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

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DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 345

energy conservation equation

H(x(t)) = u

(t)y(t) −

∂H

∂x

(x(t))R(x(t))



∂H

∂x

(x(t))



, t ≥ 0, (5.67)

which shows that the rate of change in energy, or power, is equal to the

system power input minus the internal system power dissip ated. Note that

(5.67) can be equivalently wr itten as

H(x(t)) −H(x(0)) =

(s)y(s)ds

−

∂H

∂x

(x(s))R(x(s))



∂H

∂x

(x(s))



ds. (5.68)

Equation (5.68) shows that the stored or accumulated system energy is equal

to the energy supp lied to the system via the external input u m inus the

energy dissipated over the time interval [0, t]. Since R(x) is nonnegative

deﬁnite for all x ∈ D, it follows from (5.68) that

−

(s)y(s)ds ≤ H(x(0)), (5.69)

which shows that the energy that can be extracted from the port-controlled

Hamiltonian system through the input-output ports is less than or equal to

the initial energy stored in the system. Hence, port-controlled Hamiltonian

systems are dissip ative with respect to the (power) supply rate r(u, y) =

y. Fu rthermore, note that in the case where n o system dissipation is

present, that is, R(x) ≡ 0, then port-controlled dynamical sys tems are

lossless with respect to the supply rate r(u, y) = u

Finally, we exploit the skew-symmetric structure of the internal

interconnection matrix function J(x), x ∈ D, of (5.65) to establish the

existence of Casimir functions for port-controlled Hamiltonian systems.

Speciﬁcally, let C : D → R be su ch that

∂C

∂x

(x)[J(x) − R(x)] = 0, x ∈ D. (5.70)

In this case, it follows th at

C(x) =

∂C

∂x

(x)[J(x) − R(x)]



∂H

∂x

(x)



∂C

∂x

(x)G(x)u

∂C

∂x

(x)G(x)u. (5.71)

Now, if u(t) ≡ 0 or

∂C

∂x

(x)G(x) = 0, x ∈ D, then C : D → R is conserved

along the ﬂow of (5.65) irrespective of the form of the system Hamiltonian.

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346 CHAPTER 5

Note that (5.70) is also implied by the stronger conditions

∂C

∂x

(x)J(x) = 0, x ∈ D, (5.72)

∂C

∂x

(x)R(x) = 0, x ∈ D. (5.73)

In addition, since J(x) = −J

(x) and R(x) = R

(x), (5.72) and (5.73)

imply

[J(x) − R(x)]



∂C

∂x

(x)



= 0, x ∈ D. (5.74)

Now, using the fact that the sum of a skew-symmetric m atrix and a

symmetric matrix is zero if and only if the individual matrices are zero, it

follows that (5.72) and (5.73) hold if and only if (5.70) and (5.74) hold. Next,

assuming u(t) ≡ 0, it follows from (5.71) that the α-level set of C(x) given

by C

−1

(α) = {x ∈ D : C(x) = α}, where α ∈ R, is invariant with respect

to the port-controlled Hamiltonian system (5.65). Hence, if the system

Hamiltonian H(·) is not positive deﬁnite at an equilibrium point x

∈ D,

then, constructing a shaped Hamiltonian H

(x) = H(x)+H

(C(x)) such that

(x) is positive deﬁnite at x

by properly choosing H

, it follows that H

(x)

serves as a Lyapunov function candidate for (5.65) with u(t) ≡ 0. Theorem

3.8 can be used to construct such a shaped Hamiltonian. Now, (5.62) and

(5.74) imply that

(x) ≤ 0, x ∈ D, establishing Lyapunov stability of

(5.65). More generally, in an identical fashion as above one can construct

r independent two-times continuously diﬀerentiable Casimir functions and

use Theorem 3.8 to construct shaped Hamiltonians as Lyapunov functions

for (5.65) with u(t) ≡ 0.

5.4 Extended Kalman-Yakubovich-Popov Co n d itions for

Nonlinear Dynamical Systems

In this section, we show that dissipativeness, exponential dissipativeness,

and losslessness of nonlinear aﬃne dynamical systems G of the form

˙x(t) = f(x(t)) + G(x(t))u(t), x(t

) = x

, t ≥ t

, (5.75)

y(t) = h(x(t)) + J(x(t))u(t), (5.76)

where x(t) ∈ D ⊆ R

, D is an open set with 0 ∈ D, u(t) ∈ U ⊆ R

y(t) ∈ Y ⊆ R

, f : D → R

, G : D → R

n×m

, h : D → Y , and J :

D → R

l×m

, can be characterized in terms of the system functions f (·), G(·),

h(·), and J(·). We assume that f(·), G(·), h(·), and J(·) are continuously

diﬀerentiable mappings and f (·) has at least one equilibrium s o that, without

loss of generality, f(0) = 0 and h(0) = 0. Furth ermore, for the nonlinear

dynamical system G we assume that the r equ ired properties for the existence

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DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 347

and uniqueness of solutions in forward and backward time are satisﬁed. For

the following resu lt we consider the special case of dissipative s ystems with

quadratic supply rates [457]. Speciﬁcally, set D = R

, U = R

, Y =

, let Q ∈ S

, R ∈ S

, and S ∈ R

l×m

be given, and assume r(u, y) =

Qy +2y

Su+u

Ru. Furthermore, we assume that there exists a function

κ : R

→ R

such that κ(0) = 0 and r(κ(y), y) < 0, y 6= 0, and the available

storage V

(x), x ∈ R

, f or G is a continuously diﬀerentiable function.

Theorem 5.6. Let Q ∈ S

, S ∈ R

l×m

, R ∈ S

, and let G be zero-state

observable and completely reachable. G is dissipative with respect to the

quadratic su pply rate r(u, y) = y

Qy +2y

Su + u

Ru if and only if there

exist functions V

: R

→ R, ℓ : R

→ R

, and W : R

→ R

p×m

such that

(·) is continuously diﬀerentiable and positive deﬁnite, V

(0) = 0, and, for

all x ∈ R

0 = V

′

(x)f(x) −h

(x)Qh(x) + ℓ

(x)ℓ(x), (5.77)

0 =

′

(x)G(x) − h

(x)(QJ(x) + S) + ℓ

(x)W(x), (5.78)

0 = R + S

J(x) + J

(x)S + J

(x)QJ(x) −W

(x)W(x). (5.79)

If, alternatively,

N(x)

△

= R + S

J(x) + J

(x)S + J

(x)QJ(x) > 0, x ∈ R

, (5.80)

then G is dissipative with respect to the quadratic supply rate r(u, y) =

Qy +2y

Su+u

Ru if and only if there exists a continuously diﬀerentiable

function V

: R

→ R such th at V

(·) is positive deﬁnite, V

(0) = 0, and, for

all x ∈ R

0 ≥ V

′

(x)f(x) −h

(x)Qh(x) + [

′

(x)G(x) − h

(x)(QJ(x) + S)]

·N

−1

(x)[

′

(x)G(x) − h

(x)(QJ(x) + S)]

. (5.81)

Proof. First, suppose that there exist functions V

: R

→ R, ℓ : R

→

, and W : R

→ R

p×m

such that V

(·) is continuously diﬀerentiable and

positive deﬁnite, and (5.77)–(5.79) are satisﬁed. Then for every admissible

input u(·) ∈ U, t

, t

∈ R, t

≥ t

≥ 0, it follows from (5.77)–(5.79) that

r(u, y)dt =



Qy + 2y

Su + u





(x)Qh(x) + 2h

(x)(S + QJ(x))u

(x)QJ(x) + S

J(x) + J

(x)S + R)u





′

(x)(f(x) + G(x)u) + ℓ

(x)ℓ(x) + 2ℓ

(x)W(x)u



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348 CHAPTER 5

(x) + [ℓ(x) + W(x)u]

[ℓ(x) + W(x)u]

≥ V

(x(t

)) − V

(x(t

)),

where x(t), t ≥ 0, satisﬁes (5.75) and

(·) denotes the total derivative of

the storage function along the trajectories x(t), t ≥ 0, of (5.75). Now, the

result is immediate from Corollary 5.1.

Conversely, suppose that G is dissipative with respect to a quadratic

supply rate r(u, y). Now, it follows from Theorem 5.1 that the available

storage V

(x) of G is ﬁnite for all x ∈ R

, V

(0) = 0, and

(x(t

)) ≤ V

(x(t

)) +

r(u(t), y(t))dt, t

≥ t

, (5.82)

for all admissible u(·) ∈ U. Dividing (5.82) by t

− t

and letting t

→ t

follows that

(x(t)) ≤ r(u(t), y(t)), t ≥ 0, (5.83)

where x(t), t ≥ 0, satisﬁes (5.75) and

(x(t))

△

= V

′

(x(t))(f(x(t)) + G(x(t))

·u(t)) denotes the total derivative of the available storage function along the

trajectories x(t), t ≥ 0. Now, with t = 0, it follows from (5.83) that

′

)(f(x

) + G(x

)u) ≤ r(u, y(0)), u ∈ R

. (5.84)

Next, let d : R

× R

→ R be such that

d(x, u)

△

= −

(x) + r(u, y) = −V

′

(x)(f(x) + G(x)u) + r(u, h(x) + J(x)u).

(5.85)

Now, it follows from (5.83) that d(x, u) ≥ 0, x ∈ R

, u ∈ R

. Furthermore,

note that d(x, u) given by (5.85) is quadratic in u, and hence, there exist

functions ℓ : R

→ R and W : R

→ R

p×m

such that

d(x, u) = [ℓ(x) + W(x)u]

[ℓ(x) + W(x)u]

= −V

′

(x)(f(x) + G(x)u) + r(u, h(x) + J(x)u)

= −V

′

(x)(f(x) + G(x)u) + (h(x) + J(x)u)

Q(h(x) + J(x)u)

+2(h(x) + J(x)u)

Su + u

Ru.

Now, equating coeﬃcients of equal powers yields (5.77)–(5.79) with V

(x) =

(x) and the positive deﬁniteness of V

(x), x ∈ R

, follows from Theorem

5.3.

Finally, to show (5.81) note th at (5.77)–(5.79) can be equivalently

written as



A(x) B(x)

(x) C(x)



= −



ℓ

(x)





ℓ(x) W(x)



≤ 0, x ∈ R

, (5.86)

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DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 349

where A(x)

△

= V

′

(x)f(x) −h

(x)Qh(x), B(x)

△

′

(x)G(x)−h

(x)(QJ(x)

+S), and C(x)

△

= −(R + S

J(x) + J

(x)S + J

(x)QJ(x)). Now, for all

invertible T ∈ R

(m+1)×(m+1)

(5.86) holds if and only if T

(5.86)T h olds.

Hence, the equivalence of (5.77)–(5.79) to (5.81) in the case when (5.80)

holds follows fr om the (1,1) block of T

(5.86)T , where

△



1 0

−C

−1

(x)B

(x) I



This completes the proof.

Note that the assumption of complete r eachability in Theorem 5.6 is

needed to establish the existence of a nonnegative-deﬁnite storage function

(·) while zero-state observability ensures that V

(·) is positive deﬁnite. In

the case where the existence of a continuously diﬀerentiable positive-deﬁnite

storage function V

(·) is assumed for G, then G is dissipative with respect to

the quadratic supply rate r(u, y) with storage f unction V

(·) if and only if

(5.77)–(5.79) are satisﬁed.

Next, we provide necessary and suﬃcient conditions for exponential

dissipativeness with r espect to quadratic supply rates. Here, once again we

assume that there exists a function κ : R

→ R

such that κ(0) = 0 and

r(κ(y), y) < 0, y 6= 0, and the available exponential storage V

(x), x ∈ R

for G is a continuously diﬀerentiable f unction.

Theorem 5.7. Let Q ∈ S

, S ∈ R

l×m

, R ∈ S

, and let G be zero-state

observable and completely reachable. G is exponentially dissipative with

respect to the quadratic supply rate r(u, y) = y

Qy +2y

Su + u

Ru if and

only if there exist functions V

: R

→ R, ℓ : R

→ R

, and W : R

→ R

p×m

and a scalar ε > 0 such that V

(·) is continuously diﬀerentiable and positive

deﬁnite, V

(0) = 0, and, for all x ∈ R

0 = V

′

(x)f(x) + εV

(x) − h

(x)Qh(x) + ℓ

(x)ℓ(x), (5.87)

0 =

′

(x)G(x) − h

(x)(QJ(x) + S) + ℓ

(x)W(x), (5.88)

0 = R + S

J(x) + J

(x)S + J

(x)QJ(x) −W

(x)W(x). (5.89)

If, alternatively, N(x) > 0, x ∈ R

, then G is exponentially dissipative with

respect to the quadratic supply rate r(u, y) = y

Qy +2y

Su + u

Ru if and

only if there exists a continuously diﬀerentiable function V

: R

→ R and

a scalar ε > 0 such that V

(·) is positive deﬁnite, V

(0) = 0, and , for all

x ∈ R

0 ≥ V

′

(x)f(x) + εV

(x) − h

(x)Qh(x) + [

′

(x)G(x) − h

(x)(QJ(x) + S)]

·N

−1

(x)[

′

(x)G(x) − h

(x)(QJ(x) + S)]

. (5.90)

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350 CHAPTER 5

Proof. The proof is analogous to the proof of Theorem 5.6.

Finally, we provide necessary an d suﬃcient conditions for the case

where G given by (5.75) and (5.76) is lossless with respect to a quadratic

supply rate r(u, y).

Theorem 5.8. Let Q ∈ S

, S ∈ R

l×m

, R ∈ S

, and let G be zero-

state observable an d completely r eachable. G is lossless with respect to the

quadratic supp ly rate r(u, y) = y

Qy + 2y

Su + u

Ru if and only if there

exists a function V

: R

→ R such that V

(·) is continuously diﬀerentiable

and positive deﬁnite, V

(0) = 0, and, for all x ∈ R

0 = V

′

(x)f(x) −h

(x)Qh(x), (5.91)

0 =

′

(x)G(x) − h

(x)(QJ(x) + S), (5.92)

0 = R + S

J(x) + J

(x)S + J

(x)QJ(x). (5.93)

Proof. The proof is analogous to the proof of Theorem 5.6.

For particular examples of dynamical systems with force inputs and

velocity outputs we can associate th e storage f unction with the stored or

available energy in the system and the supply rate with the net ﬂow of energy

or power into the system. However, as discussed in [320,456], the concepts

of supply rates and storage functions also apply to more general systems f or

which a physical energy interpretation is no longer valid. Speciﬁcally, using

(5.77)–(5.79) it follows that

r(u(s), y(s)ds = V

(x(t)) −V

(x(t

))

[ℓ(x(s)) + W(x(s))u(s)]

[ℓ(x(s)) + W(x(s))u(s)]ds, (5.94)

which can be interpreted as a ge neralized energy balance equation, where

(x(t)) − V

(x(t

)) is the stored or accumulated generalized energy of the

system and the second path-dependent term on the right corresponds to the

dissipated generalized energy of the system. Rewriting (5.94) as

(x) = r(u, y) − [ℓ(x) + W(x)u]

[ℓ(x) + W(x)u], (5.95)

yields a generalized energy conservation equation which shows that the rate

of change in generalized system energy, or generalized power, is equal to

the external generalized system power input minus the internal generalized

system power dissipated.

Note that if G with a continuously diﬀerentiable positive-deﬁnite

storage function is dissipative with respect to the quadratic supply rate

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DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 351

r(u, y) = y

Qy + 2y

Su + u

Ru, and if Q ≤ 0 and u(t) ≡ 0, then it follows

that

(x(t)) ≤ y

(t)Qy(t) ≤ 0, t ≥ 0. (5.96)

Hence, the zero solution x(t) ≡ 0 of the undisturbed (u(t) ≡ 0) nonlinear

system (5.75) is Lyapunov stable. Alternatively, if G with a continuously

diﬀerentiable positive-deﬁnite storage function is exponentially dissipative

with respect to the quadratic supply rate r(u, y) = y

Qy + 2y

Su + u

Ru,

and if Q ≤ 0 and u(t) ≡ 0, then it follows that

(x(t)) ≤ −εV

(x(t)) + y

(t)Qy(t) ≤ −εV

(x(t)), t ≥ 0. (5.97)

Hence, the zero solution x(t) ≡ 0 of the undisturbed (u(t) ≡ 0) nonlinear

system (5.75) is asymptotically stable. If, in addition, there exist scalars

α, β > 0 and p ≥ 1 such that

αkxk

≤ V

(x) ≤ βkxk

, x ∈ R

, (5.98)

then the zero solution x(t) ≡ 0 of the u ndisturbed (u(t) ≡ 0) nonlinear

dynamical system (5.75) is exponentially stable.

Next, we provide several deﬁnitions of nonlinear dynamical systems

which are dissipative or exponentially dissipative with respect to s upply

rates of a speciﬁc form.

Deﬁnition 5.7. A dynamical system G of the form (5.7) and (5.8)

with m = l is passive if G is dissipative with respect to the supply rate

r(u, y) = 2u

Deﬁnition 5.8. A d y namical sy s tem G of the form (5.7) and (5.8) with

m = l is strictly passive if G is strictly dissipative with respect to the supply

rate r(u, y) = 2u

Deﬁnition 5.9. A dynamical system G of the form (5.7) and (5.8) is

input strict passive if there exists ε > 0 such that G is dissipative with

respect to the supply rate r(u, y) = 2u

y − εu

Deﬁnition 5.10. A dy namical system G of the f orm (5.7) and (5.8)

is output strict passive if there exists ε > 0 such that G is dissip ative with

respect to the supply rate r(u, y) = 2u

y − εy

Deﬁnition 5.11. A dynamical system G of the form (5.7) and (5.8) is

input-output strict passive if th ere exist ε, ˆε > 0 such that G is dissipative

with respect to the supply rate r(u, y) = 2u

y − εu

u − ˆεy

Deﬁnition 5.12. A dy namical system G of the f orm (5.7) and (5.8)

is nonexpansive if G is dissipative with respect to th e supply rate r(u, y) =

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352 CHAPTER 5

u −y

y, where γ > 0 is given.

Deﬁnition 5.13. A dy namical system G of the form (5.7) and (5.8)

with m = l is exponentially passive if G is exponentially dissipative with

respect to the supply rate r(u, y) = 2u

Deﬁnition 5.14. A dynamical system G of the form (5.7) and (5.8) is

exponentially nonexpansive if G is exponentially dissipative with respect to

the supply rate r(u, y) = γ

u − y

y, where γ > 0 is given.

In light of the above deﬁnitions the following result is immediate.

Proposition 5.2. Consider the nonlinear dy namical system G given by

(5.7) and (5.8). Then th e following statements h old:

i) I f G is passive with a continuously d iﬀerentiable positive-deﬁnite

storage function V

(·), then the zero solution x(t) ≡ 0 of the

undisturbed (u(t) ≡ 0) system G is Lyapunov stable.

ii) If G is exponentially passive with a continuously diﬀerentiable positive-

deﬁnite storage function V

(·), then the zero solution x(t) ≡ 0 of

the undisturbed (u(t) ≡ 0) system G is asymptotically stable. If,

in addition, V

(·) satisﬁes (5.98), then the zero solution x(t) ≡ 0 of the

undisturbed (u(t) ≡ 0) system G is exponentially stable.

iii) If G is zero-state observable and nonexpansive with a continuously

diﬀerentiable positive-deﬁnite storage function V

(·), then the zero

solution x(t) ≡ 0 of the undisturbed (u(t) ≡ 0) system G is

asymptotically stable.

iv) If G is exponentially nonexpansive with a continuously diﬀerentiable

positive-deﬁnite storage function V

(·), then the zero solution x(t) ≡ 0

of the undisturbed (u(t) ≡ 0) system G is asymptotically stable. If, in

addition, V

(·) satisﬁes (5.98), then the zero solution x(t) ≡ 0 of the

undisturbed (u(t) ≡ 0) system G is exponentially stable.

v) If G is strictly passive w ith a continuously diﬀerentiable positive-

deﬁnite storage function V

(·), then the zero solution x(t) ≡ 0 of the

undisturbed (u(t) ≡ 0) system G is asymptotically stable.

vi) If G is input strict passive with a continuously diﬀerentiable positive-

deﬁnite storage function V

(·), then the zero solution x(t) ≡ 0 of the

undisturbed (u(t) ≡ 0) system G is Lyapunov stable.

vii) If G is zero-state observable and output strict passive with a contin-

uously diﬀerentiable positive-deﬁnite storage function V

(·), then the

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DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 353

zero solution x(t) ≡ 0 of the undisturbed (u(t) ≡ 0) system G is

asymptotically stable.

viii) If G is zero-state observable and input-output passive with a contin-

uously diﬀerentiable positive-deﬁnite storage function V

(·), then the

zero solution x(t) ≡ 0 of the undisturbed (u(t) ≡ 0) system G is

asymptotically stable.

ix) If G is output strict passive with a continuously diﬀerentiable positive-

deﬁnite storage function V

(·), then G is nonexpansive.

x) If G is inp ut strict passive and nonexpansive with a continuously

diﬀerentiable positive-deﬁnite storage function V

(·), then G is input-

output strict passive.

Proof. Statements i)–viii) are immediate and follow from (5.31)–

(5.33) using Lyapunov and invariant set stability arguments. To show i x),

note that if G is output strict passive with a continuously diﬀerentiable

positive-deﬁnite storage function V

(·) it follows that, for ε > 0,

(x) ≤ 2u

y − εy

= −

2ε

(2u − εy)

(2u − εy) +

u −

≤

u −

y, (5.99)

which implies that G is nonexpansive. Finally, to show x), it follows that if

G is input strict passive and nonexpansive with a continuously diﬀerentiable

positive-deﬁnite storage function V

(·) it follows that

(x) ≤ 2u

y − εu

u, ε > 0, (5.100)

and

(x) ≤ γ

u − y

y, γ > 0, (5.101)

which implies that for all α > 0,

(1 + α)

(x) ≤ 2u

y − ˆεu

u −αy

y, (5.102)

where ˆε = ε − αγ

. Now, the result is immediate by choosing α > 0 such

that ˆε > 0.

Example 5.2. Consider the matrix second-order nonlinear dynamical

system of an n-link robot given by [342]

M(q(t))¨q(t) + C(q(t), ˙q(t)) ˙q(t) + g(q(t)) = u(t),

q(0) = q

, ˙q(0) = ˙q

, t ≥ 0, (5.103)

y(t) = ˙q(t), (5.104)

where q, ˙q, ¨q ∈ R

represent generalized position, velocity, and acceleration

co ordinates, respectively, u ∈ R

is a force input, y = ˙q ∈ R

is a velocity

NonlinearBook10pt November 20, 2007

354 CHAPTER 5

measurement, M(q) is a positive-deﬁnite inertia matrix fu nction for all q ∈

, C(q, ˙q) is an n×n matrix function accounting for centrifugal and C oriolis

forces and has the property that

M(q(t))−2C(q(t), ˙q(t)) is skew-symmetric

for all q, ˙q ∈ R

, and g(q) is an n-dimensional vector accounting for gravity

forces and is given by g(q) =

∂V (q)

∂q

, where V (q) is the system potential.

Here, we assume V (0) = 0, V (q) is positive deﬁnite, and g(q) = 0 has an

isolated root at q = 0.

To show that (5.103) and (5.104) is lossless, consider the energy storage

function V

(q, ˙q) = ˙q

M(q) ˙q + 2V (q). Now,

(q, ˙q) = 2 ˙q

M(q)¨q + ˙q

M(q) ˙q + 2

∂V (q)

∂q

˙q

= 2 ˙q

[u −C(q, ˙q) ˙q − g(q)] + ˙q

M(q) ˙q + 2g

(q) ˙q

= 2y

u, (5.105)

and h en ce, the system is lossless. Alternatively, with u = −K

˙q + v, w here

is a positive-deﬁnite matrix, it follows that

(q, ˙q) = 2y

v − 2y

y, (5.106)

and hence, the input-output map fr om v to y is output strict passive. Note

that in the case wh ere v = 0,

(q, ˙q) = −2 ˙q

˙q ≤ 0, and hence, R

△

{(q, ˙q) :

(q, ˙q) = 0} = {(q, ˙q) : ˙q = 0}. Now, since ˙q(t) ≡ 0 it follows

that ¨q(t) ≡ 0 which, using (5.103), implies g(q(t)) ≡ 0, and hence, q(t) ≡ 0.

Thus , M = {(0, 0)} is the largest invariant set contained in R so that

the zero solution (q(t), ˙q(t)) ≡ (0, 0) is asymptotically stable. Finally, we

note that if V (q) is radially unbounded, then global asymptotic stability is

ensured. △

Example 5.3. Consider the nonlinear mass-spring-damper dynamical

system

m¨x(t) + x

(t) ˙x

(t) + x

(t) = 2u(t), x(0) = x

, ˙x(0) = ˙x

, t ≥ 0, (5.107)

y(t) = ˙x(t). (5.108)

To show that (5.107) and (5.108) is passive consider the energy storage

function V

(x, ˙x) =

m ˙x

. Now,

(x, ˙x) = 2 ˙xu − x

˙x

≤ 2yu, and

hence, (5.107) and (5.108) is passive. △

Example 5.4. Consider the nonlinear d y namical system

˙x

(t) = x

(t), x

(0) = x

, t ≥ 0, (5.109)

˙x

(t) = −g(x

(t)) − ax

(t) + u(t), x

(0) = x

, (5.110)

y(t) = bx

(t) + x

(t), (5.111)