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

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NonlinearBook10pt November 20, 2007

STABILITY OF FEEDBACK SYSTEMS 465

Problem 6.26. Show that for scalar stabilizable sy s tems V (x) =

is always a control Lyapunov function. Using this fact, obtain a stabilizing

controller for th e dynamical system considered in Example 6.6 using the

control Lyapunov function approach and compare this controller to the

optimal Hamilton-Jacobi-Bellman controller.

Problem 6.27. Consider the nonlinear dynamical system (6.76).

Assume there exists a continuously diﬀerentiable function V : D → R

such that V (·) is positive deﬁnite and

′

(x)f(x) ≤ −c(V (x))

, x ∈ R, (6.175)

where c > 0, α ∈ (0, 1), and R , {x ∈ R

, x 6= 0 : V

′

(x)G(x) = 0}.

Show that the zero solution x(t) ≡ 0 to (6.76) with the feedback controller

u = φ(x), x ∈ R

, given by

φ(x) =







−



(α(x)−w(V (x)))+

√

(α(x)−w(V (x)))

+(β

(x)β(x))

(x)β(x)



β(x), β(x) 6= 0,

0, β(x) = 0,

(6.176)

where c

> 0, α(x) , V

′

(x)f(x), x ∈ R

, β(x) , G

(x)V

′T

(x), x ∈ R

, and

w(V (x)) , −c(V (x))

, x ∈ R

, is ﬁnite-time stable with the settling-time

function T (x

) ≤

c(1−α)

(V (x

))

1−α

, x

∈ R

. Furthermore, show that V (·)

is a control Lyapunov function.

Problem 6.28. Consider the system (6.83) and (6.84) with F (x, u, t) =

Ax + Bu and L(x, u, t) = x

x + u

u, where A ∈ R

n×n

, B ∈ R

n×m

∈ R

n×n

, and R

∈ R

m×m

, such that R

≥ 0, R

> 0. Show that the

optimal control law characterized by the Hamilton-Jacobi-Bellman equ ation

(6.93) is given by

u(t) = −R

−1

P (t)x(t), (6.177)

where

−

P (t) = A

P (t)+ P (t)A+ R

−P (t)BR

−1

P (t), P (t

) = 0. (6.178)

Problem 6.29. Consider the nonlinear dynamical system

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

, t ≥ 0, (6.179)

with performance functional

J(x

, u(·)) =

∞

(x(t)) + u

(t)R

(x(t))u(t)]dt, (6.180)

where x ∈ R

, u ∈ R

, f : R

→ R

satisﬁes f (0) = 0, G : R

→ R

n×m

: R

→ R is such that L

(x) > 0, x ∈ R

, x 6= 0, and R

: R

→ R

m×m

NonlinearBook10pt November 20, 2007

466 CHAPTER 6

is su ch that R

(x) > 0, x ∈ R

. Show that the optimal control u = φ(x)

characterized by the Hamilton-Jacobi-Bellman equation (6.93) is given by

φ(x) = −

−1

(x)G

(x)V

′T

(x), (6.181)

where V (0) = 0 and

0 = L

(x) + V

′

(x)f(x) −

′

(x)G(x)R

−1

(x)G

(x)V

′T

(x), x ∈ R

(6.182)

Also show that (6.181) is a stabilizing feedback control law.

Problem 6.30. Consider the nonlinear dynamical system (6.76). Show

that the feedback control law (6.81) is an optimal stabilizing control law

minimizing the perform an ce functional

J =

∞

[2α(x(t)) −q(x(t))β

(x(t))β(x(t)) + q

−1

(x(t))u

(t)u(t)]dt,

(6.183)

where

q(x)

△

(

α(x)+

√

(x)+(β

(x)β(x))

(x)β(x)

, β(x) 6= 0,

, β(x) = 0,

(6.184)

and c

> 0.

Problem 6.31. Consider the nonlinear dynamical system given by

(6.76). Show that whenever the control Lyapunov function V satisfying

(6.77) has the same level sets as the value function V

∗

satisfying the

Hamilton-Jacobi-Bellman equation (6.182), the feedback control law (6.81)

reduces to the optimal control law given by (6.181).

Problem 6.32. Consider the nonlinear dynamical system

˙x(t) = F (x(t), u(t), t), x(t

) = x

, x(t

) = x

, t ≥ t

, (6.185)

where x(t) ∈ D ⊆ R

, t ∈ [t

, t

], x(t

) = x

is given, x(t

) = x

is ﬁxed, and

u(t) ∈ U ⊆ R

, t ∈ [t

, t

], with performance functional

J(x

, u(·), t

) =

L(x(t), u(t), t)dt. (6.186)

Let J

∗

(x, t) and u

∗

(t), t ∈ [t

, t

], denote the minimal cost and optimal

control, respectively, for the optimal control problem given by (6.185) and

(6.186) and assume that J

∗

(·, ·) is two-times continuously diﬀerentiable.

Show that

˙p(x(t), t) = −



∂

∂x

H(x(t), u

∗

(t), p(x(t), t), t)



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STABILITY OF FEEDBACK SYSTEMS 467

p(x(t

), t

) =



∂

∂x

∗

(x(t

), t

)



, (6.187)

where p(x(t), t) =



∂

∂x

∗

(x(t), t)



and H(x, u, p(x, t), t) = L(x, u, t)+ p

(x,

t)F (x, u, t).

Problem 6.33. Consider an n-degree-of-freedom dynamical system

with action integral

J =

L(q(t), ˙q(t))dt, (6.188)

where q ∈ R

denotes generalized system positions, ˙q ∈ R

denotes

generalized system velocities, L : R

× R

→ R denotes the system

Lagrangian given by L(q, ˙q) = T (q, ˙q) − V (q), where T : R

× R

→ R

is the system kinetic energy and V : R

→ R is the system potential energy.

Hamilton’s principle of least action states that among the set of all smooth

kinematically possible paths that a dynamical system may move between

two ﬁxed end points over a speciﬁed time interval, the only dynamically

possible system paths are those that render J stationary to all variations

in the shape of the paths. Use the Hamilton-Jacobi-Bellman equation to

show that Hamilton’s principle for an n-d egree-of-freedom dynamical sys tem

implies



∂L

∂ ˙q

(q, ˙q)



−



∂L

∂q

(q, ˙q)



= 0. (6.189)

Problem 6.34. Consider the nonlinear dynamical system

˙x

(t) = x

(t)x

(t), x

(0) = x

, t ≥ 0, (6.190)

˙x

(t) = u(t), x

(0) = x

, (6.191)

y(t) = x

(t). (6.192)

Show that (6.190)–(6.192) is weakly minimum phase an d u = x

+ v

renders (6.190)–(6.192) passive. In addition, show that the output feedback

controller v = −y globally stabilizes (6.190)–(6.192).

Problem 6.35. Consider the nonlinear dynamical system

˙x

(t) = −x

(t) + x

(t), x

(0) = x

, t ≥ 0, (6.193)

˙x

(t) = x

(t) + u(t), x

(0) = x

, (6.194)

y(t) = x

(t). (6.195)

Show that (6.193)–(6.195) has relative degree 1 and is minimum phase. Can

this system be rendered passive via a static output feedback controller?

Problem 6.36. Consider the nonlinear dynamical system G given by

(6.98) and (6.99) with u, y ∈ R. Show that if G has relative degree r at a

NonlinearBook10pt November 20, 2007

468 CHAPTER 6

point x

, th en the vectors {

∂

∂x

h(x),

∂

∂x

h(x)], . . . ,

∂

∂x

r−1

h(x)]} at x = x

are linearly independent.

Problem 6.37. Consider the nonlinear dynamical system G given by

(6.98) and (6.99). Show that if G is input s trictly passive, then G has relative

degree zero.

Problem 6.38. Show that every minimum phase system of the form

(6.98) and (6.99) with relative degree {1, 1, . . . , 1} and complete vector ﬁeld

G(L

−1

is feedback equivalent to a J-Q type system (see Problem 6.25).

Problem 6.39. Let

G(s) ∼



A B

C 0



be an m × m rational transfer function an d s uppose q ∈ C is not a pole

of G(s). q is a transmission zero of G(s) if rank G(q) < nrank G(s),

where nrank G(s)

△

= max

s∈C

rank G(s). Show that if ˙z = A

z denotes the

zero dynamics of G(s) and G(s) is minimum phase and h as relative degree

{1, 1, . . . , 1}, then the eigenvalues of A

are the transmission zeros of G(s).

Problem 6.40. Consider the nonlinear dynamical system G given by

(6.98) and (6.99). Show that if G is feedback linearizable, then the solutions

to the partial diﬀerential equations

0 = L

(x), 0 ≤ k ≤ r

− 2, 1 ≤ j ≤ m,

exist, where G

, j = 1, . . . , m, are the smooth column vector ﬁelds of G,

, i = 1, . . . , m, are the smooth components of h, r

+ r

+ ··· + r

= n,

and r

is the smallest integer such that at least one of th e inputs u

appears

(x). Also show that in the case where f(x) = Ax, A ∈ R

n×n

G(x) = B, B ∈ R

n×1

, h(x) = Cx, C ∈ R

1×n

, the integers r

, . . . , r

reduce to a single integer corresponding to the relative degree of the transfer

function G(s) = C(sI − A)

−1

Problem 6.41. Consider the dynamical system (6.98) and (6.99) with

f(x) = Ax, G(x) = B, and h(x) = Cx, where A ∈ R

n×n

, B ∈ R

n×m

, and

C ∈ R

l×n

. Show that in this case Cond ition i) of Theorem 6.11 reduces to

rank[B AB ··· A

n−1

B] = n.

Problem 6.42. Consider the nonlinear dynamical system (6.98). Show

that (6.98) is feedback linearizable at x

∈ R

only if the linearized system

is controllable at x

∈ R

Problem 6.43. Let f , f

, f

, g, g

, g

: R

→ R be continuously

NonlinearBook10pt November 20, 2007

STABILITY OF FEEDBACK SYSTEMS 469

diﬀerentiable vector ﬁelds, let h : R

→ R, and let α

and α

be real

constants. Show that:

i) [α

(x) + α

(x), g

(x)] = α

(x), g

(x)] + α

(x), g

(x)].

ii) [f

(x), α

(x) + α

(x)] = α

(x), g

(x)] + α

(x), g

(x)].

iii) [f(x), g(x)] = −[g(x), f (x)].

iv) L

[f,g]

h(x) = L

h(x) − L

h(x).

Problem 6.44. Let T : D → R

. Show that if T

′

(x) is invertible at

x = x

∈ D, th en there exists a neighborhood N of x

such that T (x) is a

diﬀeomorphism for all x ∈ N.

Problem 6.45. T : R

→ R

is a global diﬀeomorphism if it is a

diﬀeomorphism on R

and T (R

) = R

. Show that T (·) is a global

diﬀeomorphism if and only if T

′

(x) is invertible for all x ∈ R

and

lim

kxk→∞

kT (x)k = 0.

Problem 6.46. Consider the controlled Van der Pol oscillator (6.105)

and (6.106) given in Example 6.9 with (6.107) replaced by y(t) = x

(t).

Calculate the relative degree of this system. Is this system m inimum phase?

Is this system feedback linearizable?

Problem 6.47. Consider the nonlinear dynamical system

˙x

(t) = x

(t) + x

(t), x

(0) = x

, t ≥ 0, (6.196)

˙x

(t) = u(t), x

(0) = x

. (6.197)

Using Theorem 6.11 show that (6.196) and (6.197) is feedback linearizable.

Furth ermore, using the diﬀeomorphism T (x) = [x

+ x

]

, where x =

]

, construct a stabilizing nonlinear controller that cancels out all

nonlinearities in the sys tem. Alternatively, construct a globally stabilizing

controller based on the control Lyapun ov function V (x) = T

(x)P T (x),

where P > 0 satisﬁes (6.101). Compare the state response and the control

eﬀort versus time of both designs.

6.9 Notes and References

For a treatment of feedback interconnections of dissipative systems, see Pop-

ov [364], Zames [476,477], Desoer and Vidyasagar [104], Willems [456, 457],

and Hill and Moylan [189]. Stability of feedback interconnections involving

dissipative and exponentially dissipative systems was ﬁrst introd uced by

Chellaboina and Haddad [88]. The treatment of stability for nonlinear

dissipative feedback systems is adopted from Hill and Moylan [189] and

NonlinearBook10pt November 20, 2007

470 CHAPTER 6

Chellaboina an d Haddad [88]. Energy-based controllers for port-controlled

Hamiltonian systems were ﬁrst developed by Maschke, Ortega, and van

der Schaft [301] and Maschke, Ortega, van der Schaft, and Escobar [302].

The treatment here is adopted from Ortega, van der Schaft, Maschke, and

Escobar [343]. Gain and sector margins for nonlinear systems are motivated

by work on Nyquist stability theory and absolute stability theory and

can be found in Lur´e [291], Popov [364], Narendra and Taylor [331], and

Safonov [377]. An operator approach is given by Zames [476, 477] while a

state space approach can be found in Hill and Moylan [188,189]. The concept

of disk margins can be found in the monograph by Sepu lchre, Jankovi´c, and

Kokotovi´c [395].

The concept of control Lyapunov f unctions is du e to Artstein [13],

while the constructive feedback control law based on the control Lyapunov

function given in Section 6.5 is due to Sontag [406] and is known as Sontag’s

universal formula. The principle of optimality is due to Bellman [36]. The

Hamilton-Jacobi-Bellman equation was developed by Bellman [36] and can

be viewed as an extension of Jacobi’s work on conjugate points in ﬁelds of

extremals and Hamilton’s work on least action in mechanical systems; both

of which were developed in the nineteenth century. For a modern textbook

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

For a thorough textbook treatment of feedback linearization, zero

dynamics, and minimum -phase s ystems the reader is referred to Isidori [212]

and Nijmeijer and van der Schaft [336]. One of the ﬁrst contributions to

solving nonlinear control problems using diﬀerential geometric methods is

due to Brockett [68]. The feedback linearization problem was ﬁrst posed by

Brockett [69] and solved by Su [419] in the single-input case and by Hunt,

Su, and Meyer [210] in the multi-input case. The deﬁnitions in Section

6.7 are adopted from Is idori [212]. For an excellent treatment on zero

dynamics and minimum-phase systems see Byrnes and Isidori [74–76] and

Saberi, Kokotovi´c, and Sussmann [376]. Th e concepts of feedback passivity

equivalence and global stabilization of minimum phase systems are due to

Kokotovi´c and Sussm ann [241] for linear systems and Byrnes, Isidori, and

Willems [77] for nonlinear systems.

NonlinearBook10pt November 20, 2007

Chapter Seven

Input-Output Stability and Dissipativity

7.1 Introduction

As mentioned in Chapter 1, the main objective of this book is to present the

necessary mathematical tools for s tability and control design of nonlinear

dynamical systems, with an emphasis on Lyapunov-based methods. For

completeness of exposition, in this chapter we digress from our main

objective to present an alternative approach to the mathematical modeling

of nonlinear dynamical systems based on input-output notions. In contrast

to the input, state, and output setting for nonlinear dynamical systems

presented in the preceding chapters, an input-output representation of a

dynamical system relates the output of the system directly to the input

via an input-output mapping with no knowledge of the internal state of

the system. Th ere are numerous situations w here an input-output system

description is ap propriate. These include, for example, intelligent machines

such as computer hardware and signal processors, as well as computer

software algorithm execution.

The principal advantage of the input-output modeling approach to

dynamical systems is that it addresses inﬁnite-dimensional systems almost as

easily as ﬁnite-dimensional systems. This is also the case for continuous-time

and discrete-time systems. In addition, the input-output approach leads

to more powerful and general results. However, in most physical systems

the output of the system also depen ds on the system’s initial conditions.

In addition, an input-output system description cannot deal with physical

system interconnections. Hence, dynamical systems modeling predicated

on state space models, wherein an internal state model is used to describe

the system dynamics using physical laws and system interconnections, is of

fundamental importance in the description of physical dynamical systems.

Input-output system descriptions can be traced back to the work

of Heaviside on impedance circuit descriptions and the cybernetic force-

response systems view of Wiener. This input-output systems approach led

to the concept of input-output stability which was pioneered by Sandberg

NonlinearBook10pt November 20, 2007

472 CHAPTER 7

[387, 388, 390] and Zames [476, 477]. In this chapter, we introduce input-

output systems descriptions and deﬁne th e concept of input-output L

stability. In addition, we introduce the concepts of input-output ﬁnite-

gain, dissipativity, passivity, and nonexpansivity. Furthermore, we develop

connections between input-output stability and Lyapunov stability theory.

Finally, we develop explicit formulas for induced convolution operator norms

for linear input-outpu t dynamical systems.

7.2 Input-Output Stability

In this section, we introduce the deﬁnition of input-output stability for

general operator d ynamical systems. Let U and Y deﬁne an input and

an output spaces, respectively, consisting of continuous bounded U-valued

and Y -valued functions on the semi-inﬁnite interval [0, ∞), where U ⊆ R

and Y ⊆ R

. The set U contains the set of input values, that is, for every

u(·) ∈ U and t ∈ [0, ∞), u(t) ∈ U. The s et Y contains the set of output

values, that is, for every y(·) ∈ Y and t ∈ [0, ∞), y(t) ∈ Y . T he spaces

U and Y are assumed to be closed und er the shift operator, that is, if

u(·) ∈ U (respectively, y(·) ∈ Y), then the function deﬁned by u

△

= u(t+T )

(respectively, y

△

= y(t+T )) is contained in U (respectively, Y) for all T ≥ 0.

In this chapter, we consider operator dynamical systems G : U →

Y. For example, G may denote a linear, time-invariant, ﬁ nite-dimensional

dynamical system given by the transfer function

G(s) ∼



A B

C D



or, equivalently,

y(t) = G[u](t)

△

H(t −τ )u(τ)dτ + Du(t), t ≥ 0, (7.1)

where H(t)

△

= Ce

B is the impulse response matrix function. For

notational convenience, we denote the functional dependence of y ∈ Y on

u ∈ U given by (7.1) as y = G[u]. Similarly, consider a nonlinear dynamical

system described by

˙x(t) = f(x(t)) + G(x(t))u(t), x(t

) = 0, t ≥ t

, (7.2)

y(t) = h(x(t)) + J(x(t))u(t), (7.3)

where x(t) ∈ D ⊆ R

, D is an open set with 0 ∈ D, u(t) ∈ U ⊆ R

, y(t) ∈

Y ⊆ R

, f : D → R

, G : D → R

n×m

, h : D → Y , and J : D → R

l×m

. T he

mapping from u(·) to y(·) is also an operator dynamical system. However,

in this case, it is not possible in general to provide an explicit expression for

the operator G[u].

NonlinearBook10pt November 20, 2007

INPUT-OUTPUT STABILITY AND DISSIPATIVITY 473

In this book, we restrict our attention to causal operator dynamical

systems. In order to provide the deﬁnition for causality we need the following

deﬁnition.

Deﬁnition 7.1. Let u(·) ∈ U. Then for all T ∈ [0, ∞), the fu nction

(·) deﬁned by

(t)

△



u(t), 0 ≤ t ≤ T,

0, T < t,

is called the truncation of u(·) on the interval [0, T ].

Deﬁnition 7.2. An operator dynamical system G : U → Y is said to

be causal if for every T ∈ [0, ∞) and u(·) ∈ U,

= (G[u])

= (G[u

])

Note that if u(·) ∈ U, then u

(·) ∈ U, and hen ce, G[u

] is well deﬁned.

Furth ermore, for operator dynamical systems described by (7.1) or (7.2)

and (7.3) it is easy to show th at the G is a causal system. Th e following

proposition is now immediate.

Proposition 7.1. Let G : U → Y. G is causal if and only if for every

pair u, v ∈ U and for every T > 0 such that u

= v

, (G[u])

= (G[v])

Proof. Suppose G is causal and let u, v ∈ U be such that u

= v

for

some T > 0. Then, it follows that

(G[u])

= (G[u

])

= (G[v

])

= (G[v])

where the ﬁrst and last equalities follow from the causality of G. Conversely,

suppose for every pair u, v ∈ U and for every T > 0 such that u

= v

, G

satisﬁes (G[u])

= (G[v])

. Let u ∈ U and v = u

for some T > 0. Noting

that u

= v

and (G[u])

= (G[v])

= (G[u

])

, the causality of G follows.

Next, we deﬁne the notion of inpu t-output stability. First, however, we

restrict the input and output to belong to normed linear sp aces. Speciﬁcally,

let u(·) ∈ U be measurable

and d eﬁ ne the L

-norm as

|||u|||

△



∞

ku(t)k



1/p

, (7.4)

where p ∈ [1, ∞) and k·k denotes the Euclidean vector norm deﬁned on R

A function u : [0, ∞) → R

is measurable if it is the poi ntwise limit, except for a set of

Lebesgue measure zero, of a sequence of pi ecewise constant functions on [0, ∞).

NonlinearBook10pt November 20, 2007

474 CHAPTER 7

Furth ermore, deﬁne the L

∞

-norm as

|||u|||

∞

△

= ess sup

t≥0

ku(t)k. (7.5)

Now, for every p ∈ [1, ∞], deﬁne the Lebesgue normed space

△

= {u(·) ∈ U : |||u|||

< ∞}. (7.6)

The notion of input-output stability involves operator dynamical

systems that map L

to L

. Note th at, in general, for an arbitrary operator

dynamical system G : U → Y, y = G[u] need not belong to L

. In order

to deﬁne input-output stability, we refer to an operator dynamical sys tem

G : U → Y as well behaved if for every u(·) ∈ L

, G[u] is measurable.

Furth ermore, deﬁne the extended L

space by

△



u(·) ∈ U : u

∈ L

for every T > 0



Note that the set L

consists of all measurable functions whose truncations

belong to L

. Hence, an operator dynamical system is well behaved if and

only if G : L

→ L

Deﬁnition 7.3. Let G : L

→ L

. G is L

-stable if G[u] ∈ L

for

every u ∈ L

. G is L

-stable with ﬁnite gain γ if there exist γ, β > 0 such

that

|||G[u]|||

≤ γ|||u|||

+ β, u ∈ L

. (7.7)

G is L

-stable with ﬁnite gain and zero bias if there exists γ > 0 such that

|||G[u]|||

≤ γ|||u|||

, u ∈ L

. (7.8)

Note that if G is L

-stable with ﬁnite gain and zero bias, then G is

-stable with ﬁnite gain, and if G is L

-stable with ﬁnite gain, then G is

-stable. Furthermore, note that if G is causal, then (7.7) is equivalent to

|||(G[u])

|||

≤ γ|||u

|||

+ β, T > 0, u ∈ L

, (7.9)

and (7.8) is equivalent to

|||(G[u])

|||

≤ γ|||u

|||

, T > 0, u ∈ L

. (7.10)

Example 7.1. Consider the scalar linear dynamical s ystem given by

˙x(t) = x(t) + u(t), x(0) = 0, t ≥ 0, (7.11)

y(t) = x(t), (7.12)

For p ∈ [1, ∞), the Lebesgue normed space L

consists of all r eal-valued measurable functions

u : [0, ∞) → R

for which kuk

is Lebesgue integrable with norm deﬁned by (7.4). For p = ∞,

the Lebesgue normed space L

∞

consists of all Lebesgue measurable functions on [0, ∞) which are

bounded, except possibly on a set of measure zero.