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

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NonlinearBook10pt November 20, 2007

DISCRETE-TIME THEORY 795

deﬁnite, that is, v(0) = 0, v(x) > 0, x 6= 0. Note that v(x) = p

V (x) ≤

max

i=1,...,q

V (x), x ∈ D, and hence, the function e

V (x), x ∈ D,

is also positive deﬁnite. Thus, there exist r > 0 and class K functions

α, β : [0, r] → R

such that B

(0) ⊂ D and

α(kxk) ≤ e

V (x) ≤ β(kxk), x ∈ B

(0). (13.88)

i) Let ε > 0 and choose 0 < ˆε < min{ε, r}. It follows from Lyapunov

stability of the nonlinear comparison system (13.84) that there exists µ =

µ(ˆε) = µ(ε) > 0 such that if kz

< µ and z

∈ R

, then kz(k)k

< α(ˆε)

and z(k) ∈ R

, k ≥ k

, for every x

∈ D. Now, choose z

= V (x

) ≥≥ 0,

∈ D. Since V (x), x ∈ D, is continuous, the function e

V (x), x ∈ D, is

also continuous. Hence, for µ = µ(ˆε) > 0 there exists δ = δ(µ(ˆε)) = δ(ε) > 0

such that δ < ˆε and if kx

k < δ, then e

V (x

) = e

< µ, which implies

that e

z(k) = kz(k)k

< α(ˆε), k ≥ k

. Now, with z

= V (x

) ≥≥ 0, x

∈

D, and the assumption that w(·) ∈ W

, it follows from (13.86) and Corollary

13.8 that 0 ≤≤ V (x(k)) ≤ ≤ z(k) on any compact interval {k

, . . . , k

+ τ},

and hence, e

z(k) = kz(k)k

, k ∈ {k

, . . . , k

+ τ}. Let τ > k

be such that

z(k) ∈ B

(0), k ∈ {k

, . . . , k

+ τ}, for all x

∈ B

(0). Thus, us ing (13.88),

if kx

k < δ, then

α(kx(k)k) ≤ e

V (x(k)) ≤ e

z(k) < α(ˆε), k ≥ k

, (13.89)

which implies kx(k)k < ˆε < ε, k ≥ k

. Now, suppose, ad absurdum,

that for some x

∈ B

(0) there exists

k > k

+ τ such that kx(

k)k = ˆε.

Then, for z

= V (x

) and the compact interval {k

, . . . ,

k} it follows from

(13.86) and Corollary 13.8 that V (x(

k)) ≤≤ z(

k), which implies that

α(ˆε) = α(kx(

k)k) ≤ e

V (x(

k)) ≤ e

k) < α(ˆε). This is a contradiction,

and hence, for a given ε > 0 there exists δ = δ(ε) > 0 such that for all

∈ B

(0), kx(k)k < ε, k ≥ k

, which implies Lyapunov stability of the

zero solution x(k) ≡ 0 to (13.77).

ii) It follows f rom i) and the asymptotic stability of the nonlinear

comparison system (13.84) that the zero solution to (13.77) is Lyapunov

stable and there exists µ > 0 such that if kz

< µ and z

∈ R

, then

lim

k→∞

z(k) = 0 for every x

∈ D. As in i), choose z

= V (x

) ≥≥ 0, x

∈

D. It follows fr om Lyapun ov stability of the zero solution to (13.77) and

the continuity of V : D → Q ∩ R

that there exists δ = δ(µ) > 0 s uch that

if kx

k < δ, th en kx(k)k < r, k ≥ k

, and e

V (x

) = e

= kz

< µ.

Thus , by asymptotic stability of (13.84), for every ε > 0 there exists K =

K(ε) > k

such that e

z(k) = kz(k)k

< α(ε), k ≥ K. Thus, it follows

from (13.86) and C orollary 13.8 that V (x(k)) ≤≤ z(k), k ≥ K, and hence,

by (13.88),

α(kx(k)k) ≤ e

V (x(k)) ≤ e

z(k) < α(ε), k ≥ K. (13.90)

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796 CHAPTER 13

Now, suppose, ad absurdum, that for some x

∈ B

(0), lim

k→∞

x(k) 6= 0,

that is, there exists a sequence {k

}

∞

n=1

, with k

→ ∞ as n → ∞, such that

kx(k

)k ≥ ˆε, n ∈ Z

, for some 0 < ˆε < r. Choose ε = ˆε and the interval

{k, . . . , k+τ } such that at least one k

∈ {k, . . . , k+τ}. Then it follows from

(13.89) that α(ε) ≤ α(kx(k

)k) < α(ε), which is a contradiction. Hence,

there exists δ > 0 such that f or all x

∈ B

(0), lim

k→∞

x(k) = 0 which, along

with Lyapunov stability, implies asymptotic stability of the zero solution

x(k) ≡ 0 to (13.77).

iii) Suppose D = R

, Q = R

, v : R

→ R

is a positive deﬁnite,

radially unbounded f unction, and the nonlinear comparison system (13.84) is

globally asymptotically stable. In this case, for V : R

→ R

the inequality

(13.88) holds for all x ∈ R

where the functions α, β : R

→ R

are of

class K

∞

. Furthermore, Lyapunov stability of the zero solution x(k) ≡ 0 to

(13.77) follows from i). Next, for every x

∈ R

and z

= V (x

) ∈ R

the

identical arguments as in ii) can be us ed to show that lim

k→∞

x(k) = 0,

which proves global asymptotic stability of the zero solution x(k) ≡ 0 to

(13.77).

iv) Suppose (13.87) holds. Since p ∈ R

, then

ˆαkxk

≤ e

V (x) ≤

βkxk

, x ∈ D, (13.91)

where ˆα , α/ max

i=1,...,q

} and

β , β/ min

i=1,...,q

}. It follows from the

geometric stability of the nonlinear comparison system (13.84) that there

exist positive constants γ, µ, and η > 1 such that if kz

< µ and z

∈ R

then z(k) ∈ R

, k ≥ k

, and

kz(k)k

≤ γkz

−(k− k

)

, k ≥ k

, (13.92)

for all x

∈ D. Choose z

= V (x

) ≥≥ 0, x

∈ D. By continuity of

V : D → Q ∩ R

, there exists δ = δ(µ) > 0 s uch that for all x

∈ B

(0),

V (x

) = e

= kz

< µ. Furthermore, it follows from (13.91) and

(13.92), and Corollary 13.8 that for all x

∈ B

(0),

ˆαkx(k)k

≤ e

V (x(k)) ≤ e

z(k) ≤ γkz

−(k− k

)

≤ γ

βkx

−(k− k

)

k ≥ k

. (13.93)

This in turn implies that for every x

∈ B

(0),

kx(k)k ≤



ˆα



kη

−

k−k

, k ≥ k

, (13.94)

which establishes geometric stability of the zero solution x(k) ≡ 0 to (13.77).

v) The proof is identical to the proof of iv).

If V : D → Q ∩ R

satisﬁes the conditions of Theorem 13.15 we say

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DISCRETE-TIME THEORY 797

that V (x), x ∈ D is a vector Lyapunov fu nc tion. Note that for stability

analysis each component of a vector Lyapunov function need not be positive

deﬁnite, nor does it need to have a negative deﬁnite time diﬀerence along

the trajectories of (13.85). This provides more ﬂexibility in searching for

a vector Lyapu nov function as compared to a scalar Lyapunov function for

addressing the stability of discrete-time nonlinear dynamical systems.

Finally, we provide a time-varying extension of Theorem 13.15. In

particular, we consid er the discrete-time nonlinear time-varying dynamical

system

x(k + 1) = f(k, x(k)), x(k

) = x

, k ≥ k

, (13.95)

where x(k) ∈ D ⊆ R

, 0 ∈ D, f : {k

, . . . , k

}×D → D is such that f(·, ·) is

continuous, and, for every k ∈ {k

, . . . , k

}, f (k, 0) = 0.

Theorem 13.16. Cons ider the discrete-time nonlinear time-varying

dynamical system (13.95). Assume that there exist a continuous vector

function V : Z

× D → Q ∩ R

, a positive vector p ∈ R

, and class

K functions α, β : [0, r] → R

such that V (k, 0) = 0, k ∈ Z

, the scalar

function v : Z

×D → R

deﬁned by v(k, x)

△

= p

V (k, x), (k, x) ∈ Z

×D,

is such that

α(kxk) ≤ v(k, x) ≤ β(kxk), (k, x) ∈ Z

× B

(0), B

(0) ⊆ D, (13.96)

and

V (k + 1, f(x)) −V (k, x) ≤≤ w(k, V (k, x)), x ∈ D, k ∈ Z

, (13.97)

where w : Z

× Q → R

is continuous, w(k, ·) ∈ W

, and w(k, 0) = 0, k ∈

. Then the s tability properties of the zero solution z(k) ≡ 0 to

z(k + 1) = w(k, z(k)), z(k

) = z

, k ≥ k

, (13.98)

where z

∈ Q ∩ R

, imply the corresponding stability properties of the zero

solution x(k) ≡ 0 to (13.95). Th at is, if the zero solution z(k) ≡ 0 to (13.98)

is uniformly Lyapunov (respectively, un iform ly asymptotically) stable, then

the zero solution x(k) ≡ 0 to (13.95) is uniformly Lyapun ov (respectively,

uniformly asymptotically) stable. If, in addition, D = R

, Q = R

, and

α(·), β(·) are class K

∞

functions, then global uniform asymptotic stability

of the zero solution z(k) ≡ 0 to (13.98) implies global uniform asymptotic

stability of the zero solution x(k) ≡ 0 to (13.95). Moreover, if there exist

constants ν ≥ 1, α > 0, and β > 0 such that v : Z

× D → R

satisﬁes

αkxk

≤ v(k, x) ≤ βkxk

, (k, x) ∈ Z

× D, (13.99)

then geometric stability of the zero solution z(k) ≡ 0 to (13.98) implies

geometric stability of the zero solution x(k) ≡ 0 to (13.95). Finally, if

D = R

, Q = R

, then there exist constants ν ≥ 1, α > 0, and β > 0 such

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798 CHAPTER 13

that v : Z

× R

→ R

satisﬁes (13.99), then global geometric stability of

the zero solution z(k) ≡ 0 to (13.98) implies global geometric stability of

the zero solution x(k) ≡ 0 to (13.95).

Proof. T he proof is similar to the proof of Theorem 13.15 and is left

as an exercise for the reader.

13.9 Dissipative and Geometrically Dissipative Discrete-Time

Dynamical Systems

In this s ection, we consider the discrete-time nonlinear dynamical systems

G of the form

x(k + 1) = F (x(k), u(k)), x(k

) = x

, k ≥ k

, (13.100)

y(k) = H(x(k), u(k)), (13.101)

where x ∈ D ⊆ R

, D is open with 0 ∈ D, u(k) ∈ U ⊆ R

, y(k) ∈ Y ⊆ R

F : D × U → R

, and H : D × U → Y . We assume that F (·, ·) and H(·, ·)

are continuous mappings and F (·, ·) has at least one equilibrium so that,

without loss of generality, F (0, 0) = 0 and H(0, 0) = 0. Note that since all

input-output pairs u(·) ∈ U, y(·) ∈ Y, of the discrete-time dynamical system

G are deﬁned on Z

, the supply rate r : U ×Y → R satisfying r(0, 0) = 0 is

locally summable for all input-output pairs satisfying (13.100) and (13.101),

that is,

k=k

|r(u(k), y(k))| < ∞, k

, k

∈ Z

Deﬁnition 13.11. A nonlinear discrete-time dynamical system G of

the form (13.100) and (13.101) is dissipative with respect to the supply rate

r(u, y) if the dissipation inequality

0 ≤

k−1

i=k

r(u(i), y(i)) (13.102)

is satisﬁed for all k − 1 ≥ k

and all u(·) ∈ U with x(k

) = 0 along th e

trajectories of G. A discrete-time dyn amical system G of the form (13.100)

and (13.101) is geometrically dissipative with respect to the supply rate r(u, y)

if there exists a constant ρ > 1 such that the geometric dissipation inequality

0 ≤

k−1

i=k

i+1

r(u(i), y(i)) (13.103)

is satisﬁed for all k − 1 ≥ k

and all u(·) ∈ U with x(k

) = 0 along th e

trajectories of G. A dynamical system G of the form (13.100) and (13.101)

is lossless with respect to the supply rate r(u, y) if G is dissipative with respect

to the supply rate r(u, y) and the dissipation inequality (13.102) is satisﬁed

as an equality for all k − 1 ≥ k

and all u(·) ∈ U with x(k

) = x(k) = 0

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DISCRETE-TIME THEORY 799

along the trajectories of G.

Next, deﬁne the available storage V

) of the discrete-time nonlinear

dynamical system G by

)

△

= − inf

u(·), K≥0

K−1

k=0

r(u(k), y(k))

= sup

u(·), K≥0

−

K−1

k=0

r(u(k), y(k))

, (13.104)

where x(k), k ≥ k

, is the solution to (13.100) with admissible input u(·) ∈

U. Note that V

(x) ≥ 0 for all x ∈ D since V

(x) is the supremum over a

set of numbers containing the zero element (K = 1). Similarly, deﬁne the

available geometric storage V

) of the nonlinear dynamical system G by

)

△

= − inf

u(·), K≥0

K−1

k=0

k+1

r(u(k), y(k)), (13.105)

where x(k), k ∈ Z

, is the solution to (13.100) with x(0) = x

and admissible

input u(·) ∈ U. Note that if we d eﬁ ne the available geometric s torage as the

time-varying function

, k

) = − inf

u(·), K≥k

K−1

k=k

k+1

r(u(k), y(k)), (13.106)

where x(k), k ≥ k

, is the solution to (13.100) with x(k

) = x

and

admissible input u(·) ∈ U, it follows that, since G is time invariant,

, k

) = −ρ

inf

u(·), K≥0

K−1

k=0

k+1

r(u(k), y(k)) = ρ

). (13.107)

Hence, an alternative expression for the available geometric s torage function

) is given by

) = −ρ

−k

inf

u(·), K≥k

K−1

k=k

k+1

r(u(k), y(k)). (13.108)

Next, we show that the available storage is ﬁnite and zero at the origin

if and only if G is dissipative. For this result we require the following three

deﬁnitions.

Deﬁnition 13.12. A nonlinear dynamical system G is completely

reachable if for all x

∈ D ⊆ R

there exist a k

< k

and a square summable

input u(k) deﬁned on [k

, k

] such that the state x(k), k ≥ k

, can be driven

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800 CHAPTER 13

from x(k

) = 0 to x(k

) = x

. G is completely null controllable if for all

∈ D ⊆ R

there exist a ﬁnite time k

> k

and a square summable input

u(k) deﬁned on [k

, k

] such that x(k), k ≥ k

, can be driven from x(k

) = x

to x(k

) = 0.

Deﬁnition 13.13. Consider the discrete-time nonlinear dynamical

system G given by (13.100) and (13.101). A continuous nonnegative-deﬁnite

function V

: D → R satisfying V

(0) = 0 and

(x(k)) ≤ V

(x(k

)) +

k−1

i=k

r(u(i), y(i)), k − 1 ≥ k

, (13.109)

for all k

, k ∈ Z

, w here x(k), k ∈ Z

, is th e solution of (13.100) with

u(·) ∈ U, is called a storage function for G.

Deﬁnition 13.14. Consider the discrete-time nonlinear dynamical

system G given by (13.100) and (13.101). A continuous nonnegative-deﬁnite

function V

: D → R satisfying V

(0) = 0 and

(x(k)) ≤ ρ

(x(k

)) +

k−1

i=k

i+1

r(u(i), y(i)), k −1 ≥ k

, (13.110)

for all k

, k ∈ Z

, where x(k), k ≥ k

, is the solution of (13.100) with

u(·) ∈ U, is called an geometric storage function for G.

Theorem 13.17. Consider the discrete-time nonlinear dynamical sys-

tem G given by (13.100) and (13.101), and assume that G is completely

reachable. Then G is dissipative (respectively, geometrically dissipative)

with respect to the supp ly rate r(u, y) if and only if the available system

storage V

) given by (13.104) (resp ectively, the available geometric

storage V

(x) given by (13.105)) is ﬁnite for all x

∈ D and V

(0) = 0.

Moreover, if V

(0) = 0 and V

) is ﬁnite for all x

∈ D, then V

(x),

x ∈ D, is a storage function (respectively, geometric storage function) for

G. Finally, all storage functions (respectively, geometric storage functions)

(x), x ∈ D, for G satisfy

0 ≤ V

(x) ≤ V

(x), x ∈ D. (13.111)

Proof. Suppose V

), x

∈ D, is ﬁnite. Now, it follows from (13.104)

(with K = 1) that V

) ≥ 0, x

∈ D. Next, let x(k), k ≥ k

, satisfy

(13.100) with admissible input u(k), k ∈ [k

, K −1]. Since − V

), x

∈ D,

is given by the inﬁmum over all admissible inputs u(·) ∈ U in (13.104), it

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DISCRETE-TIME THEORY 801

follows that for all admissible inputs u(·) ∈ U and K − 1 ≥ k

−V

(x(k

)) ≤

K−1

k=k

r(u(k), y(k))

−1

k=k

r(u(k), y(k)) +

K−1

k=k

r(u(k), y(k)),

which implies

−V

(x(k

)) −

−1

k=k

r(u(k), y(k)) ≤

K−1

k=k

r(u(k), y(k)).

Hence,

(x(k

)) +

−1

k=k

r(u(k), y(k)) ≥ − inf

u(·), K≥k

K−1

k=k

r(u(k), y(k))

= V

(x(k

))

≥ 0, (13.112)

which implies that

−1

k=k

r(u(k), y(k)) ≥ −V

(x(k

)). (13.113)

Hence, since by assumption V

(0) = 0, G is d iss ipative with respect to the

supply rate r(u, y). Furthermore, V

(x), x ∈ D, is a storage function for G.

Conversely, suppose that G is dissipative with respect to the supply

rate r(u, y). Since G is completely reachable it follows that for every x

∈ R

such that x(k

) = x

, there exist

k ≤ k < k

and an adm iss ible input

u(·) ∈ U deﬁned on [

k, k

] such that x(

k) = 0 and x(k

) = x

. Furthermore,

since G is dissip ative with respect to the supply rate r(u, y) and x(

k) = 0 it

follows that

K−1

r(u(k), y(k)) ≥ 0, K − 1 ≥

k, (13.114)

or, equivalently,

K−1

k=k

r(u(k), y(k)) ≥ −

−1

r(u(k), y(k), K − 1 ≥ k

, (13.115)

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802 CHAPTER 13

which implies that there exists a function W : R

→ R such that

K−1

k=k

r(u(k), y(k)) ≥ W (x

) > −∞, K − 1 ≥ k

. (13.116)

Now, it follows from (13.116) that for all x ∈ R

(x) = − inf

u(·), K≥0

K−1

k=0

r(u(k), y(k))

≤ −W (x), (13.117)

and hence, the available storage V

(x) < ∞, x ∈ D. Furthermore, with

x(k

) = 0 and for all admissible u(k), k ≥ k

K−1

k=k

r(u(k), y(k)) ≥ 0, K − 1 ≥ k

, (13.118)

which implies that

sup

u(·), K≥k





−

K−1

k=k

r(u(k), y(k))





≤ 0, (13.119)

or, equivalently, V

(x(k

)) = V

(0) ≤ 0. However, since V

(x) ≥ 0, x ∈ D, it

follows that V

(0) = 0.

Moreover, if V

(x) is ﬁnite for all x ∈ D, it follows from (13.112) that

(x), x ∈ R

, is a storage function for G. Next, if V

(x), x ∈ D, is a storage

function then it follows that, for all K − 1 ≥ 0 and x

∈ D,

) ≥ V

(x(K)) −

K−1

k=0

r(u(k), y(k))

≥ −

K−1

k=0

r(u(k), y(k)),

which implies

) ≥ − inf

u(·), K≥0

K−1

k=0

r(u(k), y(k))

= V

Finally, the proof for the geometrically dissipative case follows an

identical construction and, hence, is omitted.

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DISCRETE-TIME THEORY 803

For the remainder of this chapter, we assume that all storage

functions of G are continuous on D. Now, the following corollary is

immediate from Theorem 13.17 and s hows that a system G is dissipative

(respectively, geometrically dissipative) with respect to the supply rate

r(·, ·) if and only if there exists a continuous storage function V

(·)

satisfying (13.109) (respectively, a continuous geometric storage f unction

(·) satisfying (13.110)).

Corollary 13.9. Consid er the discrete-time nonlinear dynamical sys-

tem G given by (13.100) and (13.101) and assume that G is completely

reachable. Then G is dissipative (respectively, geometrically dissipative)

with respect to supply rate r(·, ·) if and only if there exists a continuous

storage function (respectively, geometric storage function) V

(x), x ∈ D ,

satisfying (13.109) (respectively, (13.110)).

Proof. The result is immediate from Theorem 13.17 with V

(x) =

(x).

The following theorem provides conditions f or guaranteeing that all

storage functions (respectively, geometric storage functions) of a given

discrete-time dissipative (respectively, geometrically dissipative) nonlinear

dynamical s ystem are positive deﬁnite. For this result we requ ir e the

following deﬁnition.

Deﬁnition 13.15. A dynamical system G is zero-state observable if

u(k) ≡ 0 and y(k) ≡ 0 implies x(k) ≡ 0.

Theorem 13.18. Consider the discrete-time nonlinear dynamical sys-

tem G given by (13.100) and (13.101), and assume that G is completely

reachable and zero-state observable. Furthermore, assume that G is

dissipative (respectively, geometrically dissipative) with respect to s upply

rate r(u, y) and there exists a function κ : Y → U such that κ(0) = 0 and

r(κ(y), y) < 0, y 6= 0. Then all the storage functions (respectively, geometric

storage functions) V

(x), x ∈ D, for G are positive deﬁnite, that is, V

(0) = 0

and V

(x) > 0, x ∈ D, x 6= 0.

Proof. It follows from Theorem 13.17 that the available storage V

(x),

x ∈ D, is a storage function for G. Next, suppose there exists x ∈ D

such that V

(x) = 0, which implies that r(u(k), y(k)) = 0, k ∈ Z

, for all

admissible inputs u(·) ∈ U. Since th ere exists a function κ : Y → U such

that r(κ(y), y) < 0, y 6= 0, it follows that y(k) = 0, k ∈ Z

. Now, since G

is zero-state observable it follows that x = 0, and hence, V

(x) = 0 if and

only if x = 0. The result now follows from (13.111). Finally, the proof for

the geometrically dissipative case is identical and, hence, is omitted.

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804 CHAPTER 13

An equivalent statement for dissipativity of G with respect to the

supply rate r(u, y) is

∆V

(x(k)) ≤ r(u(k), y(k)), k ∈ Z

, (13.120)

where ∆V

(x(k))

△

= V (x(k + 1)) − V (x(k)). Alternatively, an equivalent

statement f or geometric dissipativity of G w ith respect to the su pply rate

r(u, y) is

ρV

(x(k + 1)) − V

(x(k)) ≤ ρr(u(k), y(k)), k ∈ Z

. (13.121)

Furth ermore, a system G with storage function V

(·) is strictly dissipative

with respect to the supply rate r(u, y) if and only if

(x(k)) < V

(x(k

)) +

k−1

i=k

r(u(i), y(i)), k − 1 ≥ k

. (13.122)

Note that geometric dissipativity implies strict dissipativity, however, the

converse is not necessarily true.

Next, we introduce the concept of a required supply of a discrete-time

nonlinear dynamical system. Speciﬁcally, deﬁne the requi red supply V

)

of the discrete-time nonlinear dynamical system G by

) = inf

u(·), K≥1

−1

k=−K

r(u(k), y(k)), (13.123)

where x(k), k ≥ −K, is the solution to (13.100) with x(−K) = 0 and

x(0) = x

. It follows from (13.123) that the required supply of a nonlinear

dynamical system is the minimum amount of generalized energy that has to

be delivered to the dynamical system in order to transfer it from an initial

state x(−K) = 0 to a given state x(0) = x

. Similarly, deﬁne the req uired

geometric supply of the n on linear dynamical system G by

) = inf

u(·), K≥1

−1

k=−K

k+1

r(u(k), y(k)), (13.124)

where x(k), k ≥ −K, is the solution to (13.100) with x(−K) = 0 and

x(0) = x

. Note that since, w ith x(0) = 0, the inﬁmum in (13.123) is zero

it follows that V

(0) = 0.

Next, using the notion of a required supply, we show that all storage

functions are bounded from ab ove by the required supply and bounded

from below by the available storage, and hence, a dissipative discrete-time

dynamical system can deliver to its s urround ings only a fraction of its stored

generalized energy and can store only a fraction of the generalized work done

to it.