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

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NonlinearBook10pt November 20, 2007

STABILITY OF FEEDBACK SYSTEMS 455

Conversely, assume (6.98) and (6.99) has relative d egree {1, 1, . . . , 1}

at x = 0 and is weakly minimum phase. In this case, it follows from Lemma

6.2 that (6.98) and (6.99) can be equ ivalently written as

˙y(t) = L

h(x(t)) + L

h(x(t))u(t), y(0) = y

, t ≥ 0, (6.122)

˙z(t) = f

(z(t)) + r(z(t), y(t))y(t), z(0) = z

. (6.123)

Now, since by assumption (6.98) and (6.99) or, equivalently, (6.122) and

(6.123), is weakly minimum phase there exists a continuously diﬀerentiable

positive-deﬁnite function V

(z), z ∈ R

n−m

, such that

′

(z)f

(z) ≤ 0, z ∈ R

n−m

. (6.124)

Hence,

(z) = V

′

(z)[f

(z) + r(z, y)y] ≤ V

′

(z)r(z, y)y.

Next, using the state f eedback trans formation

u(z, y) =

h(x))

−1

[−L

h(x) − r

(z, y)V

′T

(z) + v],

and the positive-deﬁnite function

(z, y) = V

(z) + y

it follows that

(z, y) ≤ 2y

v, which shows that (6.98) and (6.99) is

feedback passive.

Finally, the proof of the equivalence between feedback strict passivity

and relative degree {1, 1, . . . , 1} at x = 0 and minimum phase is identical

with (6.124) replaced by

′

(z)f

(z) < 0, z ∈ R

n−m

, z 6= 0,

since in this case the zero dynamics ˙z = f

(z) are asymptotically stable.

To specialize Theorem 6.12 to linear dynamical systems let f (x) = Ax,

G(x) = B, and h(x) = Cx so that

˙x(t) = Ax(t) + Bu(t), x(0) = x

, t ≥ 0, (6.125)

y(t) = Cx(t). (6.126)

Now, with r ank B = m, it follows from Theorem 6.12 and Theorem 5.13 that

(6.125) and (6.126) is feedback equivalent to a passive linear system with a

quadratic positive-deﬁnite storage function V

(·) if and only if det(CB) 6= 0

and (6.125) and (6.126) is weakly minimum phase. S ince a minimal linear

passive system is equ ivalent to a positive real system with a quadratic

positive-deﬁnite storage function, it follows th at any minimal linear system

is feedback positive real if and only if det(CB) 6= 0 and (6.125) and (6.126)

is weakly minimum phase.

NonlinearBook10pt November 20, 2007

456 CHAPTER 6

Next, we present suﬃ cient conditions for which a given block cascade

system is feedb ack equivalent to a passive system. Speciﬁcally, consider the

nonlinear block cascade system

˙x(t) = f (x(t)) + G(x(t))y(t), x(0) = x

, t ≥ 0, (6.127)

ˆx(t) =

f(ˆx(t)) +

G(ˆx(t))u(t), ˆx(0) = ˆx

, (6.128)

y(t) = h(ˆx(t)), (6.129)

where x ∈ R

, ˆx ∈ R

, u, y ∈ R

, f : R

→ R

satisﬁes f (0) = 0, G : R

→

n×m

f : R

→ R

satisﬁes

f(0) = 0,

G : R

→ R

q×m

, and h : R

→ R

satisﬁes h(0) = 0.

Theorem 6.13. Consider the block cascade system (6.127)–(6.129).

Assume that the input subsystem (6.128) and (6.129) is passive (respectively,

strictly passive) with a continuous ly diﬀerentiable positive-deﬁnite storage

function V

: R

→ R, and suppose the zero solution x(t) ≡ 0 to (6.127)

with y(t) ≡ 0 is globally asymptotically stable. Then (6.127)–(6.129) is

feedback equivalent to a passive (respectively, strictly passive) system with a

continuously diﬀerentiable positive-deﬁnite storage fun ction V : R

n+q

→ R.

Proof. Since the zero s olution x(t) ≡ 0 to (6.127) with y(t) ≡ 0 is

globally asymptotically s table it follows from Theorem 3.9 that there exists

a continuously diﬀerentiable positive-deﬁnite function V

sub

: R

→ R such

that V

sub

(0) = 0 and V

sub

(x)f(x) < 0, x ∈ R

, x 6= 0. Next, note that

(6.127)–(6.129) can be equivalently written as

˜x(t) =

f(˜x(t)) +

G(˜x(t))u(t), ˜x(0) = ˜x

, t ≥ 0, (6.130)

y(t) =

h(˜x(t)), (6.131)

where

˜x

△



ˆx



f(˜x)

△



f(x) + G(x)y

f(ˆx)



G(˜x)

△



G(ˆx)



h(˜x)

△

= h(ˆx).

Now, let

V (x, ˆx) = V

sub

(x) + V

(ˆx)

and note that since (6.128) and (6.129) is passive with a continuously

diﬀerentiable positive-deﬁnite storage function V

: R

→ R, it follows from

(5.116) of Corollary 5.2 with J(x) ≡ 0 that

(ˆx) =

′

(ˆx)

G(ˆx) =

′

(˜x)

G(˜x) =

(˜x). (6.132)

Next, let u = α(˜x) + v and note that

′

(˜x)[

f(˜x) +

G(˜x)α(˜x)] = V

′

sub

(x)f(x) + V

′

sub

G(x)h(ˆx)

NonlinearBook10pt November 20, 2007

STABILITY OF FEEDBACK SYSTEMS 457

′

(ˆx)

f(ˆx) + 2h

(ˆx)α(˜x).

Now, choosing α(˜x) = −

′T

sub

(x)G

(x) it follows that

′

(˜x)[

f(˜x) +

G(˜x)α(˜x)] = V

′

sub

(x)f(x) + V

′

(ˆx)

f(ˆx) ≤ 0, (x, ˆx) ∈ R

×R

(6.133)

which, with (6.132), shows that (6.127)–(6.129) is rendered passive with

u = α(˜x) + v. To show strict feedback passivity, it need only be noted

that if (6.128) and (6.129) is strictly feedback passive, then V

′

(ˆx)

f(ˆx) < 0,

ˆx ∈ R

, ˆx 6= 0, and hence, (6.133) holds w ith a strict inequ ality.

Finally, it is important to note that since exponential p assivity implies

strict passivity, the resu lts in this section also hold for exponentially passive

systems.

6.8 Problems

Problem 6.1. Consider the nonlinear zero-state observable dissipative

dynamical system G with inputs u

, outputs y

, i = 1, . . . , m, and internal

states x interconnected with m diss ipative dynamical subsystems G

, i =

1, . . . , m, given by

˙x

(t) = f

(t)) + G

(t))y

(t), x

(0) = x

, t ≥ 0, (6.134)

−u

(t) = h(x

(t)) + J

(t))y

(t). (6.135)

Let r

(−u

, y

) and V

) denote the storage functions and supply rates of

the G

th subsystem, respectively, and deﬁne the overall subsystem supply

rate R(u, y)

△

i=1

(−u

, y

), where u = [u

, . . . , u

]

, y = [y

, . . . ,

]

, and γ

> 0, i = 1, . . . , m. Show that if there exists a positive-deﬁnite

storage function V

(·) such that

(x) ≤ −R(u, y), then the zero solution

(x(t), x

(t), . . . , x

(t)) ≡ (0, 0, . . . , 0) of the interconnected system G with

the subsystems G

, i = 1, . . . , m, is Lyapu nov stable.

Problem 6.2. Let

G(s)

min

∼



A B

C 0



, G

(s)

min

∼





be asymptotically stable transfer functions. If G(s) and G

(s) are bounded

real, show that the feedback interconnection of G(s) and G

(s) is Lyapun ov

stable, that is, the linear system with d ynamics matrix

△



A BC

C A



is Lyapunov stable. If, alternatively, G

(s) is strictly bounded real, then

show that

A is asymptotically stable.

NonlinearBook10pt November 20, 2007

458 CHAPTER 6

Problem 6.3. Let

G(s)

min

∼



A B

C 0



, G

(s)

min

∼





be Lyapunov stable transfer functions. If G(s) and G

(s) are positive

real, show that the negative feedback interconnection of G(s) and G

(s)

is Lyapunov stable, that is, the linear system with dynamics matrix

△



A −BC

C A



is Lyapunov stable. If , alternatively, G

(s) is s tr ictly positive real, then

show that

A is asymptotically stable.

Problem 6.4. Consider two nonlinear dynamical systems G

and G

with inpu t-output pairs (u

, y

) and (u

, y

), respectively. Assume G

and

are passive. Show that:

i) The parallel interconnection of G

and G

is passive.

ii) The negative feedback interconnection of G

and G

is passive.

Problem 6.5. Consider two nonlinear dynamical system G

and G

with inpu t-output pairs (u

, y

) and (u

, y

), respectively. Assume G

and

are nonexpansive with gains γ

> 0 and γ

> 0, respectively. Show that

the cascade interconnection of G

and G

is nonexpansive w ith gain γ

Problem 6.6. Consider the two nonlinear dynamical systems G

and

with inp ut-output pairs (u

, y

) and (u

, y

), respectively. S how that:

i) I f G

and G

are input strict passive, then the parallel interconn ection

of G

and G

is input strict passive.

ii) If G

and G

are output strict passive, then the parallel interconnection

of G

and G

is output strict p assive.

iii) If G

and G

are strictly passive, then the parallel interconnection of

and G

is s tr ictly passive.

Problem 6.7. Consider the closed-loop dynamical system consisting

of the n onlinear system G given by (6.1) and (6.2), and the nonlinear

compensator G

given by (6.3) and (6.4). Show that if there exist functions

: R

→ R, V

: R

→ R, ℓ : R

→ R

, ℓ

: R

→ R

, W : R

→ R

p×m

and W

: R

× R

→ R

×m

, such that V

(·) and V

(·) are continuously

diﬀerentiable and positive deﬁnite, V

(0) = 0, V

(x) → ∞ as

NonlinearBook10pt November 20, 2007

STABILITY OF FEEDBACK SYSTEMS 459

kxk → ∞, V

) → ∞ as kx

k → ∞, and

0 > V

′

(x)f(x) + ℓ

(x)ℓ(x), x ∈ R

, x 6= 0, (6.136)

0 =

′

(x)G(x) − h

(x) + ℓ

(x)W(x), (6.137)

0 = J(x) + J

(x) − W

(x)W(x), (6.138)

0 > V

′

) + ℓ

)ℓ

), x

∈ R

, x

6= 0, (6.139)

0 =

′

, x

) −h

, x

) + ℓ

, x

), (6.140)

0 = J

, x

) + J

, x

) −W

, x

), (6.141)

then the negative feedb ack interconnection of G and G

is globally asymp-

totically stable.

Problem 6.8. Consider the controlled undamped oscillator

¨x(t) +

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

, ˙x(0) = ˙x

, t ≥ 0, (6.142)

y(t) = ˙x(t), (6.143)

where m, k > 0. Show that the dynamic compensator

˙x

(t) = A

(t) + B

y(t), x

(0) = x

, t ≥ 0, (6.144)

−u(t) = C

(t) + D

y(t), (6.145)

where



0 1

−k

−c



, B







−c

− c



, D

= c

emulating a dynamic (m

, k

, c

) absorber w ith m

, k

, c

> 0 guarantees

that the closed-loop system (6.142)–(6.145) is asymptotically stable.

Problem 6.9. Consider the nonlinear dynamical system

˙x

(t) = x

(t), x

(0) = x

, t ≥ 0, (6.146)

˙x

(t) = −x

(t) − f(x

(t)) + u(t), x

(0) = x

, (6.147)

y(t) = x

(t) + u(t), (6.148)

where zf(z) > 0, z ∈ R, z 6= 0, and f(0) = 0. Show that the linear dynamic

compensator

˙x

(t) = x

(t), x

(0) = x

c10

, t ≥ 0, (6.149)

˙x

(t) = −x

(t) +

(t) + y(t), x

(0) = x

c20

, (6.150)

u(t) = −x

(t), (6.151)

guarantees global stability of the closed-loop system (6.146)–(6.151).

NonlinearBook10pt November 20, 2007

460 CHAPTER 6

Problem 6.10. Consider the rotational/translational nonlinear dy-

namical system G given in Problem 5.49 and the nonlinear controller G

given by

˙x

(t) = f

(t)) + G

(t), y(t))y(t), x

(0) = x

, t ≥ 0, (6.152)

−N

(t) = h

(t), y(t)) + J

(t), y(t))y(t), (6.153)

where x

= [θ θ

]

) =







−

sin(θ

−θ)

−

g sin θ







, G

, y) =









, y) = κ sin(θ − θ

), J

, y) =

αtanh(γy)

α, γ, κ, e

, m

> 0, and y =

θ. Show that th e closed-loop system consisting

of G and G

with N (t) = −mgl sin θ(t) − N

(t) is globally asymptotically

stable.

Problem 6.11. Let ε

, δ

> 0 be su ch that ε

< 1. Consid er the

nonlinear dynamical system G given by (6.1) and (6.2) and the nonlinear

controller G

given by

˙x

(t) = [A

− P

−1

M(u

(t), x

(t)]x

(t) + (1 −2ε

)

·[B

+ N (u

(t), x

(t))]u

(t), x

(0) = x

, t ≥ 0, (6.154)

(t) = [C

+ N

(t), x

(t))P ]x

(t) + δ

(t), (6.155)

where A

∈ R

×n

, B

∈ R

×m

, C

∈ R

m×n

, N : R

×R

→ R

×m

, and

P ∈ R

×n

, with P > 0, satisfying

0 > A

P + P A

, (6.156)

= B

P, (6.157)

and

M(u

, x

)

△

= ε

+ N

(t), x

(t))P ]

+ N

(t), x

(t))P ],

, x

) ∈ R

× R

. (6.158)

Show that if G is passive and zero-state observable, then the negative

feedback interconnection of G and G

is globally asymptotically stable.

Problem 6.12. Show that if in Problem 6.11 G is strictly passive,

M(u

, x

) is an arbitrary nonnegative-deﬁnite function, and δ

= 0, then

the negative feedback interconnection of G and G

is globally asymptotically

stable.

NonlinearBook10pt November 20, 2007

STABILITY OF FEEDBACK SYSTEMS 461

Problem 6.13. Consider the square linear dyn amical system

G(s) ∼



A B

C D



Let R

∈ R

n×n

and R

∈ R

m×m

, with R

and R

positive deﬁnite, be such

that

> C

− (D + D

)]C, (6.159)

> D + D

. (6.160)

Show that th e linear dynamic controller

(s) ∼



A − BR

−1

[C + B

P − DR

−1

P ] BR

−1

P 0



, (6.161)

where P > 0 satisﬁes

0 = A

P + P A + R

− P BR

−1

P, (6.162)

is strictly positive real. Alternatively, show that if D = 0, G(s) is positive

real, and R

satisﬁes

= L

L + C

C, (6.163)

where L satisﬁes (5.151), then the dynamic controller

(s) ∼



A − 2BR

−1

C BR

−1

C 0



(6.164)

is strictly positive real.

Problem 6.14. Consider the nonlinear dynamical system G given by

(6.1) and (6.2) with J(x) ≡ 0 and rank[G(0)] = m, and the nonlinear

controller G

given by (6.3) and (6.4) with J

(x) ≡ 0. Assume that G is

completely reachable, zero-state observable, and exponentially passive with

continuously diﬀerentiable radially unbounded, positive-deﬁnite storage

function V

: R

→ R. Suppose there exists a continuously diﬀerentiable

positive-deﬁnite function V : R

→ R such that V (x) → ∞ as kxk → ∞,

and

0 = V

′

(x)f(x) + L

(x) −

′

(x)G(x)R

−1

(x)V

′T

(x), x ∈ R

, (6.165)

where L

: R

→ R is given by L

(x) = ℓ

(x)ℓ(x) + εV

(x) + h

(x)R

−1

h(x),

> 0, and ε and ℓ(·) satisfy (5.137). Show that with

) = f (x

) −2G(x

−1

h(x

), (6.166)

) = G(x

−1

(6.167)

) = R

−1

h(x

), (6.168)

the negative feedback interconnection of G and G

is globally asymptotically

stable.

NonlinearBook10pt November 20, 2007

462 CHAPTER 6

Problem 6.15. Consider the linear matrix second-order dynamical

system in energy coordinates given by



˙x

(t)

˙x

(t)





0 Ω

−Ω −2ηΩ



(t)







u(t),



(0)







, t ≥ 0, (6.169)

y(t) = b

(t), (6.170)

where x

∈ R

, x

∈ R

, u, y, ∈ R, Ω = diag[ω

, . . . , ω

], ω

> 0, i =

1, . . . , n, η = diag[η

, . . . , η

], η

> 0, i = 1, . . . , n, and b ∈ R

. Show that

the nonlinear dynamic compensator

˙x

(t) =



0 Ω

−Ω

−2η

Ω



+ 2α[0 e



(t)

+κ





(t), x

(0) = x

, t ≥ 0, (6.171)

u(t) = −κ([0 e

(t))y(t), (6.172)

where x

∈ R

, n

≤ n, κ > 0, α > 0, e

= [1, 1, . . . , 1] ∈ R

1×n

, S = −S

Ω

= diag[ω

, . . . , ω

], ω

> 0, i = 1, . . . , n, and η

= diag[η

, . . . , η

> 0, i = 1, . . . , n, guarantees asymptotic stability of the closed-loop

system (6.169)–(6.172).

Problem 6.16. Consider the nonlinear dynamical system G given by

(6.1) and (6.2) and the nonlinear controller G

given by

˙x

(t) = (A

+ S)x

(t) + B

diag(y(t))y(t), x

(0) = x

, t ≥ 0, (6.173)

u(t) = −diag(y(t))B

(t), (6.174)

where A

∈ R

×n

, B

∈ R

×m

, S ∈ R

×n

is skew symmetric, and diag(y)

is a diagonal matrix whose entries on the diagonal are the components of y.

Show that if G is passive and zero-state observable and the triple (A

, B

)

is strictly positive real and self-du al (see Problem 5.33), then the negative

feedback interconnection of G and G

is asymptotically stable.

Problem 6.17. Consider the nonlinear controlled oscillator given in

Problem 5.35. Using Corollary 6.1 design a nonlinear reduced-order dynamic

compensator that asymptotically stabilizes the closed-loop system. Compare

your results with the static linear controller u = −R

−1

y, where R

= 1, for

the initial condition [x

(0), x

(0)]

= [1, 0.5]

Problem 6.18. Consider the controlled nonlinear damped oscillator

given in Problem 5.36. Using Corollary 6.1 and Problem 6.13 design a

nonlinear full-order dyn amic compensator that asymp totically stabilizes th e

closed-loop system. Compare the open-loop and closed-loop responses for

NonlinearBook10pt November 20, 2007

STABILITY OF FEEDBACK SYSTEMS 463

the in itial condition [x

(0), x

(0)]

= [0, 1]

and R

= 0.01.

Problem 6.19. Let q ∈ R

, r ∈ R

, q

∈ R

, and r

∈ R

. Consider

the nonlinear nonnegative dynamical systems G and G

(see Problem 5.9)

given by (6.1) and (6.2), and (6.3) and (6.4), respectively, where f : D →

is essentially nonnegative (see Problem 3.7), G(x) ≥≥ 0, h(x) ≥≥ 0,

J(x) ≥≥ 0, x ∈ R

, f

: R

→ R

is essentially nonnegative, G

, x

) =

) ≥≥ 0, h

, x

) = h

) ≥≥ 0, x

∈ R

, and J

, x

) ≡ 0.

Assume that G is dissipative with respect to the linear supply rate s(u, y) =

y + r

u and with a continuously diﬀerentiable positive-deﬁnite s torage

function V

(·), and assum e that G

is dissipative with respect to the linear

supply rate s

, y

) = q

+ r

and with a continuously diﬀerentiable

positive-deﬁnite storage function V

(·). Show that the following statements

hold:

i) I f there exists a scalar σ > 0 s uch that q + σr

≤≤ 0 and r +σq

≤≤ 0,

then the positive feedback interconnection of G and G

is Lyapunov

stable.

ii) If G and G

are zero-state observable and there exists a scalar σ > 0

such that q + σr

<< 0 and r + σq

<< 0, then the positive feedback

interconnection of G and G

is asymptotically stable.

iii) If G is zero-state observable, rank G

(0) = m

, G

is exponentially

dissipative with respect to the supply rate s

, y

) = q

+ r

and there exists a scalar σ > 0 such that q + σr

≤≤ 0 and r +

σq

≤≤ 0, then the positive feedback interconnection of G and G

asymptotically stable.

iv) If G is exponentially dissipative w ith respect to the supply rate s(u, y)

= q

y +r

u, G

is exponentially d iss ipative with respect to the supply

rate s

, y

) = q

+ r

, and there exists a scalar σ > 0 such

that q + σr

≤≤ 0 and r + σq

≤≤ 0, then the positive feedback

interconnection of G and G

is asymptotically stable.

(Hint: First show that the positive feedback interconnection of G and G

gives a nonnegative closed-loop system.)

Problem 6.20. Consider the dynamical system G given by (6.62) and

(6.63) with f(x) = Ax, G(x) = B, φ(x) = Kx, wh ere A ∈ R

n×n

, B ∈

n×1

, and K ∈ R

1×n

. Suppose that G has gain margin (α, β) and let

L(ω) = −K(ωI

−A)

−1

B denote the loop gain of (6.62) and (6.63). Using

a Nyquist sketch show that the maximum amount by which the loop gain

can be increased before instability is β, that is, the upward gain margin is β.

Show the maximum amount by w hich the loop gain can be decreased before

instability is (1 − α)100, that is, the gain reduction tolerance is (1 −α)100.

NonlinearBook10pt November 20, 2007

464 CHAPTER 6

Finally, show that the maximum angle by wh ich the loop gain can lag before

instability is φ = cos

−1



αβ+1

α+β



Problem 6.21. Consider the linear dynamical system

G(s)

min

∼



A B

C D



Show that if G(s) has a disk margin (α, β), then [I + βG(s)][I + αG(s)]

−1

is positive real.

Problem 6.22. Consider the linear dynamical system

G(s)

min

∼



A B

C D



with inp ut u(·) ∈ U and output y(·) ∈ Y. Show that G(s) is dissipative with

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

y + αy

y, 0 < α < 1, if and only if

G(s) has a disk margin (α, ∞).

Problem 6.23. Consider the linear dynamical system

G(s)

min

∼



A B

C D



with inp ut u(·) ∈ U and output y(·) ∈ Y. Show that G(s) is dissipative with

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

y +

u, β > 1, if and only if G(s)

has a disk margin (0, β).

Problem 6.24. Consider the nonlinear controlled system (6.73). Sh ow

that if Condition (6.75) is satisﬁed, then there exists a feedback control law

φ : D → U, in general discontinuous, such that V

′

(x)F (x, φ(x)) < 0, x ∈ D,

x 6= 0.

Problem 6.25. A smooth system of the form (6.76) with f (0) =

0 is a Jurdjevic-Quinn (J-Q) type system if there exists a continuously

diﬀerentiable positive-deﬁnite radially unbounded function V : R

→ R

such that:

i) V

′

(x)f(x) ≤ 0, x ∈ R

ii) W

△

= {x ∈ R

: L

k+1

V (x) = L

V (x) = 0, k = 0, 1, . . . , i = 1, . . . ,

= {0}.

Show that u = −[V

′

(x)G(x)]

is a globally stabilizing controller for a J-Q

type system.