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

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NonlinearBook10pt November 20, 2007

DISTURBANCE REJECTION CONTROL 645

0 =

(ˆx)V

′T

(ˆx) −

h(ˆx), (10.264)

and the subsystem (10.259) has a globally stable equilibrium at x = 0 with

y = 0 and Lyapunov function V

sub

(x) so that

′

sub

(x)f(x) < 0, x ∈ R

, x 6= 0. (10.265)

Furth ermore, assume that there exists a function L

: R

×R

→ R

1×m

such

that L

(0, 0) = 0 and

′

sub

(x)f(x) +

4γ

′

sub

(x)J

(x)V

′T

sub

(x) < 0, x ∈ R

, x 6= 0,

(x)V

′T

sub

(x) −

−1

(x, ˆx){L

(x, ˆx)

′

(ˆx)

G(ˆx) + 2h

(x, ˆx)J(x, ˆx)}

− k(ˆx)]

+[h(x, ˆx) + J(x, ˆx)φ(x, ˆx)]

[h(x, ˆx) + J(x, ˆx)φ(x, ˆx)]

4γ

′

(ˆx)J

(ˆx)V

′T

(ˆx) ≤ 0, (x, ˆx) ∈ R

× R

, (10.266)

where R

(x, ˆx)

△

= R

(x, ˆx) + J

(x, ˆx)J(x, ˆx), φ(x, ˆx) = −

−1

(x, ˆx)[L

(x,

ˆx)+ V

′

(ˆx)

G(ˆx)+2h

(x, ˆx)J(x, ˆx)]

, and γ > 0. Show th at the zero solution

(x(t), ˆx(t)) ≡ (0, 0) of the undisturbed (w(t) ≡ 0) cascade system

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

, t ≥ 0, (10.267)

ˆx(t) =

f(ˆx) +

G(ˆx)φ(x(t), ˆx(t)), ˆx(0) = ˆx

, (10.268)

y(t) =

h(ˆx), (10.269)

is globally asymptotically stable with the feedback control law

φ(x, ˆx) = −

−1

(x, ˆx)[L

(x, ˆx) + V

′

(ˆx)

G(ˆx) + 2h

(x, ˆx)J(x, ˆx)]

(10.270)

Furth ermore, show that the performance functional (10.235) satisﬁes

J(x

, ˆx

, φ(x, ˆx)) ≤ J(x

, ˆx

, φ(x, ˆx)) = V (x

, ˆx

), (x

, ˆx

) ∈ R

× R

(10.271)

where

J(x

, ˆx

, u(·))

△

∞

[L(x(t), ˆx(t), u(t)) + Γ(x(t), ˆx(t), u(t))]dt, (10.272)

V (x, ˆx) = V

sub

(x) + V

(ˆx), (10.273)

and

Γ(x, ˆx, u) =

4γ

′

(x, ˆx)J

(x, ˆx)V

′T

(x, ˆx)

+[h(x, ˆx) + J(x, ˆx)u]

[h(x, ˆx) + J(x, ˆx)u], (10.274)

where J

(x, ˆx)

△

= [J

(x), J

(ˆx)]

and (x(t), ˆx(t)), t ≥ 0, solves (10.259) and

(10.260) with w(t) ≡ 0. In addition, show that the performance functional

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646 CHAPTER 10

(10.272), with

(x, ˆx)

= φ

(x, ˆx)R

(x, ˆx)φ(x, ˆx) −V

′

(ˆx)

f(ˆx) −V

′

sub

(x)(f(x) + G(x)

h(ˆx))

−h

(x, ˆx)h(x, ˆx) −

4γ

′

(x, ˆx)J

(x, ˆx)V

′T

(x, ˆx), (10.275)

is minimized in the sense that

J(x

, ˆx

, φ(x(·))) = min

u(·)∈S(x

,ˆx

)

J(x

, ˆx

, u(·)). (10.276)

With u(·) = φ(x(·), ˆx(·)), sh ow that the s olution (x(t), ˆx(t)), t ≥ 0, of the

cascade system (10.259) and (10.260) satisﬁes the nonexpansivity constraint

(t)z(t)dt ≤ γ

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

, ˆx

), w(·) ∈ L

, T ≥ 0.

(10.277)

Finally, for J(x, ˆx) ≡ 0, h(x, ˆx) = E(x, ˆx)

h(x), and J

(x) =

G(ˆx), wh ere

E : R

× R

→ R

p×m

, show that a particular choice satisfying (10.266) is

given by

(x, ˆx) = 2[G

(x)V

′T

sub

(x) +

4γ

h(ˆx)

(x, ˆx)E(x, ˆx)

h(ˆx) −k(ˆx)]

(x, ˆx),

yielding a feedback control law

φ(x, ˆx) = k(ˆx) −

−1

(x, ˆx)

(ˆx)V

′T

(ˆx) − V

′

sub

(x)G(x) −

4γ

h(ˆx)

−E

(x, ˆx)E(x, ˆx)

h(ˆx).

10.10 Notes and References

The distu rbance rejection problem for analysis and feedback control design

can be traced back to th e 1960s and early 1970s with the work of Cru z

[100] and Frank [124] giving a textb ook treatment on the subject. During

this period, the role of diﬀerential game theory was also recognized as a

framework for disturbance rejection control; see for example Dorato and

Drenick [108], Witsenhausen [464], and Salmon [385]. In the 1980s the

disturbance rejection problem was formulated in the frequency d omain by

the pioneering work of Zames [478] using H

∞

theory.

For linear multivariable systems, the H

∞

disturbance rejection control

problem was formulated in the fr equency domain by Francis, Helton, and

Zames [123], Chang and Pearson [82], Francis and Doyle [122], Francis [121],

and Safonov, J on ckheere, Verma, and Limebeer [383]. A state-space Riccati

equation appr oach to the H

∞

control problem was given by Petersen [353],

Glover and Doyle [137], and Doyle, Glover, Khargonekar, and Francis [111].

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DISTURBANCE REJECTION CONTROL 647

See also Khargonekar, Petersen, and Rotea [236]. The mixed-norm H

∞

control problem was ﬁrst formulated by Bernstein and Hadd ad [49] and

Haddad and Bernstein [144].

A nonlinear game-theoretic framework for the “nonlinear” H

∞

control

problem is given by Ball and Helton [22] with a textbook treatment

given in Basar and Bernhard [31]. In parallel research, Isidori [213, 214],

Isidori and Astolﬁ [215, 216], and van d er Schaft [438–441] addressed the

“nonlinear” H

∞

control problem using dissipativity theory. The optimality-

based nonlinear disturbance rejection framework presented in this chapter

is adopted from Haddad and Chellaboina [159] and Haddad, Chellaboina,

and Fausz [163].

NonlinearBook10pt November 20, 2007

Chapter Eleven

Robust Control for Nonlinear Uncertain

Systems

11.1 Introduction

Unavoidable discrepancies between system models and real-world sys tems

can result in degradation of control-system performance including instability

[109,410]. Thus, it is not su rprising that on e of the fund amental problems

in feedback control design is the ability of the control system to guarantee

robustness with respect to system uncertainties in the design model.

Although the theory of linear robust control is highly mature, nonlinear

robust control techniques remain relatively undeveloped. Traditionally,

Lyapunov function theory along with Hamilton-Jacobi-Bellman theory have

been instrumental in advancing nonlinear control theory by addr essin g

control system stability and optimality. Unfortunately, however, there does

not exist a uniﬁed procedure for ﬁnding Lyapunov function candidates that

will stabilize closed-loop nonlinear systems. Fu rthermore, computational

methods for establishing the existence of solutions to Hamilton-Jacobi-

Bellman equations involving highly complex nonlinear partial diﬀerential

equations have not been developed to a level comparable to the existence of

solutions of Riccati equations arising in linear optimal control problems.

These problems are further exacerbated w hen addressing robustness in

uncertain nonlinear systems.

Diﬀerential geometric methods [212,336,445] have made the design of

controllers for certain classes of nonlinear systems more methodical. These

methods are predicated on feedback linearization, or d y namic inversion,

wherein state feedback along with coordinate transformations are used to

transform a class of nonlinear systems to a linear time-invariant system. In

this case, the resulting linear system can be stabilized using standard linear-

quadratic design techniques. However, a serious drawback of all feedback

linearization techniques is the failure to account for system uncertainty since

exact cancelation of the nonlinear dynamics via feedback is required, and

hence, an exact knowledge of the dynamics is assumed resulting in non -

NonlinearBook10pt November 20, 2007

650 CHAPTER 11

robust designs. To see this, consider the scalar nonlinear system

˙x(t) = x

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

, t ≥ 0. (11.1)

Clearly, a feedback linearizing control law for this system is given by u

−x

− x resulting in the globally asymptotically stable closed-loop system

˙x(t) = −x(t). Now, suppose that (11.1) possesses an in put uncertainty so

that (11.1) is not valid. Rather, in place of (11.1) a more accurate model is

given by

˙x(t) = x

(t) + (1 + ε)u(t), x(0) = x

, t ≥ 0, (11.2)

where ε 6= 0. Using the feedback linearizing control law u

= − x

− x on

the actual model yields

˙x(t) = −(1 + ε)x(t) − εx

(t), x(0) = x

, t ≥ 0. (11.3)

It can be easily shown that for every ε 6= 0 there exists an initial condition

such that the solution to (11.3) has a ﬁnite escape time.

Even though robustness frameworks to parametric uncertainty via

feedback linearization techniques involving a two stage design consisting

of nominal feedback linearization followed by additional state feedback

designed to guarantee robustness have been developed [386,404,413,414], the

fact that such approaches do not directly account for system uncertainties

can result in severe robustness problems with respect to nonlinear errors

internal to the system dynamics. Furthermore, restrictive matching

conditions are imposed to the structure of the uncertainty in order to address

general f eedback linearizable systems [412]. Finally, these techniques often

lead to in eﬃcient designs since feedback linearizing controllers may generate

unnecessarily large control eﬀort to cancel beneﬁcial system nonlinearities

[127,247].

In this chapter, we extend the framework presented in Chapter 8

to develop an optimality-based framework for addressing the problem of

nonlinear-nonquadratic optimal control for uncertain nonlinear systems

with structured parametric uncertainty. Speciﬁcally, using a Lyapunov

bounding framework, the robust nonlinear control problem is transformed

into an optimal control problem by modifying a nonlinear-nonquadratic cost

functional to account for system uncertainty. Furth ermore, it is shown that

the Lyapunov function guaranteeing closed-loop robust stability is a solution

to the steady-state Hamilton-Jacobi-Bellman equation for the controlled

nominal system.

The main f ocus of this chapter is to develop a methodology for

designing nonlinear controllers which provide both robust stability and

robust performance over a prescribed range of system uncertainty. The

present framework extends the guaranteed cost control approach [50, 83]

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ROBUST NONLINEAR CONTROL 651

to n on linear systems by utilizing a performance bound to provide robust

performance in addition to robust stability. In particular, the performance

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 robust stability over a prescribed

uncertainty s et. This L yapunov function is shown to be the solution to the

steady-state form of the Hamilton-Jacobi-Bellman equation for the nominal

system and plays a key role in constructing the optimal nonlinear robust

control law. Hence, the overall framework provides for a generalization of

the Hamilton-Jacobi-Bellman conditions for addressing the design of robust

optimal controllers for nonlinear uncertain s ystems.

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

suﬃcient Hamilton-Jacobi-Bellman optimality conditions are obtained for

a modiﬁed nonlinear-nonquadratic performance functional rather than the

original performance functional, globally optimal controllers are guaranteed

to provide both robust stability and performance. Of cours e, since our

approach allows us to construct globally optimal controllers that min imize

a given Hamiltonian, the resulting robust nonlinear controllers p rovide the

best worst-case performance over the robust stability range.

11.2 Robust Stability Analysis of Non linear Uncertain Systems

In this s ection, we present suﬃcient conditions for robust stability f or a

class of nonlinear un certain systems. Speciﬁcally, we extend the analysis

framework of Chapter 8 in order to address robust stability of a class of

nonlinear uncertain systems. In the pr esent framework we consider the

problem of evaluating a performance bound for a nonlinear-nonquadratic

cost functional depending upon a class of nonlinear uncertain systems. It

turns out that the cost bound can be evaluated in closed form as long as

the cost functional is related in a speciﬁc way to an underlying L yapunov

function that guarantees robust stability over a prescribed uncertainty set.

Hence, the overall framework provides f or robust stability and performance

where robust performance here refers to a guaranteed bound on the worst-

case value of a nonlinear-nonquadratic cost criterion over a prescribed

uncertainty s et.

Once again we restrict our attention to time-invariant inﬁnite horizon

systems. Furthermore, for the class of nonlinear uncertain systems consid-

ered 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 ⊂ {f : D → R

: f (0) = 0}

denote the class of uncertain nonlinear systems with f

(·) ∈ F deﬁning the

nominal nonlinear system. Within the context of robustness analysis, it is

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652 CHAPTER 11

assumed that the zero solution x(t) ≡ 0 of the nominal nonlinear dynamical

system ˙x(t) = f

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

, is asymptotically stable.

Theorem 11.1. Consider the nonlinear uncertain dynamical system

˙x(t) = f (x(t)), x(0) = x

, t ≥ 0, (11.4)

where f (·) ∈ F, with performance functional

)

△

∞

L(x(t))dt. (11.5)

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

where V (·) is a continuously diﬀerentiable function, such that

V (0) = 0, (11.6)

V (x) > 0, x ∈ D, x 6= 0, (11.7)

′

(x)f(x) ≤ V

′

(x)f

(x) + Γ(x), x ∈ D, f(·) ∈ F,

(11.8)

′

(x)f

(x) + Γ(x) < 0, x ∈ D, x 6= 0, (11.9)

L(x) + V

′

(x)f

(x) + Γ(x) = 0, x ∈ D, (11.10)

where f

(·) ∈ F deﬁnes the nominal nonlinear system. Then there exists

a neighborhood of the origin D

⊆ D such that if x

∈ D

, then the zero

solution x(t) ≡ 0 to (11.4) is locally asymptotically stable for all f (·) ∈ F,

and

sup

f(·)∈F

) ≤ J(x

) = V (x

), (11.11)

where

J(x

)

△

∞

[L(x(t)) + Γ(x(t))]dt, (11.12)

and where x(t), t ≥ 0, is the solution to (11.4) with f (x(t)) = f

(x(t)).

Finally, if D = R

and

V (x) → ∞ as kxk → ∞, (11.13)

then the zero solution x(t) ≡ 0 to (11.4) is globally asymptotically stable

for all f(·) ∈ F.

Proof. Let f (·) ∈ F and x(t), t ≥ 0, satisfy (11.4). Then

V (x(t))

△

V (x(t)) = V

′

(x(t))f(x(t)), t ≥ 0. (11.14)

Hence, it follows from (11.8) and (11.9) that

V (x(t)) < 0, t ≥ 0, x(t) 6= 0. (11.15)

Thus , from (11.6), (11.7), and (11.15) it follows that V (·) is a L yapunov

function for (11.4), which proves local asymptotic stability of the zero

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ROBUST NONLINEAR CONTROL 653

solution x(t) ≡ 0 to (11.4) for all f (·) ∈ F. Consequently, x(t) → 0 as

t → ∞ for all in itial cond itions x

∈ D

for some neighborhood of the origin

⊆ D. Now, (11.14) implies that

0 = −

V (x(t)) + V

′

(x(t))f(x(t)), t ≥ 0,

and h en ce, using (11.8) and (11.10),

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 f or all x

∈ D

yields J

) ≤

V (x

Next, let x(t), t ≥ 0, satisfy (11.4) w ith f (x(t)) = f

(x(t)). Then,

with L(x) replaced by L(x) + Γ(x) and J(x

) rep laced by J(x

) it follows

from Theorem 8.1 that J(x

) = V (x

). Finally, for D = R

and for all

f(·) ∈ F, global asymptotic stability of the zero solution x(t) ≡ 0 to (11.4)

is a direct consequence of the radially unbounded condition (11.13) on V (x),

x ∈ R

Theorem 11.1 is an extension of Theorem 8.1. Note that conditions

(11.6) and (11.7) ensure that V (x) is a Lyapunov function candidate for

the nonlinear uncertain system (11.4). Conditions (11.8) and (11.9) imply

V (x(t)) < 0, t ≥ 0, for x(·) satisfying (11.4) for all f(·) ∈ F, and hence,

V (·) is a Lyapunov f unction guaranteeing robust stability of the nonlinear

uncertain system (11.4). It is important to note that condition (11.9)

is a veriﬁable condition since it is independent of the uncertain system

parameters f(·) ∈ F. To apply Theorem 11.1 we specify a bounding function

Γ(·) for the uncertain set F such that Γ(·) bounds F (see Propositions 11.1

and 11.2). Also note that if F consists of only the nominal nonlinear system

(·), then Γ(x) ≡ 0 satisﬁes (11.8), and h en ce, J

) = J(x

). In this

case, Theorem 11.1 specializes to Theorem 8.1. The key feature of Theorem

11.1 is that it provides suﬃcient conditions for robust stability of a class

of nonlinear uncertain systems f (·) ∈ F. Furthermore, an up per bound to

the nonlinear-nonquadratic performance functional is given in terms of a

Lyapunov function which can be interpreted in terms of an auxiliary cost

deﬁned for the nominal system in the spirit of [48,50, 51].

If Γ(·) bou nds F then clearly Γ (·) bounds the convex hull of F. Hence,

only convex uncertainty sets F need be considered. Next, we use the obvious

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654 CHAPTER 11

fact that if Γ

(·) bounds F

and Γ

(·) bounds F

, then Γ

(·) + Γ

(·) bounds

+ F

. Hence, if F can be decomposed additively then it suﬃces to

bound each compon ent separately. Finally, if Γ(·) bounds F and there exists

Γ : D → R such that Γ(x) ≤

Γ(x) for all x ∈ R

, then

Γ(·) boun ds F. That

is any overbound

Γ(·) for Γ(·) also bounds F. Of course, it is quite possible

that an overbound

Γ(·) for Γ(·) may actually bound a set

F that is larger

than the “original” uncertainty set F.

Next, we specialize Theorem 11.1 to nonlinear uncertain systems of

the form

˙x(t) = f

(x(t)) + ∆f(x(t)), x(0) = x

, t ≥ 0, (11.16)

where f

: D → R

satisﬁes f

(0) = 0 and f

+ ∆f ∈ F. Here, F is such

that

F ⊂ {f

+ ∆f : D → R

: ∆f ∈ ∆}, (11.17)

where ∆ is a given nonlinear uncertainty set of nonlinear perturbations ∆ f

of the nominal system dyn amics f

(·) ∈ F. Since F ⊂ {f : D → R

f(0) = 0} it follows that ∆f (0) = 0 for all ∆f ∈ ∆.

Corollary 11. 1. Consider the nonlinear uncertain dynamical system

(11.16) with performance functional (11.5). Furthermore, assume that there

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

diﬀerentiable function, such th at (11.6) an d (11.7) hold and

′

(x)∆f(x) ≤ Γ(x), x ∈ D, ∆f(·) ∈ ∆, (11.18)

′

(x)f

(x) + Γ(x) < 0, x ∈ D, x 6= 0, (11.19)

L(x) + V

′

(x)f

(x) + Γ(x) = 0, x ∈ D. (11.20)

Then the zero solution x(t) ≡ 0 to (11.16) is locally asymptotically stable

for all ∆f (·) ∈ ∆ and there exists a neighborhood of the origin D

⊆ D

such that the performance functional (11.5) satisﬁes

sup

∆f(·)∈∆

∆f

) ≤ J(x

) = V (x

), x

∈ D

, (11.21)

where

J(x

) =

∞

[L(x(t)) + Γ(x(t))]dt, (11.22)

and where x(t), t ≥ 0, is the solution to (11.16) with ∆ f (x) ≡ 0. Finally, if

D = R

, and V (x), x ∈ R

, satisﬁes (11.13), then the zero solution x(t) ≡ 0

to (11.16) is globally asymptotically stable for all ∆f(·) ∈ ∆.

Proof. The result is a direct consequence of T heorem 11.1 with f (x) =

(x) + ∆f (x). Speciﬁcally, in this case, it follows from (11.18) and (11.19)

that V

′

(x)f(x) ≤ V

′

(x)f

(x) + Γ(x) for all x ∈ D and ∆f (·) ∈ ∆. Hence,

all the conditions of Th eorem 11.1 are satisﬁed.