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

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DISTURBANCE REJECTION CONTROL 605

bounded) closed-loop input-output maps, respectively. In the special case

where the controlled sys tem is linear the results, with appropriate quadratic

supply rates, specialize to the mixed-norm H

∞

framework developed

in [49] and the mixed H

/positivity framework d eveloped in [146, 151].

The main focus of this chapter is a methodology for designing optimal

nonlinear controllers which guarantee disturbance rejection and minimize

a (derived) performance functional that serves as an upper bound to a

nonlinear-nonquadratic cost functional. In particular, the p erformance

bound can be evaluated in closed form as long as the nonlinear-nonquadratic

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

Lyapunov function that guarantees stability. This Lyapunov function

is shown to be the solution to the steady-state form of the Hamilton-

Jacobi-Isaacs equation for the controlled system and plays a key role

in constructing the optimal nonlinear disturbance rejection control law.

Furth ermore, since the nonlinear-nonquadratic cost functional is closely

related to the structure of the Lyapunov function the proposed framework

provides a class of feedback stabilizing controllers that minimize a derived

performance functional. Hence, the overall framework provides for a

generalization of the Hamilton-Jacobi-Isaacs conditions for addressing the

design of optimal and inverse optimal controllers for nonlinear systems with

exogenous disturbances.

A key feature of the present chapter is that since the necessary and

suﬃcient Hamilton-Jacobi-Isaacs optimality conditions are obtained for a

modiﬁed nonlinear-nonquadratic performance functional rather than the

original performance functional, globally optimal controllers are guaranteed

to provide disturbance rejection. Of course, since the approach allows us to

construct globally optimal controllers th at minimize a given Hamiltonian,

the resulting disturbance rejection controllers p rovide the best worst-case

performance over the class of admissible input disturbances.

10.2 Nonlinear Dissipative Dynamical Systems with

Bounded Disturbances

In this chapter, we consider nonlinear dynamical systems G of the form

˙x(t) = f(x(t)) + J

(x(t))w(t), x(0) = x

, t ≥ 0, (10.1)

z(t) = h(x(t)) + J

(x(t))w(t), (10.2)

where x ∈ R

, w ∈ R

, z ∈ R

, f : R

→ R

, J

: R

→ R

n×d

, h :

→ R

, and J

: R

→ R

p×d

. We assume that f (·), J

(·), h(·), and

(·) are continuous mappings and f(·) has at least one equilibrium so th at,

without loss of generality, f(0) = 0 and h(0) = 0. Furthermore, for the

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606 CHAPTER 10

nonlinear system G we assume th at the required properties for the existence

and u niqueness of solutions are satisﬁed, that is, w(·) satisﬁes suﬃcient

regularity conditions such that (10.1) has a unique solution forward in time.

In this section, we present suﬃcient conditions for dissipativity for a

class of nonlinear systems with bounded energy and bounded amplitude

disturbances. In addition, we consider the problem of evaluating a

performance bound for a nonlinear-nonquadratic cost functional. The cost

bound is evaluated in closed form by relating the cost functional to an

underlying Lyapu nov function that guarantees asymptotic stability of the

nonlinear system. Here, we restrict our attention to time-invariant, inﬁnite-

horizon systems. For the following result presenting suﬃcient conditions

under which a nonlinear system is dissipative with respect to the supply

rate r(z, w), let D ⊂ R

be an open set, assume 0 ∈ D, let f : D → R

such that f(0) = 0, h : D → R

be s uch that h(0) = 0, J

: D → R

n×d

, and

: D → R

p×d

. Finally, let W ⊂ R

and let r : R

× R

→ R be a given

function.

Lemma 10.1. Consider the nonlinear dynamical system

˙x(t) = f(x(t)) + J

(x(t))w(t), x(0) = x

, w(·) ∈ L

, t ≥ 0, (10.3)

z(t) = h(x(t)) + J

(x(t))w(t). (10.4)

Furth ermore, assume that there exist fun ctions Γ : D → R and V : D → R,

where V (·) is continuously diﬀerentiable, such that

V (0) = 0, (10.5)

V (x) ≥ 0, x ∈ D, (10.6)

′

(x)J

(x)w ≤ r(z, w) + Γ(x), x ∈ D, w ∈ R

, (10.7)

′

(x)f(x) + Γ(x) ≤ 0, x ∈ D. (10.8)

Then the solution x(t), t ≥ 0, of (10.3) satisﬁes

V (x(T )) ≤

r(z(t), w(t))dt + V (x

), w(·) ∈ L

, T ≥ 0. (10.9)

Proof. Let x(t), t ≥ 0, satisfy (10.3) an d let w(·) ∈ L

. Then it follows

from (10.7) and (10.8) that

V (x(t))

△

dV (x(t))

= V

′

(x(t))[f(x(t)) + J

(x(t))w(t)]

≤ V

′

(x(t))f(x(t)) + Γ(x(t)) + r(z(t), w(t))

≤ r(z(t), w(t)), t ≥ 0. (10.10)

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DISTURBANCE REJECTION CONTROL 607

Now, integrating over [0, T ] yields

V (x(T )) − V (x

) ≤

r(z(t), w(t))dt, w(·) ∈ L

, T ≥ 0,

which proves the result.

For the next r esult let L : D → R be given.

Theorem 10.1. Consider the nonlinear dynamical system given by

(10.3) and (10.4) with performance functional

J(x

)

△

∞

L(x(t))dt, (10.11)

where x(t), t ≥ 0, solves (10.3) with w(t) ≡ 0. Assume th at there exist

functions Γ : D → R and V : D → R, where V (·) is continuously

diﬀerentiable, such that L(x) + Γ(x) ≥ 0, x ∈ D,

V (0) = 0, (10.12)

V (x) > 0, x ∈ D, x 6= 0, (10.13)

′

(x)f(x) < 0, x ∈ D, x 6= 0, (10.14)

′

(x)J

(x)w ≤ r(z, w) + L(x) + Γ(x), x ∈ D, w ∈ W,

(10.15)

L(x) + V

′

(x)f(x) + Γ(x) = 0, x ∈ D. (10.16)

Then the zero solution x(t) ≡ 0 of the undisturbed (w(t) ≡ 0) system (10.3)

is locally asymptotically stable and there exists a neighborhood D

⊆ D of

the origin such that if Γ(x) ≥ 0, x ∈ D, then

J(x

) ≤ J(x

) = V (x

), x

∈ D

, (10.17)

where

J(x

)

△

∞

[L(x(t)) + Γ(x(t))]dt (10.18)

and where x(t), t ≥ 0, is a solution to (10.3) with w(t) ≡ 0. Furtherm ore,

the solution x(t), t ≥ 0, to (10.3) satisﬁes the dissipativity constraint

r(z(t), w(t))dt + V (x

) ≥ 0, w(·) ∈ L

, T ≥ 0. (10.19)

Finally, if D = R

, w(t) ≡ 0, and

V (x) → ∞ as kxk → ∞, (10.20)

then the zero solution x(t) ≡ 0 to (10.3) is globally asymptotically stable.

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608 CHAPTER 10

Proof. Let x(t), t ≥ 0, satisfy (10.3). Then

V (x(t))

△

V (x(t)) = V

′

(x(t))(f(x(t)) + J

(x)w(t)), t ≥ 0. (10.21)

Hence, with w(t) ≡ 0, it follows from (10.14) that

V (x(t)) < 0, t ≥ 0, x(t) 6= 0. (10.22)

Thus , from (10.12), (10.13), and (10.22) it follows that V (·) is a Lyapunov

function for (10.3), which proves local asymptotic stability of the zero

solution x(t) ≡ 0 of (10.3) with w(t) ≡ 0. Consequently, x(t) → 0 as

t → ∞ for all initial conditions x

∈ D

for some neighborhood D

⊆ D of

the origin.

Next, if Γ(x) ≥ 0, x ∈ D, and w(t) ≡ 0, (10.16) implies

L(x(t)) = −

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

′

(x(t))f(x(t))

≤ −

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

′

(x(t))f(x(t)) + Γ(x(t))

= −

V (x(t)).

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

Next, let x(t), t ≥ 0, satisfy (10.3) with w(t) ≡ 0. Then, with

L(x) replaced by L(x) + Γ(x) and J(x

) replaced by J(x

), it follows from

Theorem 8.1 that J(x

) = V (x

). Finally, since L(x) + Γ(x) ≥ 0, x ∈ D,

it follows that (10.12)–(10.16) implies (10.5)–(10.9), and hence, with Γ(x)

replaced by L(x) + Γ(x), Lemma 10.1 yields

V (x(T )) ≤

r(z(t), w(t))dt + V (x

), w(·) ∈ L

, T ≥ 0.

Now, (10.19) follows by noting that V (x(T )) ≥ 0, T ≥ 0. Finally, for D = R

global asymptotic stability of the zero solution x(t) ≡ 0 of the undisturbed

(w(t) ≡ 0) system (10.3) is a direct consequence of the radially unbounded

condition (10.20) on V (x).

10.3 Specialization to Dissipative Systems with Quadratic

Supply Rates

In this section, we consider the special case in which r(z, w) is a quadratic

functional. Speciﬁcally, let h : D → R

, J

: D → R

p×d

Q ∈ S

S ∈ R

p×d

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DISTURBANCE REJECTION CONTROL 609

R ∈ S

, and

r(z, w) = z

Qz + 2z

Sw + w

Rw, (10.23)

such that

N(x)

△

= J

(x)

(x) + J

(x)

S +

(x) +

R > 0, x ∈ D.

Furth ermore, let L(x) ≥ 0, x ∈ D. Then

Γ(x) =

(x)V

′

(x) − J

(x)

Qh(x) −

h(x)

−1

(x)

(x)V

′

(x) − J

(x)

Qh(x) −

h(x)

− h

(x)

Qh(x)

satisﬁes (10.15) since in this case

L(x) + Γ(x) −V

′

(x)J

(x)w + r(z, w)

= L(x) +

(x)V

′T

(x) − J

(x)

Qh(x) −

(x)h(x) − N(x)w

·N

−1

(x)

(x)V

′T

(x) − J

(x)

Qh(x) −

(x)h(x) − N(x)w

≥ 0. (10.24)

Corollary 10.1. Let γ > 0 and L(x) ≥ 0, x ∈ D, and consider the

nonlinear dynamical system given by (10.3) and (10.4) with performance

functional

J(x

)

△

∞

L(x(t))dt, (10.25)

where x(t), t ≥ 0, solves (10.3) with w(t) ≡ 0. Assume that there exists a

continuously diﬀerentiable function V : D → R such that

V (0) = 0, (10.26)

V (x) > 0, x ∈ D, x 6= 0, (10.27)

′

(x)f(x) < 0, x ∈ D, x 6= 0, (10.28)

L(x) + V

′

(x)f(x) + Γ(x) = 0, x ∈ D, (10.29)

where

Γ(x) =

(x)V

′

(x) + J

(x)h(x)



I − J

(x)J

(x)



−1

(x)V

′

(x) + J

(x)h(x)

+ h

(x)h(x). (10.30)

Then the zero solution x(t) ≡ 0 of the undisturbed (w(t) ≡ 0) system (10.3)

is locally asymptotically stable and there exists a neighborhood D

⊆ D of

the origin such that

J(x

) ≤ J(x

) = V (x

), x

∈ D

, (10.31)

where

J(x

)

△

∞

[L(x(t)) + Γ(x(t))]dt (10.32)

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610 CHAPTER 10

and where x(t), t ≥ 0, solves (10.3) with w(t) ≡ 0. Furthermore, the solution

x(t), t ≥ 0, of (10.3) satisﬁes the nonexpansivity constraint

(t)z(t)dt ≤ γ

(t)w(t)dt + V (x

), w(·) ∈ L

, T ≥ 0.

(10.33)

Finally, if D = R

, w(t) ≡ 0, and

V (x) → ∞ as kxk → ∞, (10.34)

then the zero solution x(t) ≡ 0 to (10.3) is globally asymptotically stable.

Proof. With

Q = −I

S = 0, and

R = γ

, it follows from (10.24)

that Γ(x) given by (10.30) satisﬁes (10.15). The r esult now follows as a

direct consequence of Theorem 10.1.

Note that if L(x) = h

(x)h(x) in Corollary 10.1, then Γ(x) can be

chosen as

Γ(x) =

(x)V

′

(x) + J

(x)h(x)



I − J

(x)J

(x)



−1

(x)V

′

(x) + J

(x)h(x)

Example 10.1. Consider the nonlinear d y namical system

˙x

(t) = −x

(t), x

(0) = x

, t ≥ 0, (10.35)

˙x

(t) = x

(t) −x

(t)tanh(x

(t) − x

(t)) + x

(t)w(t), x

(0) = x

(10.36)

˙x

(t) = x

(t)tanh(x

− x

(t)) − x

(t)w(t), x

(0) = x

, (10.37)

z(t) = x

(t) −x

(t). (10.38)

To show that (10.35)–(10.38) is nonexp ansive with gain less than or equal

to 1, note that (10.35)–(10.38) can be written in the state space form (10.3)

and (10.4) with x = [x

]

f(x) =





−x

− x

tanh(x

−x

)

tanh(x

− x

)





, J

(x) =





−x





, h(x) = x

− x

(10.39)

and J

(x) = 0. Now, with V (x) =

x it follows that

′

(x)f(x) =









−x

− x

tanh(x

− x

)

tanh(x

− x

)





= −(x

− x

)tanh(x

− x

)

= −h(x)tanh(h(x)) (10.40)

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DISTURBANCE REJECTION CONTROL 611

and

′

(x)J

(x) =









−x





= h(x). (10.41)

Next, let D

△

= {x ∈ R

: |x

− x

| ≤ 1}. In this case, it can easily be

shown that all conditions of Corollary 10.1 are satisﬁed with γ = 1 and

L(x) = h

(x)h(x), and hence, the nonlinear dynamical s ys tem (10.35)–

(10.38) is nonexp an s ive with gain less than or equal to 1. △

The framework presented in Corollary 10.1 is an extension of the

mixed-norm H

∞

framework of Bernstein and Hadd ad [49] to nonlinear

dynamical systems. Speciﬁcally, setting f(x) = Ax, J

(x) = D, h(x) = Ex,

(x) = 0, L(x) = x

Rx, and V (x) = x

P x, where A ∈ R

n×n

, D ∈ R

n×d

E ∈ R

p×n

, R

△

= E

E > 0, and P ∈ P

satisﬁes

0 = A

P + P A + γ

−2

P DD

P + R, (10.42)

it follows from Corollary 10.1, with L(x) = h

(x)h(x) = x

Rx, Γ(x) =

−2

P DD

P x, and x

= 0, that

(t)Rx(t)dt ≤ γ

(t)w(t)dt, T ≥ 0, w(·) ∈ L

(10.43)

or, equivalently, th e H

∞

norm of

G(s) ∼



A D

E 0



satisﬁes

|||G|||

∞

△

= sup

ω ∈R

max

(G(ω)) ≤ γ. (10.44)

Now, (10.31) implies

∞

(t)Rx(t)dt ≤

∞

(t)(R + γ

−2

P DD

P )x(t)dt

∞

(R + γ

−2

P DD

P )e

dt,

where x(t), t ≥ 0, solves (10.3) with w(t) ≡ 0.

As is common practice [274], to eliminate the explicit dependence of

J(x

) on the initial condition x

we assume x

has expected value V , that

is, E[x

] = V , where E denotes expectation. Invoking this step leads to



∞

(t)Rx(t)dt



= E



∞



= E[x

P x

] = tr

P V,

where

0 = A

P +

P A + R

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612 CHAPTER 10

and



∞

(t)(R + γ

−2

P DD

P )x(t)dt



= E



∞

(R + γ

−2

P DD

P )e



= E



P x



= tr P V,

where P satisﬁes (10.42). Hence, |||G|||

= tr

P V ≤ tr P V , which implies

that J(x

) given by (10.32) provides an upper bound to the H

norm of

G(s).

Corollary 10.2. Let L(x) ≥ 0, x ∈ D, p = d, and consider the

nonlinear dynamical system given by (10.3) and (10.4) with performance

functional

J(x

)

△

∞

L(x(t))dt, (10.45)

where x(t), t ≥ 0, solves (10.3) with w(t) ≡ 0. Assume that there exists a

continuously diﬀerentiable function V : D → R such that

V (0) = 0, (10.46)

V (x) > 0, x ∈ D, x 6= 0, (10.47)

′

(x)f(x) < 0, x ∈ D, x 6= 0, (10.48)

L(x) + V

′

(x)f(x) + Γ(x) = 0, x ∈ D, (10.49)

where

Γ(x) =

(x)V

′

(x) − h(x)



(x) + J

(x)



−1

(x)V

′

(x) − h(x)

. (10.50)

Then the zero solution x(t) ≡ 0 of the undisturbed (w(t) ≡ 0) system (10.3)

is locally asymptotically stable and there exists a neighborhood D

⊆ D of

the origin such that

J(x

) ≤ J(x

) = V (x

), x

∈ D

, (10.51)

where

J(x

)

△

∞

[L(x(t)) + Γ(x(t))]dt (10.52)

and where x(t), t ≥ 0, solves (10.3) with w(t) ≡ 0. Furthermore, the solution

x(t), t ≥ 0, of (10.3) satisﬁes the passivity constraint

(t)w(t)dt + V (x

) ≥ 0, w(·) ∈ L

, T ≥ 0. (10.53)

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DISTURBANCE REJECTION CONTROL 613

Finally, if D = R

, w(t) ≡ 0, and

V (x) → ∞ as kxk → ∞, (10.54)

then the zero solution x(t) ≡ 0 to (10.3) is globally asymptotically stable.

Proof. With

Q = 0,

S = I

, and

R = 0, it follows from (10.24) that

Γ(x) given by (10.50) satisﬁes (10.15). The result now follows as a direct

consequence of Theorem 10.1.

The framework presented in Corollary 10.2 is an extension of the

/positivity framework of Haddad and Bernstein [146, 151] to nonlinear

dynamical systems. Speciﬁcally, setting f(x) = Ax, J

(x) = D, h(x) = Ex,

(x) = E

∞

, L(x) = x

Rx, V (x) = x

P x, and Γ(x) = x

P −E]

∞

)

−1

P − E]x, wh ere A ∈ R

n×n

, D ∈ R

n×d

, E ∈ R

d×n

, E

∞

∈ R

d×d

R ∈ P

, and P ∈ P

satisﬁes

0 = A

P + P A + (D

P −E)

∞

+ E

∞

)

−1

P − E) + R, (10.55)

it follows fr om Corollary 10.2, with x

= 0, that

(t)z(t)dt ≥ 0, w(·) ∈ L

, T ≥ 0, (10.56)

or, equivalently,

∞

(s) + G

∗

∞

(s) ≥ 0, Re[s] > 0, (10.57)

where

∞

(s) ∼



A D

E E

∞



Now, using similar arguments as in the H

∞

case, (10.51) implies th at

P V = E



∞

(t)Rx(t)dt



≤ E



∞

(t)[R + (D

P − E)

∞

+ E

∞

)

−1

P − E)]x(t)dt



or, equivalently, s ince



∞

(t)[R + (D

P − E)

∞

+ E

∞

)

−1

P − E)]x(t)dt



= E



∞

[R + (D

P − E)

∞

+ E

∞

)

−1

P −E)]e



= E [x

P x

]

= tr P V,

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614 CHAPTER 10

|||G|||

= tr

P V ≤ tr P V . Hence, J(x

) given by (10.52) provides an upper

bound to the H

norm of G(s).

Next, deﬁne the subset of square-integrable bounded disturbances

△

= {w(·) ∈ L

∞

(t)w(t)dt ≤ β}, (10.58)

where β > 0. Furthermore, let L : D → R be given such th at L(x) ≥ 0,

x ∈ D.

Theorem 10.2. Let γ > 0 and consider the nonlinear dynamical

system (10.3) with performance fun ctional (10.11). Assume that there exists

a continuous ly diﬀerentiable function V : D → R such that

V (0) = 0, (10.59)

V (x) > 0, x ∈ D, x 6= 0, (10.60)

′

(x)f(x) < 0, x ∈ D, x 6= 0, (10.61)

L(x) + V

′

(x)f(x) +

4γ

′

(x)J

(x)V

′

(x) = 0, x ∈ D. (10.62)

Then then the zero s olution x(t) ≡ 0 of the undisturbed (w(t) ≡ 0) system

(10.3) is lo cally asymptotically stable and there exists a neighborhood D

⊆

D of the origin such that

J(x

) ≤ J(x

) = V (x

), x

∈ D

, (10.63)

where

J(x

)

△

∞

[L(x(t)) + Γ(x(t))]dt, (10.64)

Γ(x) =

4γ

′

(x)J

(x)V

′T

(x), (10.65)

and where x(t), t ≥ 0, solves (10.3) with w(t) ≡ 0. Furthermore, if x

= 0

then the solution x(t), t ≥ 0, of (10.3) satisﬁes

V (x(T )) ≤ γ, w(·) ∈ L

, T ≥ 0. (10.66)

Finally, if D = R

, w(t) ≡ 0, and

V (x) → ∞ as kxk → ∞, (10.67)

then the zero solution x(t) ≡ 0 to (10.3) is globally asymptotically stable.

Proof. The proofs for local and global asymptotic stability and the

performance bound (10.63) are identical to the p roofs of local and global

asymptotic stability given in Theorem 10.1 and the performance bound

(10.17). Next, with r(z, w) =

w and Γ(x) given by (10.65) it follows