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

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OPTIMAL NONLINEAR FEEDBACK CONTROL 525

which shows that L(x, u) may be negative. As a result, th ere may exist a

control input u for which the performance functional J(x

, u) is negative.

However, if the control u is a regulation controller, that is, u ∈ S(x

), then

it follows fr om (8.67) and (8.68) that

J(x

, u(·)) ≥ V (x

) ≥ 0, x

∈ R

, u(·) ∈ S(x

). (8.72)

Furth ermore, in this case, substituting u = φ(x) into (8.70) yields

L(x, φ(x)) = −V

′

(x)[f(x) + G(x)φ(x)], (8.73)

which, by (8.69), is positive.

Example 8.1. To illustrate the utility of Theorem 8.3 we consider a

simple example originally studied in [75] involving the two-state nonlinear

controlled system given by

˙x

(t) = x

(t) − x

(t), x

(0) = x

, t ≥ 0, (8.74)

˙x

(t) = x

(t) + u(t), x

(0) = x

. (8.75)

To construct an inverse op timal globally stabilizing control law for (8.74)

and (8.75) let V (x) be a qu ad ratic Lyapunov function of the form

V (x) = p

+ p

, (8.76)

where x

△

= [x

]

, p

> 0, and p

> 0, and let L(x, u) = L

(x) + L

(x)u +

, where R

> 0. Now, L

(x) = 2R

(

) satisﬁes (8.62) so that

the inverse optimal control law (8.65) is given by

φ(x) = −

− x

−

. (8.77)

In this case, the performance functional (8.58), with

(x) = R

[

+ x

]

− [2p

− x

) + 2p

], (8.78)

is minimized in the s en s e of (8.67). Furthermore, since V (x) given by (8.76)

is radially unbounded and

V (x) = −2p

− 2

< 0, x ∈ R

, x 6= 0, (8.79)

the f eedback control law (8.77) is globally stabilizing. △

Example 8.2. This example is adopted from [449] and considers the

global stabilization of the Lorentz equations. These equations were p roposed

by Lorentz [286] to model ﬂuid convection and can exhibit chaotic motion.

To construct inverse optimal controllers for the controlled Lorentz dynamical

system consider the sys tem

˙x

(t) = −σx

(t) + σx

(t), x

(0) = x

, t ≥ 0, (8.80)

˙x

(t) = rx

(t) −x

(t) − x

(t)x

(t) + u(t), x

(0) = x

, (8.81)

˙x

(t) = x

(t)x

(t) −bx

(t), x

(0) = x

, (8.82)

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526 CHAPTER 8

where σ, r, b > 0. Note that (8.80)–(8.82) can be written in the form of

(8.56) with

f(x) =





−σx

+ σx

− x

− bx





, G(x) =









In order to design an inverse optimal control law for the controlled

Lorentz dynamical system (8.80)–(8.82) consider the quadratic L yapunov

function candidate given by

V (x) = p

+ p

, (8.83)

where x

△

= [x

, x

]

and p

, p

> 0. Now, letting p

= p

and

L(x, u) = L

(x) + L

(x)u + R

, where R

> 0, it follows that

(x) =

(2p

σ + 2p

r)x

− 2p

, (8.84)

satisﬁes (8.62); that is,

V (x) = V

′

(x)[f(x) −

G(x)R

−1

(x) −

G(x)R

−1

(x)V

′T

(x)]

= −2(σp

+ p

)

< 0, x ∈ R

, x 6= 0.

Hence, the output feedback control law φ(x) = −(

σ + r)x

given by (8.65)

globally stabilizes the controlled Lorentz dynamical system (8.80)–(8.82).

Furth ermore, the performance functional (8.58), with

(x) = [2σp

(

σ+r)

−2(p

σ+p

r)x

+2p

, (8.85)

is minimized in the sense of (8.67). △

Next, we specialize Theorem 8.3 to linear systems controlled by

nonlinear controllers that minimize a polynomial cost functional. For the

following result let R

∈ P

, R

∈ P

, and

∈ N

, q = 2, . . . , r, be given,

where r is a positive integer, and d eﬁ ne S

△

= BR

−1

Corollary 8.3. Consider the linear controlled dynamical system

˙x(t) = Ax(t) + Bu(t), x(0) = x

, t ≥ 0, (8.86)

where u(·) is admissible. Assume that there exist P ∈ P

and M

∈ N

q = 2, . . . , r, such that

0 = A

P + P A + R

− P SP, (8.87)

and

0 = (A −SP )

+ M

(A −SP ) +

, q = 2, . . . , r. (8.88)

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OPTIMAL NONLINEAR FEEDBACK CONTROL 527

Then the zero solution x(t) ≡ 0 of the closed-loop system

˙x(t) = Ax(t) + Bφ(x(t)), x(0) = x

, t ≥ 0, (8.89)

is globally asymptotically stable with the feedback control law

φ(x) = −R

−1





P +

q=2

q−1





x, (8.90)

and the performan ce functional (8.58) with R

(x) = R

, L

(x) = 0, and

(x) = x

q=2

q−1



q=2

q−1





q=2

q−1



x, (8.91)

is minimized in the sense that

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)), x

∈ R

. (8.92)

Finally,

J(x

, φ(x(·))) = x

P x

q=2

)

, x

∈ R

. (8.93)

Proof. The result is a direct consequence of Theorem 8.3 with f (x) =

Ax, G(x) = B, L

(x) = 0, R

(x) = R

, and

V (x) = x

P x +

q=2

Speciﬁcally, (8.59)–(8.61) and (8.63) are trivially satisﬁed. Next, it follows

from (8.87), (8.88), and (8.90) that

′

(x)[f(x) −

G(x)R

−1

(x)G

(x)V

′T

(x)] =

−x

x −

q=2

q−1

x − φ

(x)R

φ(x)

−x





q=2

q−1









q=2

q−1





which implies (8.62), so that all the conditions of Theorem 8.3 are satisﬁed.

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528 CHAPTER 8

Corollary 8.3 generalizes the deterministic version of the stochastic

nonlinear-nonquadratic optimal control problem considered in [411] to

polynomial performance criteria. Speciﬁcally, unlike the results of [411],

Corollary 8.3 is not limited to sixth-order cost functionals an d cubic

nonlinear controllers since it addresses a polynomial performance criterion

of an arbitrary even order. Corollary 8.3 requires the solutions of r − 1

modiﬁed Riccati equ ations in (8.88) to obtain th e optimal controller (8.90).

However, if

, q = 3, . . . , r, then M

= M

, q = 3, . . . , r, satisﬁes

(8.88). In this case, we require the solution of one modiﬁ ed Riccati equation

in (8.88). Finally, it is important to note that the derived performance

functional weighs the state variables by arbitrary even powers. Fu rthermore,

J(x

, u(·)) has the form

J(x

, u(

)) =

∞

q=2

q−1

)x + u

+φ

(x)R

(x)

dt,

where φ

(x) is the nonlinear part of the optimal feedback control

φ(x) = φ

(x) + φ

(x),

where φ

(x)

△

= −R

−1

P x and φ

(x)

△

= −R

−1

q=2

q−1

Next, we specialize Theorem 8.3 to linear systems controlled by

nonlinear controllers that minimize a multilinear cost functional. For the

following result recall the deﬁ nition of S and let R

∈ P

, R

∈ P

, and

∈ N

(2q,n)

, q = 2, . . . , r, be given, where r is a given integer.

Corollary 8.4. Consider the linear controlled dynamical system (8.86).

Assume that there exist P ∈ P

and

∈ N

(2q,n)

, q = 2, . . . , r, such that

0 = A

P + P A + R

− P SP (8.94)

and

0 =

[

⊕ (A − SP )] +

, q = 2, . . . , r. (8.95)

Then the zero solution x(t) ≡ 0 of the closed-loop system (8.89) is globally

asymptotically stable with the feedback control law

φ(x) = −R

−1

(P x +

′

(x)), (8.96)

where g(x)

△

q=2

[2q]

and the performance fu nctional (8.58) with R

(x)

= R

, L

(x) = 0, and

(x) = x

x +

q=2

[2q]

′

(x)Sg

′

(x), (8.97)

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OPTIMAL NONLINEAR FEEDBACK CONTROL 529

is minimized in the sense that

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)), x

∈ R

. (8.98)

Finally,

J(x

, φ(x(·))) = x

P x

q=2

[2q]

, x

∈ R

. (8.99)

Proof. The result is a direct consequence of Theorem 8.3 with f (x) =

Ax, G(x) = B, L

(x) = 0, R

(x) = R

, and V (x) = x

P x +

q=2

[2q]

Speciﬁcally, (8.59)–(8.61) and (8.63) are trivially satisﬁed. Next, it follows

from (8.94)–(8.96) that

′

(x)[f(x) −

G(x)R

−1

(x)G

(x)V

′T

(x)] = −x

x −

q=2

[2q]

−φ

(x)R

φ(x) −

′

(x)Sg

′

(x),

which implies (8.62) so that all the conditions of Theorem 8.3 are satisﬁed.

Note that since g

′

(x)(A −SP )x =

q=2

[

⊕ (A −SP )]x

[2q]

it follows

that (8.95) can be equivalently written as

0 = g

′

(x)(A −SP )x +

q=2

[2q]

for all x ∈ R

, and hence, it follows from Lemma 8.1 that there exists a

unique

∈ N

(2q,n)

such that (8.95) is satisﬁed.

8.5 Gain, Sector, and Disk Margins of Nonlinear-Nonquadratic

Optimal Regulators

The gain and phase margins of state feedback linear-quadratic optimal

regulators are well known [10, 227, 379, 466]. In particular, in terms of

classical control relative stability notions, these controllers possess at least a

±60

◦

phase margin, inﬁnite gain margin, and 50 percent gain reduction

for each control channel. Alternatively, in terms of absolute stability

theory [10] th ese controllers guarantee sector margins in that the closed-

loop system will remain asymptotically stable in the face of a memoryless

static input nonlinearity each of whose components is contained in the conic

sector (

, ∞). In both cases, these results hold if the integrand of the

quadratic performance criterion is chosen to be a quadratic nonnegative-

deﬁnite function of the state and a qu adratic positive-deﬁnite function of the

NonlinearBook10pt November 20, 2007

530 CHAPTER 8

control with a diagonal weighting matrix. Gain and phase margins of state

feedback linear-quadratic optimal regulators involving cross-weighting terms

in the quadratic performance criterion were obtained in [95]. Speciﬁcally,

the authors in [95] provide explicit connections between relative stability

margins and the selection of the state, control, and cross-weighting matrices.

However, unlike the standard linear-quadratic case, no sector margin

guarantees were shown in the linear-quadratic problem w ith cross-weighting

terms.

The problem of guaranteed sector margins for state feedback nonlinear-

nonquadratic inverse optimal regulators has also been considered in the

literature [127, 217, 320, 321]. Speciﬁcally, nonlinear Hamilton-Jacobi-

Bellman inverse optimal controllers th at minimize a meaningful (in the

terminology of [127, 395]) nonlinear-nonquadr atic performance criterion

involving a nonlinear-nonquadratic, nonnegative-deﬁnite function of the

state and a quadratic positive-deﬁnite function of the f eedback control are

shown to possess sector margin guarantees to component decoupled input

nonlinearities in the conic sector (

, ∞). These results h ave been recently

extended in [395] to disk margin guarantees where asymptotic stability of

the closed-loop system is guaranteed in the face of a dissipative dynamic

input operator.

In the remainder of this chapter we derive stability margins for the

optimal and inverse optimal n onlinear regulators presented in Sections 8.3

and 8.4. Speciﬁcally, gain, sector, and disk margin guarantees are obtained

for nonlinear dynamical systems controlled by nonlinear optimal and inverse

optimal Hamilton-Jacobi-Bellman controllers that minimize a nonlinear-

nonquadratic performance criterion with cross-weighting terms. In the case

where the cross-weighting term in the performance criterion is deleted our

results recover the gain, sector, and disk margins of [395]. Alternatively,

retaining the cross-terms in the performance criterion an d specializing the

nonlinear-nonquadratic problem to a linear-quadratic problem our results

recover the gain and phase margins of [95]. Finally, we note that even though

the inclusion of cross-weighting terms in the performance criterion is shown

to degrade gain, sector, and disk margins, the extra ﬂexibility provided by

the cross-weighting terms makes it possible to guarantee optimal and inverse

optimal nonlinear controllers th at may be far superior in terms of transient

performance over meaningful inverse optimal controllers.

In this section, we start by deriving guaranteed gain, sector, and disk

margins for nonlinear optimal and inverse optimal regulators that minimize

a nonlinear-nonquadratic performance criterion. Speciﬁcally, suﬃcient

conditions that guarantee gain, sector, and disk margins are given in terms

of the state, control, and cross-weighting nonlinear-nonquadratic weighting

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OPTIMAL NONLINEAR FEEDBACK CONTROL 531

functions. In particular, we consider the nonlinear system given by

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

, t ≥ 0, (8.100)

y(t) = −φ(x(t)), (8.101)

where φ : R

→ R

, with a nonquadratic performance criterion

J(x

, u(·)) =

∞

(x(t)) + L

(x(t))u(t) + u

(t)R

(x(t))u(t)]dt, (8.102)

where L

: R

→ R, L

: R

→ R

1×m

, and R

: R

→ R

m×m

are given such

that R

(x) > 0, x ∈ R

, and L

(0) = 0. In this case, the optimal nonlinear

feedback controller u = φ(x) that minimizes the nonlinear-nonquadratic

performance criterion (8.102) is given by the following r esult.

Theorem 8.4. Consider the nonlinear dynamical system (8.100) and

(8.101) with performance functional (8.102). Assume that there exists a

continuously diﬀerentiable function V : R

→ R such that

V (0) = 0, (8.103)

V (x) > 0, x ∈ R

, x 6= 0, (8.104)

(0) = 0, (8.105)

′

(x)[f(x) −

G(x)R

−1

(x)L

(x)

−

G(x)R

−1

(x)G

(x)V

′T

(x)] < 0, x ∈ R

, x 6= 0, (8.106)

0 = L

(x) + V

′

(x)f(x) −

′

(x)G(x) + L

(x)]

·R

−1

(x)[V

′

(x)G(x) + L

(x)]

, x ∈ R

, (8.107)

and

V (x) → ∞ as kxk → ∞. (8.108)

Then the zero solution x(t) ≡ 0 of the closed-loop system

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

, t ≥ 0, (8.109)

is globally asymptotically stable with the feedback control law

φ(x) = −

−1

(x)[V

′

(x)G(x) + L

(x)]

, (8.110)

and the performan ce functional (8.102) is minimized in the sense that

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)), x

∈ R

. (8.111)

Finally,

J(x

, φ(x(·))) = V (x

), x

∈ R

. (8.112)

Proof. The proof is identical to the proof of Theorem 8.3.

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532 CHAPTER 8

Example 8.3. Consider the nonlinear d y namical system

˙x

(t) = x

(t) + x

(t)x

(t), x

(0) = x

, t ≥ 0, (8.113)

˙x

(t) = −x

(t) + x

(t)u(t), x

(0) = x

, (8.114)

with performance functional

J(x

, x

, u(·)) =

∞

[2x

(t) + 2x

(t) +

(t)]dt. (8.115)

To design an optimal control law φ(x

, x

) that minimizes (8.115) we use

Theorem 8.4 with x = [x

, x

]

, f(x) = [−x

+ x

, −x

]

, G(x) =

[0, x

]

, L

(x) = 2x

x, L

(x) = 0, and R

(x) =

. In particular, it follows

from (8.107) that

0 = V

′

(x)



−x

+ x

−x



−

′

(x))



0 0

0 x



′T

(x)+2(x

), (8.116)

which implies that V

′

(x) = [2x

, 2x

]. Fu rthermore, since V (0) = 0,

V (x) = x

+ x

. Hence, the optimal feedback control law is given by

φ(x) = −

−1

(x)G

(x)V

′T

(x) = −2x

. Finally, note that (8.106) implies







−x

+ x

−x



− 2x

= −2(x

+ x

) < 0, (8.117)

for all (x

, x

) 6= (0, 0), and hence, φ(x

, x

) = −2x

is a global stabilizer

for (8.113) and (8.114). △

The following key lemma is needed for developing the main result of

this section.

Lemma 8.2. Consider the nonlinear dynamical system G given by

(8.100) and (8.101) where φ(x) is a stabilizing feedb ack control law given by

(8.110) and where V (x), x ∈ R

, satisﬁes

0 = V

′

(x)f(x) + L

(x) −

′

(x)G(x) + L

(x)]R

−1

(x)[V

′

(x)G(x) + L

(x)]

(8.118)

Furth ermore, suppose there exists θ ∈ R such that 0 < θ < 1 and

(1 − θ

(x) −

(x)R

−1

(x)L

(x) ≥ 0, x ∈ R

. (8.119)

Then for all u(·) ∈ U and t

, t

≥ 0, t

< t

, the solution x(t), t ≥ 0, to

(8.100) satisﬁes

V (x(t

)) ≤



[u(t) + y(t)]

(x(t))[u(t) + y(t)]

−θ

(t)R

(x(t))u(t)



dt + V (x(t

)). (8.120)

Proof. Note that it follows from (8.118) and (8.119) that for all x ∈ R

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OPTIMAL NONLINEAR FEEDBACK CONTROL 533

and u ∈ R

(x)u ≤ θ

(x)u +



√

1 − θ

(x)R

−1

(x) +

1 −θ



·R

(x)



√

1 −θ

(x)R

−1

(x) +

1 −θ



= u

(x)u +

4(1 − θ

)

(x)R

−1

(x)L

(x) + L

(x)u

≤ u

(x)u + L

(x)

= u

(x)u + L

(x)u − V

′

(x)f(x) + φ

(x)R

(x)φ(x)

= [u + y]

(x)[u + y] − V

′

(x)[f(x) + G(x)u],

which implies that, for all u(·) ∈ U and t ≥ 0,

(t)R

(x(t))u(t) ≤ [u(t) + y(t)]

(x(t))[u(t) + y(t)] −

V (x(t)).

Now, integrating over [t

, t

] yields (8.120).

Next, we present disk margins for the nonlinear-nonquadratic optimal

regulator given by Theorem 8.4. First, we consider the case in which R

(x),

x ∈ R

, is a constant diagonal matrix.

Theorem 8.5. Consider the n onlinear dynamical system G given by

(8.100) and (8.101) where φ(x) is a stabilizing feedback control law given

by (8.110) and where V (x), x ∈ R

, satisﬁes (8.107). If R

(x) ≡

diag[r

, . . . , r

], where r

> 0, i = 1, . . . , m, an d there exists θ ∈ R such

that 0 < θ < 1 and (8.119) is satisﬁed, then the nonlinear system G has

a structured disk margin (

1+θ

1−θ

). If, in addition, R

(x) ≡ I and there

exists θ ∈ R s uch that 0 < θ < 1 and (8.119) is satisﬁed, then the nonlinear

system G has a disk margin (

1+θ

1−θ

Proof. Note that for all u(·) ∈ U and t

, t

≥ 0, t

< t

, it follows

from Lemma 8.2 that the solution x(t), t ≥ 0, to (8.100) s atisﬁes

V (x(t

)) − V (x(t

)) ≤



[u(t) + y(t)]

−θ

(t)R

u(t)



dt.

Hence, with the storage function V

(x) =

V (x), G is dissipative with respect

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

y +

1−θ

u + y

y. Now, the result

is a direct consequence of Corollary 6.2 and Deﬁnitions 6.4 and 6.3 with

α =

1+θ

and β =

1−θ

Next, we consider the case in which R

(x), x ∈ R

, is not a diagonal

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534 CHAPTER 8

constant m atrix. For the following result deﬁne

△

= sup

x∈R

max

(x)), γ

△

= inf

x∈R

min

(x)), (8.121)

where R

(x) is such that γ < ∞ an d γ > 0.

Theorem 8.6. Consider the n onlinear dynamical system G given by

(8.100) and (8.101) where φ(x) is a stabilizing feedb ack control law given by

(8.110) and where V (x), x ∈ R

, satisﬁes (8.107). If there exists θ ∈ R such

that 0 < θ < 1 and (8.119) is satisﬁed, then the nonlinear system G has a

disk margin (

1+ηθ

1−ηθ

), where η

△

γ/γ.

Proof. Note that for all u(·) ∈ U and t

, t

≥ 0, t

< t

, it follows

from Lemma 8.2 that the solution x(t), t ≥ 0, to (8.100) satisﬁes

V (x(t

)) − V (x(t

)) ≤



[u(t) + y(t)]

(x(t))[u(t) + y(t)]

−θ

(t)R

(x(t))u(t)



dt,

which implies that

V (x(t

)) − V (x(t

)) ≤



γ[u(t) + y(t)]

[u(t) + y(t)] −γθ

(t)u(t)



dt.

Hence, w ith the storage function V

(x) =

2γ

V (x), G is dissipative with

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

y +

1−η

u + y

y. Now, the result

is a direct consequence of Corollary 6.2 and Deﬁnition 6.3 with α =

1+ηθ

and β =

1−ηθ

Next, we provide an alternative result that guarantees sector and gain

margins for the case in which R

(x), x ∈ R

, is diagonal.

Theorem 8.7. Consider the n onlinear dynamical system G given by

(8.100) and (8.101) where φ(x) is a stabilizing feedback control law given

by (8.110) and where V (x), x ∈ R

, satisﬁes (8.107). Furth ermore, let

(x) = diag [r

(x), . . . , r

(x)], where r

: R

→ R, r

(x) > 0, i = 1, . . . , m.

If G is zero-state observable and there exists θ ∈ R such that 0 < θ < 1 and

(1 − θ

(x) −

(x)R

−1

(x)L

(x) ≥ 0, x ∈ R

, (8.122)

then the nonlinear system G has a sector (and, hence, gain) margin (

1+θ

1−θ

Proof. Let ∆(−y) = σ(−y), where σ : R

→ R

is a static

nonlinearity such that σ(0) = 0, σ(v) = [σ

), . . . , σ

)]

, and αv

< βv

, for all v

6= 0, i = 1, . . . , m, where α =

1+θ

and β =

1−θ

;