nonlinear control systems: mathematical preliminarieslemmon/courses/ee580/slides/slides2.pdf ·...

Nonlinear Control Systems:Mathematical Preliminaries

Dept. of Electrical Engineering

Department of Electrical EngineeringUniversity of Notre Dame, USA

EE60550-01

Dept. of Electrical Engineering (ND) Nonlinear Control Systems: Mathematical Preliminaries EE60550-01 1 / 45

Overview

Routine use of concepts from real analysis, topology, and modernalgebra

Need to establish notational conventions, basic concepts, and theproperties of these concepts.

Lecture notes are details (with proofs). But the lecture will just coverhighlights

Lecture notes based on Rudins text (Principles of MathematicalAnalysis)

Topics to be covered: algebras (group, ring, field, linear space),normed linear spaces (induced gain for linear operators), topologicalconcepts (open/closed sets, limit points), sequences and Cauchysequences, compactness, continuity (uniform), maximum principle.


Sets and Binary Relations

Two types of mathematical objects; sets and maps

One type of map is a binary relation R : X X {0, 1}. We say xstands in relation R to y if R(x , y) = 1 (other ways to define).

Basic properties of binary relations

reflexive xRxsymmetric xRy implies yRxtransitive xRy and yRz implies xRz

A relation having all three properties is called an equivalence relation.

An equivalence relation partitions X into mutually disjoint sets calledequivalence classes [x ].


Totally Ordered Sets

A transitive binary relation R is a total order if for all x , y X

xRy , yRx or x = y

If X is totally ordered an E X then X such that x for allx E is an upper bound for EIf is an upper bound and there is no other upper bound in X suchthat < , then is the least upper bound or supremum (sup(E )).

Dual definition for greatest lower bound or inf(E ).


Algebras

We obtain a mathematical system when we associate sets and binaryoperators

The binary operator places an algebraic structure on X

Special algebras

Group (X ,+)Ring (X ,+,)Field Ring with multiplicative inverse (division)

The most important algebra for us will be a linear space (X ,F ,+, )where X is a set of vectors, F is a set of scalars, + is vector additionand is vector-scalar dilation.


Functions as Vectors

We generalize the notion of vectors to functions x : R R.Think of vector as sequence x1, x2, . . . , xn that samples a continuousfunction

x=

x1x2

xn

and the function x (t )

t

x(t)orxk

The function x : R R is then an infinite dimensional vector and weform linear spaces of functions (aka function spaces)

Why do we do this? A dynamical system may be seen as a map froman input function space to an output function space. So it is a mapbetween two linear spaces.


Normed and Topological Structure

We also need to place a topological structure on our sets. Namelyhow close two elements are to each other.

There are three basic ways this is done

Topology is a collection of subsets that cover X . Two elementsx , y X are close if they belong to the same covering setA metric d : X X R is the distance between two elements of XA norm : X R is the length or strength of element in X

We will focus on normed linear spaces. The notion of norm abstractsour intuitive concept of strength

x 0 for all x X and x = 0 if x = 0 = || xx + y x+ y

These are satisfied by Euclidean norm, we also need norms forfunction spaces


Lp Norms and Spaces

The function space norm is

xLp =( |x(s)|pds

)1/pThe normed linear space formed by all functions with finite Lp normsis called LpThe normed linear space formed by all functions whose finitetruncations to interval [0,T ] have a finite Lp norm is called theextended Lp space (Lpe)Two special cases; When p = 2 we have space of finite energy signals.when we let p = we have space of finite amplitude signals.


Induced Gain

Consider a linear system G : X Y where X and Y are two normedlinear function spaces (like L2e).The induced gain of the system is

G = supx 6=0

G [x ]x

= supx=1

G [x ]

G is a norm for L(X ,Y ) (space of all linear transformations).Extend notion of induced gain to nonlinear systems - incremental gain

(G ) = inf{ : G [x ] G [y ] x y}

||x||

||G[x]||

greatest lower boundon all slopes

y

x

L


Topological Concepts

Topological concepts are frequently used tools in the study of nonlineardynamical systems. We now review some of these concepts.

Let X be a normed linear space with norm over a known field F .The -neighborhood of p X is the set

N(p) = {x X : x p < }

A point p X is a limit point of set S X if every -neighborhoodof p contains a point q S that is not equal to x .A set S X is closed if it contains all of its limit points.A set S X is open if for any p S , there exists an -neighborhoodsuch that N(p) S .


Open Sets

Theorem 1

Every neighborhood N(x) for x in a normed linear space X is an open set.

We can think of neighborhoods as canonical examples of open sets.Limit points of a set E X , on the other hand, represent points whereelements E tend to cluster together. Essentially, this clustering means thatin any arbitrarily small neighborhood about a limit point, one should beable to find infinitely many points of the set E .

Theorem 2

Let X be a normed linear space and let x be a limit point of a set E X .Then every neighborhood of x contains infinitely many points of E .


Proof of Theorem 2

If x is a limit point of E X , then every neighborhood of x containsinfinitely many points of E

Proof: This is proven by contradiction. Let N(x) have finite points of E ,that we denote as y1, . . . yn. Since this set is finite, there exists r > 0 suchthat

r = min1mn

x ym > 0

The neighborhood Nr (x) would therefore contain no point of E that isdistinct from x and so x is not a limit point of E , thereby contradictingthe hypothesis.


Closed Sets

A closed set consists of limit points of the set E , namely thosepoints around which points of E clustering.

It is not necessary, of course, that these clustering points actually bein E .

A good example of such a situation occurs in the set of rationalnumbers (i.e. all real numbers that can be represented by a ratio ofintegers). If we look at any irrational number such as e or , one canalways find a ratio arbitrarily close to the irrational number, but thenumber itself cannot be represented as a ratio.

The relationship between open and closed sets can be described acomplementary as seen in the following theorem.

Theorem 3

Let X be a normed linear space and let E X . This subset, E , is open ifand only if its complement, E c , is closed.


Proof of Theorem 3

E X is open if and only if its complement, E c , is closed

Proof: Suppose E c is closed and select point x E . So x cannot be alimit point of E c since it is not in E c . This means there is a neighborhoodN(x) such that E

c N(x) = . This means N(x) E and so x is aninterior point of E . Since the choice of E was arbitrary, this means everypoint of E is an interior point and so E is open.A similar argument when E is open establishes the other direction.


Sequences and Convergence

A sequence is a function x : Z X from the set of integers, Z, ontoa normed linear space X .

We denote a sequence {xi}iI where I Z is a set of indices.A sequence {xi}i=1 is convergent in normed linear space X if thereexists a point x X such that for all > 0 there is an integer N > 0such that for any n N, we can show xn x < .If {xi} is convergent then the point x is called the limit point of thesequence and we write xn x or limn xn = x .A sequence that has no limit point is said to be divergent.


Divergent Sequence of Functions

Consider a sequence of functions {fn(t)}n=0 where fn(t) = tn for0 t 1.In the L signal space (i.e. all bounded functions), this sequence isconvergent to the function

f (t) =

{0 0 t < 11 t = 1

which is also a bounded function.

If we consider this sequence to be in the linear space of continuousfunctions C [0, 1] (note that each fn is a continuous function on [0, 1])then this sequence is divergent because the limiting function isdiscontinuous and hence not in C [0, 1].


Cauchy Sequences

Convergent sequences are useful tools in the analysis of nonlinearsystems. But the definition as stated above is difficult to verifybecause one needs to know what the limit point is, x .

We therefore introduce new notion of a sequence known as a Cauchysequence. A sequence {xi} in normed linear space X is a Cauchysequence if for every > 0 there is an integer N such thatxn xm < for all n,m N.

Theorem 4

In any normed linear space, every convergent sequence is a Cauchysequence


Proof of Theorem 10

Every convergent sequence is Cauchy

Proof: By assumption xi x so for any > 0 there exists N such thatxn x < for all n N. So use the triangle inequality to conclude

xn xm xn x+ x xm < 2

for n,m N, and so {xn} is Cauchy.


Complete Normed Linear Spaces

In general not all Cauchy sequences are convergent (see aboveexample).

Normed linear spaces in which every Cauchy sequence is convergentare said to be complete. Complete normed linear spaces aresometimes called Banach spaces.

All Euclidean spaces, Rn, are completeThe functions spaces, Lp are complete.A critical topological concept used to establish whether a space iscomplete is compactness


Compactness - introduction

Compactness is an important concept especially with regard to theexistence of extreme points for continuous functions. Its importanceis based on the observation that the behavior of finite sets and infinitesets is very different.Each of the following statements, for example, is true when a set X isfinite,

All functions are bounded: If f : X R is a real valued function on Xthen there exists M > 0 such that f (x) M for all x X .All functions attain a maximum: If f : X R is a real valued functionon X , there must exist at least one point x X such thatf (x) f (x) for all x X .All sequences have constant subsequences: If {xi}iI is a sequence inX then there must exist a subsequence {xij}ijI that is constant forsome c X .All covers have finite subcovers: If {Vi}iI are subsets of X that coverX (i.e. X iIVi ) , then there exist a finite set J I of sub-indicessuch that the finite sub-collection {Vij}ijJ still covers X (i.e.X ijJVij ).



The first statement - all functions on a finite set are bounded - is anexample of a local-to-global principle. Namely that the assertion oflocal boundedness by a constant M that is a function of x impliesglobal bounded for all x X by a constant that is independent of x .The second statement - all functions have a maximum - is an exampleof the maximum principle. It clearly holds since we can againenumerate the elements of X to find the maximum.

The third statement - every sequence has a constant subsequence - issometimes called the infinite pigeonhole principle. Since the sequenceonly cycles through a finite number of elements, it obviously allows usto identify a constant subsequence.

The last statement - every cover has a finite sub-cover - is what weoften define as countable compactness and again may not be true forarbitrary infinite sets.



When we endow our set X with some topological structure, like anorm or metric, then some objects exhibit properties similar to thoseabove that are enjoyed by finite sets. For example consider X = [0, 1],the closed unit interval.

In general, the above four properties are all false for this X . But if wemodify each assertion by inserting a topological concept such ascontinuity, convergence, or open-ness, then we can establish theseassertions for [0, 1].

All continuous functions are bounded on X = [0, 1]:All continuous functions achieve a maximum on X = [0, 1]:All sequences in [0, 1] have convergent subsequences:All open covers have finite subcovers:

In contrast all four of these statements continue to be false for setslike the open unit interval, (0, 1) or the real line, R.



Compactness is used in many ways. One may use the local-to-globalprinciple to establish some local constraints on a function that canthen be used to infer a global constraint.

Another use is to establish conditions to locate function extrema; thisis done for instance in optimal control.

A third is to partially recover the notion of a limit when dealing withnon-convergent sequences, this is what is done in characterizing limitcycles. In our studies we will need compactness in a variety of places.

So compactness is a very powerful property of linear spaces. It will beuseful to state these properties in a more precise manner.


Compact Sets

Let {Vi}iI be a collection of open subsets of X where I is a set ofindices . This collection is called an open cover for E X ifE

iI Gi .

A subset K of a normed linear space X is said to be countablycompact in X or just compact in X if every open cover of K containsa finite subcover.

More explicitly this means that if were given an open cover {Gi}iIof K where I is a set of indices (possibly infinite), then there arefinitely many indices i1, . . . , in such that

K Gi1 Gi2 Gin


Compact Sets versus Closed Sets

Every compact subset of a normed linear space is closed.

But the converse relation only holds if the closed set is taken from aset that is already known to be compact.

Theorem 5

Compact subsets of normed linear spaces are closed.

Theorem 6

Closed subsets of compact sets are compact.


Proof of Theorem 5

Compact subsets of normed linear spaces areclosed

XK

x

yVy

Wy

Proof: Let K X be compact and suppose x X and x / K . If y K ,let Vy and Wy be neighborhoods of x and y , respectively with radius lessthan 12x y (so they dont intersect). Since K is compact there arefinitely many points y1, . . . , yn in K such that

K Wy1 Wyn = W

Let V = Vy1 Vyn so that V is a neighborhood of x that doesntintersect W . So V K c which means x is an interior point of K c ,thereby establishing that K c is open and so K is closed.


Compactness and Limit Points

As mentioned above, an important aspect of compact sets is their supportof the local-to-global principle and the maximum principle. The foundationfor this support is established in the following theorem that asserts thatany infinite set of a compact set K has its limit point in K . If that limitpoint is the sup of a function, then this theorem lays the foundation forestablishing the existence of global bounds on a functions value.

Theorem 7

If E is an infinite subset of a compact set K , then E has a limit point in K .


Proof of Theorem 7

If E is infinite subset of a compact set K , then E has a limit point in K

Proof: If not point of y K were a limit point of E , then each y has aneighborhood y which contains at most one point of E (namely y). Thiswould mean that no finite subcollection of {Vy} could cover E and thesame would hold for K since E K . This contradicts the assumption thatK is compact.


Heine-Borel Theorem

When the compact space, X , is actually the Euclidean space, Rk , itbecomes possible to sharpen our notion of compactness and its relation tocompact and bounded sets. To establish this relationship we first need todefine what we mean by a bounded set. In particular, A set S X isbounded if there exists L > 0 such that x y < L for all x , y S . Thefollowing theorem states that a subset of Rk is compact if and only if it isclosed and bounded.

Theorem 8

(Heine-Borel Theorem) The following three assertions are equivalent setfor subset E Rk .

1 E is closed and bounded.

2 E is compact

3 Every infinite subset of E has a limit point in E .


Sequences and Compactness

One of the important features of compact sets is that Cauchy sequencesare convergent in such sets. We prove this by first looking a sequences ofcompact sets whose radius go to zero in the limit.

Theorem 9

Let {Kii=0 be a sequence of compact sets in normed linear space X suchthat Ki+1 Ki for all i . If

limi

diam(Ki ) = 0

Then

i=1 Ki = limi Ki consists of exacty one point in X


Proof of Theorem 9

A decreasing sequence of compact sets {Ki} such thatlimi diam(Ki ) = 0 has exactly one limit point in XProof: Proof: Let K =

1 Ki . Assume it is empty but note that the

intersection of any finite subcollection of Ki must be nonempty. Fix amember K1 of {Ki} and let Gi = K ci . Assume no point of K1 belongs toevery Ki . This means the sets {Gi} form an open cover of K1. Since K1 iscompact there are finitely many indices i1, . . . , in such thatK1 Gi1 Gin . This means, however that K1 Ki1 Kin wouldbe empty which would contradict the fact that any finite subcollection ofKi s must be nonempty and so K must also be nonempty.So since K is not empty, if K contains more than one point, thendiamK > 0. But for each n, K Kn, so that diam(Kn) diam(K ),which contradicts the assumption that diam(Kn) 0.


Sequences and Compactness

Theorem 10

If X is a compact normed linear space and if {xn} is a Cauchy sequence inX , then {xn} converges to some point of X .

Proof: Let {xn} be a Cauchy sequence in compact space X . ForN = 1, 2, . . . ,, let EN be the set consisting of points xN , xN+1, . . .. Then

limN

diamEN = 0

Being a closed subset of the compact space X , each EN is compact bytheorem 6. Also EN+1 EN so that EN+1 EN . Theorem 9 shows thereis a unique x X which lies in every EN .Let > 0 be given. Since limN diamEN = 0, there is an integer N0 suchthat diam(EN) < if N N0. Since x EN , it follows that x y < for every y EN and so for all y EN . In other words, x xn < forall n N0 which is precisely xn x .Dept. of Electrical Engineering (ND) Nonlinear Control Systems: Mathematical Preliminaries EE60550-01 32 / 45

Bolzano-Weierstrass Theorem

Given a sequence {pn}, consider a sequence {nk} of positive integerssuch that n1 < n2 < n3 < . Then the sequence {pni} is called asubsequence of {pn}. If {pni} converges, its limit is called asubsequential limit of {pn}.The following theorem asserts that any sequence in a compact spacehas a convergent subsequence and the restriction of this fact to R isusually called the Bolzano-Weierstrass theorem.

Theorem 11

If {pn} is a sequence in a compact normed linear space, X , then somesubsequence of {pn} converges to a point of X . (Bolzano-WeierstrassTheorem): When X = Rk , every bounded sequence contains aconvergent subsequence.


Weierstrass Theorem

The fact that every compact set in Rk is also closed and bounded and thatCauchy sequences are convergent in compact spaces makes it easy toestablish that Rk is a complete normed linear space. The first result belowis known as the Weierstrass theorem and follows immediately fromtheorem 7. The second theorem 13 proves every Cauchy sequence in Rk isconvergent by appealing to the Heine-Borel theorem and then invokingtheorem 10.

Theorem 12

(Weierstrass Theorem) Every bounded infinite subset of Rk has a limitpoint in Rk .

Theorem 13

(Rk is Complete) In Rk , every Cauchy sequence converges.


Compactness and Continuous Functions

Continuous functions have maxima when they are defined overcompact spaces.

This is useful because it can be used to determine whether or not anoptimization problem is well posed in the sense of a solution existing.

Suppose E X and that there is a function f : E Y . Let p be alimit point of E , then we write f (x) q as x p or

limxp

f (x) = q

if there is a point q Y such that for all > 0 there exists > 0such that

f (x) qy <

for all points x E for which

0 < x px <

This is our usual notion of the limit of a function f .Dept. of Electrical Engineering (ND) Nonlinear Control Systems: Mathematical Preliminaries EE60550-01 35 / 45

Compactness and Limits

A necessary and sufficient condition for such a limit to exist is that for anysequence pn p that limn f (pn) = q. This is stated and proven in thefollowing theorem.

Theorem 14

Let X and Y be two normed linear spaces. Suppose E X and that fmaps E into Y (i.e., f : E Y ) and that p is a limit point of E . Thenlimxp f (x) = q if and only if limn f (pn) = q for every sequence {pn}in E such that pn 6= p and limn pn = p. Moreover this limit is unique.


Continuous Functions

Since the sup of a function is a limit point, we then also use this toestablish the existence of solutions to optimization problems; or whatis commonly called the maximum principle.

The principle holds if the function f is continuous. The formal definition of a continuous function is given below.

Suppose X and Y are two normed linear spaces, E X , p E and fmaps E into Y . Then f is said to be continuous at p if for every > 0 there exits a > 0 such that f (x) f (p)y < for all pointsx E for which x px < .


Topological Characterization of Continuity

In proving the maximum principle, however, we will find it convenient toestablish an alternative topological characterization of continuity interms of open sets. This characterization will be useful because it allowsus to connect the functions continuity directly to our definition for acompact set.

Theorem 15

(Alternative Definition of Continuity) A mapping f of normed linearspace X into normed linear space Y is continuous on X if and only iff 1(V ) is open in X for every open set V in Y .


Proof of Theorem 15

f : X Y is continuous if and only for every open V Y we can showf 1(V ) is open in X

Proof; Suppose f is continuous on X and V is an open set in Y . Supposep X and f (p) V . Since V is open there is 0 such that y V iff (p) y < . Since f is continuous at p there exists > 0 such thatf (x) f (p) < when x p < . Therefore x f 1(V ) as soon asx p < and so x is an interior point of f 1(V ) which means f 1(V )is open.Similar argument can be used to establish converse.


Compactness and Uniform Continuity

In our - definition for continuity of a function f at x , we requirethat given x for any , there exists a for which f (x) f (y) < for all all y such that x y < . The point here is that the weare looking for may in fact be a function of the we choose.

This can be problematic in some cases for instance, it may be possibleto consider a sequence of points {xi} such that f is continuous ateach xi , but in the limit we lose that continuity because increaseswithout bound as i .This occurs for a function such as f (x) = 1/x , which is continuouseverywhere by at 0.

The ability to manage what happens to sequences as they march offto infinity is the real reason why we introduced the topologicalconcept of compactness, so it makes sense that we would want tolimit this rate of growth in and that enforcing such limits should betied back to our notion of compactness.


Compactness and Uniform Continuity

In particular, a function f : X Y is uniformly continuous if for all > 0 there exists > 0 such that f (x) f (y) < holds wheneverx y < for all x , y X .The difference of course in this case is we for regular continuity weonly tried to enforce the bound in a neighborhood of a given x . Inthis case, the bound holds for any pair of points and so isindependent of the choice of x .

The connection of compactness to uniform continuity is made clear inthe following theorem.

Theorem 16

Let E be a bounded noncompact set in R1, then there exists a continuousfunction on E that is not uniformly continuous.


Compactness and Uniform Convergence

This uniform notion of continuity can be applied to sequences of functions{fn}. In particular, we say that a sequence of functions {fn} convergesuniformly on E to a function f if for every > 0 there is an integer N suchthat n N implies fn(x) f (x) for all x E . One importantconsequence of uniformity is that it allows us to interchange the order oflimiting operations in the function sequence,

Theorem 17

Suppose fn f uniformly on a set E in a normed linear space. Let x be alimit point of E and suppose

limyx

fn(y) = An

then {An} converges and

limyx

limn

fn(y) = limn

limyx

f (n(y)


Uniform Convergent Sequences

An immediate corrollary of the above theorem is that if {fn} is a sequenceof continuous functions on E that is uniformly convergent to f then f isalso continuous on E . Of course, we know that we can define integrationand differentiation of the function f as limits as well. So again, if we canalso conclude that if fn f uniformly then

limn

( ba

fn(s)ds

)=

ba

(limn

fn(s))ds =

ba

f (s)ds


Maximum Principle

As mentioned above, the preceding topological definition of acontinuous function connects continuity to compactness. This allows us toestablish that the range set, f (X ) Rk is compact, and so f (X ) is closedand so contains its limit points. The sup and inf of f are limits points off (X ) and so the function achieves these values.

Theorem 18

Suppose f is a continuous mapping of a compact normed space, X into anormed space Y . Then f (X ) is compact.

Theorem 19

(Maximum Principle:) Suppose f is a continuous real function on acompact normed linear space X and M = suppX f (p) andm = infpX f (p). Then there exist points p, q X such that f (p) = Mand f (q) = m.


Summary

These lectures reviewed concepts from topology, modern algebra, andreal analysis that are used in studying nonlinear control systems.

The main things to remember

Signal space and Systems spaces may all be viewed as linear spaceswith a norm. (normed linear space)The induced gain of a system is induced by the assumed norms on thesignal space. In many cases we can actually find formulas to evaluatethese norms.Compactness is an essential technical property of real sets that allow usto make assertions about infinite sets (convergence, maximum, etc.) Inour case the space Lp that we usually work in are always compact.

We will take these tools and immediately use them to study solutionsto ordinary differential equations.


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