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

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 ;

e.g. [ ): 0, mu ∞ →ℜ . Examples are the space of piecewise continuous, bounded functions;

that is, ( )0

supt

u t≥

< ∞ , and the state space of piecewise continuous, square-integrable

functions; that is, ( ) ( )0

Tu 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 1 2 1 2u u u u+ ≤ + for any signals 1u and 2u .

For the space of piecewise continuous, bounded functions, the norm is defined as

and the space is denoted by mL∞ . For the space of piecewise continuous, square-integrable functions, the norm is defined by

and the space is denoted by 2

mL . More generally, the space mpL for 1 p≤ < ∞ is defined

as

y Hu=

( )0

supL

tu u t

∞ ≥= < ∞

( ) ( )2 0

TL

u u t u t dt∞

= < ∞∫

the set of all piecewise continuous functions [ ): 0, mu ∞ →ℜ such that

The subscript p in m

pL refers to the p -norm used to define the space, while the superscript m is the dimension of the signal u . If we think of m

pu L∈ as a stable input, the question to ask is whenever the output y will

be stable in the sense that qy L∈ , allowing for different numbers of outputs to inputs. A system having the property that any stable input will generate a stable output is defined as a stable system. However, we cannot define H as a mapping from mL to qL , because we have to deal with systems which are unstable, in that an input mu L∈ may generate an output y that does not belong to qL . Therefore, H is usually defined as a mapping from an extended space m

eL to an extended space qeL , where m

eL is defined by

and uτ is a truncation of u defined by

( ) ( ) ,0,

u tu tτ

⎧= ⎨⎩

0 t

tτ

τ≤ ≤>

The extended space m

eL is a linear space that contains the unextended space mL as a subset. It allows us to deal with unbounded ever-growing signals. Example 1.1: The signal ( )u t t= does not belongs to the space L∞ , but its truncation

( ),

0,t

u tτ⎧

= ⎨⎩

0 t

tτ

τ≤ ≤>

belongs to L∞ for every finite τ . Hence, ( )u t t= belongs to the extended space eL∞ . ∆ Definition 1.1: A system is causal if the value of the output is only determined by past inputs, that is

( )( )1

0p

pp

Lu u t dt

∞= < ∞∫

[ ){ }, 0,m meL u u Lτ τ= ∈ ∀ ∈ ∞

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

from editing field codes. Remark 1.1: All systems of the form ( ),x f x u=& , ( ),y h x u= are causal Definition 1.2: [ ) [ ): 0, 0,aα → ∞ , continuous is a class K function if ( )0 0α = and is

monotone increasing. [ ) [ ): 0, 0,aα → ∞ is a class K∞ if α is class K , a = ∞ and

( )rα →∞ as r →∞ . Definition 1.3: A mapping : m q

e eH L L→ is L stable if there exists a class K function α , defined on [ )0,∞ , and a nonnegative constant β such that

for all m

eu L∈ and [ )0,τ ∈ ∞ . It is finite-gain L stable if there exists nonnegative constants γ and β such that

for all m

eu L∈ and [ )0,τ ∈ ∞ . 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 [ ): 0,h ∞ ×ℜ→ℜ can be

viewed as an operator H that assigns to every input signal ( )u t the output signal

( ) ( )( ),y t h t u t= . We use this simple operator to illustrate the definition of L stability. Let

for some nonnegative constants a , b and c . Using the fact

( )( )

'2

4cu cu

bch u bce e−

= ≤+

, u∀ ∈ℜ

( ) ( )Hu Huττ τ=

( ) ( )LLHu uττ

α β≤ +

( ) LLHu uττ

γ β≤ +

( ) tanhcu cu

cu cu

e eh u a b cu a be e

−

−

−= + = +

+

we have ( )h u a bc u≤ + , u∀ ∈ℜ Hence, H is finite-gain L∞ stable with bcγ = and aβ = . Furthermore, if 0a = , then for each [ )1,p∈ ∞ ,

Thus, for each [ ]1,p∈ ∞ , the operator H is finite-gain pL stable with zero bias and

bcγ = . Let h be a time-varying function that satisfies ( ),h t u a u≤ , 0t∀ ≥ , u∀ ∈ℜ for some positive constant a . For each [ ]1,p∈ ∞ , the operator H is finite-gain pL stable with zero bias and aγ = . Finally let

( ) 2h u u= Since

H and L∞ stable with zero bias and ( ) 2r rα = . It is not finite-gain L∞ stable because the

function ( ) 2h u u= cannot be bounded by a straight line of the form ( )h u uγ β≤ + , for all u∈ℜ . ∆

1.2 The Small-Gain Theorem The formalism of input-output stability is particular useful in studying stability of interconnected systems, since the gain of a system allows us to track how the norm of a signal increases or decreases as it passes through the system. This is particularly so for the feedback connection in the next figure.

( )( ) ( ) ( )0 0

p pph u t dt bc u t dt∞ ∞

≤∫ ∫

( )( ) ( )2

0 0sup supt t

h u t u t≥ ≥

⎛ ⎞≤ ⎜ ⎟⎝ ⎠

Here, we have two systems 1 : m q

e eH L L→ and 2 : q me eH L L→ . Suppose both systems are

finite-gain L stable; that is

1 1 1 1

2 2 2 2

,

,L L

L L

y e

y eτ τ

τ τ

γ β

γ β

≤ +

≤ +

[ )[ )

1

2

, 0,

, 0,

me

qe

e L

e L

τ

τ

∀ ∈ ∀ ∈ ∞

∀ ∈ ∀ ∈ ∞

Theorem 1.1: Under the preceding assumptions, the feedback connection is finite-gain L stable if 1 2 1γ γ < . Proof Assuming existence of the solution we can write

( )1 1 2 2e u H eτ τ τ= − , ( )2 2 1 1e u H eτ τ τ

= − Then,

Since 1 2 1γ γ < ,

for all [ )0,τ ∈ ∞ . Similarly,

( )

( )( )

1 1 2 2 1 2 2 2

1 2 2 1 1 1 2

1 2 1 1 2 2 2 2 1

L L L LL

L L L

L L L

e u H e u e

u u e

u u u

τ τ τ ττ

τ τ τ

τ τ τ

γ β

γ γ β β

γ γ γ β γ β

≤ + ≤ + +

≤ + + + +

= + + + +

( )1 1 2 2 2 2 11 2

11 L L

e u uτ τ ττγ β γ β

γ γ≤ + + +

−

for all [ )0,τ ∈ ∞ . The proof is completed by noting that 1 2L L

e e eτ≤ + , which follows

from the triangle inequality.

■ Remark 1.2: This is a sufficient but not necessary condition. Remark 1.3: Restricting ourselves to linear systems, there is a converse result. For any linear system 1H with gain 1γ , there exists a destabilizing linear system 2H with gain

12

1γγ = .

The feedback connection in the figure provides a convenient setup for studying robustness issues in dynamical systems. Quite often, dynamical systems subject to model

uncertainties can be represented in the form of a feedback connection with 1H , say, as a stable nominal system and 2H as a stable perturbation. Then, the requirement 1 2 1γ γ < is satisfied whenever 2γ is small enough, respectively, by controlling the system 1H to have the smallest gain, it becomes robust to the largest purturbations. Thus, the small-gain theorem provides a conceptual framework for understanding many of the robustness results that arise in the study of dynamical systems, especially when feedback is used.

( )1 1 2 2 2 2 11 2

11 L L

e u uτ τ ττγ β γ β

γ γ≤ + + +

−