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

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NonlinearBook10pt November 20, 2007

DISCRETE-TIME THEORY 835

where R ∈ R

n×n

is a positive-deﬁnite matrix. Furthermore, s how if A is

asymptotically stable, then P is the u nique solution to (13.244) and is given

P =

∞

k=0

. (13.245)

Problem 13.5. Consider the dynamical s ys tem (13.1) with f (x) =

Ax, where A ∈ R

n×n

Let R = C

C, where C ∈ R

l×n

, and assume

(A, C) is obs ervable. Show if there exists a positive-deﬁnite matrix P ∈

n×n

satisfying the discrete-time Lyapunov equation (13.244), then A is

asymptotically stable.

Problem 13.6. Let D

⊂ D be a compact positively invariant set

for the nonlinear dynamical system (13.1). Show that if there exists a

continuous function V : D → R such that

V (x) = 0, x ∈ D

, (13.246)

V (x) > 0, x ∈ D, x 6∈ D

, (13.247)

V (f (x)) −V (x) ≤ 0, x ∈ D, (13.248)

then D

is Lyapunov stable (see Section 4.9). If, in addition,

V (f (x)) −V (x) < 0, x ∈ D, x 6∈ D

, (13.249)

show that D

is asymptotically stable (see Section 4.9). Finally, if D = R

and V (x) → ∞ as kxk → ∞ show th at D

is globally asymptotically stable

(see Section 4.9).

Problem 13.7. Consider the nonlinear dynamical s ystem (13.1) and

recall the deﬁnition of a semistable equilibrium point x ∈ D (see Problem

3.44). Suppose the orbit O

of (13.1) is bounded for all x ∈ D and assume

that there exists a continuous function V : D → R such that

V (f (x)) −V (x) ≤ 0, x ∈ D. (13.250)

Furth ermore, let M denote the largest invariant set contained in R

△

{x ∈ D : V (f(x)) − V (x) = 0}. Show that if M ⊆ {x ∈ D : f (x) =

x and x is Lyapunov stable}, then (13.1) is semistable.

Problem 13.8. Consider the dyn amical system (13.1) with f (x) = Ax,

where A ∈ R

n×n

. Using the results of Problem 13.7 show that the following

statements are equ ivalent:

i) The zero solution x(k) ≡ 0 to (13.1) is semistable.

ii) For every initial condition x

∈ R

, lim

k→∞

x(k) exists.

NonlinearBook10pt November 20, 2007

836 CHAPTER 13

iii) If λ ∈ spec(A), th en either |λ| < 1, or both λ = 1 and λ is semisimple.

iv) lim

k→∞

exists.

Problem 13.9. Consider the dynamical system (13.1) with f(x) = Ax,

where A ∈ R

n×n

. Show that if there exist n ×n matrices P ≥ 0 and R ≥ 0

such that

P = A

P A + R, (13.251)













R(A −I

)

R(A −I

)

n−1













= N(A −I

), (13.252)

then the zero solution x(t) ≡ 0 to (13.1) is semistable (see Problem 13.8).

(Hint: First show that N(P ) ⊆ N(A −I

) ⊆ N(R) and N(A −I

) ∩R(A −

) = {0}.)

Problem 13.10 (Lyapunov’s Indirect Method). Let x(k) ≡ 0 be an

equilibrium point f or the nonlinear dynamical system

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

, k ∈ Z

, (13.253)

where f : D → D is continuously diﬀerentiable and D is an open set with

0 ∈ D. Furthermore, let

A =

∂f

∂x

(x)



x=0

Show that:

i) I f |λ| < 1, where λ ∈ spec(A), then the origin of the nonlinear

dynamical system (13.253) is asymptotically stable.

ii) If there exists λ ∈ spec(A) such that |λ| > 1, where λ ∈ spec(A), then

the origin of the nonlinear dynamical system (13.253) is unstable.

Problem 13.11. Let A ∈ R

n×n

. A is nonnegative if A

(i,j)

≥ 0, i, j =

1, . . . , n. Consider the dynamical system (13.1) with f (x) = Ax, where

A ∈ R

n×n

. Show that R

is an invariant set with respect to (13.1) if and

only if A is nonnegative.

Problem 13.12. Consider the nonlinear dynamical system

(k + 1) =

αx

(k)

1 + x

(k)

, x

(0) = x

, k ∈ Z

, (13.254)

(k + 1) =

βx

(k)

1 + x

(k)

, x

(0) = x

. (13.255)

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DISCRETE-TIME THEORY 837

Analyze th e stability of the zero solution (x

(k), x

(k)) ≡ (0, 0) to (13.254)

and (13.255) using the Lyapunov function candidate V (x

, x

) = x

+ x

for

i) α

< 1 and β

< 1, ii) α

≤ 1, β

≤ 1, and α

+ β

< 2, iii) α

= β

= 1,

and iv) α

> 1 and β

> 1.

Problem 13.13. The nonlinear dynamical system (13.1) is nonneg-

ative if for every x(0) ∈ R

, the solution x(k), k ∈ Z

, to (13.1) is

nonnegative, that is, x(k) ≥≥ 0, k ∈ Z

. The equilibrium solution x(k) ≡ x

of the nonnegative dynamical system (13.1) is Lyapunov stable if for every

ε > 0 there exists δ = δ(ε) > 0 such that if x

∈ B

) ∩ R

, then

x(k) ∈ B

) ∩ R

, k ∈ Z

. The equilibrium solution x(k) ≡ x

the nonnegative dynamical system (13.1) is asymptotically stable if it is

Lyapunov stable and there exists δ > 0 such that if x

∈ B

) ∩R

, then

lim

k→∞

x(k) = x

. Consider the dynamical system (13.1) with f (x) = Ax,

where A ∈ R

n×n

is n on negative (see Pr ob lem 13.11). Show that the following

statements hold:

i) I f there exist vectors p, r ∈ R

such that p >> 0 and r ≥≥ 0 satisfy

p = A

p + r, (13.256)

then A is Lyapunov stable.

ii) If there exist vectors p, r ∈ R

such that p ≥≥ 0 and r ≥≥ 0

satisfy (13.256) and (A, r

) is observable, then p >> 0 and A is

asymptotically stable.

Furth ermore, show that the f ollowing statements are equ ivalent:

iii) A is asymptotically stable.

iv) There exist vectors p, r ∈ R

such that p >> 0 and r >> 0 satisfy

(13.256).

v) There exist vectors p, r ∈ R

such that p ≥≥ 0 and r >> 0 satisfy

(13.256).

vi) For every r ∈ R

such that r >> 0, there exists p ∈ R

such that

p >> 0 satisﬁes (13.256).

(Hint: Use the Lyapunov function candidate V (x) = p

x in your analysis

and show that the Lyapunov stability theorems of Section 13.2 and th e

invariant set theorem of Section 13.3 can be used dir ectly for nonnegative

systems with the required s uﬃcient conditions veriﬁed in R

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838 CHAPTER 13

Problem 13.14. Prove Lemma 13.2.

Problem 13.15. Prove Theorem 13.22.

Problem 13.16. Prove Theorem 13.25.

Problem 13.17. Let f = [f

, . . . , f

]

: D → D, where D is an open

subset of R

that contains R

. Then f is nonnegative if f

(x) ≥ 0, for all

i = 1, . . . , n, and x ∈ R

. Consider the nonlinear dynamical system (13.1).

Show that th e following statements hold:

i) Suppose R

⊂ D. Then R

is an invariant set with respect to (13.1)

if and only if f : D → D is nonnegative.

ii) Suppose f(0) = 0 and f : D → D is nonnegative and continuously

diﬀerentiable on R

. Then A

△

∂f

∂x

(x)



x=0

is nonnegative (see Problem

13.11).

iii) If f (x) = Ax, where A ∈ R

n×n

, then f is nonnegative if and only if A

is nonnegative.

Problem 13.18. Consider a discrete epidemic model involving two

distinct populations where infected members of one population can trans mit

a disease to a sus ceptible in the other population. Letting x

denote the

infected fraction of the ith population and (1 − x

) denote the fraction of

susceptibles, the epidemic model can be characterized as

(k + 1) = a

(k)[1 −x

(k)] + (1 −b

(k), x

(0) = x

, k ∈ Z

(13.257)

(k + 1) = a

(k)[1 −x

(k)] + (1 −b

(k), x

(0) = x

, (13.258)

where a

∈ (0, 1) and b

∈ (0, 1) for i = 1, 2. C haracterize all the equilibria of

(13.257) and (13.258). Using the Lyapunov function candidate V (x) = p

where x = [x

, x

]

and p ∈ R

, analyze the stability of the equilibria

of (13.257) and (13.258). Make sure to justify that V (x) = p

x is a valid

Lyapunov function. (Hint: See Problem 13.17.)

Problem 13. 19. The nonlinear dynamical system G given by (13.135)

and (13.136) is nonnegative if for every x(0) ∈ R

and u(k) ≥≥ 0, k ∈ Z

the solution x(k), k ∈ Z

, to (13.100) and the output y(k), k ∈ Z

, are

nonnegative, that is, x(k) ≥≥ 0, k ∈ Z

, and y(k) ≥≥ 0, k ∈ Z

. Consider

the nonlinear dynamical system G given by (13.135) and (13.136). Show th at

if f : D → R

is nonnegative (see Problem 13.17), G(x) ≥≥ 0, h(x) ≥≥ 0,

and J(x) ≥≥ 0, x ∈ R

, then G is nonnegative.

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DISCRETE-TIME THEORY 839

Problem 13.20. Let q ∈ R

and r ∈ R

. Consider the nonlinear

nonnegative dynamical system G (see P roblem 13.19) given by (13.135) and

(13.136) where f : D → R

is nonnegative (see Problem 13.17), G(x) ≥≥ 0,

h(x) ≥≥ 0, and J(x) ≥≥ 0, x ∈ R

. Show that if there exist functions

: R

→ R

, ℓ : R

→ R

, W : R

→ R

, P

: R

→ R

1×m

, and a scalar

ρ > 1 (respectively, ρ = 1) such that V

(·) is continuous and nonnegative

deﬁnite, V

(0) = 0,

(f(x) + G(x)u) = V

(f(x)) + P

(x)u, x ∈ R

, u ∈ R

, (13.259)

and, for all x ∈ R

0 = V

(f(x)) −

(x) − q

h(x) + ℓ(x), (13.260)

0 = P

(x) − q

J(x) −r

+ W

(x), (13.261)

then G is geometrically d issipative (respectively, dissipative) w ith respect to

the supply rate s(u, y) = q

y + r

u. (Hint: The deﬁnition of dissipativity

and geometric d issipativity should be modiﬁed to reﬂect the fact that x

∈

, and u(k), k ∈ Z

, and y(k), k ∈ Z

, are nonnegative.)

Problem 13.21. Let q ∈ R

and r ∈ R

. Consider the nonlinear

nonnegative dynamical system G (see P roblem 13.19) given by (13.135) and

(13.136) where f : D → R

is nonnegative (see Problem 13.17), G(x) ≥≥ 0,

h(x) ≥≥ 0, and J(x) ≥≥ 0, x ∈ R

. Show that G is lossless with respect

to the supply rate s(u, y) = q

y + r

u , u ∈ R

, if and only if there exist

functions V

: R

→ R

and P

: R

→ R

1×m

such that V

(·) is continuous,

(0) = 0, and for all x ∈ R

, (13.259) holds and

0 = V

(f(x)) −V

(x) − q

h(x), (13.262)

0 = P

(x) − q

J(x) − r

. (13.263)

If, in addition, V

(·) is continuously diﬀerentiable sh ow that

(x) = V

′

(f(x))G(x). (13.264)

Problem 13.22. Let q ∈ R

and r ∈ R

and assum e G given by

(13.135) and (13.136) is such that f : D → R

is nonnegative (see Problem

13.17), G(x) ≥≥ 0, h(x) ≥≥ 0, and J(x) ≥≥ 0, x ∈ R

. Suppose G

is geometrically dissipative (respectively, dissipative) with respect to the

supply rate s(u, y) = q

y + r

u. Show th at there exist p ∈ R

, l ∈ R

, and

w ∈ R

and a scalar ρ > 1 (respectively, ρ = 1) such that

0 = A

p −

p − C

q + l, (13.265)

0 = B

p − D

q − r + w, (13.266)

NonlinearBook10pt November 20, 2007

840 CHAPTER 13

where

A =

∂f

∂x



x=0

, B = G(0), C =

∂h

∂x



x=0

, D = J(0). (13.267)

If, in addition, (A, C) is observable, show that p >> 0.

Problem 13.23. Let q ∈ R

and r ∈ R

. Consider the nonnegative

dynamical system G given by (13.198) an d (13.199) where A ≥≥ 0, B ≥≥ 0,

C ≥≥ 0, and D ≥≥ 0. Show that G is geometrically dissipative (respectively,

dissipative) with respect to the supply rate s(u, y) = q

y + r

u if and only

if there exist p ∈ R

, l ∈ R

, and w ∈ R

, and a scalar ρ > 1 (respectively,

ρ = 1) such that

0 = A

p −

p − C

q + l, (13.268)

0 = B

p − D

q − r + w. (13.269)

(Hint: Use Problems 13.21 and 13.22 to show the result.)

Problem 13.24 (Parseval’s Theorem). Let u : Z

→ R

and

y : Z

→ R

be in ℓ

, p ∈ [0, ∞), and let u(z) and y(z) denote their Z-

transforms, respectively. Show that

∞

k=0

(k)y(k) =

2π

−π

∗

θ

)y(e

θ

)dθ. (13.270)

Problem 13.25 (Positivity Theorem). Consider the controllable and

observable system

x(k + 1) = Ax(k) + Bu(k), x(0) = x

, k ∈ Z

, (13.271)

y(k) = Cx(k) + Du(k), (13.272)

u(k) = −σ(y(k), k), (13.273)

where x ∈ R

, u, y ∈ R

, and

σ(·, ·) ∈ Φ

△

= {σ : R

×Z

→ R

: σ(0, ·) = 0,

(y, k)y ≥ 0, y ∈ R

, k ∈ Z

Furth ermore, suppose

G(z) ∼



A B

C D



is strictly positive real. Show that the negative feedb ack interconnection of

(13.271)–(13.273) is globally uniform ly asymptotically stable for all σ(·, ·) ∈

Problem 13.26 (Small Gain Theorem). Consider the controllable and

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DISCRETE-TIME THEORY 841

observable system

x(k + 1) = Ax(k) + Bu(k), x(0) = x

, k ∈ Z

, (13.274)

y(k) = Cx(k) + Du(k), (13.275)

u(k) = σ(y(k), k), (13.276)

where x ∈ R

, u ∈ R

, y ∈ R

, and

σ(·, ·) ∈ Φ

△

= {σ : R

× Z

→ R

: σ(0, ·) = 0, kσ(y, k)k

≤ γ

−1

kyk

y ∈ R

, k ∈ Z

and where γ > 0. Furthermore, suppose

G(z) ∼



A B

C D



is strictly positive real. Show that the negative feedb ack interconnection of

(13.274)–(13.276) is globally uniform ly asymptotically stable for all σ(·, ·) ∈

Problem 13.27 (Circle Criterion). Consider the controllable and

observable system

x(k + 1) = Ax(k) + Bu(k), x(0) = x

, k ∈ Z

, (13.277)

y(k) = Cx(k) + Du(k), (13.278)

u(k) = −σ(y(k), k), (13.279)

where x ∈ R

, u ∈ R

, y ∈ R

, and

σ(·, ·) ∈ Φ

△

= {σ : R

× Z

→ R

: σ(0, ·) = 0,

[σ(y, k) − M

y] ≤ 0, y ∈ R

, k ∈ Z

and where M

, M

∈ R

m×l

. Furthermore, suppose [I + M

G(z)][I +

G(z)]

−1

is strictly positive real, where

G(z) ∼



A B

C D



and det[I + M

G(x)] 6= 0, |z| ≥ 1. Show that the negative feedback

interconnection of (13.277)–(13.279) is globally uniformly asymptotically

stable for all σ(·, ·) ∈ Φ

Problem 13.28 (Szeg¨o Criterion). Consider the controllable and

observable system

x(k + 1) = Ax(k) + Bu(k), x(0) = x

, k ∈ Z

, (13.280)

y(k) = Cx(k), (13.281)

u(k) = −σ(y(k)), (13.282)

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842 CHAPTER 13

where x ∈ R

, u, y ∈ R

, and

σ(·) ∈ Φ

△

= {σ : R

→ R

: σ(0) = 0, σ

(y)[M

−1

σ(y) − y] ≤ 0, y ∈ R

σ(y) = [σ

), σ

), . . . , σ

)]

, and

0 <

(ν) − σ

(ˆν)

ν − ˆν

< µ

, ν, ˆν ∈ R, i = 1, . . . , m},

and where M ∈ R

m×m

is a positive-deﬁnite matrix and ν 6= ˆν. Furthermore,

suppose there exists a positive-deﬁnite diagonal matrix N such that M

−1

[I + (z − 1)N]G(z) −

|z − 1|

∗

(z)µNG(z) is strictly positive real, where

G(z) ∼



A B

C 0



and µ = diag[µ

, µ

, . . . , µ

]. Show th at the n egative feedback interconnec-

tion (13.280)–(13.282) is globally asymptotically stable for all σ(·) ∈ Φ

Problem 13.29. Show that the results of Problems 6.4–6.6 also hold

in the case where G

and G

are d iscrete-time dynamical systems.

Problem 13.30. Consider the nonnegative dynamical system G (see

Problem 13.19) given by (13.198) and (13.199), and assume that (A, C) is

observable and G is geometrically dissipative with respect to the supply

rate s(u, y) = e

u − e

My, where M >> 0. Show that the positive

feedback interconnection of G and σ(·, ·) is globally asymptotically stable

for all σ(·, ·) ∈ Φ, where

△

= {σ : Z

× R

→ R

: σ(·, 0) = 0, 0 ≤≤ σ(k, y) ≤≤ My,

y ∈ R

, k ∈ Z

}, (13.283)

M >> 0, and M ∈ R

m×l

. (Hint: Use Problem 13.23 to show that if G is

geometrically dissipative with respect to the supply rate s(u, y) = e

u −

My, then there exist p ∈ R

, l ∈ R

, and w ∈ R

, and a scalar ρ > 1

such that

0 = A

p −

p + C

e + l, (13.284)

0 = B

p + D

e − e + w. (13.285)

Now, use the Lyapunov function candidate V (x) = p

x.)

Problem 13.31. Let q ∈ R

, r ∈ R

, q

∈ R

, and r

∈ R

. Consider

the nonlinear nonnegative dynamical s ystems G and G

(see Prob lem 13.19)

given by (13.135) and (13.136), and (13.225) and (13.226), respectively,

where f : D → R

is nonnegative, G(x) ≥≥ 0, h(x) ≥≥ 0, J(x) ≥≥ 0,

x ∈ R

, f

: R

→ R

is nonnegative, G

, x

) = G

) ≥≥ 0,

, x

) = h

) ≥≥ 0, x

∈ R

, and J

, x

) ≡ 0. Assume that G

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DISCRETE-TIME THEORY 843

is dissipative with respect to the linear supply rate s(u, y) = q

y + r

u and

with a continuous positive-deﬁnite storage function V

(·), and assume that G

is dissipative with respect to the linear supply rate s

, y

) = q

+ r

and with a continuous positive-deﬁnite storage function V

(·). Show that

the f ollowing statements hold:

i) I f there exists a scalar σ > 0 s uch that q + σr

≤≤ 0 and r +σq

≤≤ 0,

then the positive feedback interconnection of G and G

is Lyapunov

stable.

ii) If G and G

are zero-state observable and there exists a scalar σ > 0

such that q + σr

<< 0 and r + σq

<< 0, then the positive feedback

interconnection of G and G

is asymptotically stable.

iii) If G is zero-state ob s ervable, rank G

(0) = m

, G

is geometrically

dissipative with respect to the supply rate s

, y

) = q

+ r

and there exists a scalar σ > 0 such that q + σr

≤≤ 0 and r +

σq

≤≤ 0, then the positive feedback interconnection of G and G

asymptotically stable.

iv) If G is geometrically dissipative with respect to the sup ply r ate

s(u, y) = q

y + r

u, G

is geometrically dissipative with respect to

the s upply rate s

, y

) = q

+ r

, and there exists a scalar

σ > 0 such that q + σr

≤≤ 0 and r + σq

≤≤ 0, then the positive

feedback interconnection of G and G

is asymptotically stable.

(Hint: First show that the positive feedback interconnection of G and G

gives a nonnegative closed-loop system.)

13.17 Notes and References

In comparison to the stability theory of continuous-time dynamical systems,

there is very little literature on the stability theory of discrete-time systems.

This is due to the fact that discrete-time stability theory very closely

parallels continuous-time stability theory and is often p resented as a footnote

to the continuous-time theory. Among the earliest pap ers on discrete-

time stability theory is due to Li [276]. A self-contained summary of

the application of Lyapunov stability theory to discrete-time systems is

given by Hahn [177]. See also Kalman and Bertram [229]. The invariance

principle and the invariant set theorems for discrete-time systems is due

to LaSalle; see for example [259]. Nonlinear discrete-time extensions

of passivity and losslessness are du e to Byrnes and Lin [78] and Lin

and Byrnes [281, 282]. Extended Kalman-Yakubovich-Popov equations for

nonlinear discrete-time dissipative systems are given in Chellaboina and

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844 CHAPTER 13

Haddad [86]. The concepts of geometric dissipativity, geometric passivity,

and geometric non exp ansivity are a discrete-time analog of exponential

dissipativity introd uced by Chellab oina and Haddad [88] f or continuous-

time systems. Th e discrete-time positive real and bounded real lemmas

for single-input, single-output s ystems are due to Szeg¨o and Kalman [422]

and Szeg¨o [421], respectively. Multivariable generalizations are given in Hitz

and Anderson [197] an d Vaidyanathan [437]. A textbook treatment of linear

discrete-time passivity is given by Cains [80].

Gain, sector, and d isk margins for nonlinear discrete-time systems are

due to Chellaboina and Haddad [86] and can be viewed as a generalization

of the work by Lee and Lee [262] on gain and phase margins for d iscrete-

time linear systems. Finally, the concept of discrete-time control Lyapunov

functions introduced in this chapter is a generalization of the single-input,

discrete-time control Lyapunov function introduced by Amicucci, Monaco,

and Normand-Cyrot [6] to multi-inp ut systems.