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

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STRUCTURED PARAMETRIC UNCERTAINTY 755

Then, with the feedback control law u = φ(x)

△

= − R

−1

x +

′

(x)),

where g(x) =

ν=2

[2ν]

, the zero solution x(t) ≡ 0 of the uncertain

system (12.162) is globally asymptotically stable for all x

∈ R

and ∆A ∈

∆

, and the performance functional (12.144) satisﬁes

∆A

, φ(x(·))) ≤ J(x

, φ(x(·))) = x

(P + P

(∆A)x

k=2

[2ν]

∆A ∈ ∆

, (12.167)

where

J(x

, u(·))

△

∞

[L(x, u) + Γ(x, u) −x

(t)(A

(∆A) + P

(∆A)A)x(t)]dt,

(12.168)

and where u(·) is admissible, and x(t), t ≥ 0, solves (12.162) with ∆A = 0

and

Γ(x, u) = x

Ω

(P )x + 2x

Ω

(P )u + u

Ω

(P )u +

ν=2

Ω

(

[2ν]

where u(·) is admissible, ∆A ∈ ∆

. In addition, the performance functional

(12.144), with R

(x) = R

and

L(x, u) = x

x +

ν=2

2ν

[2ν]

′

(x)BR

−1

′

(x)

is minimized in the sense that

J(x

, φ(x(·)) = min

u(·)∈S(x

)

J(x

, u(·)), (12.169)

where S(x

) is the set of regulation controllers for the nominal system and

∈ R

Proof. The result is a direct consequence of Corollary 12.4 with

(x) = Ax, ∆f(x) = ∆Ax, G

(x) = B, V

(x) = x

P x +

ν=2

[2ν]

and V

∆f

(x) = x

(∆A)x.

12.8 Problems

Problem 12.1. Consider the subset ∆ ⊆ ∆

consisting of sector-

bounded matrices

∆

△

= {∆ ∈ ∆

: 2(∆−M

)

∗

−M

)

−1

(∆−M

) ≤ (∆−M

)+(∆−M

)

∗

(12.170)

where M

, M

∈ ∆

are Hermitian matrices such that M

△

= M

− M

positive deﬁnite. Let ∆ ∈ ∆

. Show that the following statements are

equivalent:

NonlinearBook10pt November 20, 2007

756 CHAPTER 12

i) ∆ ∈ ∆.

ii)



He ∆ −M

∆

∗

− M

∆ − M



≥ 0.

iii)



He ∆ −M

∆ − M

∆

∗

− M



≥ 0.

iv)



He ∆ −M Sh ∆

−Sh ∆ M

− He ∆



≥ 0.

If, in addition, det(M

− He ∆) 6= 0, show that

v) M

−Sh ∆(M

−He ∆)

−1

Sh ∆ ≤ He ∆ ≤ M

, is equivalent to i)–iv).

Furth ermore, show that if ∆ ∈ ∆, then the following statements hold:

vi) M

≤ He ∆ ≤ M

vii) σ

max

(∆) ≤ σ

max

(M) + σ

max

Finally, show that ∆ ∈ ∆ if and only if ∆

∗

∈ ∆ and in the case ∆ = ∆

∗

for

all ∆ ∈ ∆ then ∆ ∈ ∆ if and only if M

≤ ∆ ≤ M

Problem 12.2. Let Z(·) ∈ Z, let ω ∈ R ∪ ∞, and sup pose det(I +

G(ω)M

) 6= 0. Show that if (12.17) holds, then det(I + G(ω)∆) 6= 0 for

all ∆ ∈ ∆, where ∆ is given by (12.170).

Problem 12.3. Consider ∆

given by (12.24) with r = 0. Let

G(ω) ∈ C

m×m

be such that G(ω)∆ = ∆G(ω) for all ∆ ∈ ∆

. Show

that µ(G(ω) = ρ(G(ω)), where ρ(·) denotes th e spectral radius.

Problem 12.4. Let G(ω) ∈ C

m×m

. Show that

ρ(G(ω)) = inf

D∈D

max

(DG(ω)D

−1

where D

△

= {D ∈ C

m×m

: det D 6= 0}. Using this result show th at for

∆

= {∆ ∈ C

m×m

: ∆ = δI

, δ ∈ C}, µ(G(ω)) given by (12.29) is

nonconservative.

Problem 12.5. Let G(ω) ∈ C

m×m

be such that G(ω) ≥ 0, ω ∈ R.

Show that µ(G(ω)) = inf

D∈D

max

(DG(ω)D

−1

) for every D.

NonlinearBook10pt November 20, 2007

STRUCTURED PARAMETRIC UNCERTAINTY 757

Problem 12.6. Let ω ∈ R and deﬁne

(G(ω))

△



max{|λ| : λ ∈ spec(G(ω)) ∩ R}, if spec(G(ω)) ∩ R 6= Ø,

0, otherwise.

Show that ρ

(G(ω)) ≤ µ(G(ω)). Furthermore, s how that if r = 0 in ∆

then ρ(G(ω)) ≤ µ(G(ω)).

Problem 12.7. Let ∆ ∈ ∆

, where

∆

= {∆ ∈ R

m×m

: ∆ = diag[δ

, δ

, . . . , δ

], δ

∈ R, i = 1, . . . , m}.

(12.171)

Deﬁne the frequency-dependent scaling matrix functions in D and N corre-

spondin g to small gain (D

, N

), Popov (D

, N

) [151, 172], monotonic

, N

) [172, 328, 331], odd-monotonic (D

RLC

, N

RLC

) [172, 328, 331],

generalized odd-monotonic (D

GRLC

, N

GRLC

) [425, 428], and LC multipliers

, N

) [71], respectively, by

△

= {D ∈ R

m×m

: D = I},

△

= {N ∈ R

m×m

: N = 0},

△

= {D ∈ R

m×m

: D = diag(α

), 0 < α

∈ R, i = 1, . . . , m},

△

= {N : R → R

m×m

: N (ω) = diag(−ωβ

), β

∈ R, i = 1, . . . , m},

△

= {D : R → R

m×m

: D(ω) = D

+ diag



j=1

(1 −

(ω

+η

)



0 ≤ α

, β

, η

∈ R, η

− α

≤ 0,

i = 1, . . . , m; j = 1, . . . , m

, D

∈ D

△

= {N : R → R

m×m

: N (ω) = N

(ω) + diag



j=1

−

(ω

+η

)



0 ≤ α

, β

, η

∈ R, η

− α

≤ 0,

i = 1, . . . , m; j = 1, . . . , m

, N

(·) ∈ N

RLC

△

= {D : R → R

m×m

: D(ω) = diag



j=m

(1 +

(ω

+η

)



(ω), 0 ≤ α

, β

, η

∈ R,

i = 1, . . . , m; j = m

+ 1, . . . , m

, D

(·) ∈ D

RLC

△

= {N : R → R

m×m

: N(ω) = diag



j=m

(ω

+η

)



(ω), 0 ≤ α

, β

, η

∈ R,

i = 1, . . . , m; j = m

+ 1, . . . , m

, N

(·) ∈ N

GRLC

△

= {D : R → R

m×m

: D(ω) = D

RLC

(ω)

NonlinearBook10pt November 20, 2007

758 CHAPTER 12

+ diag



j=m

+ω

−b

−η

)+b

(η

−ω

)

+λ



∈ R, 0 ≤ α

, β

, η

∈ R, a

− b

− η

≥ 0,

i = 1, . . . , m; j = m

+ 1, . . . , m

, D

RLC

(·) ∈ D

RLC

GRLC

△

= {N : R → R

m×m

: N (ω) = N

RLC

(ω)

+ diag



j=m

−ωα

−λ

)+(a

−λ

)

(η

−ω

)

+λ



∈ R, 0 ≤ α

, β

, η

∈ R, a

− b

− η

≥ 0,

i = 1, . . . , m; j = m

+ 1, . . . , m

, N

RLC

(·) ∈ N

RLC

△

= {D ∈ R

m×m

: D = 0},

△

= {N : R → R

m×m

: N (ω) = diag(∓ω

±1

Π(α

−ω

)

Π(β

−ω

)

, β

∈ R, β

6= 0, i = 1, . . . , m}.

Show that for G(ω)) ∈ C

m×m

µ(G(ω)) ≤ µ(G(ω)) ≤ µ

GRLC

(G(ω)) ≤ µ

RLC

(G(ω))

≤ µ

(G(ω)) ≤ µ

(G(ω)), (12.172)

where µ

(G(ω)), for i = GRLC , RLC, P, and sg, correspond s to µ bounds

predicated on ﬁxed frequency-dependent scaling functions D

and N

. What

can you say about µ

(G(ω)) with respect to the above ordering?

Problem 12.8. Let β > 0, G(ω) ∈ C

m×m

, D(·) ∈ D, N (·) ∈ N, and

deﬁne

X(D, N, β)

△

= G

∗

(ω)DG(ω) + (NG(ω) − G

∗

(ω)N) − βD,

Φ(D, N)

△

= inf{η > 0 : X(D, N, η) < 0}.

Show that th ere exist D(·) ∈ D and N(·) ∈ N such that Φ(D, N) < η if and

only if X(D, N, η) < 0.

Problem 12.9. A functional f on a vector space H is quasiconvex if,

for all α ∈ [0, 1] and H,

H ∈ H, f (αH + (1 − α)

H) ≤ max{f(H), f(

H)}.

Show that Φ(D, N) deﬁned by (12.173) is quasiconvex on D × N.

Problem 12.10. Consider the linear uncertain system (12.74) where

∆A ∈ ∆

and ∆

is given by

∆

△

= {∆A ∈ R

n×n

: ∆ A = B

F C

, M

≤ F ≤ M

}. (12.173)

Let X ∈ R

p×p

and Y ∈ N

be s uch that

(F − M

+ C

(F −M

)XB

≤ Y, (12.174)

NonlinearBook10pt November 20, 2007

STRUCTURED PARAMETRIC UNCERTAINTY 759

and let H ∈ H

and N ∈ N

be s uch that

△

= [HM

−1

−NC

] + [HM

−1

− N C

]

> 0, (12.175)

where M

△

= M

− M

. Show that the functions

Ω

(P ) = [HC

+ N C

(A + B

) + B

P − XB

]

−1

[HC

+ N C

) + B

P − XB

] + P B

+ C

P + Y,

(12.176)

(F ) = C

(F − M

)NC

, (12.177)

satisfy (12.76) with ∆A given by (12.173). Furthermore, show th at

Y = B

MXB

+ C

, (12.178)

satisﬁes (12.174) for all X ∈ R

p×p

and F ∈ S

such that M

≤ F ≤ M

Finally, for the special case of diagonal uncertainty F show that Y =

+ C

also satisﬁes (12.174).

Problem 12.11. Consider the linear uncertain system (12.74) where

∆A ∈ ∆

and ∆

is given by (12.173). Let X ∈ R

p×p

, Y ∈ N

, and

H ∈ H

be s uch that

(F − M

)HC

+ C

H(F − M

)XB

≤ Y, (12.179)

and let N ∈ N

be s uch that (12.175) holds. Show that the fu nctions

Ω

(P ) = [HC

+ N C

(A + B

) + H

−1

P − XB

]

−1

[HC

+NC

(A + B

) + H

−1

P − XB

]

+P B

+ C

P + Y, (12.180)

(F ) = C

(F − M

)NC

, (12.181)

satisfy (12.76) with ∆A given by (12.173).

Problem 12.12. Consider the linear uncertain system (12.122) with

∆A ∈ ∆

, where ∆

is given by (12.173), and the performance functional

(12.123). Let H ∈ H

and N ∈ N

be such that (12.175) holds, and let

X ∈ R

p×p

and Y ∈ N

be such that (12.174) is satisﬁed. Show that the

zero solution x(t) ≡ 0 to (12.122) is globally asymptotically stable for all

∆A ∈ ∆

with the feedback control φ(x) = − R

−1

x, where

△

= R

+ B

−1

△

= B

P + B

−1

[HC

+ N C

(A + B

) + B

P − XB

and P > 0 satisﬁes

0 = A

P + P A

+ R

+ Y + [HC

+ NC

(A + B

) − XB

]

−1

·[HC

+ N C

(A + B

) − XB

] + P B

−1

P − P

−1

NonlinearBook10pt November 20, 2007

760 CHAPTER 12

where A

△

= A + B

+ B

−1

[HC

+ N C

(A + B

) −XB

Problem 12.13. Consider the linear dynamical system with state

delay given by

˙x(t) = Ax(t) + A

x(t −τ

), x(θ) = φ(θ), −τ

≤ θ ≤ 0, (12.182)

where x(t) ∈ R

, A ∈ R

n×n

, A

∈ R

n×n

, and φ : [−τ

, 0] → R

is a

continuous vector-valued function specifying the initial state of the system.

Show that th e zero solution x

≡ 0 to (12.182) is globally asymptotically

stable (in the sense of Problem 3.65) for all τ

∈ [0, τ ] if |||DG(s)D

−1

|||

∞

< 1,

where

G(s) ∼





A + QA

τQ (I

− Q)A

A 0 A

0 0





D = diag(D

, D

), D

> 0, D

> 0, and Q ∈ R

n×n

. Show that the result

also holds if G(s) is replaced by

G(s) ∼





A + A

τA

A 0 0

0 0





Finally, show that in both cases the problem can be represented as a

feedback pr oblem involving an uncertain block-structured operator ∆(s) =

diag(∆

(s), ∆

(s)) satisfying |||∆

(s)|||

∞

≤ 1, i = 1, 2.

12.9 Notes and References

The eﬃcacy of parameter-dependent Lyapunov f unctions for nonconser-

vatively addressing real parameter uncertainty was ﬁrs t conjectured by

Narendra [325] and later rigorously proven by Th athachar and Srinath

[426]. Speciﬁcally, Thathachar and Srin ath [426] constructed a parameter-

dependent Lyapunov function for proving necessary and suﬃcient conditions

for robust stability of a linear time-invariant system with a single constant

uncertainty. The proof of this fundamental result stemmed from absolute

stability theory [5, 64, 65, 67, 70, 71, 94, 326, 328–331, 361, 362, 364, 425, 427,

428, 458, 477, 479] and was based on the fact that if the uncertain system

is robustly stable, then there always exists a plant-dependent inductor-

capacitor multiplier s uch that th e tandem connection of the nominal plant

and multiplier is positive real. For further details on this f act see Narendra

and Taylor [331] and How and Haddad [204]. Building on the wealth of

knowledge of absolute stability theory, Haddad and Bernstein [147–149,151,

152] uniﬁed and extended classical absolute stability theory for linear time-

invariant dynamical systems with loop nonlinearities to address th e modern-

day robust analysis and synthesis problem via parameter-depend ent Lya-

punov functions. Potential advantages of parameter-dependent Lyapunov

NonlinearBook10pt November 20, 2007

STRUCTURED PARAMETRIC UNCERTAINTY 761

functions over “ﬁxed” Lyapunov functions were also discussed by Barmish

and DeMarco [27] and Leal and Gibson [261].

Connections between absolute stability theory and mixed-µ bounds for

real parameter uncertainty were ﬁrst reported by Haddad, How, Hall, and

Bernstein [170–172] and How and Hall [206], with further extensions given

by Safonov and Chiang [381] and Haddad, Bern stein, and Chellaboina [153].

A direct beneﬁt of this un iﬁcation resulted in new machinery for mixed-µ

controller synthesis by providing an alternative to the standard multiplier-

controller iteration and curve-ﬁtting procedure [97,203, 205,207, 381].

Wong [465] was the ﬁrst to consider the robustness pr oblem with

structured uncertainty, laying down the framework for multivariable stability

and µ theory. Doyle [110] was the ﬁrst to introdu ce the term structured

singular value. Bu ilding on th e work of Wong [465], Safonov and Athans

[380] and Safonov [378] addressed stability margins of diagonally perturbed

multivariable feedback systems and introduced the notion of excess stability

margin; the r eciprocal of the structured singular value. The mixed-µ bounds

for diagonal real parameters are due to Fan, Tits, and Doyle [117] while

the mixed-µ bounds f or real and complex multiple-block uncertainty with

internal matrix structure are due to Haddad, Bernstein, and Chellaboina

[153]. For a complete historical perspective on the structured singular value

the reader is referred to the editorial by Safonov and Fan [382].

Finally, the presentation of the str uctured singular value in this

chapter is adopted from Haddad, Bernstein, and Chellaboina [153], while the

presentation of the robust nonlinear-nonquadratic feedback control prob lem

via parameter-dependent Lyapunov functions is adop ted from Haddad and

Chellaboina [156].

NonlinearBook10pt November 20, 2007

Chapter Thirteen

Stability and Dissipativity Theory for

Discrete-Time Nonlinear

Dynamical Systems

13.1 Introduction

In the ﬁrst twelve chapters of this book we presented a thorough treatment

of nonlinear analysis and control for continuous-time dynamical systems

using a Lyapunov and dissipative dynamical s ystems approach. In the next

two chapters we give a condensed review of the same theory for discrete-

time s ys tems. Since the theory for nonlinear discrete-time dynamical

systems closely parallels the theory of nonlinear continuous-time dynamical

systems, many of th e results are similar. Hence, th e contents in this and

the next chapter are brief, except in those cases where th e discrete-time

results deviate markedly from their continuous -time counterparts. Even

though many of the p roofs are similar to the continuous-time proofs, for

completeness of exposition we provide a self-contained treatment of the

fundamental discrete-time resu lts.

13.2 Discrete-Time Lyapunov Stability The ory

We begin by considering the general discrete-time nonlinear dynamical

system

x(k + 1) = f (x(k)), x(0) = x

, k ∈ Z

, (13.1)

where x(k) ∈ D ⊆ R

, k ∈ Z

, is the system state vector, D is an open

set, 0 ∈ D, f : D → D, and f (0) = 0. We assume that f(·) is continuous

on D. Furthermore, we denote the solution to (13.1) with initial condition

x(0) = x

by s(·, x

), so that the map of the dynamical system given by

s : Z

× D → D is continuous on D and satisﬁes the consistency property

s(0, x

) = x

and the semigroup property s(κ, s(k, x

)) = s(k + κ, x

), for

all x

∈ D and k, κ ∈ Z

. For this and the next chapter we use the notation

s(k, x

), k ∈ Z

, and x(k), k ∈ Z

, interchangeably as the solution of the

NonlinearBook10pt November 20, 2007

764 CHAPTER 13

nonlinear discrete-time system (13.1) with initial cond ition x(0) = x

Unlike continuous-time dynamical systems, establishing existence and

uniqueness of solutions for discrete-time dynamical systems is straightfor-

ward. To see this, consider th e discrete-time dynamical system (13.1) and let

⊆ Z

be th e maximal interval of existence for the solution x(·) of (13.1).

Now, to construct a solution to (13.1) we can construct the solution sequence

or discrete trajectory x(k) = s(k, x

) iteratively by s etting x(0) = x

and

using f (·) to deﬁne x(k) recursively by x(k + 1) = f(x(k)). Speciﬁcally,

s(0, x

) = x

s(1, x

) = f (s(0, x

)) = f(x

)

s(2, x

) = f (s(1, x

))

s(k, x

) = f (s(k − 1, x

)). (13.2)

If f(·) is continuous, it follows that f(s(k − 1, ·) is also continuous sin ce it

is constructed as a composition of continuous fun ctions. Hence, s(k, ·) is

continuous. If f(·) is such that f : R

→ R

, this iterative process can be

continued indeﬁnitely, and hence, a solution to (13.1) exists for all k ≥ 0.

Alternatively, if f (·) is such th at f : D → R

, the solution may cease to exist

at some point if f (·) maps x(k) into some point x(k + 1) outside the domain

of f (·). In this case, the solution sequence x(k) = s(k, x

) will be d eﬁ ned

on the maximal interval of existence x(k), k ∈ I

⊂ Z

. Furthermore, note

that the solution sequence x(k), k ∈ I

, is uniquely deﬁned for a given x

if f (·) is a continuous function. That is, any other solution sequence y(k)

starting from x

at k = 0 will take exactly the same values as x(k) an d can

be continued to the same interval as x(k). It is important to note that if

k ∈ Z

, uniqueness of solutions backward in time need not necessarily hold.

This is due to the fact that (k, x

) = f

−1

(s(k + 1, x

)), k ∈ Z

, and there is

no gu arantee that f (·) is invertible for all k ∈ Z

. However, if f : D → D

is a homeomorphism for all k ∈ Z

, then the solution sequence is unique

for all k ∈ Z. Identical arguments can be used to establish existence and

uniqueness of solutions for time-varying discrete-time systems. In light of

the above d iscus sion the following theorem is immediate.

Theorem 13.1. Consider the nonlinear dynamical system (13.1).

Assume that f : D → D is continuous on D. Then for every x

∈ D,

there exists I

⊆ Z

such that (13.1) has a unique solution x : I

→ R

Moreover, for each k ∈ I

, the solution s(k, ·) is continuous. If, in addition,

f(·) is a homeomorphism of D onto R

, then the solution x : I

→ R

unique in all I

∈ Z and s(k, ·) is continuous for all k ∈ I

. Finally, if

D = R

, then I

= Z.