Input-Output Stability 输入输出稳定

h ( u ) = a + b tanh cu = a + b
ecu − e− cu ecu + e− cu
for some nonnegative constants a , b and c . Using the fact
h' ( u ) =
4bc
(e
cu
+ e− cu
)
2
≤ bc , ∀u ∈ℜ
u
L∞
= sup u ( t ) < ∞
t ≥0
and the space is denoted by L∞ m . For the space of piecewise continuous, square-integrable functions, the norm is defined by
u
function h ( u ) = u 2 cannot be bounded by a straight line of the form h ( u ) ≤ γ u + β , for all u ∈ ℜ . ∆
Example 1.1: The signal u ( t ) = t does not belongs to the space L∞ , but its truncation
⎧ t, 0 ≤ t ≤ τ uτ ( t ) = ⎨ ⎩0, t > τ
belongs to L∞ for every finite τ . Hence, u ( t ) = t belongs to the extended space L∞e . ∆
1 Input-Output Stability
Input-output stability analysis allows us to analyze the stability of a given system without knowing the internal state x of the system. Before going forward, we have to introduce some input-output mathematical models.
1.1 L Stability
We consider a system whose input-output relation is represented by
y = Hu
where H is some mapping or operator that specifies y in terms of u . The input u belongs to a space of signals that map the time interval [ 0, ∞ ) into the Euclidian space v ;
( Hu )τ
for all u ∈ Le m and τ ∈ [ 0, ∞ ) .
L
≤ γ uτ
L
+β
Note that β is called bias term and the smallest possible γ is called the gain of H .
Example 1.2: A memoryless, possibly time-varying, function h : [ 0, ∞ ) × ℜ → ℜ can be
h ( t , u ) ≤ a u , ∀t ≥ 0 , ∀u ∈ ℜ for some positive constant a . For each p ∈ [1, ∞ ] , the operator H is finite-gain L p stable with zero bias and γ = a . Finally let
monotone increasing. α : [ 0, a ) → [ 0, ∞ ) is a class K ∞ if α is class K , a = ∞ and
α ( r ) → ∞ as r → ∞ .
Definition 1.3: A mapping H : Le m → Le q is L stable if there exists a class K function
h (u ) = u 2
Since
sup h ( u ( t ) )
t ≥0
⎞ ≤⎛ ⎜ sup u ( t ) ⎟ ⎝ t ≥0 ⎠
2
H and L∞ stable with zero bias and α ( r ) = r 2 . It is not finite-gain L∞ stable because the
Definition 1.1: A system is causal if the value of the output is only determined by past inputs, that is
Or equivalently, that for all times τ , for any input signals u, v, if Error! Objects cannot be created from editing field codes., then Error! Objects cannot be created
viewed as an operator H that assigns to every input signal u ( t ) the output signal
Let
y ( t ) = h ( t , u ( t ) ) . We use this simple operator to illustrate the definition of L stability.
L2
=
∫
∞
0
uT ( t ) u ( t ) dt < ∞
and the space is denoted by L2 m . More generally, the space L p m for 1 ≤ p < ∞ is defined
as
the set of all piecewise continuous functions u : [ 0, ∞ ) → ℜm such that
e.g. u : [ 0, ∞ ) → ℜm . Examples are the space of piecewise continuous, bounded functions; that is, sup u ( t ) < ∞ , and the state space of piecewise continuous, square-integrable
Le m = u uτ ∈ Lm , ∀τ ∈ [ 0, ∞ )
{
}
and uτ is a truncation of u defined by
⎧u ( t ) , 0 ≤ t ≤ τ uτ ( t ) = ⎨ t >τ ⎩ 0,
The extended space Le m is a linear space that contains the unextended space Lm as a subset. It allows us to deal with unbounded ever-growing signals.
we have h ( u ) ≤ a + bc u , ∀u ∈ℜ
Hence, H is finite-gain L∞ stable with γ = bc and β = a . Furthermore, if a = 0 , then for each p ∈ [1, ∞ ) ,
∫ h (u (t ))
tis,
∫
0
u T ( t ) u ( t ) dt < ∞ . To measure the size of a signal, we introduce the
norm function u , which satisfies the following three properties: 1) The norm of a signal is zero if and only if the signal is identical zero and is strictly positive otherwise 2) Scaling a signal results in a corresponding scaling of the norm; that is, au = a u for any positive constant a and every signal u 3) The norm satisfies the triangle inequality u1 + u2 ≤ u1 + u2 for any signals u1 and u2 . For the space of piecewise continuous, bounded functions, the norm is defined as
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( Hu )τ = ( Huτ )τ
& = f ( x, u ) , y = h ( x, u ) are causal Remark 1.1: All systems of the form x
Definition 1.2: α : [ 0, a ) → [ 0, ∞ ) , continuous is a class K function if α ( 0 ) = 0 and is
u =
Lp
(∫
∞
0
u ( t ) dt
p
)
1
p
<∞