for all i.
Because f is uniformly continuous, there exists a number δ( 0) > 0 such that
|f ( t) − f ( ti) | < 0 for every t ∈ [ t
2
i, ti + δ( 0)]
Hence, for every t ∈ [ ti, ti + δ( 0)], we have
|f ( t) | = |f ( t) − f ( ti) + f( ti) | ≥ |f( ti) | − |f( t) − f( ti) |
≥
0 −
0 = 0
2
2
which implies that
ti+ δ( 0)
ti+ δ( 0)
f ( τ ) dτ =
|f ( τ ) | dτ > 0 δ( 0)
(3.2.4)
t
2
i
ti
where the first equality holds because f ( t) retains the same sign for t ∈ [ ti, ti+ δ( 0)].
On the other hand, g( t) = t f ( τ ) dτ has a limit as t → ∞ implies that there exists 0
a T ( 0) > 0 such that for any t 2 > t 1 > T ( 0) we have
|g( t 1) − g( t 2) | < 0 δ( 0)
2
i.e.,
t 2
f ( τ ) dτ < 0 δ( 0)
t
2
1
which for t 2 = ti + δ( 0) , t 1 = ti contradicts (3.2.4), and, therefore, lim t→∞ f( t) = 0.
✷
3.2. PRELIMINARIES
77
The proof of Lemma 3.2.5 follows directly from that of Lemma 3.2.6 by
noting that the function f p( t) is uniformly continuous for any p ∈ [1 , ∞)
because f, ˙
f ∈ L∞.
The condition that f ( t) is uniformly continuous is crucial for the results
of Lemma 3.2.6 to hold as demonstrated by the following example.
Example 3.2.4 Consider the following function described by a sequence of isosceles
triangles of base length 1 and height equal to 1 centered at n where n = 1 , 2 , . . . ∞
n 2
as shown in the figure below:
f ( t)
✻
1
❅
❈✄
❈✄
❅
✄ ❈
· · · · · ·
✄ ❈
· · · · · ·
❅
✄ ❈
✄ ❈
✲
✛
1
✲
✲
✛2
✲
✛
n
2
1
1
4
n 2
This function is continuous but not uniformly continuous. It satisfies
t
1 ∞ 1
π 2
lim
f ( τ ) dτ =
=
t→∞ 0
2
n 2
12
n=1
but lim t→∞ f( t) does not exist.
The above example also serves as a counter example to the following situ-
ation that arises in the analysis of adaptive systems: We have a function V ( t)
with the following properties: V ( t) ≥ 0, ˙
V ≤ 0. As shown by Lemma 3.2.3
these properties imply that lim t→∞ V ( t) = V∞ exists. However, there is no
guarantee that ˙
V ( t) → 0 as t → ∞. For example consider the function
t
V ( t) = π −
f ( τ ) dτ
0
where f ( t) is as defined in Example 3.2.4. Clearly,
V ( t) ≥ 0 ,
˙
V = −f ( t) ≤ 0 ,
∀t ≥ 0
and
π 2
lim V ( t) = V∞ = π −
t→∞
12
78
CHAPTER 3. STABILITY
but lim
˙
t→∞ V ( t) = − lim t→∞ f ( t) does not exist. According to Barb˘
alat’s
lemma, a sufficient condition for ˙
V ( t) → 0 as t → ∞ is that ˙
V is uniformly
continuous.
3.2.3
Positive Definite Matrices
A square matrix A ∈ Rn×n is called symmetric if A = A . A symmet-
ric matrix A is called positive semidefinite if for every x ∈ Rn, x Ax ≥ 0
and positive definite if x Ax > 0 ∀x ∈ Rn with |x| = 0. It is called neg-
ative semidefinite ( negative definite) if −A is positive semidefinite (positive
definite).
The definition of a positive definite matrix can be generalized to non-
symmetric matrices. In this book we will always assume that the matrix
is symmetric when we consider positive or negative definite or semidefinite
properties.
We write A ≥ 0 if A is positive semidefinite, and A > 0 if A is positive
definite. We write A ≥ B and A > B if A − B ≥ 0 and A − B > 0,
respectively.
A symmetric matrix A ∈ Rn×n is positive definite if and only if any one
of the following conditions holds:
(i) λi( A) > 0 , i = 1 , 2 , . . . , n where λi( A) denotes the i th eigenvalue of A, which is real because A = A .
(ii) There exists a nonsingular matrix A 1 such that A = A 1 A 1 .
(iii) Every principal minor of A is positive.
(iv) x Ax ≥ α|x| 2 for some α > 0 and ∀x ∈ Rn.
The decomposition A = A 1 A 1 in (ii) is unique when A 1 is also symmetric.
In this case, A 1 is positive definite, it has the same eigenvectors as A, and
its eigenvalues are equal to the square roots of the corresponding eigenvalues
1
of A. We specify this unique decomposition of A by denoting A 1 as A 2 , i.e.,
1
1
A = A 2 A 2 where A 2 is a positive definite matrix and A / 2 denotes the
transpose of A 1 / 2.
A symmetric matrix A ∈ Rn×n has n orthogonal eigenvectors and can
be decomposed as
A = U Λ U
(3.2.5)
where U is a unitary (orthogonal) matrix (i.e., U U = I) with the eigen-
3.3. INPUT/OUTPUT STABILITY
79
vectors of A, and Λ is a diagonal matrix composed of the eigenvalues of A.
Using (3.2.5), it follows that if A ≥ 0, then for any vector x ∈ Rn
λmin( A) |x| 2 ≤ x Ax ≤ λmax( A) |x| 2
Furthermore, if A ≥ 0 then
A 2 = λmax( A)
and if A > 0 we also have
1
A− 1 2 = λmin( A)
where λmax( A) , λmin( A) is the maximum and minimum eigenvalue of A,
respectively.
We should note that if A > 0 and B ≥ 0, then A + B > 0, but it is not
true in general that AB ≥ 0.
3.3
Input/Output Stability
The systems encountered in this book can be described by an I/O mapping
that assigns to each input a corresponding output, or by a state variable
representation. In this section we shall present some basic results concerning
I/O stability. These results are based on techniques from functional analysis
[42], and most of them can be applied to both continuous- and discrete-
time systems. Similar results are developed in Section 3.4 by using the state
variable approach and Lyapunov theory.
3.3.1
Lp Stability
We consider an LTI system described by the convolution of two functions
u, h : R+ → R defined as
t
t
y( t) = u ∗ h =
h( t − τ ) u( τ ) dτ =
u( t − τ ) h( τ ) dτ
(3.3.1)
0
0
where u, y is the input and output of the system, respectively. Let H( s) be
the Laplace transform of the I/O operator h( ·). H( s) is called the transfer
80
CHAPTER 3. STABILITY
function and h( t) the impulse response of the system (3.3.1). The system
(3.3.1) may also be represented in the form
Y ( s) = H( s) U ( s)
(3.3.2)
where Y ( s) , U ( s) is the Laplace transform of y, u respectively.
We say that the system represented by (3.3.1) or (3.3.2) is Lp stable if
u ∈ Lp ⇒ y ∈ Lp and y p ≤ c u p for some constant c ≥ 0 and any
u ∈ Lp. When p = ∞, Lp stability, i.e., L∞ stability, is also referred to as
bounded-input bounded-output (BIBO) stability.
The following results hold for the system (3.3.1).
Theorem 3.3.1 If u ∈ Lp and h ∈ L 1 then
y p ≤ h 1 u p
(3.3.3)
where p ∈ [1 , ∞] .
When p = 2 we have a sharper bound for y p than that of (3.3.3) given by
the following Lemma.
Lemma 3.3.1 If u ∈ L 2 and h ∈ L 1 , then
y 2 ≤ sup |H( jω) | u 2
(3.3.4)
ω
For the proofs of Theorem 3.3.1, Lemma 3.3.1 see [42].
Remark 3.3.1 It can be shown that (3.3.4) also holds [232] when h( ·) is of
the form
0
t < 0
h( t) =
∞
i=0 fiδ( t − ti) + fa( t)
t ≥ 0
where fa ∈ L 1 , ∞
i=0 |fi| < ∞ and ti are nonnegative finite constants.
The Laplace transform of h( t) is now given by
∞
H( s) =
fie−sti + Ha( s)
i=0
which is not a rational function of s. The biproper transfer functions
that are of interest in this book belong to the above class.
3.3. INPUT/OUTPUT STABILITY
81
Remark 3.3.2 We should also note that (3.3.3) and (3.3.4) hold for the
truncated functions of u, y, i.e.,
yt p ≤ h 1 ut p
for any t ∈ [0 , ∞) provided u ∈ Lpe. Similarly,
yt 2 ≤ sup |H( jω) | ut 2
ω
for any t ∈ [0 , ∞) provided u ∈ L 2 e. This is clearly seen by noticing
that u ∈ Lpe ⇒ ut ∈ Lp for any finite t ≥ 0.
It can be shown [42] that inequality (3.3.3) is sharp for p = ∞ because
h 1 is the induced norm of the map T : u → T u = y from L∞ into L∞, i.e.,
T ∞ = h 1. Similarly for (3.3.4) it can be shown that the induced norm
of the linear map T : L 2 → L 2 is given by
T 2 = sup |H( jω) |
(3.3.5)
ω∈R
i.e., the bound (3.3.4) is also sharp.
The induced L 2 norm in (3.3.5) is referred to as the H∞ norm for the
transfer function H( s) and is denoted by
H( s) ∞ = sup |H( jω) |
ω∈R
Let us consider the simple case where h( t) in (3.3.1) is the impulse response
of an LTI system whose transfer function H( s) is a rational function of s.
The following theorem and corollaries hold.
Theorem 3.3.2 Let H( s) be a strictly proper rational function of s. Then
H( s) is analytic in Re[ s] ≥ 0 if and only if h ∈ L 1 .
Corollary 3.3.1 If h ∈ L 1 , then
(i) h decays exponentially , i.e., |h( t) | ≤ α 1 e−α 0 t for some α 1 , α 0 > 0
(ii) u ∈ L 1 ⇒ y ∈ L 1 L∞, ˙ y ∈ L 1 , y is continuous and lim t→∞ |y( t) | = 0
(iii) u ∈ L 2 ⇒ y ∈ L 2 L∞, ˙ y ∈ L 2 , y is continuous and lim t→∞ |y( t) | = 0
(iv) For p ∈ [1 , ∞] , u ∈ Lp ⇒ y, ˙ y ∈ Lp and y is continuous
82
CHAPTER 3. STABILITY
For proofs of Theorem 3.3.2 and Corollary 3.3.1, see [42].
Corollary 3.3.2 Let H( s) be biproper and analytic in Re[ s] ≥ 0 . Then u ∈
L 2 L∞ and lim t→∞ |u( t) | = 0 imply that y ∈ L 2 L∞ and lim t→∞ |y( t) | =
0 .
Proof H( s) may be expressed as
H( s) = d + Ha( s)
where d is a constant and Ha( s) is strictly proper and analytic in Re[ s] ≥ 0. We have
y = du + ya,
ya = Ha( s) u
where, by Corollary 3.3.1, ya ∈ L 2
L∞ and |ya( t) | → 0 as t → ∞. Because
u ∈ L 2
L∞ and u( t) → 0 as t → ∞, it follows that y ∈ L 2
L∞ and |y( t) | → 0
as t → ∞.
✷
Example 3.3.1 Consider the system described by
e−αs
y = H( s) u,
H( s) = s + β
for some constant α > 0. For β > 0, H( s) is analytic in Re[ s] ≥ 0. The impulse response of the system is given by
e−β( t−α)
t ≥ α
h( t) =
0
t < α
and h ∈ L 1 if and only if β > 0. We have
∞
∞
1
h 1 =
|h( t) |dt =
e−β( t−α) dt =
0
α
β
and
e−αjω
1
H( s) ∞ = sup
=
ω
jω + β
β
Example 3.3.2 Consider the system described by
2 s + 1
y = H( s) u,
H( s) = s + 5
3.3. INPUT/OUTPUT STABILITY
83
The impulse response of the system is given by
2 δ
h( t) =
∆( t) − 9 e− 5 t
t ≥ 0
0
t < 0
where ha = − 9 e− 5 t ∈ L 1. This system belongs to the class described in Re-
mark 3.3.1. We have
1
1 + 2 jω
1 + 4 ω 2 2
H( s) ∞ = sup
= sup
= 2
ω
5 + jω
ω
25 + ω 2
Hence, according to (3.3.4) and Remarks 3.3.1 and 3.3.2, for any u ∈ L 2 e, we have