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

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NonlinearBook10pt November 20, 2007

DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 335

The following corollary to Theorem 5.1 shows that the nonlinear

dynamical system G is dissipative (respectively, exponentially dissipative)

with respect to the supply rate r(·, ·) if and only if there exists a continuous

storage function (respectively, exponential storage fu nction) V

(·) satisfying

(5.16) (respectively, (5.17)).

Corollary 5.1. Consid er the nonlinear 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 ≥ t

, satisfy (5.7), and G is locally controllable at every

ˆx ∈ D. Then G is dissipative (respectively, exponentially dissipative) with

respect to the supply rate r(u, y) if and only if there exists a continuous

storage function (respectively, exponential storage function) V

(x), x ∈ D,

satisfying (5.16) (respectively, (5.17)).

Proof. The result is immediate from Theorems 5.1 and 5.2 with

(x) = V

(x).

In this book, we assume that G is locally controllable at every

ˆx ∈ D and, for every ˆx ∈ D, there exists ˆu ∈ R

such that x(t) ≡ ˆx

and u(t) ≡ ˆu, t ≥ t

, satisfy (5.7), and hence, all storage functions of

G are continuous on D. The following theorem provides conditions for

guaranteeing that all storage functions (respectively, exponential storage

functions) of a given dissipative (respectively, exponentially dissipative)

nonlinear dynamical system are positive deﬁnite. For this result we require

the f ollowing deﬁnition.

Deﬁnition 5.6. A nonlinear dynamical system G is zero-state observ-

able if u(t) ≡ 0 and y(t) ≡ 0 implies x(t) ≡ 0.

Theorem 5.3. Consider the n onlinear dynamical system G given by

(5.7) and (5.8), and assume that G is completely reachable and zero-

state observable. Furthermore, assume that G is dissipative (respectively,

exponentially dissipative) with respect to the supply rate r(u, y) and there

exists a function κ : Y → U su ch that κ(0) = 0 and r(κ(y), y) < 0, y 6= 0.

Then all th e storage functions (respectively, exponential storage functions)

(x), x ∈ D, for G are positive deﬁnite, that is, V

(0) = 0 and V

(x) > 0,

x ∈ D, x 6= 0.

Proof. It follows from Theorem 5.1 that the available storage V

(x),

x ∈ D, is a storage function for G. Next, suppose there exists x ∈ D such

that V

(x) = 0, which implies that r(u(t), y(t)) = 0 almost everywhere t ≥ 0,

for all admissible inputs u(·) ∈ U. Since there exists a function κ : Y → U

such that κ(0) = 0 and r(κ(y), y) < 0, y 6= 0, it follows that y(t) = 0

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336 CHAPTER 5

almost everywhere t ≥ 0. Now, since G is zero-state observable it follows

that x = 0, and hence, V

(x) = 0 if and only if x = 0. Th e result now

follows from (5.18). Finally, the proof for the exponentially dissipative case

is identical.

If V

(·) is continuously diﬀerentiable in Corollary 5.1, then an equiv-

alent statement for the dissipativeness of G with respect to the supply rate

r(u, y) is

(x(t)) ≤ r(u(t), y(t)), t ≥ 0, (5.31)

or, equivalently,

(x) ≤ r(u, y), where

(x) =

V (s(t, x, u))



t=0

denotes

the total derivative of V

(x) along the state trajectories s(t, x, u) of (5.7)

through x ∈ D with u(·) ∈ U at t = 0. Alternatively, an equivalent statement

for exponential dissipativeness of G with respect to the supply rate r(u, y)

(x(t)) + εV

(x(t)) ≤ r(u(t), y(t)), t ≥ 0. (5.32)

Furth ermore, a system G with storage function V

(·) is strictly dissipative

with respect to the supply rate r(u, y) if and only if

(x(t)) < V

(x(t

)) +

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

. (5.33)

Note that exponential dissipativity implies strict dissipativity; however, the

converse is not necessarily true.

Next, we introduce the concept of a required supply of a nonlinear

dynamical system. Speciﬁcally, deﬁne the required supply V

) of the

nonlinear dynamical system G by

) = inf

u(·), T ≥0

−T

r(u(t), y(t))dt, (5.34)

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

The in ﬁmum in (5.34) is taken over all system trajectories starting from

x(−T ) = 0 and time t = −T and ending at x(0) = x

at time t = 0, and all

times t ≥ 0 or, equivalently, over all admissible inputs u(·) which drive the

dynamical system G from the origin to x

over the time interval [−T, 0]. If

the system is not reachable from the origin, then we deﬁne V

) = ∞. It

follows from (5.34) that the required supply of a nonlinear dynamical system

is the minimum amount of generalized en ergy that has to be delivered to

the dynamical system in order to transfer it from an initial state x(−T ) = 0

to a given state x(0) = x

. Similarly, d eﬁ ne the required exponential supply

of the nonlinear dynamical system G by

) = inf

u(·), T ≥0

−T

εt

r(u(t), y(t))dt, (5.35)

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DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 337

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

Note that since, with x(0) = 0, the inﬁmum in (5.34) is zero it follows that

(0) = 0.

Next, using the notion of a required supply, we show that all storage

functions are bounded from above by the required supp ly and bounded f rom

below by the available storage, and hence, a dissipative dynamical system

can deliver to its su rroundings only a fraction of its generalized stored energy

and can store only a fraction of the generalized work done to it.

Theorem 5.4. 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 0 ≤ V

(x) < ∞, x ∈ D. Moreover, if V

(x)

is ﬁnite and nonnegative 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. (5.36)

Proof. Suppose 0 ≤ V

(x) < ∞, x ∈ D. Next, let x(t), t ∈ R, satisfy

(5.7) and (5.8) with admissible inputs u(t), t ∈ R, and x(0) = x

. Since

(x), x ∈ D, is given by the inﬁmum over all admissible inputs u(·) ∈ U and

T > 0 in (5.34), it follows that for all admissible inputs u(·) and −T ≤ t ≤ 0,

) ≤

−T

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

−T

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

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

and h en ce,

) ≤ inf

u(·), T ≥0



−T

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



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

= V

(x(t)) +

r(u(s), y(s))ds, (5.37)

which shows that V

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

Corollary 5.1 implies that G is dissipative.

Conversely, suppose G is dissipative with respect to the supply rate

r(u, y) and let x

∈ D. Since G is completely reachable it follows that there

exist T > 0 and u(t), t ∈ [−T, 0], such that x(−T ) = 0 and x(0) = x

Hence, s ince G is dissipative with r espect to the supply rate r(u, y) it follows

that, for all T ≥ 0,

0 ≤

−T

r(u(t), y(t))dt, (5.38)

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338 CHAPTER 5

and h en ce,

0 ≤ in f

u(·), T ≥0



−T

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



, (5.39)

which implies that

0 ≤ V

) < ∞, x

∈ D. (5.40)

Next, if V

(·) is a storage function for G, then it follows from Theorem

5.1 that

0 ≤ V

(x) ≤ V

(x), x ∈ D. (5.41)

Furth ermore, for all T ≥ 0 such that x(T ) = 0 it follows that

) ≤ V

(0) +

−T

r(u(t), y(t))dt, (5.42)

and h en ce,

) ≤ inf

u(·), T ≥0



−T

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



= V

) < ∞,

which implies (5.36).

Finally, the proof for the exponentially dissipative case follows a similar

construction and, hence, is omitted.

As a direct consequence of Theorems 5.1 and 5.4, we show that the

set of all possible storage functions of a dynamical system forms a convex

set parameterized by the system available storage and the system required

supply. An identical resu lt holds for exponential storage fun ctions.

Proposition 5.1. Consider the nonlinear dynamical system G given

by (5.7) and (5.8) with available storage V

(x), x ∈ D, and required s upply

(x), x ∈ D, and assume G is completely reachable. Then for every α ∈

[0, 1],

(x) = αV

(x) + (1 −α)V

(x), x ∈ D, (5.43)

is a storage function for G.

Proof. The result is a direct consequence of the dissipation inequ ality

(5.16) by noting that if V

(x) and V

(x) satisfy (5.16), then V

(x) satisﬁes

(5.16).

In light of Theorems 5.1 and 5.4 we have the following result on lossless

dynamical systems.

Theorem 5.5. Consider the n onlinear dynamical system G given by

(5.7) and (5.8), and assume G is completely reachable to and from the origin.

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DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 339

Then G is lossless with respect to the supply rate r(u, y) if and only if there

exists a continuous storage fu nction V

(x), x ∈ D, satisfying (5.16) as an

equality. Furthermore, if G is lossless with respect to the supply rate r(u, y),

then V

(x) = V

(x), and hence, the storage function V

(x), x ∈ D, is unique

and is given by

) = −

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

−T

−

r(u(t), y(t))dt, (5.44)

where x(t), t ≥ 0, is the solution to (5.7) with admissible u(·) ∈ U and

x(0) = x

, x

∈ D, for every T

−

, T

> 0 such that x(−T

−

) = 0 and

x(T

) = 0.

Proof. Sup pose G is lossless with respect to the supply rate r(u, y).

Since G is completely reachable to and from the origin it follows that, for

every x

∈ D, there exist T

−

, T

> 0, and u(t) ∈ U , t ∈ [−T

−

, T

], such

that x(−T

−

) = 0, x(T

) = 0, an d x(0) = x

. Now, it follows that

0 =

−T

−

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

−T

−

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

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

≥ inf

u(·), T ≥0

−T

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

u(·), T ≥0

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

= V

) − V

), (5.45)

which implies that V

) ≤ V

), x

∈ D. However, since by deﬁnition G

is dissipative with respect to the supply rate r(u, y) it follows from Theorem

5.4 that V

) ≤ V

), x

∈ D, and hence, every storage function V

∈ D, satisﬁes V

) = V

). Furthermore, it follows that the

inequality in (5.45) is indeed an equality, w hich implies (5.44).

Next, let t

, t, T ≥ 0 be such that t

< t < T , x(T ) = 0. Hence, it

follows from (5.44) that

0 = V

(x(t

)) +

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

= V

(x(t

)) +

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

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

= V

(x(t

)) +

r(u(s), y(s))ds − V

(x(t)),

which implies that (5.16) is satisﬁed as an equality.

Conversely, if there exists a storage function V

(x), x ∈ D, satisfying

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340 CHAPTER 5

(5.16) as an equality, it follows from Corollary 5.1 that G is dissipative with

respect to the supply rate r(u, y). Furthermore, for every u(·) ∈ U, t ≥ 0,

and x(t

) = x(t) = 0, it follows from (5.16) (with an equality) that

r(u(s), y(s))ds = 0,

which implies that G is lossless with respect to the supply rate r(u, y).

Example 5.1. Consider the integrator scalar system

˙x(t) = u(t), x(0) = x

, t ≥ 0, (5.46)

y(t) = x(t). (5.47)

To show that an integrator is th e simplest storage element note th at with

(x) =

and r(u, y) = uy it follows that

(x(t)) = u(t)y(t), t ≥ 0,

and hence,

u(s)y(s)ds = 0 for all t ≥ 0 with x(0) = x(t) = 0. Hence,

(5.46) and (5.47) is lossless with respect to the supply rate r(u, y) = uy.

Furth ermore, the available storage for (5.46) and (5.47) is given by

) = sup

u(·), T ≥0



−

u(t)y(t)dt



≥

∞

(t)dt = x

∞

−2t

dt =

(5.48)

where the above inequality follows by choosing u = −y and T = ∞. Now,

since V

) ≥ V

), it follows that V

) =

. △

5.3 Lagrangian and Hamiltonian Dynamical S y stems

In this section, we show that two of the fundamental dynamical system

formulations of analytical mechanics, namely, Lagrangian and Hamiltonian

dynamical systems, can be formulated as special cases of dissipative

dynamical system theory. In particular, we show that conservation of energy,

internal sy s tem interaction, and interaction with the environment through

input-output ports, inherent in Lagrangian and Hamiltonian dynamical

system formulations, can be captured as a special case of dissipative

dynamical system theory w ith appropriate storage functions and supply

rates corresponding to p hysical system energy an d supplied system power,

respectively.

To begin, consider the governing equations of motion of an n degree-

of-freedom dynamical system given by the Euler-Lagrange eq uation



∂L

∂ ˙q

(q, ˙q)



−



∂L

∂q

(q, ˙q)



= u, (5.49)

where q ∈ R

represents the generalized system positions, ˙q ∈ R

represents

the generalized system velocities, L : R

× R

→ R denotes the system

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DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 341

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 sys tem potential energy, and

u ∈ R

is the vector of generalized forces acting on th e system. Furthermore,

let H : R

×R

→ R denote the Legendre transformation of the Lagrangian

function L(q, ˙q) with respect to the generalized velocity ˙q deﬁned by

H(q, p)

△

= ˙q

p − L(q, ˙q)



˙q=Φ

−1

(p)

. (5.50)

Here p denotes the vector of generalized momenta given by

p(q, ˙q) =



∂L

∂ ˙q

(q, ˙q)



(5.51)

and Φ : R

→ R

is a bijective map from the generalized velocities ˙q to

the generalized momenta p. Now, if H(q, p) is lower bounded, then we can

always shift H(q, p) so that, with minor abus e of notation, H(q, p) ≥ 0,

(q, p) ∈ R

×R

, and H(0, 0) = 0.

In this case, using (5.49) and the fact that

[L(q, ˙q)] =

∂L

∂q

(q, ˙q) ˙q +

∂L

∂ ˙q

(q, ˙q)¨q, (5.52)

it follows that

˙q =





∂L

∂ ˙q

(q, ˙q)



−

∂L

∂q

(q, ˙q)



˙q

[p(q, ˙q)]

˙q −

∂L

∂q

(q, ˙q) ˙q

(q, ˙q) ˙q] −p

(q, ˙q)¨q +

∂L

∂ ˙q

(q, ˙q)¨q −

L(q, ˙q)

(q, ˙q) ˙q − L(q, ˙q)]

H(q, p). (5.53)

Hence, with V

(q, ˙q) = H(q, p(q, ˙q)) and y = ˙q, it follows from Theorem 5.5

that the Euler-Lagrange s ystem (5.49) is lossless with respect to the supply

rate r(u, y) = u

y. This gives a power balance equation that states that the

increase in system en ergy H(q, p) is equal to the supplied work due to the

generalized force u.

Alternatively, if the n degree-of-freedom dynamical s y s tem possesses

internal dissipation, th en the E uler-Lagrange equation takes the form



∂L

∂ ˙q

(q, ˙q)



−



∂L

∂q

(q, ˙q)





∂R

∂ ˙q

( ˙q)



= u, (5.54)

where R : R

→ R r ep resents the Rayleigh dissipation f unction satisfying

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342 CHAPTER 5

∂R

∂ ˙q

( ˙q) ˙q ≥ 0, ˙q ∈ R

. In this case, u s ing (5.52) and (5.54), it follows that

˙q =





∂L

∂ ˙q

(q, ˙q)



−

∂L

∂q

(q, ˙q) +

∂R

∂ ˙q

( ˙q)



˙q

[p(q, ˙q)]

˙q −

∂L

∂q

(q, ˙q) ˙q +

∂R

∂ ˙q

( ˙q) ˙q

(q, ˙q) ˙q] −p

(q, ˙q)¨q +

∂L

∂ ˙q

(q, ˙q)¨q −

L(q, ˙q) +

∂R

∂ ˙q

( ˙q) ˙q

(q, ˙q) ˙q − L(q, ˙q)] +

∂R

∂ ˙q

( ˙q) ˙q

H(q, p) +

∂R

∂ ˙q

( ˙q) ˙q, (5.55)

or, equivalently, integrating over the interval [0, T ],

H(q(T ), p(T )) = H(q(0), p(0)) +

(s) ˙q(s)ds −

∂R

∂ ˙q

( ˙q(s)) ˙q(s)ds.

(5.56)

Equation (5.56) shows that the system energy at time t = T is equal to the

initial energy stored in the s y s tem p lus the energy supplied to the system via

the external force u minus the internal energy dissipated. Since

∂R

∂ ˙q

( ˙q) ˙q ≥ 0,

˙q ∈ R

, it follows from (5.56) and Corollary 5.1 that the Eu ler-Lagrange

system (5.54) is dissipative with respect to the supply rate r(u, y) = u

where y = ˙q, and with storage function V

(q, ˙q) = H(q, p(q, ˙q)).

Next, we compute the available storage and the required supply for

the Euler-Lagrange d y namical system (5.54). Speciﬁcally, using (5.11) and

the fact that the map between the generalized velocities to the generalized

momenta is bijective we obtain

, ˙q

)

= sup

u(·), T ≥0



−

(t) ˙q(t)dt



= sup

u(·), T ≥0



−



H(q(t), p(t)) +

∂R

∂ ˙q

( ˙q(t)) ˙q(t)





= sup

u(·), T ≥0



−H(q(T ), p(T )) + H(q(0), p(0)) −

∂R

∂ ˙q

( ˙q(t)) ˙q(t)dt



≤ H(q(0), p(0)). (5.57)

Note that in the case where

∂R

∂ ˙q

( ˙q) ≡ 0 it follows that V

, ˙q

) = H(q(0),

p(0)), which shows that the available storage is simply the total initial system

energy.

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DISSIPATIVITY THEORY FOR NONLINEAR SYSTEMS 343

Alternatively, using (5.34) we obtain

, ˙q

)

= inf

u(·), T ≥0

−T

(t) ˙q(t)dt

= inf

u(·), T ≥0



−T



H(q(t), p(t)) +

∂R

∂ ˙q

( ˙q(t)) ˙q(t)





= inf

u(·), T ≥0



H(q(0), p(0)) − H(q(−T ), p(−T )) +

−T

∂R

∂ ˙q

( ˙q(t)) ˙q(t)dt



≥ H(q(0), p(0)), (5.58)

where (q(−T ), p(−T )) = (0, 0). Hence, V

, ˙q

) ≤ H(q(0), p(0)) ≤ V

˙q

). Note that in the case where

∂R

∂ ˙q

( ˙q) ≡ 0, it follows from (5.58) that

, ˙q

) = H(q(0), p(0)) −H(q(−T ), p(−T )) = H(q(0), p(0)), (5.59)

and h en ce, V

, ˙q

) = V

, ˙q

) = H(q(0), p(0)).

Next, we transform the Euler-Lagrange equations to the H amiltonian

equations of motion. These equations provide a fu ndamental structure of

the mathematical description of numerous physical dynamical systems by

capturing energy conservation as well as internal interconnection structural

properties of physical dynamical systems. To reduce the Euler-Lagrange

equations (5.54) to a Hamiltonian system of equations consider the Legendre

transformation H(q, p) given by (5.50) of the Lagrangian function L(q, ˙q),

called th e H amiltonian function, and the vector of generalized momenta

given by (5.51). Now, it follows from (5.50), (5.51), and (5.54) that

p(q, ˙q) =



∂L

∂ ˙q

(q, ˙q)





∂L

∂q

(q, ˙q)



−



∂R

∂ ˙q

( ˙q)



+ u

= −



∂H

∂q

(q, p)



−



∂R

∂ ˙q

( ˙q)



+ u. (5.60)

Assuming that the Rayleigh d iss ipation fun ction R : R

→ R is

quadratic, that is, R( ˙q) =

˙q

D ˙q, where D is a nonnegative-deﬁnite system

dissipation matrix, (5.60) implies

˙p = −



∂H

∂q

(q, p)



− D ˙q + u. (5.61)

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344 CHAPTER 5

Next, it follows from (5.50) that

˙q =



∂H

∂p

(q, p)



. (5.62)

Hence, (5.61) and (5.62) can be equivalently written as

˙x(t) = [J − R]



∂H

∂x

(x(t))



+ Gu(t), x(0) = x

, t ≥ 0, (5.63)

where x = [q

, p

]

and

△



−I



, R

△



0 0

0 D



, G

△





. (5.64)

The dynamical system (5.63) with outputs y(t) = G

(x(t))



∂H

∂x

(x(t))



called a Hamiltonian dynamical system.

The Hamiltonian dynamical system (5.63) can be further generalized

to what is commonly referred to as port-controlled Hamiltonian dynamical

systems described in local coordinates x ∈ D ⊆ R

. In particular, port-

controlled Hamiltonian dynamical systems are given by

˙x(t) = [J(x(t)) − R(x(t))]



∂H

∂x

(x(t))



+ G(x(t))u(t),

x(0) = x

, t ≥ 0, (5.65)

y(t) = G

(x(t))



∂H

∂x

(x(t))



, (5.66)

where x(t) ∈ D ⊆ R

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

, y(t) ∈

Y ⊆ R

, H : D → R is a continuously diﬀerentiable Hamiltonian function for

the system (5.65) and (5.66), J : D → R

n×n

is such that J(x) = − J

(x),

R : D → S

is such that R(x) ≥ 0, x ∈ D, [J(x) − R(x)]



∂H

∂x

(x)



x ∈ D, is Lipschitz continuous on D, and G : D → R

n×m

. Th e skew-

symmetric matrix function J(x), x ∈ D, captures the internal system

interconnection structur e, the input matrix function G(x), x ∈ D, captures

interconnections with the environment, and the symmetric nonnegative-

deﬁnite matrix function R(x), x ∈ D, captures system dissipation. Note,

that the in puts and outputs are dual (conjugated) variables. Here, we

assume that u(·) is restricted to the class of admissible inputs consisting

of measurable functions such that u(t) ∈ U for all t ≥ 0.

Assuming that the Hamiltonian energy function H(·) is lower boun ded,

it can be s hown that port-controlled Hamiltonian systems provide an energy

balance in terms of the stored or accumulated energy, supplied system

energy, and dissipated energy. Speciﬁcally, computing the rate of change

of the Hamiltonian along the system state trajectories x(t), t ≥ 0, yields the