ xt 2 δ, it follows that for all t ≥ t 1
|¯
x( t) | ≤ c 1 ut,t 1 2 δ,
¯
xt,t 1 2 δ ≤ c 2 ut,t 1 2 δ
(3.3.27)
From (3.3.22) we have
|x( t) | ≤ λ 0 e−α 0( t−t 1) |x( t 1) | + |¯ x( t) |
∀t ≥ t 1
94
CHAPTER 3. STABILITY
which together with (3.3.27) imply ( i)( a). Using (3.3.22) we have
xt,t 1 2 δ ≤ ( eA( t−t 1) x( t 1)) t,t 1 2 δ + ¯ xt,t 1 2 δ
which implies that
1
t
2
xt,t
e−δ( t−τ) e− 2 α 0( τ−t 1) dτ
λ
1
2 δ
≤
0 |x( t 1) | +
¯
xt,t 1 2 δ
t 1
λ
( t−t 1)
≤
0 e− δ 2
√
|x( t
2 α
1) | +
¯
xt,t 1 2 δ
(3.3.28)
0 − δ
From (3.3.27) and (3.3.28), (i)(b) follows.
Let us now apply the results of Lemma 3.3.2 to the system (3.3.26), also de-
scribed by
¯
y = H( s) v
we have
¯
yt 2 δ ≤ H( s) ∞δ vt 2 δ
and for H( s) strictly proper
|¯
y( t) | ≤ H( s) 2 δ vt 2 δ
for any 0 ≤ δ < 2 α 0. Since vt 2 <