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

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NonlinearBook10pt November 20, 2007

DISCRETE-TIME NONLINEAR CONTROL 895

v) There exists a (un ique) minimal solution to (14.307) in the class of

real symmetric solutions.

vi) P is the minimal solution to (14.307) if and only if ρ(A + D(γ

I −

P D)

−1

P A) < 1.

vii) |||G|||

∞

< γ if and only if A+ D(γ

I −D

P D)

−1

P A is S chur, where

P is the minimal solution to (14.307).

viii) If P is th e minimal solution to (14.307) and |||G|||

∞

< γ, then I(G, γ) =

−ln d et(γ

I − D

P D).

Problem 14.5. Consider the controlled nonlinear dynamical system

x(k + 1) = F (x(k), u(k)) + J

(x(k))w(k), x(0) = x

, w(·) ∈ ℓ

, k ∈ Z

(14.308)

z(k) = h(x(k), u(k)) + J

(x(k))w(k), (14.309)

with performance functional

J(x

, u(·))

△

∞

k=0

L(x(k), u(k)), (14.310)

where F : D × U → R

is continuous and satisﬁes F (0, 0) = 0, J

: D →

n×d

, h : D × U → R

satisﬁes h(0, 0) = 0, J

: D → R

p×d

, L : D ×U → R,

and u(·) ∈ U is an admissible control. Assume there exist fun ctions V : D →

R, Γ : D × U → R, P

: D × U → R

1×d

, P

: D × U → N

, and a control

law φ : D → U s uch that V (·) is continuous and

(0, 0) = 0, (14.311)

V (0) = 0, (14.312)

V (x) > 0, x ∈ D, x 6= 0, (14.313)

φ(0) = 0, (14.314)

V (F (x, φ(x))) − V (x) < 0, x ∈ D, x 6= 0, (14.315)

(x, φ(x))w + w

(x, φ(x))w ≤ r(z, w) + L(x, φ(x)) + Γ(x, φ(x)),

x ∈ D, w ∈ W, (14.316)

V (F (x, u) + J

(x)w) = V (F (x, u)) + P

(x, u)w

(x, u)w,

x ∈ D, u ∈ U, w ∈ W, (14.317)

H(x, φ(x)) = 0, x ∈ D, (14.318)

H(x, u) ≥ 0, x ∈ D, u ∈ U, (14.319)

where

H(x, u)

△

= V (F (x, u)) − V (x) + L(x, u) + Γ(x, u). (14.320)

NonlinearBook10pt November 20, 2007

896 CHAPTER 14

Show, with the feedback control u(·) = φ(x(·)), that there exists a

neighborhood D

⊆ D of the origin such that if x

∈ D

and w(k) ≡ 0, the

solution zero x(k) ≡ 0 of the closed-loop system is locally asymptotically

stable. If, in addition, Γ(x, φ(x)) ≥ 0, x ∈ D, then show

J(x

, φ(x(·))) ≤ J(x

, φ(x(·))) = V (x

), (14.321)

where

J(x

, u(·))

△

∞

k=0

[L(x(k), u(k)) + Γ(x(k), u(k))] (14.322)

and where u(·) is admissible and x(k), k ∈ Z

, solves (14.308) with w(k) ≡ 0.

Furth ermore, if x

∈ D

show that the feedback control u(·) = φ(x(·))

minimizes J(x

, u(·)) in the s en se that

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)), (14.323)

where S(x

) is the set of regulation controllers for the system (14.308) with

w(k) ≡ 0 and x

∈ D. Finally, show that the solution x(k), k ∈ Z

, of

(14.308) satisﬁes the dissip ativity constraint

i=0

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

) ≥ 0, w(·) ∈ ℓ

, k ∈ Z

. (14.324)

Problem 14.6. Consider the linear discrete-time system

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

, k ∈ Z

, w(·) ∈ ℓ

(14.325)

z(k) = E

x(k) + E

u(k), (14.326)

with performance functional

J(x

, u(·)) =

∞

k=0

(k)R

x(k) + u

(k)R

u(k)], (14.327)

where A ∈ R

n×n

, B ∈ R

n×m

, D ∈ R

n×d

, E

∈ R

p×n

, E

∈ R

p×m

, R

△

> 0, R

△

= E

> 0, and u(·) is admissible. Assume there exists

P ∈ P

such that

P = A

P A + R

+ A

P D(γ

−D

P D)

−1

P A − P

−1

, (14.328)

where γ > 0 and

△

= R

+ B

P B + B

P D(γ

− D

P D)

−1

P B,

△

= B

P A + B

P D(γ

− D

P D)

−1

P A.

Show that, with the feedback control law u = φ(x) = −R

−1

x, the closed-

loop undisturbed (w(k) ≡ 0) system (14.325) is globally asymptotically

NonlinearBook10pt November 20, 2007

DISCRETE-TIME NONLINEAR CONTROL 897

stable for all x

∈ R

. Furthermore, show

J(x

, φ(x(·))) ≤ J(x

, φ(x(·))) = x

P x

, (14.329)

where

J(x

, u(·)) =

∞

k=0

[(D

P (Ax(k) + Bu(k)))

(γ

− D

P D)

−1

P (Ax(k)

+Bu(k))) + x

(k)R

x(k) + u

(k)R

u(k)], (14.330)

and where u(·) is admissible and x(k), k ∈ Z

, solves (14.325) with w(k) ≡

0. In addition, show

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)), (14.331)

where S(x

) is the set of regulation controllers for the system (14.325) with

w(k) ≡ 0 and x

∈ R

. Finally, if x

= 0 show that, with u = φ(x), the

solution x(k), k ∈ Z

, of (14.325) satisﬁes the nonexpansivity constraint

i=0

(i)z(i) ≤ γ

i=0

(i)w(i), w(·) ∈ ℓ

, k ∈ Z

, (14.332)

or, equivalently, |||

G(z)|||

∞

≤ γ, where K

△

= −R

−1

and

G(z) ∼



A + BK D

+ E

K 0



Problem 14.7. Let p = d and consider the linear system (14.325) with

performance variables

z(k) = E

x(k) + E

u(k) + E

∞

w(k), (14.333)

and performance functional (14.327). Ass ume there exists P ∈ P

such that

P = A

P A + R

+ (D

P A −E

)

−1

P A −E

) −P

−1

, (14.334)

where

△

= E

∞

+ E

∞

− D

P D,

△

= B

P B + R

+ (D

P B − E

)

−1

P B − E

△

= B

P A + (D

P B − E

)

−1

P A − E

Show that, with the feedback control law u = φ(x) = −R

−1

x, the closed-

loop undisturbed (w(k) ≡ 0) sys tem (14.325) is globally asymptotically

stable for all x

∈ R

. Furthermore, show

J(x

, φ(x(·))) ≤ J(x

, φ(x(·))) = x

P x

, (14.335)

where

J(x

, u(·))

NonlinearBook10pt November 20, 2007

898 CHAPTER 14

∞

k=0

(k)(R

+ (D

P A − E

)

−1

P A −E

))x(k) + u

(k)

·(R

− B

P B)u(k) + 2x

(k)(D

P A −E

)

−1

P B − E

)u(k)],

(14.336)

and where u(·) is admissible and x(k), k ∈ Z

, solves (14.325) with w(k) ≡

0. In addition, s how

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)), (14.337)

where S(x

) is the set of regulation controllers for the system (14.325) with

w(k) ≡ 0 and x

∈ R

. Finally, if x

= 0 show that, with u = φ(x), the

solution x(k), k ∈ Z

, of (14.325) satisﬁes the passivity constraint

i=0

(i)w(i) ≥ 0, w(·) ∈ ℓ

, k ∈ Z

, (14.338)

or, equivalently,

∞

(z) +

∗

∞

(z) ≥ 0, |z| > 1, where K = −R

−1

and

∞

(z) ∼



A + BK D

+ E

K E

∞



Problem 14.8. Consider the controlled uncertain dynamical system

x(k + 1) = F (x(k), u(k)), x(0) = x

, k ∈ Z

, (14.339)

with performance functional

J(x

, u(·))

△

∞

k=0

L(x(k), u(k)), (14.340)

where F (·, ·) ∈ F ⊂ {F : D×U → R

: F (0, 0) = 0}, F denotes the class of

uncertain closed-loop systems with F

(·, ·) ∈ F deﬁning the n ominal closed-

loop system, L : D×U → R, and u(·) is an admissible control. Assume there

exist functions V : D → R, Γ : D × U → R, and a control law φ : D → U

such that V (·) is continuous and

V (0) = 0, (14.341)

V (x) > 0, x ∈ D, x 6= 0, (14.342)

φ(0) = 0, (14.343)

V (F (x, φ(x))) ≤ V (F

(x, φ(x))) + Γ(x, φ(x)), x ∈ D, F(·, ·) ∈ F,

(14.344)

V (F

(x, φ(x))) − V (x) + Γ(x, φ(x)) < 0, x ∈ D, x 6= 0, (14.345)

H(x, φ(x)) = 0, x ∈ D, (14.346)

H(x, u) ≥ 0, x ∈ D, u ∈ U, (14.347)

NonlinearBook10pt November 20, 2007

DISCRETE-TIME NONLINEAR CONTROL 899

where

H(x, u)

△

= L(x, u) + V (F

(x, u)) − V (x) + Γ(x, u). (14.348)

Show that, with the feedback control u(·) = φ(x(·)), there exists a

neighborhood D

⊆ D of the origin such that if x

∈ D

, the zero solution

x(k) ≡ 0 of the closed-loop system is locally asymptotically stable for all

F (·, ·) ∈ F. Furthermore, s how

sup

F (·,·)∈F

J(x

, φ(x(·))) ≤ J(x

, φ(x(·))) = V (x

), (14.349)

where

J(x

, u(·))

△

∞

k=0

[L(x(k), u(k)) + Γ(x(k), u(k))], (14.350)

and where u(·) is admissible and x(k), k ∈ Z

, solves (14.339) w ith

F (x(k), u(k)) = F

(x(k), u(k)). In addition, if x

∈ D

show that the

feedback control u(·) = φ(x(·)) minimizes J(x

, u(·)) in the sense th at

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)), (14.351)

where S(x

) is the set of regulation controllers for (14.339) w ith F (·, ·) =

(·, ·).

14.14 Notes and References

The principle of optimality is due to Bellman [36]. The Bellman equation was

developed by Richard E. Bellman [36] in the late 1950s and is the functional

equation of dynamic programming. For a modern textbook treatment, see

Kirk [239] and Bryson and Ho [72].

The resu lts in Sections 11.2–11.6 on stability analysis and optimal

and inverse optimal control of nonlinear discrete-time systems are due

to Haddad and Chellaboina [155] and Haddad, Chellaboina, Fausz, and

Abdallah [165]. In particular, [155,165] give an overview of the discrete-time,

nonlinear-nonquadratic control problem with polynomial and multilinear

cost functionals. A brief treatment of nonlinear-nonquadratic discrete-

time optimal control is also given by Jacobson [217]. For a treatment

on stabilization of discrete-time nonlinear systems see Tsinias [432] and

Byrnes, Lin, and Ghosh [79]. Gain, sector, and disk margins of nonlinear-

nonquadratic optimal regulators are due to Chellaboina and Haddad [86].

The original work on gain and phase margins of linear-quadratic discrete-

time optimal regulators is due to Willems and van de Voorde [459] and

Shaked [397]. Later extensions were presented by Lee and Lee [262]. The

discrete-time nonlinear disturbance rejection framework using dissipativity,

NonlinearBook10pt November 20, 2007

900 CHAPTER 14

passivity, and n on exp ansivity concepts presented in Sections 11.10–11.12 is

due to Haddad, Chellaboina, and Wu [168]. See also Lin and Byrnes [280].

The linear discrete-time mixed-norm H

∞

problem was formulated by

Haddad, Bernstein, and Mustafa [154]. A state space approach to linear

discrete-time H

∞

can be found in S toorvogel [418] and I glesias and Glover

[211].

Finally, the stability analysis of nonlinear uncertain discrete-time

systems presented in Section 14.12 is a generalization of the linear discrete-

time guaranteed cost control framework developed by Haddad, Huang, and

Bernstein [173] and Haddad and Bernstein [148,149].

NonlinearBook10pt November 20, 2007

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