Design Methods of Nonlinear Dynamic Compensator Based Upon Canonical Forms

Copyright © IFAC Design Methods of Control Systems. Zurich. Switzerland. 1991

DESIGN METHODS OF NONLINEAR DYNAMIC COMPENSATOR BASED UPON CANONICAL FORMS T. Ohtani* and M. Masubuchi** *Yokogawa Electric Corporation. R&D IV. Tokyo . Japan **Hosei University. College of Engineering. Tokyo. Japan

Abstract. The design methods of nonlinear dynamic compensator using a nonlinear observer and feedback linear izat ion are discussed in a class of nonlinear systems in which the states are not measurable. The state feedback linearization is not exact owing to the estimated states, but output feedback linearization is always exact if it exists. Furthermore, we cons id er a class of nonlinear systems in which a nonlinear dynamic compensator can be desIgned so that the resulting closed loop system contains linearized output relations. Our formulations are based upon canonical form representaions , which are useful in understanding the system structures including the controllability and observability [12, 18]. Finally, our des ign met hods are demonstrated by a simple example. Keywords. Non linear observer; state feedback linea rization; nonlinear dynamic compensator; nonlinear canonical form ; differential geometry.

1

Introduction

M.Zeitz [17] surveyed nonlinear canonical forms, which are useful in understanding the system structures.

In linear system theory, there is a design method of dynamic compensator in which an observer estimates the states and state feedback loop is constructed by using these estimated states (Fig.l). However , such a theory does not exist in nonlinear systems. Since 1970's , the differential geometric approach for nonlinear control systems has been progressing and yeilding some concepts and design methods [18, 19] ; the early successful works were studies on nonlinear controllability and observability [1], [4] . In 1980's, equivalence of nonlinear systems to linear systems under coordinate transformations was extensively discussed. Among them, the exact state space linearization technique by a coordinate transformation and state feedback ( also called 'slale feedback linearization') [5, 8, 9, 10] has been one of the most remarkable results and proved useful in some applications; flight control, chemical plant control, robotics arms. . . . The other available result is the non linear observer form which linearize the observer error dynamics by a coordinate transformation [11, 12, 15] . Corresponding to the linear dynamic compensator for linear systems, the authers [20,21] have shown that these two theories are comb ined to be a nonlinear dynamic compensator. On the other hand,

In this paper, the designs of nonlinear dynamic compensator using a nonlinear observer and feedback linearization are discussed. It is shown that linearization is either exact or not according to process models and output feedback control system can be designed systematically in a special case. We present the classes of nonlinear systems in which these designs are possi ble by canonical form representations. In Sec.2,3,4 the system descriptions and previous works are summarized, and the designs of nonlinear dynamic compensator and resulted control systems are described in Sec.5. They are demonstrated in Sec.6, and some concluding remarks are shown in Sec.7.

2

System Description and Some Mathematical Notations

In this paper, we consider a SISO nonlinear system described by

J(x)+g(x)u (1) h(x) where input u E R, state x E Er', output y E R, J,9 are Coo mappings on Rn, and h is Coo function on Rn. i: = { y =

External Input

Some notations are summarized below [3, 18, 19J. Lie bracket of vector fields , [J,gJ = ~J

atf}g

= g,

Lfh = ~J L~h = h, L}h

Linear Dynamic Compensator

365

9 :

- ~g ad}g = [J,adJ-lg], k

Lie derivative of function Fig.1

J and

= 1,2, ... h along vector field J:

= Lf(L~-lh),

k

= 1,2, ...

(2)

(3)

Olle-fo rm of fu nct ion h :

=[

dil (x)

Brunovsk.y Canonical Fo rm

.. . .!!.!!..] OXn

(4)

" [0'. D J " [OJ

X-~O-'lX+~V

IIl\'oluti"ity: a set of Coo vector fie lds {Id x ), f2(x), . . , fm(x)} o n Rn is ca lled invol ut ive if, an d on ly if, there ex ist ex> fu nctio ns ')"i)dx) such t hat

[f;. f)](·r)

L

=

o

0'" 0

1

Nonlinear Plant /ij

d :r)fdx), I :::; i, j :::; m, i

i-

j.

input tra nsform ation

k= 1

(5)

3

State Feedback Linearization

['i. 8. 9, 10.1 7.1 8) Suppose there exis ts a change of sta te coord inate x" = T(x)

(6)

F ig.2 State Feedback Lineariza tion llnde r whi ch system (I) is tra nsfo rnwd to a non linear troller canonica l form [7, 17) .;." { !J

A " ~' "+b" {(( (x " )+r " (~' " )Il} h "(.r" )

=

=

CO I1 -

( 7)

:\ s Ih(' lill(,Miz('d ~yst(, 1ll (0) has all poles at t he or igin. additional linciH state feedback in 1he t ransforllled coordi nal('

where (I' , ,." arc scalar functions on Rn an d

.1' "

(J:3 ) where k"T = [k j k~J is a gain n?Ctor mu st be p<'rfor!ll('d suc h tha t t he closed loop system is governcd by .i· " = (A " - b" /..- " ) J' " + b" t'c . ( 14)

T hen, a nonli near st ate feedbac k

(8)

u=o(x)+p(x)v where

o(x)

=

q"(T(x)) -r"(T(J:))' p(x)

Thell , the characteris ti c ('qllat io n wi ll be

p"(.5)

I

= r"(T(x))

s"

4

(9)

Th eo re m 1 [5, 9, IS} Suppose system (1) is gil'en. The

Nonlinear Observer .r

. . . adj-lg(xO))

10

a nonlinear ob-

( 17)

where

ii) the set of l'ECiorjields {g(x) , ad}g(.t), is int'olutil'e near xo.

0

.. , ad'F2g(:r)}

T(x)

LA(X) f A(X)

= [

0

eT

1

O.

1 0

= [ 0

i

where A (x) is t he non zero so lution of part ial d ifferent ia l eq uat ions

f.

b(y,u) = b"(y,ll)

0

T hen , the nonl inea r observer t hat estimates i can be designed as

( 11 )

L'FiA(X)

= Lad/g A(X) = ... = Lad ,-2gA(X) = 0

0 0 0 0

A=

If t he co nd it ions of Th eo rem 1 a re sat isfied a coord in ate trans format ion T(x) of eq . (6) is obtained by

Lad';-' gA( X)

(16)

x=/li-b(y,u) { Y = C- T .T- = .rn

{l0)

has mnk n .

Lg A(X)

[ll, 12, 15, IS)

= 11' (.1')

under which system (1) is transfor!llf?d sef\'cr ca no nica l fo rm

i) the control/ability matrix

= [g(X O) ad f 9(x o )

(1.5 )

Suppose t here exists a change of state coord inates

coordinate transforma tion (6) under which system (1) is tm.nsfomud to nonlinear controller canonical form (7) exists near a point X O if, and on ly if, the f ollowing conditions aI'e satisjied

At(xo )

+ /..-~ ,n- l + .. + k; s + kj

and the po les can be assigned a rbitrar ily by gain vector k" ( Fig.2 ).

compensates t he non linearit ies a nd the relation between the external input v a nd the state x" is lin ea ri zed to !3rul1ovsky ca non ical form

x"=Ax"+bv.

dct[ s l - ( /\ " - b" k")) =

=

.H -

b(y , u) -

9 . (In -

y)

where 9 = [91 . . . 9n]T is a ga in vector. From eqs. (17) ,(IS) the dynami cs of estimation er ror i - i is governed by

( 12)

~

366

= (A - 9 . i?) . e

(IS)

e= ( 19)

and the characteristic equation will be p(s)

= det[s] -

(A - 9 eT)]

Nonlinear Dynamic Compensator

5

= sn + 9n sn-l + ... + 92 S + 91.

(20) Finally, the state estimates in original coordinatei: is obtained by (21)

The authers [20, 21] have already proposed the design of nonlinear dynamic compensator by using a nonlinear observer and state feedback linearization technique. In this Section, the original design method and its drawbacks are summarized first, then it is shown that the drawbacks are removed in a certain class of nonlinear systems.

Theorem 2 [11, 1S} There exists a coordinate transformation (16) near a point X O unde,' which system (1) withoul inpul (g(x) = 0) is lransform ed to a nonlinear obse,'Ver canonical form

5.1

(22) where //(y) = [b;(y) ... b:,(yW, /I,e is same as eqs. (17) if, and only if, the following relations hold

Design (I) : Nonlinear dynamic compensator using state feedback linearization

Suppose the system of the form (I) sat isfi es the conditions for state feedback lin earization and nonlill car observcr designs described in Sec.3 and 4. A nonlincar dynamic com· pensator can be composed by an non linear observer for state estimation and state feedback linea rization with poleassignment usi ng the estimated states (Fig.4). [20,2 1]

i) th e obse/'uability ma/,-;x

(23)

Two drawbacks arises in this case: has rank n.

ii) The llniqlle vec/o,' field solution of

N(x)

r(.T) = [0 ... 0 1

r

• Although the nonlinear observer works with linear error dynamics as shown in Sec.3 , the state feedback linearization using the 'eslimaled values' of the states in cq. (8) results in approximate one.

(24)

• The state feedback linearization technique linearizes only the relation between input and statcs. However, also the linearization of output equatioll is required in most actual plant because the control system is usu ally constructed for the purpose of output feedback control.

is such thal

[ad/r, adJr] = 0, for all 0:::: i, j ::::

11 -

1. (25)

If conditions in Theorem 2 are sat isfied, coordinate transformation W(x) is obtained by [r(x) - adfr(x)

... (_1),,-1 ad'tlr(x)

Then, we show that there exists a class of nonlinear systems in which these drawbacks are removed.

(26)

] x= W -1(r)

Generally, when g(x) # 0 hold s, it is necessary to confirm that the conditions of Theorem 2 is satisfied when g(1') = o and that the system (1) is transformable into system (17) by the coordinate tran sformation IV .

Nonlinear Dynamic Compensator

v

Vc

input transformation for linearization

Non linear Plant

+ Non linear Observer Form y

$

x -w

Nonlinear Plant u -~---i~

x- f (XI + glxI U

~_[O"'O 1 0 00J~".: o "

"

{xI

coordinate . transformation

x

Fig.4 Design (I) Nonlinear Dynamic Compensator (using state feedback linearization)

a ly ' ul

-

+ G'(y - Xn)

Nonlinear Observer

Fig.3 Nonlinear Observer

367

In Design (I) , state feedback linearization is not exact because of using estimated states. Therefore, the output feedback linearization using only meas ured output will compose an exact nonlinear dynamic compensator. The class in which output feedback lineari zat ion can be performed is presented as follows.

The main target of actual control systems is output feedback control. Therefore, it is desired that the output equation is linearized at the same time with state space linearization. Then , the output feedback loop can be designed with state feedback compensator by pole-assignment. In this section the class in which such a design is poss ible is shown.

Definition 1 Th e system (1) is O"tp"t f eedback lineal';:ablt if it is transformed to th e special for m of 1I0nlincal' control/er canonical form Aa Ia + ba {qa(Y) ha (J' a )

+ "a(Y) "l

Design (Ill) : N onlinear dynamic compensator with linearized output equation in transformed coordinate

5.3

Design (11) : Nonlinear dynamic compensator using output feedback linearization

5.2

Definition 2 System (1) is output feedback linearizable to lineal' systcm with output if it is trallsfoT'med by the change of stat e coordinate Xb = T(x) obtain ed by ['1.(11) to the special form of nOll/incal' controller canonical form

(27)

where 'la , ra are sca lar functions all R, ha is any function all Rn, and

0 Aa

=

0 -a,

0 0 -a2

.i:b = { Y =

0

, ba =

A b Xb

cr Ib

+ bb {'!b(y) + I'b(Y) ll}

(30)

WhCT' f Ab , hb is th e same as Aa. ba in [q. (27) , = [Cb, Cb2 ..• Cbn], and qb, I'b are scalar functions on R.

er

0 ['1.( 11) is always transformed by a nonsingular linear transformation I e = 5 Tb to

-an

by the coordinate transfol'matioll l'a = 7'(.1') from ( 'I . (11). A, Ie c~ Xc

In comparison with eq. (7) , the n - Ih column of eq.(2/) ha s the non lin ear term depending only on 1l and y. Then, nonlinear output feedback

+ be {qb(Y) + I'b(Y) ll}

(31)

where

(28)

l Ie =

where %(y) Da(Y) = - - (- , j3(:r) = l'a y) l'a(Y)

T Cc

:

[

[0 ..

0 -a, 0 -a2 •

1 0

-~n

lin ear ize the system (1) in tran sfo rmed coordinate to

1 1

bC --

b,2 b"•

[ 1 b~n

l·

This form is the special form of nonlinear observer canonical form. Hence, a nonlinear observer can be designed in transformed coordinate Xc' Consequently, the nonlinear observer directly estimates the lin ea rized states Xc = SXb , i.e. the unique nonlinear coordinate transformation T(x) is required.

(29)

The non linear observer and linear state feedback for pole assignment in transformed coordinate aTe the same as Design (I), and total system is shown in Fig ..5.

Desired

Nonlinear dynamic compensator

Non li near dynamic compensator Vc

V

y

+

Linear state !feedback eq . (35) Output feedback loop Linear state feedback L-_ _-j ka

Fig.6 Desin(lIl) Nonlinear Dynamic Compensator (with linearized output equation in transformed coordinate)

Fig.S Design (Il) Nonlinear dynamic compensator (using output feedback linearization)

368

The nonlinear dynamic compensator and output feedback control loop is composed by the following steps;

• Nonlinear observer canonical form

1) Nonlinear observer Xc

{

Acxc+bc{qb(y)+rb(Y)u}

Y

= = =

~a12a21 - alla22)Y - all~Y U XI + (all + a22)Y + b2y U X2

(42)

where the coordinate transformation is

(32)

-9c' (xcn - y)

~I X2

(33)

(43) where 9c = [9q ... 9cnV is observer gain vector.

2) Output feedback linearization

Therefore, the Design (I) is possible. However, set Xb = x·, t hen (40) can be shown to be the special form of nonlinear contro ller canonical form (30) as

(34) where Qb(Y)

qb(Y)

= - -rb (-Y )'

/3b(Y)

1

Xb,

= -(-) rb Y

(a12a21 - a'lan)Xb,

-~ Xbl +

then the system in the transformed coordinate Xb = T(x) is linearized.

(44) Thus, th e output feedback con trol system with a nonlinear dynamic compensator can be designed following Design (lIl) as below:

3) Linear state feedback using estimated state v

= Vc - k[ Xb

+ (all + an)Xb, + a12~Y U

a!2Xb2

(35)

where k[ = [kb, ... kbn ] is a feedback gain veclr.

1) Nonlinear observer

4) Output feedback control

±c, =

Vc = I con(Yd,Y)

(~12a21 -alla22)X c, -allb2yu -:9c,' (X C2 - y) XC2 = XC2 + (all + a22)X C2 + b2yu -9C2 . (XC2 - y)

(36)

1

where Yd is the desired value of output, I con is the controller function or dynamical system , for example, integrator f(Yd - y )/ T/ dt (See Sec.6)

linear coordinate transformation Xb

Consequently, the dynami cs of closed-loop system (except for output feedback control loop) and observer error ec = Xc - Xc is governed by

(45)

= S-I xc

][ !:: ]

(46)

2) Output feedback linearization

(47) Therefore, the characteristic equation becomes p( s) = det[(s I - Ad bb k[)· (s I - Ac + 9c c~)] = {sn + (an + k b.)· sn-I + ... + (a2 + kb,)S + (al X {sn + (an + 9c')' sn-I + ... + (a2 + 9c,)S + (al

3) Linear state feed back for pole assignment

+ k b,)} + 9q)}

(48)

(38) which is the same results as the lin ear dynamic com pensator.

6

4) Output feedback control

(49)

Example

As a result, an easy calculation shows that the characteristic equation of overall system in transformed coordinate Xc is described by

The design methods of non linear dynamic compensator described in Sec.5 is demonstrated by a simple example. We consider the 2nd-order model of simple thermal system described by the following equations. XI X2 Y

1

= =

=

allXI a21 XI X2

+ al2 x 2 + a22X2 + ~X2 U

p(s)

where al = allan - a12a21, a2 = -(all the linear system.

Eq.(39) can be transformed to two canonical forms:

(40) where the coordinate transformation is

x~

xi

+ a22) just

like in

The classes in which nonlinear dynamic compensators can be designed are restricted as [the class of Design (I)] J [Design (Il)] J [Design (III)]. Since all design methods can be applied to this example, Design method (III) is applied. Notes that if Design (I) is applied, it does not automatically equal Design (Ill). Consequently, it is very important that the best design method applicable to the system should be found by canonical form representations.

• Nonlinear controller canonical form

{

(50)

X

(39)

(41)

369

7

[11] A.J.Krener, A.lsidori, "Linearizat ion by output injection and nonlinear observsers", Syst. Contr. Lett. 3, 47/52 (1983)

Conclusions

The class of nonlinear systems in which the nonlinear dynamic compensators can be constructed are shown and the design methods are proposed. The 'equivalence' theory has been studied since A.J.Krener [2] began and he sloved the problem when a non linear system without input can be transformed into a linear controllable system by a state space coordinate transformation. R.W.Brockett [5] proposed 'feedback lineari zation' using the state feedback, and solved it. Since then, {static / dynamic} {state / output} feedback linearization of {state space / input-output} and the lineaization of observer error dynamics have been studied by many authors. In this paper, static {state / output} feedback linerization of {state space / input-output} and the linearization of observer error dynamics are used in the design of nonlinear dynamic compensators. Th e class of nonlinear systems in which the nonlinear dynamic compensators can be constructed is extremely restricted mainly by the restriction of nonlinear observer designs. Some approximate methods for lineari zat ion and nonlinear observer are proposed by D.Bestle, M.Zeitz [12] , A ..l.Krener [13], C.Reboulet, C.Champetier [14] etc. We may apply these approximate theories to the nonlinear dynamic compensators.

[12] D.Bestle, M.Zeitz, "Canonical form observer des ign for non-linear time-variab le systems", In t. J . of Control, Vo1.38, No.2 , 419/431 (1983) [13] A.J.Krener, " Approximate linear ization by state feedback and coordinate change", Syst. Contr. Le tt. 5, 181/185 (1984) [14] C.Reboulet, C.Champct ier, "A new method for linearizing non-linear systems: t he pseudo lincarizat ion", Int . J. of Cont rol , Vo1.40 , No..!, 631 /638 (1984) [15] A.J.Krener, W.Respondek, " Non linear observers with li nearl izable error dynamics" , SIAM J. Control and Optimization, Vo1.23 , No.2, 197/2 16 (1985) [16] J.C.Kantor, "A finite dimensional nonlinear observer for an exot hermic stirred-tan k rcactor", Chemical Engineeri ng Science, Vo1A4 , No.7, 1503/1510 (1989) [17] M.Zeitz, "Canonical forms for nonlinear systems", IFA C Nonlinear Cont rol Systems Des ign Symposium, Capri, Italy, 472/477 (1989) [18] A.lsidori, "Nonlinear Control Systems: an introduction", Lecture Notes in Control and Information Sciences, Vo1.72, Springer-Verlag (1985), 2nd Edit ion (1989)

References [I] R.W Brockett , " System theory on group manifolds and cosel spaces", SIAM J.Control, 10-2,265/284 (1972)

[19] H.Nijmeijer, A.J.van der Schaft, "Non linear Dynamical Control Systems" , Springer- Verlag (1990)

[2] A.J.I
[20] T.Ohtani, M.Masubuchi, " A note on nonlin ear dynamic compensator", to appear in Trans. of S.I.C.E. in Japan

[3] R.W.Brockett, "Nonlinear systems and differential geometry", Proc.IEEE, Vo1.64, 61/72 (1976)

[21] T.Ohtani, M.Masubuchi, " A new design met hod of nonlinear dynamic compensator", prese nted to 1l\1A CS Simposium on Modeling and Control of Tcchnological Systems, Casablanca, Morocco (1991)

[4] Il.Hermann, A.J.I
370