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

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NonlinearBook10pt November 20, 2007

DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 385

0 = B

P −(MC + MN CA) + W

L, (5.228)

0 = (I

+ M NCB) + (I

+ M NCB)

− W

W. (5.229)

Now, consider the Lur´e-Postnikov Lyapun ov function candidate

V (x) = x

P x + 2

i=1

(s)M

ds, (5.230)

where P satisﬁes (5.227)–(5.229). Note that s ince P is positive deﬁn ite and

σ(·) ∈ Φ

, V (x) is positive deﬁnite for all nonzero x ∈ R

Using Leibnitz’s integral rule,

the corresponding Lyapunov derivative

of V (·) is given by

V (x) = 2x

P [Ax −Bσ(y)] + 2

i=1

˙y

, (5.231)

or, equivalently, using (5.227),

V (x) = −x

(εP + L

L)x −2σ

(y)B

P x + 2σ

(y)MN ˙y. (5.232)

Next, since ˙y = C ˙x = CAx − CBσ(y), (5.232) becomes

V (x) = −εx

P x − x

Lx − σ

(y)(B

P −MN CA)x

−x

P − MNCA)

σ(y) − σ

(y)[MNCB + B

NM]σ(y).

(5.233)

Adding and su btracting 2σ

(y)MCx and 2σ

(y)σ(y) to and from (5.233)

and u sing (5.228) and (5.229) yields

V (x) = −εx

P x−[Lx−W σ(y)]

[Lx− W σ(y)]+2σ

(y)[σ(y)−M y]. (5.234)

Since σ

(y)[σ(y) − My] ≤ 0 for all σ(·) ∈ Φ

, it follows that

V (x) < 0,

x ∈ R

, x 6= 0, which shows that the zero solution x(t) ≡ 0 of the negative

feedback interconnection (5.219)–(5.221) is globally asymptotically stable

for all σ(·) ∈ Φ

As in the positivity case, the Popov criterion also has an interesting

geometric interpretation in the Nyquist plane and in a modiﬁed plane called

the Popov plane. To see this, let m = 1, set G(ω) = x + y, and require

that 1 + M(1 + Ns)G(s) be strictly positive real. In this case, we obtain

ωy <

x +

, ω ∈ R. (5.235)

Recall that for f : R

→ R, g : R → R, and h : R → R, Leibnitz’s integral rule is

h(t)

g(t)

f(t, s)ds =

h(t)

g(t)

∂

∂t

f(t, s)ds + f (t, h(t))

dh(t)

− f (t, g(t))

dg(t)

whenever the above integrals exist.

NonlinearBook10pt November 20, 2007

386 CHAPTER 5

In the Nyquist plane, (5.235) requires the Nyquist plot at each ω ∈ R

be to the right of a rotating line that is a linear function of ω ∈ R.

Alternatively, (5.235) is a frequency d omain stability criterion with a

graphical interpretation in a modiﬁed Nyquist plane, known as the Popov

plane involving Re G and ωIm G, in terms of a ﬁxed straight line (Popov

line) with real axis intercept −1/M and slope 1/N (see Figure 5.9). It is

ω Im G

Re G

−

Popov slope = −

Popov line

Figure 5.9 Popov plot.

interesting to note that, unlike the Nyquist plot, the Popov plot is an odd

plot, and hence, we need only consider ω ∈ [0, ∞) in generating the plot.

Furth ermore, setting N = 0 the Pop ov criterion collapses to the positivity

theorem. Finally, we note that as for the positivity theorem, the restriction

on A being asymptotically stable can be removed using the loop sifting

techniques as discus s ed in Section 5.8. However, in this case the resulting

frequency domain stability conditions do not provide a simple graphical test

as in the case of the single s ector Popov criterion. See [331] and Problem

5.59 for further details.

Example 5.9. Consider the linear dynamical system with the satura-

tion feedback nonlinearity add ressed in Example 5.7. Figure 5.10 shows the

Popov plot of G(ω). Note that the Popov plot lies to the right of any line

of slope 1/N ≤ 0.5 that intersects the real axis to the left of −1/M = −0.5.

Hence, the maximum possible value of M is 2, which is substantially less

conservative than the result arrived at by the positivity theorem in Example

5.7. △

The strict positive real conditions appearing in Theorem 5.20 can be

written as I

+ MZ(s)G(s), where Z(s) = Z

(s)

△

= I

+ N s is known as

the stability Popov multiplier. Further reﬁnements of the absolute stability

NonlinearBook10pt November 20, 2007

DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 387

−2.5 −2 −1.5 −1 −0.5 0 0.5

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0.2

0.4

0.6

Re G(jω)

ω Im G(jω)

Figure 5.10 Popov plot for Example 5.9.

Popov criterion can be developed by considering extended Lur´e-Postnikov

Lyapunov functions. Speciﬁcally, absolute stability criteria can be derived

that extend the Popov criterion for sector-bounded, time-invariant nonlinear

functions to monotonic and odd monotonic nonlinearities. In particular,

suitable positive real stability multipliers Z(s) can be constructed as driving-

point impedances of passive electrical networks involving resistor-inductor

(RL), resistor-capacitor (RC), and inductor-capacitor (LC) combinations

which exhibit interlacing pole-zero patterns on the negative real axis and

imaginary axis [71, 328, 330, 479]. These stability multipliers eﬀectively

place less restrictive conditions on the linear part of the system and more

restrictive conditions on the allowable class of feedback nonlinearities. In

addition, the stability criteria for these reﬁned class of nonlinearities are

predicated on extended Lu r´e-Postnikov Lyapunov functions involving real

signals obtained by passing the system outputs (y = Cx) through a parallel

bank of decoupled low pass ﬁlters with s peciﬁed time constants and positive

gains corresponding to the RL, RC, and LC networks. However, as a result

of th e more involved multiplier construction, the resulting frequ en cy domain

conditions do not provide a simple graphical test involving ﬁxed shapes in

the Nyquist and Popov planes as in the case of the circle and Popov criteria.

NonlinearBook10pt November 20, 2007

388 CHAPTER 5

5.10 Prob lems

Problem 5.1. Let H(s) ∈ C

m×m

. Sup pose |||H(s)|||

∞

≤ γ, where γ >

0, and det[I − γ

−1

H(s)] 6= 0, Re[s] ≥ 0. Show that

G(s) = [I + γ

−1

H(s)][I − γ

−1

H(s)]

−1

, (5.236)

is positive real. Conversely, show that if G(s) ∈ C

m×m

is positive real, then

−1

H(s) = [G(s) −I][G(s) + I]

−1

, (5.237)

is bounded real (i.e., |||H(s)|||

∞

≤ γ).

Problem 5.2. Consider the nonlinear dynamical system G given by

(5.75) and (5.76). Show that:

i) I f G is dissipative with respect to the supply rate r(u, y) = u

y −εy

where ε > 0, then αG is dissipative with respect to the supply r ate

r(u, y) =

√

y −

y, where α > 0. Here, αG d en otes a non linear

system with output y

√

αy, where y is the output of G.

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

y −εu

where ε > 0, then αG is dissipative with respect to the supply r ate

r(u, y) =

√

αu

y − αεu

u, where α > 0.

Problem 5.3. The nonlinear dynamical system G given by (5.7) and

(5.8) is cyclo-dissipative (respectively, exponentially cyclo-dissipative) with

re spect to the supply rate r(u, y) if (5.9) (respectively, (5.10)) is satisﬁed for

all t ≥ t

and all u(·) ∈ U with x(t

) = x(t) = 0. In this case, V

)

given by (5.11) (respectively, (5.12)) is called the virtual available storage

(respectively, virtual available exponential storage) of G. Show that G is

cyclo-dissipative (respectively, exponentially cyclo-dissipative) with respect

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

(x) is ﬁnite for all x

∈ D and

(0) = 0.

Problem 5.4. A function V

: D → R is a virtual storage function

(respectively, virtual exponential storage function) of the nonlinear dynam-

ical system G given by (5.7) and (5.8) if it satisﬁes V

(0) = 0 an d (5.16)

(respectively, (5.17)). Show that if the virtual available storage (respectively,

virtual available exponential storage) V

) is ﬁnite for all x

∈ D and

(0) = 0, then V

(·) is a virtual storage function (respectively, virtual

exponential storage function) for G. Furthermore, show that every virtual

storage function (respectively, virtual exponential storage f unction) V

(·) for

G satisﬁes V

(x) ≤ V

(x), x ∈ D.

Problem 5.5. The nonlinear dynamical system G given by (5.7) and

NonlinearBook10pt November 20, 2007

DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 389

(5.8) is cyclo-lossless with respect to the supply rate r(u, y) if (5.9) is satisﬁed

as an equality for all t ≥ t

and all u(·) ∈ U with x(t

) = x(t) = 0. Ass ume G

is completely reachable to and from the origin. Show that G is cyclo-lossless

with respect to the supply rate r(u, y) if and only if there exists a virtual

storage function V

(x), x ∈ D, satisfying (5.16) as an equality. Furthermore,

show that if G is cyclo-lossless with respect to the supply rate r(u, y), then

the virtual storage function V

(x), x ∈ D, is unique and is given by (5.44)

where x(t), t ≥ 0, is the solution to (5.7) with admissible u(·) ∈ U, t ≥ 0,

x(−T ) = 0, x(T ) = 0, and x(0) = x

, x

∈ D.

Problem 5.6. Let Q ∈ S

, S ∈ R

l×m

, R ∈ S

, and let G given

by (5.75) and (5.76) be zero-state observable and completely reachable.

Furth ermore, assume that all virtual storage functions (see Problem 5.4)

of G are continuously diﬀerentiable. Show that G is cyclo-dissipative (see

Problem 5.3) with respect to the quadratic supply rate r(u, y) = y

+2y

Su + u

Ru if and only if there exist functions V

: R

→ R, ℓ : R

→

, and W : R

→ R

p×m

such that V

(·) is continuously diﬀerentiable,

(0) = 0, and, for all x ∈ R

, (5.77)–(5.79) hold.

Problem 5.7. Let Q ∈ S

, S ∈ R

l×m

, R ∈ S

, and let G given by (5.75)

and (5.76) be zero-state observable and completely reachable. Furthermore,

assume that all virtual exponential storage functions (see Problem 5.4)

of G are continuously diﬀerentiable. Show that G is exponentially cyclo-

dissipative (see Problem 5.3) with respect to the quadratic supply rate

r(u, y) = y

Qy +2y

Su + u

Ru if and on ly if there exist functions

: R

→ R, ℓ : R

→ R

, W : R

→ R

p×m

, and a scalar ε > 0 such

that V

(·) is continuously diﬀerentiable, V

(0) = 0, and, for all x ∈ R

(5.87)–(5.89) hold.

Problem 5.8. Consider the nonlinear dynamical system G given by

(5.7) and (5.8). Deﬁne the minimum input ene rgy of G by

) = inf

u(·), T ≥0

r(u(t), y(t))dt, (5.238)

where x(t), t ≥ 0, is the solution to (5.7) with x(0) = 0 and x(T ) = x

. The

inﬁmum in (5.238) is taken over all time t ≥ 0 and all admissible inputs u(·)

which drive G from x(0) = 0 to x(T ) = x

. It follows from (5.238) that the

minimum inpu t energy is the minimum energy it takes to drive G from the

origin to a given state x

. Assuming G is completely reachable, show that

) + V

) = V

Problem 5.9. The nonlinear dynamical system (5.75) and (5.76) is

nonnegative if for every x(0) ∈ R

and u(t) ≥≥ 0, t ≥ 0, the solution

x(t), t ≥ 0, to (5.75) and the output y(t), t ≥ 0, are nonnegative, that is,

NonlinearBook10pt November 20, 2007

390 CHAPTER 5

x(t) ≥≥ 0, t ≥ 0, and y(t) ≥ ≥ 0, t ≥ 0. Consider the nonlinear dynamical

system G given by (5.75) and (5.76). Show that if f : D → R

is essentially

nonnegative (see Problem 3.8), h(x) ≥≥ 0, G(x) ≥≥ 0, and J(x) ≥≥ 0,

x ∈ R

, then G is nonnegative.

Problem 5.10. Let q ∈ R

and r ∈ R

. Consider the nonlinear

nonnegative dynamical system G given by (5.75) and (5.76) where f : D →

is essentially nonnegative (see Problem 3.8), G(x) ≥≥ 0, h(x) ≥≥ 0, and

J(x) ≥≥ 0, x ∈ R

. Show that G is exponentially dissipative (respectively,

dissipative) with respect to the supply rate s(u, y) = q

y + r

u if and only

if there exist functions V

: R

→ R

, ℓ : R

→ R

, and W : R

→ R

and a scalar ε > 0 (respectively, ε = 0) such that V

(·) is continuously

diﬀerentiable and nonnegative d eﬁ nite, V

(0) = 0, and for all x ∈ R

0 = V

′

(x)f(x) + εV

(x) − q

h(x) + ℓ(x), (5.239)

0 = V

′

(x)G(x) − q

J(x) − r

+ W

(x). (5.240)

(Hint: The deﬁnition of dissipativity and exponential dissipativity s hould

be modiﬁed to reﬂect the fact that x

∈ R

, and u(t), t ≥ 0, and y(t), t ≥ 0,

are n on negative.)

Problem 5.11. Let q ∈ R

and r ∈ R

and consider the nonlinear

nonnegative dynamical sys tem G (see Problem 5.9) given by (5.75) and (5.76)

where f : D → R

is essentially nonnegative (see Pr oblem 3.8), G(x) ≥≥ 0,

h(x) ≥≥ 0, and J(x) ≥≥ 0, x ∈ R

. Suppose G is exponentially dissipative

(respectively, dissipative) with respect to the supply rate s(u, y) = q

y+r

Show that there exist p ∈ R

, l ∈ R

, and w ∈ R

, and a scalar ε > 0

(respectively, ε = 0) such that

0 = A

p + εp −C

q + l, (5.241)

0 = B

p − D

q − r + w, (5.242)

where

A =

∂f

∂x



x=0

, B = G(0), C =

∂h

∂x



x=0

, D = J(0). (5.243)

If, in addition, (A, C) is observable, show that p >> 0.

Problem 5.12. Let q ∈ R

and r ∈ R

. Consider the nonnegative

dynamical system G (see Problem 5.9) given by (5.160) and (5.161) where

A is essentially nonnegative (see Problem 3.7), B ≥≥ 0, C ≥≥ 0, and

D ≥≥ 0. Show that G is exponentially dissipative (respectively, dissipative)

with respect to the supply rate s(u, y) = q

y + r

u if an d only if there exist

p ∈ R

, l ∈ R

, and w ∈ R

, and a scalar ε > 0 (respectively, ε = 0) such

NonlinearBook10pt November 20, 2007

DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 391

that

0 = A

p + εp −C

q + l, (5.244)

0 = B

p − D

q − r + w. (5.245)

(Hint: Use Problems 5.10 and 5.11 to show this r esult.)

Problem 5.13. Consider the controlled rigid spacecraft given in

Problem 3.15. Show that this system is a port-controlled Hamiltonian

dynamical system. What does the output have to be in this case?

Problem 5.14. Consider the dynamical sys tem

G(s)

min

∼



A B

C D



with in put u(·) ∈ U and output y(·) ∈ Y. Let Q ∈ R

l×l

, R ∈ R

m×m

, and

S ∈ R

l×m

be such that Q and R are symmetric. Show that the following

statements are equ ivalent:

i) G

∗

(s)QG(s) + G

∗

(s)S + S

G(s) + R ≥ 0, Re [s] > 0.

ii)

(t)Qy(t) + 2y

(t)Su(t) + u

(t)Ru(t)]dt ≥ 0, T ≥ 0.

iii) There exist matrices P ∈ R

n×n

, L ∈ R

p×n

, and W ∈ R

p×m

, with P

positive deﬁnite, such that (5.145)–(5.147) are satisﬁed.

If, alternatively, R + S

D + D

S + D

QD > 0, show that i) holds if and

only if there exists an n ×n positive-deﬁnite matrix P such th at

0 ≥ A

P + P A − C

QC + [B

P − (QD + S)

(R + S

D + D

QD)

−1

P − (QD + S)

C]. (5.246)

Problem 5.15. Consider the dynamical sys tem

G(s)

min

∼



A B

C D



Let Q ∈ R

l×l

, R ∈ R

m×m

, and S ∈ R

l×m

be such that Q and R are

symmetric, Q > 0, and R < S

−1

S. Show that the following statements

are equivalent:

i) G

∗

(s)QG(s) + G

∗

(s)S + S

G(s) + R ≥ 0, Re [s] > 0.

ii) L

1/2

G(s) + Q

−1/2

−1

is bounded real, where L

△

= S

−1

S −R.

iii) [QG(s) + S − Q

1/2

][QG(s) + S + Q

1/2

]

−1

Q is positive real.

NonlinearBook10pt November 20, 2007

392 CHAPTER 5

Problem 5.16. Consider the dynamical sys tem

G(s)

min

∼



A B

C D



Let Q ∈ R

l×l

, R ∈ R

m×m

, and S ∈ R

l×m

be such that Q and R are

symmetric, Q > 0, and R + S

−1

S > 0. S how that the following

statements are equ ivalent:

i) −G

∗

(s)QG(s) + G

∗

(s)S + S

G(s) + R ≥ 0, Re [s] > 0.

ii) [Q

1/2

G(s) − Q

−1/2

S]L

−1/2

is bounded real, where L

△

= R + S

−1

iii) −[QG(s) − S − Q

1/2

][QG(s) − S + Q

1/2

]

−1

Q is positive real.

Problem 5.17. Consider the dynamical sys tem

G(s)

min

∼



A B

C D



let Q ∈ R

l×l

, R ∈ R

m×m

, and S ∈ R

l×m

be such that Q and R are

symmetric, det R 6= 0, and either one of the following assumptions is

satisﬁed:

i) A has no eigenvalues on the ω-axis.

ii) Q is sign d eﬁ nite, that is, Q ≥ 0 or Q ≤ 0, (A, B) has no uncontrollable

eigenvalues on the ω-axis, and (A, C) has no unobservable eigenvalues

on the ω-axis.

Show that th e following statements are equivalent:

i) G

∗

(ˆω)QG(ˆω) + G

∗

(ˆω)S + S

G(ˆω) + R is singu lar for some ˆω ∈ R.

ii) The Hamiltonian matrix

H =



A − B

−1

(QD + S)

C −B

−1

−[CQC

− C

(QD + S)

−1

(QD + S)

C] −[A − B

−1

(QD + S)



(5.247)

where

R = R + S

D + D

S + D

QD, has no eigenvalues at ˆω.

Problem 5.18. Consider the dynamical sys tem

G(s)

min

∼



A B

C D



let Q ∈ R

l×l

, R ∈ R

m×m

, and S ∈ R

l×m

be such that Q and R are

symmetric, R > 0, and either one of the assumptions i) or ii) of Problem

5.17 is satisﬁed. Show that the following statements are equivalent:

NonlinearBook10pt November 20, 2007

DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 393

i) G

∗

(ω)QG(ω) + G

∗

(ω)S + S

G(ω) + R > 0, ω ∈ R.

ii) The Hamiltonian matrix (5.247) has no eigenvalue on the ω-axis.

If, in addition, (A, B) is stabilizable show that:

iii) There exists a unique P = P

such that

0 = [A − B

−1

(QD + S)

P + P [A −B

−1

(QD + S)

+P B

−1

P − C

QC + C

(QD + S)

−1

(QD + S)

(5.248)

and spec(A − B

−1

(QD + S)

C + B

−1

P ) ⊂ C

−

, where

△

R + S

D + D

S + D

QD, if and only if i) holds.

Finally, show that the following statements are also equivalent:

iv) G

∗

(ω)QG(ω) + G

∗

(ω)S + S

G(ω)R ≥ 0, ω ∈ R.

v) There exists a unique P = P

such that (5.248) holds and spec(A −

−1

(QD + S)

C + B

−1

P ) ⊂ C

−

Problem 5.19. Let γ > 0,

G(s) ∼



A B

C D



∈ RH

∞

and d eﬁ ne

△



A + B

−1

C B

−1

−C

(I + D

−1

)C −(A + B

−1



, (5.249)

where

△

= γ

−D

D. Show that the following statements are equivalent:

i) |||G(s)|||

∞

< γ.

ii) σ

max

(D) < γ and H has no eigenvalues on the ω-axis.

iii) There exists P ≥ 0 such th at

0 = A

P + P A + C

C + (B

P + D

−1

P + D

C). (5.250)

If, in addition, (A, C) is observable show that P > 0.

NonlinearBook10pt November 20, 2007

394 CHAPTER 5

Problem 5.20. Let α ∈ (0, π], let G(s) be a scalar transf er function,

and let q, r ∈ R be such that q, r < 0 and qr = cos

α. Show that if

∗

(ω)G(ω) + G(ω) + G

∗

(ω) + r ≥ 0, ω ∈ R, (5.251)

then ∠G(ω) ∈ [−α, α], ω ∈ R.

Problem 5.21. Let α ∈ (0, π] and let

G(s) ∼



A B

C D



be a scalar transfer function, where A ∈ R

n×n

, B ∈ R

n×1

, C ∈ R

1×n

, and

D ∈ R. Show that if there exist a scalar µ > 0 and matrices P ∈ S

L ∈ R

p×n

, and W ∈ R

p×1

such that

0 = A

P + P A + µC

C + L

L, (5.252)

0 = B

P + (µD − λ)C + W

L, (5.253)

0 = 2D − µD

−

cos

α −W

W, (5.254)

then ∠G(ω) ∈ [−α, α], ω ∈ R.

Problem 5.22. Prove Theorem 5.7.

Problem 5.23. Prove Theorem 5.8.

Problem 5.24. Prove Theorem 5.10.

Problem 5.25. Consider the dynamical system G given by (5.75) and

(5.76) with f (x) = Ax, G(x) = B, h(x) = Cx, and J(x) ≡ 0. Assume G is

passive. Show that G is stabilizable if and only if G is detectable.

Problem 5.26. Consider the dynamical system G given by (5.75) and

(5.76) w ith f (x) = Ax, G(x) = B, h(x) = Cx, and J(x) = D. Assume G is

minimal and passive with storage function V

(x), x ∈ R

. Show that there

exists a constant ε > 0 such that V

(x) > εkxk

+ inf

x∈R

(x), x ∈ R

Problem 5.27. Consider the dynamical system G given by (5.75) and

(5.76) with f (x) = Ax, G(x) = B, h(x) = Cx, and J(x) = D. Show

that if G is passive, then the available storage is given by V

(x) = x

P x,

where P = lim

ε→0

and where P

is the nonnegative-deﬁnite solution to

the Riccati equation

0 = A

+ P

A + (B

− C)

(D + D

+ εI)

−1

− C). (5.255)

Problem 5.28. Consider the dynamical system G given by (5.75) and

(5.76) with f (x) = Ax, G(x) = B, h(x) = Cx, and J(x) = D. Assum e G

is minimal and passive. Show that the available storage is given by V

(x) =