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

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NonlinearBook10pt November 20, 2007

DISCRETE-TIME NONLINEAR CONTROL 865

Lemma 14.3. Consider the nonlinear system G given by (14.97) and

(14.98), where φ(x) is a stabilizing feedback control law given by (14.104)

and where V (x), P

(x), and P

(x) satisfy (14.73), (14.74), (14.76),

(14.100), and (14.101). Then for all u(·) ∈ U and k

, k

∈ Z

, k

< k

the solution x(k), k ∈ Z

, of the closed-loop system (14.103) satisﬁes

V (x(k

)) ≤

−1

k=k



(u(k) + y(k))

(x(k)) + P

(x(k)))(u(k) + y(k))

−u

(k)R

(x(k))u(k)



+ V (x(k

)). (14.107)

Proof. Note that it follows from (14.101) that for all u(·) ∈ U and

k ∈ Z

(k)R

(x(k))u(k)

≤ L

(x(k)) + u

(k)R

(x(k))u(k)

= −V (f(x(k)) + V (x(k)) +

(x)(R

(x) + P

(x))

−1

(x)

(k)R

(x(k))u(k)

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

(x)(R

(x) + P

(x))

−1

(x) + u

(k)R

(x(k))u(k)

(x(k))u(k) + u

(k)P

(x(k))u(k)

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

+(u(k) + y(k))

(x(k)) + P

(x(k)))(u(k) + y(k))

= −V (x(k + 1)) + V (x(k)) + (u(k) + y(k))

(x(k))

(x(k)))(u(k) + y(k)).

Now, summing over [k

, k

] yields (14.107).

Note that with R

(x) ≡ I condition (14.107) is precisely the discrete-

time counterpart of the return diﬀerence condition given by (8.120) f or

continuous-time sys tems. However, as shown in Theorem 8.5, in the

continuous-time case a feedback law φ(x) satisfying the return diﬀerence

condition is equivalent to the fact that a continuous-time nonlinear aﬃne

system with inp ut u and outpu t y = − φ(x) is dissipative with respect to

the quadratic supp ly rate [u + y]

[u + y] −u

u. Hence, using the nonlinear

Kalman-Yakubovich-Popov lemma one can show that a feedback control

law φ(x) satisﬁes the return diﬀerence inequality if and only if φ(x) is

optimal with respect to a performance criterion involving a nonnegative-

deﬁnite weighting function on the s tate. Alternatively, in the discrete-time

case (14.107) is not equivalent to the dissipativity property of (14.97) and

(14.98) du e to the presence of P

(x). However, as will be shown below,

(14.107) does imply that G is dissipative with respect to a quadratic supply

rate.

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866 CHAPTER 14

Now, we present our main result, wh ich provides disk margins for the

nonlinear-nonquadratic optimal regulator given by Theorem 14.6. For the

following result deﬁne

△

= sup

x∈R

max

(x) + P

(x)), γ

△

= inf

x∈R

min

(x)). (14.108)

Theorem 14.7. Consider the nonlinear s ystem G given by (14.97) and

(14.98) where φ(x) is a stabilizing feedback control law given by (14.104) and

where V (x), P

(x), and P

(x) satisfy (14.73), (14.74), (14.76), (14.100),

and (14.101). Then the nonlinear system G has a disk margin (

1+θ

1−θ

where θ

△

γ/γ.

Proof. Note that for all u(·) ∈ U and k

, k

∈ Z

, k

< k

, it follows

from Lemma 14.3 that the solution x(k), k ∈ Z

, of the closed-loop system

(14.103) satisﬁes

V (x(k

)) − V (x(k

)) ≤

−1

k=k



(u(k) + y(k))

(x(k))

(x(k)))(u(k) + y(k)) − u

(k)R

(x(k))u(k)



which implies that

V (x(k

)) − V (x(k

)) ≤

−1

k=k

[γ(u(k) + y(k))

(u(k) + y(k)) − γu

(k)u(k)].

Hence, w ith the storage function V

(x) =

2γ

V (x), G is dissipative with

respect to supply rate r(u, y) = u

y +

(1−θ

)

u +

y. Now, the result

follows immediately from Corollary 13.17 and Deﬁnition 6.3 with α =

1+θ

and β =

1−θ

14.7 Linear-Quadratic Optimal Regulators

In this section, we specialize Theorem 14.7 to the case of linear discrete-time

systems. Speciﬁcally, consider th e stabilizable linear system given by

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

, k ∈ Z

, (14.109)

y(k) = −Kx(k), (14.110)

where A ∈ R

n×n

, B ∈ R

n×m

, and K ∈ R

m×n

, and assum e that (A, K)

is detectable and the linear system (14.109) and (14.110) is asymptotically

stable with the feedback u = −y. Furtherm ore, assume that K is an optimal

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DISCRETE-TIME NONLINEAR CONTROL 867

regulator which minimizes the quadratic perf ormance functional given by

J(x

, u(·)) =

∞

k=0

(k)R

x(k) + u

(k)R

u(k)], (14.111)

where R

∈ R

n×n

and R

∈ R

m×m

are such that R

≥ 0, R

> 0, and (A, R

)

is observable. In this case, it follows from Theorem 14.6 with f (x) = Ax,

G(x) = B, L

(x) = x

x, R

(x) = R

, φ(x) = Kx, and V (x) = x

P x

that the optimal control law K is given by K = −(R

+ B

P B)

−1

P A,

where P > 0 is the solution to the discrete-time algebraic regulator Riccati

equation given by

P = A

P A + R

− A

P B(R

+ B

P B)

−1

P A. (14.112)

The following result provides guarantees of gain and sector margins for the

linear system (14.109) and (14.110).

Corollary 14.4. Consider the linear system (14.109) and (14.110) with

performance functional (14.111). Then, with K = −(R

P B)

−1

P A,

where P > 0 solves (14.112), the linear system (14.109) and (14.110) has

sector (and, hence, gain) margin (

1+θ

1−θ

), where

θ =



min

)

max

+ B

P B)



1/2

. (14.113)

Proof. The result is a direct consequence of Theorem 14.7 with f (x) =

Ax, G(x) = B, φ(x) = Kx, V (x) = x

P x, and L

(x) = x

x. Speciﬁcally,

(14.73), (14.74), and (14.76) are trivially satisﬁed and note that (14.112) is

equivalent to (14.101). Finally, since (A, R

) is observable (14.112) implies

(14.100) so that all the conditions of Theorem 14.7 are satisﬁed.

Note that for single-input systems

+ B

P B

= 1 −(R

+ B

P B)

−1

P B

= det(I − B(R

+ B

P B)

−1

P )

det(A − B(R

+ B

P B)

−1

P A)

detA

det(A + BK)

detA

where K = −(R

P B)

−1

P A. In this case, the gain margins obtained

in Corollary 14.4 are precisely the gain margins given in [459] for discrete-

time linear-quadratic optimal regulators.

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868 CHAPTER 14

The following result specializes Lemma 13.35 to discrete-time linear

systems and recovers Theorem 1 of [262].

Corollary 14.5. Consider the linear system (14.109) and (14.110) and

let α, β ∈ R be s uch that 0 ≤ α < 1 < β < ∞. Suppose there exist a

positive-deﬁnite matrix P ∈ R

n×n

and a scalar q > 0 such that

0 < 2qI − B

P B, (14.114)

0 ≥ A

P A −P −

2αβ

q(α+β)

P BB

P A. (14.115)

Then, with K = −

q(α+β)

P A, the linear system (14.109) and (14.110)

has a disk margin (α, β).

Proof. The result is a direct consequence of T heorem 13.35 with

f(x) = Ax, G(x) = B, φ(x) = Kx, V

(x) = x

P x, P

(x) = 2x

P B,

(x) = B

P B, and Z = 2qI.

Note that since the controller in Corollary 14.5 guarantees a disk

margin of (α, β) it also guarantees sector and gain margins of (α, β) and

phase margin φ given by (see Problem 6.20)

cos(φ) =

1 + αβ

α + β

. (14.116)

14.8 Stability Margins, Meaningful Inverse Optimality, and

Control Lyapunov Functions

In this section, we give suﬃcient conditions that guarantee that a given

nonlinear feedback controller of a speciﬁc form is inverse optimal and has

certain disk, sector, and gain margins.

Theorem 14.8. Consider the nonlinear s ystem G given by (14.97) and

(14.98). Assume there exist fu nctions V : R

→ R, P

: R

→ R

1×m

: R

→ N

, and γ : R

→ R such that V (·) is continuous and satisﬁes

(14.73), (14.74), (14.76), and for all x ∈ R

0 > V (f(x) −

−1

(x)G(x)P

(x)) − V (x), x 6= 0, (14.117)

0 < γ(x)I − P

(x), (14.118)

0 ≥ V (f(x)) − V (x) −

−1

(x)P

(x), (14.119)

and

V (x) → ∞ as kxk → ∞. (14.120)

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

−1

(x)P

(x), the nonlinear

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DISCRETE-TIME NONLINEAR CONTROL 869

system G has a disk margin (

1+θ

1−θ

), where

θ =



inf

x∈R

min

(γ(x)I − P

(x))

sup

x∈R

γ(x)



1/2

. (14.121)

Furth ermore, with the feedback control law φ(x) the performance functional

J(x

, u(·)) =

∞

k=0

[−V (f (x(k)) + G(x(k))u(k)) + V (x(k))

+γ(x(k))(u(k) − φ(x(k)))

(u(k) − φ(x(k)))



, (14.122)

is minimized in the sense that

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)), x

∈ R

. (14.123)

Proof. The r esult is direct consequence of Theorem 14.6 and Theorem

14.7 with R

(x) = γ(x)I − P

(x) and L

(x) = −V (f(x)) + V (x) +

γ(x)φ

(x)φ(x).

Example 14.1. Consider the discrete-time nonlinear system given by

(k + 1) =

(k), x

(0) = x

, k ∈ Z

, (14.124)

(k + 1) =

(k) + u(k), x

(0) = x

, (14.125)

y(k) = 2(x

(k) + u(k)). (14.126)

With x = [x

, x

]

, f(x) = [

]

, G(x) = [0, 1]

, h(x) = 2x

J(x) = 2, V

(x) = x

+ 4x

(x) = 4x

(x) = 4, ℓ(x) = [

, 2x

]

W(x) = 0, Q = R = 0, and S = I

, it follows from Theorem 13.21 that

the discrete-time nonlinear system G is passive, that is, G is diss ipative with

respect to the sup ply rate r(u, y) = 2yu. Speciﬁcally, note that V

(0) = 0,

(x) > 0, x 6= 0, and

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

+ 4x

u + 4u

= V

(f(x)) +

(x)u +

(x)u

, (14.127)

so that (13.137) is satisﬁed. Now, (13.138)–(13.140) can be easily veriﬁed

to show the passivity of the discrete-time system G.

Next, with V (x) = x

+ 4x

, P

(x) = 4x

, P

(x) = 4, L

(x) =

+ 4x

, and R

(x) = 60, it follows from Theorem 14.7 that the feedback

control law φ(x) = −

(x) + P

(x))

−1

(x) = −

minimizes the

performance functional (14.99) in the sense of (14.105). Furthermore, it

follows that θ =

15/16, and hence, the nonlinear system G has a disk

margin (0.5081, 31.4919). Finally, by choosing γ(x) to be a constant such

that γ(x) > 4, a whole class of inverse optimal controllers can be constructed

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870 CHAPTER 14

using Theorem 14.8. In particular, setting γ(x) to be very large we can

obtain an inverse optimal controller that has a disk m argin (

1+θ

1−θ

), where

θ is arbitrarily close to 1 so as to maximize disk margin guarantees. △

Next, we give suﬃcient conditions that guarantee that a given

nonlinear feedback controller of a speciﬁc form is inverse optimal and has

prespeciﬁed stability margins.

Corollary 14.6. Consider the nonlinear system G given by (14.97) and

(14.98). Let θ ∈ (0, 1) and assume there exist functions V : R

→ R,

: R

→ R

1×m

, P

: R

→ N

, and a scalar q > 0 such that V (·) is

continuous and satisﬁes (14.73), (14.74), and (14.76), and for all x ∈ R

0 > V (f (x) −

1−θ

G(x)P

(x)) −V (x), x 6= 0, (14.128)

0 < qI − P

(x), (14.129)

0 ≥ V (f (x)) −V (x) −

1−θ

(x)P

(x), (14.130)

and

V (x) → ∞ as kxk → ∞. (14.131)

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

1−θ

(x), the nonlinear

system G has a disk margin (

1+θ

1−θ

). Furthermore, with the feedback

control law φ(x) the performance functional

J(x

, u(·)) =

∞

k=0

[−V (f (x(k)) + G(x(k))u(k)) + V (x(k))

1−θ

(u(k) − φ(x(k)))

, (14.132)

is minimized in the sense that

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)), x

∈ R

. (14.133)

Proof. The result is a d ir ect consequence of Theorem 14.8 with γ(x) =

1−θ

I. Speciﬁcally, s ince θ ∈ (0, 1), (14.129) implies (14.118) so that all the

conditions of Theorem 14.8 are satisﬁed.

Note that (14.129) implies 0 <

1−θ

I − P

(x), x ∈ R

and θ ∈ (0, 1).

Hence, if there exist functions V : R

→ R, P

: R

→ R

1×m

, P

: R

→

, and a scalar q > 0 such that (14.73), (14.74), (14.76) and (14.128)–

(14.130) are satisﬁed, then it follows f rom Theorem 14.8 with γ(x) =

1−θ

that the nonlinear system G has a d isk margin of (

1−

), where

△



1 −

1−θ

sup

x∈R

min

(x))



1/2

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DISCRETE-TIME NONLINEAR CONTROL 871

In this case, since 0 < qI −P

(x), x ∈ R

, it follows that θ ≤

θ so that the

disk margins provided by Theorem 14.8 are always greater than or equal

to the disk margins provided by Corollary 14.6. However, in the latter

case considerable numerical simpliﬁcation in computing the disk margins is

achieved in comparison to Theorem 14.8.

The following result specializes Corollary 14.6 to discrete-time linear

systems and recovers Theorem 3 of [262].

Corollary 14.7. Consider the linear system (14.109) and (14.110), and

let θ ∈ R be such that 0 < θ < 1. Suppose there exist a positive-deﬁnite

matrix P ∈ R

n×n

, a nonnegative-deﬁnite matrix R

∈ R

n×n

, and a scalar

q > 0 such that

0 < 2qI − B

P B, (14.134)

0 = A

P A − P + R

−

1−θ

P BB

P A. (14.135)

If (A, R

) is observable then, with K = −

1−θ

P A, the linear system

(14.109) and (14.110) has a disk margin (

1+θ

1−θ

). In addition, with th e

feedback control law φ(x) = Kx, the performance functional

J(x

, u(·)) =

∞

k=0

(k)R

x(k) + u

(k)(

1−θ

I − B

P B)u(k)

, (14.136)

is minimized in the sense that

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)), x

∈ R

. (14.137)

Proof. The result is a dir ect consequence of Corollary 14.6 with f(x) =

Ax, G(x) = B, φ(x) = Kx, V (x) = x

P x, P

(x) = 2x

P B, P

(x) =

P B, and q replaced by 2q.

Next, since φ(x) given by (13.243) has no a priori guarantees of

stability margins we use Theorem 14.8 to obtain a feedback control law

that has stability margins. Let γ(x) = σ

max

(x))+ η(x), where η(x) > 0,

x ∈ R

, so that γ(x)I > P

(x). In this case, it follows from Theorem 14.8

that if γ(·), V (·), and P

(·) satisfy (14.119) then the feedback control law

φ(x) = −

−1

(x)P

(x) is inverse optimal with respect to the perf ormance

functional (14.122) and the nonlinear system G possesses certain disk

margins. Next, note that (14.119) is equivalent to

(x)P

†

(x)P

(x) − γ

−1

(x)P

(x) ≤ ε(x), x ∈ R

, (14.138)

where ε(x)

△

= 4(V (x) − V (f(x))) + P

(x)P

†

(x)P

(x) > 0, x 6= 0, since

V (x) is a control Lyapunov function. Note that (14.138) can equivalently

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872 CHAPTER 14

be written as

η(x)(P

(x)P

†

(x)P

(x) − ε(x)) ≤ P

(x)P

(x)

−σ

max

(x))(P

(x)P

†

(x)P

(x) − ε(x)). (14.139)

Now, if P

(x)P

†

(x)P

(x) − ε(x) < 0, then η(x) can be chosen to be an

arbitrary positive-deﬁnite function. However, if P

(x)P

†

(x)P

(x)−ε(x) >

0, then there may not exist a function η(x) such that (14.139) is satisﬁed,

which implies that the control laws obtained by using the above control

Lyapunov function do not possess any disk margins. This is in contrast to

the continuous-time case where it was shown in Section 8.8 that if there

exists a control Lyapunov function, then there always exist feedback control

laws that have guaranteed sector and gain margins of (

, ∞).

In order to d emonstrate the above observations consider the linear

discrete-time system given by



(k + 1)





2 0

0 2



(k)





1 0

0 0.1



(k)





(0)







, (14.140)

with control Lyapunov function candidate V (x) = x

x, where x

△

= [x

, x

]

In this case, f (x) = 2x, G(x) = diag[1, 0.1], u = [u

, u

]

, P

(x) =

[4x

, 0.4x

]

, and P

(x) = diag[1, 0.01]. Hence,

V (f (x)) −V (x) −

(x)P

†

(x)P

(x) = −x

x < 0, x ∈ R

, x 6= 0,

which shows that V (x) = x

x is a control Lyapunov function for (14.140).

In this case, (14.139) becomes

η(x)[12x

+ 12x

] ≤ 4x

− 11.84x

. (14.141)

Now, taking x = [1, 1]

, (14.141) is equivalent to

η(x) ≤ −0.326 < 0, (14.142)

which shows that it is not possib le to construct η(x) > 0 such that (14.139)

is satisﬁed.

14.9 Nonlinear Discrete-Time Dynamical Systems with

Bounded Disturbances

In this and the next two sections we extend the analysis results of Chapter

10 to discrete-time dyn amical systems. Speciﬁcally, we present suﬃcient

conditions for dissipativity for a class of nonlinear discrete-time systems

with bounded energy and boun ded amplitude disturbances. In addition, we

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DISCRETE-TIME NONLINEAR CONTROL 873

consider the p roblem of evaluating a performance bound for a nonlinear-

nonquadratic cost functional. The cost bound is evaluated in closed form

by relating the cost functional to an underlying Lyapunov function that

guarantees asymptotic stability of the nonlinear system. For the following

result, let D ⊂ R

be an open set, assume 0 ∈ D, let f : D → D be su ch

that f (0) = 0, h : D → R

be such that h(0) = 0, J

: D → R

n×d

, and

: D → R

p×d

. Finally, let W ⊂ R

and let r : R

× R

→ R be a given

function.

Lemma 14.4. Consider the nonlinear discrete-time system

x(k + 1) = f (x(k)) + J

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

, k ∈ Z

, w(·) ∈ ℓ

(14.143)

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

(x(k))w(k). (14.144)

Furth ermore, assume there exist functions Γ : D → R, P

: D → R

1×d

: D → N

, and V : D → R such th at V (·) is continuous and

(0) = 0, (14.145)

V (0) = 0, (14.146)

V (x) ≥ 0, x ∈ D, (14.147)

(x)w + w

(x)w ≤ r(z, w) + Γ (x), x ∈ D, w ∈ W, (14.148)

V (f (x) + J

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

(x)w + w

(x)w,

x ∈ D, w ∈ W, (14.149)

V (f(x)) − V (x) + Γ(x) ≤ 0, x ∈ D. (14.150)

Then the solution x(k), k ∈ Z

, of (14.143) satisﬁes

V (x(k + 1)) ≤

i=0

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

), w(·) ∈ ℓ

, k ∈ Z

(14.151)

Proof. Let x(k), k ∈ Z

, satisfy (14.143) and let w(·) ∈ ℓ

. Then it

follows from (14.148), (14.149) and (14.150) that for all k ∈ Z

∆V (x(k))

△

= V (x(k + 1)) − V (x(k))

= V (f(x(k)) + J

(x(k))w(k)) − V (x(k))

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

(x(k))w(k) + w

(k)P

(x(k))w(k) − V (x(k))

= V (f(x(k))) − V (x(k)) + Γ(x(k)) + P

(x(k))w(k)

(k)P

(x(k))w(k) − Γ(x(k))

≤ r(z(k), w(k)), k ∈ Z

(14.152)

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874 CHAPTER 14

Now, summing over [0, k] yields

V (x(k + 1)) − V (x

) ≤

i=0

r(z(i), w(i)), k ∈ Z

which proves the result.

For the next r esult let L : D → R be given.

Theorem 14.9. Consider the nonlinear system given by (14.143) and

(14.144) with performance functional

J(x

)

△

∞

k=0

L(x(k)), (14.153)

where x(k), k ∈ Z

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

functions Γ : D → R, P

: D → R

1×d

, P

: D → N

, and V : D → R such

that V (·) is continuous and

(0) = 0, (14.154)

V (0) = 0, (14.155)

V (x) > 0, x ∈ D, x 6= 0, (14.156)

(x)w + w

(x)w ≤ r(z, w) + L(x) + Γ(x), x ∈ D, w ∈ W,

(14.157)

V (f(x)) − V (x) < 0, x ∈ D, x 6= 0, (14.158)

V (f (x) + J

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

(x)w + w

(x)w,

x ∈ D, w ∈ W, (14.159)

L(x) + V (f(x)) − V (x) + Γ(x) = 0, x ∈ D. (14.160)

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 if Γ (x) ≥ 0, x ∈ D, then

J(x

) ≤ J(x

) = V (x

), x

∈ D

, (14.161)

where

J(x

)

△

∞

k=0

[L(x(k)) + Γ(x(k))] (14.162)

and where x(k), k ∈ Z

, is a solution to (14.143) with w(k) ≡ 0.

Furth ermore, the solution x(k), k ∈ Z

, to (14.143) satisﬁes the dissipativity

constraint

i=0

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

) ≥ 0, w(·) ∈ ℓ

, k ∈ Z

. (14.163)