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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 2003

2191

Technical Notes and Correspondence_______________________________ Direct Transformation of Nonlinear Systems Into State Affine MISO Form for Observer Design I. Souleiman, A. Glumineau, and G. Schreier

Abstract—This paper gives necessary and sufficient conditions to solve in a constructive way the transformation of a general multiple-input–singleoutput nonlinear system into a state affine system. This result is direct i.e., without solving the state elimination problem when computing the inputoutput differential equation. A Kalman-like observer can be designed for the obtained state affine system. Index Terms—Direct transformation, observer, state affine system.

I. INTRODUCTION In the linear case, the observer design has its standard Kalman and Luenberger solutions. In the nonlinear case, a fruitful observer design technique consists in transforming the nonlinear system (NLS) into an equivalent system for which the observer design is easier. Some authors have studied the transformation of a general nonlinear system into a linear system via an input–output injection; see [10] and [16] for preliminary works and [5], [11], and [13] for more complete results. A problem with fewer constraints consists in transforming the NLS into state affine form for which a Kalman observer can be designed; see, for example, [9], [1], and [12]. In the last decade, besides the geometric approach in [7] and [8] an alternative algebraic approach was considered in [12] to tackle this problem. In this case, input–output differential equations are required; thus the preliminary problem of state elimination has to be solved. Whereas in [9], abstract and concise conditions are given for the existence of a solution, an effective algorithm is displayed in [12] to derive the state affine form. The procedure is effective up to standard integration of one form. The goal of this paper is to give a constructive algorithm that locally transforms a nonlinear multiple-input–single-output (MISO) system into a state affine system without solving the state elimination problem. The class of considered state affine systems is general enough to include the MISO systems studied in [7]–[9], and [12]. Models of practical examples as the inverted pendulum [12], the induction motor [6], the series dc motor [3], the bioreactor process in [2], and the synchronous generator [9] for which Kalman like observers can be designed. The paper is organized as follows. Some preliminaries are given in Section II. In Section III, the problem under interest is stated. In Section IV, necessary and sufficient conditions as well as a constructive algorithm are given. The algorithm is illustrated by an example in Section V.

where x 2 IRn ; u 2 IRm is the input, y 2 IR is the measured output, f and h are meromorphic functions of their arguments. There exists an open dense submanifold M de IRn so that any point x0 of M is a regular point for the considered computations. All definitions and results given here can be rewritten locally around any point x0 of M . If a property is generically satisfied, this property is supposed to be satisfied locally around a regular point x0 of M . A. Exterior Differential Systems and Poincaré’s Lemma Consider ! 2 SpanK (a;b) fda1 ; . . . ; da ; db1 ; . . . ; db g, a differential 1-form with a 2 IR ; b 2 IR ; K is the field of meromorphic functions of a and b. The differential 1-form ! is a closed form if d! = 0. A differential 1-form ! is an exact form if there exists a function (a; b) so that ! = d . The exterior product is denoted ^, and ^db will be used for ^db1 ^ 1 1 1 ^ db . Lemma 1 (Poincaré’s Lemma): A differential 1-form ! 2 SpanK (a;b) fda1 ; . . . ; da ; db1 ; . . . ; db g (a 2 IR and b 2 IR ) is locally exact if and only if d! = 0. Lemma 2 (Frobenius Theorem): [4] Consider a differential 1-form ! 2 SpanK (a;b) fda1 ; . . . ; da ; db1 ; . . . ; db g. There exist locally two functions (a; b) and T (a; b) so that T 1 d = ! if and only if d! ^ ! = 0. Lemma 3: Consider a differential 1-form ! 2 SpanK (a;b) fda1 ; . . . ; da g. There exists locally a function (a; b) so that  i=1

@ 1 dai = ! @ai

if and only if d! ^ db1 ^ 1 1 1 ^ db = 0: (2) In Section IV, the given algorithm uses the spaces defined as follows. Definition 1: For all k 2 [1; n], the subspace E k is formally defined

E k = SpanK d x; dy; du; . . . ; dy (k01) ; du(k01) : Definition 2: For all k

2 [1; n], the subspace E k is formally defined

as

E k = SpanK dy; du; . . . ; dy(k01) ; du(k01) : E k is used to identify the elements of the matrix A(u; y ) and to compute the state coordinate transformation (5). E k is used to find the different elements of the input–output injection (u; y ). III. PROBLEM STATEMENT Consider the nonlinear system

II. SOME PRELIMINARIES

x~_ = f~(~x; u) y = h(~x)

Consider the nonlinear system

x_ = f (x; u) y = h(x)

(1)

Manuscript received July 17, 2002; revised May 15, 2003. Recommended by Associate Editor J. M. A. Scherpen. The authors are with IRCCyN Institut de Recherche en Communications et Cybernétique de Nantes, UMR CNRS 6597, Ecole Centrale de Nantes, 44312 Nantes Cedex 3, France. Digital Object Identifier 10.1109/TAC.2003.820071

(3)

where x ~ 2 M is the state, u 2 IRm is the input, y 2 IR is the measured output. In the sequel, (3) is assumed to be generically observable. We make the assumption that the output and the input are known. First, without loss of generality to simplify the study, we rewrite the nonlinear system (3) using a state coordinate transformation x := X (~ x) (i.e., rang((@X )=(@ x~)) = n)

0018-9286/03$17.00 © 2003 IEEE

x :=

h(~x) x

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where x T := [x2 ; . . . ; xn ]. The system equations can be expressed by using the new coordinates

From (8), the differential dy (n) can be computed. Then, this differential can be written as

6 : fx_ = f (x; y; u) y = x1 (4)  _ with the dynamics definitions: x_ 1 := f1 ( x; y; u); x := f ( x; y; u).

= P10 + P + P by using the following definitions with y ( 1 ) := (y; . . . ; y ( 01) ) and u( 1 ) := (u; . . . ; u( 01) ):

These notations are introduced to explicitly express the known and measured variables: (u; y). This will be used in Section IV to establish the main results. The problem under interest is the search for a state coordinate transformation z1 := x1 = h(~ x) = T1 (x) z2 = T2 (x2 ; . . . ; xn ) .. . ~T : (5) T : zk = Tk (x2 ; . . . ; xn ) .. . zn = Tn (x2 ; . . . ; xn )

dy (n)

n

P10

n

=

j =2

Pu

0 0

a1;2 (u; y ) 0 a2;3 (u; y ) a2;2 (u; y ) .. .. .. . . . 111 0 an01;2 (u; y) 111 0 an;2 (u; y) '1 (u; y ) .. C = [1 0 . 'n (u; y ) n21

A=

=

..

.

111

i=1 j =0

01

n

:=

dy (n)

2n

P11

n

=

Pjy x; y ( 1 ) ; u( 1 ) dy (j )

01 @y(n)

n

@y (j )

01

n

dy (j ) :

jy x; y ( 1 ) ; u( 1 ) dy (j )

j =0

n:

T

where R is a real symmetric, positive–definite matrix.  is a positive constant so that observer (7) is converging. The details and proof can be found in [9]. Thus, the original state x can be estimated by inverse transformation T 01 .

P2

=

uj

y ( 1 ) ; u( 1 ) dui(j ) :

Note that the functions ju and jy depend explicitly on x . We will provide a way to separate P11 from P2 because it is easier to find P10 since it involves all terms in dxj for j 2 [2; n]. 2) Second: We derive similar computations for the state affine system (6) and we obtain

dya(n)

= P1

a

T ( x); u; . . . ; u(n01) ; ya ; . . . ; ya(n01)

u; . . . ; u(n01) ; ya ; . . . ; ya(n01) : (10)

Remark 1: Note that the differential forms P1 ; P2 ; P1a , and P2a are exact. Remark 2: A solution to the state transformation problem exists, means: P1 = P1a ; P2 = P2a . Theorem 1 (Necessary Condition): If the nonlinear system (4) is locally equivalent to system (6) by a coordinates transformation (5), then the differential term dy (n) can be decomposed as follows:

= P10 + P11 + P2 with dP11 = 0dP10 P11 =  ( x) y ( 1 ) ; u( 1 )

dy (n)

n

i

i

i=2

P2

=

01

n

yj y ( 1 ) ; u( 1 ) dy (j )

j =0

(8)

(9)

i=1 j =0

a

= f1 (x; y; u) @f1 @f1  @f1 f ( x; y; u) + y = y_ + u_ @ x @y @u := f2 (x; y; y;_ u; u_ )

= @f@nx01 f(x; y; u) n02 m @fn01 (k+1) @fn01 (k+1) + y + ul : ( k) @y @ul(k) k=0 l=1

01

n

m

+ P2

y_

x; y ( 1 ) ; u( 1 ) dui(j )

yj y ( 1 ) ; u( 1 ) dy (j )

j =0

IV. MAIN RESULTS In this paper, the main tool consists in analyzing the differential of the nth time derivative of the output y . This differential can be computed without solving the state elimination problem in the following way. 1) First: We compute the nth time derivative of y with respect to 6 (4) as a function of x

ju

i=1 j =0

01

n

+ (7)

01

n

m

+

T

y

dui(j )

= P1 + P2 with P1 = P10 + P11 and

1 1 1 0] 2

z^_ = A(#)^ z + (#) 0 R01 C (C z^ 0 y ) O : R_ = 0R 0 A (#)R 0 RA(#) + C C with # = (u; y)

(n)

@ui(j )

j =0

Note that z 2 IRn and that T~T = [T1 ; T T ]. The stated problem is of major importance since a Kalman-like observer for (6) can be designed [9]

.. .

x; y ( 1 ) ; u( 1 ) dui(j )

In the sequel, the decomposition of the differential of y (n) is given by the equation

an01;n (u; y ) an;n (u; y )

T

Pju

01 @y(n)

j =0

.. .

1

01

n

n

m

=

=

0 0

@y (n) dxj @xj

i=1 j =0

(6)

111 111

&j x; y ( 1 ) ; u( 1 ) dxj

m

:=

a

where

n

:=

j =2

Py

a

y

n

so that 6 is locally equivalent to the MISO state affine system

)z + (u; y) 6 : zy_ =:=A(yu;=yCz

u

+

m

01

n

i=1 j =0

uj

y ( 1 ) ; u( 1 ) dui(j )

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where i ( x)’s are non constant functions (i.e., di (x) 6= 0) and i ’s exact differential forms d i = 0. Proof: If the nonlinear system (4) is locally equivalent to (6) by a coordinates transformation (5), then the differential term dy (n) can be written (cf. Appendix A) as n

dy( ) =

if so that

!

H y( 1 ) ; u( 1 ) dz

n

i

i;j

i

0

Compute functions m

and

 du( 0 ) 0  dy( 0 ) 2 E 0 : n

u i;j

i

y i;j

l

n

i

n

i

l=1

i=2

+ z dH y( 1 ) ; u( 1 ) + dG y( 1 ) ; u( 1 ) i

n

=

Define a differential one-form

i

as

H y( 1 ) ; u( 1 ) dT (x) i

. Check

i

i=2

+ T (x)dH y( 1 ) ; u( 1 ) + dG y( 1 ) ; u( 1 ) : i

i

This implies n

P10 =

(H y( 1 ) ; u( 1 ) dT (x)) i

i

i=2 n

P11 =

T (x)dH y( 1 ) ; u( 1 ) i

i

i=2

P2 = dG y( 1 ) ; u( 1 ) : The necessity of Theorem 1 is then proved. In the sequel, we will decompose dy (n) as dy (n) = P1 + P2 where P1 = P10 + P11 . Our previous result [14] gives a direct approach to the search for a transformation from a general nonlinear system into a special state affine system where the matrix A is up-diagonal. Here, by using a similar approach, we develop the transformation into a more general class of state affine form (6). In this section, we present an algorithm which enables to know if (4) is locally equivalent to (6). For (6), we write

P1 x; u; y; . . . ; u( 01) ; y( 01) = a

n

n

(H dT + T dH );

n

i

i

i

i

i=2

n

Algorithm for a general class of state affine systems Initial step: compute

dy( ) = P1 x; u; . . . ; u( 01) ; y; . . . ; y( 01) n

n

+ P2 u; . . . ; u( 01) ; y; . . . ; y( 01) n

so that and formal system (6), we yield the same partition

n

a

n

a

a

j;n

i;j

n

m

i

@H 0 du : @u( 0 ) j;n

i

n

l=1

i

l

i

(11)

l

Check . If not, stop! The same computations are . can be identified made on . with the equation If the given conditions are satisfied, we substitute elements a in P2 . Then, in the following step we try to check the existence of the input–output injections ' in order to compute them. j

2nd Step: Computation of Set and . compute functions so that

 =  = j

n

1st Step: Computation of Set and

a

and

n a

m

a j

l=1

If follows: n a

and

. Define a differential as . one-form Check and . If not, stop! Deterin the same way. Then, is a mine the solution of

. With the

dy( ) = P1 T (x); u; . . . ; u( 01) ; y ; . . . ; y( 01) + P2 u; . . . ; u( 01) ; y ; . . . ; y( 01) : n a

! = @H( 00) dy + @y

a

n

Detailed definitions of the functions G; Hi and their respective decomposition in functions Gi and Hj;i are reported in Appendix A. In the sequel, we give an algorithm with two steps. The first step allows to check the existence of functions aij of (6) and Tj . If the first step is completed, the second step will be to check the existence of input–output functions 'i (u; y ).

n

If not, stop! is computed on in From the same way. By can be comidentification puted with the equation . is also useful to . (See Appendix compute A2 for more details). , then if If In this case, we need the functions . is computed using the following one-forms:

i;j

P2 u; y; . . . ; u( 01) ; y( 01) = dG: a

and

.

@G 0 du + @G 0 dy: @y( 0 ) @u( 0 ) n

j

n

j

l

n

j

n

j

l

, then

is determined as

. From this algorithm, we state the main theorem. Remark 3: The elements ai;j of the matrix A are computed in the precise order given by the algorithm. Details are given in Appendix A. Let ^du denote du1 ^ 1 1 1 ^ dum .

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Theorem 2: The system 6 is locally equivalent to 6a if and only if for k; i 2 [1; n] and j 2 [2; min(n; i + 1)]

d d! d!

k

^ du = 0 d ^ dy = 0 ^ ! ^ du = 0 d! ^ du = 0 if j 6= i + 1: k

i;i+1

i;i+1

i;i+1

^ !

i;i+1

^ dy = 0

i;j

Remark 4: If these conditions are not fulfilled at an iteration of the algorithm, then the algorithm stops and the system 6 is not locally equivalent to 6a . Proof: A sketch of the proof of this theorem is developed in Appendix B2. A complete proof is available in [15]. V. EXAMPLE This example is given to illustrate Theorem 2 and the associated algorithm. Consider the following observable system 6ex :

x_ 1 = u(x2 + ln x3 ) + y x_ 2 = (y 0 uy)(x2 + ln x3 ) + (u2 0 u) ln x3 0 u x_ 3 = x3 (uy(x2 + ln x3 ) + u ln x3 + u) y = x1

(12)

We search for the diffeomorphism so that 6ex is locally equivalent to

6exa

Fig. 1. Computation of a

so that !1a;2 0 au  0 ay  1;2 du 1;2 dy a one-form ! 1;2 as

2E

2

. We also define a differential

! 1 2 = 1 2 du + 1 2 dy = T2 (x) @a1 2 du + @a1 2 dy : @u @y From the equality ! 1 2 = ! 1 2 , a trivial solution is a ;

au ;

;

ay ;

a ;

;

;

T2 (x) = x2 + ln x3 a1 2 = u _ + uy _ + uy2 + u4 y: H2 = u + uy ;

0 a1 2 (u; y) 0 '1 (u; y) z_ = 0 a2 2 (u; y) a2 3 (u; y) z + '2 (u; y) 0 a3 2 (u; y) a3 3 (u; y) '3 (u; y) y = y = z1 : ; ;

;

;

;

a

Now, the algorithm is applied. 1) Initial Step: From 6ex , we have

According to different monomial derivation orders, we separate H2

_ + uy H2 2 = u H2 1 = uy _ H2 0 = uy2 + u4 y: ;

;

;

T2 and H2 ; !2 3 is determined as follows: !2 3 = !1 2 0 d(H2 T2 (x))a2 1 ; H2 and its decompositions are known. As mentioned in Appendix A and Fig. 1, a2 2 will be obtained in the next iteration.

Knowing

_ + uy dy(3) = df(u + uy _ + uy2 + u4 y)(x2 + ln x3 ) + (4uu _ 2 + u3 y + u4 )(ln x3 )g + dfu4 + yg

;

;

;

;

;

Now, we only give the results of each step. The details can be found in [15].

i = 2j = 2 : a2 2 = y; i = 2j = 3 : a2 3 = u2 ; T3 (x) = _ 2 + u3 y + u4 . H3 1 = 4uu _ 2 H3 0 = ln x3 ; H3 = 4uu 3 4 u y + u i = 3j = 2 : a3 2 = uy; i = 3j = 3 : a3 3 = u. After this first step, we substitute the elements a in 6 and we seek to identify in the following step the input–output injections ' . 3) Second Step: Set 1 = P2 = u4 + y(1 2 E 3 ) and 1 = P2 . We compute the input-output injection ' :j = 1 : '1 = y; j = 2 : '2 = 0; j = 3 : '2 = u. 6ex is then diffeomorphic to the following ;

thus

;

;

P1 x; u ; y = df(u + uy_ + uy _ + uy2 + u4 y)(x2 + ln x3 ) + (4uu _ 2 + u3 y + u4 )(ln x3 )g _ y;_ u; y) = dfu4 + yg: P2 (u; y; u; (1)

(1)

The third-order output time derivative of 6ex2a can be expressed as

dy(3) = P1 (z = T (x); u; y; . . . ; u; y) + P2 (u; y; . . . ; u; y): a

a

a

To distinguish the formal state affine system from the 6ex2 , we use the index a, even if y = ya . The first step is to compute the elements of the matrix A and the transformation Ti ; (i = 2; . . . ; n). 2) First Step: Set !1;2 = P1 and !1a;2 = P1a . Then, we can search for element a2;1 , transformation T2 and function H2 . i = 1 j = 2 first, we compute u1;2 and 1y;2 so that

!1 2 0 1 2 du(2) 0 1 2 dy(2) 2 E 2 = Span fdx; du; dy; du; _ dy_ g: ;

functions in the algorithm.

u ;

y ;

K

The differential one-form !  1;2 can be expressed

! 1 2 = 1 2 du + 1 2 dy = (x2 + ln x3 ) du: ;

u ;

y ;

Conditions d!  1;2 ^ ! 1;2 ^ du = 0 and d! 1;2 ^ ! 1;2 ^ dy = 0 are satisfied. In the same way, by using the possible equivalent state affine ay system, we have !1a;2 = P1a and we compute functions au 1;2 and 1;2

;

;

;

i;j

a

j a

a

j

state affine system:

6exa :

0 u 0 y z_ = 0 y u2 z + 0 : 0 uy u u y = z1 with z1 = x1 ; z2 = x2 + ln x3 ; z3 = ln x3 : VI. CONCLUSION

New necessary and sufficient conditions for local transformation of a nonlinear system into a general state affine form have been obtained. The class of MISO state affine systems which has been considered is the most general class with respect to previous results in the current literature. Finding such transformation is of major importance since Kalman-like observers can be designed for state affine systems [9]. Differential algebraic tools have been used. As opposed to previous results [12], the main result is stated without solving the state elimination problem. It is stated in a constructive way and the given algorithm can be implemented in symbolic computation systems. An open problem consists in extending this result to the multiple-output case. The use of

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the extended Kalman filter does not need to assume a scalar output and can be used for the extension of the approach to the multi-output case.

and T3 (x) (if n > 2). In the same manner, H3 is found with the functional division (13), etc.

APPENDIX A ABOUT THE STATE AFFINE DIFFERENTIAL EQUATIONS

APPENDIX B MAIN THEOREM: SKETCH OF PROOF

Differential equations associated with the considered state affine system are developed in Appendix A1. The necessary order to compute the matrix elements ai;j is explained in A2.

1) Necessity: The details of the proof is available in [15]. Let suppose the transformation z = T (x) exists. The following equation is verified by both systems:

A1 Following the definitions in step 1 of the given algorithm, each one-form !i;j is used to find ai;j . Thus, each one-form !i01;i (for i = 2; . . . ; n) is used to identify ai01;i and transformation Ti (x) which appears at each iteration on the highest derivation order terms. (n) n From (6), we have dya = i=2 (Hi dzi + zi dHi ) + dG where z = T (x), and from (10)

P1a x ; u; y; . . . ; u(n01) ; y (n01) =

n i=2

(Hi dTi + Ti dHi )

P2a u; y; . . . ; u(n01) ; y (n01) = dG: After identifying the transformation Ti (x), a functional division can be done as follows to have Hi .

y (n) 0

i01 j =2 Hj Tj

Ti ( x)

= Hi u; y; . . . ; u(n01) ; y (n01) + Oi2 x; u; y; . . . ; u(n01) ; y (n01)

(13)

where Hi contains all terms depending only on u; y; . . . ; u(n01) ; y (n01) and Oi2 is the rest of the functional division which involves all terms depending on x . Hi will be split up into components with different derivation orders as follows: Hi = Hi;n0i+1 + 1 1 1 + Hi;0 , where the functions Hi;j involves ( ) monomials in Hi depending on (y (r) )q and (ul )s so that m rq + l=1 l sl = j . Afterwards, each Hi (u; y; . . . ; u(n01) ; y (n01) ) is used to identify ai;i ; . . . ; an;i . However, it also depends on other functions ai;j as follows: Hi depends on (a1;2 ; . . . ; an;2 ; . . . ; ai01;i ; . . . ; an;i ; ai;i+1 ; . . . ; an01;i+1 ; . . . ; an01;n ) and their time derivatives for i < n; Hn could depend on all functions ai;j and their time derivatives. The second step of the algorithm uses dG which is defined from

G=

n k=1

Gn0k

(14)

where Gn0k ; k > 1 depends on ('k ; . . . ; '2 ; a1;2 ; . . . ; ak01;2 ; . . . ; ak01;k ) and their time derivatives. Gn0k involves all monomials in G depending on (y (r) )q and (ul( ) )s so that rq + m l=1 l sl = n 0 k . k = 1 (n01) represents a special case: Gn01 = '1 . Remark 5: For the special state affine system considered in [14], only upper-diagonal terms of matrix A(u; y ) are nonzero. Then, Hi is limited to Hi;n0i+1 . A2 At the first step of the algorithm the computation of functions ai;j follows the scheme given in Fig. 1. With !1;2 , one gets the function a1;2 (u; y ) and the transformation T2 (x). By using the given functional division, one finds H2 . After the decomposition of H2 ; a2;2 can be computed with H2;n02 . In a similar way, the one-form !2;3 defined in the algorithm as !2;3 = !1;2 0 d(H2 T2 ) is applied to find a2;3 (u; y )

dy (n) =

n i=2

(Hi dzi + zi dHi ) + dG:

(15)

After defining Hi and Gi as given in Appendix A, the algorithm can be applied. By selecting Hi , we can identify ai01;i ; . . . ; an;i . In this way, ai;j is identified by selecting Hj;n0i . Thus, ai01;i is found with Hi;n0i+1 ; . . . ; and an;i with Hi;0 . As mentioned in (9), P1 explicitly depends on x  and dy (n) can be distinguished in two parts: P1 (depending on x ) and P2 . By applying the algorithm, we can verify the necessity of Theorem 2 conditions. 2) Sufficiency: Suppose that conditions of Theorem 2 are verified. By applying the algorithm, the first step of the algorithm enables to compute the ai;j (u; y ) and Ti (x). After substitution of these components in target system (6), the second step of the algorithm allows the computation of the 'i (u; y ). The input–output injections ('2 ; . . . ; 'j 01 ) as well as all elements ai;j of matrix A are known at the j th iteration, and the function Gn0j only depends on ('2 ; . . . ; 'j ; a1;2 ; . . . ; aj 01;2 ; . . . ; aj 01;j ) and their time derivatives. So, only the input–ouput injection 'j is unknown and can be determined at this step. Consequently, if the conditions of Theorem 2 are satisfied, the functions ai;j ; Ti and 'i can be computed: the sufficiency of Theorem 2 is proved. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers and C. Moog for their helpful comments. REFERENCES [1] G. Bornard, N. Couenne, and F. Celle, “Regularly persistent observer for bilinear systems,” in New Trends in Nonlinear System Theory, ser. Lecture Notes in Control and Info. Sciences. New York: Springer-Verlag, 1988, vol. 122, pp. 130–140. [2] K. Busawon, M. Farza, and H. Hammouri, “Observers’ synthesis for a class of nonlinear systems with application to state and parameter estimation in bioreactors,” Proc. 36th IEEE Conf. Decision Control, pp. 5060–5061, 1997. [3] J. N. Chiasson, “Nonlinear differential-geometric techniques for control of a serie DC motor,” IEEE Trans. Contr. Syst. Technol., vol. 2, pp. 35–42, Jan. 1994. [4] G. Conte, C. H. Moog, and A. M. Perdon, Nonlinear Control Systems—An Algebraic Setting, ser. Lecture Notes in Control and Info. Sciences. New York: Springer-Verlag, 1999, vol. 242. [5] A. Glumineau, C. H. Moog, and F. Plestan, “New algebro-geometric conditions for the linearization by input-output injection,” IEEE Trans. Automat. Contr., vol. 41, pp. 598–603, Apr. 1996. [6] J. de Leon-M, A. Glumineau, and I. Souleiman, “Nonlinear observer for induction motors: A benchmark test,” presented at the Proc. IFAC Control Systems Design, Bratislava, Slovakia, 2000. [7] H. Hammouri and J. P. Gauthier, “Global time varying linearization up to output injection,” SIAM J. Control Optim., vol. 30, pp. 1295–1310, 1992. [8] H. Hammouri and M. Kinnaert, “A new procedure for time-varying linearization up to output injection,” Syst. Control Lett., vol. 28, pp. 151–157, 1996. [9] H. Hammouri and J. De Leon Morales, “Observer synthesis for state affine systems,” Proc. 29th IEEE Conf. Decision Control, pp. 784–785, 1990.

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[10] A. J. Krener and W. Respondek, “Nonlinear observers with linearizable error dynamics,” SIAM J. Control Optim., vol. 23, no. 2, pp. 197–216, 1985. [11] V. López-M and A. Glumineau, “Further results on linearization of nonlinear systems by input-output injection,” Proc. 36th IEEE Conf. Decision Control, pp. 10–12, 1997. [12] V. Lòpez-M, J. de Leon Morales, and A. Glumineau, “Transformation of nonlinear systems into state affine control systems and observer synthesis,” in Proc. IFAC Conf. System Structure and Control, Nantes, France, 1998, pp. 771–776. [13] F. Plestan and A. Glumineau, “Linearization by generalized input–output injection,” Syst. Control Lett., vol. 31, pp. 115–128, 1997. [14] I. Souleiman, A. Glumineau, and G. Schreier, “Direct transformation of nonlinear systems into state affine systems,” in Proc. NOLCOS, St. Petersburg, Russia, 2001, pp. 545–551. , “Direct transformation of nonlinear systems into state affine sys[15] tems,”, Nantes, France, IRCCyN Report RI2002-12, 2002. [16] X. H. Xia and W. B. Gao, “Nonlinear observer design by observer error linearization,” SIAM J. Control Optim., vol. 1, pp. 199–216, 1989.

Finite Gain Stabilization of Discrete-Time Linear Systems Subject to Actuator Saturation: The Case of Yacine Chitour and Zongli Lin

Abstract—It has been established by Bao, Lin, and Sontag (2000) that, for neutrally stable discrete-time linear systems subject to actuator saturation, finite gain stabilization can be achieved by linear output feedback, for every (1 ] except = 1. An explicit construction of the corresponding feedback laws was given. The feedback laws constructed also resulted in a closed-loop system that is globally asymptotically stable. This note complements the results of Bao, Lin, and Sontag (2000) by showing that they also hold for the case of = 1. In establishing our results for = 1, we also allow the presence of certain additional external disturbance. Index Terms—Actuator saturation, discrete-time systems, finite gain stabilization, Lyapunov functions.

Standard closed-loop connection.

is well-defined and finite gain stable. This problem was first studied for continuous-time systems and various results have been established for such systems. It was shown in [8] that, for neutrally stable open loop systems, linear feedback laws can be used to achieve finite gain Lp stabilization, for all p 2 [1; 1]. Various continuity and incremental-gain properties of the resulting closed-loop system were discussed in detail in [4]. For a neutrally stable system, all open-loop poles are located in the closed left-half plane, with those on the j! axis having Jordan blocks of size one. In the case that full state is available for feedback (i.e., y1 = x and u2 = 0), it was shown in [7] that, if the external input signal is uniformly bounded, then finite-gain Lp -stabilization, for any p 2 (1; 1] except p = 1, and local asymptotic stabilization can always be achieved simultaneously by linear feedback, no matter where the poles of the open loop system are. The uniform boundedness condition of [7] was later removed for the case p = 2 by resorting to nonlinear feedback [6]. More recently, the problem of Lp stabilization for a double integrator system subject to input saturation feedback and disturbances that are not input additive was investigated in [3]. In particular, [3] considers the control system (DI ), x  =  (0x 0 x_ + u)+ v , where x 2 IR and (u; v ) are disturbances. For v = 0 and zero initial conditions, it was established, among other results, that the L2 -gain from u to the output of the saturation nonlinearity to was finite. This partially solved [2, Prob. 36]. For nonzero v , an L -bound is of course necessary for getting any positive result regarding Lp -stabilization. It was shown that (DI ) is Lp -stable (see [3] for the precise definition of Lp -stability) if and only if p  2. In other words, one can construct, for p > 2, a disturbance v with finite Lp -norm and arbitrarily small L -norm that results in an unbounded trajectory of (DI ). Examples of other works related to the topic are [5], [9], [10], and the references therein. The extension of the results of [8] to discrete-time systems was made in [1]. In particular, it was shown in [1] that, for neutrally stable discrete-time linear systems subject to actuator saturation, finite gain lp stabilization can be achieved by linear output feedback, for every p 2 (1; 1] except p = 1. An explicit construction of the corresponding feedback laws was given. The feedback laws constructed also result in a closed-loop system that is globally asymptotically stable. The objective of this note is to complement the results of [1] by showing that they also hold true for the case of p = 1. In establishing our results for p = 1, we also allow the presence of certain additional external disturbance.

1

1

I. INTRODUCTION This short note revisits the problem of simultaneous global asymptotic stabilization and finite gain Lp (lp ) stabilization of a linear system in the presence of actuator saturation and measurement and actuator noises (see [8], [1], and the references therein). More specifically, we would like to construct a controller C so that the operator (u1 ; u2 ) 7! (y1 ; y2 ) as defined by the following standard systems interconnection (see Fig. 1):

y1 = P (u1 + y2 ) y2 = C (u2 + y1 )

Fig. 1.

(1.1)

Manuscript received April 9, 2003. Recommended by Associate Editor Z.-P. Jiang. The work of Z. Lin was supported in part by the National Science Foundation under Grant CMS-0324329. Y. Chitour is with the Département de Mathématiques, Université de Paris-Sud, Orsay 91405, France (e-mail: [email protected]). Z. Lin is with Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904-4743 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2003.820142

II. STATEMENT AND PROOF OF THE RESULT We consider a discrete-time linear system subject to actuator saturation, external disturbances and sensor noises

P:

x+ = Ax + B(u + u1 ) + v; x; v 2 IRn ; u; u1 2 IRm y = Cx + u2 ; y 2 IRr

0018-9286/03$17.00 © 2003 IEEE

(2.1)