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

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NonlinearBook10pt November 20, 2007

STABILITY OF FEEDBACK SYSTEMS 425



θ(t)

u(t)

Figure 6.7 Inverted pendulum.

system. Furthermore, we set

(x) = 0, R

(x) =



0 0

0 1



, x ∈ D. (6.36)

In this case, it follows f rom (6.27) that the feedback controller is given by

u = φ(x) = x

+(x

−θ

)+g sin x

, x ∈ D. Next, note that

(x) = −x

≤ 0,

x ∈ D. Hence, R

△

= {x ∈ D :

= 0} = {x ∈ D : x

= 0}. Finally, since for

every x ∈ R, ˙x

6= 0 if and only x

6= θ

, it follows that the largest invariant

set contained in R is given by M = {x

}, and hence, the equilibrium solution

x(t) ≡ [θ

, 0]

is asymptotically stable. With θ

= 15

, Figure 6.8 shows the

phase portrait of the port-controlled Hamiltonian system. Figure 6.9 shows

the control force versus time and the shaped Hamiltonian versu s time. △

Example 6.4. Consider the two-mass, two-spring system shown in

Figure 6.10. A control force ˆu(·) acts on mass 2 with the goal to stabilize

the position of the second mass. The system dynamics, with state variables

deﬁned in Figure 6.10, are given by

¨q

(t) + (k

+ k

(t) − k

(t) = 0,

(0) = q

, ˙q

(0) = ˙q

, t ≥ 0, (6.37)

¨q

(t) −k

(t) + k

(t) = ˆu(t),

(0) = q

, ˙q

(0) = ˙q

. (6.38)

Deﬁning x

= q

, x

= ˙q

, x

= q

, and x

= ˙q

, we can rewrite (6.37)

and (6.38) in state space form (6.22) with x = [x

, x

]

, R(x) = 0,

NonlinearBook10pt November 20, 2007

426 CHAPTER 6

-6 -5 -4 -3 -2 -1 0 1 2

-10

-5

θ(t)

Figure 6.8 Phase portrait of the inverted pendulum.

G(x) = [0, 0, 0, 1]

, u =

ˆu

J(x) =







0 0

−

0 0 0

0 0 −







, (6.39)

D = {x ∈ R

: x

≥ 0, x

≥ 0}, and Hamiltonian function H(·)

corresponding to the total energy in the system given by H(x) =

−x

)

Next, to stabilize the equilibrium point x

= [x

, 0, x

, 0]

, where

)

, w ith steady-state control value of u

c ss

)

we assign the shaped Hamiltonian fun ction H

(x) =

−x

)

−

)

for the closed-loop system. Furthermore, we set

(x) ≡ 0 and

(x) =







0 0 0 0

0 0 0







, x ∈ D. (6.40)

In this case, it follows f rom (6.27) that the feedback controller is given by

NonlinearBook10pt November 20, 2007

STABILITY OF FEEDBACK SYSTEMS 427

0 5 10 15 20 25 30

−20

−15

−10

−5

Time

Control force

0 5 10 15 20 25 30

Time

Shaped Hamiltonian

Figure 6.9 Control force and shaped Hamiltonian versus time.



(t) q

(t)

ˆu(t)

Figure 6.10 Two-mass, two-spring system.

u = φ(x) =

)

− x

, x ∈ D. Next, note that

(x) = −m

≤

0, x ∈ D. Hence, R

△

= {x ∈ D :

= 0} = {x ∈ D : x

= 0}. Now, if

M ⊆ R is the largest invariant set contained in R, then f or every x

∈ M,

(t) ≡ 0, wh ich implies that x

(t)−x

(t)+

= 0 and ˙x

(t) = 0, t ≥

0. In this case, it follows that ˙x

(t) = 0, and hence, ˙x

(t) = 0, t ≥ 0. Hence,

the only point that belongs to M is x

= [

)

, 0, x

, 0]

, which

implies that x

is an asymptotically stable equilibrium point of the closed-

loop system. With m

= 1.5 kg, m

= 0.8 kg, k

= 0.1 N/m, k

= 0.3 N/m,

L = 0.4 m, and x

= 3 m, Figure 6.11 shows the phase portrait of x

versus

of the port-controlled Hamiltonian system. Figures 6.12 and 6.13 show,

respectively, the positions and velocities of the masses versus time. Finally,

Figure 6.14 sh ows the control force versus time and the shaped Hamiltonian

versus time. △

Next, we consider energy-based dynamic control for port-controlled

Hamiltonian systems wherein energy shaping is achieved by combining the

physical energy of the plant and the emulated energy of the controller.

NonlinearBook10pt November 20, 2007

428 CHAPTER 6

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0.5

1.5

Velocity of mass 1

Velocity of mass 2

Figure 6.11 Phase portrait of x

versus x

This approach has been extensively s tudied by Ortega et al. [340, 341] to

design Euler-Lagrange controllers for potential energy sh aping of m echanical

systems.

We begin by considering the port-controlled Hamiltonian system G

given by (6.22) and (6.23) with m = l. Fu rthermore, we consider the port-

controlled Hamiltonian feedback control system G

given by

˙x

(t) = [J

(t)) − R

(t))]



∂H

∂x

(t))



+ G

(t))u

(t),

(0) = x

, t ≥ 0, (6.41)

(t) = G

(t))



∂H

∂x

(t))



, (6.42)

where x

(t) ∈ R

, u

(t) ∈ U

⊆ R

, y

(t) ∈ Y

⊆ R

, m

= l

, H

: R

→

R is a continuously diﬀerentiable Hamiltonian function of the feedback

control system G

, J

: R

→ R

×n

is such that J

) = −J

), R

→ S

is such that R

) ≥ 0, x

∈ R

, [J

) −R

)]



∂H

∂x

)



∈ R

, is Lipschitz continuous on R

, G

: R

→ R

×m

, m

= l, and

= m. Here, we assume that u

(·) is restricted to the class of admissible

inputs consisting of measurable functions su ch that u

(t) ∈ U

for all t ≥ 0.

Note that with the f eedback interconnection given by Figure 6.1, u

= y and

= −u. Hence, the closed-loop dynamics can be written in Hamiltonian

NonlinearBook10pt November 20, 2007

STABILITY OF FEEDBACK SYSTEMS 429

0 5 10 15 20 25 30 35 40 45 50

−6

−4

−2

Time

Position of mass 1

0 5 10 15 20 25 30 35 40 45 50

−2

−1

Time

Position of mass 2

Figure 6.12 Mass positions versus time.

form given by

˜x(t) =



J(x(t)) −G(x(t))G

(t))

(t))G

(x(t)) J

(t))



−



R(x(t)) 0

0 R

(t))









∂H

∂x

(x(t))





∂H

∂x

(t))







˜x(0) = ˜x

, t ≥ 0, (6.43)

where ˜x

△

= [x

, x

]

It can be seen f rom (6.43) that by relating the controller state

variables x

to the plant state variables x, one can shape the Hamiltonian

function H(·) + H

(·) so as to preserve the Hamiltonian structure under

dynamic feedback for part of the closed-loop system associated with

the plant dynamics. Since the closed-loop dynamical system (6.43) is

Hamiltonian involving skew-symmetric interconnection matrix function

terms and nonnegative-deﬁnite dissipation matrix function terms, we can

establish the existence of energy-Casimir functions [63,441] (i.e., dynamical

invariants) that are independent of the closed-loop Hamiltonian and relate

the controller states to the plant states. Since, as shown in Section 3.4,

energy-Casimir functions are comp osed of integrals of motion, it follows

that these functions are constant along the trajectories of the closed-loop

system (6.43). Fu rthermore, since the controller Hamiltonian H

(·) can

NonlinearBook10pt November 20, 2007

430 CHAPTER 6

0 5 10 15 20 25 30 35 40 45 50

−3

−2

−1

Time

Velocity of mass 1

0 5 10 15 20 25 30 35 40 45 50

−2

−1

Time

Velocity of mass 2

Figure 6.13 Mass velocities versus time.

be assigned, the en ergy-Casimir method can be used to construct suitable

Lyapunov functions for the closed-loop system.

To proceed, consider the candidate vector energy-Casimir fun ction E :

D×R

→ R

, where E(·, ·) is continuously diﬀerentiable and has the form

E(x, x

) = x

− F (x), (x, x

) ∈ D × R

, (6.44)

where F : D → R

is a continuously diﬀerentiable function. To ensure that

the candidate vector energy-Casimir fun ction E(·, ·) is constant along the

trajectories of (6.43) we require th at

E(x(t), x

(t)) = ˙x

(t) −

∂F

∂x

(x(t)) ˙x(t) = 0, t ≥ 0. (6.45)

Now, we can arrive at a set of suﬃcient conditions which guarantee that

(6.45) holds. Speciﬁcally, it follows from (6.43) that (6.45) can be rewritten

E(x(t), x

(t)) =



(x) −

∂F

∂x

(x)(J(x) − R(x))





) −R

) +

∂F

∂x

(x)G(x)G

)









∂H

∂x

(x(t))





∂H

∂x

(t))







. (6.46)

NonlinearBook10pt November 20, 2007

STABILITY OF FEEDBACK SYSTEMS 431

0 5 10 15 20 25 30 35 40 45 50

−2

−1

Time

Control signal

0 5 10 15 20 25 30 35 40 45 50

−2

Time

Shaped Hamiltonian

Figure 6.14 Control signal and shaped Hamiltonian versus time.

Hence, a set of suﬃcient conditions such that (6.45) holds is given by

(x) −

∂F

∂x

(x)(J(x) − R(x)) = 0, (x, x

) ∈ D × R

, (6.47)

) −R

) +

∂F

∂x

(x)G(x)G

) = 0, (x, x

) ∈ D × R

. (6.48)

The following proposition su mmarizes the above results.

Proposition 6.1. Consider the feedback interconnection of the port-

controlled Hamiltonian systems G and G

given by (6.22) and (6.23), and

(6.41) and (6.42), respectively. If there exists a continuously diﬀerentiable

function F : D → R

such that for all (x, x

) ∈ D × R

∂F

∂x

(x)J(x)



∂F

∂x

(x)



− J

) = 0, (6.49)

) = 0, (6.50)

R(x)



∂F

∂x

(x)



= 0, (6.51)

∂F

∂x

(x)J(x) − G

(x) = 0, (6.52)

then

E(˜x(t)) = x

(t) − F (x(t)) = c, t ≥ 0, (6.53)

where c ∈ R

and ˜x(t) = [x

(t), x

(t)]

satisﬁes (6.43).

NonlinearBook10pt November 20, 2007

432 CHAPTER 6

Proof. Postmultiplying (6.47) by



∂F

∂x

(x)



, it follows from (6.47) and

(6.48) that

∂F

∂x

(x)[J(x) − R(x)]



∂F

∂x

(x)



= J

) + R

), (x, x

) ∈ D × R

(6.54)

Next, using the fact that the sum of a skew-symmetric and symmetric matrix

is zero if and only if the individual matrices are zero, it follows that (6.54)

is equivalent to

∂F

∂x

(x)J(x)



∂F

∂x

(x)



−J

) = 0, (x, x

) ∈ D × R

, (6.55)

) +

∂F

∂x

(x)R(x)



∂F

∂x

(x)



= 0, (x, x

) ∈ D × R

. (6.56)

Now, since R(x) ≥ 0, x ∈ D, and R

) ≥ 0, x ∈ R

, it follows that (6.55)

and (6.56) are equivalent to (6.49)–(6.51). Hence, it follows that (6.47) can

be rewritten as (6.52). The equivalence between (6.47)–(6.48) and (6.49)–

(6.52) proves the result.

Note that conditions (6.49)–(6.52) are necessary and suﬃcient for

(6.47)–(6.48) to hold, which, in tu rn, provide suﬃcient conditions f or

guaranteeing that the vector energy-Casimir function E(·, ·) is constant

along the trajectories of the closed-loop system (6.43). The constant

vector c ∈ R

in (6.53) depends on the in itial conditions for the plant

and controller states. If conditions (6.49)–(6.52) are satisﬁed, then the

controller state variables along the trajectories of the closed-loop system

given by (6.43) can be represented in terms of the plant s tate variables as

(t) = F (x(t)) + c, t ≥ 0, x(t) ∈ D, c ∈ R

. In this case, it follows that

the closed-loop system associated with the plant dynamics is given by

˙x(t) = [J(x(t)) − R(x(t))]



∂H

∂x

(x(t))



−G(x(t))G

(t))



∂H

∂x

(t))



= [J(x(t)) − R(x(t))]



∂H

∂x

(x(t)) +

∂H

∂x

(t))

∂F

∂x

(x(t))



= [J(x(t)) − R(x(t))]



∂H

∂x

(x(t))



, x(0) = x

, t ≥ 0, (6.57)

where H

(x) = H(x) + H

(F (x) + c), x ∈ D, is the shaped Hamiltonian

function for the closed-loop system (6.57).

Next, we use the existence of the vector energy-Casimir function to

NonlinearBook10pt November 20, 2007

STABILITY OF FEEDBACK SYSTEMS 433

construct stabilizing dynamic controllers that guarantee that the closed-

loop system associated with the plant dynamics preserves the Hamiltonian

structure without the need for solving a set of partial d iﬀerential equations.

Theorem 6.4. Consider the feedback interconnection of the port-

cont-rolled Hamiltonian systems G and G

given by (6.22) and (6.23), and

(6.41) and (6.42), respectively. Assume that there exists a continuously

diﬀerentiable function F : D → R

such that conditions (6.49)–(6.52)

hold for all (x, x

) ∈ D × R

, and assume that the Hamiltonian fu nction

: R

→ R of the feedback controller G

is such that H

: D → R is given

by H

(x) = H(x) + H

(F (x) + c), x ∈ D. If

∂H

∂x

(F (x

) + c) = −

∂H

∂x

), x

∈ D, (6.58)

∂

∂x

(F (x

) + c) > −

∂

∂x

), x

∈ D, (6.59)

then the equilibrium solution x(t) ≡ x

of the system (6.57) is Lyapunov

stable. If, in addition, D

⊆ D is a compact positively invariant set with

respect to (6.57) and the largest invariant set contained in R

△

= {x ∈ D

∂H

∂x

(x)R(x)



∂H

∂x

(x)



= 0} is M = {x

}, then the equilibrium solution

x(t) ≡ x

of the closed-loop system (6.57) is locally asymptotically stable.

Proof. Conditions (6.49)–(6.52) imply that the closed-loop dynamics

of the port-controlled Hamiltonian system G and th e controller G

associated

with the plant states can be written in the form given by (6.57). Now,

using identical arguments as in the proof of Theorem 6.3, conditions (6.58)

and (6.59) guarantee the existence of the Lyapunov function candidate

V (x) = H

(x) − H

), x ∈ D, which guarantees Lyapunov stability of the

equilibrium solution x(t) ≡ x

of the closed-loop system (6.57). Asymptotic

stability of x(t) ≡ x

follows from Corollary 3.1.

As in the static controller case, the dynamic controller given by

Theorem 6.4 also provides an energy balance interpretation over the

trajectories of the controlled system. To see this, note th at since by (6.50),

) = 0, x

∈ R

, it follows that th e controller Hamiltonian H

(·)

satisﬁes

(F (x(t)) + c) = y

(t)y(t) = −u

(t)y(t), t ≥ 0. (6.60)

Now, it follows that

(x(t)) =

H(x(t)) +

(F (x(t)) + c)

H(x(t)) −u

(t)y(t), t ≥ 0, (6.61)

which yields (6.33).

NonlinearBook10pt November 20, 2007

434 CHAPTER 6

6.4 Stability Margins for Nonlinear Feedback Regulators

To develop relative s tability margins for nonlinear regulators consider th e

nonlinear dynamical system G given by

˙x(t) = f (x(t)) + G(x(t))u(t), x(0) = x

, t ≥ 0, (6.62)

y(t) = −φ(x(t)), (6.63)

where φ : R

→ R

is such that G is asymptotically stable with u = −y.

Furth ermore, assume that the system G is zero-state observable. Next,

we deﬁne the relative stability margins for G given by (6.62) and (6.63).

Speciﬁcally, let u

△

= −y, y

△

= u, and consider the negative f eedback

interconnection u = ∆(−y) of G and ∆(·) given in Figure 6.15, where

∆(·) is either a linear operator ∆(u

) = ∆u

, a nonlinear static operator

∆(u

) = σ(u

), or a dynamic nonlinear operator ∆(·) with input u

and

output y

. Furth ermore, we assume that in the nominal case ∆(·) = I(·) so

that the nominal closed-loop system is asymptotically stable.

∆(·)

- -

−

Figure 6.15 Multiplicative input uncertainty of G and input operator ∆(·).

Deﬁnition 6.1. Let α, β ∈ R be such that 0 < α ≤ 1 ≤ β < ∞.

Then the nonlinear d ynamical system G given by (6.62) an d (6.63) is said to

have a gain margin (α, β) if the negative feedback interconnection of G and

∆(u

) = ∆u

is globally asymptotically stable for all ∆ = diag[k

, . . . , k

where k

∈ (α, β), i = 1, . . . , m.

Deﬁnition 6.2. Let α, β ∈ R be such that 0 < α ≤ 1 ≤ β < ∞.

Then the nonlinear dynamical system G given by (6.62) and (6.63) is said

to have a sector margin (α, β) if the negative feedback interconnection of

G and ∆(u

) = σ(u

) is globally asymptotically stable for all nonlinearities

σ : R

→ R

such that σ(0) = 0, σ(u

) = [σ

), . . . , σ

)]

, and

αu

< σ

< βu

, f or all u

6= 0, i = 1, . . . , m.

Deﬁnition 6.3. Let α, β ∈ R be such that 0 < α ≤ 1 ≤ β < ∞. Then

the nonlinear dynamical system G given by (6.62) and (6.63) is said to have

a disk marg in (α, β) if the negative feedback interconnection of G and ∆(·)

is globally asymptotically stable for all dynamic operators ∆(·) such that

∆(·) is zero-state observable and dissipative with respect to the supply rate