Haddad W.M. Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach

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NonlinearBook10pt November 20, 2007

STRUCTURED PARAMETRIC UNCERTAINTY 725

equivalent to

He[Z(ω)(M

−1

+ G(ω)(I + M

G(ω))

−1

)] > 0.

Now, premultiplying and postmultiplying the above inequality by I +

∗

(ω)M

and I + M

G(ω) yields

He[(I + G

∗

(ω)M

)Z(ω)(M

−1

+ M

−1

G(ω) + G(ω))] > 0. (12.21)

Since N(· ) ∈ N and M

, M

∈ ∆ it f ollows that N M

= M

N and NM

N. Thus, NM = M N, and hence, N M

−1

= M

−1

N. Using these

relations (12.21) simpliﬁes to

He[DM

−1

+ 2DM

−1

G(ω) + G

∗

(ω)M

−1

G(ω) + DG(ω)

∗

(ω)DM

G(ω)] > He[NG(ω)]. (12.22)

Now, forming x

∗

(12.22)x yields

∗

He[DM

−1

+ 2DM

−1

G(ω) + G

∗

(ω)M

−1

G(ω)

+DG(ω) + G

∗

(ω)DM

G(ω)]x

> He[x

∗

NG(ω)x]

= −x

∗

(ω)He[∆N]G(ω)x.

Since N(· ) ∈ N, it follows that He[∆N ] =



(∆N − N∆) = 0, and hence,

∗

He[DM

−1

+ 2DM

−1

G(ω) + G

∗

(ω)M

−1

G(ω)

+DG(ω) + G

∗

(ω)DM

G(ω)]x > 0,

which contradicts (12.20). Consequently, det (I + G(ω)∆) 6= 0 for all ∆ ∈

∆.

For the next result we assume that the feedback interconnection of

G(s) and ∆ is well posed, that is, det[I + G(∞)∆] 6= 0 for all ∆ ∈ ∆.

Theorem 12.1. Suppose G

(s)

△

= (I + G(s)M

)

−1

G(s) is asymptoti-

cally stable. If there exists Z(·) ∈ Z such that

He [Z(s)(M

−1

+ G

(s))] > 0, (12.23)

for all s = ω, ω ∈ R ∪ ∞, then the negative feedback interconn ection of

G(s) and ∆ is asymptotically stable for all ∆ ∈ ∆.

Proof. Let ∆ ∈ ∆ and

G(s) ∼



A B

C D



be minimal so that the negative f eedback interconnection of G(s) and ∆ is

NonlinearBook10pt November 20, 2007

726 CHAPTER 12

given by

(I + G(s)∆)

−1

G(s) ∼



A − B∆(I + D∆)

−1

C B − B∆(I + D∆)

−1

(I + D∆)

−1

C (I + D∆)

−1



Suppose that, ad absurdum, (I +G(s)∆)

−1

G(s) is not asymptotically stable

so that A − B∆(I + D∆)

−1

C is not Hurwitz. Sin ce by assump tion G

(s)

is asymptotically stable it follows that A −BM

(I + DM

)

−1

C is Hurwitz.

Hence, there exists ε ∈ (0, 1) such that A − B∆

(I + D∆

)

−1

C has an

eigenvalue ˆω on the imaginary axis, where ∆

△

= ε∆ + (1 − ε)M

Next, note that

det(I + G(ˆω)∆

)

= det[I + C(ˆωI − A)

−1

B∆

+ D∆

]

= det(I + D∆

)

−1

det[I + (I + D∆

)

−1

C(ˆωI − A)

−1

B∆

]

= det(I + D∆

)

−1

det(ˆωI − A)

−1

det[ˆωI − (A −B∆

(I + D∆

)

−1

= 0.

However, since ∆

∈ ∆ and det(I + G(ˆω)M

) 6= 0, Lemma 12.1 with ω = ˆω

implies that det(I + G(ˆω)∆

) 6= 0, which is a contradiction.

Next, we specialize Theorem 12.1 to the case of norm-bounded

uncertainty in order to draw connections with th e structured singular value

for real and complex block-structured uncertainty. Letting M

= −γ

−1

I and

= γ

−1

I, where γ > 0, it follows that M = 2γ

−1

I so that M

−1

γI.

The set ∆ thus becomes

∆

= {∆ ∈ ∆

: −γ

−1

I ≤ ∆ ≤ γ

−1

I}.

Now, ∆ ∈ ∆

if and only if σ

max

(∆) ≤ γ

−1

. Therefore, ∆

is given by

∆

= {∆ ∈ ∆

: σ

max

(∆) ≤ γ

−1

Alternatively, one can also consider the case where there may exist ∆ ∈ ∆

such that ∆ 6= ∆

∗

and still arrive at ∆ ∈ ∆

for the case where M

−γ

−1

I and M

= γ

−1

I. In this case, it can be shown that Lemma 12.1

also holds (see P roblem 12.2). For the remainder of this section we consider

uncertainties ∆ ∈ ∆

such that ∆ 6= ∆

∗

Now, we consider a special case of the sets ∆

, D, and N where

∆ ∈ ∆

. In particular, let ∆

be th e set of block-structured matrices w ith

possibly repeated real scalar entries, complex scalar entries, and complex

blocks given by

∆

= {∆ ∈ C

m×m

: ∆= block−d iag[δ

, . . . , δ

; δ

r+1

, . . . , δ

r+q

;

∆

r+q+1

, . . . , ∆

r+c

], δ

∈ R, i = 1, . . . , r; δ

∈ C, i = r + 1, . . . , r + q;

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STRUCTURED PARAMETRIC UNCERTAINTY 727

∆

∈ C

ˆm

×ˆm

, i = r + q + 1, . . . , r + c}. (12.24)

Then D and N are the sets of frequency-dependent positive-deﬁnite and

Hermitian matrices, respectively, given by

D = {D : R → C

m×m

: D = block−diag[D

, . . . , D

r+q

, d

r+q+1

ˆm

r+q +1

, . . . ,

r+c

ˆm

r+c

], 0 < D

∈ C

×l

, i = 1, . . . , r + q; 0 < d

∈ R,

i = r + q + 1, . . . , r + c}, (12.25)

N = {N : R → C

m×m

: N = block−diag[N

, . . . , N

, 0

r+1

, . . . , 0

r+c

= N

∗

∈ C

×l

, i = 1, . . . , r}. (12.26)

Note that this special case is equivalent to the mixed-µ set considered in

[117]. Furthermore, with D(·) ∈ D and N (·) ∈ N given by (12.25) and

(12.26), respectively, the compatibility conditions required in D and N are

automatically satisﬁed for ∆

given by (12.24).

Alternatively, let ∆

be given by (12.24) with the additional con-

straint that the complex blocks possess internal matrix structur e. Then D

and N are given by

D = {D : R → C

m×m

: D = block−diag[D

, . . . , D

r+c

], 0 < D

∈ C

×l

i = 1 . . . r + c; D

∆

= ∆

, i = r + q + 1, . . . , r + c},

N = {N : R → C

m×m

: N = block−diag[N

, . . . , N

, 0

r+1

, . . . , 0

r+q

r+q+1

, . . . , N

r+c

], N

= N

∗

∈ C

×l

, i = 1, . . . , r + c;

∆

= ∆

∗

, i = r + q + 1, . . . , r + c}.

For example, if ∆

= ∆

∗

∈ ∆

then we can choose D

= d

I, d

∈ R and

= n

I, n

∈ R, i = r + q + 1, . . . , r + c.

Next, we present a key lemma which is important in connecting the

absolute stability criterion given in Theorem 12.1 to the mixed-µ upper

bounds.

Proposition 12.1. Let Z(·) ∈ Z, D(·) ∈ D, N(·) ∈ N, and ω ∈ R ∪∞.

Then the following statements are equivalent:

i) He [Z(ω)(

γI + G

(ω))] > 0.

ii) G

∗

(ω)DG(ω) + γ(NG(ω) − G

∗

(ω)N) − γ

D < 0.

Proof. Note that He [Z(ω)(

γI + G

(ω))] > 0 is equivalent to

(D − N)(

γI + G(ω)(I − γ

−1

G(ω))

−1

)

γI + (I − γ

−1

∗

(ω))

−1

∗

(ω))(D + N) > 0.

NonlinearBook10pt November 20, 2007

728 CHAPTER 12

Now, premultiplying and postmultiplying the above inequality by I −

−1

∗

(ω) and I − γ

−1

G(ω), respectively, we obtain

∗

(ω)D(I − γ

−1

G(ω)) −(I − γ

−1

∗

(ω))NG(ω)

+G

∗

(ω)N(I − γ

−1

G(ω)) + γ(I − γ

−1

∗

(ω))D(I − γ

−1

G(ω))

+(I − γ

−1

∗

(ω))DG(ω) > 0,

which, upon collecting terms, further simpliﬁes to

γD − γ

−1

∗

(ω)DG(ω) − (NG(ω) − G

∗

(ω)N) > 0,

which is equivalent to (ii).

We now obtain upper boun ds for the structured singular value for real

and complex multiple-block structured uncertainty. These bounds are based

upon the absolute stability criterion of Theorem 12.1 for norm-bounded,

block-structured uncertain matrices. The stru ctured singular value [117] of

a complex matrix G(ω) for mixed real and complex uncertainty is deﬁned

by µ(G(ω))

△

= 0 if det(I + G(ω)∆) 6= 0, ∆ ∈ ∆

, and

µ(G(ω))

△



min

∆∈∆

{σ

max

(∆) : det (I + G(ω)∆) = 0}



−1

, (12.27)

otherwise. Hence, a necessary and suﬃcient condition for robust stability

of th e feedb ack interconn ection of G(s) and ∆ is given by the following

theorem. For the statement of this result we assume that the feedback

interconnection of G(s) and ∆ is well posed for all ∆ ∈ ∆

Theorem 12.2. Let γ > 0 and s uppose G(s) is asymptotically stable.

Then the negative feedback interconnection of G(s) and ∆ is asymptotically

stable for all ∆ ∈ ∆

if and only if

µ(G(ω)) < γ, ω ∈ R ∪ ∞. (12.28)

Proof. Let

G(s) ∼



A B

C D



where A is Hurwitz, and suppose the negative f eedback interconn ection of

G(s) and ∆ given by

(I + G(s)∆)

−1

G(s) ∼



A − B∆(I + D∆)

−1

C B − B∆(I + D∆)

−1

(I + D∆)

−1

C (I + D∆)

−1



is asymptotically stable for all ∆ ∈ ∆

. Next, note that, for all ∆ ∈ ∆

and ω ∈ R ∪ ∞

det[I + G(ω)∆]

= det[I + (C(ωI − A)

−1

B + D)∆]

NonlinearBook10pt November 20, 2007

STRUCTURED PARAMETRIC UNCERTAINTY 729

= det(I + D∆)det[I + (ωI − A)

−1

B∆(I + D∆)

−1

= det(I + D∆)det(ωI − A)

−1

det[ωI − (A − B∆(I + D∆)

−1

C)]

6= 0.

Hence, by deﬁnition µ(G(ω)) < γ for all ω ∈ R ∪∞.

Conversely, suppose µ(G(ω)) < γ for all ω ∈ R ∪ ∞ and assu me that

G(s) ∼



A B

C D



is minimal. Then, by assumption, det(I + G(∞)∆) = det(I + D∆) 6= 0 for

all ∆ ∈ ∆

. Now, ad absurdum, su ppose there exists ∆ ∈ ∆

such that

(I + G(s)∆)

−1

G(s) is not asymptotically stable, and hence, A − B∆(I +

D∆)

−1

C is not Hurwitz. Since G(s) is assumed to be asymptotically stable

it follows that A is Hurwitz, and hence, there exists ε ∈ (0, 1) such that

A − εB∆(I + εD∆)

−1

C has an imaginary eigenvalue ˆω. Hence,

det[I + εG(ˆω)∆]

= det(I + εD∆)det(ˆωI − A)

−1

det[ˆωI − (A − εB∆(I + εD∆)

−1

C)]

= 0.

However, since ε∆ ∈ ∆

it follows from the deﬁnition of µ(G(ω)) th at

det[I + εG(ˆω)∆] 6= 0, w hich is a contradiction.

Next, deﬁne µ(G(ω)) by

µ(G(ω))

△

= inf{γ > 0 : there exists Z(·) ∈ Z such that

He [Z(ω)(

γI + G

(ω))] > 0}, (12.29)

or, equivalently, using Proposition 12.1

µ(G(ω)) = inf{γ > 0 : there exist D(·) ∈ D and N(·) ∈ N such that

∗

(ω)DG(ω) + γ(NG(ω) − G

∗

(ω)N) − γ

D < 0}.

(12.30)

To show that µ(G(ω)) is an upper bound to µ(G(ω)), we require the

following immediate result.

Lemma 12.2. Let ω ∈ R ∪ ∞. If th ere exists Z(·) ∈ Z such that

He[Z(ω)(

γI + G

(ω))] > 0, (12.31)

then γ ≥ µ(G(ω)) and det(I + G(ω)∆) 6= 0 for all ∆ ∈ ∆

. Conversely,

if γ > µ(G(ω)) then there exists Z(·) ∈ Z such that (12.31) holds and

det (I + G(ω)∆) 6= 0 for all ∆ ∈ ∆

Proof. Suppose there exists Z(·) ∈ Z such that (12.31) holds. Sin ce

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730 CHAPTER 12

µ(G(ω)) is the inﬁmum over all γ such that there exists Z(·) ∈ Z and (12.31)

holds, it follows that γ ≥ µ(G(ω)). Conversely, suppose that γ > µ(G(ω)).

Then there exists ˆγ satisfying µ(G(ω)) ≤ ˆγ < γ and Z(·) ∈ Z such that

He [Z(ω)(

ˆγI+G

(ω))] > 0. Now, using the fact that D(ω) = He Z(ω) ≥

0, it follows that

He [Z(ω)(

γI + G

(ω))] =

(γ − ˆγ)He Z(ω) + He [Z(ω)(

ˆγI + G

(ω))]

> 0.

Finally, applying L emm a 12.1 with M

−1

γI and ∆ = ∆

, it follows

that det (I + G(ω)∆) 6= 0 for all ∆ ∈ ∆

Theorem 12.3. Let ω ∈ R ∪ ∞ and let G(ω) be a complex matrix.

Then

µ(G(ω)) ≤ µ(G(ω)). (12.32)

Proof. Suppose,ad absurdum, that µ(G(ω)) < µ(G(ω)) and let γ > 0

satisfy µ(G(ω)) < γ ≤ µ(G(ω)). Then, from the deﬁnition of µ(G(ω))

it follows that min

∆∈∆

{σ

max

(∆) : det(I + G(ω)∆) = 0} ≤ γ

−1

. It thus

follows that there exists ∆ ∈ ∆

such that det (I + G(ω)∆) = 0. However,

using Lemma 12.2 we know if µ(G(ω)) < γ, then det (I + G(ω)∆) 6= 0 for

all ∆ ∈ ∆

, which is a contradiction. Hence, µ(G(ω)) ≤ µ(G(ω)).

Next, in order to provide a s ystematic comparison of mixed-µ bounds

for a ﬁxed, internally block-structured u ncertainty set ∆

deﬁne µ

(G(ω))

△

= inf{γ > 0 : there exist D(·) ∈ D

and N (·) ∈ N

such that

∗

(ω)DG(ω) + γ(NG(ω) − G

∗

(ω)N) −γ

D < 0}, (12.33)

where D

, N

correspond to the pairs of frequency-dependent scaling matrix

sets tailored to a ﬁxed uncertainty structure ∆

. The following result is

immediate.

Proposition 12.2. Let G(ω) be a complex matrix and let D

, N

and

, N

be frequency-dependent scaling matrix sets associated with a ﬁxed

uncertainty structure ∆

. Suppose D

⊆ D

and N

⊆ N

. Then

(G(ω)) ≤ µ

(G(ω)). (12.34)

Proof. S uppose, ad absurdum, µ

(G(ω)) < µ

(G(ω)) and let some

γ > 0 satisfy µ

(G(ω)) < γ ≤ µ

(G(ω)). Then by deﬁnition of µ

(G(ω))

there exist D(·) ∈ D

and N(·) ∈ N

such that G

∗

(ω)DG(ω)+γ(NG(ω)−

∗

(ω)N) − γ

D < 0. Now, since D

⊆ D

and N

⊆ N

, it follows that

D(·) ∈ D

and N(·) ∈ N

, and hence, µ

(G(ω) < γ, which is a contradiction.

Hence, µ

(G(ω)) ≤ µ

(G(ω)).

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STRUCTURED PARAMETRIC UNCERTAINTY 731

Finally, note that in the complex uncertainty case, that is, r = 0, we

take D > 0 and N = 0 so that the complex-µ upper bound [110] given by

µ(G(ω)) =

inf{γ > 0 : there exists D(·) ∈ D such that G

∗

(ω)DG(ω) − γ

D < 0},

or, equivalently,

µ(G(ω)) = inf

D(·)∈D

max

G(ω)D

−

) (12.35)

is recovered. In this case, the upper bound can be computed via a convex

optimization problem [396].

12.3 Robust Stability Analysis of Non linear Uncertain Systems

via Parameter-Dependent Lyapunov Functions

In this s ection, we present suﬃcient conditions for robust stability f or a

class of nonlinear uncertain systems. Speciﬁcally, we extend the framework

of Chapter 11 in order to address stability of a class of nonlinear systems

with time-invariant structured uncertainty. Here, we restrict our attention

to time-invariant inﬁnite-horizon systems. Once again, for the class

of nonlinear uncertain systems considered we assume that the required

properties for the existence and uniqueness of s olutions are satisﬁed. For the

following result, let D ⊂ R

be an open set, assume 0 ∈ D, let L : D → R,

and let F ⊂ {f : D → R

: f(0) = 0} denote the class of uncertain n onlinear

systems with f

(·) ∈ F deﬁning the nominal nonlinear system.

Theorem 12.4. Consider the nonlinear uncertain dynamical system

˙x(t) = f(x(t)), x(0) = x

, t ≥ 0, (12.36)

where f (·) ∈ F, with performance functional

)

△

∞

L(x(t))dt. (12.37)

Furth ermore, assume there exist functions Γ : D → R and V

, V

∆f

, V : D →

R, where V

(·) and V

∆f

(·) are continuously diﬀerentiable functions such that

(x) + V

∆f

(x) = V (x) for all x ∈ D and

V (0) = 0, (12.38)

V (x) > 0, x ∈ D, x 6= 0, (12.39)

′

(x)f(x) ≤ V

′

(x)f

(x) + Γ(x), x ∈ D, f(·) ∈ F,

(12.40)

′

(x)f

(x) + Γ(x) < 0, x ∈ D, x 6= 0, (12.41)

L(x) + V

′

(x)f

(x) + Γ(x) = 0, x ∈ D, (12.42)

NonlinearBook10pt November 20, 2007

732 CHAPTER 12

where f

(·) ∈ F deﬁnes the nominal nonlinear system. Then th e zero

solution x(t) ≡ 0 to (12.36) is locally asymptotically stable for all f(·) ∈ F,

and there exists a neighborhood of the origin D

⊆ D such that

) ≤ J(x

) = V (x

), f(·) ∈ F, x

∈ D

, (12.43)

where

J(x

)

△

∞

[L(x(t)) + Γ(x(t)) −V

′

∆f

(x(t))f

(x(t))]dt, (12.44)

and where x(t), t ≥ 0, is the solution to (12.36) with f (x(t)) = f

(x(t)).

Finally, if D = R

and

V (x) → ∞ as kxk → ∞, (12.45)

then the zero solution x(t) ≡ 0 to (12.36) is globally asymp totically stable

for all f(·) ∈ F.

Proof. Let f (·) ∈ F and x(t), t ≥ 0, satisfy (12.36). Then

V (x(t))

△

V (x(t)) = V

′

(x(t))f(x(t)), t ≥ 0. (12.46)

Hence, it follows from (12.40) and (12.41) that

V (x(t)) < 0, t ≥ 0, x(t) 6= 0. (12.47)

Thus , from (12.38), (12.39), and (12.47) it follows that V (·) is a Lyapunov

function for (12.36), which proves local asymp totic stability of the zero

solution x(t) ≡ 0 to (12.36) for all f(·) ∈ F. Consequently, x(t) → 0 as

t → ∞ for all initial conditions x

∈ D

for some neighborhood of the origin

⊂ D. Now, (12.46) implies that

0 = −

V (x(t)) + V

′

(x(t))f(x(t)), t ≥ 0,

and h en ce, using (12.40) and (12.42),

L(x(t)) = −

V (x(t)) + L(x(t)) + V

′

(x(t))f(x(t))

≤ −

V (x(t)) + L(x(t)) + V

′

(x(t))f

(x(t)) + Γ(x(t))

= −

V (x(t)).

Now, integrating over [0, t) yields

L(x(s))ds ≤ −V (x(t)) + V (x

Letting t → ∞ and noting that V (x(t)) → 0 f or all x

∈ D

yields J

) ≤

V (x

Next, let x(t), t ≥ 0, satisfy (12.36) with f(x(t)) = f

(x(t)). Then it

NonlinearBook10pt November 20, 2007

STRUCTURED PARAMETRIC UNCERTAINTY 733

follows from (12.42) that

L(x(t)) + Γ(x(t)) −V

′

∆f

(x(t))f

(x(t)) = −

V (x(t)) + L(x(t))

′

(x(t))f

(x(t)) + Γ(x(t))

= −

V (x(t)).

Integrating over [0, t) yields

[L(x(s)) + Γ(x(s)) −V

′

∆f

(s)f

(x(t))]ds = −V (x(t)) + V (x

Now, letting t → ∞ yields J(x

) = V (x

). Finally, for D = R

global

asymptotic stability of the solution x(t) = 0, t ≥ 0, for all f(·) ∈ F is a

direct consequence of the radially unbounded condition (12.45) on V (x).

Theorem 12.4 is an extension of Theorem 8.1. Speciﬁcally, if V

∆f

(x) =

0 and F consists of only the nominal nonlinear system f

(·), then Γ(x) = 0

for all x ∈ D satisﬁes (12.40), and hence, J

) = J(x

). In this case,

Theorem 12.4 specializes to Theorem 8.1. Alternatively, s etting V

∆f

(x) = 0

and allowing f (·) ∈ F we recover Theorem 11.1.

Next, we specialize Theorem 12.4 to nonlinear uncertain systems of

the f orm

˙x(t) = f

(x(t)) + ∆f(x(t)), x(0) = x

, t ≥ 0, (12.48)

where f

: D → R

satisﬁes f

(0) = 0 and f

+ ∆f ∈ F. Here, F is such

that

F ⊂ {f

+ ∆f : D → R

: ∆f ∈ ∆}, (12.49)

where ∆ is a given nonlinear uncertainty set of nonlinear perturbations ∆f

of the nominal system dyn amics f

(·) ∈ F. Since F ⊂ {f : D → R

f(0) = 0} it follows that ∆f (0) = 0 for all ∆f ∈ ∆.

Corollary 12.1. Consider the dynamical system given by (12.48) where

∆f(·) ∈ ∆, with performance functional (12.37). Furthermore, assume there

exist functions Γ : R

→ R and V

, V

∆f

, V : R

→ R, where V

(·) and V

∆f

(·)

are continuously diﬀerentiable functions such that V

(x) + V

∆f

(x) = V (x)

for all x ∈ R

and

V (0) = 0, (12.50)

V (x) > 0, x ∈ R

, x 6= 0, (12.51)

′

(x)∆f(x) ≤ Γ(x) −V

′

∆f

(x)(f

(x) + ∆f(x)), x ∈ R

, ∆f(·) ∈ ∆,

(12.52)

′

(x)f

(x) + Γ(x) < 0, x ∈ R

, x 6= 0, (12.53)

L(x) + V

′

(x)f

(x) + Γ(x) = 0, x ∈ R

, (12.54)

NonlinearBook10pt November 20, 2007

734 CHAPTER 12

and

V (x) → ∞ as kxk → ∞. (12.55)

Then the zero solution x(t) ≡ 0 to (12.48) is globally asymptotically stable

for all ∆f(·) ∈ ∆, and

∆f

) ≤ J(x

) = V (x

), ∆f(·) ∈ ∆, (12.56)

where

J(x

)

△

∞

[L(x(t)) + Γ(x(t)) −V

′

∆f

(x(t))f

(x(t))]dt, (12.57)

and where x(t), t ≥ 0, is the solution to (12.48) with ∆f (x) ≡ 0.

Proof. The result is a direct consequence of T heorem 12.4 with f (x) =

(x) + ∆f(x).

Having established the theoretical basis of our approach, we now assign

explicit structure to the set ∆ and the bounding function Γ(·). Speciﬁcally,

the uncertainty set ∆ is assum ed to be of the form

F = {f

(x)+∆f (x) : ∆f(x) = G

(x)δ(h

(x)), x ∈ R

, δ(·) ∈ ∆}, (12.58)

where ∆ satisﬁes

∆ = {δ(·) ∈ ∆

: [δ(y) − m

(y)]

[δ(y) − m

(y)] ≤ 0, y ∈ R

}, (12.59)

and where m

(·), m

(·) ∈ ∆

are such that m

(y)m

(y) ≤ 0, y ∈ R

: R

→ R

n×m

, h

: R

→ R

is continuously diﬀerentiable and

satisﬁes h

(0) = 0, and

∆

⊂

(

δ : R

→ R

: δ(0) = 0, δ(·) is continuously diﬀerentiable, and

dδ(y)



dδ(y)



)

. (12.60)

For the structure of ∆ as s peciﬁed by (12.58), the bounding f unction

Γ(·) satisfying (12.52) can now be given a concrete form. For the following

result deﬁne the set of compatible scaling matrices H

and N

△

= {H ∈ P

: Hδ(y) = δ(Hy), y ∈ R

, δ(·) ∈ ∆

, }, (12.61)

△

= {N ∈ S

: Nδ(y) = δ(Ny), N (δ(y) − m

(y)) ≥ 0, y ∈ R

δ(·) ∈ ∆

, }. (12.62)

Furth ermore, deﬁne

h(x)

△

= H[m

(x)) + m

(x))] + Nh

′

(x)f

(x) and

(x)

△

= 2H − Nh

′

(x)G

(x) − G

(x)h

′T

(x)N.