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

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Chapter Five

Dissipativity Theory for Nonlinear

Dynamical Systems

5.1 Introduction

In control engineering, dissipativity theory provides a fundamental frame-

work for the analysis and control design of dynamical systems using an

input, state, and output system description based on system-energy-related

considerations. The notion of energy here refers to abstract energy notions

for which a physical system energy interpretation is not necessary. The

dissipation hypothesis on dynamical systems results in a fundamental

constraint on their d ynamic behavior, wherein a dissipative dynamical

system can deliver only a fraction of its energy to its surroundings and can

store only a fraction of the work done to it. Many of the great landmarks

of feedback control th eory are associated with d issipativity theory. In

particular, dissipativity theory provides the foundation for absolute stability

theory which in turn forms the basis of the Lur´e problem, as well as the

circle and Popov criteria, which are extensively developed in the classical

monographs of Aizerman and Gantmacher [5], Lefschetz [265], and Popov

[364]. Since absolute stability theory concerns the stability of a dynamical

system for classes of f eedback nonlinearities which, as noted in [147, 148],

can readily be interpreted as an uncertainty model, it is not surprising that

absolute stability theory (and, hence, dissipativity theory) also forms the

basis of modern-day robust stability analysis and s ynthesis [147,151,172].

The key foundation in developing dissipativity theory for general

nonlinear dynamical systems was presented by J. C. Willems [456, 457] in

his semin al two-part paper on dissipative dynamical systems. In particular,

Willems [456] introduced the deﬁnition of dissip ativity for general dynamical

systems in terms of a dissipation inequality involving a generalized system

power input, or supply rate, and a generalized energy function, or storage

function. The dissipation inequ ality implies that the increase in generalized

system energy over a given time interval cannot exceed the generalized

energy supply delivered to the system during this time interval. The

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326 CHAPTER 5

set of all possible system storage functions is convex and every system

storage function is bounded from below by the available system storage and

bounded from above by the requir ed supply. The available storage is the

amount of internal generalized stored energy which can be extracted from

the dynamical system and the required supply is the amount of generalized

energy which can be delivered to the dynamical system to transfer it from

a state of minimum potential to a given state. Hence, as noted above,

a dissipative dynamical system can deliver only a fraction of its stored

generalized energy to its sur roundings and can store only a fraction of

generalized work done to it.

Dissipativity theory is a system theoretic concept that provides a

powerful framework for the analysis and control design of dynamical systems

based on generalized energy considerations. In particular, dissipativity

theory exploits the notion that numerous physical dynamical sys tems have

certain input-output system properties related to conservation, dissipation,

and transport of mass and energy. Such conservation laws are prevalent

in dynamical systems such as mechanical systems, ﬂuid systems, elec-

tromechanical systems, electrical systems, combustion systems, s tructural

systems, biological systems, physiological systems, biomedical systems,

ecological systems, economic systems, as well as feedback control systems.

On the level of analysis, dissipativity can involve conditions on system

parameters that render an input, state, output system dissipative. Or,

alternatively, analyzing system stability robustness by viewing a dynamical

system as an interconnection of dissipative dynamical subsystems. On the

synthesis level, dissipativity can be used to design feedback controllers that

add dissipation and guarantee stability robustness allowing stabilization to

be understood in physical terms.

To elucidate the notion of dissipativity of an input, state, and

output system, consider the single-degree-of-freedom spring-mass-damper

mechanical system given by

M ¨x(t) + C ˙x(t) + Kx(t) = u(t), x(0) = x

, ˙x(0) = ˙x

, t ≥ 0, (5.1)

where M > 0 is the system mass, C ≥ 0 is the system damping constant,

K ≥ 0 is the system stiﬀness, x(t), t ≥ 0, is the position of the mass M, and

u(t), t ≥ 0, is an external force acting on the mass M. The energy of this

system is given by

(x, ˙x) =

M ˙x

. (5.2)

Now, assu ming that the measured output of this system is the system

velocity, that is, y(t) = ˙x(t), it follows that the time rate of change of

the system energy along the system trajectories is given by

(x, ˙x) = M ¨x ˙x + Kx ˙x = uy − C ˙x

. (5.3)

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DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 327

Integrating (5.3) over the time interval [0, T ], it follows that

(x(T ), ˙x(T )) = V

(x(0), ˙x(0)) +

u(t)y(t)dt −

C ˙x

(t)dt, (5.4)

which shows that the system energy at time t = T is equal to the initial

energy stored in the system plus th e energy supplied to the system via

the external force u minus the energy dissipated by the system damper.

Equivalently, it follows from (5.3) that the rate of change in the system

energy, or system power, is equal to the external supplied system power

through the input port u minus the internal system power dissipated by the

viscous damper.

Note that in the case where the external input force u is zero and C =

0, that is, no system supply or dissipation is present, (5.3) or, equivalently,

(5.4) shows that the system energy is constant. Furthermore, note that since

C ≥ 0 and V (x(T ), ˙x(T )) ≥ 0, T ≥ 0, it follows from (5.4) that

u(t)y(t)dt ≥ −V

, ˙x

), (5.5)

or, equivalently,

−

u(t)y(t)dt ≤ V

, ˙x

). (5.6)

Equation (5.6) shows that the energy that can be extracted from the system

through its input-output ports is less than or equal to the initial energy

stored in the system. As will be seen in this chapter, this is precisely the

notion of dissipativity.

Since, as discussed in Chapter 3, Lyapunov functions can be viewed

as generalizations of energy functions f or nonlinear dynamical systems,

the notion of diss ipativity, with appr opriate storage functions and supply

rates, can be used to construct Lyapunov functions for nonlinear feedback

systems by appropriately combining storage functions for each subsystem.

Even though the original work on dissipative dynamical systems was

formulated in the state space setting, describing the system dynamics

in terms of continuous ﬂows on appropr iate manifolds, an input-output

formulation for dissipative dynamical systems extending the notions of

passivity [476], nonexpansivity [477], and conicity [377,476] was presented in

[188,191,320]. In this chapter, we introduce precise mathematical deﬁnitions

for dissipativity as well as develop a general framework for characterizing

system dissipativity in terms of system storage functions and supply rates.

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328 CHAPTER 5

5.2 Dissipative and Exponentially Dissipative Dynamical Systems

In this section, we introd uce the deﬁnition of dissipativity for general

dynamical systems in terms of an inequality involving generalized system

power input, or supply rate, and a generalized energy function, or storage

function. In Chapters 2–4, we considered closed dynamical systems wherein

each system trajectory is determined by the system initial conditions and

driven by the internal dynamics of the system without any inﬂuence f rom

the environment. Alternatively, in this chapter we consider open dynamical

systems wherein the system interaction with the environment is explicitly

taken into account through the system inputs and outputs. Speciﬁcally, the

environment acts on the dynamical system through the system inputs, and

the dynamical system reacts through the system outpu ts.

We begin by considering nonlinear dynamical systems G of the form

˙x(t) = F (x(t), u(t)), x(t

) = x

, t ≥ t

, (5.7)

y(t) = H(x(t), u(t)), (5.8)

where x(t) ∈ D ⊆ R

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

, y(t) ∈

Y ⊆ R

, F : D×U → R

, and H : D×U → Y . For the dynamical system G

given by (5.7) and (5.8) deﬁned on th e state space D ⊆ R

, U and Y deﬁne

an input and output space, respectively, consisting of continuous bounded

U-valued and Y -valued functions on the semi-inﬁnite interval [0, ∞). The set

U contains th e set of input values, that is, for every u(·) ∈ U and t ∈ [0, ∞),

u(t) ∈ U. T he set Y contains the set of outp ut values, that is, for every

y(·) ∈ Y and t ∈ [0, ∞), y(t) ∈ Y . T he spaces U and Y are assumed to be

closed under 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. We assume that F (·, ·)

and H(·, ·) are continuously diﬀerentiable mappings in (x, u) and F (·, ·) has

at least one equilibrium so that, without loss of generality, F (0, 0) = 0 and

H(0, 0) = 0. Furthermore, for the nonlinear dynamical system G we assume

that th e required properties for the existence and uniqueness of solutions

are satisﬁed, that is, u(·) satisﬁes suﬃ cient regularity conditions such that

the system (5.7) has a unique solution forward and backward in time. For

the dynamical system G given by (5.7) and (5.8), a function r : U ×Y → R

such that r(0, 0) = 0 is called a supply rate if r(u, y) is locally integrable

for all inp ut-output pairs satisfying (5.7) and (5.8), that is, for all input-

output pairs u(·) ∈ U and y(·) ∈ Y satisfying (5.7) and (5.8), r(·, ·) satisﬁes

|r(u(s), y(s))|ds < ∞, t

, t

≥ 0.

Deﬁnition 5.1. A dynamical system G of the form (5.7) and (5.8) is

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DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 329

dissipative with respect to the supply rate r(u, y) if the dissipation inequality

0 ≤

r(u(s), y(s))ds (5.9)

is satisﬁed for all t ≥ t

and all u(·) ∈ U with x(t

) = 0 along the trajectories

of G. A dynamical system G of the form (5.7) and (5.8) is exponentially

dissipative with respect to the supply rate r(u, y) if there exists a constant

ε > 0 such that the exponential dissipation inequality

0 ≤

εs

r(u(s), y(s))ds (5.10)

is satisﬁed for all t ≥ t

and all u(·) ∈ U with x(t

) = 0 along the trajectories

of G. A dynamical system G of the form (5.7) and (5.8) is lossless with respect

to the supply rate r(u, y) if G is dissipative with respect to the supply rate

r(u, y) and the dissipation inequality (5.9) is satisﬁed as an equality for all

t ≥ t

and all u(·) ∈ U with x(t

) = x(t) = 0 along the trajectories of G.

In the following we shall use either 0 or t

to denote the initial time

for G. Next, deﬁne the available storage V

) of the nonlinear dynamical

system G by

)

△

= − inf

u(·), T ≥0

r(u(t), y(t))dt

= sup

u(·), T ≥0



−

r(u(t), y(t))dt



, (5.11)

where x(t), t ≥ 0, is the solution to (5.7) with x(0) = x

and admissible input

u(·) ∈ U. The supremum in (5.11) is taken over all admissible in puts u(·),

all time t ≥ 0, an d all system trajectories with initial value x(0) = x

and

terminal value left f ree. Note that V

(x) ≥ 0 for all x ∈ D since V

(x) is the

supremum over a s et of numbers containing the zero element (T = 0). When

the ﬁnal state is not free b ut rather constrained to x(t) = 0 corresponding

to the equilibrium of the uncontrolled system, then V

) corresponds to

the virtual available storage. For details, see Problem 5.3. It follows from

(5.11) that the available storage of a nonlinear dynamical system G is the

maximum amount of storage, or generalized stored energy, which can be

extracted from the nonlinear dynamical system G at any time T . Similarly,

deﬁne the available exponential storage V

) of the nonlinear dynamical

system G by

)

△

= − inf

u(·), T ≥0

εt

r(u(t), y(t))dt, (5.12)

where x(t), t ≥ 0, is the solution to (5.7) with x(0) = x

and admissible

input u(·) ∈ U. Note that if we deﬁne the available exponential storage as

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330 CHAPTER 5

the time-varying function

, t

) = − inf

u(·), T ≥t

εt

r(u(t), y(t))dt, (5.13)

where x(t), t ≥ t

, is the solution to (5.7) with x(t

) = x

and admissible

input u(·), it follows that, since G is time invariant,

, t

) = −e

εt

inf

u(·), T ≥0

εt

r(u(t), y(t))dt = e

εt

). (5.14)

Hence, an alternative expression for the available exponential storage

function V

) is given by

) = −e

−εt

inf

u(·), T ≥t

εt

r(u(t), y(t))dt. (5.15)

, t

) given by (5.13) deﬁnes the available storage function for nons ta-

tionary (time-varying) dynamical systems [191,456]. As shown above, in the

case of exponentially dissipative sys tems,

, t

) = e

εt

Next, we show that the available storage (respectively, available

exponential storage) is ﬁnite and zero at the origin if and only if G is

dissipative (respectively, exponentially d issipative). For this result we

require three more deﬁnitions.

Deﬁnition 5.2. A n onlinear dynamical system G is completely reach-

able if for all x

∈ D ⊆ R

there exist a ﬁnite time t

< t

and a square

integrable input u(t) deﬁned on [t

, t

] such that the state x(t), t ≥ t

, can be

driven from x(t

) = 0 to x(t

) = x

. G is completely null controllable if for all

∈ D ⊆ R

there exist a ﬁnite time t

> t

and a square integrable input

u(t) deﬁned on [t

, t

] such that x(t), t ≥ t

, can be driven from x(t

) = x

to x(t

) = 0.

Deﬁnition 5.3. Consider the nonlinear dynamical system G given by

(5.7) and (5.8). A continuous, nonnegative-deﬁnite function V

: D → R

satisfying V

(0) = 0 and

(x(t)) ≤ V

(x(t

)) +

r(u(s), y(s))ds, t ≥ t

, (5.16)

for all t

, t ≥ 0, where x(t), t ≥ t

, is the solution of (5.7) with u(·) ∈ U, is

called a storage function for G.

Inequality (5.16) is known as th e dissipation inequality and reﬂects

the fact that some of the supplied generalized energy to the open dynamical

system G is stored, and some is dissipated. The dissipated generalized energy

is nonnegative and is given by the diﬀerence of what is s upplied and what

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DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 331

is stored. In addition, the amount of generalized stored energy is a function

of the state of the dynamical system.

Deﬁnition 5.4. Consider the nonlinear dynamical sy s tem G given by

(5.7) and (5.8). A continuous, nonnegative-deﬁnite function V

: D → R

satisfying V

(0) = 0 and

εt

(x(t)) ≤ e

εt

(x(t

)) +

εs

r(u(s), y(s))ds, t ≥ t

, (5.17)

for all t

, t ≥ 0, where x(t), t ≥ t

, is the solution of (5.7) with u(·) ∈ U, is

called an exponential storage function for G.

Theorem 5.1. Consider the n onlinear dynamical system G given by

(5.7) and (5.8), and assume that G is completely reachable. Then G

is dissipative (respectively, exponentially dissipative) w ith respect to the

supply rate r(u, y) if and only if the available system storage V

) given

by (5.11) (respectively, the available exponential storage V

(x) given by

(5.12)) is ﬁnite for all x

∈ D and V

(0) = 0. Moreover, if V

(0) = 0

and V

) is ﬁnite for all x

∈ D, then V

(x), x ∈ D, is a storage

function (respectively, exponential storage function) for G. Finally, all

storage functions (respectively, exponential storage functions) V

(x), x ∈ D,

for G satisfy

0 ≤ V

(x) ≤ V

(x), x ∈ D. (5.18)

Proof. Suppose V

(0) = 0 and V

), x

∈ D, is ﬁ nite. Now, it

follows from (5.11) (with T = 0) that V

) ≥ 0, x

∈ D. Next, let x(t),

t ≥ t

, satisfy (5.7) with admissible input u(t), t ∈ [t

, T ]. Since −V

∈ D, is given by the inﬁmum over all admissible inputs u(·) in (5.11), it

follows that for all admissible inputs u(·) ∈ U and T > t

−V

(x(t

)) ≤

r(u(t), y(t))dt

r(u(t), y(t))dt +

r(u(t), y(t))dt,

which implies

−V

(x(t

)) −

r(u(t), y(t))dt ≤

r(u(t), y(t))dt.

Hence,

(x(t

)) +

r(u(t), y(t))dt ≥ − inf

u(·), T ≥t

r(u(t), y(t))dt

= V

(x(t

))

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332 CHAPTER 5

≥ 0, (5.19)

which implies that

r(u(t), y(t))dt ≥ − V

(x(t

)). (5.20)

Hence, since by assumption V

(0) = 0, G is d iss ipative with respect to the

supply rate r(u, y). Furthermore, V

(x), x ∈ D, is a storage function for G.

Conversely, suppose G is dissipative with respect to the su pply rate

r(u, y). Since G is completely reachable it follows that for every x

∈ D

such that x(t

) = x

, there exist

t ≤ t < t

and an admissible input u(·) ∈

U deﬁned on [

t, t

] such that x(

t) = 0 and x(t

) = x

. Now, since G is

dissipative with respect to the supply rate and x(

t) = 0 it follows that

r(u(t), y(t))dt ≥ 0, T >

or, equivalently,

r(u(t), y(t))dt ≥ −

r(u(t), y(t))dt, T > t

which implies that there exists a function W : D → R such that

r(u(t), y(t))dt ≥ W (x

) > −∞, T > t

. (5.21)

Now, it follows from (5.21) that f or all x ∈ D,

(x) = − inf

u(·), T ≥t

r(u(t), y(t))dt

≤ −W (x), (5.22)

and hence, the available storage V

(x) < ∞, x ∈ D. Furthermore, with

x(t

) = 0, it follows that for all admissible u(t), t ≥ t

r(u(t), y(t))dt ≥ 0, T ≥ t

, (5.23)

which implies that

sup

u(·), T ≥t



−

r(u(t), y(t))dt



≤ 0, (5.24)

or, equivalently, V

(x(t

)) = V

(0) ≤ 0. However, since V

(x) ≥ 0, x ∈ D, it

follows that V

(0) = 0.

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DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 333

Moreover, if V

(x) is ﬁnite for all x ∈ D, it follows from (5.19) that

(x), x ∈ D, is a storage f unction for G. Next, if V

(x), x ∈ D, is a storage

function, then it f ollows that, for all T > 0 and x

∈ D,

) ≥ V

(x(T )) −

r(u(t), y(t))dt

≥ −

r(u(t), y(t))dt,

which implies

) ≥ − in f

u(·), T ≥0

r(u(t), y(t))dt

= V

yielding (5.18).

Finally, the proof for the exponentially dissipative case follows an

identical construction and, hence, is omitted.

Theorem 5.1 presents necessary and suﬃcient conditions for the

existence of an available storage function for a nonlinear dynamical system

G. The following result presents suﬃcient conditions for guaranteeing that

the system storage function is a continuous function. First, however, th e

following deﬁnition is needed.

Deﬁnition 5.5. Consider the nonlinear dynamical sy s tem G given by

(5.7) and (5.8) and let ˆx ∈ R

and ˆu ∈ R

be such that x(t) ≡ ˆx and

u(t) ≡ ˆu, t ≥ 0, satisfy (5.7). G is locally controllable at ˆx if, for every T > 0

and ε > 0, the set of points that can be reached from and to ˆx in ﬁnite time

T using admissible inputs u : [0, T ] → U, satisfying ku(t) − ˆuk < ε, contains

a neighborhood of ˆx.

Consider the linearization of (5.7) at x = ˆx and u = ˆu given by

˙x(t) = A(x(t) − ˆx) + B(u(t) − ˆu), x(t

) = x

, t ≥ 0, (5.25)

where A =

∂F

∂x



x=ˆx,u=ˆu

and B =

∂F

∂u



x=ˆx,u=ˆu

. Now, it follows from

Proposition 3.3 of [336] that if the pair (A, B) is controllable, then (5.7)

is locally controllable.

Theorem 5.2. Consider the n onlinear dynamical system G given by

(5.7) and (5.8), and assume that G is completely reachable. Furthermore,

assume that for every ˆx ∈ D, there exists ˆu ∈ R

such that x(t) ≡ ˆx and

u(t) ≡ ˆu, t ≥ 0, satisfy (5.7), and G is locally controllable at every ˆx ∈ D. I f

G is dissipative (respectively, exponentially dissip ative) with respect to the

supply rate r(u, y), then every storage function (respectively, exponential

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334 CHAPTER 5

storage function) V

(x), x ∈ D, is continuous on D.

Proof. Let ˆx ∈ D and ˆu ∈ R

be su ch that x(t) ≡ ˆx and u(t) ≡ ˆu,

t ≥ 0, satisfy (5.7). Now, let δ > 0 and note that it follows from the

continuity of F (·, ·) that there exist T > 0 and ε > 0 such that for every

u : [0, T ) → U and ku(t) − ˆuk < ε, kx(t) − ˆxk < δ, t ∈ [0, T ), where u(·) ∈ U

and x(t), t ∈ [0, T ), denotes the solution to (5.7) with the initial condition

ˆx. Furthermore, it follows from the local controllability assumption that for

every

T ∈ (0, T ], there exists a strictly increasing, continuous function γ :

R → R such that γ(0) = 0, and for every x

∈ D such that kx

−ˆxk ≤ γ(

T ),

there exists

t ∈ [0,

T ) and an input u : [0,

T ] → R

such that ku(t)−ˆuk < ε,

t ∈ [0,

t), and x(

t) = x

. Hence, there exists β > 0 such that for every

∈ D such that kx

− ˆxk ≤ β, there exists

t ∈ [0, γ

−1

(kx

− ˆxk)] and input

u : [0,

t] → R

such that ku(t) − ˆuk < ε, t ∈ [0,

t], and x(

t) = x

Next, since r(·, ·) is locally integrable for all input-output pairs

satisfying (5.7) and (5.8), it follows that there exists M ∈ (0, ∞) such that

sup

kx−ˆxk<δ,ku−ˆuk<ε

|r(u, y)| = M, (5.26)

and h en ce, it follows that



r(u(s), y(s))ds



≤

|r(u(s), y(s))|ds

≤ M

≤ Mγ

−1

(kx

− ˆxk). (5.27)

Now, if V

(·) is a storage function of G, then

(x(

t)) ≤ V

(ˆx) +

r(u(s), y(s))ds, (5.28)

or, equivalently,

−

r(u(s), y(s))ds ≤ V

(ˆx) − V

(x(

t)). (5.29)

If V

(ˆx) ≤ V

(x(

t)), then combining (5.27) and (5.29) yields

(ˆx) − V

(x(

t))| ≤ Mγ

−1

(kx

− ˆxk). (5.30)

Alternatively, if V

(x(

t)) ≥ V

(ˆx), then (5.30) can be derived by reversing

the roles of ˆx and x(

t). Hence, since γ(·) is continuous and x(

t) is arbitrary,

it follows that V

(·) is continuous on D.

Finally, the proof for the exponentially dissipative case follows an

identical construction and, hence, is omitted.