co-ordinate transformation in back-stepping design for a class of non-linear systems

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, VOL. 9.433-442 (1995)

CO-ORDINATE TRANSFORMATION IN BACK-STEPPING DESIGN FOR A CLASS OF NON-LINEAR SYSTEMS

WE1 ZHAN AND LE YI WANG Department of Electrical and Computer Engineering. Wayne State University, Detroit, MI 48202, U.S.A

SUMMARY In this paper the problem of existence and construction of a co-ordinate transformation is investigated for non-linear systems appearing in feedback linearization and back-stepping adaptive control problems. Conditions are derived to completely characterize the classes of non-linear systems that are transformable to the simple triangular and parametric simple triangular forms via linear transformations. The conditions are shown to be invariant under linear co-ordinate transformations and non-linear feedback. For systems satisfying these conditions, the development of the co-ordinate transformation is shown to be equivalent to that of their first-order approximations and is straightforward.

KEY WORDS: co-ordinate transformation; adaptive control; non-linear systems; feedback linearization

1. INTRODUCTION

This paper is concerned with the problem of co-ordinate transformation for a class of non- linear systems appearing in feedback linearization and back-stepping adaptive control problems. In particular we pursue characterizations of non-linear systems which are transformable to the simple triangular and parametric simple triangular forms via linear co-ordinate transformations, as well as the construction of such transformations.

During the past decade there has been renewed interest in non-linear systems in triangular or parametric strict feedback forms, mainly owing to their accessibility to powerful design tools such as feedback lineari~ation'-~ and the back-stepping adaptive control design methodology introduced in Reference 5. It was shown in Reference 5 that for non-linear systems in the parametric strict feedback form the back-stepping design guarantees global regulation and tracking. The success of the back-stepping design has attracted much attention to adaptive control for non-linear systems. Much effort has since been devoted to extending the main results of Reference 5 to more general situations."'

A very appealing aspect of the back-stepping design methodology is that in principle it can be applied to the class of non-linear systems which are transformable to the parametric strict feedback form via co-ordinate transformations. Certain necessary and sufficient conditions are obtained in Reference 5 to characterize this class of non-linear systems. However, to apply the back-stepping methodology to these systems, one faces very challenging problems. How should the co-ordinate transformation T ( . ) be constructed? When is such a transformation a global diffeomorphism? This is exactly the same difficulty one encounters in feedback linearization. In general the construction of co-ordinate transformations for feedback linearization amounts to

This paper was recornrnended for publication by editor J . P . Norton

CCC 0890-6327/95/050433-10 0 1995 by John Wiley & Sons, Ltd.

Received 7 April I995 Revised 19 May 1995

434 W. ZHAN AND L. Y. WANG

solving partial differential equations, which can be extremely difficult. Without the explicit construction of the co-ordinate transformation the back-stepping methodology is then not applicable to such systems. To avoid this difficulty, Set0 et aL6 generalized the back-stepping methodology to triangular systems, for which the co-ordinate transformations mapping the non- linear system to the parametric pure feedback or the parametric strict feedback form can be easily constructed.

Motivated by these developments, we study in this paper the problem of the construction of co-ordinate transformations which map non-linear systems to the parametric strict feedback form. Instead of pursuing general solutions, we concentrate on the characterization of non- linear systems which are transformable to the parametric strict feedback form via linear co- ordinate transformations. Conditions are derived to completely characterize such non-linear systems. These conditions are shown to be invariant under linear co-ordinate transformation and non-linear feedback. An algorithm for the construction of the co-ordinate transformation is developed and demonstrated with an example. The algorithm can be applied to all non-linear systems satisfying the conditions.

The paper is organized as follows. In Section 2 we introduce the notation used and formally state the problem studied. The main results are presented in Section 3. First, conditions are derived to characterize non-linear systems transformable to the simple triangular form (Theorem 1 ) . Then the result of Theorem 1 is extended to the case of the parametric simple triangular form (Theorem 2 ) . The conditions in Theorems 1 and 2 are shown to be invariant under linear co-ordinate transformations and non-linear feedback (Proposition 1). An example is given in Section 4 to demonstrate the construction of the transformation that maps the non- linear system into the parametric strict feedback form. Finally, some conclusions are drawn in Section 5 .

2 . PRELIMINARIES

2.1. Notatioii

In this paper we will generally follow the notation used in Reference 4. R" denotes the n- dimensional Euclidean space. For a vector v we denote its transpose by vT. A non-linear mapping is said to be C' or smooth if every component of the mapping is continuously differentiable up to any order. A mapping of the form

is referred to as a vector field. The Lie derivative of a vector field f with respect to a function h is defined to be

ax, ax* ax,, The Lie bracket of two vector fields f(x) and g(x>, denoted by [ f, g], can be calculated in local co-ordinates as

CO-ORDINATE TRANSFORMATION IN BACK-STEPPING DESIGN 435

where df/dx and dg/dx are the Jacobian matrices of the mappings f and g respectively. We use C"span( v l , v2, ..., v i ) to denote the set of vector fields which can be written as linear combinations of the vector fields v l , v2, . . . , u; , with the coefficients being C" functions.

2.2. Problem formulation

A non-linear system is said to be in the parametric strict feedback form' if it is expressed as

where z = (z , ,z2, ..., z,)'E R " and y j , j = O , ..., n , and are C" functions, with yO(0) = y , (0) = = y,(O) = 0. Also /!?o(Z) # 0, Vz E R". Systems in this form are feedback- linearizable. In the case of unknown parameters 8 an adaptive control methodology, namely the back-stepping design, has been developed, (see e.g. Reference 5 for details). The back-stepping design is closely related to the feedback linearization problem, where the parameter 8 is known. In particular, when 8 = 0, the system (1) is a special case of triangular systems, I for which feedback linearization can be easily achieved.

Owing to their accessibility to the powerful design tools for non-linear systems, such as feedback linearization as well as the back-stepping methodology in adaptive control, it is of great importance to characterize non-linear systems which are transformable to either triangular systems or parametric strict feedback forms.

As was shown in Reference 5, under certain conditions, non-linear systems of the form

can be transformed into the parametric strict feedback form via a parameter-independent diffeomorphism z = @ ( x ) . It was also proved that if the system (2) is transformable to the parametric strict feedback form, then g i ( x ) = O for 1 c i c p . For this reason, without loss of generality, we will concentrate on the system

While the existence of a diffeomorphsm which transforms the system (2) into the parametric strict feedback form has been established,' its construction remains a challenging open problem. In general the construction amounts to solving certain partial differential equations, which can be extremely difficult. In this paper we will investigate the problem of characterizing the class of non- linear systems which are transformable to the simple triangular or the parametric simple triangular form via linear co-ordinate transformation and constructing the transformations when they exist.

Consider the non-linear system P

.f = f o ( x ) + C e,f;(x) + bu (4) i= I

where x ( t ) E [w" is the state, u ( t ) E R1 is the control, 8 = [ O , , 02, ..., 0,1] is the vector of unknown constant parameters and f i (x ) , 0 d i 6 p , are smooth vector fields, with f,(O) = 0 for 0 s id p. The main constraint here is that the vector field go(x) = b is constant. It will be shown later that for the method developed in this paper, the assumption go(x) = b is in fact necessary. The following problems are to


be investigated. Under what conditions can the system (4) be transformed into the parametric strict feedback form (l)? How should such transformations be constructed when they exist?

3. MAIN RESULTS The back-stepping design can be easily modified to be applicable to systems of the more general form

ii = z i + l + di(zl , . .., z i ) + e T Y i ( Z I , .. . , z i ) , i = 1, .. ., n - 1 (5 )

2" = YO(Z) + eTy,(z) + Bo(z)u

where d i ( z l , . . . , z i ) , i = 1, . . ., n - 1 , are C- functions. One can modify the back-stepping design procedure by taking into account the extra terms di(zl , ..., 2;). An alternative is to apply a non- linear coordinate transformation to transform the system into the parametric strict feedback form before the back-stepping design is applied. Specifically, we define the non-linear co- ordinate transformation

W I = ZI

~ ~ = ~ ~ - ~ l 8 - o , i = 2 , 3 ,..., n

It can be readily verified that the above non-linear mapping is a diffeomorphism and, in w-co- ordinates, the system ( 5 ) takes the parametric strict feedback form. It is worth noting that both the co-ordinate transformation and the inverse transformation can be constructed explicitly. Therefore a non-linear system is transformable to the parametric strict feedback form if and only if the system is transformable to the form (5). Moreover, the corresponding co-ordinate transformation is given by (6). It turns out that it is much simpler to study systems of the form (5 ) , which we shall refer to as the parametric simple triangular form. When 0 = 0, the system ( 5 ) is said to be in the simple triangular form.

We start with the feedback linearization problem for the system (4) with 0; = 0, 1 6 i 6 p , i.e.

i = fo (x ) + bu (7) The Jacobian matrix of f o evaluated at x = 0 is denoted by

A,= -

Theorem 1

Suppose (Ao, b ) is controllable. Then there exists a linear co-ordinate transformation which transforms the system (7) into the simple triangular form under the new co-ordinate system if and only if there exist C- functions k i I ( x ) such that

i

[ f o ( x ) , A b b l = - A b " b + C k i I ( x ) A : , b , i = O , 1 , ..., n - 2 (8) I - 0

Proof. First we demonstrate that the condition (8) is invariant under linear co-ordinate transformations.

Let T be any non-singular matrix which defines a co-ordinate transformation z = Tx. We would like to show that the system (7) satisfies the condition (8) under the new co-ordinate system if and only if it satisfies the condition (8) under the old co-ordinate system.

CO-ORDINATE TRANSFORMATION IN BACK-STEPPING DESIGN

In z-co-ordinates the system takes the form

i = T f o v - ' z ) + Tbu

We introduce the notation

A,= TAoT- ' , 6=Tb , A&) = Tfov-'z) Since T is invertible, it suffices to show that (8) implies that there exist some C" functions & ( z ) such that the following holds:

437

(9)

(10) 1-0

We will prove (10) by induction. For i = 0 we have

Suppose (10) holds for i = j - 1 s n - 3. Then

= -T -

=-(TAoT-') '+'Tb + k;.,(z)Tb + k;.,(z)(TA,T-')Tb +

A i b = - T ( A c ' b + k;.,(z)b + k;.,(z)Aob + + k;.i(z)A', b)

+ k;,(z)(TA,T-')'Tb i

= -A< I 6 + c 4/ (z)AL6 / - 0

Therefore, by the induction principle, (10) holds for i = 0, 1, .. . , n - 2. Now we prove the necessity. Suppose there exists a non-singular matrix T such that after the

co-ordinate transformation z = Tx the system is in the simple triangular form, i.e. it takes the form ( 5 ) with 8 = 0. To verify that (10) is satisfied, we first calculate A, A, and 6. Since & ( z ) takes the special form


- - x 1 0 ... 0 x x 1 ... 0

x x x ... 1

. . . . . . . . .

x x x ... x -

where x denotes some unspecified constants. satisfied in z-co-ordinates.

..

It is easy to verify that the condition (10) is

To prove the sufficiency, we first apply a linear co-ordinate transformation To such that the linear system (ToAnT,', Tnb) is in the simple triangular form. Such a linear transformation always exists, since (An, b) is controllable. Now define z = Tox. In the new co-ordinates the matrix A, = T,A,T,' and the vector 6= Tnb have the forms

where a, and b, are non-zero constants and x denotes some unspecified constants. Now it can be readily verified that the condition (10) forces &(z) to have the simple triangular structure. Therefore the co-ordinate transformation z = Tnx transforms the system (7) into the simple triangular form. 0

Remark. It is interesting to observe from the proof that if there exists a linear co-ordinate transformation which maps the system (7) into the simple triangular form, then any linear co- ordinate transformation which maps the first-order approximation of the system into the simple triangular form will map the non-linear system itself into the simple triangular form. It is well- known that systems in the simple triangular form are feedback-linearizable and the non-linear co-ordinate transformation can be easily constructed. Thus Theorem 1 gives a very interesting result on feedback linearization for systems of the form (7).

Now we turn to the construction of a co-ordinate transformation in order to apply the back- stepping design to the system (4).

Theorem 2

Suppose (Ao, b) is controllable. Then there exists a linear co-ordinate transformation z = Tx which transforms the system (4) into the parametric simple triangular form ( 5 ) if and only if


(8) and the following condition hold:

Lf;(x),A',b] E C"span{b,A,b, ..., A',b], i = 1, ..., p , j = O , ..., n - 2 (11)

Proof. The proof is just a slightly modified version of the proof for Theorem 1 and will be sketched here only briefly. First, using induction, it can be readily shown that the condition (1 1) is also invariant under linear co-ordinate transformation. As a result, to prove necessity, we only need to verify, in addition to the condition (8), that in the new co-ordinates (where (Ao, 6) is in the simple triangular form), (1 1) is satisfied. For sufficiency we need to check that

?

where x denotes some unspecified C" functions of z. This will imply that the vector fields f;(z), i = 1, . . . , p , have the required form. The verification of these conclusions is

0

Now let us consider the system (3) where the vector field g, (x ) is not necessarily a constant vector field. Suppose there exists a linear co-ordinate transformation z = T x such that in the new co-ordinates z the system is in the parametric simple triangular form. Then we must have

straightforward and is omitted for brevity.

Tgo(T-'z) = B ( Z ) e n

where en is the unit vector (O,O, ..., 0, and B(z) is a scalar function. This implies that the vector field y o ( x ) must have a constant direction. The scalar ,8(z) can be incorporated into the input by the transformation v = B(z)u. Since B(z)+O, the input transformation is invertible. Therefore up to the above simple input transformation the conditions in Theorem 2 characterize the class of non-linear systems which are transformable to the parametric simple triangular form via linear co-ordinate transformations.

It should be emphasized here that to apply the results in Theorem 2, it is not necessary to check the conditions (8) and (1 1) in x-co-ordinates. From the proofs of Theorems 1 and 2 the conditions are invariant under linear co-ordinate transformations. It is much easier to check these conditions in a co-ordinate system under which the pair (Ao, 6) is in the simple triangular form. Therefore, instead of checking the conditions (8) and (11) first and then trying to construct the linear co-ordinate transformation when the conditions are satisfied, we first construct the linear co-ordinate transformation which transforms (Ao, b ) into the simple triangular form. Then we apply this co-ordinate transformation to the given system. If the system is transformed into the parametric simple triangular form, then we are done. Otherwise we conclude that the system cannot be transformed into the parametric simple triangular form via any linear co-ordinate transformation. The following proposition summarizes this idea and extends the results in Theorem 2 to the case where some non-linear feedback is also allowed.

Proposition I Suppose (A,,, b ) is controllable. Then the system (4) can be transformed into the parametric

simple triangular form via the linear co-ordinate transformation z = Sx and non-linear feedback u = a(x ) + v, where a ( x ) is a C" function, if and only if the system (4) takes the parametric


simple triangular form in w-co-ordinates, where

w = sox, so = [SI, s2, . .. , s,] - I

s, = b, s,-~ = Abb + u,Ag-'b + + u,b, i = 1, ..., n - 1

and the ai are the coefficients in the characteristic polynomial of A,, i.e.

I I1 - A0 I = 1" + u 11 + * * * U,

Proof. It is a well-known result in linear system theory that if (Ao, b) is controllable, then So transforms (Ao, b) into the simple triangular form. Since the conditions (8) and (1 1) are invariant under linear co-ordinate transformations and since feedback and co-ordinate transformations are commutative, it suffices to show that these conditions are also invariant under the non-linear feedback u = a(x) + v. Define f , ( x ) = f o ( x ) + ba(x) and A, = A. + b[aa(x)/ax] I +-,,. Noticing that the feedback is invertible, we need to show only that

i

~ ~ ( x ) , A b b ] = - A ~ ' b + C Z , ( x ) A : , b , i = O , l , ..., n - 2 (12) 1-0

and

If i (x) ,di ,b]€C~span(b,Aob ,..., A',b], i = l , ..., p , j = O ,..., n - 2 (13)

hold when the conditions (8) and (1 1) hold. Indeed, for i = 0,

ui, b] = [f+ ba, bl = [f, bl + [ba, b ] = [f, b] + Lba. b = -Aob + (koob + &a)b = -A,b + Loo b

Thus (12) holds for i = 0. Suppose (12) holds for i - 1 ; then

vo ,A;b]= If+ ab,Abb]+ c(x)b

Since

where cI is a constant, we have

By induction, (12) holds. The equivalence of (11) and (13) is straightforward owing to the relationship

If;, Lob] = If;, Aib] + C I If;, AA-'b] + + V;., b] Therefore Proposition 1 follows. 0

Remark. In the above theorem we use the feedback u = a ( x ) + v instead of u = a ( x ) + B (x)v. The scalar /? ( x ) can always be incorporated into the input when B ( x ) # 0.


4. ANEXAMF'LE

We illustrate our results by the following example:

i, = x2 + (x, - x ~ ) ~ + ( x , - x213 + e [ X , + ( x , - x213] + u +X:U

i2 = (x, - x2)3 + ex2 + u +& First we note that the system is not in the parametric strict feedback form. To apply our results, we introduce the input transformation v = (1 + x:)u to get a constant go(x) . A, and b can be easily computed as

0 1 0]' b = [ ; ]

Clearly (A,, b) is controllable. To verify the conditions (8) and ( l l ) , we calculate the following:

As a result, all conditions in Theorem 2 are satisfied. Therefore there exists a linear co- ordinate transformation So which transforms the system into the parametric simple triangular form. From Proposition 1 we conclude that the verification of the conditions (8) and (11) is not necessary. All we need to do is to find a linear co-ordinate transformation which transforms the pair (Ao, b) into the simple triangular form. Then the original non-linear system is transformable to the parametric simple triangular form via linear co-ordinate transformation if and only if it is transformed into the parametric simple triangular form via So. Now define

S o = [ A , b + ~ , b , b ] - '

where a , is the coefficient in the characteristic polynomial of the matrix A,, i.e.

det(iE1 -Ao) = ,I2+ a l l + a,

Clearly for this example we have a, =a2 = 0. Hence the co-ordinate transformation can be chosen as z = Sox, where

i.e.

z , = X I - X 2 , 22 = X2

In the new co-ordinates the system has the form

i, = z2 + z:+ e(z, + z;), i2 = z:+ ez2 + Apparently the non-linear co-ordinate transformation

w , = z , , w2 = z :+ z2


will transform the system into the parametric strict feedback form and the back-stepping methodology can then be applied.

5 . CONCLUSIONS

The problem of existence and construction of co-ordinate transformations is studied for applying the back-stepping methodology to feedback-linearizable non-linear systems. Classes of non-linear systems which are transformable via linear co-ordinate transformations to the simple triangular or parametric simple triangular form are completely characterized by certain conditions. The construction of such transformations for non-linear systems is shown to be equivalent to that of their first-order approximations and hence is straightforward. A three-step design procedure is proposed for adaptive control: (i) use a linear co-ordinate transformation to transform the system into the parametric simple triangular form; (ii) use a non-linear co- ordinate transformation to transform the system in the parametric simple triangular form into the parametric strict feedback form; (iii) apply the back-stepping design to the resulting system. This has been demonstrated by an example (where step (iii) was omitted). The first two steps also give a simple method for the construction of the co-ordinate transformation in feedback linearization. Extensions of the main results to other types of non-linear systems are under further investigation.

ACKNOWLEDGEMENTS

The authors would like to thank Professor Jing Sun and Professor Yong Liu for helpful discussions.

This research is supported in part by the National Science Foundation under grant ECS- 9209001.

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co-ordinate transformation in back-stepping design for a class of non-linear systems

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