introduction to geometric nonlinear control; linearization, observability, decoupling

Introduction to Geometric Nonlinear Control; Linearization, Observability, Decoupling

W ito ld Respondek*

Laboratoire de Mathmatiques, IN SA de Rouen, France

Lectures given at the Summ er School on Mathematical Control Theory

Trieste, 3-28 September 2001

LNS028004

* wrespQmsa-rouen.fr

A bstract

These notes are devoted to the problems of linearization, observability, and decoupling of nonlinear control systems. Together with notes of Bronislaw Jakubczyk in the same volume, they form an introduction to geometric methods in nonlinear control theory. In the first part we discuss equivalence of control systems. We consider various aspects of the problem: state-space and feedback equivalence, local and global equivalence, equivalence to linear and partially linear systems. In the second part we present the notion of observability and give a geometric rank condition for local observability and an algebraic characterization of local observability. We discuss uniform observability, decompositions of nonobservable systems, and properties of generic observable systems. In the third part we introduce the notion of invariant distributions and discuss disturbance decoupling and input-output decoupling. Many concepts and results are illustrated with examples.

Contents

1 I n tr o d u c t io n 173

2 F e e d b a c k l in e a r iz a tio n : a n in t r o d u c t io n 173

3 E q u iv a le n c e o f c o n tro l s y s te m s 1813.1 State space equivalence ........................................................................1813.2 Feedback equ ivalence............................................................................... 184

4 F e e d b a c k l in e a r iz a t io n 1864.1 Static feedback lin e a riz a tio n .................................................................1864.2 Restricted feedback l in e a r iz a t io n ....................................................... 1924.3 P artia l linearization ............................................................................... 194

5 O b s e rv a b il ity 1965.1 Nonlinear observab ility ............................................................................1965.2 Local decom positions.............................................................................. 2035.3 Uniform observability ............................................................................2055.4 Local observability: a necessary and sufficient condition . . . 2065.5 Generic observability p ro p e r tie s ...........................................................207

6 D e c o u p lin g 2106.1 Invariant d is t r ib u t io n s ............................................................................2116.2 D isturbance decoupling ........................................................................ 2126.3 Inpu t-ou tpu t d e c o u p lin g ........................................................................ 214

R eferences 219

Linearization, Observability, Decoupling 173

1 Introduction

These notes, together w ith notes of Jakubczyk [26] of the same volume, form an in troduction to geometric nonlinear control. Section 2 has an elem entary and in troductory character. We form ulate the problem of feedback linearization, show why the concept of Lie bracket appears naturally, and give necessary and sufficient conditions for feedback linearization in the single-input case. In Section 3 we introduce two concepts of equivalence of control systems: sta te space equivalence and feedback equivalence. We also sta te a result th a t any nonlinear control system is (locally) determ ined by iterated Lie brackets of vector fields corresponding to constant controls. In Section 4 we discuss various aspects of the feedback linearization problem. In particular, we consider the m ulti-input as well as non control-affine system s and the problems of global feedback linearization, restricted feedback linearization, and partial linearization. Section 5 is concerned w ith the concept of observability. We introduce observability rank condition and then discuss Kalman-like decomposition of nonlinear non observable systems, uniform observability, and generic properties of observable systems. Finally, in Section 6 we introduce te concept of invariant and controlled invariant distributions and, based on it, discuss solutions to the disturbance decoupling and inpu t-ou tpu t decoupling problems.

We do not provide proofs of the presented results and send the reader to the literature on geometric control theory (see the list of references) and, in particular, to m onographs [18], [23], [29], [37]. As a small recompense , we illustrate many notions, concepts, and results by simple, m ainly mechanical, examples.

2 Feedback linearization: an introduction

The aim of th is prelim inary section is to introduce the concept of feedback linearization and a fundam ental geometric tool of nonlinear control theory, which is the Lie bracket. Feedback linearization is a procedure of transform ing a nonlinear system into the simplest possible form, th a t is, into a linear system. Necessary and sufficient conditions for this to be possible will be expressed using the notion of Lie bracket, which is om nipresent in very many nonlinear control problems.

The problem of feedback linearization is to transform the nonlinear con

174 W. Respondek

tro l systemX = f ( x , u )

into a linear system of the form

x = A x + B u

via a diffeomorhism(x ,u ) = ($(a:), ^ ( x , )),

called feedback transform ation. We will s ta rt w ith an in troductory example.

E x a m p le 2.1 Consider a nonlinear pendulum (rigid one-link m anipulator) consisting of a mass m w ith control torque u.

The evolution of the pendulum is described by the Euler-Lagrange equation w ith external force

m l 2 9 + m gl sin 9 = u .

We rew rite it as9 = ujdj = ^ f s i n 9 + ^ p .

Denote x \ = 9 and X2 = to and consider the evolution of the pendulum on the sta te space R2, th a t is x = (XI,X 2 )T E R2. We get the system E

v , Xl = x 2 . Q i 'ifx 2 = +

Replace the control u by

u = m l2u + m lg sinari,

which can be in terpreted as a transform ation in the control space U depending on the sta te x C X . We get the linear control system

XI = X22 = u.

Using a simple transform ation in the control space we thus brought the system into the simplest possible form: a linear one. Notice th a t the families of all trajectories of bo th systems coincide although they are param etrized (with respect to the control param eters u and u, respectively) in two different ways.


Now fix an angle 9q. The goal is to stabilize the system around xq =(xiq ,X 2o)t } where io = Go and X20 = 0. Introduce new coordinates

x i = x i - a:ioX2 = x 2.

and apply the controlu = k ix i + k 2x 2,

where hi, *2 are real param eters to be chosen. We get a closed loop system described by the system of linear differential equations

x i = x 2x 2 = klX l + k 2 x 2,

whose characteristic polynom ial is given by

p( A) = A2 A&2 ki.

Let Ai, A2 G C be any pair of conjugated complex numbers. Take

k\ = ^AiA2 k 2 = Ai + A2 ,

then the eigenvalues of the closed loop system are Ai and A2. In particular, by choosing Ai and A2 in the left half plane we stabilize exponentially the pendulum around an arb itrary angle 0 q and a stabilizing control can be taken as

u = k im l2(xi xio) + k 2m l 2 X2 + m gl sinari.

Now fix for the system E an initial point xq = (x iq 7X2o)t K2 and a term inal point x t = ( it, 2t ) T G R2 and consider the problem of finding a control u(t), 0 < i < T , which generates a tra jectory x(t) , 0 < t < T , such th a t a:(0) = xq and x (T ) = x j \ This is the controllability problem, called also motion planning problem. Due to the above described linearization, we get the following simple solution of the problem. Choose an arb itrary C 2-function ip(t), 0 < t < T , such th a t

176 W. Respondek

and apply to the system the control

m =


which can be viewed at as a sta te depending transform ation in the control space U, we bring any single-input controllable linear system into the n-fold integrator

Xi = X2 , X2 = 3, , Xni = Xn , Xn = U.

We will consider the problem of w hether and when such a transform ation is possible in the nonlinear case. Consider a single-input control affine system of the form

E : x = f ( x ) + g (x )u ,

where x G X , an open subset of W 1, and / and g are C-sm ooth vector fields on X .

Recall th a t L vip denotes the derivative of a function ip w ith respect to a vector field v, th a t is

n L vip{x) = j^ -{ x )v i{ x ) .

i=l OXi

Fix a point xq E X and assume th a t there exist a C-sm ooth function ip on X such th a t (compare the linear case)

Lg(p = LgLfCp = = LgL n^ 2ip = 0

andL gL nf l ip{x) = d(x),

where d(x) is a sm ooth function such th a t d(xo) ^ 0. If around the point xq, the functions cp, Lf(p, . . . are independent (in the sense th a t theirdifferentials are linearly independent around xq), then in a neighborhood V of xq the map

x i = = Lfip + uLgip = 2

x n- i = = L 7j ^ l cp + u L gL n^ 2ip = x nx n = < d L n^ l i p ,x > = L'jip + u L gL n^ l ip = L rjip + ud(x).

178 W. Respondek

By introducing a new control variable

U = UfLp + uLgl/J ip,

which can be viewed at as a transform ation in the control space U, depending nonlinearly on the sta te x, we bring our single-input nonlinear system into the n-fold integrator

X i = X 2 , X 2 = 3 , , X n i = X n , X n = U.

The proposed m ethod works under two assum ptions. Firstly, we assumed the existence of a function ip such th a t L gip = LgLfip = = L gL n^~2ip = 0. Secondly, we assumed th a t the functions ip, Lfip, . . . , L n^ l ip are independent in a neighborhood of xq. The former is a system of it 1 first order partial differential equations. In order to see it, let us consider the two first equations L gip = 0 and L gLfip = 0, which imply th a t

LfLgip LgLfip = 0.

A lthough the expression on the left hand side involves a priori partia l derivatives of order two, it depends on partial derivatives of ip of order one only and a direct calculation shows th a t we can represent it as

LfLgip - LgL f ip = L y ig] R" (resp. / : X > R "). We will call [ f , g ] the Lie bracket of the vector fields / and g. We would like to emphasize two im portan t aspects of the nature of Lie bracket. Firstly, it is a vector field, because if we change coordinates then the Lie bracket is m ultiplied on the left by the Jacobi m atrix of the derivative of the coordinate change. This shows its vector, i.e., contravariant, nature. Secondly, a Lie bracket [/, g] acts on a function ip by the formula L y ^ ip , th a t is, acts as a first order differential operator. Notice th a t, as we have already said, the expression LfLgip LgLfip involves, a priori, second order derivatives of ip bu t all of them are mixed partials th a t m utually cancel due to Schwarz lemma.


Introduce the notationad f g = [/, g]

and, inductively,ad3f +lg = [/, ad3f g\,

for any integer j > 1. P u t adg = g. It can be shown by an induction argum ent th a t the existence of a function ip such th a t L gip = L gLfip = = L gL n^ 2ip = 0 is equivalent to the solvability of the following system of first order partial differential equations

L gip = 0 Ladjgip = 0

L &dnf 2glP ~ >

(2.1)

which in coordinates is expressed asn r, dip

( a d } g ) j= 0 , for 0 < j < n 2 ,

Tvn)where (ad^c?)* denotes the i-th component, in the coordinates (i, . . . ,x.

of the vector field adjg.

It can be shown th a t the requirem ent th a t the differentials dL^ip, for 0 < j < n 1 , where ip is a nontrivial solution of the system (2 .1), are linearly independent a t xq is equivalent to the linear independence of ad^g a t ./(). for 0 < j < n l.

We will show th a t a necessary condition for the above system of first order P D E s to adm it a nontrivial solution is th a t for any 0 < i, j < n 2 the Lie bracket [ad^g, ad^g] (x) belongs to the linear space generated by (ad jg (x ) , 0 < q < n 2}. In view of the linear independence of the ad^gs,

this is equivalent to the existence of sm ooth functions dq such th a t

n - 2

[aA)g,ad3f g] = ]P a ^ a d |g > .q=0

To prove it, assume th a t there exists a vector field v of the form v = [ad^g, adj-g], for some 0 < i , j < n 2 , and a point x C X . such th a t v(x) ^ span {ad^g(x), 0 < q < n 2}. We have

L v ip ^ [a d ^ .a d ^ j ]^ L - ^ g L ^ g i p L ^ g L ^ g i p 0.

180 W. Respondek

The n vector fields v and ad^g, for ( ) < < / < it 2. are linearly independent in a neighborhood of x E X and therefore the only solutions of the system of n first order P D E s

f L vip = 0{ L *d?f g for 0 < i < n - 2 ,

are ip =constan t.It tu rns out th a t the two above necessary conditions are also sufficient

for the solvability of the problem. Indeed, we have the following result.

T h e o re m 2 .2 There exist a local change of coordinates x = (f>{x) and afeedback of the form u = a (x) + (3(x)u, where (3(x) ^ 0, transforming,locally around xq X , the nonlinear system

E : x = f (x) + g(x)u

into a linear controllable system of the form

A : x = A x + bu

i f and only i f the system E satisfies in a neighborhood o f xq :(C l) g(x), a,dfg(x) , . . . , ad- 1g(:r) are linearly independent;(C2) for any 0 < i , j < n 2, there exist smooth functions dq such that

n - 2

[ad%f g,a,d3f g] = ^ a * Jad jg.9=0

The condition (C2), called involutivity, is discussed in the general context in the section devoted to Frobenius theorem of [26] in this volume and in the context of feedback linearization in Section 4. It has a clear geometric interpretation. If the above defined system of P D E s L gip = = L &dn-2gip = 0 adm its a nontrivial solution then for any constant c 6 l the equation ip = c defines a hypersurface in X . The vectors g(x), a,d/g(x) , . . . , ad n^ 2 g{x) form at any x G {ip(x) = c} the tangent space to th a t hypersurface. In general, such a hypersurface need not exist; the involutivity condition (C2) guarantees its existence.

Especially simple is the p lanar case, th a t is, n = 2, in which the involutiv ity follows autom atically from the linear independence condition.


C o ro lla ry 2 .3 A control-affine planar system

x = f ( x ) + g (x )u ,

where il2, is locally feedback linearizable at x q i f and only i f g and ad g are independent at x q .

E x a m p le 2 .4 (Example 2.1 cont.) We have / = 2g^ and g =Thus the vector fields g and adg = are independent and hence,by Corollary 2.3, we can conclude feedback linearization of the pendulum , a property which we have established by a direct calculation in Exam ple 2.1.

3 Equivalence of control system s

The question of feedback linearization discussed in Section 2 is a subproblem of a more general problem of feedback equivalence. In this section we study equivalence of control systems. We s ta rt w ith sta te space equivalence in Section 3.1 and then we define feedback equivalence in Section 3.2. Various aspects of the problem of feedback linearization will be discussed in Section 4.

3.1 S ta te sp a ce eq u iv a len ce

Two systems are state-space equivalent if they are related by a diffeomorphism (and then also their trajectories, corresponding to the same controls, are related by th a t diffeomorphism). A question of particular interest is th a t of when a nonlinear system is equivalent to a linear one. If this is the case the nonlinearities of the considered system are not intrinsic, they appear because of a wrong choice of coordinates, and the nonlinear system shares all properties of its linear equivalent.

Consider a sm ooth nonlinear control system of the form

E : x = f ( x , u ),

where x G X , an open subset of R (or an n-dim ensional manifold) and u G U, an open subset of Rm (or an m -dimensional manifold). The class of admissible controls U is fixed and VC C W C M , where VC denotes the class of piece-wise constant controls w ith values in U and M the class of m easurable controls w ith values in U.

182 W. Respondek

Consider another control system of the same form w ith the same control space U and the same class of admissible controls U

S : x = f ( x , u ),

where x G X , an open subset of W 1 (or an n-dim ensional manifold) and u G U. Analogously to the transform ation of a vector field g(-) by a diffeomorphism $ , we define the transform ation of / ( , ) by $ . P u t

($ /) (? , ) = m $ - l (P)) f ( $ - l (P),u).

We say th a t control systems S and S are state space equivalent (respectively, locally state space equivalent at points p and p ) if there exists a diffeomorphism $ : X X (respectively, a local diffeomorphism $ : Xo X , $ (p ) = p , where Xo is a neighborhood ofp) such th a t

* . / = /

P u tT = { f u \ u e U } and T = { f u \ u e U } ,

where f u = and f u = th a t is, T (resp. T ) stands for thefamily of all vector fields corresponding to constant controls of S (resp. of S ). (Local) sta te space equivalence of S and S means simply th a t

$* fu = fu for any u e U ,

i.e., th a t $ establishes a correspondence between vector fields defined by constant controls.

Recall the notion of the Lie algebra C of the system, see the section on controllability and accessibility of [26] in this volume. Assume d im (p ) = dim(j5) = n, which implies th a t S and S are accessible a t p and p, respectively.

The following observation shows th a t (local) sta te space equivalence is very natural.

P r o p o s i t io n 3 .1 S and E are (locally) state space equivalent i f and only i f there exists a (local) diffeomorphism $ which (locally, in neighborhoods of p and p ) preserves trajectories corresponding to the same controls () G U, %. 6 .


for any () U and any t for which both sides exist, where 7 f (p ) (resp. JtiP)) denotes the trajectory of E (resp. T,) corresponding to the control function u(-) U and passing by p (resp. by p ) for t = 0 .

Introduce the following notation for left iterated Lie brackets

/ [ M lli2 ...U fc ] [fui 1 [fu21 > [fuk-1 5 / t t f c ] ' ' ' ]]

and analogous for the tilded family. In particu lar /[M1] = f u i .The following result was established by Krener [32] (see also Sussmann [42]).

T h e o re m 3 .2 Assume that the systems E and E are analytic and that dim C(p) = n and dim C(p) = n.

(i) E and E are locally equivalent at p and p i f and only i f there exists a linear isomorphism of the tangent spaces F : TpX T pX such that

P f{uiU2--Mk ] (? ) = / [ 141U2Mfc ](P ), (3-1)

for any k > 1 and any 1, . . . Uf. U.(ii) Assume, moreover, that X and X are simply connected and that the

Lie algebras C and C of E and E, respectively, consist of complete vector fields and satisfy Lie rank condition everywhere. I f there exist points p X and p X and a linear isomorphism F : TpX ^ TpX satisfying (3.1) then E and E are state space equivalent.

This theorem shows th a t all inform ation concerning (local) behavior is contained in the values a t the in itial condition of Lie brackets from C. In a sense (iterative) Lie brackets form invariant (higher order) derivatives of the dynamics of the system and in the analytic case they completely determ ine its local properties as (higher order) derivatives do for analytic functions.

Consider a control-affine system of the form

m

Saff : i = f { x ) + Y ^ g i ( x )u i-i= 1

Denote go = f . Using the above theorem we obtain the following linearization result (compare [38], [42]).

P r o p o s i t io n 3 .3 Consider a control-affine analytic system E afj.

184 W. Respondek

(i) The system E ag- is locally state space equivalent at p X to a linear controllable system of the form

mAc : x = A x + c + B u = A x + c + 6** , x R , u Rm ,

i=l

at xq R i f a n d only i f

(E l) fan, ]](?) = fo r any k > 2 and any 0 < ij < m , 1 < j < k, provided that at least two i j s are different from zero and

(E2) dim span {ad^gdp) | 1 < i < m . 0 < j < n l}(p) = n.(ii) The system E aff is locally state space equivalent at p X to a linear

controllable system of the form

mA : x = A x + B u = A x + 6** , x I , u Rm ,

i=i

at 0 G R i f and only i fH satisfies (E l), (E2) and f (p ) = 0.(iii) The system E afj is globally state space equivalent to a controllable linear

system A on R i f and only i f it satisfies (E l) , (E2), there exists p X such that f (p ) = 0, the state space X is simply connected and, moreover,(E3) the vector fields f and g i , . . . ,gm are complete.

Recall th a t a vector field / is complete if its flow 7 / (p) is defined for any (t , p ) G R x X .

3 .2 F eedb ack eq u iv a len ce

The role of the concept of feedback in control cannot be overestim ated and is very well understood, bo th in the linear and nonlinear cases. We would like to consider it as a way of transform ing nonlinear systems in order to achieve desired properties. W hen considering state-space equivalence the controls rem ain unchanged. The idea of feedback equivalence is to enlarge state-space transform ations by allowing to transform controls as well and to transform them in a way which depends on the state: thus feeding the state back to the system.

Consider two general control systems E and E given respectively by x = f ( x , u ), x X , u U and x = f ( x , u ), x X , -u U. Assume th a t U and


U are open subsets of Rm . We say th a t S and S are feedback equivalent if there exists a diffeomorphism x ' X x U ^ X x U of the form

(x, u) = x (x , u) = ($(ar), ^ ( x , u))

which transform s the first system into the second, i.e.,

D $ ( x ) f ( x , u ) = f ( $ ( x ) , * ( x , u ) ) -

Observe th a t $ plays the role of a coordinate change in X and W, called feedback transform ation, changes coordinates in the control space in a way which is sta te dependent.

W hen studying dynam ical control systems w ith param eters and their bifurcations, the situation is opposite: coordinate changes in the param eters space are state-independent, while coordinate changes in the sta te space may depend on the param eters.

For the control-affine case, i.e., for systems of the form

m

Saff : i = / O ) + ^ 2 g i ( x ) u i = f ( x ) + g(x )u ,i= 1

where g = ( g \ , . . . ,gm) and u = (1, . . . , urn)T , in order to preserve the control affine form of the system, we will restrict feedback transform ations to control affine ones

u = ^ ( x , u ) = a (x) + $(x)u ,

where f3(x) is an invertible m x m m atrix and u = (1, . . . , m)T. Denote the inverse feedback transform ation by u = a (x) + f3(x)u. Then feedback equivalence means th a t

/ = $ * ( / + ga) and g = $ * ( # ) ,

where g = ( g i , . . . , g m ).For control linear systems of the form x = g(x)u = YliL i g%ix )u %y (local)

feedback equivalence coincides w ith (local) equivalence of d istributions Q spanned by the vector fields

186 W. Respondek

4 Feedback linearization

Since feedback transform ations change dynam ical behavior of a system they are used to achieve some required properties of the system. In Sections 6.2 and 6.3 we will show how feedback transform ations are used to synthesize controls w ith decoupling properties. In this Section we will study the problem of when a nonlinear system can be transform ed to a linear form via feedback. A particu lar case of feedback linearization of single-input control affine systems has been discussed in Section 2. The interest in feedback linearization is two-fold. Firstly, if one is able to com pensate nonlinearities by feedback then the modified system possesses all control properties of its linear equivalent and linear control theory can be used in order to study it an d /o r to achieve the desired control properties. This shows possible engineering applications of feedback linearization, compare Exam ple 2.1. From m athem atical (or system theory) viewpoint, if we would like to classify nonlinear systems under feedback transform ations (which define a group action on the space of all systems) then one of the m ost na tu ra l problems is to characterize those nonlinear systems which are feedback equivalent to linear ones. In Section 4.1 we will study feedback linearization of m ulti-input and general nonlinear systems. In Section 4.2 we will consider linearization using feedback which changes the drift vector field only. Finally, in Section 4.3 we will study the problem of finding the largest possible linearizable subsystem of the given system.

4 .1 S ta tic feed b ack lin e a r iza tio n

A general nonlinear control system

E : x = f ( x , u ),

is (locally at (x q 7 u q )) feed b a ck lin ea riza b le if it is (locally a t (x q 7 uq)) feedback equivalent to a controllable linear system Ac of the form

Ac : x = A x + c + B u .

Recall the notation

? = {iu I u e u } .

For any u G U, define the following distributions on X which will play the


fundam ental role in solving the feedback linearization problem.

Q jA i ( x ,u ) = I m (x,u )

ouA 2 (x ,u ) = A i ( x ,u ) + span [T, A i] (x, u)

= A i (x ,u ) + span {[fu,g] (x ,u ) | f u G T , g G A i}and, inductively,

A j ( x ,u ) = A j - i ( x , u ) + span [T, A j_ i] (x, u)

= A j - i ( x , u ) + span {[fu,g] (x ,u ) | / G g G A 5-_i}.

R e m a rk 4 .1 For the linear system Ac we have

A | = Ijii I! A j = Im ( B , . . . , A 1 1 B ) , j > 0.

In the control-affine case, the feedback linearization problem was solved by Jakubczyk and Respondek [27], and independently by Hunt and Su [22] (see Theorem 4.5 below). In the general case we have the follwing result.

T h e o re m 4 .2 E is locally feedback linearizable at (xq, uq) i f and only i f it satisfies in a neighborhood of (xq, uq) the following conditions

(AO) A i does not depend on u,

(AI) dim A j ( x ,u ) =const, j = l , . . . , n ,

(A2) A j are involutive, j = l , . . . , n ,

(A3) dim A(a:o5 0) = n .

R e m a rk 4 .3 One can show th a t if A i is involutive, of constant rank, and does not depend on u then the successive distributions A j, for j > 2, do not depend on u either. Thus we can check the involutivity condition (A2) for them for a single value u only (for example for 0).

In applications, one is often interested in points of equilibria. Denote by A a linear system of the form

A : x = A x + B u,

th a t is, the system Ac w ith c = 0.

188 W. Respondek

C o ro lla ry 4 .4 E afj is locally feedback equivalent at (xq, uq) to a controllable linear system A at (0,0) i f and only i f it satisfies the conditions (A0)-(A3) and moreover f (xo ,u o ) A i(xo ,ug)-

Consider feedback equivalence of linear controllable m ulti-input systems A of the form x = A x + B u (in th is case the diffeomorphism $(a:) and feedback ^ ( x , u) are taken to be linear w ith respect to the sta te and control). As shown by Brunovsky [5], complete feedback invariants are the dimensions m j of Im M . where the m ap M 1 : > K is defined as [B, A B , . . . , A 1 l I f .P u t no = 0 and n j = m j for 1 < j < n. Define

Kj = max{n* | n* > j } . (4.1)

Observe th a t k \ > > Krn and Y1T= l = n - The integers ;*, called controllability (or Brunovsky) indices, form another set of complete invariants of feedback equivalence of linear controllable systems.

Every controllable system A w ith indices k \ > > nm is feedback equivalent to the system

i i , j = Xij+i , for 1 < j < Ki - 1 , ,4 2,i i ,Ki = Ui ,

where 1 < i < m, called Brunovsky canonical fo rm , which consists of m independent series of k* integrators.

Very often we deal w ith control-affine systems Eafj. To sta te a feedback linearization result for Lag, we define the following distributions

V l (x) = span{gj(:r), 1 < i < m }

V 3 {x) = s p a n la d ^ 1^ ^), 1 < q < j, 1 < i < m },

for j > 2. If the dimensions dj(x) of (x) are constant (see (A l) and (B l) below) we denote them by dj and we define indices pj as follows. Define do = 0 and pu t rj = dj d j - 1 for 1 < j < n . Then (compare (4.1))

Pj = max{r* | r* > j } . (4.3)

If the distributions A j defined earlier in th is section are involutive, thenthey are feedback invariant. If, moreover, the system is affine w ith respect to controls then, clearly, A j = X>J , for j > 1 and, in particular, pi >> pm are feedback invariant. In this case the indices pj coincide w ith Kj , the controllability indices of the linear equivalent of E.

The following result (see [27] and [22]) describes linearizable control-affine systems.


T h e o re m 4 .5 The following conditions are equivalent.(i) E is locally feedback linearizable at x q 6 l n .(ii) E satisfies in a neighborhood of x q

(A l) dim V 3 (x) =const, for 1 < j < n,

(A2) V 3 are involutive, for 1 < j < n,

(A3) dim V n (xo) = n.(iii) E satisfies in a neighborhood of x q

(B l) dim V 3 (x) =const, for 1 < j < n,

(B2) V pi ^ 1 are involutive, for 1 < j < m ,

(B3) dim V p1 (xq) = n, where pi is the largest controllability index.

In the single-input ease m = 1, the condition (A3) (or, equivalently,(B3)) states th a t g(xo) , . . . , a,d^l g(xo) are independent, which implies th a t all d istributions X>J , for 1 < j < n, are of constant rank. In the following Corollary of Theorem 4.5 we thus rediscover Theorem 2.2.

C o ro lla ry 4 .6 A scalar input system E is feedback linearizable i f and only i f it satisfies

(C l) g(xo) , . . . , ad - 1g(:ro) are independent,

(C2) V n -1 is involutive.

E x a m p le 4 .7 Consider the following rigid two-link robot m anipulator (double pendulum); compare, e.g., [6] or [37].

1 2 x = x

x 2 = M ^ l (x l ) (C (x l , x 2) + k ( x 1)) + M ^ l (x l )u ,

where 0 i and 6 2 represent the angles (between the horizontal and the first arm and between the arms) and x l = (^1,^ 2), x 2 = {0 1 , 6 2 ). The contro l torques applied to the joints are u = (1, ^ 2) and the positive definite sym m etric m atrix M ( x l ) is given by

m i l 2 + m 2 l2 + + t2m 2 l i h cos 6 2 + m 2 h h cos 6 2m 2^| + ^ h h cos 6 2 m 2l |

The term k ( 6 ) represents the gravitational force and the term C (6 , 6 ) reflects the centripetal and Coriolis force.

190 W. Respondek

We have th a t V 1 = span {gfy} = span -J-} is involutive and dim V 2 (x 1 , x 2) = 4. Hence the double pendulum is feedback linearizable. A linearizing feedback is given, e.g., by u = C ( x l , x 2) + k ( x l ) + M ( x l )u.

E x a m p le 4 .8 Consider the following model of a perm anent m agnet stepper m otor [46]

1 = K \ x \ + K 2 X%sin{KX4) + u\

X 2 = K \ . / '2 + K 2 X ^ C O . s ( K ^ X 4) + 2

3 = ^K% xisin{KX 4) + K%X2 Cos{KX4) K ^xs + K ^ s in ^ K ^ x ^ ) t l / J

4 = x 3 ,

where x i , x 2 denote currents, X3 denotes the ro tor speed and X4 its position, J is the ro tor inertia, and t l is the load torque. We see the distributions V 1 = span{g|^-, and V 2 = span{g|^-, are involutive and th a tdimX>3(:r) = 4 and thus the system is locally (and even globally!) feedback linearizable.

E x a m p le 4 .9 The goal of this example is to show how to solve nonlinear problems by transform ing the system to an equivalent linear system and solving the linear version of the problem for the linear system. Consider the following system

x = y + yz y = z z = -u + sina:,

where (x, y, z) R3. We want to stabilize it exponentially globally on R3. Firstly, we show th a t the system is feedback linearizable. To simplify calculations, replace / by f = f + ag = f (sin x)g = (y + yz , z, 0)T . We have g = (0 ,0 ,1)T , ad jg = (y, 1,0)T , and [g, ad^g] = 0. Thus the distributions V 1 = s p a n { ^ } and V 2 = span{y j^ + are involutive. We seek for afunction ip whose differential anihilates V 2 which means to find a solution of the following system of 1-st order partial differential equations (compare Section 2)

f ^ 0I dz u

I - n{ y dx ^ dy u -


We conclude th a t ip can be an arb itra ry function of x \ and we choose 2 2

ip = x Therefore we pu t, see Section 2, x = x y = Lfip = y,z = L 2ip = z and, finally, u = u sin a:. This yields the following linearsystem

x = yy = zz = u,

which we stabilize on R3 globally and exponentially via a linear feedback of the form u = k x + ly + m z , where the m atrix

/ 0 1 0 \0 0 1

\ k I m /

is Hurwitz. Therefore the nonlinear feedback

y 2u = k (x ) + ly + m z sinLi

stabilizes globally and asym ptotically on R3 the original system.

E x a m p le 4 .10 Consider the following model of the rigid body whose gas jets control the rotations around the two first principal axes.

Wi = a\Ul2 Ul^ + 1 2 = 02W1W3 + 23 = a 3w 1W2.

We have / = (aiw2W3 , a2 u iw3, a 3wiw2)T , 5 i = (1, 0 ,0)T and g>2 = (0 ,1 ,0)T . We calculate ad /

192 W. Respondek

E x a m p le 4 .11 Consider the following model of unicycle

i i = ui cos 9

x 2 = u 2 sin 9

9 = u 2 ,

where ( x i , x 2 ,9) G R2 x S 1. We have

gi = (cos 9, s in 9 ,0)T , g2 = (0 ,0 ,1)T ,

thus [


and study their equivalence to linear single-input systems of the form

A,; : x = A x + c + bu , x R -u R.

We have the following result [1].

T h e o re m 4 .12 E afj is locally restricted feedback linearizable at xq i f and only i f it satisfies in a neighborhood of xq the following conditions

(RC1) g(xo) , . . . ,a d -1 g(:ro) are independent.

(RC1) [a,djg, ad/#] C V n f or any 0 < q ,r < n - 1 ,

R e m a rk 4 .13 Like in the case of feedback linearization (compare Corollary 4.4), afjf is restricted feedback equivalent a t xq to Ac, w ith c = 0, a t 0 if and only if f ( x 0) T>1 (xq).

In the single-input case all linearizable systems are equivalent to the Brunovsky canonical form (compare (4.2))

i i = x i+i , for 1 < i < n - 1 , , .x n = u .

If afj is (locally) feedback linearizable, then there are many pairs (a , ft) and many (local) diffeomorphisms which transform afj into its Brunovsky canonical form. However, if we allow for restricted feedback only, then a transform ing afj into the canonical form is unique and is given by

a = (-l)"-1L r17, (4-6)where L / stands for the Lie derivative along / and the sm ooth function 7 is uniquely defined by

nf =

i=iThis observation is crucial for establishing the following result on restricted feedback linearization [9], [39].

T h e o re m 4 .14 E is restricted feedback globally linearizable, that is, globally equivalent via a restricted feedback to a linear system on R , i f and only i f it satisfies the conditions (RC1),(RC2) and, moreover,

(RC3) the vector fields f and g are complete, where f = f + ga and a is defined by (4.6),

(RC4) the state space X is simply connected.

194 W. Respondek

E x a m p le 4 .15 (Continuation of Examples 2.1 and 2.4). We have / = ~ f and 9 = Therefore [adf g,g] = 0 and since g and

ad g are independent everywhere, the system is restricted feedback linearizable. Indeed, it is im m ediate to see th a t the feedback u = m gl sin B + u brings the system to a linear form (no action of diffeomorphism is needed).

The nonlinear pendulum defined on S 1 x R1 is globally equivalent to a linear system evolving on S1 x R1. If we enlarge the class of linear systems to include systems of the form x = A x + B u , where each component X{ of x is either a global coordinate on R1 or a global coordinate (angle) on 5 1, then Theorem 4.14 rem ains true if we drop the assum ption (RC4). This includes many mechanical control systems.

4 .3 P a r tia l lin e a r iza tio n

The linearizability conditions are restrictive (except for the scalar input affine systems on the plane, compare Corollary 2.3). Given a nonlinearizable system it is therefore na tu ra l to ask w hat is its largest linearizable subsystem. Consider a partially linear system Apart of the form

, T - /17 I < Y J i}H

p t ' i 2 = / 2( i \ i 2) + Er=i!7i!( i i , i 2)s ..

w ith x l , x 2 being possibly vectors. Recall the notion of the Lie ideal o of the system (see [26] of this volum e), which is defined as the Lie ideal generated by g i , . . . , gm in , or, in other words, o = Lie {ad^ft | 1 < i < to, q > 0}. W ith the help of Cq we define another Lie ideal by pu tting

2 = [o,o\ = { [ f i , f 2 \ | / i , / 2 e 0}.

It is 2 which contains all intrinsic nonlinearities not removable by the action of diffeomorphisms as the following result [40] shows.

T h e o re m 4 .16 Consider a control affine system Saff.(i) I f E afj is locally state space equivalent at xq to a partially linear system

Apart then dim 2 (x) < n in a neighborhood of x q .(ii) Assume that Eaff satisfies dim o(o) = n and that dim 2(a:) =

a=const. in a neighborhood o f x q . Then E aff is locally state space equivalent to a partially linear system Apart> such that the dimension


of the linear subsystem is dim x l = n a and, moreover, the linear subsystem is controllable.

C o ro lla ry 4 .17 Let an analytic system E ag- satisfies dim Cq(xq) = n. I t is locally state space equivalent at xq to a partially linear system Apart i f and only i f

dim C2(xo) < n.

Moreover, there exists a system Apart, with n a-dimensional linear controllable subsystem, where a = dim C2(xo), which is state space equivalent to aff.

Now we consider the problem of transform ing a nonlinear system to a partially linear one via feedback. This problem has been studied and solved in the scalar-input case in [33] and in the m ulti-input case in [34] and [40]. Recall th a t for a sm ooth d istribution V we denote by V its involutive closure, th a t is, the smallest d istribution containing V and closed under the Lie bracket.

T h e o re m 4 .18 Consider a single-input system Eaff.(i) I f E afj is locally feedback equivalent at x q to a partially linear Apart

with p-dimensional linear controllable subsystem then satisfies the following conditions:

(P C I) g (xo) , . . . , adpj ^ l (xo) are independent,

(PC2) dim V p l (x) < n in a neighborhood o f x o,

(PC3) a,dpf l g(x) $ V p^ l (x).

(ii) Assume that there exists an integer p such that d i m V p l (x)=const. and that (P C I), (PC2), (PC3) are satisfied. Then E afj is locally feedback equivalent to a partially linear system Apart with p-dimensional linear controllable subsystem, that is dim x 1 = p. Moreover, the largest p satisfying the above conditions gives the largest dimension of linear subsystem among all possible partial linearizations.

E x a m p le 4 .19 Consider a sym m etric rigid body (two inertia m om enta are equal) w ith one pair of jets

u)i = aw2^3 + e\u2 = 1W3 + 62 u0)3 = 63 u

196 W. Respondek

Com pute g = ( e i , e2 , e 3)T , adg = a(e2W3 + 63012, 61013 63011, 0)T . Hence for V 2 to be involutive, th a t is, [g, ad/g] = 2063( ^ 62, 61, 0 ) C V 2 span{g ,ad /g} , we need either 63 = 0 or ei = 62 = 0. In the former case, u>3 rem ains constant, in the latter, the sym m etric spacecraft is controlled in a sym m etric way: the angular m om entum of the je t is parallel to the th ird principal axis. Notice th a t for all values of the control vector e = (ei, 62, 63)T , the system is not feedback linearizable. Indeed, either V 2 is not involutive or the system is not accessible. On the other hand, for all values of the control vector field e ^ 0 , the system contains a 2-dimensional linear subsystem for an open and dense set of in itial conditions.

5 Observability

In th is chapter we consider briefly the concept of nonlinear observability. We s ta rt w ith geometric approach to the observability problem and in Section 5.1 we sta te a sufficient condition, called observability rank condition, based on successive Lie derivatives of the ou tpu t along the dynamics. In Section 5.2 we discuss (local) decompositions into observable and completely unobservable parts which generalize the classical K alm an decomposition. Then in Section5.3 we consider the problem of uniform observability, which means th a t we can observe the system for any input. In Section 5.4 we give a necessary and sufficient condition for local observability. Finally, in Section 5.5 we discuss generic properties: we give norm al forms for generic systems and recall results concerning genericity of observability.

5.1 N o n lin ea r o b serv a b ility

Consider the class of nonlinear systems w ith ou tputs (measurements) of the form

x = f ( x , u ) ,y = H x ),

where x C X . u C V. y C Y . Here X , U, and Y are open subsets of K , Rm , and Rp , respectively (or differentiable manifolds of dimensions n, to, and p, respectively)1. The m ap h : X ^ Y represents the vector of p measurem ents (observations), where h% C (X ) , for 1 < i < p, and h = (h i , . . . , hp)T .

1Except for the second part of Section 5.5, where we assume U to be J m, w ith J being a com pact subinterval of R,


Throughout th is section, E will denote the above described nonlinear system w ith ou tput.

The class of admissible controls U is fixed and VC c W c M , where VC denotes the class of piece-wise constant controls w ith values in U and M the class of m easurable controls w ith values in U.

Let y denote the space of absolutely continuous functions on X w ith values in Y . For the system E we define the response map , called also inputoutput map ,

: X x U > y ,

which to any initial condition 0 such th a t

yqi,u(t) ~f~ yq2 ,u(t)-

meaning th a t the states qi and q2 are distinguishable.

D e fin it io n 5 .2 E is called locally observable a t q X if there is a neighborhood V of q such th a t for any

198 W. Respondek

Given a system E and an open set V C X , by the restriction E |y we will m ean a control system w ith the sta te space V, defined by the restrictions of / and h to V x U and V , respectively.

D efin ition 5.3 E is called strongly locally observable a t a point q X if there exists a neighborhood V of q such th a t the restricted system E jy is observable.

We would like to emphasize some features of the introduced concepts of observability. Strong local observability is a local concept in two aspects. Firstly, strong local observability means th a t, in general, we are able to distinguish neighboring points only. Secondly, we are able to do so considering trajectories which stay close to the initial condition. O f course, observability implies strong local observability a t any point (we can take V = X ), which, in tu rn , implies local observability a t any point (for each point we take the neighborhood V existing due to the strong local observability). In general, the reversed implications do not hold, see Examples 5.6 and 5.7 below.

E xam ple 5.4 Consider a mechanical system evolving according to Newto n s law

1 = x 2X 2 = U,

where x \ denotes the position, x 2 the velocity, and u is the control force. We observe the position

y = i-

This system is clearly observable. Indeed, let yq,u (t) and yq,u(t) be the ou tputs of the system initialized, respectively, a t q = (xio ,X 2o)T and at q = (zioi%2o)T and governed by a control (). Assume th a t yq,u(t) = yq,u(t) for any t. Then comparing at t = 0 bo th sides of the above equality as well as derivatives a t t = 0 of bo th sides, we get io = io and X20 = 20? which proves the observability.

Now assume th a t, for the same control system, we observe the velocity

y = x 2.

The system is not observable. Indeed, the in itial conditions q = (x iq ,X 2o)t and q = (xio ,X 2o)T , such th a t io ^ 10 bu t X20 = 20? produce the same ou tpu t y(i) = J l u(s)ds + X2o- Mechanically, this is obvious: we cannot estim ate the position if we observe the velocity only.


E x a m p le 5.5 Consider the linear oscillator (linear pendulum ) given by

i i = x 2 i'2 = - X l ,

where x \ denotes the position and x 2 the velocity. Assume th a t we observe the position

y = i-

Then the system is observable and, given the ou tpu t function y(-), we can deduce the in itial condition (xiq7x 2q)t by taking, like in Exam ple 5.4, the ou tpu t and its first derivative w ith respect to tim e at t = 0.

Now assume th a t we observe the velocity

V = x 2.

This system is also observable and once again we can deduce the initial condition by looking at the values a t t = 0 of the ou tpu t and its first tim e derivative. The reason for which observing the velocity renders the system observable is th a t the evolution of the velocity x 2 depends on the position x \ y which is not the case of the system of Exam ple 5.4.

E x a m p le 5 .6 Consider the unicycle

i i = i cos 9 , y\ = x \x 2 = i s i n0 , y 2 = x 29 = u2,

where (x i , x 2)t 6 1 x 1 is the position of the center of the mass of the unicycle and 9 E S l is the angle between the horizontal and the axis of the unicycle. We observe the position of the center of the mass.

The unicycle is observable. To see it, consider the ou tputs yq,u (t) and yq,u (t) of the system controlled by u(t) = ( i(i) , u 2 (t))T , such th a t u\( t) = 1, passing for t = 0 by q = (xw , x 2(h6 o)T and q = ( io , 20, ^o)T , respectively. Assume th a t yq,u (t) = yq,u(t) Thus ario = i io , %2o = 20, and moreover s in 9(t) = s in 0{i) and cos 6 {i) = cos 6 {i). Hence we conclude th a t 9q = 9q, where 9q, 9q G S 1.

Now consider the unicycle, w ith the same observations yi = x \ and y 2 = x 2y evolving on R3, th a t is, we consider 9 G R. It tu rns out th a t the system is not observable. To see this, we will show th a t the outputs yq,u (t) and yq,u(t) of the system coincide for q = (x io ,x 2o,9q)t and q =

200 W. Respondek

(x io ,x 2o ,6 o)t such th a t x w = io, x 2o = 20Land 90 = &o + 2kir. We have 9(i) = / 0* U2 (s)ds + 9q and 9(t) = / oi 2(s)efs + 0o and hence 9(t) = 9(t) + 2k-K. Thus sin0(i) = sin0(i) and cos 9{i) = cos 9{i) implying th a t yq,u (t) = yq,u(t), for any control u(-) G U and the in itial conditions as above. In other words, the points q = (x iq 7X2o79 q)t and q = ( iio , 20, ^o)T such th a t io = iio? %20 = 20? and 9q = 9q + 2kir are indistinguishable. O f course, the system is strongly locally observable a t any q G R3.

E x a m p le 5 .7 Consider the system

x = 0 , y = x 2 ,

where x G R and y G R . The system is not observable because the initial conditions xq and xq give the same ou tpu t trajectories. This system is strongly locally observable a t any xq ^ 0. Notice th a t it is locally observable a t any point, in particu lar a t 0 G R2, although in any neighborhood of 0 there are indistinguishable states. This shows th a t local observability is indeed a weaker property th an strong local observability.

We will give now a sufficient condition for strong local observability. To this end, we will introduce the following concepts.

D e f in it io n 5 .8 The observation space of S is defined as

11 spav I 1 < i < p , k > 0, 1, . . . , * G 17},

where /() = / ( , % ) and L gip stands for the Lie derivative of a sm ooth function ip w ith respect to a sm ooth vector field g, i.e.,

L gip(x) = dip(x) -g(x).

Observe th a t H is the smallest linear subspace of C (X ) containing the observations h \ , . . . , h p and closed w ith respect to Lie differentiation by all elements of T = { / ( , ) , G U}, i.e., all vector fields corresponding to constan t controls. Using functions from H we define the following codistribution

Ti = span {d : G H } .

Notice th a t, in general, Ti is not of constant rank.


In the ease of control affine systems of the form

f ( x ) + YT= i 9i(x )u t

we have

= span | dLgjh L 9h hi : 1 < i < p, 0 < j i < m j ,

where gQ = f .The following result of H erm ann and Krener [21] gives a fundam ental

criterion for nonlinear observability.

T h e o re m 5.9 Assume that the system E satisfies

Then E is strongly locally observable at q.

The condition (5.1) will be called observability rank condition. It can be considered as a counterpart of the accessibility and strong accessibility rank conditions (see, e.g., the survey [26] of this volume), although the duality is not perfect, as we will see in the next example.

E x a m p le 5 .10 The converse of Theorem 5.9 does not hold (even in the analytic case) as the following simple example shows. Consider

where x G R, y G R. O f course, the system is strongly locally observable at any i G R (even observable on R) bu t it does not satisfy the rank condition at 0 G R, since we have Ti = span{a;2cfcc }. This shows also th a t the rank of Ti need not be constant. This is to be compared w ith the accessibility rank condition, which, in the analytic case, is necessary and sufficient for accessibility.

E x a m p le 5.11 Consider a linear control system w ith ou tpu ts of the form

dim Hiq) = n. (5.1)

x = 0 , y = x 3 ,

x = A x + B u , y = C x ,

202 W. Respondek

where x G W 1 , u G W n , y W . We have / = A x , = b/., for 1 < k < m , and hi = C%x, for 1 < i < p, where C* denotes the i-th row of the m atrix C. We calculate

L^hi = CiA3x and L gkL^hi = CiA^bk-

Thus 7i(q) = span {C* A3 | 1 < i < p, 0 < j < n 1} and dim 7i(q) = rank O, where O is the Kalm an observability m atrix

f c \

0 = CA .

\ C A n - 1 )

Therefore a linear system satisfies the observability rank condition if and only if it satisfies K alm an observability condition rank O = n. In th is case, as it follows from Theorem 5.9, the system is strongly locally observable. Moreover, we know from the linear control theory, see e.g. [30], th a t the system is observable. Indeed, the response m ap R \ of the linear system A given by

y(t) = C x ( t ) = C eAtx o + f C eA^ ~ s^Bu(s)dsJ o

and associating to an initial condition xq the ou tpu t trajectory, is affine with respect to the in itial condition xq and thus local injectivity implies global injectivity. Notice th a t observability properties of a linear system do not depend on the chosen control; indeed, they depend only on the injectivity of the m ap

xq i > C eAtx o-

In the next example we will show th a t this no longer true in the nonlinear case.

E x a m p le 5 .12 The aim of th is example is to show th a t, contrary to the linear case, controls play an im portant role in the nonlinear observability. In general, there may exist controls which do not distinguish points nevertheless the system can be observable if other controls distinguish. To illustrate th a t phenomenon, consider the bilinear system

11 = X2 X2u , y = x 11 2 = 0 ,


where ( x i , x 2 )r G M2. This system is observable, because if we pu t u(t) = 0 we get an observable linear system. Notice, however, th a t the constant control u(t) = 1 does not distinguish xq and xq such th a t a?io = 10 and X20 ~f~ 20 (we will come back to th is phenomenon in Section 5.3). Of course, we can deduce strong local observability a t any point from the rank condition. Indeed, we have h = x i , f = x 2 - t^ , and L f h = x 2. Hence Ti = span{cfcci,cfcc2}- D

E xam ple 5.13 Consider the unicycle, see Example, 5.6, for which we observe i/i = x i and y 2 = x 2. We have hi = x i , h 2 = x 2, gi = cos0-^ + s in 0 g |^ , and g2 = J j . Hence L gih\ = cos9 and L gih 2 = s in 9. Thus Ti = span { d x i ,d x 2 ,d s m 9 , dcos 9} implying th a t dim Hiq) = 3, for any 5 G K2 x S 1. Therefore the unicycle satisfies the observability rank condition at any point of its configuration space.

5 .2 L oca l d e c o m p o s it io n s

Let us s ta rt w ith linear systems of the form

. x = A x + B u ,: y = C x ,

where i G l " , G Rm , y E W . Denote by W , the kernel of the linear m ap defined by the Kalm an observability m atrix O (see Exam ple 5.11). If A is not observable then we can find new coordinates (x l , x 2)y w ith a;1 and x 2 being possibly vectors and dim x l = k, where dim W = n k, such th a t a- G W7 if and only if x = (0, x 2). Then A reads

i 1 = A l x l + + B l u , y = C l x l ,x 2 = ^421a:1 + ^422a;2 + B 2u ,

where the pair {Cl , A l ) is observable.As a consequence, any two initial states whose difference is not in W are

distinguishable from each other, in particular, by means of the ou tpu t produced by the zero input. Contrary, if their difference is in W , then they are indistinguishable. The factor system A// , where I is the indistinguishability equivalence relation, is observable and is given by

A1 : x l = A l x l + B l u , y = C l x l .

Geometrically, A1 is obtained by factoring the system through the subspace W and the factor system is well defined since W is invariant under A. A

204 W. Respondek

natu ra l question is whether we can proceed similarly for the nonlinear system E?

T h e o re m 5 .14 Consider the nonlinear system E . Assume that the distribution % is of constant rank equal to k locally around q. Then we have.

(i) The codistribution % is integrable and there exist local coordinates (x l , x 2)T defined in a neighborhood V of q, with x l , x 2 possibly being vectors, such that % = span {d x \ , . . . , x^}.

(ii) In the local coordinates (x l 7x 2)T , the system E takes the form

1 = , y = h l ( x l ) ,i 2 = f 2 (x l , x 2 ,u).

(iii) By taking V sufficiently small, two pints q, q V are indistinguishable for E jy i f and only i f q S q, where S q is the integral leaf, passing through q, of the codistribution % restricted to V .

(iv) In V , factoring the system through the foliation of the integrable codistribution %, produces the strongly locally observable system E 1 which, in (x l , x 2)T - coordinates, is given by

E 1 : ./1 = / 1(a:1,u ) , y = h 1 (x1).

This result says th a t locally and under the constant rank assum ption, the leaves of the foliation of the integrable codistribution T consist of indistinguishable points and th a t, on the other hand, we can distinguish the leaves.

From Theorem 5.14 we im m ediately get the two following corollaries.

C o ro lla ry 5 .15 I f % is of constant rank in a neighborhood of q then the following conditions are equivalent.

(i) E is locally observable at q.(ii) E is strongly locally observable at q.(iii) dim 7i(q) = n .

C o ro lla ry 5 .16 / / E is locally observable at any point of X then dim 'H{q) = n, for q X ' , an open and dense subset of X .

An im portant case when the observability rank is constant is given by the following.


P r o p o s i t io n 5 .17 Assume that an analytic control system E satisfies the accessibility Lie rank condition everywhere on X . Then % is of constant rank on X . In particular, the system is locally observable at q (or, equivalently, strongly locally observable at q) i f and only i f dim Tiiq) = n.

To illustrate the decomposition result of this section we consider the following example.

E x a m p le 5 .18 Consider the unicycle, see Exam ple 5.6, for which we measure the angle 9 only, th a t is h = 9. We have L gih = L g2h = 0. Thus Ti = span {d9} defines the foliation

{9 = const.} ,

whose leaves consist of indistinguishable points. Indeed, if 9q = 9q then the points q = (xio7X2o,Go)t and q = ( i io,20,^o)T are indistinguishable. The obvious reason for this is th a t the evolution of the observed variable y(t) = 9(t) is independent of th a t of x \( t ) and x 2 (t).

5 .3 U n ifo rm o b serv a b ility

In Exam ple 5.12 we pointed out th a t for observable nonlinear systems there may exist controls th a t render the system unobservable. In th is section we describe a class of systems, for which all controls distinguish points.

D e f in it io n 5 .19 The system E is called uniformly observable, w ith respect to the inputs, if for any two states qi,q 2 X , such th a t qi ^ q>2> and any control u(-) U

yqi,u(t) 7^ yq%,u(t)-

E is uniformly locally observable a t q X , if there exists a neighborhood V of q, such th a t E restricted to V is uniformly observable.

E x a m p le 5 .20 Exam ple 5.12 illustrates the existence of nonlinear systems th a t are not uniformly observable. Another example is the unicycle, see Exam ple 5.6, for which we observe y\ = x 2 and j/2 = 2- The system is observable, nevertheless, for the control u\( t) = 0, any two points q = O io, 20, 90)T and q = ( i0, 20, ^o)T , such th a t zio = 10 and x 2o = x 2o are indistinguishable.

206 W. Respondek

O f course, linear observable systems are uniformly observable. We will describe now a class of nonlinear uniformly observable systems. Consider a single-input single-output control-affine system of the form

x = f { x ) + g(x)u L a f f y = h(x),

where x E X , u E R , y E R and / , g are sm ooth vector fields on X .The following result is due to G authier and Bornard [14].

T h e o re m 5.21 For the system we have:(i) I f afjf is uniformly locally observable at any q E X , then around any

point of on an open and dense submanifold X ' of X there exist local coordinates ( x i , . . . , x n)T in which the system takes the following normal form

x i = x 2 + ug i(x i ) , y = x iX2 = x 3 + ug2(x i , x 2)

(U O )

Xni En I- ugni ( x i , . . . , x nijX n = f n ( x i , . . . , X n ) + U g n ( x i , . . . , x n )

(ii) I f afjf admits, locally at q, the form (UO) then it is uniformly locally observable at q.

(iii) A necessary and sufficient condition for afj to admit locally at q the normal form (UO) is that dim span {dh , . . . , dLn^ l h}{q) = n and that in a neighborhood of q

[Djid] C D j ,

for any 1 < j < n, where D j = ker {dh , . . . , dL 3^ l h}.

5 .4 L oca l ob serv a b ility : a n ecessa ry an d su ffic ien t c o n d itio n

Recall th a t the H erm ann-Krener observability rank condition gives only a sufficient condition for (strong) local observability (compare Exam ple 5.10). Following Bartosiewicz [1] we will provide in this section a necessary and sufficient condition for local observability.

Consider a nonlinear system E and assume th a t it is analytic, th a t is, X is an analytic manifold, the vector fields f u are analytic and h is an analytic map.

We s ta rt w ith the following simple observation.


Proposition 5.22 T h e p o in ts q\ a n d q2 are in d is tin g u ish a b le i f and only i f fo r any 7i we have (qi) = (q2)-

Introduce now the observa tio n algebra of S . It is the smallest subalgebra over R of C W( X ) , the algebra of analytic functions on X , which contains hi and is closed under Lie derivatives w ith respect to f u, u e U . We denote it by H a - Observe th a t H ,\ consists of all elements of H and of all constant functions.

For x G X , by O x we denote the algebra over R of germs of analytic functions a t x . Denote by m x the unique m axim al ideal of O x . It consists of all germs th a t vanish a t x . For ./ C X we define I x to be the ideal in O x generated by germs of those functions from H a which vanish a t x . Of course, I x C m x . T he real radical of an ideal I in a com m utative ring R is

V i = {a G R | a 2m + bf-\ G I for some m > 0, k > 0, b i , . . . ,bf. G R } .

Clearly, the real radical is an ideal.

T heorem 5.23 T h e sy s te m E is locally observable a t x i f a n d only i f

E xam ple 5.24 We can easily see th a t for the system x = 0 , y = x 3 (compare Exam ple 5.10), which is clearly locally observable a t any x G R, we have = m x for any x G R, in particular, for x = 0.

5 .5 G en er ic o b serv a b ility p ro p ertie s

In this section we discuss the problem of w hat observability properties are shared by generic control systems. We consider C-W hitney topology for sm ooth systems. Recall th a t a sequence of sm ooth function ipn on a manifold X converges in C-W hitney topology to a sm ooth function ip if there exists a compact subset C C X such th a t the all derivatives ip f f , for i > 0, converge uniformly on C to the corresponding and (pn = (p on X \ C, for all n sufficiently large. In the case of a compact sta te space X, it is ju st the topology of C uniform convergence on X. We s ta rt by presenting results of Jakubczyk and Tchon [28] who classified uncontrolled observed dynamics

208 W. Respondek

of the form

x = f ( x ) , y = h ( x ) ,

where x C X and y R, / is a sm ooth vector field and h is a sm ooth Unvalued function. Let E denote the family of all systems E of the above form equipped w ith the C 00-W hitney topology.

Theorem 5.25 There exists an open and dense subset So C S such that any E G So is locally equivalent at any q X to one of the following normal forms.

(i) ///(


T heorem 5.26 I f E satisfies the observability rank condition at q then it is locally equivalent either to the form (5.4)-(5.5) i f f ( q) = 0 or, otherwise, to one of the following normal forms

h(x) = a^+1 + (pr-ix^ 1 + b (pixi + 4>q + c, (5.6)

f { x ) = f i ( x i , . . . , Xn)~^~, (5.7)

where r > 0, and fa are C-functions of x 2 , . . . , x n , for 0 < i < r 1, satisfying = 0, and f \ is a C-function such that f \ > 0.

If r = 1 we can always take h = x \ + (f)2, while for r = 0 we take h = x 1 + c.

We end up th is chapter by sta ting some results of G authier and Kupka devoted to the genericity of uniform observability. Consider an observed sm ooth control system of the form

s . % = f { x , u ),y = h(x, u)

where x G X , u G U, and y G Y . Notice th a t we assume the output y = h(x, u) to depend explicitly on the control u.

Recall th a t for the system E we define the response map, called also inpu t-ou tpu t map

R s : X x U ^ y 7

which to any initial condition xq G X and any admissible control u(-) G U attaches the ou tpu t of the system

y ( t , x 0 ,u(-)) = h ( x ( t , x 0 ,u (-)),u ( t)) .

The control u(-) being defined on an interval 0 G I u C R, we consider the ou tpu t y(-) on the m axim al interval I y C Iu C R on which it exists.

In the rem aining part of th is section, we will assume th a t the sta te space X is a compact manifold and U = J m, where J is some compact interval of R. We denote by E the class of such systems equipped w ith the topology of C uniform convergence on X x I m .

For any C^-function w(t) of tim e we will denote w k (t) = ( w ( t ) , w ' ( t ) , , w W ()). For the system E, for any integer k and for a C^-differentiable input u(t), we define the fc-prolongation of the response m ap as

210 W. Respondek

th a t is, as the vector formed by the ou tpu t and its first k derivatives with respect to tim e t.

For an open subset W of R9 (or a differential manifold), and for a C k- differentiable function w of I C R into W, such th a t 0 G I , we denote by j kw the fc-jet a t 0 G R of w. We will denote by J kW the space of fc-jets at 0 G R of m aps from I into W . Now we consider the fc-jet

j kR s : X x J kU > J kY

of the m ap defined by

j kR x ( x o , j ku ) = j ky,

where j ky = y k (0), y k (t) = R ^ ( x o ,u k (t)), and u(t) is any C^-control such th a t u k (0) = j ku.

The following fundam ental result has been proved by G authier and Kupka[16], [17], [18],

Theorem 5.27 Assume p > m, that is, the number of outputs is greater than that of inputs. Fix a sufficiently large positive integer k.

(i) The set of systems such that j kR ^ ( - , j ku) is an immersion of X into R ^ +1), for all j k u G J kU , contains an open dense subset of E.

(ii) The set of systems E such that j kR s ( - , j ku) is an embedding of X into RpO+i) , for all j ku G J kU , is a residual subset of E.

(iii) For any compact subset C of J kU , the set of systems E such that j kR ^ ( - , j ku) is an embedding, for all j ku G C, is open dense in E.

The above result implies th a t, in the case p > m , the set of systems th a t are observable for all C k inputs is residual, th a t is, it is a countable intersection of open dense sets. If a bound on the derivatives of the controls is given a-priori, th a t is | u W (t) |< M , for some constant M and any 0 < i < k. then th is set is open dense. If the num ber of ou tpu ts is not greater than th a t of controls all statem ents of the above theorem are false.

6 Decoupling

In this section we show how sta tic feedback allows to transform the dynamics of a nonlinear system in order to achieve desired decoupling properties. In Section 6.1 we will introduce a crucial concept of invariant distributions. In Section 6.2 we consider disturbance decoupling while in Section 6.3 we deal w ith inpu t-ou tpu t decoupling.


6 .1 In varian t d is tr ib u tio n s

Consider a sm ooth nonlinear control system of the form

: x = g0( x ) + Y ^ g i ( x )u i = go(x ) + g(x )u7i=1

where x G X , u G Rm , g = ( g i , . . . ,gm) and u = ( u i , . . . , u m)T . Notice th a t for simplicity we denote the drift of the system by / = go

A distribution V is called invariant for if

[gi, T> C V . for 0 < i < m.

If a d istribution is not invariant for E it may become invariant under a suitable feedback modification. A d istribution V is called controlled invariant if there exists an invertible feedback of the form

u = a (x) + (3(x)u, /?() invertible,

such th a t V is invariant under the feedback modified dynamics

%=ith a t is,

= 9o(x) + Y ^ 9 i ( x )u i ,i= 1

[gh V\ C V ,

for 0 < i < in.. where

g0 = g0 + got, g = gj3 .

E xam ple 6.1 In the case of a linear system of the form

A : x = A x + B u , i G f 1, u G Rm .

a subspace V C K is said to be invariant for A if A V C V. We say th a t V is controlled invariant (or (A, B )-invariant) if there exists a linear feedback of the form u = F x + Gu such th a t

(A + B F ) V C V.

Observe th a t in the linear case (A, B)-invariance does not depend on G so one can take G = Id or G-noninvertible.

212 W. Respondek

One can cheek by a direct calculation th a t (A, B ) -invariance is equivalentto

A V C V + Im B . (6.1)

We refer to [45] for an extensive treatm ent of the concept of invariance in the linear case.

P u t Q = span {g\ , . . . , grn}. In the nonlinear case, controlled invariance of a d istribution V implies the following property of local controlled invariance (compare (6.1))

[(/,:. P] C V + Q. for 0 < i < m .

For involutive d istributions the converse holds locally under regularity assum ptions, see [20],[25],[35].

Proposition 6.2 Assume that the distributions V , Q, and V fl Q are of constant rank. I f V is involutive and locally controlled invariant then it is controlled invariant, locally at any point x X .

6 .2 D is tu r b a n c e d eco u p lin g

In th is Section we apply the concept of controlled invariant distributions to solve the nonlinear disturbance decoupling problem. Consider the following nonlinear system w ith ou tpu t affected by disturbances d = (d\ , . . . ,


does not depend on the disturbances d(t). By the la tte r we mean th a t

y(t, q, u(-),d(-)) = y(t, q, (),

214 W. Respondek

E xam ple 6.5 Consider the linear system w ith disturbances

x = A x + B u + E d-A-dist y = C x ,

where x K , u Rm , y W , d Rk , and d denotes the disturbances. DD P is solvable if and only if Im E C V"!. where V* is the largest controlled invariant subspace in ker C (compare [45]).

E xam ple 6.6 Consider a particle of unit mass moving on the surface of a cylinder according to a potential force given by the potential function V (see[37])

qi = p i = P2^ = ~ ^ (Q u Q 2 ) + u p 2 = - f ^ ( g i , 2 ) + d ,

where (qi,q2 iPi->P2 ) T ( S l x K). Let the ou tpu t be given as y = q\. We can see th a t V* = span {^ . Moreover, the disturbance vector field ^ V* and hence D D P is solvable by the feedback u = ?2) + u.

6 .3 In p u t-o u tp u t d eco u p lin g

Consider a sm ooth nonlinear control affine system w ith ou tputs of the form

x = f ( x ) + Y 7 = i u i9i(x ) s S ~ V = h ( x ) ,

where x X , u Rm , and y R p.We say th a t the input-output decoupling problem (called also 1-0 de

coupling problem or noninteracting problem) is solvable for E if there exists an invertible feedback of the form u = a(x) + f3(x)u such th a t the feedback modified system x = f ( x ) + Y1 T=i w ith y = h(x), where/ = / + got, g = gfiy satisfies

y(kt) _ (br 1 < i < p. (6.2)

for suitable nonnegative integers k i . Observe th a t we assume th a t the inpu tou tpu t m ap of the modified system is linear. Therefore there is no lose of generality in assuming the form (6.2) because if the transfer m atrix of the inpu t-ou tpu t response is diagonal (which is the usual definition of noninteracting) we can always achieve (6.2) by applying a suitable linear feedback.


Fix an initial condition xq X . For each ou tpu t channel we define its relative degree pi , called also characteristic number, to be the smallest integer such th a t for any neighborhood VXQ of xq

L gjL pf - l hi(x) + 0,

for some 1 < j < to and for some x Vxo. By L ph we will m ean the vector of p sm ooth functions whose -entry is L pi h i .

Define the (p x to) decoupling m atrix D ( x ), denoted also by L gL ph , whose (,j)-en try is

L gjL f - l hi(x).

T heorem 6.7 Consider a control affine system S aff-(i) The system Lag is input-output decouplable at xq via an invertible

feedback of the form u = a(x) + 3(x)u i f and only i f

ra,nkD(xo) = p .

(ii) Moreover, for the square system, i.e., m = p, the feedback

u = - ( L gL pf l h ) - l L pf h + [hgL pf l h ) - l u (6.3)

yields y \kt ^ = Ui, where hi = pi, for 1 < i < p.

R em ark 6.8 Inverting formula (6.3), we get the following expression for the new controls

mU i = L pihi + Y ^ U j L gjL pf ^ l hi,

3=1

for 1 < i < m = p. An analogous formula holds also in the non-square case.Indeed, if the system satisfies the decoupling condition rankD(a:o) = P? thenwe can assume after a perm utation of controls, if necessary, th a t the first p columns of the m atrix D ( x o) are independent. Then a decoupling feedback can be taken as

u,i = L f h i + j y jL i u j L g X f ^ h i , for 1 < i < p, ^u,i = Ui, for p + 1 < i < to.

216 W. Respondek

E xam ple 6.9 Consider the following rigid two-link robot m anipulator or, in other words, double pendulum , (see [37], compare also Exam ple 4.7)

i 1 = x 2

x 2 = M ( x 1 ) ^ 1 ( C( x l , x 2) + k ( x 1)) + M ( x l ) ^ l u ,

where x l = 6 = (9i,92)t , x 2 = 6 = {9i,92)t y u = ( u i ,u 2)t . The term k{9) represents the gravitational force and the term C{9 , 9) reflects the centripetal and Coriolis forces, and the positive definite sym m etric m atrix M ( x l ) is given by

m i l 2 + m 2 l2 + m 2 l2 + 2m 2h l 2 cos 92 m 2 l2 + rn2 l i l2 cos 92 m 2 l2 + fn2 h l 2 cos 92 m 2 l2

As the ou tputs we take the cartesian coordinates of the endpoint

Vi = h i (9 i ,9 2) = h s in^ i + l2 sin(9i + 92)V2 = h 2 (9 i,92) = h c o s 9 i + l2 cos(9i + 92).

By a direct com putation we get pi = p2 = 2 and rank D( x) = 2 if and only if h l 2 sin 92 ^ 0. Thus the system is inpu t-ou tpu t decouplable if 92 ^ ki r, th a t is, we have to exclude configurations a t which the two robot arm s are parallel.

E xam ple 6.10 Consider the unicycle (compare Examples 4.11 and 5.6) and assume th a t we observe the x i -cartesian coordinate and the angle 9

i i = ui cos 9 yi = xi x 2 = i sin 9 9 = u 2 y 2 = 9.

The control u 2 has a direct im pact on the second component y 2 of the ou tpu t as well as, through cos 9, on the first component. Thus the system is not inpu t-ou tpu t decoupled b u t it can be decoupled via a static feedback. We obviously have pi = p2 = 1 and the decoupling m atrix is

cos 9 0 \0 1 ) '

Therefore the system is inpu t-ou tpu t decouplable a t all points such th a t 9 ^ | + kn.


E xam ple 6.11 Consider the same dynamics of the unicycle and suppose th a t th is tim e we observe x \ and X2

x \ = u\ cos 9 yi = x \2 = ui sin 9 V2 =9 = 2-

Obviously, we have pi = p2 = 1 bu t this tim e the decoupling m atrix

cos 9 0D = i - a nsin0 u

is of rank one everywhere and thus the system is not 1 - 0 decouplable.

If the system is 1-0 decouplable then it is straightforw ard to calculate V*, the m axim al locally controlled invariant d istribution in ker dh (compare Section 6.2 and, consequently, solve the DD P problem.

Proposition 6.12 Consider the system Edist- Assume that the undisturbed system, that is, when di = 0, for 1 < i < k, is input-output decouplable. Then

(i) V* = V , where V = span {dL^hi, 1 < i < p, 0 < j < pi 1}.(ii) I f moreover, Q C V 1' then the D D P problem is solvable and the feed

back (6.4) simultaneously decouples the disturbances and renders the system input-output decoupled and input-output linear.

E xam ple 6.13 To illustrate th is result let us consider the following model of the unicycle. We suppose th a t the dynamics is affected by a disturbing ro tation (of un unknown varying strength d(t)) and th a t we m easure the angle and the square of the distance from the origin:

1 = i cos 9 + X2 d yi = x f + x \2 = i sin 9 x \ d 9 = U2 y 2 = 9.

The decoupling m atrix is

2a; i cos 9 + 2./'2 sin 9 0D ' 0 1

and is of rank two at any point away from N = { ( x \ , X2 , 9 ) | a;icos0 + a;2 sin 9 = 0} . Notice th a t N consists of points where the direction of the

218 W. Respondek

unicycle is perpendicular to the ray from the origin passing through the center of the unicycle. At points of (M2 x S 1) \ N , the system is inpu tou tpu t decouplable and, moreover, V = span { x \d x \ + x 2d x 2 , dB}. The vector field q = is annihilated by V and thus the feedback1 = (2a;i cos 6 + 2 x 2 sin 0 ) ^ l u \ and 2 = 2 decouples the disturbances from the ou tpu t yielding an 1 - 0 decoupled and 1 - 0 linear system expressed, in (R, (p, 0)-coordinates, where R = r 2 = x 2 + x \ , x \ = r cos (p, X2 = r s in


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[45] W. M. W onham, Linear Multivariable Control: A Geometric Approach, Springer-Verlag, Berlin, 1979.

[46] M. Zribi and J. Chiasson, Exact linearization control of a PM stepper motor, Proc. American Control Conference, P ittsburgh , 1989.

introduction to geometric nonlinear control; linearization, observability, decoupling

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