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

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DISCRETE-TIME NONLINEAR CONTROL 855

Lyapunov function for (14.21). Hence, Lemma 14.2 can be used to generate

Lyapunov functions of speciﬁc structure.

Suppose now that h(x) in (14.31) is of the more general form

h(x) =

ν=2

2ν

(x), (14.37)

where, for ν = 2, . . . , r, h

2ν

: R

→ R is a nonnegative-deﬁnite 2ν-

multilinear function. Now, using Lemma 14.2, let g

2ν

: R

→ R be the

nonnegative-deﬁnite 2ν-multilinear functions satisfying

0 = g

2ν

(Ax) − g

2ν

(x) + h

2ν

(x), x ∈ R

, ν = 2, . . . , r, (14.38)

and d eﬁ ne

g(x)

△

ν=2

2ν

(x). (14.39)

Now, summing (14.38) over ν yields (14.30). Since (14.17) is satisﬁed with

L(x) an d V (x) given by (14.31) and (14.33), respectively, (14.18) implies

that

J(x

) = x

P x

+ g(x

). (14.40)

As another illustration of condition (14.30), suppose that V (x) is

constrained to be of the form

V (x) = x

P x + (x

Mx)

, (14.41)

where P satisﬁes (14.23) and M is an n ×n symmetric matrix. In this case,

g(x) = (x

Mx)

is a nonnegative-deﬁnite 4-multilinear function. Then

(14.30) yields

h(x) = −(x

MA + M )x)(x

MA − M )x). (14.42)

R is an n ×n symmetric matrix and M is chosen to satisfy

M = A

MA +

R, (14.43)

then (14.42) implies that h(x) satisfying (14.30) is of th e form

h(x) = (x

MA + M)x)(x

Rx). (14.44)

Note that if

R is nonnegative deﬁnite, then M is also nonnegative deﬁnite,

and hence, h(x) is a nonnegative-deﬁnite 4-multilinear function. Thus, if

V (x) is of the form (14.41), and L(x) is given by

L(x) = x

Rx + (x

MA + M )x)(x

Rx), (14.45)

where M and

R satisfy (14.43), then cond ition (14.30), and hence, (14.17) is

satisﬁed. The following proposition generalizes the above resu lts to general

polynomial functionals.

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856 CHAPTER 14

Proposition 14.1. Let A ∈ R

n×n

be Schur, R ∈ P

, and

∈ N

q = 2, . . . , r. Consid er the linear sy s tem (14.21) with perform an ce fu nctional

J(x

)

△

∞

k=0

(k)Rx(k) +

q=2

(k)

x(k))

j=1

(k)M

x(k))

j−1

·(x

(k)A

Ax(k))

q−j

. (14.46)

Furth ermore, assume that there exist P ∈ P

and M

∈ N

, q = 2, . . . , r,

such that

P = A

P A + R, (14.47)

= A

A +

, q = 2, . . . , r. (14.48)

Then the zero solution x(k) ≡ 0 to (14.21) is globally asymptotically stable

and

J(x

) = x

P x

q=2

)

. (14.49)

Proof. The resu lt is direct consequence of Theorem 14.3 with f(x) =

Ax, L(x) = x

Rx +

q=2

j=1

j−1

Ax)

q−j

V (x) = x

P x +

q=2

, and D = R

. Speciﬁcally, conditions

(14.14) and (14.15) are trivially satisﬁed. Now,

V (f (x)) −V (x) = x

P A − P )x +

q=2

A −M

j=1

j−1

Ax)

q−j

and hence, it follows from (14.47) and (14.48) that L(x)+V (f(x))−V (x) = 0

so that all the conditions of Theorem 14.3 are satisﬁed. Finally, since V (x)

is radially unbounded (14.21) is globally asymptotically stable.

Proposition 14.1 requires the solutions of r −1 Lyapun ov equations in

(14.48) to obtain the nonquadratic cost. However, if

, q = 3, . . . , r,

then M

= M

, q = 3, . . . , r, satisﬁes (14.48). In this case, we require the

solution of one Lyapunov equation in (14.48).

14.4 Optimal Discrete-Time Nonlinear Control

In this section, we consider the discrete-time control problem involving

a notion of optimality with respect to a nonlinear-nonquadratic cost

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DISCRETE-TIME NONLINEAR CONTROL 857

criterion. The optimal feedback controllers are derived as a direct con-

sequence of Theorem 14.3 and pr ovide a discrete-time analog of the

continuous-time Hamilton-Jacobi-Bellman conditions f or time-invariant,

inﬁnite-horizon problems for addressing optimal controllers addressed in

Section 8.3. To address the discrete-time optimal control problem let

D ⊂ R

be an open set and let U ⊆ R

, where 0 ∈ D and 0 ∈ U, and

let F : D × U → R

such that F (0, 0) = 0. Next, consider the controlled

system

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

, k ∈ Z

, (14.50)

where the control u(·) is restricted to the class of admissible controls

consisting of measurable functions u(·) such that u(k) ∈ U for all k ∈ Z

where the control constraint set U is given. Given a control law φ(·) and a

feedback control law u(k) = φ(x(k)), the closed-loop system has the form

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

, k ∈ Z

. (14.51)

Next, we present a key theorem for characterizing discrete-time

feedback controllers that guarantee stability for a nonlinear discrete-time

system and minimize a nonlinear-nonquadratic performance functional. For

the statement of this result let L : D×U → R and deﬁne the set of regulation

controllers for the nonlinear system F (·, ·) by

S(x

)

△

= {u(·) : u(·) is admissible and x(·) given by (14.50)

satisﬁes x(k) → 0 as k → ∞}.

Theorem 14.4. Consider the nonlinear controlled system (14.50) with

performance functional

J(x

, u(·))

△

∞

k=0

L(x(k), u(k)), (14.52)

where u(·) is an admissible control. Assume that there exist a continuous

function V : D → R and control law φ : D → U such that

V (0) = 0, (14.53)

V (x) > 0, x ∈ D, x 6= 0, (14.54)

φ(0) = 0, (14.55)

V (F (x, φ(x))) − V (x) < 0, x ∈ D, x 6= 0, (14.56)

H(x, φ(x)) = 0, x ∈ D, (14.57)

H(x, u) ≥ 0, x ∈ D, u ∈ U, (14.58)

where

H(x, u)

△

= L(x, u) + V (f(x, u)) − V (x). (14.59)

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858 CHAPTER 14

Then, with the feedback control u(·) = φ(x(·)), the zero solution x(k) ≡ 0

of the closed-loop system (14.51) is locally asymptotically stable and there

exists a neighborhood of the origin D

⊆ D such that

J(x

, φ(x(·))) = V (x

), x

∈ D

. (14.60)

In addition, if x

∈ D

then 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(·)). (14.61)

Finally, if D = R

and

V (x) → ∞ as kxk → ∞, (14.62)

then the zero solution x(k) ≡ of the closed-loop system (14.51) is globally

asymptotically stable.

Proof. Local and global asymptotic stability is a direct consequence

of (14.53)–(14.56) and (14.62) by applying Theorem 14.3 to the closed-

loop system (14.51). Furthermore, using (14.57), condition (14.60) is a

restatement of (14.18) as app lied to the closed-loop system. Next, let

∈ D

, let u(·) ∈ S(x

), and let x(·) be the solution of (14.50). T hen

it follows that

0 = −∆V (x(k)) + V (F (x(k), u(k))) − V (x(k)).

Hence,

L(x(k), u(k)) = −∆V (x(k)) + L(x(k), u(k)) + V (F (x(k), u(k))) − V (x(k))

= −∆V (x(k)) + H(x(k), u(k)).

Now, using (14.59) and the fact that u(·) ∈ S(x

), it follows that

J(x

, u(·)) =

∞

k=0

[−∆V (x(k)) + H(x(k), u(k))]

= − lim

k→∞

V (x(k)) + V (x

) +

∞

k=0

H(x(k), u(k))

= V (x

) +

∞

k=0

H(x(k), u(k))

≥ V (x

)

= J(x

, φ(x(·)),

which yields (14.61).

Next, we specialize Theorem 14.4 to discrete-time linear systems and

provide connections to the discrete-time, linear-quadratic-regulator problem.

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DISCRETE-TIME NONLINEAR CONTROL 859

For the following result let A ∈ R

n×n

, B ∈ R

n×m

, R

∈ P

, and R

∈ P

be given.

Corollary 14.2. Consider the discrete-time linear controlled system

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

, k ∈ Z

, (14.63)

with performance functional

J(x

, u(·))

△

∞

k=0

(k)R

x(k) + u

(k)R

u(k)], (14.64)

where u(·) is an admissible control. Furthermore, assume th at there exists

a positive-deﬁnite matrix P ∈ R

n×n

such that

P = A

P A + R

− A

P B(R

+ B

P B)

−1

P A. (14.65)

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

△

= −(R

+ B

P B)

−1

P Ax, the

zero solution x(k) ≡ 0 to (14.63) is globally asymptotically stable and

J(x

, φ(x(·))) = x

P x

. (14.66)

Furth ermore,

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)), (14.67)

where S(x

) is the set of r egulation controllers for (14.63) and x

∈ R

Proof. The result is a direct consequence of Theorem 14.4 with

F (x, u) = Ax + Bu, L(x, u) = x

x + u

u, V (x) = x

P x, D = R

and U = R

. Speciﬁcally, conditions (14.53) and (14.54) are trivially

satisﬁed. Next, it follows from (14.65) that H(x, φ(x)) = 0, and hence,

V (F (x, φ(x))) − V (x) < 0 for all x 6= 0. Thus, H(x, u) = H(x, u) −

H(x, φ(x)) = [u − φ(x)]

+ B

P B)[u − φ(x)] ≥ 0 so that all the

conditions of Theorem 14.4 are satisﬁed. Finally, since V (x) is radially

unbounded the solution x(k) = 0, k ∈ Z

, of (14.63) with u(k) = φ(x(k)) =

−(R

+ B

P B)

−1

P Ax(k), is globally asymptotically stable.

The optimal f eedback control law φ(x) in Corollary 14.2 is derived

using the properties of H(x, u) as deﬁned in Theorem 14.4. Speciﬁcally, since

H(x, u) = x

x + u

u + (Ax + Bu)

P (Ax + Bu) −x

P x it follows that

∂

H/∂u

= R

P B > 0. Now, ∂H/∂u = 2(R

P B)u+2B

P Ax =

0 gives the unique global minimum of H(x, u). Hence, since φ(x) minimizes

H(x, u) it follows that φ(x) satisﬁes ∂H/∂u = 0 or, equivalently, φ(x) =

−(R

+ B

P B)

−1

P Ax.

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860 CHAPTER 14

14.5 Inverse Optimal Control for Nonlinear Aﬃne Systems

In this section, we s pecialize Theorem 14.4 to aﬃne systems. As in the

continuous-time case, in order to avoid the complexity in solving th e Bellman

equation, we consider an inverse optimal control problem. Consider the

discrete-time nonlinear system given by

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

, k ∈ Z

, (14.68)

where f : R

→ R

satisﬁes f (0) = 0, G : R

→ R

n×m

, D = R

, and

U = R

. Furthermore, we consider performance summands L(x, u) of the

form

L(x, u) = L

(x) + L

(x)u + u

(x)u, (14.69)

where L

: R

→ R, L

: R

→ R

1×m

, and R

: R

→ P

so that (14.52)

becomes

J(x

, u(·)) =

∞

k=0

(x(k)) + L

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

(k)R

(x)u(k)]. (14.70)

Theorem 14.5. Consider the discrete-time nonlinear controlled aﬃne

system (14.68) with performance functional (14.70). Assume that there

exist functions V : R

→ R, L

: R

→ R

1×m

, P

: R

→ R

1×m

, and a

nonnegative-deﬁnite function P

: R

→ N

such that V (·) is continuous,

(0) = 0, (14.71)

(0) = 0, (14.72)

V (0) = 0, (14.73)

V (x) > 0, x ∈ R

, x 6= 0, (14.74)

V [f (x) −

G(x)(R

(x) + P

(x))

−1

(x)] − V (x) < 0, x ∈ R

, x 6= 0,

(14.75)

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

(x)u + u

(x)u, x ∈ R

, u ∈ R

(14.76)

and

V (x) → ∞ as kxk → ∞. (14.77)

Then the zero solution x(k) ≡ 0 of the closed-loop system

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

, k ∈ Z

, (14.78)

is globally asymptotically stable with the feedback control law

φ(x) = −

(x) + P

(x))

−1

(x) + P

(x)]

, (14.79)

and the performan ce functional (14.70), with

(x) = φ

(x)(R

(x) + P

(x))φ(x) − V (f(x)) + V (x), (14.80)

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DISCRETE-TIME NONLINEAR CONTROL 861

is minimized in the sense that

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)), x

∈ R

. (14.81)

Finally,

J(x

, φ(x(·))) = V (x

), x

∈ R

. (14.82)

Proof. The result is a direct consequence of Theorem 14.4 with

F (x, u) = f (x)+G(x)u, L(x, u) = L

(x)+L

(x)u+u

(x)u, D = R

, and

U = R

. Speciﬁcally, with (14.68), (14.69), and (14.76) the Hamiltonian

(14.59) has the form

H(x, u) = L

(x)+V (f (x))+[L

(x)+P

(x)]u+ u

(x)+P

(x))u−V (x).

(14.83)

Now, the feedback control law (14.79) is obtained by setting

∂H(x,u)

∂u

= 0.

With (14.79), it follows that (14.71)–(14.75) imply (14.53)–(14.56). Next,

with L

(x) given by (14.80) and φ(x) given by (14.79) and (14.57) holds.

Finally, since

H(x, u) = H(x, u) − H(x, φ(x)) = (u −φ(x))

(x) + P

(x))(u −φ(x)),

(14.84)

and R

(x) + P

(x) > 0, x ∈ R

, condition (14.58) holds. The result now

follows as a direct consequence of Theorem 14.4.

Note that (14.75) is equivalent to

∆V (x)

△

= V (f(x) + G(x)φ(x)) − V (x) < 0, x ∈ R

, x 6= 0, (14.85)

with φ(x) given by (14.79). Furthermore, conditions (14.73), (14.74), and

(14.85) ensure that V (x) is a L yapunov fu nction f or the closed-loop sys tem

(14.78). Furthermore, as in the continuous-time case, it is important to

recognize that the function L

(x) which appears in the summand of the

performance functional (14.69) is an arbitrary function of x ∈ R

subject to

conditions (14.71) and (14.75). Thus, L

(x) provides ﬂexibility in choosing

the control law.

With L

(x) given by (14.80) and φ(x) given by (14.79), L(x, u) can be

expressed as

L(x, u)

= (u − φ(x))

(x) + P

(x))(u −φ(x)) −V (f(x) + G(x)u) + V (x)

= [u +

(x) + P

(x))

−1

(x)]

(x) + P

(x))[u +

(x)

(x))

−1

(x)] − [V (f (x) + G(x)φ(x)) −V (x)] + φ

(x)P

(x)φ(x)

−

(x)(R

(x) + P

(x))P

(x) − u

(x)u. (14.86)

Since R

(x) + P

(x) ≥ R

(x) > 0 for all x ∈ R

the ﬁrst and third terms

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862 CHAPTER 14

of the right-hand side of (14.86) are nonnegative, while (14.85) implies that

the second term is n on negative. T hus, we h ave

L(x, u) ≥ −

(x)(R

(x) + P

(x))P

(x) − u

(x)u, (14.87)

which s hows that L(x, u) may be negative. As a result, there may exist a

control inpu t u for which the performance functional (14.70) is n egative.

However, if the control u is a regulation controller, that is, u ∈ S(x

), then

it follows fr om (14.81) and (14.82) that

J(x

, u) ≥ V (x

) ≥ 0, x

∈ R

, u(·) ∈ S(x

). (14.88)

Furth ermore, in this case substituting u = φ(x) into (14.86) yields

L(x, φ(x)) = −[V (f(x) + G(x)φ(x)) −V (x)], (14.89)

which, by (14.85), is positive.

Next, we specialize Theorem 14.5 to the case of quadratic Lyapunov

functions.

Corollary 14.3. Consider the discrete-time nonlinear controlled aﬃne

system (14.68) with performance functional (14.70). Assume that there exist

a function L

: R

→ R

1×m

and a positive-deﬁnite matrix P ∈ P

such that

(0) = 0, (14.90)

V [f (x) −

G(x)

−1

(x)(L

(x) + f

(x)P G(x))

] −V (x) < 0,

x ∈ R

, x 6= 0, (14.91)

where

V (x) = x

P x (14.92)

and

(x)

△

= R

(x) + G

(x)P G(x). Then the zero solution x(k) ≡ 0 of

the closed-loop s ystem (14.78) is globally asymptotically stable with the

feedback control law

φ(x) = −

−1

(x)



(x) + 2f

(x)P G(x)



, (14.93)

and the performan ce functional (14.70), with

(x) = φ

(x)

(x)φ(x) − f

(x)P f(x) + x

P x, (14.94)

is minimized in the sense that

J(x

, φ(x(·))) = min

u∈S(x

)

J(x

, u(·)), x

∈ R

. (14.95)

Finally,

J(x

, φ(x(·))) = V (x

), x

∈ R

. (14.96)

Proof. The result is a direct consequence of Theorem 14.5 with V (x) =

P x, P

(x) = 2f

(x)P G(x), and P

(x) = G

(x)P G(x). Speciﬁcally,

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DISCRETE-TIME NONLINEAR CONTROL 863

conditions (14.73), (14.74), (14.76), and (14.77) are trivially satisﬁed by

(14.92). Next, conditions (14.90) and (14.91) imply (14.71) and (14.75).

Now, (14.93) and (14.94) are immediate from Theorem 14.5.

14.6 Gain, Sector, and Disk Margins of Discrete-Time

Optimal Regulators

Gain and ph ase margins of continuous-time state feedback linear-quadratic

optimal regulators were extensively discussed in Chapter 8. In particular, in

terms of classical control relative stability notions, these controllers possess

at least a ±60

◦

phase margin, inﬁnite gain margin, and 50 percent gain

reduction for each control channel. Alternatively, in terms of absolute

stability theory [10] these controllers guarantee sector margins in that

the closed-loop system will remain asymptotically stable in the face of a

memoryless static input nonlinearity contained in the conic sector (

, ∞).

In contrast, the stability margins of discrete-time linear-quadratic optimal

regulators are not as well known and depend on the open-loop and closed-

loop poles of the discrete-time dynamical system [397, 459].

Synthesis techniques for discrete-time linear state feedback control

laws guaranteeing that the closed-lo op system possesses p respeciﬁed sec-

tor, gain, and phase margins were developed in [262]. However, un-

like continuous-time nonlinear-nonquadratic inverse optimal state feedback

regulators possessing guaranteed sector and disk margins to component

decoupled input nonlinearities in the conic sector (

, ∞) [321] an d dissipative

dynamic input operators [395], sector and disk margin guarantees f or

discrete-time nonlinear-nonquadratic regulators have not been addressed in

the literature.

In this section, we develop suﬃcient conditions for gain, sector, and

disk margin guarantees for discrete-time n on linear systems controlled by

optimal and inverse optimal nonlinear regulators that minimize a nonlinear-

nonquadratic performance criterion involving a nonlinear-nonquadr atic fun-

ction of the state and a quadratic function of the feedback control. In

the case where we specialize our results to the linear-quadratic case, we

recover the classical discrete-time linear-quadratic optimal regulator gain

and phase margin guarantees obtained in [262, 459]. Speciﬁcally, we derive

the relative stability margins for a discrete-time nonlinear optimal regulator

that minimizes a nonlinear-nonquadratic performance criterion involving a

nonlinear-nonquadratic f unction of the state and a quadratic function of the

feedback control.

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864 CHAPTER 14

Consider the nonlinear sys tem given by

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

, k ∈ Z

, (14.97)

y(k) = −φ(x(k)), (14.98)

where φ : R

→ R

, with a nonlinear-nonquadratic performance criterion

J(x

, u(·))

△

∞

k=0

(x(k)) + u

(k)R

(x(k))u(k)], (14.99)

where L

: R

→ R and R

: R

→ R

m×m

are given such that L

(x) ≥ 0 and

(x) > 0, x ∈ R

. In th is case, the optimal nonlinear feedback controller

u = φ(x) that min imizes the nonlinear-nonquadratic perf ormance criterion

(14.99) is given by the following result.

Theorem 14.6. Consider the nonlinear system (14.97) with perfor-

mance functional (14.99). Assume th ere exist functions V : R

→ R,

: R

→ R

1×m

, and P

: R

→ N

, with V (·) continuous such that

(14.73), (14.74), and (14.76) are satisﬁed and

V (f (x) −

G(x)(R

(x) + P

(x))

−1

(x)) −V (x) < 0, x ∈ R

, x 6= 0,

(14.100)

0 = L

(x) + V (f(x)) −V (x) −

(x)(R

(x) + P

(x))

−1

(x), x ∈ R

(14.101)

and

V (x) → ∞ as kxk → ∞. (14.102)

Then the zero solution x(k) ≡ 0 of the closed-loop system

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

, k ∈ Z

, (14.103)

is globally asymptotically stable with the feedback control law

φ(x) = −

(x) + P

(x))

−1

(x), (14.104)

and the performan ce functional (14.99) is minimized in the sense that

J(x

, φ(x(·))) = min

u(·)∈S(x

)

J(x

, u(·)), x

∈ R

. (14.105)

Finally,

J(x

, φ(x(·))) = V (x

), x

∈ R

. (14.106)

Proof. The proof is identical to the proof of Theorem 14.5.

The following key lemma is needed for developing the main result of

this section.