NASA Technical Memorandum84249

NASA-TM-84249 19820022137

Applicationsto Aeronauticsof theTheory of TransformationsofNonlinear Systems

George Meyer, Renjeng Su and L. R. Hunt

May 1982t_,,i_!::;il)1_82

LAI_,GLEYRFSEARS_ CENTERL1SR;,RY, I'_ASA

HALiPTON, VIRGINIA

NASANationalAeronautics andSpace Administration

https://ntrs.nasa.gov/search.jsp?R=19820022137 2018-06-10T06:59:07+00:00Z

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NASA Technical Memorandum 84249

Applicationsto Aeronauticsof theTheoryof TransformationsofNonlinear SystemsGeorge MeyerRenjeng Su, Ames ResearchCenter, Moffett Field, CaliforniaL. R. Hunt, Texas Tech University, Lubbock, Texas

NASANationalAeronautics andSpace Administration

AmesResearchCenterMoffett Field. California 94035

N_ 9 -3oo t3""d_

APPLICATIONS TO AERONAUTICS OF THE THEORY OF TRANSFORMATIONS OF NONLINEAR SYSTEMS

George Meyer,* Renjeng Su,** and L. R. Hunt t

NASA Ames Research Center, Moffett Field, California, U.S.A.

ABSTRACT

We discuss the development of a theory, its application to the control design of nonlinear sys-tems, and results Concerning the use of this design technique for automatic flight control of air-craft. The theory examines the transformation of nonlinear systems to linear systems. We show how

to apply this in practice, in particular, the tracking of linear models by nonlinear plants. Resultsof manned simulation are also presented.

INTRODUCTION

Suppose we model a physical plant by a nonlinear system

m

_(t) = f[x(t)] +E ui(t)gi[x(t)] ' (i)i=l

where f' gl ' "''' g- are C== vector fields on ]Rn and f(0) = 0. If we are to have the output ofthis plant follow a _articular path, then we have a difficult problem to consider. However, if thereare new state space coordinates and new controls under which equation (1) becomes a linear system,then our task appears to be much easier because of the known results for controller design on linearsystems.

We feel that the following problems are thus of interest:

(a) Find necessary and sufficient conditions for the system (i) to be transformable toa controllable linear systems.

(h) Show how to use these transformations so that the controller design for nonlinear systemscan be reduced to that of linear systems.

(c) Apply the above theory to the field of aeronautics.

In the next three sections of this paper we discuss the solutions of these problems.

TRANSFORMATION THEORY

The classification of those nonlinear systems that can be transformed to linear systems is actu-

ally a subproblem of a much deeper result, the construction of canonical forms for nonlinear system_.We are presently developing a theory for such canonical forms, and in the case that a nonlinear sys-tem is transformable as in this paper, the canonical form is actually the Brunovsky (ref. i) form fora linear system.

Here we concentrate on the transformation theory developed in references 2, 3 and 4. Other

significant research in this area is due to Krener (ref. 5), Brockett (ref. 6), Jakubcyzk andRespondek (ref. 7), and Hermann (The Theory of Equivalence of Pfaffian Systems and Input Systemsunder Feedback). We also refer to the early work of the first author in references 8 and 9.

If we are to map our nonlinear system (i) to a controllable linear system, we may aswell assume that this linear system is in Brunovksy (ref. I) canonical form with Kronecker indices

m

KI, K2, ..., Km satisfying E Ki = n and _I • K2 > "'" • _ " Hence this system isi=l .... m

= Ay + Bw , (2)

where A is n × n, B is n × m, w = (Wl,W2, ..., wm) are the new controls, A is equal to

*Research Scientist at NASA Ames Research Center.

**Research Associate of National Research Council.

_Research supported by NASA Ames Research Center under the IPA Program and the Joint Services

Electronics Program at Texas Tech University, Lubbock, TX and under ONR Contract N00014-76-C-I136.

This paper is declared a work of the U.S. Government and therefore is in the public domain.

_io .... oloo ..... ol oo .....oolo...oloo ..... ol ..• . I.. I .....

_I ! II I

xl Ioo ..... oloo ..... ol oo ..... ooo ...... o OlO .... o oo ..... o

I .. ooio...o ..

• , • .e, oo

1

O0 ..... 0 O0 ..... 0 O0 ..... 0

I

O0 ..... 0 O0 ..... 0 010 .... 0

.... ooio...o

., °, .,,

Km

1

o0 ..... 0 00 ..... 0 00 ..... 0 ,

and B is given as

-. bo .,... (_

00 .... 0

K I

0 . • • • 1_

00 .... 0

00 .... 0

K2

010 . . . 0

O0 .... 0

0 0 0

I<m

O0 .... 1

The transformatio.: results we present are actually local (in some open neighborhood of the

origin in (xI, x2, ..., xn) space), and global theorems are found in reference 3. We simplify nota-tlon by saying R n whe_t we actually mean an open neighborhood of (0, O, ..., O) in _n. However,

Rm means (Ul, u2 ..... um) space (or (wl, w2, ..., wm) space) and this is not local.

We discuss the allowable transformations mapping system (i) to system (2). We want a C_ map

Y = (Yl, Y2, .... Yn, wl' w2 ..... Wm) mapping _n × Rm[(xl, x2..... Xn, ul, u2, .... Um) space]to _n × Rm [(Yl, Y2, "''' Yn, wl, w2' "''' Wm) space] that satisfies the following conditions:

J

I. Y maps the origin to the origin,

2. Yl, Y2, "'', Yn are functions of Xl, x2, ..., xn only and have a nonsingular Jacobianmatrix,

3. Wl, w2, ..., wm are functions of xl, x2, ..., Xn, Ul, u2, ..., um and for fixed x l, x2,..., Xn, the m × m Jacobian matrix of wl, w2, ..., wm with respect to ul, u2, ..., umis nonsingular,

4. Y maps system (i) to system (2),

5. Y is a one-to-one map of _n x R m onto _n × Rm.

Next we introduce some basic definitions from differential geometry.

If f and g are C_ vector fields on _n, the Lie bracket of f and g is

_ _f[f'g] = f - 7x g

where and _x are Jacobian matrices. We let

(ad°f,g) z g

(adlf,g) = [f,gl

(ad2f,g) = [f,(f,g)]

(adkf,g) = [f(adk-lf,g)]

A collection of C_ vector fields hl, h2, ..., h r is involutive if there exists C_ func-

tions Yijk such thatr

=_Yijk(X)hk(X)_ , 1 _ i, j _ r, i # j

m

[hi,hj| (x)k=l

Let <.,.> denote the duality between one forms and vector fields. If _ = _I dxl + _2dxp +

+ c_ndxn is a differentiable one form and f a vector field on _n then

(_l,f> = _ifl + w2f 2 + ... + w fnn

To state the main result from reference 4 givin_ necessary and sufficient conditions for trans-

forming system (l) to system (2) we need tile following sets:

C = Igl,[f,gl] .....(adKl-lf,gl),g2,[f,g2] ..... (ad*:2-1f,g2).....

gm,[f,gm] ..... (ad_m-lf,gm) }

Cj = gl,[f,gl] .....(ad 3 "f,gl),gs,[f,g2] .....(ad_:j-2f,g?) .....

_'-2 1gm,[f,gm] .....(ad 3 f,gm) for j=l,2 .....m

Theorem 2.1 There exists a transformation Y = (Yl, Y2, -.., Yn, Wl, w?, ..., wm) satisfying condi-tions i) through v) above if and only if on _n

I) the set C spans an n dimensional space,

2) each set C. is involutive for j = 1,2 .....m, and,2

3) the span of Cj equals the span of Cj _ C for j = 1,2 .... ,m.

Let °l = _I, 02 = KI + _2, "'', O = KI + _2 + "'" + _ = n. Then the transformation is con-m . . m .structed in reference 4 by solving the partial dlfferentlal equatlons

<dYl,(adJf,gi) ) = O,j=O,l ....._:i-2 and i=l,2 .....m,

<dYol+l,(adJf,gi) ) = O,j=O,l ....._:2-2 and i=1,2 .....m,

3

<dYOm_l +l,(adJf,gl)> = O,J=0,1....,Km-2 and i=l,2,...,m, (3)

m

<dYol,f) +_ ui<dYol'gi> = w Ii=l

m

<dYo2'f) +_ ui<dYa2'gl) = w2i=l

m

<dYn,f) +_ ui<dYn,gi> = wm ,i=i

where the matrix

<dyt,(adKl-lf,gl)) . <dy l,(ad_l-lf,gm)>

_dy °1 +l'(ad_2-1f'gl)> • (dYol +l'(ad_2-1f'gm)

(4)

(dYom_l+l, (ad_'m-lf,gI)) . .(dV.om_l+l'(adKm-lf'gm) )I

is nonsingular.

It can be shown that matrix (4) being invertible means we can solve for Ul, u2, ..., Urnin terms of wl, w2, ..., wm in the last m equations in equation (3). \

Equation (3) can be formally solved by considering a sequence of ordinary" differential equationsas in reference 4, but we shall not mention details here.

If a nonlinear system is transformable to a linear system, we study' tile process of using tilL'transformation to construct a controller for the nonlinear system.

TRANSFOPJ_TIONS IN CONTROLLER DESIGN

Let Y = (Yl' Y2' ---, Yn, Wl, w2, ..., wm) be the transformation from system (I) to system (2)as before. The structure of the control system using transformation theory is illustrated infigure 1. Tile design scheme is implemented on the "linear part" of the diagram, and this systemis in Brunovskv form.

We ask that the output of the nonlinear system follow a particular path which corresponds to atrajectory for the output of the linear model. If we know how to design for the linear system, thenwe actually have a tracking of a linear model by a nonlinear plant.

Linear design is used to generate an open loop command Wc, for the system (2), and we findthe corresponding y coordinates Yc by plugging w c into equation (2). The transformation Ymaps the measured x space to y space and v is compared to Yc and the difference is an

error e v. The regulator yields a control 6w which sends ey to zero, and variations in plantdynamics'and disturbances are compensated for in this way.

The controls wc and 6w are added and transformed through the inverse map R (actually

wc + _w is substituted into the last m equations in equation (3) and u = (ul, u2, ..., u m) isgenerated) to obtain a control which is applied to the plant. Thus we have an exact model follower,:::_d the difficult problem of finding an open loop control and the regulator control are constrained tothe linear system.

The remainder of this paper contains the application of the transformation theory to aeronautics.

AUTOmaTIC FLICHT CONTROLLER DESIGN

The a_rcraft will be represented by a rigid body moving in 3-dimensiona] space in response togravity, aerodynamics and propulsion.

4

The state

x = _ XC _3 x _3 x SO(3)x _3 (5)

where r and v are inertial coordinates of body center of mass position and velocity, respectively;C is the direction cosine matrix of body fixed axes relative to the runway fixed axes (inertial),and _ is the angularvelocity.

The controls,

Mwhere u is the B-axis moment control such as ailerons, elevator and rudder in a conventional air-

craft or roll cyclic, pitch cyclic and tall rotor collective in a helicopter; and uP controlspower - throttle in a conventional aircraft, and the main rotor collective in a helicopter. The

state equation consists of the translational and rotational kinematic and dynamic equations:

= v

= fF(x,u)(7)

=s(_)c

= fH(x,u)

where fF and fH are the total force and moment generation processes and x£X. We wish Lu transformequation (7) into a linear system.

In genera], fH is Invertible wlth respect to the (vector) pair (_,uH), and, for the specificclass of helicopter maneuvers being considered (i.e., no 360° rolls), fF is invertlble with respect

to thu (scalar) palr (v3,uP). Thus, a function h:X × R_ _ U can be constructed such that if

<uM) = h(r,v,C,_,_0,_30) (8)uP

then

= _)o(9)

_3 = _3o

for all admJssH)le maneuvers. That is, angular and vertical accelerations can be chosen as the newset of independent controls in which case the state equation may be written as follows

= v

= f0(r,v,C,O30) + cfl(r,v,C,O30,_,_,0)(in)

= s(_)c

= _o

where _ = l, and fl(r,v,C,O,O,O) : 0 for all admissible maneuvers.

The function f0 Is invertlble with respect to ((_I,_2,E3(_)),C) where E3(_) is an elementaryrotation about the z-axls and represents the heading of the helicopter. Thus, a function hf:_ B× SO(2) * SO(3) can be constructed such that if

CO = hf(r,v,_0,E3(_0)) (ii)

then

= 4 0 (12)

Equations(8) and (ii)are the trim equationsof the process equation(10)(with c = 0). That is,for a given path (r(t), E3(_(t))), t 2 0 with _3(t) = O, the corresponding state and control may beconstructed as follows

r0= r(t)

vo = _(t)

CO = hf(r0,v0,_(t) ,E3, (@(t))

_o = q(_Ct) (13)

_o = (_o)"

u0 = h(r0,v0,C0,m0,_o,O)

where the function q extracts to from CCt = S(oo) . The required time derivatives in equation (13)

can be computed provided that the path (r,E 3) i_ generated by the system diagrammed in figure 2where • represent_ a scalar integrator and Y0 the control (w in the previous section is },5 here).

We construct an approximation to the linearizing transformation as follows: YI, RI, Q are con-structed so that

y = Y(x) _ Y0(x0) + Y16x = Y0 + Y1_x(14)

u = R(x,y 5) _ u0 + Rl6y 5 + Q_x

Here Y0 is the transformation when _ = 0, and 6x (and 6y 5) is the perturbation about the nominal

xr_ (and y0 ) given in equation (13) (and figure 2).

From equation (I0) with c = 0, it follows that (C = (I + S(_)C 0)

(6r)" = _vO 0 0 0

(6v)" = _r +-_v _v+ c +_.a (15)

(_) = _

(_)" = _o ,

where _ is attitude perturbation.

The pattern of equation (15) after .qome rearrangement ef coordinates is shown in equation (lb).

10 I 0 0 0I

r2 _ r2 !

vll vl0 C 1 C2 C 3 0 C4 0 Cb

v2] v2

• i

_°1]_:21 c2

= 0 0 0 l + 0 i (16)

t3l _23 u2

r3l rl &3

toll Wl LvCO2 ] 0 0 0 0 to2 I

_31 m3

v3J V3.

In the present case of the helicopter, CI, C_, and C5 are negligible. Their effect will he con-trolled by the regulator.

The transformations

I

6yI 6rIlI 0 0 0

_y_ 6r212

gYl 6ri!0 I 0 0

6Y22. 6r2

6Y_ €l

3 C2 C36y2 €2

= o o (]7)

_y33 • c 30 I

3 6r36Y4

146Yl 6_°1

C2 C3

4 6_a26Y2 0 0

0 I I

_r _6r3.,Yh

6_1[ Yl

6321 C21 -C2]C3 16vll: " (18)

0 I 6 5

take the system in equation (16)(with C I, C4, C,, = 0) into the canonic system:

0 I 0 0 0

I

0 0 It 0 0 0I

(&y)" = 0 0 0 1 6y + 0 6y 5 (19)

0 0 0 0 __

Thus, the linearlzlng transformation (Y and R in figure 1) is constructed.

That the accuracy of the transformation is adequate may be seen from the results of the simula-tion of the flight experiment to be briefly summarized next.

The test consists in automatically flying a trajectory which exercises the system over a widerauge of flight conditions as shown in figures 4 and 5.

Thus, the test takes the helicopter from hover (WPI) to high speed (150 ft/sec) turning ac_!elera-

tion, ascending flight.

Figure 6 shows the resulting tracking errors.

Is quite small. Tile accelerntlon errors e., whichAs can be seen, position tracking error er v.

is due to the neglected terms in the construction of the linearizing transformation is also qultesmall. In summary, the resulting performance of the system is good.

REFERENCES

I. Brunovsky, P.: A Classification of Linear Controllable Systems. Kibernetika (Praha). Vol. 6,1970, pp. 173-188.

2. Su, R.: On the Linear Equivalents of Nonlinear Systems, Systems and Control Letters. Vol. 2,No. i. (To appear in print 1982.)

3. Hunt, L. R.; Su, R.; and Meyer, G.: Global Transformations of Nonlinear Systems. IEEE Trans.on Automatic Control. Vol. 27, No. 6. (To appear in print 1982.)

4. Hunt, L. R.; Su, R.; and Meyer, G.: Multi-input Nonlinear Systems. Differential Geometric

Control Theory Conference. Birkhauser Boston, Cambridge. (To appear in print 1982.)

5. Krener, A. J.: On the Equivalence of Control Systems and Linearization of Nonlinear Systems.SlAM Journal of Control. Vol. 11, 1973, pp. 670-676.

6. Brockett, R. W.: Feedback Invariants for Nonlinear Systems. IFAC Congress, Helsinki, 1978.

7. Jakubcyzk, B.; and Respondek, W.: On Linearization of Control Systems. Bull. Acad. Pol. Sc_.,Ser. Sci., Math. Astronom, Phys., Vol. 28, 1980, pp. 517-522.

8. Meyer, (;.;and Cicolani, L.: A Formal Structure for Advanced Flight Control Systems.NASA TN D-7940, 1975.

9. Meyer, G.; and Cicolani, L.: Applications of Nonlinear System Inverses to Automatic FllghtControl Design - System Concepts and Flight Evaluations. ACARDograph on Theory and Applica-tion of Optimal Control in Aerospace Systems, P. Kant, ed., 1980.

REFERENCE=INPUT Y.___.c__ Y [_ x

m

=AYc+Bwc =f, uigi. i=1

; w c _ u

LINEAR SYSTEM REGULATOR TRANS- NONLINEARFORMATION PLANT

ANDINVERSE

Figure I. Structure of the Control System

Y_ Y_ Y_ Y_ Y_

_. . E3

_ _- r3

Figure 2. Reference Canonic System

Ez" 500

400I

4 m/s

_<_300200

100

Figure 3. Experimental Flightpath Shown in Horizontaland Vertical Planes

10 IIvN/5

- 10 t 'v -j

-_ ,__,;_ •< .2r t-I- = P_ v2. / i 1. _ t \ 4_-I r I I: \ ' . _-_ l,

" "7o '< -.2 "/

'-1- 0 90 180 270 360 450TIME, sec

Figure 4. Speed and Acceleration

of Experimental Trajectory

Z E 2 er(1) er(2)

_, - -2_

_" e_121e_(31e_(l)

_o o .,''-:" ____ __r,,,,

"' -.1 I I I I i = i i I I0 90 180 270 360 450

TIME, sec

Figure 5. Tracking Errors

10

I Report No. 2. Government Accession No. 3. Recipient's Catalug No.

NASA TM 89249

4. ratle and Subtitle 5. Report Date

APPLICATIONS TO AERONAUTICS OF THE THEORY OF . May 1982TRANSFOR>L_TIONSOF NONLINEARSYSTEMS 6.Perforn.nnOrqaniPa,,onCode

7. Au_,hor(s) 8. Performing Organization Report No.

George Meyer, Renjeng Su and L. R. Hunt A-8943

10. Work Unit No.

9. Performin 9 Organization Name and Address

, T-Sg_TvAmes Research Center, Moffett Field, CA 94035 tt Con,tactorGrant No

13 Type of Report and Period Cov-'red

12.Sponso,'in9 AgencyName and Address TechnicalMemorandum

14. Sponsoring Agency Code

National Aeronautics and Space AdministrationWashin_ton_ DC 20546 505-34-31

15 Supplementary Notes

Point of contact: Dr. George Meyer. Ames Research Center, MS 210-3,Moffett Field, CA 94035. (415) 965-5444 or FTS 448-5444.

15 _bstract

We discuss the development of a theory, its application to the control

design of nonlinear systems, and results concerning the use of this designtechnique for automatic flight control of aircraft. The theory examinesthe transformation of nonlinear systems to linear systems. We show how

to apply this in practice; in particular, the tracking of linear models

by nonlinear plants. Results of manned simulation are also presented.

17. Key Words (Sug_t_ by Authorls)) 18. DIstributton Statement

Nonlinear SystemsTransformations Unlimited.

Flight Control

Design Subject Category: 66

19S_urityClassif.(ofth,sre_r:,UnclassifiedI20SecUritVClassif'(°fthis_!Unclassified21.No.ofPages13 I22.An2_,ce"

"For sale by theNat,o,,:_l ]P,.Fm¢_l Inforq_Jtpon Serv,ce. Sp=,ngfield, Virg;m,= 22161