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

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NonlinearBook10pt November 20, 2007

INPUT-OUTPUT STABILITY AND DISSIPATIVITY 505

it follows fr om (7.100) with H

(s) = H

(s) =

s+1−

that

|||G|||

(∞,∞),(∞,∞)

≤

|||G

|||

(∞,2),(2,2)

|||G

|||

(2,2),(1,2)

Hence,

|||G|||

(∞,∞),(∞,∞)

≤

∞

−(2−α)t

dt =

α(2 −α)

, 0 < α < 2,

which implies

|||G|||

(∞,∞),(∞,∞)

≤ inf

0<α<2

α(2 − α)

= 2.

△

7.11 Problems

Problem 7.1. For each of th e following functions f

: [0, ∞) → R,

i = 1, . . . , 7, state whether f

∈ L

, f

∈ L

, and f

∈ L

∞

i) f

(t) = 1.

ii) f

(t) =

1+t

iii) f

(t) =

1+t

1/4

(1+t)

iv) f

(t) = e

−t

v) f

(t) =

1+t

1/4

(1+t

)

vi) f

(t) =

1+t

1/2

(1+t

)

vii) f

(t) = sin t.

Problem 7.2. Let f : [0, ∞) → R. Show that if

|f(s)|ds converges,

then

f(s)ds converges.

Problem 7.3. Let f : [0, ∞) → R and let f ∈ L

. Show that

lim

t→∞

f(t) exists.

Problem 7.4. Let f : [0, ∞) → R. Show that if f ∈ L

and

f ∈ L

then f ∈ L

∞

and lim

t→∞

f(t) = 0.

Problem 7.5. Let f : [0, ∞) → R. Show that if f ∈ L

and

f ∈ L

then lim

t→∞

f(t) = 0.

NonlinearBook10pt November 20, 2007

506 CHAPTER 7

Problem 7.6. Let f : [0, ∞) → R. Show that if

f ∈ L

∞

, then f is

uniformly continuous on R.

Problem 7.7. Let f : [0, ∞) → R. Show that if f ∈ L

and f is

Lipschitz continuous on R, then lim

t→∞

f(t) = 0.

Problem 7.8. Let f : [0, ∞) → R and p ∈ [1, ∞]. Sh ow that if f ∈ L

and f is uniformly continuous on R, then lim

t→∞

f(t) = 0.

Problem 7.9. Let f

: [0, ∞) → R and f

: [0, ∞) → R. Show that if

∈ L

and f

∈ L

, then f

+ f

∈ L

Problem 7.10. Let f : [0, ∞) → R be continuously diﬀerentiable and

square integrable on [0, ∞). Furthermore, assume

f is bounded on [0, ∞).

Show that lim

t→∞

f(t) = 0.

Problem 7.11. Let f : [0, ∞) → R be continuously diﬀerentiable on

[0, ∞). Prove or refute that if lim

t→∞

f(t) = 0, then lim

t→∞

f(t) exists.

Conversely, prove or refute that if lim

t→∞

f(t) exists, then lim

t→∞

f(t) = 0.

Problem 7.12. Consider the linear dynamical system given by

˙x(t) = Ax(t) + Bu(t), x(0) = x

, t ≥ 0, (7.106)

y(t) = Cx(t), (7.107)

where x(t) ∈ R

, u(t) ∈ R

, y(t) ∈ R

, t ≥ 0, and A is Hur witz. Show

that if u(·) ∈ L

, then x(·) ∈ L

∩ L

∞

, ˙x(·) ∈ L

, and lim

t→∞

x(t) = 0.

Furth ermore, sh ow that if u(·) ∈ L

, then y(·) ∈ L

∩ L

∞

and ˙y(·) ∈ L

for

p = 1, 2, and ∞. In addition, for p = 1 and 2, show that lim

t→∞

y(t) = 0.

Problem 7.13. Consider the operator dynamical system G : L

→ L

given by

y(t) = G[u](t) =

∞

h(t, τ)u(τ)dτ. (7.108)

Show that G is causal if and only if h(t, τ) = 0 whenever t < τ.

Problem 7.14 (Minkowski’s Inequality). Let p ∈ [1, ∞] and suppose

that f, g ∈ L

. Show that f + g ∈ L

and |||f + g|||

≤ |||f|||

+ |||g|||

Problem 7.15. Show that f or each p ∈ [1, ∞], the pair (L

, k·k

) is a

normed linear space. Furth ermore, show that (L

, k·k

) is a Banach space.

(Hint: Use Minkowski’s inequality.)

Problem 7.16 (H¨older’s Inequality). Let p, q ∈ [1, ∞] be such that

1/p+1/q = 1. S uppose f ∈ L

and g ∈ L

. Show that h : [0, ∞ → R deﬁned

NonlinearBook10pt November 20, 2007

INPUT-OUTPUT STABILITY AND DISSIPATIVITY 507

by h(t)

△

= f (t)g(t) belongs to L

. In addition, show that |||fg|||

≤ |||f|||

|||g|||

Problem 7.17. Consider the nonlinear feedback system

˙x(t) = Ax(t) + Bu(t), x(0) = x

, t ≥ 0, (7.109)

y(t) = Cx(t), (7.110)

u(t) = r(t) − σ(t, y(t)), (7.111)

where x(t) ∈ R

, u(t) ∈ R

, y(t) ∈ R

, r(t) ∈ R

, and σ : R

× R

→ R

satisﬁes σ(t, 0) = 0, t ≥ 0. Show that if σ(·, ·) is globally Lipschitz continuous

on R

and the zero solution x(t) ≡ 0 of the undisturbed system (7.109) is

globally exponentially s table, then (7.109) is L

-stable for all p ∈ [1, ∞].

Conversely, show that if (7.109) and (7.110) is min imal and (7.109) is L

stable, then lim

t→∞

x(t) = 0 for all x

∈ R

and u(t) ≡ 0.

Problem 7.18. Consider the f eedback system shown in Figure 7.1

where G

is a linear single-input, single-output time-invariant dynamical

system with transfer function G(s) ∈ H

∞

and G

is a memoryless time-

varying nonlinearity σ(·, ·) ∈ Φ

. Show that if |||G(s)|||

∞

< γ, where γ > 0,

then the feedback interconn ection of G

and G

is L

-stable with ﬁnite gain

and zero bias.

Problem 7.19 (Circle Cr iterion). Consider the feedback system shown

in Figure 7.1 where G

is a linear s ingle-input, single-output time-invariant

dynamical system with transfer function G(s) ∈ H

∞

and G

is a memoryless

time-varying n onlinearity σ(·, ·) ∈ Φ. Show that if |||G

(s)|||

∞

< r

−1

, where

(s)

△

= G(s)/(1 + cG(s)), c

△

= (M

+ M

)/2, and r

△

= (M

− M

)/2, then

the feedb ack interconnection of G

and G

is L

-stable with ﬁnite gain and

zero bias. Connect this result to Theorem 5.19.

Problem 7.20 (Popov Criterion). Consider the feedback system

shown in Figure 7.1 where G

is a strictly proper linear single-input, single-

output time-invariant dynamical system with transfer function G(s) ∈ H

∞

and G

is a memoryless time-invariant nonlinearity σ(·) ∈ Φ

. Assume that

sG(s) ∈ H

∞

. Show that if there exists N ≥ 0 such that

△

= inf

ω ∈R

Re[(1 + ωN)G(ω)] +

> 0, (7.112)

then there exists γ > 0 s uch that

|||y

||| ≤ γ(|||u

|||

+ |||u

|||

+ |||˙u

|||

), i = 1, 2. (7.113)

Problem 7.21. Consider the nonlinear dynamical s ystem (7.40) and

(7.41) where F (·, ·) is Lipschitz continuous in (x, u), H(·, ·) is continuous,

F (0, 0) = 0, and H(0, 0) = 0. Assume there exists a continuously

diﬀerentiable function V : R

→ R such that V (0) = 0, V (x) > 0, x ∈ R

NonlinearBook10pt November 20, 2007

508 CHAPTER 7

x 6= 0, V (x) → ∞ as kxk → ∞, and

′

(x)F (x, u) ≤ −W (x) + φ(u), (x, u) ∈ R

× R

, (7.114)

where W (x) is continuous on R

, W (x) > 0, x ∈ R

, x 6= 0, W (x) → ∞ as

kxk → ∞, φ(·) is continuous on R

, and φ(0) = 0. Show that in this case

(7.40) and (7.41) is L

∞

-stable.

Problem 7.22. Consider the nonlinear d ynamical s ystem (7.2) and

(7.3) where f(·) is Lipschitz continuous on R

, G(·), h(·), and J(·) are

continuous on R

, and f (0) = 0 and h(0) = 0. Let γ > 0 and assume there

exists a continuously diﬀerentiable function V : R

→ R such that V (·) is

positive deﬁnite, V (0) = 0, and, for all x ∈ R

0 ≥ V

′

(x)f(x) + h

(x)h(x) + [

′

(x)G(x) + h

(x)J(x)]

·[γ

− J

(x)J(x)]

−1

[

′

(x)G(x) + h

(x)J(x)]

. (7.115)

Show that in this case, for each x

∈ R

, (7.2) and (7.3) is L

-stable and

the L

gain is less than or equal to γ.

Problem 7.23. Consider the ﬁnite gain operator dynamical systems

: L

→ L

and G

: L

→ L

. Show that G

◦ G

is a ﬁn ite gain

operator dynamical system.

Problem 7.24. Let G

: L

→ L

, i = 1, . . . , q, be a collection of

causal operator dynamical systems that are (Q

, R

, S

)-dissipative for each

i ∈ {1, . . . , q}. Deﬁne the input vector u

△

= [u

, . . . , u

]

, the output vector

△

= [y

, . . . , y

]

, and the system interconnection by

u = v − Hy, (7.116)

where H ∈ R

+···+m

)×(l

+···+l

)

and v ∈ L

+···+m

)

. In addition, deﬁne

Q = diag[Q

, . . . , Q

], R = d iag[R

, . . . , R

], and S = diag[S

, . . . , S

]. Sh ow

that if

Q = Q + H

RH − SH − H

< 0, (7.117)

then the interconnected system with in put v and output y is ﬁnite-gain

-stable.

Problem 7.25. For discrete-time nonlinear operator dynamical sys-

tems the input and output spaces are spaces of sequences, denoted by ℓ

and ℓ

, respectively. In particular, ℓ

is th e set of sequences such that

|||f |||

p,q

< ∞, q ∈ [1, ∞], w here f : Z

→ R

, that is,

ℓ

△

= {f : Z

→ R

: |||f |||

p,q

< ∞, q ∈ [1, ∞]},

NonlinearBook10pt November 20, 2007

INPUT-OUTPUT STABILITY AND DISSIPATIVITY 509

where

|||f |||

p,q

△

∞

k=0

kf(k)k

1/p

, 1 ≤ p < ∞, (7.118)

|||f |||

∞,q

△

= sup

k∈Z

kf(k)k

. (7.119)

Show that the main results of this chapter including the small gain theorem

and the (Q, R, S)-dissipativity theorem go through unchanged for discrete-

time nonlinear operator dynamical s ys tems.

7.12 Notes and References

Input-output system theory can be traced back to Norbert Wiener. In

the late 1950s, Wiener used Volterra integral series to represent input-

output maps of nonlinear dynamical systems [453]. The foundations of

input-output stability theory were developed by S an dberg [387, 388, 390]

and Zames [476, 477]. An excellent textbook treatment of input-output

stability of feedback systems is given by Desoer and Vidyasagar [104]. Input-

output dissipativity theory is due to Moylan and Hill [322], and Hill and

Moylan [192]. Connections between input-output stability and Lyapunov

stability theory were developed by Willems [455], Hill and Moylan [190], and

Vidyasagar and Vannelli [446]. Finally, the induced convolution operator

norms of linear dynamical systems pr esented in Sections 7.8–7.10 are d ue to

Chellaboina, Haddad, Bernstein, and Wilson [89].

NonlinearBook10pt November 20, 2007

Chapter Eight

Optimal Nonlinear Feedback Control

8.1 Introduction

Under certain conditions n on linear controllers oﬀer signiﬁcant advantages

over linear controllers. In particular, if the plant dynamics and/or system

measurements are nonlinear [34, 212, 398, 452], the plant/measurement

disturbances are either nonadditive or non-Gaussian, the performance

measure considered in nonquadratic [14, 33, 133, 244, 275, 3 66 , 384, 391, 399,

411,429], the plant model is uncertain [24,28,99,142,268,351], or the control

signals/state amplitudes are constrained [62, 125, 234, 375], then nonlinear

controllers yield better performance than the best linear controllers. In a

paper by Bernstein [43] the current s tatus of continuous-time, nonlinear-

nonquadratic p roblems was presented in a simpliﬁed and tutorial manner.

The basic underlying ideas of the results in [43] are based on the fact

that the steady-state solution of the Hamilton-Jacobi-Bellman equation is

a Lyapunov function for the nonlinear s y s tem and thus guaranteeing both

stability and op timality [217].

Building on the results of [43], in this chapter, we present a f ramework

for analyzing and designing feedback controllers for nonlinear systems.

Speciﬁcally, we consider a feedback control problem over an inﬁnite horizon

involving a n onlinear-nonquadratic performance functional. The perfor-

mance functional can be evaluated in closed f orm as long as the nonlinear-

nonquadratic cost functional considered is related in a speciﬁc way to an

underlying Lyapunov function that guarantees asymp totic stability of the

nonlinear closed-loop system. This Lyapunov function is shown to be the

solution of the steady-state Hamilton-Jacobi-Bellman equation. The overall

framework provides the foun dation for extending linear-quadratic control to

nonlinear-nonquadratic problems.

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512 CHAPTER 8

8.2 Stability Analysis of Nonlinear Systems

In this section, we pr esent suﬃcient conditions for stability and performance

for a given nonlinear system with a nonlinear-nonquadratic performance

functional. For the class of nonlinear systems considered we assume that

the required properties for the existence and uniqueness of solutions are

satisﬁed. For the following result, let D ⊆ R

be an open set, assume 0 ∈ D,

let L : D → R, and let f : D → R

be s uch that f (0) = 0.

Theorem 8.1. Consider the nonlinear dynamical system

˙x(t) = f (x(t)), x(0) = x

, t ≥ 0, (8.1)

with nonlinear-nonquadratic performance functional

J(x

)

△

∞

L(x(t)) dt. (8.2)

Furth ermore, assume that there exists a continuously diﬀerentiable function

V : D → R such that

V (0) = 0, (8.3)

V (x) > 0, x ∈ D, x 6= 0, (8.4)

′

(x)f(x) < 0, x ∈ D, x 6= 0, (8.5)

L(x) + V

′

(x)f(x) = 0, x ∈ D. (8.6)

Then the zero solution x(t) ≡ 0 to (8.1) is locally asymptotically stable and

there exists a neighborhood of the origin D

⊆ D such that

J(x

) = V (x

), x

∈ D

. (8.7)

Finally, if D = R

and

V (x) → ∞ as kxk → ∞, (8.8)

then the zero solution x(t) ≡ 0 to (8.1) is globally asymptotically stable.

Proof. Let x(t), t ≥ 0, satisfy (8.1). Then it follows from (8.5) that

V (x(t)) = V

′

(x(t))f(x(t)) < 0, t ≥ 0, x(t) 6= 0. (8.9)

Thus , from (8.3), (8.4), and (8.9) it follows from Theorem 3.1 that V (·)

is a Lyapunov function for the nonlinear dynamical system (8.1), which

proves local asymptotic stability of the zero solution x(t) ≡ 0 to (8.1).

Consequently, x(t) → 0 as t → ∞ for all initial conditions x

∈ D

for some

neighborhood of the origin D

⊆ D. Now, since

0 = −

V (x(t)) + V

′

(x(t))f(x(t)), t ≥ 0,

it follows fr om (8.6) that

L(x(t)) = −

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

′

(x(t))f(x(t)) = −

V (x(t)).

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OPTIMAL NONLINEAR FEEDBACK CONTROL 513

Now, integrating over [0, t] yields

L(x(s))ds = −V (x(t)) + V (x

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

∈ D

yields

J(x

) = V (x

). Finally, for D = R

global asymptotic stability is a d irect

consequence of the radially unbounded condition (8.8) on V (·).

It is important to note that if (8.6) holds, then (8.5) is equivalent to

L(x) > 0, x ∈ D, x 6= 0. Next, we specialize Theorem 8.1 to linear systems.

For this result let A ∈ R

n×n

and let R ∈ R

n×n

be a positive-deﬁnite matrix.

Corollary 8.1. Consider the linear dynamical system

˙x(t) = Ax(t), x(0) = x

, t ≥ 0, (8.10)

with quadratic performance functional

J(x

)

△

∞

(t)Rx(t)dt. (8.11)

Furth ermore, assume that there exists a positive-deﬁnite matrix P ∈ R

n×n

such that

0 = A

P + P A + R. (8.12)

Then, the zero solution x(t) ≡ 0 to (8.10) is globally asymptotically stable

and

J(x

) = x

P x

, x

∈ R

. (8.13)

Proof. The result is a direct consequence of Theorem 8.1 with f (x) =

Ax, L(x) = x

Rx, V (x) = x

P x, and D = R

. Speciﬁcally, conditions

(8.3) and (8.4) are trivially satisﬁed. Now, V

′

(x)f(x) = x

P + P A)x,

and hence, it follows from (8.12) that L(x) + V

′

(x)f(x) = 0, x ∈ R

, so that

all the conditions of Theorem 8.1 are satisﬁed. Finally, since V (·) is radially

unbounded, the zero solution x(t) ≡ 0 to (8.10) is globally asymptotically

stable.

It follows from Corollary 8.1 that Theorem 8.1 is an extension of the

analysis framework to nonlinear systems. Recall that the H

Hardy

space consists of complex matrix-valued fun ctions G(s) ∈ C

l×m

that are

analytic in the open right half p lane and satisfy

sup

η>0

2π

∞

−∞

kG(η + ω)k

dω < ∞. (8.14)

NonlinearBook10pt November 20, 2007

514 CHAPTER 8

The norm of an H

function G(s) is deﬁned by

|||G|||

△



2π

∞

−∞

kG(ω)k

dω



1/2

. (8.15)

Alternatively, using Parseval’s theorem we can express the H

norm as an

norm of the impulse response H(t) =

2π

∞

−∞

G(ω)e

ωt

dω. In particular,

the L

norm of the matrix-valued impulse response fun ction H(t) ∈ R

l×m

t ≥ 0, is deﬁned by

|||H|||

△



∞

kH(t)k



1/2

. (8.16)

Now, letting R = E

E and deﬁning the free response z(t)

△

= Ex(t) =

, t ≥ 0, it follows that the performance functional (8.11) can be

written as

J(x

) =

∞

(t)z(t)dt

∞

= x

P x

∞

kH(t)k

2π

∞

−∞

kG(ω)k

dω

= |||G|||

, (8.17)

where H(t) = Ee

and

G(s) ∼



A x

E 0



Alternatively, assuming x

has an expected value V , that is, E[x

]

= V , where E denotes expectation, and letting V = DD

, it follows that

the averaged perform an ce functional is given by

E[J(x

)] = E[x

P x

] = tr D

P D = |||G|||

, (8.18)

where P satisﬁes (8.12) and

G(s) ∼



A D

E 0



Next, we specialize Theorem 8.1 to linear and n on linear systems with

multilinear cost functionals. First, however, we give several deﬁnitions

involving multilinear functions and a key lemma establishing the existence