# Approximate reduction of dynamical systems

**ABSTRACT** The reduction of dynamical systems has a rich history, with many important applications related to stability, control and verification. Reduction of nonlinear systems is typically performed in an exact manner - as is the case with mechanical systems with symmetry--which, unfortunately, limits the type of systems to which it can be applied. The goal of this paper is to consider a more general form of reduction, termed approximate reduction, in order to extend the class of systems that can be reduced. Using notions related to incremental stability, we give conditions on when a dynamical system can be projected to a lower dimensional space while providing hard bounds on the induced errors, i.e., when it is behaviorally similar to a dynamical system on a lower dimensional space. These concepts are illustrated on a series of examples.

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**ABSTRACT:**We consider the problem of approximating plants with discrete sensors and actuators (termed 'systems over finite alphabets') by deterministic finite memory systems for the purpose of certified-by-design controller synthesis. We propose a new, control-oriented notion of input/output approximation for these systems, that builds on ideas from robust control theory and behavioral systems theory. We conclude with a brief discussion of the key features of the proposed notion of approximation relative to those of two existing notions of finite state approximation and abstraction.CoRR. 01/2011; abs/1105.3788. - SourceAvailable from: Hamid Reza Shaker
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##### Conference Paper: Generalized Gramian Framework for Model Reduction of Switched Systems

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**ABSTRACT:**In this paper, a general method for model order reduction of switched linear dynamical systems is presented. The proposed technique is based on the generalized gramian framework for model reduction. It is shown that different classical reduction methods can be developed into generalized gramian framework. Balanced reduction within specified frequency bound is developed within this framework. In order to avoid numerical instability and also to increase the numerical efficiency, generalized gramian based Petrov- Galerkin projection is constructed instead of the similarity transform approach for reduction. The method preserves the stability of the original switched system under arbitrary switching signal and is applicable to both continuous and discrete time systems. The performance of the proposed method is illustrated by numerical examples.European Control Conference; 01/2009

Page 1

APPROXIMATE REDUCTION OF DYNAMICAL SYSTEMS

PAULO TABUADA, AARON D. AMES, AGUNG JULIUS, AND GEORGE J. PAPPAS

Abstract. The reduction of dynamical systems has a rich history, with many important applications related

to stability, control and verification. Reduction of nonlinear systems is typically performed in an “exact”

manner—as is the case with mechanical systems with symmetry—which, unfortunately, limits the type of

systems to which it can be applied. The goal of this paper is to consider a more general form of reduction,

termed approximate reduction, in order to extend the class of systems that can be reduced. Using notions

related to incremental stability, we give conditions on when a dynamical system can be projected to a lower

dimensional space while providing hard bounds on the induced errors, i.e., when it is behaviorally similar to

a dynamical system on a lower dimensional space. These concepts are illustrated on a series of examples.

1. Introduction

Modeling is an essential part of many engineering disciplines and often a key ingredient for successful designs.

Although it is widely recognized that models are only approximate descriptions of reality, their value lies

precisely on the ability to describe, within certain bounds, the modeled phenomena. In this paper we consider

modeling of closed-loop nonlinear control systems, i.e., differential equations, with the purpose of simplifying

the analysis of these systems. The goal of this paper is to reduce the dimensionality of the differential

equations being analyzed while providing hard bounds on the introduced errors. One promising application

of these techniques is to the verification of hybrid systems, which is currently constrained by the complexity

of high dimensional differential equations.

Reducing differential equations—and in particular mechanical systems—is a subject with a long and rich

history. The first form of reduction was discovered by Routh in the 1860’s; over the years, geometric reduction

has become an academic field in itself. One begins with a differential equation with certain symmetries, i.e.,

it is invariant under the action of a Lie group on the phase space. Using these symmetries, one can reduce

the dimensionality of the phase space (by “dividing” out by the symmetry group) and define a corresponding

differential equation on this reduced phase space. The main result of geometric reduction is that one can

understand the behavior of the full-order system in terms of the behavior of the reduced system and vice

versa [MW74, vdS81, BKMM96]. While this form of “exact” reduction is very elegant, the class of systems

for which this procedure can be applied is actually quite small. This indicates the need for a form of reduction

that is applicable to a wider class of systems and, while not being exact, is “close enough”.

In systems theory, reduced order modeling has also been extensively studied under the name of model re-

duction [BDG96, ASG00]. Contrary to model reduction where approximation is measured using L2norms

we are interested in L∞norms. The guarantees provided by L∞norms are more natural when applica-

tions to safety verification are of interest. More recent work considered the exact reduction of control sys-

tems [vdS04, TP04] based on the notion of bisimulation which was later generalized to approximate bisimu-

lation [GP05, Tab07, GP07].

We develop our results in the framework of incremental stability and our main result is in the spirit of existing

stability results for cascade systems that proliferate the Input-to-State Stability (ISS) literature. See for

example [Son06] and the references therein. A preliminary version of our results appeared in the conference

paper [TAJP06].

This research was partially supported by the National Science Foundation, EHS award 0712502.

1

arXiv:0707.3804v1 [math.OC] 25 Jul 2007

Page 2

2PAULO TABUADA, AARON D. AMES, AGUNG JULIUS, AND GEORGE J. PAPPAS

2. Preliminaries

A continuous function γ : R+

γ(r) → ∞ as r → ∞. A continuous function β : R+

s, the map β(r,s) belongs to class K∞with respect to r and, for each fixed r, the map β(r,s) is decreasing

with respect to s and β(r,s) → 0 as s → ∞.

For a smooth function φ : Rn→ Rmwe denote by Tφ the tangent map to φ and by Txφ the tangent map to

φ at x ∈ Rn. We will say that φ is a submersion at x ∈ Rnif Txφ is surjective and that Tφ is a submersion if

it is a submersion at every x ∈ Rn. When φ is a submersion we will also use the notation ker(Tφ) to denote

the distribution:

ker(Tφ) = {X : Rn→ Rn| Tφ · X = 0}.

The Lie bracket of vector fields X and Y will be denotes by [X,Y ].

0→ R+

0, is said to belong to class K∞if it is strictly increasing, γ(0) = 0 and

0×R+

0→ R+

0is said to belong to class KL if, for each fixed

Given a point x ∈ Rn, |x| will denote the usual Euclidean norm while ||f|| will denote esssupt∈[0,τ]|f(t)| for

any given function f : [0,τ] → Rn, τ ∈ R+.

2.1. Dynamical and control systems. In this paper we shall restrict our attention to dynamical and control

systems defined on Euclidean spaces.

Definition 2.1. A vector field is a pair (Rn,X) where X is a smooth map X : Rn→ Rn. A smooth curve

x(·,x) : I → Rn, defined on an open subset I of R including the origin, is said to be a trajectory of (Rn,X)

if the following two conditions hold:

(1) x(0,x) = x;

(2)

d

dtx(t,x) = X(x(t,x)) for all t ∈ I.

A control system can be seen as an under-determined vector field.

Definition 2.2. A control system is a triple (Rn,Rm,F) where F is a smooth map F : Rn× Rm→ Rn. A

smooth curve xu(·,x) : I → Rn, defined on an open subset I of R including the origin, is said to be a trajectory

of (Rn,Rm,F) if there exists a smooth curve u : I → Rmsuch that the following two conditions hold:

(1) xu(0,x) = x;

(2)

d

dtxu(t,x) = F(xu(t,x),u(t)) for all t ∈ I.

We have defined trajectories based on smooth input curves mainly for simplicity since the presented results

hold under weaker regularity assumptions.

3. Exact reduction

For some dynamical systems described by a vector field X on Rnit is possible to replace X by a vector

field Y describing the dynamics of the system on a lower dimensional space, Rm, while retaining much of the

information in X. When this is the case we say that X can be reduced to Y . This idea of (exact) reduction

is captured by the notion of φ-related vector fields.

Definition 3.1. Let φ : Rn→ Rmbe a smooth map. The vector field (Rn,X) is said to be φ-related to the

vector field (Rm,Y ) if for every x ∈ Rnwe have:

(3.1)

Txφ · X(x) = Y ◦ φ(x).

The following proposition, proved in [AMR88], characterizes φ-related vector fields in terms of their trajecto-

ries.

Page 3

APPROXIMATE REDUCTION OF DYNAMICAL SYSTEMS3

Proposition 3.2. The vector field (Rn,X) is φ-related to the vector field (Rm,Y ) for some smooth map

φ : Rn→ Rmiff for every x ∈ Rnwe have:

(3.2)

φ ◦ x(t,x) = y(t,φ(x)),

where x(t,x) and y(t,y) are the trajectories of vector fields X and Y , respectively.

For φ-related vector fields, we can replace the study of trajectories x(·,x) with the study of trajectories

y(·,φ(x)) living on the lower dimension space Rm. In particular, formal verification of X can be performed

on Y whenever the relevant sets describing the verification problem can also be reduced to Rm.

If a vector field and a submersion φ are given we can use the following result, proved in [MSVS85], to determine

the existence of φ-related vector fields.

Proposition 3.3. Let (Rn,X) be a vector field and let φ : Rn→ Rmbe a smooth submersion. There exists a

vector field (Rm,Y ) that is φ-related to (Rn,X) iff:

[ker(Tφ),X] ⊆ ker(Tφ).

In an attempt to enlarge the class of vector fields that can be reduced we introduce, in the next section, an

approximate notion of reduction.

4. Approximate Reduction

The generalization of Definition 3.1 proposed in this section requires a decomposition of Rnof the form

Rn= Rm× Rk. Associated with this decomposition are the canonical projections πm: Rn→ Rmand πk :

Rn→ Rktaking Rn? x = (y,z) ∈ Rm× Rkto πm(x) = y and πk(x) = z, respectively.

Definition 4.1. The vector field (Rn,X) is said to be approximately πm-related to the vector field (Rm,Y )

if there exist a class K∞function γ and a constant c ∈ R+

x ∈ Rn:

(4.1)

|πm◦ x(t,x) − y(t,πm(x))| ≤ γ(|πk(x)|) + c.

0such that the following estimate holds for every

Note that when X and Y are πm-related we have:

|πm◦ x(t,x) − y(t,πm(x))| = 0,

which implies (4.1).

Definition 3.1. Similar ideas have been used in the context of approximate notions of equivalence for control

systems [GP07].

Definition 4.1 can thus be seen as a generalization of exact reduction captured by

Although the bound on the gap between the projection of the original trajectory x and the trajectory y of

the approximate reduced system is a function of x, in concrete applications the initial conditions are typically

restricted to a bounded set of interest. The following result has interesting implications in these situations.

Proposition 4.2. If (Rn,X) is approximately πm-related to (Rm,Y ) then for any compact set S ⊆ Rnthere

exists a δ ∈ R+such that for all x ∈ S the following estimate holds:

(4.2)

|πm◦ x(t,x) − y(t,πm(x))| ≤ δ.

Proof: Let δ = maxx∈Cγ(|πk(x)|) + c. The scalar δ is well defined since γ(|πk( · )|) + c is a continuous map

and C is compact.

?

From a practical point of view, approximate reduction is only a useful concept if it admits characterizations

that are simple to check. In order to derive such characterizations we need to review several notions of

incremental stability.

Page 4

4PAULO TABUADA, AARON D. AMES, AGUNG JULIUS, AND GEORGE J. PAPPAS

4.1. Incremental stability. In this subsection we review two notions of incremental stability which will be

fundamental in proving the main contribution of this paper. We follow [BM00] and [Ang02].

Definition 4.3. A control system (Rn,Rm,F) is said to be incrementally uniformly bounded-input-bounded-

state stable (IUBIBSS) if there exist two class K∞functions γ1and γ2such that for each x1,x2∈ Rnand for

each pair of smooth curves u1,u2: I → Rmthe following estimate holds:

(4.3)

|xu1(t,x1) − xu2(t,x2)| ≤ γ1(|x1− x2|) + γ2(?u1− u2?)

for all t ∈ I.

In general it is difficult to establish IUBIBSS directly. A sufficient condition is given by the existence of an

IUBIBSS Lyapunov function. Note, however, that IUBIBSS only implies the existence of a IUBIBSS Lyapunov

function with very weak regularity conditions [BM00].

Definition 4.4. A C1function V : Rn× Rn→ R+

system (Rn,Rm,F) if there exist a ξ ∈ R+and class K∞functions α,α, and µ such that for every x1,x2∈ Rn

and u1,u2∈ Rmthe following holds:

(1) |x1− x2| ≥ ξ

(2) µ(r) ≥ r + ξ for r ∈ R+

(3) |x1− x2| ≥ µ(|u1− u2|)

A stronger notion than IUBIBSS is incremental input-to-state stability.

0is said to be an IUBIBSS Lyapunov function for control

=⇒

α(|x1− x2|) ≤ V (x1,x2) ≤ α(|x1− x2|);

0;

=⇒

˙V ≤ 0.

Definition 4.5. A control system (Rn,Rm,F) is said to be incrementally input-to-state stable (IISS) if there

exist a class KL function β and a class K∞function γ such that for each x1,x2∈ Rnand for each pair of

smooth curves u1,u2: I → Rmthe following estimate holds:

(4.4)

|xu1(t,x1) − xu2(t,x2)| ≤ β(|x1− x2|,t) + γ(?u1− u2?)

Since β is a decreasing function of t we immediately see that (4.4) implies (4.3) with γ1(r) = β(r,0) and

γ2(r) = γ(r), r ∈ R+

for a converse result when the inputs take values in a compact set.

0. Once again, IISS is implied by the existence of an IISS Lyapunov function. See [Ang02]

Definition 4.6. A C1function V : Rn× Rn→ R+

system (Rn,Rm,F) if there exist class K∞functions α,α,α, and µ such that:

(1) α(|x1− x2|) ≤ V (x1,x2) ≤ α(|x1− x2|);

(2) |x1− x2| ≥ µ(|u1− u2|)

4.2. Fiberwise stability. In addition to incremental stability we will also need a notion of partial practical

stability.

0is said to be an IISS Lyapunov function for the control

=⇒

˙V ≤ −α(|x1− x2|).

Definition 4.7. A vector field (Rn,X) is said to be fiberwise practically stable with respect to πkif there

exist a class K∞function γ and a constant c ∈ R+

?πk(x(·,x))? ≤ γ(|πk(x)|) + c.

Fiberwise practical stability can be checked with the help of the following result.

0such that the following estimate holds:

Lemma 4.8. A vector field (Rn,X) is fiberwise practically stable with respect to πk if there exist two K∞

functions, α and α, a constant d ∈ R+

|πk(x)| ≥ d we have:

(1) α(|πk(x)|) ≤ V (x) ≤ α(|πk(x)|),

(2)˙V ≤ 0.

0, and a function V : Rn→ R such that for every x ∈ Rnsatisfying

Page 5

APPROXIMATE REDUCTION OF DYNAMICAL SYSTEMS5

4.3. Existence of approximate reductions. In this subsection we prove the main result providing sufficient

conditions for the existence of approximate reductions.

Theorem 4.9. Let (Rn,X) be a fiberwise practically stable vector field with respect to πkand let F = Tπm·X :

Rm×Rk→ Rm, viewed as a control system with state space Rm, be IUBIBSS. Then, the vector field (Rm,Y )

defined by:

Y (y) = T(y,0)πm· X(y,0) = F(y,0)

for every y ∈ Rmis approximately πm-related to (Rn,X).

Proof: By assumption, control system (Rm,Rk,F = Tπm◦X) is IUBIBSS. If we denote by y a trajectory of

F we have:

|yv1(t,y1) − yv2(t,y2)| ≤ γ1(|y1− y2|) + γ2(?v1− v2?).

In particular, we can take:

y1= y2= πm(x),

v1= πk◦ x(·,x),

to get:

v2= 0,

|πm◦ x(t,x) − y(t,πm(x))|

=

=

≤

|yπk◦x(t,x)(t,πm(x)) − y0(t,πm(x))|

|yv1(t,πm(x)) − y0(t,πm(x))|

γ2(?v1?) = γ2(?πk◦ x(·,x)?).

But it follows from fiber practical stability of X with respect to πkthat:

?πk◦ x(·,x)? ≤ γ(|πk(x)|) + c.

We thus have:

|πm◦ x(t,x) − y(t,πm(x))|≤

≤

γ2

?γ(|πk(x)|) + c?

γ2

?λ1γ(|πk(x)|)?+ γ2

?λ2c?,

for some constants λ1,λ2 ∈ R+

γ2(λ2c) ∈ R+

Theorem 4.9 shows that sufficient conditions for approximate reduction can be given in terms of ISS-like

Lyapunov functions and how reduced system can be constructed. Before illustrating Theorem 4.9 with several

examples in the next section we present an important corollary.

0. This concludes the proof since γ2(λ1γ(| · |)) is a class K∞ function and

0.

?

Corollary 4.10. Let (Rn,X) and (Rm,Y ) be vector fields satisfying the assumptions of Theorem 4.9. Then,

for any compact set S ⊆ Rnthere exists a δ > 0 such that for any x ∈ S and y ∈ πm(S) the following estimate

holds:

|πm◦ x(t,x) − y(t,y)| ≤ δ

Proof: Using the same proof as for Theorem 4.9, except picking y1= πm(x) and y2= y, it follows that:

|πm◦ x(t,x) − y(t,y)| ≤ γ1(|πm(x) − y|) + γ2

The bound δ is now given by:

?λ1γ(|πk(x)|)?+ γ2

?λ1γ(|πk(x)|)?+ γ2

?λ2c?.

?λ2c?

δ =max

(x,y)∈S×πm(S)γ1(|πm(x) − y|) + γ2

which is well defined since S × πm(S) is compact.

?

Page 6

6PAULO TABUADA, AARON D. AMES, AGUNG JULIUS, AND GEORGE J. PAPPAS

Figure 1. Ball in a rotating hoop.

02

4

6

8

10

0

5

10

15

20

02

4

6

8

10

0

5

10

15

20

02

4

6

8

10

0

5

10

15

20

02

4

6

8

10

0

5

10

15

20

Figure 2. A trajectory of the full order system (red) vs. a trajectory for the reduced system

(blue) for R = 5,10,20,40 (from left to right and top to bottom, respectively).

5. Examples

In this section, we consider examples that illustrate the usefulness of approximate reduction.

Page 7

APPROXIMATE REDUCTION OF DYNAMICAL SYSTEMS7

Example 5.1. As a first example we consider the ball in a rotating hoop with friction, as described in Chapter

2 of [MR99] and displayed in Figure 1. For this example, there are the following parameters:

= mass of the ball,

= Radius of the hoop,

= acceleration due to gravity,

= friction constant for the ball.

The equations of motion are given by:

−µ

˙θ

=

ω

(5.1)

where θ is the angular position of the ball and ω is its angular velocity.

m

R

g

µ

˙ ω

=

mω + ξ2sinθcosθ −g

Rsinθ

If πω: R2→ R is the projection πω(ω,θ) = ω, then according to Proposition 3.3 there exists no vector field Y

on R which is πω-related to X (as defined by (5.1)). However, we will show that Y (ω) = T(ω,0)πω· X(ω,0) is

approximate πω-related to X.

First, we use:

V =1

2mR2ω2+ mgR(1 − cosθ) −1

2mR2ξ2sin2θ

as a Lyapunov function to show that (5.1) is stable. Note that V (ω,θ) = 0 for (ω,θ) = (0,0) and V (ω,θ) > 0

for (ω,θ) ?= (0,0) provided that Rξ2< g, which we assume. Computing the time derivative of V we obtain:

˙V = −µR2ω2≤ 0,

thus showing stability of (5.1). We now consider a compact set C invariant under the dynamics and restrict

our analysis to initial conditions in this set. Such a set can be constructed, for example, by taking {x ∈

R2| V (x) ≤ c} for some positive constant c. Note that stability of (5.1) implies fiberwise stability on C since

πm(C) is compact.

To apply Theorem 4.9 we only need to show that:

T(ω,θ)πω· X(ω,θ) = −µ

is IUBIBSS on C with θ seen as an input. We will conclude IUBIBSS by proving the stronger property of

IISS. Consider the function:

U =1

2(ω1− ω2)2.

Its time derivative is given by:

˙U

=(ω1− ω2)

−g

−µ

−g

≤

=

2m(ω1− ω2)2

+

−

where the second inequality follows from the fact that ξ2sinθcosθ −

the compact set πθ(C) and is thus globally Lipschitz on πθ(C) (since its derivative is continuous and thus

mω + ξ2sinθcosθ −g

Rsinθ

?

−µ

m(ω1− ω2) + ξ2sinθ1cosθ1

Rsinθ1− ξ2sinθ2cosθ2+g

m(ω1− ω2)2+ |ω1− ω2|

Rsinθ1− ξ2sinθ2cosθ2+g

−µ

−µ

?

Rsinθ2

???ξ2sinθ1cosθ1

?

≤

Rsinθ2

???

m(ω1− ω2)2+ |ω1− ω2|L|θ1− θ2|

(5.2)

µ

2m(ω1− ω2)2+ |ω1− ω2|L|θ1− θ2|

?

,

g

Rsinθ is a smooth function defined on

Page 8

8PAULO TABUADA, AARON D. AMES, AGUNG JULIUS, AND GEORGE J. PAPPAS

Figure 3. A graphical representation of the pendulum on a cart mounted to a spring.

bounded on any compact convex set containing πθ(C)) with Lipschitz constant L. We now note that the

condition:

|ω1− ω2| >2mL

µ

|θ1− θ2|

makes the second term in (5.2) negative from which we conclude the following implication:

|ω1− ω2| >2mL

µ

|θ1− θ2|

=⇒

˙U ≤ −µ

2m(ω1− ω2)2

showing that U is an IISS Lyapunov function. We can thus reduce (5.1) to:

˙ ω = −µ

mω.

Projected trajectories of the full-order system as compared with trajectories of the reduced system can be

seen in Figure 2; here µ = m = 1 and ξ = 0.1. Note that as R → ∞, the reduced system converges to the

full-order system (or the full-order system effectively becomes decoupled).

Example 5.2. We now consider a pendulum attached to a cart which is mounted to a spring (see Figure 3).

For this example, there are the following parameters:

M

=

=

=

=

=

=

=

mass of the cart,

mass of the pendulum,

length of the rod,

spring stiffness,

acceleration due to gravity,

friction constant for the cart,

friction constant for the pendulum.

m

R

k

g

d

b

Page 9

APPROXIMATE REDUCTION OF DYNAMICAL SYSTEMS9

-2-1012

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1012

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2-1012

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1012

-1.5

-1

-0.5

0

0.5

1

1.5

2

Figure 4. A projected trajectory of the full-order system (red) and a trajectory for the

reduced system (blue) for d = 0.001,0.01,0.1,1 (from left to right and top to bottom, respec-

tively).

The equations of motion are given by:

˙ x

˙θ

=

=

v

ω

˙ v

=

1

M + msin2θ

?

mRω2sinθ + mg sinθcosθ − kx − dv +b

?

−(m + M)g sinθ + kxcosθ + dv cosθ −

Rcosθ

?

˙ ω

=

1

R(M + msin2θ)

− mRω2sinθcosθ

?

1 +M

m

?b

Rω

?

(5.3)

where x is the position of the cart, v its velocity, θ is the angular position of the pendulum and ω its angular

velocity.

If π(x,v): R4→ R2is the projection π(x,v)(x,θ,v,ω) = (x,v) and X is the vector field as defined in (5.3), the

goal is to reduce X to R2by eliminating the θ and ω variables and thus obtaining Y defined by:

?

˙ x

˙ v

?

=

Y (x,v) = T(x,θ,v,ω)π(x,v)· X(x,0,v,0) =

?

v

−1

M(dv + kx)

?

.

Page 10

10PAULO TABUADA, AARON D. AMES, AGUNG JULIUS, AND GEORGE J. PAPPAS

The objective is now to show that X and Y are approximately π(x,v)-related. In particular, note that the

reduced system, Y , is linear while the full-order system, X, is very nonlinear. This will be discussed in more

detail after proving that they are in fact approximately related.

Stability of X, and in particular fiberwise stability, can be proven as in the previous example by noting that

X is Hamiltonian for d = b = 0 and using the Hamiltonian as a Lyapunov function V . Consider now the

control system:

F ((x,v),(θ,ω))=

Tπ(x,v)· X(x,θ,v,ω)

1

M + msin2θ

+mg sinθcosθ − dv +b

(5.4)

=

?

mRω2sinθ − kx

Rcosθ

?

with θ and ω regarded as inputs. To show that F is IUBIBSS we first rewrite (5.4) in the form:

1

M + m

and consider the following IISS candidate Lyapunov function:

1

2(m + M)(x1− x2)2+1

F ((x,v),(θ,ω)) =

?

mRω2sinθ − kx − dv − mR˙ ω cosθ

?

U =

2(v1− v2)2.

Its time derivative is given by:

˙U

=

−

d

m + M(v1− v2)2

mR

m + M

+

?

ω2

1sinθ1− ˙ ω1cosθ1− ω2

2sinθ2+ ˙ ω2cosθ2

?

(v1− v2).

Using an argument similar to the one used for the previous example, we conclude that:

|v1− v2| ≥2mRL

with L the Lipschitz constant of the function ω2sinθ − ˙ ω cosθ, implies:

˙U ≤ −

d

|(θ1,ω1, ˙ ω1) − (θ2,ω2, ˙ ω2)|,

d

2(m + M)(v1− v2)2,

thus showing that X is IISS and in particular also IUBIBSS. That is, we have established that X and Y are

approximately π(x,v)-related.

In order to illustrate some of the interesting implications of approximate reduction, we will compare the

reduced system, Y , and the full-order system, X, in the case when R = m = k = b = 1 and M = 2. It follows

that the equations of motion for the reduced system are given by the linear system:

?

so we can completely characterize the dynamics of the reduced system: every solution spirals into the origin.

This is in stark contrast to the dynamics of X (see (5.3)) which are very complex. The fact that X and Y are

approximately related, and more specifically Theorem 4.9, allows us to understand the dynamics of X through

the simple dynamics of Y . To be more specific, because the distance between the projected trajectories of X

and the trajectories of Y is bounded, we know that the projected trajectories of X will “essentially” be spirals.

Moreover, the friction constant d will directly affect the rate of convergence of these spirals. Examples of this

can be seen in Figure 4 where d is varied to affect the convergence of the reduced system, and hence the full

order system.

˙ x

˙ v

?

=

?

01

d

−1

2

??

x

v

?

,

Page 11

APPROXIMATE REDUCTION OF DYNAMICAL SYSTEMS 11

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[Ang02]

[ASG00]

[BDG96]

Department of Electrical Engineering, University of California at Los Angeles,, Los Angeles, CA 90095

E-mail address: tabuada@ee.ucla.edu

Control and Dynamical Systems Department, California Institute of Technology, Pasadena, CA 91125

E-mail address: ames@cds.caltech.edu

Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104

E-mail address: agung@seas.upenn.edu

Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104

E-mail address: pappasg@ee.upenn.edu

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