Model Matching of Nonlinear Systems

CopHiKllI © IFA C 9th Triennial World Congre"s Bud.lpe"!' I-Iungan. 19X4

MODEL MATCHING OF NONLINEAR SYSTEMS T . Okutani* and K. Furuta** *TU
Abstract. This papaer investigates the model matching problem for nonlinear multi-input/ output systems. The model matching problem (M.M.P . ) is in general to fit the external descrip'tion (for example, the transfer function in linear systems) of a system to that of a model system with desired properties by state feedback, dynamic compensator, etc. Several studies on M.M.P. have been done for linear systems. However, there has been no discussion about M.M. P. for nonlinear systems. The difficulty in the nonlinear case is that there is no simple external description . So in this paper M.M.P . is defined as fitting the output of a system to that of a model system for the same input. First, zeroing the output for any disturbance inputs is discussed by use of the output zero condition. Next, sufficient condition for achieving the model matcing by state feedback with the above definition of M.M. P . is derived . Finally, the approximate calculation method of the state feedback for M. M. P. is given for a special case . In the process of this calculation the existence of the feedback is verified. Keywords .

Nonlinear control systems; multivariable systems; control theory

INTRODUCTION

denotes the set of piecewise continuous and bounded time functions defined in [O,T). System (1) will be denoted by (A(x,t),B(x,t),C(x,t),D(x,t». Let the model system be

This paper is concerned with the model matching of a nonlinear system to a linear system by state feedback.

z=AM(t)z+BM(t)v

On the other hand, the model matching problem for a nonlinear system has not been studied, since there is no appropriate external discription for a nonlinear system. This paper considers the problem of model matching as the problem to find a control system such that the output of the compensated system is equivalent to that of the model. This concept is similar to the equivalence of external discriptions for linear systems. The approach to the problem is presented in the paper based on the design technique developed by Freund (1975, 1977), where the model system to which the closed loop system is matched is assumed linear. The necessary and sufficient condition of the model matching the nonlinear system to the linear system is given and the algorithm to calculate the approximate compensator is also presented.

(2.a)

where F(x,t) is an m dimensional vector and G(x,t) is an mXr dimensional matrix. It is assumed that F(x,t) is analytic in Rand F(O,t)=O . When the output of the system (A,B,C,D) coincides with that of the system (A' ,B' ,C' ,0') for tE[O,T) to the same input yEn and zero initial states, we will write

(A,B,C,D)v '\, (A' ,B' ,C' ,D')v Then the model matching problem considered in the paper is determining F(x,t) and G(x,t) such that (A (x , t) + B(x , t) F (x , t) , B(x, t ) G(x , t) ,C (x , t) + D(x,t)F(x,t),D(x,t)G(x,t»v '\, (AM(t) , BM(t) ,CM(t), DM(t»v (4) To treat the model matching, this paper considers the zeroing of the output for the augmented system represented by

DEFINITIONS AND PRELIMINARIES The nonlinear system considered is represented by x=A(x, t)+B(x,t)u x(O)=O y=C(x,t)+D(x,t) where yERp xERn

z(O)=O

w=CM(t)z+DM(t)v (2 . b) where v=R n ', vER r , wER P . Matrices AM(t), BM(t), CM(t) and DM(t) are n'xn', n'xm, pxn' and pXr respectively and these matrices are assumed to be analytic in [O,T). The following type feedback law is considered for the plant (1) in the paper u=F(x,t)+G(x,t)v (3)

For a linear system, model matching problem is the designing of a control system for a system so that of the prespecified model, and many papers have delt the model matching problem of linear systems (Silverman 1971, Wolovich 1972).

x*=Ax(x*,t)+B*(x*,t)u* y*=C*(x*,t)+D*(x*,t)u*

(La) (l.b)

x*(O)=O

(5. a)

(s.b)

where the dimension of the system is n*=n+n', x*=[x) z '

A(x,t), B(x,t), C(x,t) and D(x,t) are respectively nxl, nXm, mxl and nXm matrices, and assumed to be analytic with respect of x and t in the region L= XX[O,T) inclusing the origin. X is an open subset in Rn . It is also assumed A(O,t)=O, C(O,t)=O. n

u*=[u) v '

y*=y- w

° ° )f

A*(x* T)=[A(X , t») B*(x* t)=[B(x,t), 'AM(t)z ' BM(t) C*(x*,t)=C(x,t)-CM(t)z D*(x*,t)=[D(x,t), -DM(t») 399

1 (6)

T. Okutani and K. Furuta

4::0

and the state feedback law employed is given by u*=F*(x*,t)+G*(x*,t)v where F* (x*, t) = [F(Xo' t)] , G*(x* ,t)=[ G(~ ,t)]

(7)

[Theorem 1] Suppose that the mapping from u t o y has the maximal rank q. Then there exists the control law F(x,t) and G(x ,t) such that

r

The relation (4) can be rewritten as (A*+B*F*, B*G*, C*+D*F*, D*G*)v '0 0

For the system with the maximal rank, the next theorem is gives the output zeroing solvability.

(8)

for the augmented system, and it is found that the model matching can be treated by the zeroing of the output, where '00 of (8) denotes that the output of the sys tem y*=O.

(A+BF, BG, C+DF, DG)v '0 i)

ii(x(t,v),t)G(x(t,v),t)=O

ii)

a ax

I

In this section, the condition under which the equa tion (8) is statisfied is to be shown. This output zeroing is considered for the nonlinear system (1). Let the nonlinear operator Ni be defined by

°

NAC(x,t)=C(x,t)

(9.a)

k a k-l a k-l NAC(x,t)= at NA C(x ,t)+ ~ NA C(x,t)'A(x,t) k=1,2,·· (9.b) The differential order of system di(i=1,2, . . p) is defined similar to Singh (1972) and Freund (1975) such that for Di(X,t)'I'O di= fO a · -1 lmin(jl~ Ni Ci(x ,t)·B( x ,t)=O) for Di(x,t)=O (10) where N~Ci' Di are the i-th component of N:C, and the i-th row vector of D respectively. d~ are assumed constant in [ . T Let y#=[y(dl),y(d2),· · · ,yp(d p )] , then y# can be expressed by using (1) in [ , as y# =CII (x,t)+D II (x,t)u

(ll)

where cl and Df are the i-th row vectors of given by

C~(X ,t)=N~iCi(X,t)

clI,

DII

(12)

lD~(X,t)

for di=O d.-l ~ NA1 Ci(x,t) ·B(x,t) for di>O

Di(x,t) = a

i=1,2,'"

l C (x, t)

P

+ [O(X,t»)u

°

(14)

where O(x,t) is the matrix which is constructed from linearly independent rows of matrix DII(x,t) in [ . Matrices C(x,t), ~(x,t) and O(x,t) are also consi~ered analytic in [ . Let yO =M(x,t)y ll and CO(x,t)= C(x,t), and partition yO in accordance with O(x,t) as yO= [yo)}q y~ }p-q

so that YO =C o(x ,t). If the relation yk=~k(x,t) is satisfied and there exists the analytic matrix Sk(x,t) such that

a

~

-k

C (x,t)·B(x,t)=Sk (x,t)D(x,t)

(15)

then yk+l(x,t) can be ob tained as ~k+l(x,t), where yk+l and Ck+ l are defined as -k+l Y ~

k+l

d -k k - 0 dt y - S (x,t)y for k=O,1,2 , 3, '" 1 k

=NA~

(19)

k (x,t) - S (x,t)C(x,t)

MODEL MATCHING The model matching problem by state feedback can be solved by using theorem. This leads t o the following theorem. [Theorem 2] Suppose that the mapping from u* to y* of the aubmented system (5) has the maximal rank q in [0. The model matching problem by state feedback so that (A+BF ,BG ,C+DF, DG)v '0 (AM, BM ,CM, DM)v is solvable if and only if 1) O(x(t,v) ,t)G(x(t,v) ,t)=iiM(t) 2)

(20)

k axa NA+BF{C(x,t)+D(x,t)F(x ,t ) } .B(x,t)G(x,t) Ix=

x(t;v)=

a a;-

k-M M NAC (t)z'B (t)

k=O,l,·· ,n*-l

(21)

where x(t;v)=[In O]x*(t;v). It can be easily shown that condition 1) and 2) are derived from Theorem 1 by the using relation (6) . The condition 2) seems to include the state z but actually is independent of z. Therefore th e dynamic behavior of the model is not r equ ired . Specially if the model system is time - invariant, then the right hand side of condition 2) is equi valent to the Markov parameter of the model.

(13)

If rank DII=q~m and q is constant in [ then there exists a nonsingular and analytic matrix M(x,t) such that

M(X,t)yll={~(X ,t» )

(k=O,1.2 .... )

where x is the xolution of (1) and (3) co responding to input v. [The proof is given in Appendix.]

OUTPUT ZEROING

11

(18)

k } ·B (x,t)G(x,t) X= NA+BF{C(x,t)+D(x,t)F(x,t)

x(t,v)=o

and

°

i f and only i f

(16)

Since the above two conditions include the trajectory of solution x depending on v, Theorem 2 may not be practical . If these conditions are satisfied for any x on X, they are sufficient conditions for the solvability of the model matching problem. A computable procedure of F(x,t), G(x ,t) will be given hereafter. In order to simplify the discu ssion it is assumed that both systems are time invariant and the maximal rank of the augmented system q* is equal to m. Letting Ao(X)=A(x) - B(x)O-l(x)C(x)

(22a)

Bo (x)=B(x)ii-l(x) F(x)=C(x)+O(x)F(x)

(22b)

The following equations are obtained from Theorem O(x)G(x)=iiM (23) a/dx N:o+BoFF(x) . B(x)G(x)=CM(AM)kBM

The right hand of (24) denotes Hk afterwards . Since Ao(x), Bo(x) and F(x) with Ao(O) =O are analvtic, the y can be expanded bv Taylor series in the neighbourhood the origin as Ao(x)= JlA Bo(x)=i~OB -

[Defitition] If the above operation can be conti nued for k=O,1,2,"', we say that q is the maximal rank of the mapping from input u to the output y.

(24)

(k=O,l, .. · n*-1)



(17)

(22c)



(i) (i)

F(x)=i~lFiJ

(x)

(25a)

(x)

(25b)

(i)

(x)

(25c)

where positive integer i denotes the o rder of x and J(i)(x) denotes a column vector which consists of

Model Matching of Nonlinear Sys tems all i-th order of x . k -

and

For convenience let

d

a/ax NAoF(X)=Yk(x)

(26a)

a/ax f(x)~fx(x)

(26b)

(b)JOYk (i) (x)B(l-i) (x)=O

(27)

then equation (24) is satisfied. Where (O)-M Qo=B D (1)

Qj=Ax (x)Qj _l+B Hk=CM(AM)kBM

PAPA'Y(x) =PA(PA , Y(x» Let . j pJ (l) = 1: IT P (l ) A ll +l2 + .. +l j =l i=l A i

(0) -

Hj _ l

(j=k.·· .n*-l)

(29) (30)

[Proof) Firstly. it will be shown that the condition (b) the following equation k k-l (0) (0) i NAo+BoFF(X)=Yk _l (X)Ao(x)+i~OYk_l_iB NAo+BoFF(X) (k=l •..• n*- l)

where P1(0)=I . P1(l)=0 (l~l) and P!(3)=P A(2)PA(1) P (1)P (2) . A A It can be proved that the next equality holds by induction.

y~j)(X)=F1P~(i+k)J~1)(x)+F2P~(i+k-l)J~2)(x)+ ...

+Fi+1P~(k)J;i+l) (x)

-

NAo+BoFF(X)= ax F(x) ' {Ao(x)+Bo(x)F(x)}

[Theorem 4)

FdQo .Ql • ooQn*_l) =[H o • ooHn* _l)

(6)

F

l +l

'{Ao(x)+Bo(x)F(x)} k-l (0) (0) i =Y k - l (X)Ao(x)+i~OYk _ l_iB NAo+BoFF(x) Therefore relation (31) holds for eve ry k . Secondly the Theorem is proved by induction . For k=O a/ax N!0 +B oFF(x) ' Bo(x)6M=Yo(X)Bo(x)6M

can easily be obtained using (28). If above relation exists for 0 .1.··· .k-l. then using the relation (31)

(i+l) (l -i) -j~OFj+l i~j [PA(i-j)J x (x)·B (x).···

- ( ) B ( )-H =y (0)B(0)6 H+ k f l y(0) . B(O)h i k 1=0 k-l-l ax Ao+BoFF x . 0 x D =y(0)B(0)6N+~fly(0) .B(O)H i k 1=0 k- l - l

=[A~l)(x»)kQo+~~~[A~l)(x»)k-l-iB(O)Hi

-k

ax ~Ao+BoFF(X)'Bo(x)D =F1Qk=H

-d x= dt

]"I 1*,,

The nonlinear system represented by 0

["

l+Xl - Xl + - 2X2

0

xo=O

1+x2

1 O)x

y=[l 1

d

(33)

-1

o

1

1 w(t)=[ 1

(34)

Then we can express Yl (x) =PA oYo(X) .Y 2(X) =PAoYl (x).

0 l)v.

zo=O

1 O)z

The nonlinear system is sufficiently smooth in X= {xER2Ixl ~ - l} and dl=d2=d~=d~=1 . So Xl

(k=O .· · . v*-l)

and eq. (27) derived . An operator P is introduced about matricies Y(x) and A(x) such that

~ (Y(x)A(x» oX

Remark 2 . In the case that Fj has parameters which can be determined arbitrarily. th e existance of Fk (k~j) depends on choice of the parameters .

dt z=[ 0 _2)z+[0

B\' equations (32) and (33). conditions (a) and (b) impl,' that

PAY(x)=

and G(x) can also be obta ined by eq. (23).

to match the linear model

=A~l)(x)' (A;l) (x)Qk _2+B(0)Hk- 2)+B(0)Hk- l

- ~l

F(x)=6- 1 (x){F(X)-C(X)}

u=F(x)+G(x)v

Qk=A~l)(x)Qk_l+~(O)Hk -l

-

[Proof) Substituting equation (35) into condition (b) of Theorem 3 . we obtain condition (6). Once F(x) is determined by Theorem 4. F(x) can easily be computed by the following

is conside red to be cont r olled by

-

On the other hand. from the difinition of Qj .

k

then the eq . (24) holds .

(32)

is obtained. since y(O) =FdA (l{x»)j (j>0). x

0

... pn*- l(i+n*_l_ j)J(j+l) (x)B(l-i) (x») (37) A x

Example:

-M

a/ax NAo+BoFF(x) ' Bo(X)D

a

,X

l

Remark 1. The existance of Fl .F2.··· . Fl can be investigated one after another . When they exist. the y can be computed by solving eqs. (36) and (37) . Being app r oximated by a finite number of the terms up to Fl . eq.(19) will have the error corresponding to the high order terms.

=Yo(0)B(0)6 M

J

(36)

[po(O)J(l+l)( )'B(O) . . pn*-l J(l+l) )B(O») A x x • • A (n-l) x

l-l

(0) a j _ k- 2 (0) -[Y k - l (x)+i~OYk_2_iB ~Ao+BoFF(X»)

=Fl{[A~l)(x)kQo+~~~[A~l)(x»)k-l - iB(O)Hi}

If

(a)

Let us suppose that the relation (31) is satisfied for k-l. then k a k-l NAo+BoFF(x)= a; NAo+BoFF(x) ' {Ao(X) +Bo(X)F(X)}

-

(35)

Using this relation. we conclude the computabl e procedure in the next Theorem .

=Yo(X)Ao(x)+Yo(O)B(O)F(X)

k

(i=O.l.oo ;k=O.l. 00)

(31)

can be proved as follows . In the case of k=l the relation (31) holds . since

a -

(l>j)

lj~l

(l=1.2.·· ;k=O.· 'n*-l)

(28)

-

The . sum and product of operator P are defined as follows. (PA+PA ' )Y(x)=PAY(x)+PA'Y(x)

[Theorem 3) If -0 -1 -n*-l (a) FdQo.Ql.· ·. Qn* _l)=[H .H . · · . H )

1

401

c*"=

- - - xl - 2x 2,

l+x l [ -xl- - Xl l+Xl 1 1

T . Okutani and K. Furuta

402

q*=2 is the maximal rank for the input- output map and constant. and also D(x) is nonsingular in X. From eq. (23)

~) 1+x2

G(x)= [ :

elements in l: until all independent elements are chose . vi- l Let NA Ci be the highest o rder of the function independent element for Ci(x.t). i.e. a V' p s(j a k at NA1Ci(X.t) = j~1 k~O Clij(x .t)~ NACj(x.t)

.i\

and the following results are given from (22).

o

Ao(x)=O.

+i:1: l ViNviC. j=lCl ij A J Vj>vi

1

Bo(x)=[l _1]

Therefore the higher order terms of F(x) are not necessary to be determined and

and

(A. Sa)

(i' i\

a V' p s a k a;z NA1Ci(X.t)=j~1 k=O Clij(X .t) ax NACj(x.t) i - l Vi a V· +jhClij ax NA 1C j (A.5b) Vj>vi

yields where s(j .i)=min(vj.vi)' then from (A. Sa) and (A.5b)

F(x)=

v'+l p s(j .i) k k+l NA1 Ci(X.t)=j~l k~O ClijNA Cj CONCLUSION In this paper. we clarify the condition under which the model matching problem for a certain class of nonlinear systems can be solved by the state feed back. and show an algorithm to compute the compensating functions to be determined in the problem. The model matching approach described here can be used for the linealization of nonlinear systems. In this case the condition (1) of the Theorem 4 is the most important. The relation of the approach proposed here to the geometric approach should be studied further.

(A.B.C.D)u is equivalent to the following nonlinear system in the sense that the same output is given for the same input. d -

(A.8a)

where A(x. t)=

[Theorem A.l](Output zeroing condition) (A.B.C.D)u~O if and only if for VtE[to.t] and vu. the following relations exist.

iii)

a ax

_

Apl (x. t).

o

1 0 ..... 0

o

1

o

-.--....----

NACi(X.t)·B(x.t)

_

A.. 1J

= [ ~"'''''''O ) . 0 X

0 x····· · ·· x

(A.2)

(i=1.2. "p; k=0.1.2.")

k

vi+l

(A .l )

D(x(t;xo.u).t)=O NACi(XO .t)=O

[

~I~:(X.t) .

x ......... x

To prove Theorem 1. the output zeroing conditions for general nonlinear systems should be proved first.

ii)

-

(A . 8b)

APPENDIX

k

-

y(t)=C:L+D(x. t)u

Authors appreciate the discussion and comment given by Messers K.Kosuge and Y.G.Barraquand.

i)

-

(ftL =A(x.t)L+B(x.t)u

Ix=x(t;xo.u)=O

(A.3)

(;=1.2." ' p; k=0 . 1.2 ... ·) whe r e the operato r N is defined by k a k-l a k-l N Ci(X t)= N Ci(x.t)+ NA Ci(x.t)·A(x.t) A t A x (A.4) with N1=Ci(X .t) and Ci(X.t) denotes the i-th component of C(x .t ) .

a-

Proof)

a-

.1... ax

Let

L(x. t)~[N1Cl (x. t)

NVPC A p

000 010 .... 0

o .. .. . ..

.N~C2 (x. t) . ' .• N~Cp(x. t) .N~Cl (x. t) C=

... ]T

and sequentially choose elements which are function independent~ of the set of precedently chosen

0 ...... · .... · 1 0 · 0 vit2

The function g(x.t) is analytically function dependent of f 1 (x.t).f 2 (x.t) ... · f n (x.t) in l: if and only if g can be expressed as a n a a n a ax g(x.t)=ihClia;zfi(X.t). at g (x.t)=ihCli i (X.t) for any (x.t)El:

3tf

with Cl i(x.t) in R.

Eq. (A.8) gives

f

1

y(t) =C{I+J1(X(TO) .T l )dTl+f ;;:(X(Tl) . Tl) to to to XA(X(T2).T2)dT2 dTl+··· )L(xo.to) +f ii (X(Tl)' Tl )u(Tl )dTl to

Model Matching of Nonlinear Systems

403

and u(t) is the arbitrary input. (A.ln to (la), + ... }+D(x(t), t)u(t)

(A.9)

Therefore the necessary and sufficient condition for (A,B,C,D)u~O is given by i)

(A.IO)

ii)

(A.Il)

_ k _ Hi) CjUOA(xCr ;xo,u) ,Tj) .B(X(Tj+l;xo,u) ,Tj+l)=O j ~

k=0,1,2,'"

(A.12)

where t..> Tl

~ T2~

'"

d dt

By substituting

-

-

x=A(x,t)+Bo(x,t)y+B(x,t)H(x,t)u, x(to)=xo (A. LSa) y(t) =C(x,t) (A.lSb)

(A. 15) and (.5) which tells that q is the maximal rank yields d

k-

k

-

dX NAC(x,t)'B(x,t)H(x,t)zS (x,t)D(x,t)H(x,t)=O (A.19) From [Corollary A.l], (Ao,BH,C)~(Ao,O,C)u tells that y(t) is independent of a(t), so (A.16) is derived. From [Theorem A.l] and [Theorem A.2], Theorem 1 is proved.

~to

Letting

A(

x (TO;Xo,U),To)=I

and

v=max{vi} ,

REFERENCES

then

rank

(A.l3)

and the theorem A.l is proved. From the proof, it is found that k is not necessary to take until infinite but vi-I. The theorem gives the following corollary. [Corollary A.I] The necessary and sufficient condition for (A,B,C,D)u~ (A,O,C,O) is that conditions i) and iii) of [Theorem A.l] are satisfied for arbitrary u. Letting Ao(x,t) and Bo(x,t) be Ao(x,t)=A(x,t)-B(x,t)D+(x,t)C(x,t)

(A.14a)

Bo(x,t)=B(x,t)D+(x,t)

(A.14b)

then for a system with the maximal rank q k -

-k

NA/(x,t)=C (x,t)

k=0,1,2,'"

(A. 15)

This relation can be proved by the mathematical induction as follows. (A.16) exists for k=O and is assumed to exist for k-l as k-l-k-l NAo C(x,t)=C (x,t) then by using (15) and (17), k d k-ld k-lNAoC(x,t)= at NAo C(x,t)+ dX NAo C(x,t)·Ao(x,t) d at

- k-l d - k-l -+ C (x,t)+a,zC (x,t)·(A(x,t) -B (x,t)D (x,t)

xC (x, t» k-l -+ l - k- l =NAC (x,t)-S (x,t)D(x,t)D (x,t)C(x,t)

=~ k(x, t) Then using above results, the following theorem is derived. [Theorem A.2] The following relation exists if the direct part of the mapping from u to y has the constant maximal rank q. d dt

-

x=Ao(x,t)+Bo(x,t)y(t), =C (x, t)

x(to)=xo

y( t)

Pr oof)

(A.16b)

From the equation (14),

u(t)=D+(x,t)(y(t)-C(x,t»+H(x,t)u(t) where H(x,t) is the matrix such that D(x, t)H(x, t)=O

IWC1-N

(A .16a)

(A. In

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