On <span class="mathjax-formula">$\mathcal {L}_2$</span> performance induced by feedbacks with multiple saturations

ON L2 PERFORMANCE INDUCED BY FEEDBACKS WITH MULTIPLE SATURATIONS

ANDREW R. TEEL

Abstract. Multi-level saturation feedbacks induce nonlinear distur- bance-to-state ^L2 stability for nonlinear systems in feedforward form.

This class of systems includes linear systems with actuator constraints.

NotationA function : IR 0 ^!IR 0 is said to be of class-^K+ ( ^2K+) if it is continuous and nondecreasing. It is of class-^K if, in addition, it is zero at zero. It is of class-^K1 if, moreover, it is strictly increasing and unbounded. For ^2K1, its inverse is another function of class-^K1 and is denoted ^;1.

By abuse of notation we will often write the vector (

x

^T

u

^T)^T as (

xu

).

A measurable signal

v

: 0

¹) ^! IR^m is said to belong to ^L2 or

v

^{2 L}2

(respectively, belong to^L1 or

v

^2L1) if the quantity

jj

v

^jj2:=

s

Z

1

0 ^j

v

(

t

)^j2

dt

(resp. ^jj

v

^jj1:= ess. sup_{t 0}^j

v

(

t

)^j) is nite.

The function sat : IR^{n !}IRⁿ is dened as sat(

x

) =

x

max^f1

^j

x

^jg.

More generally, a function

: IR^{m !} IR^m is said to be a saturation function if it is dierentiable at the origin and there exist

K >

0,

b >

0 such that, for all

uv

²IR^m,

1. ^j

(

u

+

v

)^;

(

u

)^jmin^f

K

^j

v

^j

b

^g, 2. ^j

(

u

)^;

u

^j

Ku

^T

(

u

) .

For a nonlinear control system with state space

x

²IRⁿ, an

m

-level satu- ration feedback is any feedback of the form

u

=

(

Fx

+

v

) (0.1)

where

F

is a matrix of appropriate dimension,

is a saturation function and

v

is an (

m

^;1)-level saturation feedback. A zero-level saturation feedback is the identically zero function.

For a strictly positive real number

a

and a vector

v

²IR^m, where

m

is an arbitrary strictly positive integer,

(

va

) =

v

^;

a

sat

v

a

. Also,

(

v

0) =

v

.

Department of Electrical Engineering, University of Minnesota, 4-174 EE/CS Building, 200 Union St. SE, Minneapolis, MN 55455. Email: [email protected].

Received by the journal April 2, 1996. Accepted for publication September 6, 1996.

Research supported in part by the NSF under grants ECS-9309523 and ECS-9502034.

c

Societe de Mathematiques Appliquees et Industrielles.

1. Introduction

Recently, it has been shown that multi-level saturation feedback (see the notation section) can be eectively used to stabilize the origin of linear systems subjected to actuator constraints (see 5], 4], 10]). In fact, these feedbacks have been shown to induce the property that, in the presence of additive disturbances converging to a suciently small ball, the state converges to a proportionally small ball.

In this paper, we will establish nonlinear disturbance-to-state^L2stability using multi-level saturation feedback. This duplicates the result of 3] where nonlinear ^L2 stability is established using a nonlinear controller that stati- cally schedules a family of linear ^H1 controllers (c.f. 9]). Our result is a corollary of a nonlinear^L2 performance result for a general class of so-called nonlinear feedforward systems. This class of systems includes, for example,

\the ball and beam" (see 6]), the \PVTOL" and \inverted pendulum on a cart" (see 8]), and linear systems with limits on the magnitude and

n

derivatives of the input (see 7]).

The proof of our main result is obtained with two tools. Initially, we use a Lyapunov argument to establish nonlinear disturbance-to-state^L2 stability for critically stable, stabilizable linear control systems with actuator satura- tion when a certain passive linear feedback is used. This result draws on the proof techniques used in 2]. A small gain argument is then used to show that the stability is robust to input-driven additive, dynamic perturbations that satisfy certain^L2 stability properties. Ultimately, this robust stabiliza- tion result is used iteratively in controller design for nonlinear feedforward systems.

The paper is organized as follows. In section 2 we summarize our main results on ^L2 performance while the subsequent sections are dedicated to the proofs. In section 3 we present a robust stabilization result (lemma 3.2) for critically stable, stabilizable linear systems with additive, dynamic perturbations driven by the input. This result is the basis for an inductive proof of our main results (theorems 2.2 and 2.3). Using the proof, we are able to show how the main results extend to cover, for example, results for linear systems with exponentially unstable modes. This is done at the end of section 3. The proof of the robust stabilization result (lemma 3.2) is given in section 4. Its proof uses a small gain argument together with a result (lemma 4.1) concerning nonlinear disturbance-to-state ^L2 stabilization by passive linear feedback for critically stable, stabilizable linear systems with actuator saturation. The proof of lemma 4.1 is given in section 5. Finally, we will provide some concluding remarks.

Esaim: Cocv, September 1996, Vol. 1, pp. 225{240

2. Main Results

Nonlinear feedforward systems: The main result of this paper applies to nonlinear feedforward control systems, i.e. systems of the form

x

_1 =

A

1

x

1+

f

1(

x

2

::: x

p

ud

)

x

_2 =

A

2

x

2+

f

2(

x

3

::: x

p

ud

)

...

x

_p^;1 =

A

p^;1

x

p^;1+

f

p^;1(

x

p

ud

)

x

_p =

A

p

x

p+

f

p(

ud

)

(2.1)

where

x

i ² IRⁿⁱ. Dene

n

=

n

1 +

:::

+

n

p and

X

i = (

x

_Ti

::: x

_Tp)^T. For system (2.1), our standing assumption will be the following.

Assumption 2.1. The

f

iare locally Lipschitz and zero at zero, the Jacobian linearization at the origin and with

d

0 exists and is stabilizable, the

A

i

are critically stable, and for each

i

there exists a class-^K+ function

'

i such that¹

j

f

i(

X

i+1

ud

)^;

f

i(

X

i+1

u

0)^j

'

i(^j(

X

i+1

u

)^j)^j

d

^j

:

(2.2) With this assumption, the class of systems we are considering is slightly less general than that considered in 10]. In particular, the condition (2.2) was not assumed and the

x

p subsystem was allowed to be more general in 10]. (See the end of section 3 for a discussion of the case where the

x

p

subsystem is more general.) Under assumption 2.1, we can show that there exists a

p

-level saturation feedback that induces nonlinear^L2 stability from

d

to the state

x

. For simplicity, we will only consider multiple saturation feedbacks that are nested, as dened in the notation section of this paper.

But, more general combinations (see 4] and 10]) could also be used to give the same result. Our result is summarized in the theorem below. The proof will be given in section 3. A control synthesis algorithm is given after the statement of the theorem.

Theorem 2.2. For the system in (2.1) satisfying assumption 2.1, there ex- ist a_d

p

-level saturation feedback

() and class-^K functions ₂, _d2, ¹ and

1such that, for each

d

^2L2 and each

x

²IRⁿ, the trajectory of the system (2.1) with

u

=

(

x

) and

x

(0) =

x

exists for all

t

0 and satis es:

jj

x

^jj2 max 2(^j

x

^j)

_d2(^jj

d

^jj2)

jj

x

^jj1 max ¹ (^j

x

^j)

^1d (^jj

d

^jj2)

:

^(2.3) This result implies that the origin is globally asymptotically stable when

d

0. Indeed, global stability follows from the second inequality. Then, with

x x

_ ^2L1 and

x

^{2 L}2 (from the rst inequality) it follows from Bar- balat's lemma (see 1, Lemma 4.4]) that

x

converges to zero.

We now summarize the control synthesis algorithm. Throughout,

i

= 1

::: p

. The matrices (^Ai

^Bi) will represent the linear approximation of

1To make sense out of the caseⁱ=^p, dene^Xp+1 to be a vector of dimension zero.

Esaim: Cocv, September 1996, Vol. 1, pp. 225{240

the

X

i subsystem with

d

0 (where

X

i is dened above assumption 2.1), i.e., with

d

0 and ignoring higher order terms in

X

i and

u

we have

X

_i =^Ai

X

i+^Bi

u :

(2.4) Let

v

i(

x

) = ^;B_Ti

P

i

X

i where

P

i is a positive denite, symmetric matrix satisfying

ATi

P

i+

P

i^A_i 0 (2.5)

with

Ai =

@

@X

i

2

4

Ai

X

i+^Bi Xp j=i+1

v

j(

x

)

3

5

:

(2.6)

(One rst determines

P

p, which depends only on (^Ap

^Bp), then

v

p, then

P

p^;1, then

v

p^;1, etc.) Next let

w

0(

x

)0 and

w

i(

x

) =

i

i

v

i(

x

) +

w

i^;1(

x

)

i

(2.7) where the

i are saturation functions. With the parameters

i

>

0 adjusted appropriately (guided by the proof of theorem 2.2), the control law

(

x

) is the

p

-level saturation feedback

w

p(

x

).

Linear systems with actuator saturation: We now state the corollary of the above theorem for linear control systems of the form

x

_ =

Ax

+

Bu

+

w

1

y

=

Cx

+

w

2 (2.8)

where

x

² IRⁿ,

w

= (

w

_T1

w

_T2)^{T 2 L}2. The result is for stabilizable, de- tectable systems that may be open loop unstable but are not open loop exponentially unstable. Open loop exponentially unstable systems will be discussed at the end of section 3.

Theorem 2.3. Consider the system (2.8). If (

AB

) is stabilizable, (

CA

) is detectable and the eigenvalues of

A

have nonpositive real part then, with

L

chosen so that

A

+

LC

is Hurwitz, there exists a

p

-level saturation feedback

() (

p

n

) and class-^K functions ₂, _e2, _w2, ¹ , ^1e and ^1w such that, using the dynamic feedback

_^

x

=

A x

^+

Bu

+

L

(

C x

^^;

y

)

u

=

(^

x

)

^(2.9)

for each

w

^2L2, each

x

²IRⁿ and each

x

^ ²IRⁿ, the trajectory of the sys- tem (2.8)-(2.9) with(

x

(0)

^

x

(0)) = (

x

x

^) exists for all

t

0 and satis es

jj

x

^jj2 max^f 2(^j

x

^j)

_e2(^j

x

^;

x

^^j)

_w2(^jj

w

^jj2) ^g

jj

x

^jj1 max^{f 1} (^j

x

^j)

^1e (^j

x

^;

x

^^j)

^1w(^jj

w

^jj2) ^g

:

^(2.10)

Proof. Dening

e

=

x

^;

x

^, we get

e

_= (

A

+

LC

)

e

+

w

1^;

Lw

2

:

(2.11)

Esaim: Cocv, September 1996, Vol. 1, pp. 225{240

Since (

A

+

LC

) is Hurwitz, there exist strictly positive real numbers

and such that

jj

e

^jj2max^f

^j

e

(0)^j

^jj

w

^jj2^g

:

(2.12) Next, since (

AB

) is stabilizable and the eigenvalues of

A

have nonpositive real part, there exists a coordinate transformation

z

=

Tx

so that

TAT

^;1 is upper triangular with

p

critically stable matrices on the diagonal for some

p

n

. In the

z

coordinates, the system is in the form of system (2.1) and satises assumption 2.1. Picking

u

=

(^

x

) where

is an appropriate

p

-level saturation feedback and using the fact that

is globally Lipschitz, the result

follows from theorem 2.2. ²

Remark 2.4. As pointed out in 4], if the result holds for

u

=

(^

x

) then the same results holds (qualitatively) for

u

=

(^

x=

). This follows by working in the coordinates

x

=

x=

. The consequence of this observation is that nonlinear^L2 stability can be achieved with a

p

-level saturation that is arbitrarily small in magnitude. In fact, the general feedforward result could be used to prove a similar result with arbitrarily small bounds on the input magnitude and any number of its derivatives (c.f. 10]).

Remark 2.5. The stability gain from

w

to the state must be, in general, super-linear at innity when using a bounded control. Otherwise, a small gain argument would give ^L2 stability for small perturbations including those that moved the open loop poles from the imaginary axis into the open right half plane. This is not possible since even global asymptotic stabilization with bounded controls is not possible in this case.

Remark 2.6. The result is robust to small (dynamic) uncertainty in how the input aects the dynamics. (Such perturbations cannot move the open loop pole locations.) We have chosen not to state this result here but the results should be transparent after digesting the proof. The analogous (re- stricted)^L1 stability results have been presented in 10].

3. Proving theorem 2.2

The proof of theorem 2.2 is by induction on a robust stabilization result for critically stable linear systems with additive, dynamic disturbances driven by the input. To state this result compactly, we make a preliminary denition.

In the denition, the ^L2-norm of a disturbance's distance to a ball of a certain radius is related to the^L2-norm of the output's distance to a ball of a related radius.

Definition 3.1. The output

y

of a dynamical system

x

_ =

f

(

xd

1

d

2)

y

=

h

(

xd

1

d

2) (3.1)

with

x

²IRⁿ,

y

²IR^m,

d

1 ² IR^p1,

d

2 ²IR^p2 is said to satisfy the induction hypothesis if there exist strictly positive real numbers

,

M

,

L

and and class-^Kfunctions ₂, ₂^d2, ¹ and ^1d2 such that, for all

a

0 and

d

1 satis- fying ^jj

d

1^jj1

M

and ^jj

(

d

1

a

)^jj2

<

¹, all

d

2 ^{2 L}2 and all

x

²IRⁿ, the

Esaim: Cocv, September 1996, Vol. 1, pp. 225{240

trajectories of the system (3.1) with

x

(0) =

x

satisfy²:

jj

(

ya

)^jj2 maxⁿ ₂(^j

x

^j)

^jj

(

d

1

a

)^jj2

₂^d2(^jj

d

2^jj2) ^o

jj

y

^jj1 max ¹ (^j

x

^j)

L

^jj

d

1^jj1

^1d2(^jj

d

2^jj2)

:

^(3.2) The following lemma on robust stabilization will be used to prove theorem 2.2. The result parallels 10, Theorem 3] where an analogous result is stated in terms of^L1 properties. The proof of lemma 3.2 is in section 4.

Lemma 3.2. Consider the locally Lipschitz control system

x

_1 =

Ax

1+

Bu

+

g

(

x

2

ud

)

x

_2 =

f

(

x

2

ud

)

:

^(3.3)

where

x

1 ²IRⁿ¹ and

x

2 ²IRⁿ². Suppose

1. (

AB

) is stabilizable and there exists

P

=

P

^T

>

0 such that

A

^T

P

+

PA

0, i.e.,

A

is critically stable,

2. the state

x

2 satis es the induction hypothesis with

d

1 :=

u

and

d

2:=

d

, 3. there exists a function

'

of class-^K+ such that

j

g

(

x

2

ud

)^;

g

(

x

2

u

0)^j

'

(^j(

x

2

u

)^j)^j

d

^j

4. lim

j(x²u)^j!0

j

g

(

x

2

u

0)^j

j(

x

2

u

)^j = 0 .

Let

be a saturation function. Then there exists a strictly positive real number

such that, with the control

u

=

^;

B

^T

Px

1+

v

(3.4) where

²(0

], the output

x

1 for the(

x

1

x

2) system satis es the induction hypothesis with

d

1:=

v

and

d

2:=

d

.

Proof of theorem 2.2.

The proof is by induction. Apply lemma 3.2 to the

x

p subsystem (

x

pis to be identied with

x

1in the lemma and there is no

x

2 subsystem in this case) to see that, with a control of the form (3.4) where

B

=

B

p= ^@f_@u^pjdu=0 and

=

psuciently small, the state

x

psatises the induction hypothesis with

d

1 :=

v

and

d

2 :=

d

. It can be easily shown that

A

p^;

B

p

B

_Tp

P

p is Hurwitz and, with the properties of a saturation function and the fact that

A

p^;1 is critically stable, it follows that the linearization of the

X

p^;1 subsystem with

v

as control is stabilizable and open loop critically stable.

We analyze the

X

i subsystem, for

i

= 1

::: p

^;1, by rst making a copy of the

X

i+1 subsystem, the state of which we denote by ~

X

i+1, i.e.,

X

_~i+1=

F

i+1( ~

X

i+1

vd

)

X

^~i+1(0) =

X

i+1(0)

:

(3.5) Let (

A

i

B

i) represent the Jacobian linearization of the

X

i subsystem and let the function

g

i(

X

i+1

vd

) be given by

g

i(

X

i+1

vd

) :=

F

i(

X

i+1

vd

)^;(

A

i

X

i+

B

i

v

)

:

(3.6)

2See the notation section for the denition of(^{v a}).

Esaim: Cocv, September 1996, Vol. 1, pp. 225{240

The fact that

g

iis independent of

x

ifollows from the structure of feedforward systems. Notice that lim

j(Xⁱ⁺¹v)^j!0

j

g

i(

X

i+1

v

0)^j

j(

X

i+1

v

)^j = 0. We can now write the

X

i system as

X

_i =

A

i

X

i+

B

i

v

+

g

i( ~

X

i+1

vd

)

X

_~i+1 =

F

i+1( ~

X

i+1

vd

)

:

^(3.7) Since (

A

i

B

i) is stabilizable and

A

i is a critically stable matrix, we can again apply lemma 3.2 (

X

i is associated with

x

1, ~

X

i+1is associated with

x

2

and

v

is associated with

u

) to get that, under a control

v

of the form (3.4) with

=

i suciently small (and with

v

on the right hand side of (3.4) replaced by

w

), the state

X

i satises the induction hypothesis with

d

1 :=

w

and

d

2 :=

d

. When

i

2, it follows from the structure of the system (2.1) and the properties of

that the linearization of the

X

i^;1 subsystem with

w

as control is stabilizable and open loop critically stable. So, theorem 2.2

follows by induction. ²

Extensions of theorem 2.2.

From the proof of theorem 2.2, we see all that is required for the

x

p

subsystem is the existence of a control

u

=

(

x

p

v

) that is dierentiable at the origin, locally exponentially stabilizes the origin of the

x

p subsystem when

d

0, the linearization is controllable through

v

, and the state

x

p

satises the induction hypothesis. So, if we rewrite the

x

p subsystem in the more general form

x

_p= ~

f

p(

x

p

ud

p) (3.8) and assume the existence of such an

(

x

p

v

) then we arrive at the conclusion of theorem 2.2. We will see from the proof of lemma 3.2 that if this feedback is such that

x

p only satises the induction hypothesis for^j

x

p(0)^jsuciently small and

d

pwith suciently small^L2norm, those restrictions can be carried through during the iteration to get the conclusion of theorem 2.2 but with restrictions on^j

x

p(0)^jand the ^L2-norm of

d

p.

A special case where this discussion is relevant is when the

x

p subsystem contains the exponentially unstable open loop modes of a linear system and when the actuators of the linear system saturate, i.e.,

x

_p =

A

p

x

p+

B

p

(

u

) +

d

p (3.9) where the eigenvalues of

A

p have positive real part and

is a saturation function. In this case, because

is bounded, no feedback exists so that

x

p satises the induction hypothesis for all

x

p(0) and all

d

p ^2L2. But,

x

p

does satisfy the hypothesis, at least for^j

x

p(0)^jsuciently small and

d

p with suciently small ^L2 norm, when

u

=

Fx

p +

v

where

u

=

Fx

p is locally exponentially stabilizing when

d

p 0. So, a local version of theorem 2.3 holds when

A

has eigenvalues with positive real part.

Esaim: Cocv, September 1996, Vol. 1, pp. 225{240

4. Proving lemma 3.2

The key piece in proving lemma 3.2 is the following result for critically stable, stabilizable linear systems with input saturation and additive dis- turbances. It shows that a particular passive feedback induces the type of stability described in the induction hypothesis.

Lemma 4.1. Let (

AB

) be stabilizable, let

A

be such that there exists

P

=

P

^T

>

0 satisfying

A

^T

P

+

PA

0 and let

be a saturation function. Then the state of the system

x

_ =

Ax

+

B

^;;

B

^T

Px

+

d

1+

d

2 (4.1) satis es the induction hypothesis.

Remark 4.2. This result for the case where

d

2 0 and

a

= 0 (in the denition of the induction hypothesis) was already reported in 2]. The proof of lemma 4.1, given in section 5, draws on the proof technique used in 2].We will combine lemma 4.1 with small gain arguments to prove lemma 3.2. Before we do, we need some preliminary facts.

Fact 1. Let

c

0. If ^j

w

^jmax^fj

w

1^j

^j

w

2^j

^j

w

3^jgthen

jj

(

wc

)^jj2^p3 max

jj

(

w

1

c

)^jj2

^jj

(

w

2

c

)^jj2

^jj

(

w

3

c

)^jj2

:

Proof. We have

j

(

wc

)^j = ^j

w

^j

1^;

c

max^f

c

^j

w

^jg

max

j

w

1^j

^j

w

2^j

^j

w

3^j

1^;

c

max^f

c

^j

w

1^j

^j

w

2^j

^j

w

3^jg

= max

j

(

w

1

c

)^j

^j

(

w

2

c

)^j

^j

(

w

3

c

)^j

:

(4.2)

Thus

j

(

wc

)^j2 max

j

(

w

1

c

)^j2

^j

(

w

2

c

)^j2

^j

(

w

3

c

)^j2

j

(

w

1

c

)^j2+^j

(

w

2

c

)^j2+^j

(

w

3

c

)^j2

:

^(4.3) We integrate both sides, use the fact that

a

+

b

+

c

3 max^f

abc

^g for positive numbers

abc

and then take the square root on both sides to obtain the result.

2

Fact 2. If 0

< a

1

< a

2 then^jj

(

wa

2)^jj2^jj

(

wa

1)^jj2. Proof. We have

j

(

wa

2)^j = ^j

w

^j1^; 1

max^f1

^j

w

^j

=a

2^g

j

w

^j

1^; 1

max^f1

^j

w

^j

=a

1^g

= ^j

(

wa

1)^j

:

(4.4)

Esaim: Cocv, September 1996, Vol. 1, pp. 225{240

2

Fact 3. Let

c

0 and

k >

0. Then

(

kwc

) =

k

(

wc=k

).

Proof. We have

(

kwc

) =

kw

^;

c

sat

kw c

=

k

w

^;

c=k

sat

w c=k

=

k

(

wc=k

)

:

(4.5)

2

Proof of lemma 3.2.

Let the given saturation function

be parameterized by the strictly pos- itive real numbers

K

and

b

(see notation section). For the given

A

,

B

and

, let (

1

M

1

L

1

1) be the set of constants of the induction hypothesis that follows from lemma 4.1. For the given

x

2 subsystem, let (

2

M

2

L

2

2) be the set of constants of the induction hypothesis given by assumption. We will use as a generic class-^Kfunction. All norms below should be thought of as norms on truncated signals. The norms on 0

¹), once this is shown to be the maximal interval of denition, can be obtained from the limiting process as the truncation time goes to innity.

We write the

x

1 subsystem as

x

_1=

Ax

1+

B

;

B

^T

Px

1

+

d

1+

d

2 (4.6) where

d

1 =

B

;

B

^T

Px

1+

v

;

;

B

^T

Px

1

+

g

2

b

sat

x

2

2

b

u

0

d

2 =

g

(

x

2

ud

)^;

g

2

b

sat

x

2

2

b

u

0

:

(4.7)

Our goal is to nd a suitable

>

0 so that the lemma holds. This will be achieved by choosing

:= min^f

1

2

3

4^g (4.8)

where the

i are strictly positive real numbers to be specied.

Let

1=

M

2

=b

. Then

²(0

] guarantees that, on the maximal interval of denition,^jj

u

^jj1

M

2. Then, since

x

2 satises the induction hypothesis and since

d

^{2 L}2,

x

2 ^{2 L1} on the maximal interval of denition. This, together with the form of the dierential equation, implies that the maximal interval of denition is 0

¹).

Let

2 be a strictly positive real number such that, for all 0

<

2,

j

x

2^j

2

b

^j

u

^j

b

=^)j

g

(

x

2

u

0)^j

< M

1

:

(4.9) Such a

2 exists from the fourth assumption of the lemma. Then

²(0

] guarantees that there exists ~

M >

0 such that ^jj

v

^jj1

M

^~ implies ^jj

d

1^jj1

M

1.

Esaim: Cocv, September 1996, Vol. 1, pp. 225{240

Now, with

u

given in (3.4), there exist a strictly positive real number

K

, class-^K+ functions

'

and

'

2 and a class-^Kfunction

such that

j

d

1^j max

K

^j

v

^j

(

b

)^j

x

2^j

(

b

)^j

x

1^j

j

d

2^j

'

(^j(

x

2

u

)^j)^j

d

^j+

'

2(^j(

x

2

u

)^j)^j

(

x

2

2

b

)^j

:

^(4.10) So, using facts 1 and 3,

jj

(

d

1

a

)^jj2^p3 max

K

v aK

2

(4.11)

(

b

)

x

2

a

(

b

)

2 (

b

)

x

1

a

(

b

)

2

:

Since

x

2 satises the induction hypothesis,

x

2

a

(

b

)

2 max

(^j

x

2 ^j)

2

u a

(

b

)

2

2

(^jj

d

^jj2)

:

(4.12) Then, combining (4.11) and (4.12) and using (3.4), the fact that

is globally Lipschitz and fact 2, there exist strictly positive real numbers

K

1 and

K

2

such that

jj

(

d

1

a

)^jj2max

(^j

x

2 ^j)

K

1

v aK

1

2

(4.13)

K

2

(

b

)

x

1

a K

2

(

b

)

2

(^jj

d

^jj2)

:

Now, using the scalings

x

^!

x=

,

d

1 ^!

d

1

=

,

d

2 ^!

d

2

=

, and using lemma 4.1 and fact 3 we have, for^jj

v

^jj1

M

^~,

jj

(

x

1

1

a

)^jj2 max

^j

x

1 ^j

1^jj

(

d

1

a

)^jj2

jj

d

2^jj2

jj

x

1^jj1 max

^j

x

1 ^j

L

1^jj

d

1^jj1

jj

d

2^jj2

:

(4.14) Let

3 satisfy

(

3

b

)

K

2

<

min

1 1

¹1

:

(4.15)

Then, combining (4.13) and the rst inequality of (4.14) and using fact 2, we have for

²(0

] that

jj

(

x

1

1

a

)^jj2 max

^j

x

1 ^j

1 (^j

x

2 ^j)

(4.16)

1

K

1

v aK

1

2

1 (^jj

d

^jj2)

jj

d

2^jj2

:

Using (4.10) and the bound on^jj

(

x

2

2

b

)^jj2from the induction hypothesis satised by

x

2 we have

jj

d

2^jj2

'

^jj(

x

2

u

)^jj1

jj

d

^jj2+

'

2

jj(

x

2

u

)^jj1

max

(^j

x

2^j)

(^jj

d

^jj2)

:

(4.17)

Esaim: Cocv, September 1996, Vol. 1, pp. 225{240

First inserting into (4.17) the bound on^jj

x

2^jj1from the induction hypothesis and, in turn, the bound^jj

u

^jj1

b

and then inserting the resulting bound on^jj

d

2^jj2 into (4.16), we get the form of bound on^jj

(

x

1

1

a

)^jj2 claimed in the lemma.

For the bound on^jj

x

1^jj1, we repeat the above argument using

jj

d

1^jj1max

K

^jj

v

^jj1 (

b

)^jj

x

2^jj1 (

b

)^jj

x

1^jj1

(4.18) instead of (4.11) and using, from the induction hypothesis on

x

2,

jj

x

2^jj1max

(^j

x

2 ^j)

L

2^jj

u

^jj1

(^jj

d

^jj2)

(4.19) instead of (4.12). We then can assert, as we did before (4.13), the existence of strictly positive real numbers

K

1and

K

2 such that (compare with (4.13))

jj

d

1^jj1max

(^j

x

2 ^j)

K

1^jj

v

^jj1

K

2

(

b

)^jj

x

1^jj1

(^jj

d

^jj2)

:

(4.20) Using the second of the inequalities in (4.14) and letting

4 satisfy

(

4

b

)

K

2

< L

¹1 (4.21) we have that

²(0

] implies (compare with (4.16))

jj

x

1^jj1max

j

x

1 ^j

L

1 (^j

x

2 ^j)

(4.22)

L

1

K

1^jj

v

^jj1

L

1 (^jj

d

^jj1)

jj

d

2^jj2

:

Then using the bound on ^jj

d

2^jj2 established above we get the form of the bound on^jj

x

1^jj1claimed in the lemma.

We conclude that the induction hypothesis is satised by ~

M

, ~

L

=

L

1

K

1,

~ = 1

K

1 and ~

=

1

K

1. ²

5. Proof of lemma 4.1

Fact 4. Under the conditions of lemma 4.1, there exists

P

2 =

P

_T2

>

0 such that

(

A

^;

BB

^T

P

)^T

P

2+

P

2(

A

^;

BB

^T

P

) =^;

I :

A preliminary energy function: Consider a function of the form

V

(

x

) =

Z x^TPx

0

1(

s

)

ds

+

Z x^TP²x

0

2(

s

)

ds

(5.1)

where

1 and

2 are continuous, nonnegative functions to be specied.

When we consider the time derivative of

V

(

x

) along the trajectories of (4.1) we nd, using ^jj

d

1^jj1

M

, completely squares and using the second in- equality in the denition of a saturation function, that there exists strictly

Esaim: Cocv, September 1996, Vol. 1, pp. 225{240

positive real numbers

c

1,

c

2 and ~

K

such that

V

_

1(

x

^T

Px

)2

x

^T

PB

(^;

B

^T

Px

) + 2

x

^T

P

(

d

1+

d

2)+

2(

x

^T

P

2

x

)^;

x

^T

x

+ 2

x

^T

P

2

B

^;

(^;

B

^T

Px

) +

B

^T

Px

+ 2

x

^T

P

2(

d

1+

d

2)

;

1(

x

^T

Px

)^;;2

x

^T

PB

(^;

B

^T

Px

)

+2

M

max(

P

)^j

x

^j

1(

x

^T

Px

) +^j

x

^j2

₂₁(

x

^T

Px

) +

c

1^j

d

2^j2

;

x

^T

x

2(

x

^T

P

2

x

) + ~

K

^j

x

^j;;2

x

^T

PB

^;;

B

^T

Px

2^;

x

^T

P

2

x

+

2^;

x

^T

P

2

x

2

max(

P

2)^j

x

^jj

d

1^j+

₂₂(

x

^T

P

2

x

)^j

x

^j2+

c

2^j

d

2^j2

:

(5.2) If

M

,

1 and

2 can be chosen so that the following three inequalities are satised:

K

~^j

x

^j

2(

x

^T

P

2

x

)

1(

x

^T

Px

) (5.3) 2

M

max(

P

)

1(

x

^T

Px

) 0

:

5^j

x

^j

2(

x

^T

P

2

x

) (5.4)

₂₁(

x

^T

Px

) +

₂₂(

x

^T

P

2

x

) 0

:

25

2(

x

^T

P

2

x

) (5.5) then

V

_

2(

x

^T

P

2

x

)

;0

:

25^j

x

^j2+ 2

max(

P

2)^j

x

^jj

d

1^j

+ ~

c

^j

d

2^j2 (5.6) where ~

c

=

c

1+

c

2. To satisfy inequalities (5.3)-(5.5), we choose

2(

s

) = 1 4

K

~²

s

³

max(

P

2) + 1

! (5.7)

where

=

min(

P

)

min(

P

2)

max(

P

)

max(

P

2) (5.8) and we choose

1(

s

) = ~

K

^r

s

min(

P

)

2

min(

P

2)

s

max(

P

)

:

(5.9)

Notice that if 0

b

a

then

2(

a

)

2(

b

) but also

2(

s

) 1

²⁽

s

) (since

1). We then have

K

~^j

x

^j

2(

x

^T

P

2

x

)

K

~

s

x

^T

Px

min(

P

)

2

min(

P

2)

max(

P

)

x

^T

Px

=

1(

x

^T

Px

) (5.10) so that (5.3) is satised. Also,

1(

x

^T

Px

)

K

~

s

max(

P

)

min(

P

)^j

x

^j

2(

x

^T

P

2

x

)

K

~

s

max(

P

)

min(

P

)^j

x

^j

^{1 2(}

x

^T

P

2

x

)

:

^(5.11)

Esaim: Cocv, September 1996, Vol. 1, pp. 225{240