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

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DISCRETE-TIME NONLINEAR CONTROL 875

Finally, if D = R

, w(k) ≡ 0, and

V (x) → ∞ as kxk → ∞, (14.164)

then the solution x(k) = 0, k ∈ Z

, of (14.143) is globally asymptotically

stable.

Proof. Let x(k), k ∈ Z

, satisfy (14.143). Then

∆V (x(k))

△

= V (x(k + 1)) − V (x(k))

= V (f(x(k))) − V (x(k)) + P

(x(k))w(k)

(k)P

(x(k))w(k), k ∈ Z

. (14.165)

Hence, with w(k) ≡ 0, it follows from (14.158) that

∆V (x(k)) < 0, k ∈ Z

, x(k) 6= 0. (14.166)

Thus , from (14.154), (14.156), and (14.166) it follows that V (·) is a Lyapunov

function for (14.143), which proves local asymptotic stability of the solution

x(k) ≡ 0 with w(k) ≡ 0. Consequently, x(k) → 0 as k → ∞ for all initial

conditions x

∈ D

for some neighborhood D

⊆ D of the origin.

Next, if Γ(x) ≥ 0, x ∈ D, and w(k) ≡ 0, (14.160) implies

L(x(k)) = − ∆V (x(k)) + L(x(k)) + V (f(x(k))) − V (x(k))

≤ −∆V (x(k)) + L(x(k)) + V (f (x(k))) − V (x(k)) + Γ(x(k))

= −∆V (x(k)).

Now, summing over [0, k] yields

i=0

L(x(i)) ≤ −V (x(k + 1)) + V (x

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

∈ D

yields J(x

) ≤ V (x

Next, let x(k), k ∈ Z

, satisfy (14.143) w ith w(k) ≡ 0. Then it follows

from (14.160) that

L(x(k)) + Γ(x(k))

= −∆V (x(k)) + L(x(k)) + V (f (x(k))) − V (x(k)) + Γ(x(k))

= −∆V (x(k)).

Summing over [0, k] yields

i=0

[L(x(i)) + Γ(x(i))] = −V (x(k + 1)) + V (x

NonlinearBook10pt November 20, 2007

876 CHAPTER 14

Now, letting k → ∞ yields J(x

) = V (x

). Finally, it follows that

(14.154), (14.155), (14.157), (14.159), and (14.160) imply (14.145)–(14.150),

and h en ce, with Γ(x) replaced by L(x) + Γ(x), Lemma 14.4 yields

V (x(k + 1)) ≤

i=0

r(z(i), w(i)) + V (x

), w(·) ∈ ℓ

, k ∈ Z

Now, (14.163) f ollows by noting that V (x(k + 1)) ≥ 0, k ∈ Z

. Finally, for

D = R

global asymptotic stability of the solution x(k) = 0, k ∈ Z

, is a

direct consequence of the radially unbounded condition (14.164) on V (x),

x ∈ R

14.10 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

R ∈ S

, and

r(z, w) = z

Qz + 2z

Sw + w

Rw, (14.167)

such that

N(x)

△

= J

(x)

(x) + J

(x)

S +

(x) +

R > P

(x), x ∈ D.

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

Γ(x) =

(x) − J

(x)

Qh(x) −

h(x)

(N(x) −P

(x))

−1

(x) − J

(x)

Qh(x) −

h(x)

− h

(x)

Qh(x),

satisﬁes (14.157) since in this case

L(x) + Γ(x) + r(z, w) − P

(x)w − w

(x)w

= L(x) + [

(x) − J

(x)

Q(x)h(x) −

(x)h(x)

−(N(x) −P

(x))w]

(N(x) −P

(x))

−1

[

(x)

−J

(x)

Q(x)h(x) −

(x)h(x) − (N(x) −P

(x))w]

≥ 0. (14.168)

Corollary 14.8. Let L(x) ≥ 0, x ∈ D, and consid er the nonlinear

system given by (14.143) and (14.144) with perform ance functional

J(x

)

△

∞

k=0

L(x(k)), (14.169)

where x(k), k ∈ Z

, solves (14.143) with w(k) ≡ 0. Assume there exist

functions P

: D → R

1×d

, P

: D → N

, and V : D → R such that V (·) is

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DISCRETE-TIME NONLINEAR CONTROL 877

continuous and

(0) = 0, (14.170)

V (0) = 0, (14.171)

V (x) > 0, x ∈ D, x 6= 0, (14.172)

V (f (x)) −V (x) < 0, x ∈ D, x 6= 0, (14.173)

V (f (x) + J

(x)w) = V (f (x)) + P

(x)w + w

(x)w,

x ∈ D, w ∈ W, (14.174)

− J

(x)J

(x) − P

(x) > 0, x ∈ D, (14.175)

L(x) + V (f(x)) − V (x) + Γ(x) = 0, x ∈ D, (14.176)

where γ > 0 and

Γ(x) =



(x) + J

(x)h(x)





− J

(x)J

(x) − P

(x)



−1



(x) + J

(x)h(x)



+ h

(x)h(x). (14.177)

Then the zero solution x(k) ≡ 0 to the undisturbed (w(k) ≡ 0) system

(14.143) is locally asymptotically stable and there exists a neighborhood

⊆ D of the origin such that

J(x

) ≤ J(x

) = V (x

), x

∈ D

, (14.178)

where

J(x

)

△

∞

k=0

[L(x(k)) + Γ(x(k))] (14.179)

and where x(k), k ∈ Z

, solves (14.143) with w(k) ≡ 0. Fu rthermore, the

solution x(k), k ∈ Z

, of (14.143) satisﬁes the nonexpansivity constraint

i=0

(i)z(i) ≤ γ

i=0

(i)w(i) + V (x

), w(·) ∈ ℓ

, k ∈ Z

(14.180)

Finally, if D = R

, w(k) ≡ 0, and

V (x) → ∞ as kxk → ∞, (14.181)

then the solution x(k) = 0, k ∈ Z

, of (14.143) is globally asymptotically

stable.

Proof. With

Q = −I

S = 0, and

R = γ

, it follows from (14.168)

that Γ(x) given by (14.177) satisﬁes (14.157). The result n ow f ollows as a

direct consequence of Theorem 14.9.

Note that if L(x) = h

(x)h(x) in Corollary 14.8 then Γ(x) can be

chosen as

Γ(x) =



(x) + J

(x)h(x)





− J

(x)J

(x) − P

(x)



−1

NonlinearBook10pt November 20, 2007

878 CHAPTER 14



(x) + J

(x)h(x)



The framework presented in Corollary 14.8 is an extension of the

mixed-norm H

∞

framework of Haddad et al. [154] to nonlinear systems.

Speciﬁcally, letting f (x) = Ax, J

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

(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

△

= E

E > 0, and P ∈ P

satisﬁes

P = A

P A + A

P D(γ

− D

P D)

−1

P A + R, (14.182)

it follows from Corollary 14.8 with L(x) = h

(x)h(x) = x

Rx, Γ(x) =

P D(γ

−D

P D)

−1

P Ax, where γ

− D

P D > 0, and x

= 0

that

i=0

(i)Rx(i) ≤ γ

i=0

(i)w(i), w(·) ∈ ℓ

, k ∈ Z

, (14.183)

or, equivalently, th e H

∞

norm of

G(z) ∼



A D

E 0



satisﬁes

|||G|||

∞

△

= sup

θ∈[0,2π]

max

(G(e

θ

)) ≤ γ, (14.184)

where σ

max

(·) denotes the maximum singular value. Now, (14.178) implies

∞

k=0

(k)Rx(k) ≤

∞

k=0

(k)[R + A

P D(γ

− D

P D)

−1

P A]x(k)

∞

k=0

)

[R + A

P D(γ

− D

P D)

−1

P A]A

where x(k), k ∈ Z

, solves (14.143) with w(k) ≡ 0.

To eliminate the explicit dependence on the initial condition x

assume x

has expected value V , that is, E[x

] = V , where E denotes

expectation. Invoking this step leads to

∞

k=0

(k)Rx(k)

= E

∞

k=0

)

= E[x

P x

] = tr

P V,

where

P = A

P A + R

and

∞

k=0

(k)(R + A

P D(γ

− D

P D)

−1

P A)x(k)

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DISCRETE-TIME NONLINEAR CONTROL 879

= E

∞

k=0

)

[R + A

P D(γ

−D

P D)

−1

P A]A

= E[x

P x

]

= tr P V, (14.185)

where P satisﬁes (14.182). Hence, |||G|||

= tr

P V ≤ tr P V , which implies

that J(x

) given by (14.179) provides an upper bound to the H

norm of

G(z).

Corollary 14.9. Let L(x) ≥ 0, x ∈ D, p = d, and consider the

nonlinear system given by (14.143) and (14.144) with performance functional

J(x

)

△

∞

k=0

L(x(k)), (14.186)

where x(k), k ∈ Z

, solves (14.143) with w(k) ≡ 0. Assume there exist

functions P

: D → R

1×d

, P

: D → N

, and V : D → R such that V (·) is

continuous and

(0) = 0, (14.187)

V (0) = 0, (14.188)

V (x) > 0, x ∈ D, x 6= 0, (14.189)

V (f (x)) −V (x) < 0, x ∈ D, x 6= 0, (14.190)

V (f (x) + J

(x)w) = V (f (x)) + P

(x)w + w

(x)w,

x ∈ D, w ∈ W, (14.191)

(x) + J

(x) − P

(x) > 0, x ∈ D, (14.192)

L(x) + V (f(x)) − V (x) + Γ(x) = 0, x ∈ D, (14.193)

where

Γ(x) =



(x) − h(x)





(x) + J

(x) − P

(x)



−1



(x) − h(x)



(14.194)

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

(14.143) is locally asymptotically stable and there exists a neighborhood

⊆ D of the origin such that

J(x

) ≤ J(x

) = V (x

), x

∈ D

, (14.195)

where

J(x

)

△

∞

k=0

[L(x(k)) + Γ(x(k))] (14.196)

and where x(k), k ∈ Z

, solves (14.143) with w(k) ≡ 0. Fu rthermore, the

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880 CHAPTER 14

solution x(k), k ∈ Z

, of (14.143) satisﬁes the passivity constraint

i=0

(i)w(i) + V (x

) ≥ 0, w(·) ∈ ℓ

, k ∈ Z

. (14.197)

Finally, if D = R

, w(k) ≡ 0, and

V (x) → ∞ as kxk → ∞, (14.198)

then the solution x(k) = 0, k ∈ Z

, of (14.143) is globally asymptotically

stable.

Proof. With

Q = 0,

S = I, and

R = 0, it follows from (14.168) that

Γ(x) given by (14.194) satisﬁes (14.157). The result now follows as a direct

consequence of Theorem 14.9.

The framework presented in Corollary 14.9 is an extension of the

continuous-time H

/positivity framework of Hadd ad and Bernstein [146] to

nonlinear discrete-time systems. Speciﬁcally, letting f (x) = Ax, J

(x) =

D, h(x) = Ex, J

(x) = E

∞

, L(x) = x

Rx, V (x) = x

P x, and

Γ(x) = x

P A − E)

∞

+ E

∞

− D

P D)

−1

P A − E)x, where

∞

+ E

∞

− D

P D > 0 and where A ∈ R

n×n

, D ∈ R

n×d

, E ∈ R

d×n

∞

∈ S

, R ∈ P

, and P ∈ P

satisﬁes

P = A

P A + (D

P A −E)

∞

+ E

∞

−D

P D)

−1

P A −E) + R,

(14.199)

it follows fr om Corollary 14.9, with x

= 0, that

i=0

(i)z(i) ≥ 0, w(·) ∈ ℓ

, k ∈ Z

, (14.200)

or, equivalently,

∞

(z) + G

∗

∞

(z) ≥ 0, |z| > 1, (14.201)

where

∞

(z) ∼



A D

E E

∞



Now, using similar arguments as in the H

∞

case, (14.195) implies

P V = E

∞

k=0

(k)Rx(k)

≤ E

∞

k=0

(k)[R + (D

P A −E)

∞

+ E

∞

−D

P D)

−1

·(D

P A −E)]x(k)

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DISCRETE-TIME NONLINEAR CONTROL 881

or, equivalently, s ince

∞

k=0

(k)[R + (D

P A −E)

∞

+ E

∞

− D

P D)

−1

·(D

P A −E)]x(k)

= E

∞

k=0

)

[R + (D

P A − E)

∞

+ E

∞

− D

P D)

−1

·(D

P A −E)]A

= E[x

P x

]

= tr P V,

where P satisﬁes (14.199). Hence, |||G|||

= tr

P V ≤ tr P V , which implies

that J(x

) given by (14.196) provides an upper bound to the H

norm of

G(z).

Next, deﬁne the subset of square-summable bounded disturbances

△

(

w(·) ∈ ℓ

∞

i=0

(i)w(i) ≤ β

)

, (14.202)

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

x ∈ D.

Theorem 14.10. Let γ > 0, L(x) ≥ 0, x ∈ D, and consider th e

nonlinear system (14.143) with performance functional (14.153). Assu me

that there exist functions P

: D → R

1×d

, P

: D → N

, and V : D → R

such that V (·) is continuous and

(0) = 0, (14.203)

V (0) = 0, (14.204)

V (x) > 0, x ∈ D, x 6= 0, (14.205)

V (f (x)) −V (x) < 0, x ∈ D, x 6= 0, (14.206)

V (f (x) + J

(x)w) = V (f (x)) + P

(x)w + w

(x)w, x ∈ D, w ∈ W,

(14.207)

− P

(x) > 0, x ∈ D, (14.208)

L(x)+V (f(x))−V (x)+

(x)



− P

(x)



−1

(x) = 0, x ∈ D.

(14.209)

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882 CHAPTER 14

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

(14.143) is locally asymptotically stable and there exists a neighborhood

⊆ D of the origin such that

J(x

) ≤ J(x

) = V (x

), x

∈ D

, (14.210)

where

J(x

)

△

∞

k=0

[L(x(k)) + Γ(x(k))], (14.211)

Γ(x) =

(x)



−P

(x)



−1

(x), (14.212)

and where x(k), k ∈ Z

, solves (14.143) with w(k) ≡ 0. Fur th ermore, if

= 0 then the solution x(k), k ∈ Z

, of (14.143) satisﬁes

V (x(k)) ≤ γ, w(·) ∈ W

, k ∈ Z

. (14.213)

Finally, if D = R

, w(k) ≡ 0, and

V (x) → ∞ as kxk → ∞, (14.214)

then the solution x(k) = 0, k ∈ Z

, of (14.143) is globally asymptotically

stable.

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

performance bound (14.210) are identical to the proofs of local and global

asymptotic stability given in Theorem 14.9 and the performance bound

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

w an d Γ(x) given by (14.212), it follows

from Lemma 14.4 that

V (x(k)) ≤

k−1

i=0

(i)w(i), k > 0, w(·) ∈ ℓ

, k ∈ Z

which yields (14.213).

14.11 A Riccati Equation Characterization for

Mixed H

/ℓ

Performance

In this s ection, we consider the dynamical system (14.143) and (14.144)

with f(x) = Ax, J

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

(x) = E

∞

, wh ere A ∈ R

n×n

D ∈ R

n×d

, E ∈ R

p×n

, E

∞

∈ R

p×d

, and A is asymptotically s table so that

x(k + 1) = Ax(k) + Dw(k), x(0) = 0, k ∈ Z

, w(·) ∈

W, (14.215)

z(k) = Ex(k) + E

∞

w(k), (14.216)

where

W consists of unit-peak input signals deﬁned by

△



w(·) : w

(k)w(k) ≤ 1, k ∈ Z



. (14.217)

NonlinearBook10pt November 20, 2007

DISCRETE-TIME NONLINEAR CONTROL 883

The following result pr ovides an upper bound to the ℓ

norm (ℓ

∞

equi-

induced norm) of the convolution operator G of the linear time-invariant

system (14.215) and (14.216) given by

|||G|||

△

= sup

w(·)∈

(

sup

k∈Z

kz(k)k

)

where k · k denotes the Euclidean vector norm.

Theorem 14.11. Let α > 1 and consider the linear system (14.215)

and (14.216). Then

|||G|||

≤ σ

max

(EP

−1

) + σ

max

∞

), (14.218)

where P > 0 satisﬁes

(α −1)I

− αD

P D > 0, (14.219)

P ≥ αA

P A + α

P D



(α −1)I

− αD

P D



−1

P A. (14.220)

Proof. Let N > 0, N ∈ Z

, and consider the dilated linear system

˜x(k + 1) = A

˜x(k) + D

v(k), ˜x(0) = 0, k ∈ Z

, (14.221)

z(k) = α

−

k−N

(E˜x(k) + E

∞

v(k)), (14.222)

where A

△

√

αA, D

△

√

αD, ˜x(k)

△

= α

k−N

x(k), and v(k)

△

k−N

w(k). Note that (14.221) and (14.222) are equivalent to (14.215)

and (14.216). Furthermore, n ote that if w(·) ∈

W then v(·) ∈ V, where

△

v(·) :

N−1

k=0

(k)v(k) ≤

α−1

. Hence,

|||G|||

≤ sup

N≥0

sup

v(·)∈V

kz(N)k.

Next, w ith f(x) = A

˜x(k), J

(x) = D

, V (x) = ˜x

P ˜x, P

(x) =

2˜x

P D

, P

(x) = D

P D

, W = V, β =

α−1

, γ = 1, and L(x) =

˜x

R˜x, where R ∈ R

n×n

is an arbitrary positive-deﬁnite matrix, it follows

from Theorem 14.10 that if there exists P > 0 such that

P = αA

P A + α

P D



(α −1)I

− αD

P D



−1

P A + R, (14.223)

then

˜x

(N)P ˜x(N) ≤ 1,

and h en ce, since x(N ) = ˜x(N), for all v(·) ∈ V,

kz(N)k = kEx(N) + E

∞

w(N )k ≤ σ

max

(EP

−1

) + σ

max

∞

The result is now immediate by noting that (14.223) is equivalent to (14.220).

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884 CHAPTER 14

Note that the R iccati inequality (14.220) is equivalent to the linear

matrix inequality



P 0

0 (α − 1)I



≥ α







A D



, (14.224)

or, equivalently,



P 0

0 (α − 1)I



≥ α



A D

0 0





P 0

0 (α − 1)I



A D

0 0



. (14.225)

Next, since α > 1, using Schur complements, (14.225) is equivalent to



−1

(α−1)



≥ α



A D

0 0



−1

(α−1)



A D

0 0



. (14.226)

Now, letting Q = P

−1

in (14.226) yields

Q ≥ αAQA

α −1

. (14.227)

Hence, Theorem 14.11 yields

|||G|||

≤ σ

max

(EQE

) + σ

max

∞

), (14.228)

where Q satisﬁes (14.227). It is interesting to note that in the case where

∞

= 0 the solution Q to (14.227) satisﬁes the bound

Q ≤ Q, (14.229)

where Q satisﬁes

Q = AQA

+ DD

, (14.230)

and h en ce,

|||G|||

= tr EQE

≤ tr EQE

. (14.231)

Thus , (14.227) can be used to pr ovide a trade-oﬀ between H

and mixed

/ℓ

performance. For further details see [157].

14.12 Robust Stability Analysis of Nonlinear Uncertain

Discrete-Time Systems

In this section, we extend the analysis results of Chapter 11 to discrete-

time uncertain dynamical systems. Speciﬁcally, we consider the problem of

evaluating a performance bound f or a nonlinear-nonquadratic cost functional

dependin g upon a class of nonlinear uncertain systems. As in the continuous-

time case, the cost bound can be evaluated in closed form as long as the cost

functional is related in a speciﬁc way to an underlying Lyapunov function

that guarantees robust stability over a prescribed uncertainty set. Hence,

the overall framework provides for robust stability and performance where