On Feedback Linearization of Nonlinear SISO Systems

Copyright © ) 996 IFAC 13th Triennial World Ct)ngrc"", San Pmnclsco, USA

21>-084

ON FEEDBACK LINEARlZATION OF NONLINEAR SISO SYSTEMS A. Astolfl and L. Del Re Automatic Control Laborotory, Swiss FederalI""titute of Technology (ETH) CH-B092 Zurich, Switzerland, FAX: +41-1- 63212 JI , E-mail: astoIIW aut.ee.e thz.ch

Abstract. This work discusses the state space exact linearization problelns, both locally and globally, for single-input single-output nonlinear systems. First it is shown that if the input vector field has a particular structure I ;, e. if it is a con:-;tant vector field , t hen local or global feedback Iinearizability can be determined by inspection. Then it is shown that, lUlder mild a..~sumptionsl there always exists a suitable (local) coordinates transformation which transforms a general nonlinear plant into a system with constant input vector. Using these two facts , simple conditiclns for local and global feedback linearizabilit.y of SISO Ilonlinear systems are derived. Finally, an explicit formula to locally rectify an affine vector field is presented. Keywords. !\onlinf'.ar control systems, Bilinear systems, Linearization. I. INTRODCCTIOI\

In this paper, wc r.onsider a Ilonlinear plant, with state x defined in a neighborhood X of the origin in lR n and control input u E JR, modeled by equations of the form :i:

= f (x) + y(x)" .

and be = ( 0 . ,. 0 I ] T . In I he second step, the control (1)

We assume tbat the vector fields I (x) a nd y(",) are smooth vector fie lds defined 011 X, that x = 0 is an equilibrium point. of the system, i. e. 1 (0) = 0, and that the distrihution span {y (x) } is nonsingular in a neighhorhood of x = 0, i. e. y(O) O. We also assume t hroughout t his paper that the linear approximation of (1) at x 0 is controllable . Feedback tinearization (Isidori, 1989) is a well known technique to force a nonlinear plant to behave like a linear one, something which may be very useful for control tasks in which t.he predictability of the behavior of the controlled system is import.ant. (Del Re and Isidori, 1995). It can be described as a two steps procedure. In the first one, wc determine a local diffeolllorphism z 4>(x) defined in a neighborhood U of the origin ) such that , in the npw coord inatf~S, the original plant is described by equations of the form (normal form)

t-

=

=

law" = u°Pc~I" is a pplied to !be system (2), yielding a new system described hy equations of the form = AbZ + bCu Cl i.e. a linear and controllable system wi th state Z, whose dynamic :~ can be arbitrarily assigned through a suitable choice of u c .

z

While it is possible to confirm or exclude the existence of feedbar.k linearizing control laws by testing some propertie. of t be vector fields I (.c) and y( x) (see (Ha user et al., 1992) for an interesting case study); unfortunat.ely, there. is no general systematic approach to constnlct analytically the local diffeomorphism 1>(x), hence the feedback linearizing control. In practice, it is CIL'"itomary to approach the problem by trial and error , llsing the notion of relative degree (see (Isidori, 1989) for a precise definition). In t he case of single-input single-output systems, it is well known (Isidori, 1989, Lemma 4.2.4) t.hat t he pla nt. (1) is feedback linearizable in a neigbborhood U of x = 0 if and only if there exists a real-valued function hex), defined on U, such that the system :i;

(2)

with iJ(z)

i

0 for all z in a neighlMlrhood of z = 0,

y

= 1(') + y(x)u = hex)

(3)

has relative degree" at x = O. Without loss of generality, we shall assume that. h(O) = O. Tbe knowledge of the

2144

function h(x) is sufficient to design the corresponding linearizing contra) law, aud is equivalent to the knowledge of the local diffeomorphism ",(x) requested in the other approacli, but it~ search is much easier, as it is better suited to a trial and error approal~h. However, the determination of such a function remains a not trivial task, as it involves the solution of a system of partial differential equations. From the above discussion , we conclude that, even if t.here exists a complete characterization for the problem of feedback linearizatioll, i. c. necessary and sufficient conditions, the explici t computation of a feedback Unearizing control law is mostly quite involved and therefore it is feasible only in special situations. In this work we approach the problem by first restricti ng our attention to a particular class of single-input singleoutput systems, described by equations of the form i:

= f (x) + bcu ,

Then there exists a (local) coordinates transformation such that" in the new coordinates, the system is described by equations of the form (4).

Proof The result is a simpl" ronsequence of Frobenius Theorem (Isirlori, 1989), keeping in mind that g(x) is a nonsingular, one dimensional distribution and therefore is always (locally) integrabl... As we have assumed that the linear approximation at x 0 of system (4) is controllable, it is always possible to find a linear coordinates transformation and a linear feedback law 1 i. e. a linear fet·dback transformation, such that, ill the new coordinates, the system is described by equations of the form

=

(6)

(4)

where h(x) is zero, at x with f( O) = 0, and we show that , for SUGh a class of systems, the state space exact linea.cization problem (".an be easily solved by inspection. Then we show that every single-input single-output systern described by equatiolL' of the form (1) and with g(O) " 0
= 1, + Sx,

(5)

i.e. the input vector g(x) is affine in the state variable, we provide explidtly die coordinates change which transforms the vp.\.t.or fiehi (5) iuto tIle vec.t or field bc . As a Lyprod uct wc obtain (constructive) conditions to solve the state spaee exact lin ~a ri zati on problem for bilinear systems.

Remark 1. It must be noticed that the state spar,e exact linearization problem for bilinear system has already been addressed in (Del Re and Gu.zella, 1994), for the r..a.se of dyadi c systems, whereas a first analysis of the general case was performed in (Celikovsky, 1990; Celikovsky, 1992). However, therein the author considered only the problem of state equivultmce, while in this note we are iuterestetJ in a prohlem of feedback equivalence.

2. ON LOCA L FEEDBACK LINEARIZABILITY

"0.

0, together with its first

Lemma 2. Consider a nonlillear SISO system described by equations of the form (6) and an output funct ion y = h(x) such t hat span{d"} is nonsingular at x O. AsstUne that the system has a well defined relative degree at x = O.

=

Then t he system can have rdative degree exactly n only if the output map is any smooth function h(x) such that dh(O)

= [1.1

0 ... 0] ,

(7)

... * 0].

(8)

with hi " 0, and dh(x) = [ *

Proof. By nonsingularity of span{d"} at x write h(x) c'" + h, (x)

= 0 we can

=

where c E III xn is a nonzerO covector and h2(X) is zero, 0, together with its first order partial derivatives. Straightforward calculations show that

at x ::::-

(9) when~ (}1 (x) is at least linear in x . As the system posse;ses a weU defined relative degree the firSt function of the sequence

In order to establish t.he results of this paper, we first lI~d two lemmas.

Lemma 1. Consider a nonlinear SISO syst.em described by equations of the form (J). Assume t.hat x ::::- 0 is an equilibrilUn, i.e. f rO) = 0 , anrl t hat g(O)

,~

orde-..I' partial derivatives.

Lgh(x) , LgLth(x) , ... , LgL'h(x), ... which is not identically zern (in a neighborhood of X == 0), say the i-th term of the sequence, must be such that

CA;-'b, " o.

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Hence, we conclude that system (6) can have relative degree equal to n only if

=

cA~b, 0 cAb'-' bc ~ o.

the transformation is immediately more involved if g(x) is affine, i.e.

Ifk < n - 1

From the structure of A. and be we deduce that the system can have relative degree equal to n only if c has the form c [Cl 0 ...

01

=

with C, ~ 0 by nOIlsingularity of ,paIl{dh} at x = o. Finally, by feedback linearizability we have £,h(x) = 0, which implies (8). " 'e can now state the following result.

y(x)

Lemma 3. Consider a (affin.,) vector field described by equations of the form (10) '\Dd assume that g(O) ~ 0, i. e. b is a nonzero vector. Then it is always possible to find explicitly n smooth functions A,(x) , A2(x) , ... , .In(x) such that (locally)

8.\ ,(x) ( x ) =, 0

a;;-g

fi(x)

,p2 (x" "2, X3)

j,(x)=

=

I; - '(x)

H(x) where th~
, :1:2 , .

cpn-l (J;l , X2! .. " Xn) dln(xJ, X2,"'! Xn)

. ,xd

are smooth functions of

Moreover: one output function for which the system has relative degree exactly n is y = X l . Proof. It is a matt~..r of straightforward computation to show that the system (6) with hex) in triangular form and with output y = Xl has relative degree equal to n; hence the result follows from the if part of (Isidori , 1989, Lemma 4.2.4).

Proof For simplicity we asS' ,me that the matrix N has simple eigenvaluesJ i .e. characteristic polynomial a nd minimal polynomial coincidp.s. The general case can be dealt with in the same manner, but more complicated formulae must be worked out. To have an idea of the general case, the reader is referred to the first example in Sect.ion 6.

=

Consider a linear coordina1'es transformation z Tx such that the matrix TNT - I is diagonal (this is always possible if the eigenvalues of" are simple) . In the new coordinates the vector field (10) is described hy equations of t.he form

with

0

3. ON THE TRANSFORMATION OF G(X)

=

ve


b;

Theorem 1 provides a simple test to check the (local) feedback Iinearizability of a nonlinf'ar syst.p.m described by equation of th e form (6). As already discussed any nonlinear system with equilibrium x 0 and controllable linear approximation at x = 0 can be transformed , via nonlinear coordinates transformation, into a system described hy equations of ttif' form (6). Such a transformation always exi~ts, as explainf'd in Lemma I! howvery difficult or ever its explicit computation may even impossible. Obviously, t.he t.r:msformation is trivial if g(x) is constant or if y(x) has only one entry which is not identicaJly zero. However, the problem of compnting

Vi

8A n ("') y( x ) =, l. 8x

The system is feed hack linearizable if the vector field hex) has a triangular form, i.e. ha.< the following form

,p1(X,,:<.)

(10)

with b E /Rn and N E JR:'''''. Nevertheless, also ill this case it is possible to find explicitly the desired transformation, as explained in the following result.

Theore.m 1. Consider a non linear system describerl by equations of the form (6).

f~(x)

= Id Nx,

[Ji

and

il =

1-'2

[1

0

. .. :

,i 0 .

.

~n

In what follows we assume that for all i == I",,} n at least one between bt and /Ji is nonzero. If this is not the case for some i, the problem of finding n functions which fulfiD the set of equations ( 11) can be reduced. In fact, if for a certain i both bi ru Id Jli are zero, the function .\,(.) ='i is such that dA ,g(z) = O.

b is nOnzero. 'VVe have to consider two (".ases: 1) there exists all index i such t hat b, '" 0 and I-'i = 0; 2) t here exists no index i such that bi I- 0 and Pi O. Case 1. Wit.hout lack of generality assume i == n . Then the n im.tependent functions Dy hypothesis} at least OIle entry of the vector

=

2146

is defined and the origin of the coordinates system is exactly e
transformation carrying the vector (12) to the constant vector be such a distance is not bigger.

~, ) =exp (- ••• (b, + I',x,) - b,

(13)

4. FEEDBACK LINEARIZATION OF SYSTEMS ,VITH AFFIKE INPUT VECTOR The results in Sections 2 and 3 leJld themselves to establish simple conditions for the feedback Iinearizability of nonHncar systems with affine input vector I i.e. of systerns deseri beel hy equations of the form

fulfill the set of conditions (11).

Case 2. Without lack of generality assume bn of o. Then the n incip.pendent functions

+ I',Z, (b n + ll.n Zn) "'" 1"

b2 + 1'2 Z 2 ~

+C2

(H)

hn - 1 + 11 rl-1 Z n-l I (f)n + J.'nzn)"""'-;;An{Z)

;:::

1'"

Proposition 1. Consider a non linear system described by equations of the form (15) . Assume that. x = 0 is an equilibriwlI point, i.e. !(O) 0 and that. it.s linear Apply a nonlinear coordinates transformation such that, in the new coordinates, the I"iystem is described by equations of the form

(16) +Cn_1

log(b,! + P-n=/J) +C n I'n

where the ca's an~ such that A;(O) = 0, i.e. the coor~ dinates transfonnation maintains tile equilihrium, fulfil I 'he set of conditions (11). Remark 2.

It must be noted that the equations (13) de-

fine a global diffeomorphism, whereas the equations (14) define only a local cliffeomorphism. It is possible to give

where ]2«) is zero, at ( = O. together with its first order partial derivatives.

Then the system (15) is feedback Iinearizable if the vector field ],«) has a triangular form . As a direct consequence of the previous Proposition it is possible to derive necessary and sufficient condit.ions for

t.he feedback linearizability of general bilinear systems, i.e. of syste.lIls described by equations of the form

:i:=Ax+(b+Nx)u.

a simple geomet.ric interpretation of the two considered

cases. Case 1 arises if b rt span{N}. or what is the same if the equation ;; + N x = 0 has 110 solution; whereas Case 2 arises if bE span{N}. Moreover, in this second case, even if it is not. possible to define a global diffeomorphism, it is possible to maximize the region where the transformation is defined. For. let (}:i =

ll,

I

/'.i

I

and let k be such that 01;

(15)

approximation , at x = 0 , is controllable.

(b n + Jlnzn) 1'.,

_

= !(x) + (b + Nz)".

=

= -'_0-'---"'-'-'-'"",,,,-" + c, _

:i:

(17)

As a matter of fact the following result holds.

Proposition 2. Consider a bilinear system described by equations of the form (17). Assume that the pair {A, b} is controllable.

Apply the nonlinear coordinates transformation (13) or (14) and a linear coordinat.", transformation such that, in the new coordinates, the :iystem is described by equations of the form

~ O'i

(18)

for a.ll i = 1,"" n. Suppose, without lack of generality, k = n and observe that the distance between the

where ]2 {() is zero, at (

boundary of the region wherc thc transformation (14)

partial derivat.ives.

2147

= 0, together with its first order

Then the system (17) is feedback linearizable if the vector field j,(,) has a triangular form.

where the ek(Xl ,'" ,Xk) are smooth functions of their argwnents defined on ./Rn . HpJlce

L 9 L''' (x) for all

5. FURTHER RESULTS

=0

E fRn and for all k = 1, ·· ·,n -2; and

I

n-l

In the present Section we discuss under what extent the conditions of Theore.m 1 and Proposition 1 must be strengthened to ~olve the state space f''xact linearization problem globally. The main result of this Section, which provides a ~l1fficient condition for the solution of the global exact linearization problem , is contained ill the following statement.

The07-em 2. Consider a non linear system described by equations of the form

L,L'j- lh(x)

= IT (1 + '(Xl,X2 ,···,I,) ) 10 i= l

for all x E R n; which prov", the claim. Consider now the coordinates transformation

z,

h(x)

=

==

Z, =

Lfh(x)

= x, (1 + 'P'(X l1 )

+ O'(I,)

Z3

L}h(x)

= X3(1+'01(X l1 )

(1 + 2(X1,X

h(x)

= [fi(x)

= L'j- 'h(x) = Xn IT (1+'(Xl,X2,·· · ,X,))+ i= l

8n(XbX~,···,

tions of the form

+ ~; ' (x,) + ",'(x"x,) i n- 1

xnljJn - l (Xl, X2, .. -, Xll, - l)

xn-d

and observe that it defines a global diffeomorphism. In the new coordiuates the Systf!ffi is described by equa-

fi(x) ... I;' - I(x) f2'(x)]" =

X,'(I1,X2)

+

,.-1

Zn

Then the system is globally feedback linearizable if the vector field "(x) has a strict triangular form, i.e. has the following form

2 ))

03(XI,X" )

(19) in which J,(x) is a smooth vect.or field defined on JRn which is zero, at :1: = 0, tugether with its first order partial derivatives.

Xl

+ Wn -

ypfl(X l' X'l , ' . "

l

(Xl, X2, " ', Xn_l)

in

X'I - J, Xn)

=

2n

-' (1+ '(x1(z) , x,(z),··· , x,(z)) ) IL = b(z) + 0IT l= 1

where the 4>i(XI' X2, . ··,:r.d are smooth flIDctions oftheir arguments defined on lRn such that

1/>,(I1 , I,,· .. , x;)

with b(z) smoot.h nmction. Hence, it is globally feedback lineariza ule.

+ 1 10

for all x E fRn, and the ",'(Xl, Xz,· . .. x,) are also smooth functions of their arguments defined On Rn.

6. SOME EXAMPLES

Moreover, one output function for which the system Las uniform relative degree exact ly n is

In this Section we present some interesting examples illustrating the theoretical ""ults discussed in the pre-

Y ==

Xl·

(20)

vious Sections.

Example 1. Consid_x the vector field P,voj. We first prove that the systE,m (19) with J,(x) in strict triangular form and with the output (20) has uniform relative degree equal to n. For, observe that, for all k:::;: 1, '" , n - lone ha..;;

L}h(x)

= "'HI

II• (1 +
ek(Xl,X2 , ···,X"J,

g(x) =

[

b1] b, b,

b.

+

[1'1 1 0 1'1

0]

o0 o

and assume that at least om' bi is nonzero J e.g. 1'1 I 0 and 1'2 10.

2148

(21)

0 0 1'2 0 0 0 0 ~

:F

0,

we conclude that. the feedback linearizability of the considered system can be decided, by inspection, and without any c,olIlput.ation. For example, if h (x) is any function of the form

Then the four smooth functions

x. -

b, + IJI X, +

~W=

b,+ /11X,

In(h. +!1-1 Xt) !1-1

~~'-'-(b"''C-+,--,--IJ.:..l : "'--='''-) x; -In --+C2 !1-1 b; { x,

A2(x) =

A3(X) =.p( (b" +1""'3) (b. +/iJ

In(b, +!1-l xt)

\()

A4,X==

/i l

+~

if b"i' 0

X.)-;:;)

=0

i = 1, 2,3

Observe that, as Jll oF 0, the transformation is defined in a complete suospace of F, namely in the subspace

I x, < -

u=

1

{x E JR

if b, < 0 /i l

1>2 1'1

I X2 > - b, -

4

if

~

>0.

1'1

1'1

Example 2. Consider the nonlinear

x=

Assume that 1;(0) = 0 for i = 1,2,3 and that its linear approximation at x = 0 is controllable, Applying the coordinates transformation

= XI

x:~

1 + Xz

Z3

= In (I + x ,)

we obtain

i

=

[~:~;~ ] + [~J /3( z )

u.

From Theorem 1 we infer that the system is locally feeubac.k lineari.able if =

aZ

-

3

8. REFERENCES Celikovsky, S. (1990). On the global Iinearization of bilinear systems. S.v st.ms fj Control utter" 15(5), 433-439. Celikovsky, S, (1992). On the global linearization of non homogeneous bilinear systems. Systems f:j Control utters 18(5), 397-402. Del Re, L. and A. Isidori (1995), Performance enhancement of nonlinear drives by feedback linearization of linear-bilinear cascade models . IEEE T'1Unsaction,~ on Control Systems Technology

3,299-308.

I

ai,(·)

In this paper a simple appl'Oach to the design of feedback linearizing control law, for nOIllinear SISO systerus has been presented. In particular, it has been shown that the feedbarl< lineanzahility of a general nonlinear system with constant input vector may be decided by illspection. Moreover 1 the problem of integrating a onedimensiollal, nonsingular and affine distribution has also been addressed and explicitlv solved. Using these results the prohlem of exact Jinearization of nonlinear systems with affine input vector havl~ been considered. As all interesting application, the case of bilinear systems has also been disclIs..~. Moreover! a sufficient condition to solve the glohal exact lillea.rjzation problem for systems with coustant input vector has ahlO been derived. Finally, it must be noted thht the considered approach can be ex tended w MIMO ,ystems.

~yst.em

[j:~:l ] + [11."],,. j,(x) x, Zl

1

7. CONCLUSIONS

+C4

aA,(x) ( ) _ 1 ax 9 x - ~

{x E Hr'

(Xl, :3",,)

+C3

=

a",

II

the system is feedback lin"arizable (provided that its linear approximation is comrollable).

where Cl, C:2, C:l and C4 are COIl~tants such that Ai(O) 0, and i",&(z) is any smooth function of its argulUent such that .p(O) = 0 and .p,(O) ¥ 0, are ' \lch that aA; (x) g(x )

h(x) =

ifb,=O

0

Del Re, L. and L, Guzzella (1994). Stability of feedback linearizatioIl of bilincar plants with bounded inputs. Kybernetika 30(2), 141- 152. Hauser, J" S, Sastry and P. Kokotovic (1992). Noulinear control via approximate input.output linearization: The ball and beam exam ple. IEEE Transactions on

Automatic CnntroI37(3 ), 392-398.

.

Finally, noting that

Isidori, A. (1989) . Nonlinea>' control systems, 2nd ed .. Springer Verlag.

il(z) = It (x(Z) ) = I, ( zl ,exP (Z3 ) -1, Z2exP(Z3)),

2149