Capítulos-1-y-2.pdf

Nonlinear Control Design I Geometric, Adaptive and Robust 1 Riccardo Marino

Patrizio Tomei University of Rome 'Tor Vergata' Rome, Italy t

PRENTICE HALL London New York Toronto Sydney Tokyo Singapore Madrid Mexico City Munich

I i I 1 Contents I

Preface

1 Introduction 1.1 Nonlinear feedback cokrol 1.2 Towards modern nonlinear control 1.3 Linearization by state feedback 1 .4 Inverse systems and zero dynamics 1.5 St,ate feedback for uncertain systems 1.6 Sonlinear observers 1.7 Output feedback 1.8 Output feedback for uncertain systems 1.9 Outline of t,he book 1.10 Physical cont,rol problems 1.11 Exercises

I STATE FEEDBACK

2 Feedback linearization 2.1 Pole placement for linear syst.ems 2.2 Feedback linearization 2 .3 Linearization by change of coordinates 2.4 Partial feedback linearization 2.5 Stabilization of t,riangular syst,ems 2.6 Global feedback lineari~at~ion 2.7 Ext,ension t o mult,i-input systems 2.8 Physical examples 2.9 Conclusions 2.10 Exercises

viii

3 Adapt ive feedback linearization 3.1 Matching and triangular conditions 3.2 Robust stabilization 3.3 Self-tuning regulator 3.4 Adaptive feedback linearization 3.5 Extension to mult,i-input systems 3 6 Physical examples 3.7 Conclusions 3.8 Exercises

4 O u t p u t t racking 1 1 I n v e r s ~ systems and tracklng dynamics 4 2 Input-output feedback linearization 4 3 Disturbance rejection 4 4 Disturbance attenuation 3 5 Adaptive tracklng w ~ t h transient specifications 1 6 Extension to multivariable systems $ 1 7 Phyri, a1 t ~ x a m p l ~ s 4 8 Conclusioris 3 9 Exerci.rs

I1 O U T P U T F E E D B A C K

5 Adapt ive observers 3 1 Observers for linrclr systems - 2 ()lwervcrs rvit1-i linear error dynaniics 5 3 :lcl;ij)t~\.c~ ob,c-.r\.rrs 5 1 F:xti>ris~c>~\ t ~ , n~:lltivarldl,lr? sys t e~us 5 5 Plivslcai rxaniples 5 6 C ~ ~ ~ l ~ l i l ~ l ~ ~ ~ l ~ 5 7 Exercisps

6 S t a b ~ l i z a t i o n and exponential t racking t, 1 'it ~t lc 111it p ~ 1 1 f t > r d \ ~ ( h lincar~zation 6 2 Dl rinrriit o u t p u t feedba~k Iinear~zation 6 3 0 1 1 t p 1 l t f r ~ ( i b a ~ k stiit)ilization 6 1 t:xponf,ntlal tracking 1, output feedback b 5 Phi ~ i c c l l c,xarnples 6 b C'onclusinns 6 7 Exrrc~.rs

Con tents

7 Robust regulation and adaptive tracking 7.1 St,ruct,ural geomet,ric conditions 7.2 Robust st,abilization 7.3 Self-t,uning set point regulation 7.4 Adapt,ivc tracking 7.5 Physical exanlples 7.6 Conclusiolls 7.7 Exercises

A Basics of differential geometry A. 1 Diffeomorphisms A.2 Vect,or fields A . 3 Lie derivat,ives A.4 Manifolds and distribut,ions

B Basics of stability theory B . l Stability theorems f B.2 Tools for adapt,ive cont,rol design

Bibliographical notes

References

This book is intended as a text for graduate courses or final year undergraduate courses on nonlinear feedback control design including geometric, adaptive, and robust techniques. It is also intended as a reference for practising engineers and applied mathematicians. Severa applications from electric machines to spacecraft i and aircraft control, from robotics t,o power syst,ems are included and worked in detail to appeal to the physical int,uition of engineers and to the curiosity of mathematicians. The assumed background is the capability of designing, or at least understanding, simple control syst,ems and an elementary knowledge of differential equations, stability t,heory, linear algebra, and functions of several variables. Notat,ion and concepts from differential geometry are used to state results in a coordinate free fashion, which is essential when dealing with nonlinear systems. Basic notation, terminology and results from differential geometry and stability theory which are used in the text are summarized in two appendices.

The book presents a self contained introduction to nonlinear feedback control design for continuous time, finit,e dimensional, uncertain systems. It is focused on design, both st,ate and output feedback, and not on analysis. Systems may be affected by uncert,ainties, such as unknown constant parameters and time-varying disturbances. Differential geomet,ric techniques are used to identify the classes of nonlinear systems considered and to develop nonlinear design t,echniques when unknown parameters or dist,urbances are not present. They also provide the frame- work on which robust and adaptive algorithms are designed for uncertain systems. A feedback control is called adaptive when it guarantees asymptotic output tracking for systems with unknown constant parameters and no disturbances. When unknown time-varying dist,urbances are present, controls achieving stabilization and arbitrary dist,urbance attenuation on t,he output are called robust,.

At the present stage of the research on nonlinear control this monograph presents a coherent collection of global design techniques basic geonetric algorithms. including feedback linearization and observers. and global output feedback exponential tracking controls, an adaptive control theory which redesigns for classes of nonlinear systems some of the basic state and output feedback adaptive algorithms (including adapt~ve obserxers) obtained for linear systems, robust controls

xii Preface

which achieve dist,urbance at,t,enuat,ion by st,at,e feedback; worst case robust st,a- bilizing algorithms which generalize earlier result,s for linear systems. All resulk are firmly root,ed in differential geometric conditions guaranteeing global transformations int,o certain canonical forms. Most of the control algorit,hms presented have been established since 1980. At t,he beginning of the eight,ies t,he stability proofs of adaptive control algorithms for linear systems appeared and robust con- t,rols began to appear. At t,he same time generalizat,ions of pole placement and observer design techniques for nonlinear syst,erns were obt,ained, using tools from different,ial geomet,ry which were introduced during the sevent,ies in t,he s tudy of c~ont,rollability and ob~ervabilit~y. Adaptive versions of those nonlinear algorit,hms were announced startsing from 1986. Since 1990 output feedback algorit,hms (adap- tivc. robust and geomet,ric) have been given for classes of nonlinear systems. The research effort on nonlinear cont,rol has been motivat,ed by the increasing availabil- ~ t y of low cost powerful dedicat,ed digital signal processors (DSP) and by t,he more dernanding performarlce required in robotics, motor drives, aircraft and spacecraft, control which typically involve nonlinear dynamics. Since 1985 several books have appeared on analysis and design of nonlinear systems wiih known paramet,ers and on adaptive cont,rol of linear systems with unknown parameters. This book reports r ~ ( . m t rtisillts on feedback control design for uncertain nonlinear systems: much of the mat,erial included has never before been published in book form. Only single ~ r ipu t , s:ngle output (SISO) syst,ems are dealt with in derail, even though most sig- nlficant applications admittedly involve multivarlable ( ~ ~ I I ~ I O ) models. However, once the SISO results are fully mastered, significant ;LIIMO applications can be carr~t ,d out . Tlie complete multlvariahle ext,ension of SISO results is not straight- forward and, in su:ne instances. not yet, available. Several output feedback SISO problc~rns are also st111 operi. One of the objectives of the book is to provide the framr3work in which new researc ti problerris car1 be formulated.

Thc book is divided into two part.s. Part I (st,at,e f eedbxk) consists of Chapters 2 . 3 ar:d 4 while, Part I1 (observers anti output feedback) colisists of Chapters 5 (on ohsc~rvcr.;i. 6 and 7 Chaptvr 1 is itn introtiuctory one and provides an informal .iir\.c,v 1 1 i r l l c ' tif,sign t c c l ~ ~ i l q u c ~ ~ prrsc.ntcd throiigh simple esamples: at the end o f t hi, t .l~~iptr-r 12 phplcal c,orlrro! problems. w11ic.11 arc, tlisc,ussed In the text . are for~~~i l lar t~ci C'llapter 2 dcbals wit11 s ta te fc~t~cll)ac,k gt>ornt1trlc. rontrol for single input .;ystf3rils i~ lc~l i l ( i~~ig fcrcil)ac.k li~iearizatiori ariti pal-tial feedback l inear iza t io~~. Robust a~ici ;tdaptl\.r >rat(, fertibac,k linearization algoritlirris are developed In Chapter 3. 111 ('1:uj)tr~r 1 ti s11lgle output to 1)r. c.ontrollrd is introtiuced and the output tracking ;)~ol)l(~rri i h \tittt'(i T h r first half of Chapter 4 is devoteti t o s ta te feedback design for SISO ystcJrrls w ~ t h no uncertainties and deals with zero and tracking dynamics. rrlln:1riulrl phase qystrrns and iriput-output feedback liriearizat,ion. The second half of C'haptc~r 4 adtiresst~s adaptive ~npu t -ou tpu t lirlearizat,ion. disturbance decoupling ar~ci cilst i ~ r h t ~ r i c . ~ ~ atteliuation. Chapter 5 deals with nonlinear observer design, bot,h c~claptivc~ alid r io~l-adaptl \-(~ Chapt,er 6 adtlresses output feedback linearization, sta- t j 1 1 1 ~ ~ i t 1 0 1 i arid c~xporlential tracking Chapter 7 addresses both robust and adapt,ive o l ~ t p ~ i t fc~r~dbwc,k tlrslgn. The theory is illustrated through many examples (more

... Preface XIII

than 70) and physical appl i~at~ions (more tjhan 20) which are worked in detail in the text and many suggest,ed exercises (more t,han 120). Two appendices collect the basics of differential geometry and stability theory act,ually used in the book. The Bibliography, which is by no means complete, includes all the sources which were consult,ed and suggests further reading on related topics. The material cov- ered in t,he text allows adequat,e selection of t,opics to meet different course content requirements. Chapters 2, 6 and the first halves of Chapt,ers 4 and 5 provide an introduct,ion to nonlinear geometric control design (including s ta te feedback, observers and out,put feedback) for syst>ems with no uncertainties. This material was used for a 30 hour graduat,e course taught by t,he first, author a t the Universit,y of Illinois in Urbana-Champaign arid at, the U~iiversity of Rome "Tor Vergat,an. A more elementary course providing an introduction t.o adaptive and robust cont,rol can be carved out from Chapt,ers 3, 4, 5 and 7. T h e mat,erial includes t,he design of ~ilodel reference adapt,ive controls, adapt,ive and robust feedback linearizing controls, adaptive observers, and adaptive and robust output feedback algorithms for minimuni phase syst,ems wit,h known r e h i v e degree.

This book sumrnarizes a coyect,ive research effort which we have had the plea- sure to contribut,e to by exchanging ideas and t,echniqlles wit,h colleagues during congresses, workshops and joint research projects. Many rcslilts reported in t,he book arc due t,o our co-authors of several papers, W. Boot,hby, D. Elliot,t,, J . Levinc, I . Kanellakopoulos, P. Kokotovic. S. Nicosiai S. Peresada, N J . Respondek, A . van d r r Scliaft,. D. Taylor, and to co-authors of our co-authors. We would like t,o thank all of t ,hen~ and in particular Petar Kokotjovic for the enthusiasm and knowledge that he con\:eycd in Joint research efforts, Salvat,ore Nicosia who established in 1981 the co~ltrol group at t,he Universi t ,~ of Romr "Tor Vergata" and int,roduced bot,h of 11s to control theory, and A1hert.o Isidori who opened up t,o bot,h of us as graduate stlitle~its t l i t 11u11lint:ar c:ont,rol world and has been over the years a reference of

R Marino and P Tonie~ Rome, Italy

Introduction

1.1 NONLINEAR FEEDBACK CONTROL

This book deals with the design of feedback controls for nonlinear systems with unknown constant parameters or time-varying disturbances. The n o n l i n e a r con- t r o l s y s t e m s considered are moMelled by finite dimensional, deterministic ordinary differential equations

in which x E Rn is the s t a t e , xo is the initial condition, u ( t ) : R+ + R m is the c o n t r o l input. Q ( t ) : R+ -+ RP is the d i s t u r b a n c e input, y E RS is the o u t p u t vector. L'l'hen m = s = 1 we speak of s ing le i n p u t s ingle o u t p u t systems; we speak of m u l t i v a r i a b l e systems when either m > 1 (multi-input) or s > 1 (multi- output) . Cont,rols are at the designer's disposal while disturbances may be either totally unknown or generated by a known model ( e x o s ~ s t e m )

wlth unknown inltlal conditions Bo Modelled disturbances become u n k n o w n pa- r a m e t e r s when in the exosystem v(B, t ) = 0 for every 6 and t > 0 i e they are constant u i t h respect to tlme and equal to an unknown lnltial condltlon ( Q = 0 B(0) = B o ) The output variables y are to be controlled, they are requlred to track a r e fe rence s igna l y , ( t ) this is called a t r a c k i n g p r o b l e m Ljrhen the reference signal y, IS constant we speak of s e t p o i n t r e g u l a t i o n

The object of this book is to deslgn feedback control algorithms uhich s o l ~ e the tracklng problem feeding back the measured variables Two cases are considered the state x IS alallable for measurement and a s t a t e f eedback c o n t r o l is to be deslgned (Part I Chapters 2 . 3 and 4 ) , only the output y IS abailable for measurement and an o u t p u t f eedback c o n t r o l IS to be designed (Par t I1 Chapters 6 and 7 possibly but not necessarily on the baas of state o b s e r v e r s (Par t I1 Chapter 5 ) .A d y n a m i c o u t p u t f eedback control algorithm of order r IS a nonlinear

Introduction

system described by ordinary differential equations

When y is replaced by x, i.e

w = P(W, 5 , Y,, t ) , w(O) = wo, w E RT ti = t1(w, x , y,, t)

we speak of d y n a m i c s t a t e f e e d b a c k control algorit,hms. If we simply have

t we speak of s t a t i c o u t p u t f eedback and of s t a t i c s t a t e f e e d b a c k control, respectively. T h e control is to be designed so that the tracking problem is solved for t,he c losed l o o p syst,em

A d d ~ t ~ o n a l spec~f ica t~ons are usual11 imposed the vectors x ( t ) and & ( t ) are requlred to be bounded. when d~sturbances B(t) are present t h e ~ r effect on the output ma\ be required to be rejectcd ( d i s t u r b a n c e r e j ec t ion problem) or at tenuated to dri arbitrnr? degree ( d i s t u r b a n c e a t t e n u a t i o n proble~n) , a sympto t~c s tab~l i ty of ~ t n e q u ~ l ~ b r ~ u r n p a n t of Interest (x, u,) w h ~ c h correspoilds to a des~red coristant rt,fere~icc output y, 1s u5uallj requlred In the special case In which there are no d~atur '~arices B(t) = 0, the referenct. o u t p t ~ t y, is Lero and 2 1s measured, the control prot)leni becomes s t a t e f eedback s t a b i l i z a t i o n I e t l l ~ design of a s ta te feed- t)ac I\ control wlllth r n a k ~ s (x,, a,) a s j ~ ~ l p t o t i c a l l ~ b t a b l ~ . locally or globallj, when on l \ 2 I > d \ a l l~ i ) l e for measurrrnent WP speak of o u t p u t f e e d b a c k s t a b i l i z a t i o n

1 . 2 TOWARDS MODERN NONLINEAR CONTROL

Noril~nenr c.oritro1 systems with known pararnet,ers and 110 dist,urbances

Towards rnodern nonlinear control 3

have been traditionally approached via their linear approximations about equilibrium points x,, corresponding t o constant inputs ir ,

in which (, = x - x,,, n, = 71 - 1 1 , and

are Jacobian matrices wit,h I,, and 7 1 , such that,

This approach involves the solut,ion of several linear control problems and makes the transition from one t,o another rather critical. Good performance, or even s tabi l i t ,~ , may not be maintained over wide ranges of variations of st,at,e variables. Tlie set of linear syst,ems arising from such appr~x imat~ ions may be viewed either as a single linear system wit,h unknown paramet,ers (adaptive approach) or as a farnily of linear syst.ems subject to disturbances (robust approach). B0t.h adaptive and robust algorithms have been developed to control t,he set of linear syst,ems arising from linear appro~ imat~ ions .

The availability of increasingly more powerful and less expensive microproces- sors and t,he need for bett,er performance st,imulated cont,rol engirieers to design innovat,ive norllinear corit,rol algorithins for advanced applications such as robots. aircraft and spacecraft, and elect,ric machines used both as motors and generators. This happenc>tl in the sevent,~es, while nonlinear cont,rollability and observabilit,y begari to be studied using diffc:rt,r~t,ial geometric: t,ools, arid niotivat,ed a theory of nonlinear feedback control design which was developed in t,he eight,ies. As often happens practice preceded theory. In tliese applications. nonlinearities suc:h as electric torques, ccnt,ripet,al and Coriolis forces, and irlert,ial forces play a significant role and can be exactly modelled using well knowri physical laws. Engineers. on the basis of a deep uriderstarlding of the physics of the syst,em and of accurate nonli~lear n~odels . designed nonlinear cont.ro1 algorit,hms in different cont.exts in order to meet tle~nandirig specifications which could not be met by means of linear c.oritrol techniques: field oriented cont,rol for ind~iction mot,ors in 1972, autopilot for I~elicopters in 1975. and corriputed torque for high speed rigid robots In 1976. These algorithms share a comnion innovative feature: they make use of rionlinear c.hanges of st,ate coordinates and of nonlinear stat,e feedback (which aims at nonlinearity cancellat.ion) t,o make the closed loop syst,em hopefully linear. or at least

4 Introduction

simpler, in new coordinates. For in~ t~ance , in the context of electric drives, the field oriented control algorithm for induction motors forces the closed loop system to coincide with the dynamics of a d.c. motor, which is well behaved and easy to cont,rol. This is not far from the spirit of the pole placement theorem which states that any linear, cont,rollable system (stable or unstable) is transformable into a system with desired eigenvalues by means of state feedback.

The above applications may be viewed as the beginning of modern nonlinear feedback design. The novelt,y of these t,echniques may be appreciated by considering simple examples, such as the first order system (0 # 0)

If y, = 0, so that the purpose of the cont,rol is to drive tlhe output to zero, on the basis of the linear approximat,ion about t,he origin

we design the control (k > 0)

The closed loop system

has three equilibrium points: an asymptotically stable one at x = 0 with domain of attraction -@ < x < VG and t,wo unstable ones a t x = *LlklB The larger the gain k, the larger the domain of attraction of t,he origin. However, no matter how large k is, global asympt,ot,ic stability is not achieved while it is guarant,eed by the nonlinear control (k > 0)

which makes the closed loop system linear and asymptot,ically st,able. The nonlinear term ox3 is cancelled and t,his requires exact knowledge both of the parameter b' arid of the nonlinearity x3.

While (1.1) illustrat,es the benefits of nonlinear feedback. the importance of using the right coordinat,es in nonlinear control design is clarified by the second order example

Towards modern nonlinear control

which is expressed in the new coordinates zl = X I , z2 = x2 + ox: as

The control (kl > 0 , k2 > 0)

guarantees global asympt,ot,ic ~ t~ab i l i ty . Notice tha t also in this case the term 0xf has to be exactly known.

The advantage of combining both nonlinear change of coordinates and nonlinear state feedback is apparent in the second order example

X I = x2 + ox: 52 = U

Y = Xl

in which no linear control

designed on the basis of the linear approximation about the origin makes the origin globally asymptotically stable when 0 # 0. On the other hand the nonlinear control (ki > 0, k2 > 0)

makes the orlgln globally asymptot~cal l j stable In fact the closed loop system becomes in new global coordinates zl = x l 2 2 = x2 - Ox:

There are cases in which a linear design based on linear approximations may cause serious drawbacks due to neglected nonlinear terms. Consider the system

.As in example (1 5 ) . a linear control

6 Introduction

based on t,he linear approximation about the origin does not guarantee, when B > 0, global asymptotic stability since the syst,em

x1 = x2 3 x 2 = - k l x l - k 2 x 2 + Ox2

Y = Xl

has an unstable limit cycle. If the linear gains are chose11 as k l = k', k2 = k , it turns out that, for instance when 0 = 113, all t,raject,ories starting in t,he region { x E R~ : kx: + ( I l k ) x; > 9) tend t,o infinity. This is shown by defining T = k t , z l = z2/&, 2 2 = f i x 1 which transforms t,he system into a reversed-time van der Pol equation. While on the basis of the linear approximation increasing k seems to improve the response, the initial conditions are to be closer arid closer t,o the x2-axis in order to guarantee that the corresponding t ra je~t~ories t,end t,o tjhe origin. Such problems do not arise if we use the nonlinear cont,rol ( k l > 0, k2 > 0)

which cancels the nonlinearity O X ; . 4 The examples (1.1), (1 .3) , (1.5). and (1.7) recnpt,ure only some of t,he feat,ures of

the above mentioned pioneering pliysical applications which act,ually involve mul- t,ivariable models. However, they show t,hat there are classes of nonlinear systems for which nonlinear controls outperform linear controls designed on the basis of linear approximations. This is reasonable and to be expected since the former use more information on the syst,em than the latter do: in fact, all of t,he information cont.airied in nonlinear models may be used by nonlinear control algorithms.

1.3 LINEARIZATION BY STATE FEEDBACK

As we have seen, the examples (1.1), (1.3), (1.5), and (1.7) are t,ransformable by st,at,e space change of coordinat,es and riorllirlear st,ate feedback into linear controllable svst,ems having linear input-out,put maps. Since not every nonlinear syst.em enjoys such advant.agcous propert,irs a classification is n r e d ~ d . t,he system

z-1 = T?

S? = ,; + ,, r { = J , - o r ;

Y = X I (1 9) zhows that ~t nid) riot be poss~hle to ach~eve s~multaneously both goals b\ state feedback In fa( t (1 9) is transformed by the change of coordinates

Linearization by state feedback

and the nonlinear s ta te feedback

which is linear and controllable but, does not have a linear input-output m a p when B # 0. On t,he other hand (1.9) is t,ransformed by the stat,e feedback

which has a linear input-output map but i ts s ta te equations are only partially linear when B # 0.

A theory is available which characterizes in terms of necessary and sufficient conditions those syst,ems which are s t a t e f eedback l inea r i zab le (i.e. transformable into linear and cont,rollable syst,ems by st,at>e space change of coordinates and nonlinear stat,e feedback) and those which are i n p u t - o u t p u t f e e d b a c k lin- e a r i z a b l e (i.e. t,ransformable intjo a syst,em with linear input,-output map) . These two result,s were obt,ained a t the beginning of the eighties for multivariable systems. They allow us t,o establish for inst,ance that,:

(i) the syst,em

is riot feedback linearizable for any constant parameter B but is input-output feedback linearizable for any 0;

(ii) the syst,em

is feedback linearizable but is not input-out,put feedback linearizable:

Introduction

(iii) t,he system

is neit,her linearizable about the origin nor input-output linearizable by st,ate feed back.

The results are constructjive. For instance, the linearizing transformation for system (1.13) is given by t,he change of ~oordinat~es

and the state feedback control

The closed loop system (1.13)-(1.15) becomes in z-coordinates

Note that t,he input-output map is still nonlinear. For those systeins which are not feedback linearizable a nat>ural problem is the

characterization of the part which can be made linear by stlate feedback. System (1.12) is not feedback linearizable but the st,ate feedback

makes the input-output map linear and the closed loop system (22 = x2 -I- 8 2 : )

partially linear, that is the cascade connection of a nonlinear system and a linear controllable system

Inverse systems and zero dynamics 9

1.4 INVERSE SYSTEMS A N D ZERO DYNAMICS

While feedback linearization is the generalization of the pole placement theorem for linear systems, input-output feedback linearization is the generalization of the zero-pole cancellation technique Consider the linear system

I 5 1 = 5 2 ! 5 2 = 5 3 + 71

x 3 = X I + 2 2 + 2 3 Y = X l

whose transfer function is (s E C)

The linear system (1.17) and the nonlinear one (1.12) have a similar structure when 0 = 0 in (1 .12) . The state feedtack (compare with (1.16) with 0 = 0)

applied to (1.17) makes the closed loop syst,em

unobservable: its t>ransfer function is

The control (1.18) places the poles in s = 1 and in s = 0 with multiplicity 2 and causes a zero-pole cancellation. The unstable dynamics

evolve in the unobservable subspace S = {x E R" X I = 0, x 2 = 0 ) and are characterized by an eigenvalue which is equal to the cancelled zero. The inverse s y s t e m for (1.17) is

LVhen it is driven by y = y = y = 0 its dynamics coincide with the dynamics made unobservable b) a state feedback w h ~ c h causes a zero-pole cancellation .As is well known for linear systems the zero-pole cancellation technique is not feasible for n o n - m i n i m u m p h a s e qs t e rns (1 e with unstable zeros)

10 Introduction

While transfer functions, poles and zeros are not. defined for nonlinear systems, observabilit,~ is a well defined concept: the closed loop syst,em (1.12) cont,rolled by ( 1.16) wit,h 0 = 0 is not observable since tlhe dynamics

which are unstable and evolve on the subspace S = {s E R? X I = 0, x:! = 0) are made unobservable from the output by the st,ate feedback (1.16). The dynamics which are made unobservable by the input,-out,put linearizing stat,e feedback, are called zero dynamics and generalize t o nonlinear syst,ems the notion of zeros developed for linear ones. More precisely, they generalize the not,ion of dynamics evolving on urlobservable subspaces.

Inverses for nonlinear syst,ems may be defined and are re1at)ed t>o t.he zero dynamics. The inverse syst,em for (1.12), with 0 = 0, is

If yr ( t ) 1s the signal to be t,racked, the dynamlcs f

are called tracking dynamics. When the tracking dynamics are driven by y = y = ,y .= 0, t,hey coincide wit,h the zero dynamics; in this case t,he inverse system generates the input signals which are needed to maintain the output equal t,o zero. Sint r system (1.12) is non-mini~num phase ( the zero dynamics are unstable) the linearizing s ta te feedback (1 16) is not feasible. In fact, even t,hough v is designed to drive y = X I and y = x2 to zero for any init,ial condit,ion different than zero, ruit) (and the control ~ ( t ) as well) tends to infinity as t tends to infinity.

1.5 STATE FEEDBACK FOR UNCERTAIN SYSTEMS

The 11:1c~~~r1zirig tec l i~l~ques so far discussed deal with norlliric?ar systems with no untertainties tile\- rrquire stat,? measurt3rnents and involve t,he cancellation of i i on l~ l l t~ i t r~ t l e~ . LVhilc t h t w are significant systems whose nonlinear rriodels may be exactly obtained from physical laws and whose states are measured, it. is much less r(.allstic to assume thitt all the parameters involved in the models (such as niasses, ~ntlrtias. resistar~c,rs. and ~nductances) are exactly known or t,hat the system is not, affected at all by disturbances

I t is the11 ~ la tu ra l to invest,igate what happens to c.ontrol systems involving uriknow11, or not precisely known, paramc,ters. T h e control algorit,hms should be dc,signcd so that the control problems are strill solved in spite of paramet,er unc,ertaint!.. This problem was addressed during t,he eight,ies for linear systenls and led to adaptive and robust controls. Since 1986 adapt,ive control of nonlinear

State feedback for uncer-tain systems 11

systems has ?merged as a very active field. Adapt,ive feedt~ac~k linearizat,iori arid adapt,ive input,-o~itput. li~iearizat,io~i algorit,lirns have been developed for classes of systkms which are rest,ric,t,ed by t.he ass~lrrlpt,ion of linear pararrieterization

arid by st,ruct,ural triangularity c,oliditiorls on the vect,or fields l lZ(x) which multi- ply the para~nctcrs. Those result,^ arc* sigliificarit since a fccdbac~k lirirarizi~lg control applied t,o systcrrls with l~rlk~iowri pararnctcJrs may rlot ~rlcct, the control ot)jec,tivc. To see this corlsidcr again syst,rrrl (1 .7 ) anti suppose tliat R > 0 is llnknowrl arid that. rl^ > 0 is t.hc cst,i~nat,e of B so that t,llc coritrol (1.8) becomes

arid the corresponding closeti loop systcrrl 1s f

X i = 5 2

j 2 = ( B - ) - k - k2.r.' If B - B < 0, t.liat is 0 is ~ve res t~ i~na te t i , t,hr origirl is globally asyrnpt,ot,ic:ally stahlr . If R - d > 0, tliat is B is ~lriderest,irriated, a llrnit cycle t:xists, which is the h o ~ ~ n d a r y of t,he dorriain of at,ttraction, arid global asyrrlpt,otic stability is riot ac,hicved. This drawback is overcwrrie t ~ y nonlinear aciaptivc t,echnicll~es. In fact, t,he adaptive feedback linearizing control ( k l > 0, k2 > 0 )

wit 11 itciapt at i o ~ i tbrinrnics

assures that I~rn~-., ( l . r ( t ) J l = 0 for any i~li t ial corlciit,ion ~ ( 0 ) . B(0) and for ally I I I I ~ I I O ~ I I pararrietcr B , positive or ~iegativc, This co~lc l~~sior l rriay be tiraw11 ~isirlg t hr Lyapuriov f ~ ~ n c t ~ o r l

I.' = T - .r .L

12 Introduction

Similar problems arise for linear or affine systems with unknown parameters. Con- sider for instance the system

wit,h (Bo, 81, B2) unknown parameters which may be positive or negative. In this special case, adaptive feedback linearization techniques lead to the design of clas- sical PID (proportional int,egral derivative) controls. There are systems such as

which do not satisfy the triangular conditions even though they are feedback linearizable for every value of 0. At the present state of t,he ar t , feedback linearizability for any parameter value is not a sufficient condition for the existence of adaptive controls.

Uncertainties due to unknown parameters may be call@ structured. Uncertain- t,ies may also be unstruct,ured: for instance a nonlinear function may be bounded by a known polynomial or the nonlinearity may be expressed by look-up tables of experiment,al data confined in a cert,ain region. These situations cannot be handled by adaptive algorithms and require robust and worst case techniques.

In those cases in which the system is affected by disturbances the problem of disturbance rejection or, at least, disturbance attenuation on t,he output to be controlled naturally arises. Such problems have been widely st,udied for linear systems and recently for nonlinear ones. The disturbance rejection (or decoupling) problem for nonlinear syst,ems is completely solved in t,erms of necessary and sufficient, conditions. The di~t~urbance at,tenuation problem, which was posed and solved for linear systems at t,he beginning of the eight,ies, has been recently addressed for nonlinear systems and sufficient condit,ions are available. For instance, suppose that a reference signal y,(f) is to be t,racked for the linear system

If 0 is a constant and unknown parameter, the adaptive cont,rol ( A > 0) u = - X ( y - y,) - By + y, B = Y I Y - Y,) (1.23)

guarantees as>mpotic tracking for any in~t ia l cond~t lon, i e limt,K(y(t) - y , ( t ) ) = 0 On the other hand, a robust control does not aim at asymptotic tracking but only a t a rb~ t ra r \ disturbance attenuation on the ;racking error, according to the formula

Nonlinear observers 13

which has to hold for any positive t and for any arbitrarily large positive k, whenever y(0) = yr(0). Note that B ( t ) is an unknown disturbance which need not be const,ant but only square integrable. The robust control ( k > 0)

solves the disturbance attenuation problem for system (1.22).

1.6 NONLINEAR OBSERVERS

When measurements of the whole state are not available in linear observable systems and only some states, called the outputs, are measured, output feedback algorithms are based on observers which provide state estimates tending asymptotically to the unmeasured states Observability guarantees the existence of a linear 9 change of coordinates z = T x transforming a linear observable system x = A x + b u , y = cx into an observer f o r m

The system

with k = [ k l . . . . . k,IT and s n + k,sn-I + . . . + k 2 s + k1 a Hurwitz polynomial. is an observer since the error dynamics (2 = z - i)

Z =

are asymptotically stable. It is then natural to look for those nonlinear systems transformable by a nonlinear change of coordinates z = T ( x ) into

z +

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

. . ,

0 . . . 1 0

-

a1 a2 ,

- an -

A y + Tbu = A,z + ay + Tbu

14 Introduction

with 7 and 6 vector fields depending on t,he output y only. Necessary and sufficient. condit,ions which ident,ify such systems were obtained in 1983. Once the t,ransformat,ion is det,ermined, the nonlinear observer is given by

which has t,he sarne linear error dynamics as in the linear case. This result is a non-trivial one as simple examples such as

show System (1.25) is transformed by the global change of coordinates

As in ft,edhack linearization, nonlincar observers are based or1 nonlinearity can- cc.llation which may not be feasible when unknown parameters are i n ~ o l v e d . For 1ristanc.e if fl 1s unknou.11 in (1.25) and is its estimate in t,he observer ( k l > 0 . k 2 > 0)

the error dvnarnits arc no 1orlgc.r l ~ n c a r

If the output y i t j docs not tend to zero as t goes to ~ n h n ~ t y . the error 5 ( t ) does not t m d t o L P I O e ~ t h e r unless 6' = 4 Iri those cases obserlers are to be redesigned to obtairi convcrgllig s ta te Psrlmates also In presence of unknown constant param- eteri such oi)sc>r\rlrs are called adaptive observers Uritier the restriction of I in~ar parar1ietc~i7arlon thls p r o b l ~ ~ r i has been iolbed both for l ~ n e a r and nolillnear \ \ i tcms

Nonlinear observers

Consider for instance the syst,em

The observer is (kI > 0, k2 > 0)

wit,h R being t,he est,imat,e of the unknown constant, pararnet,er 19. We look for an updat,e law

such that t.he est,imat,e error 2 = x - x , whose dynamics are (I9 = Q - 0 )

t,ends t,o zero as t --t cc for any unknown value of 8. In t,his case we choose k l and k 2 as follows ( A > 0)

so tha t , denoting A = A, - kc,, we obtain

Thls lrnpl~es that thcie cxist pos i t~ve definite matrices P and Q such that , accord~rig to . \ l rvt~~-E;alnlan-Yac~~bovlch Lemma B 2 2

Thc ~ l p d a t f , law

glvrs. togct her with (1.28). an adaptive observer. 111 fact. since B is constant.

and t hc error dyriani~cs are

16

Differentiating the function

V = z T p 5 + R 2 with respect to time and recalling tha t Pb = cT, we have

d V - = - z T ~ 5 + 2 z T p b y 2 j - 2 y 2 i ~ 1 = - Z T ~ 5 . d t

Introduction

If the output, y ( t ) is bounded, it follows that

lim llZ(t)ll = 0 t-oo

and (1.28), (1.29) define an adaptive observer.

1.7 OUTPUT FEEDBACK \

The assumption that all the states are measured and available for feedback is often unrealistic: for instance some states may not be accessible for sensors (such as rotor currents in a squirrel cage induction motor) or t,he required sensors may be too expensive or unreliable. For linear systems this difficulty is overcome by using s ta te observers and the separation theorem which establishes tha t any linear stabilizing s ta te feedback algorithm globally stabilizes when the states are replaced in the algorithm by their estimat,es provided by a linear observer.

For nonlinear systems global st,abilization by out,put feedback is, in general, a difficult problem. However, there are classes of systems for which global stability 1s achieved by means of observer-based controls. Consider the syst,em

If HZ = 0. in the coordinates 21 = 2 1 , 22 = xl - x2. (1.30) becomes

If the parameter is known and only y 1s measured, an output feedback stabillzing control is designed using the function

Output feedback 17

with i2 being the estimate of t,he unmeasured state 2 2 . Differentiating V with respect to time we obtain

3 ' + ( 2 2 - i2)(21 - 22 - 81z1 - 22)

Choosing the control (k > 0)

with the state estimate dynamics i2 given by

3 i2 = rl - i2 - 2612,

and substituting in v we obt,ain

The closed loop system (1.30)-(1.31) is such that 2 tends exponentially to zero and i2 e~ponent~ially to 22 as t goes to infinity. The dynamic compensator (1.31) is based on a reduced order observer (1.32) which observes only in closed loop: it is an o u t p u t feedback s tabi l iz ing con t ro l . Note that the correction term -O1z; is added to the open loop reduced order observer given by

System (1.30) (with O2 = 0) is also globally stabilized by a static output feedback ( k l > 0: k2 > 0)

which is not based on nonlinear it,^ cancellation. The closed loop system becomes in z-coordinates (zl = X I , 22 = X I - x2)

Global stability is proved by using the Lyapunov function

whose time derivative is negative definite for kl > 114 and k2 2 1

Introduction

1.8 O U T P U T FEEDBACK FOR UNCERTAIN SYSTEMS

Syst,ems (1.27) and (1.30) belong t,o t,he class of observable minimum phase nonlinear systems with linear a~ympt~otically st,able zero dynamics and nonlinearities depending, in suit,able st,ate coordinat,es, on t,he measured out.put y only. For these systems output feedback controls can be designed provided tha t parameters and n~nlineari t~ies are exactly known. When t h ~ syst,ems are uncertain, for the same class of nonlinear systems additional out,put feedback cont,rol algorithms are available:

(i) adaptive tracking ones, which are observer-based and require exact knowledge of nonlinearities and linear paramet,erization with respect t,o unknown constant parameters;

(ii) robust stabilizing ones, which are not observer-based, require only functional bounds on the nonlinearities, and do not need linear parameter i~at~ion.

For instance the system

belongs to such a (.lass: it has relative degree 1 and linear asympt,otically stable z rm dynamics as can he seen by performirig the linear change of coordinat,es

n.hlch arc, obtained by sett,ing y = 23 = 0, are asymptotically stable so t,liat the b\..st

Output feedback for uncertain systems 19

and 82 is unknown. an adaptive output feedback contxol may be designed. Rewrite (1.30) as

In x-coordinates an adapt,ive observer is

i1 = .C2 + ( I + 4 a y 2 + k l ( y - 21) i2 = u + $ a Y 2 + k 2 ( y - 21) f 1 2 = y 2 ( y - 21)

with ( A , positive real)

so t,hat t,he charact,eristic polynomial of ( A , - k,cc) is ( s + l ) ( s + A,), i.e. t,here is a zero-pole cancellation in t,he t,ransfer function

Let

be the reference model which generat,es the reference outsput y,. The gains kc = (i.,3, k q ) are chosen so t,hat the charact,erist,ic polynomial of (A,-bk,) is (s+l)(s+A,), i.e. there is a zero-pole cancellat,ion in t,he transfer function c c [ s I - ('4, - hk,)j-'b = l / ( . s + A,) (A, is a positjive real). In both zero-pole cancellat,ions we rnake use of tho crucial property enjoyed by the triples ( A , - k,c,, b , c,) and ( A , - bk,, b , c,) of being minirrlum phase and of relative degree 1 . T h e cont,rol is designed as

in which the st,ate estimat,es are provided by the observer. Both the cont,rol and t,he observer t,ry t,o cancel the nonlinearity f l a y 2 by using the same est,irnat,e i2 of the unknown parameter f 1 2 which is now adjusted according to the update law (7 s~lficient,ly large positive r ~ a l )

which reacts both to t,he tracking error y - y, and to the observer error y - The stabilit,y of t,he closed loop syst,em can be assessed by expressing the dynamics

20 Introduction

of t,he tracking error e = x - x,, observer error 2 = x - i and estimation error O2 = O2 - B2

2 = ( A , - bk,)e + by2& + bkc2 = ( A , - kocc)Z + by2i2

and using the function

with PC and Po positive definite solutions of

( A , - b k c ) T ~ c + P,(A, - hk,) = -Q, T P,h = C ,

( A , - ~ , c , ) ~ P , + Po(Ac - k,c,) = -Q, T Pob = Cc

with Q , and Q , suitable positive definite matrices, accqd ing to bleyer-Kalman- Yacubovich Lemma B.2.2. The t,ime derivative of V is in fact

The adaptive tracking control (1.36) is based on a n adaptive observer and solves, for any value of the unknown parameter B 2 , the tracking problem. The design of a tracking control for (1.30) is simple since (1.30) has relative degree 1. More complex adaptive control algorit,hms solve the tracking problem for systems with relative degree greater t,han 1. System (1.30) belongs t,o the class of observable nonlinear systems which are minimum phase, have linear zero dynamics, a well defined relative degree independent of uncertainties and nonlinearities which depend, in suitable coordinates, on the output only. Linear systems characterized by the transfer function

~ i t h unknown but constant coefficients ( a o , , an - I , bo , bn-,) belong to such a C ! ~ S S provlded that they are minimum phase (all 77 - p zeros of h,-,~"-~+ +bls+ho ha\? negatiLP redl part) and the s~gri of hn-, (usuallj called high-frequency galn) 1s known

U'heri parameters d o not enter linearly but a prior1 bounds are known, robust rionlinear algoritlirns can stabilize not only system ( 1 30) but also the more general one

Outline of the book 21

in which the nonlinearity $12F is uncertain and its graph is only known to belong t,o a region bounded by known nonlinear f ~ n c t ~ i o n s . In new coordinates t = x 2 - X I , y = x l syst,em (1.38) becomes

Assuming t,hat )81ye2p11 < y8, i.e. bounds on 81 and $2 are available, t,he st,atic out,put feedback control

globally st,abilizes the family of syst,ems (1.38): in fact the function

has a negative definit,e time d e r i ~ a t ~ i v e

The cont,rol (1.39) is designed on the basis of the worst case: it is a robust output feedback stabilizing control. Since syst,em (1.38) is of relative degree 1 and minimum phase, the design of (1.39) is rather simple and is a generalizat,ion of linear high-gain control techniques. More complex dynamic output feedback algorithms have to be resorted t,o for systems wit,h relative degree greater than 1. These results generalize t,o a class of nonlinear syst,ems t,he simultaneous stabilization results obtained for linear systems with uncertain parameters belonging to bounded sets.

1.9 OUTLINE OF THE BOOK

This book int,roduces nonlinear design t,echniques which led t,o and were motivat,ed by applications in elect,ric machines, robotics, aircraft and aerospace control, and power systems. It is divided into t,wo parts: s ta te feedback design (Chapters 2, 3: 3) arid output feedback design (Chapters 6 and 7 ) , including observer design (Chapter 5 ) . Basic result,s from differential geometry and ~ t~ah i l i ty theory which are actually used are recalled in Appendix A and Appendix B, respectively, mainly to clarify t,he notation arid the terminology: the reader is, however, referred to several excellent books on these subjects which are listed in the Bibliography. Each chapter contains worked examples, physical examples and suggested exercises. The results presented are non-local in the sense t,hat they are global whenever the change of coordinates ilivolved is global. Co~ltrol algorit,hms which only yield local results are outside the scope of the book. All results are p re~en t~ed for single input single output syst,ems

22 Introduction

even though most relevant applicat,ions involve mult.ivariable models. E ~ t ~ e n s i o n s of some of t.he main result,^ t,o multivariable syst,erris are only st,at,ed, when available, a t t,he end of each chapter. Worked physical examples involving multivariable systems are also given.

Chapt,ers 2 and 3 deal wit,h nonlinear systems with no output,s Chapter 2 deals wit,h systems with no uncert,ainties, i.e, not affecred by disturbances and wit,h paramet,ers and n~nlineari t~ies exactly k~lown. The main result is the feedback liriearizat,ion theorem which is present.ed as a generalization of the pole placement t,heorern for linear systems t80 a class of nonlinear systems ident,ified by different,ial geometric conditions. Part,ial feedback linearization is also introduced. Chapter 3 deals wit,h uncerthin feedback linearizable syst,ems which contain unknown paramet,ers or are affected by time-varying disturbances: adaptive and robust versions of the feedback linearization algorit,hrri are developed which include model reference adaptjive cont,rol and high-gain techniques for linear systems.

In Chapt,er 4 a single output variable t,o be cont,rolled is int.roduced for the first time in the book: inverse systems, relative degree* zero dynamics, t,racking dynamics, and rni~iimum phase syst,ems are defined and the d i ~ t ~ u r b a n c e rejection and the disturbance at,tenuation problems are formulat,ed. The main result is t he input,-output feedback linearization t,heorem, which is a generalization of the zero- polr, canr~l la t ion technique for linear systems. It,s relationships wit,h zero dynamics and part>ial feedback l inrar i~at~ion are illustrat,ed; it,s adaptive version, for systems with unknown paramet,ers, is developed. Necessary and sufficient conditions are given for t,he solvabilit,y of t,he dist,urbance rejection problern. Sufficient conditions are given for the solvability of t,he disturbance attenuation problem and for the design of adaptive tracking controls wit11 guaranteed transient specificat,ions.

Chapter 5 deals with observers axid adapt,ive observers. T h e main result provides sufficient coriditions for the exist,ence of observers wit,h linear asympt,ot,ically st,able error dl-narnics. This result is viewed as an extension of t,he construction of observers for linear systems which involves the t,ransformat,ion into observer forms. Xtlaptivr versions of observers wit,h linear error dynanlics are developed which in- c.ludp :tdaptivc obs~rvc r s for linear sys t en~s with unknown paramet,ers as special ('ases.

Chapters G and 7 deal u i rh output feedback design Chapter 6 addresses sys- t e m ~ ul rh n o ilrlc ertaintles Conditions for linearization are glven both by s t a t ~ c axid b\ ti\:iarnic output ferdbatk For a clasq of rionllnear s ls tems, identified by di f fcr~nt i~i l georncrric c o ~ i d l t ~ o ~ ~ s , output feedback control algorithms are deslgned Chapter 7 addre5ses the tracking problern for systems with unknown constant pardmeters and distinguishes two cases nonlinear parameterization arid linear pa- rarneteri7ation In the first case a robust stabllizlng control ls deslgned while In the second rare an ddaptlre tracking control 1s developed Both algorithms apply to 11rit.dr ~rllrilrnl~in phase s ~ s t e m s with unknown coefficients they generalize known rr,sults for lir~car 71 stems to a class of nonlinear systems

Physical control problems 23

1.10 PHYSICAL CONTROL PROBLEMS

Physical syst,ems which have nonlinear models and are widely studied are reported below. These rnodels and t8he corresponding c o n t ~ o l problerns will be used in the t,ext to illust,rat,e the t,echniques developed.

1.10.1 (Induction motor) A detailed model of an unsat,urated induction rnotor is given by

n'wh - -

-

d t d i ,

- -

-

dt 12.1 R, n , it1

*a + d l j l b (L, L, - i ~ i 1 2 ) E , L,L, - 121 2 -

, v f 2 ~ , + L;R, L, 7, + a (L, L, - iW2)L, L, L, - 121 2

i2.1 R, n,hl - rL'h - d dl, (L,L, - 1212)~, L, L, - 111

in which rotor speed d, rotor fluxes (u,,. L L ~ ) , and stator currents ( 7 , , zbi are th r states. rotor inertia J , stator and rotor inductances (L,, L,), mutual inductance All, stator and rotor resistancrs (R,. R,), and tht, number of pole pair5 n , the palarneters T h t voltages ( I I , , 1 1 ~ ) are to be desig~led as a fu~lction of the whole state (& (1,. w b , I , , 7 h ) (or , prc,ferablv as a function of measured xariables ( d , ?,, l b ) since (LS, ~ h ) are difficult to measure) so that the speed , tiacks the reference 5ignal ,,(t) and a: + W ; i y regulated to t hc~ desired t onstant valut, @, (typically chosen to maximize motor tfficiencg) in spite of load torque TL xariations and possibly rotol resistance R, variations due to ohni~c hcating A simplified model (current-fed motor) is gixen b\

In which ( I , , 1 6 ) are the control variables

24 Introduction

1.10.2 (Rigid robot) A two link planar robot, with t,wo revolute joint,s is modelled, assuming rigid joints and rigid links, by

(81 + 203 cos q2)(i'l + (02 + O3 cos q2)q2 - 283 sin 42(11(12 - 03 sin q2$ + o5 cos ql + e6 cos(ql + qz) = vr 1 (02 + 83 cos q2)ijl + o4q2 + O3 sin q 2 i T + 196 cos(q1 + q2) = u2 Y l = 11 cos q1 + 12 cos(q1 + q2) y2 = 11 sin ql + 12 sin(q1 + (12)

in which the parameters 0,, 1 5 I, 5 6 depend on physical ones

where g is the gravity const,ant, I,, m,,, l i are inertia, mass and 1engt)h of link i , respectively, and d, = 1,/2. The torque inputs (?I1, u,z) are to be designed as a function of the st,ate (ql ,q l , q2, i2) or as a function of the measured variables (ql, q2) so that the outputs, which are the end point coordinates (yl , y2), follow a reference trajectory (y,l ( t ) , y,2(t)) . Parameters 8i may be unknown. More generally, an N-link robot is modelled by

in which q E R." is the vector of link displacements, 11 E R~ is the vector of torques applied t,o each link, y E R" is the output vect,or, B(q) is the iv x N inertia matrix. C ( q , q)rj are Coriolis and centripetal forces, g(q) is t,he vector of gravity forces. The general control problem is t o design 71 as a function of t,he state [ q . 4 ) or as a funct,ion of the output y so that y tracks a desired reference signal 9 , .

1.10.3 (Rigid body) Consider the dynamics of a rigid body. Let ( x , y. z ) be the coordinates of t,he center of mass in an absolut,e frame, (@, 0, O ) be the roll, pitch and yaw angles, and ( 1 1 . z.. L ~ ) the velocity component,s in a relative frame att,achec-l to the rigid body. The cont,rol inputs are t,he torque components ( T ~ . T,,. r 2 ) with respect to the same relat,ive frame and the thrust p. The equations of motion are

o = d, + tan O(d, sin d + J, cos 4)

&,/ S I I ~ d + &, cos @ L =

cos e

Physical control problems

JzGz + ( J z - J,)wvwz = ~z J?,G?l + ( J z - Jz)wzwz = 7 , J z ~ z + (J1 / - J z ) w z ~ ! / = T Z

x = 1 1 . c o s ~ c o s t 9 + v ( c o s ~ h s i n t 9 s i n ~ - s i n $ c o s ~ ) t w (cos $J sin 6' cos C#I + sin 1L sin 4 )

y = ~1.sin~~cost9+v(sin$sint9sind+cos.rl tcosC#I) +w (sin IC/ sin t9 cos 6 - cos w sin 4)

Y v = g cos t9 sin C#J - w,v + w z w + -

rn,

Z w = g COS 0 COS 4 + W ~ / 1 1 - w,u + -

m,

in which (z, y , z ) are the cockdinates of the center of mass in an absolute frame with t,he vert,ical z-axis oriented downwards, ( v , v , w ) the velocity components in a relative frame attached to the rigid body, (c$ ,O, d) are the roll, pitch and yaw angles, (w,, w,, w,) are the components of the angular velocity with respect to t,he principal axes of inertia, (X + J p , Y , Z ) are the components of the force vector except gravity, is the mass, and (J,, J,, J,) are the moment of inertia with respect to the principal axes of inertia. The control problem is to design (r,, r7/, r,, p ) so that a reference traject,ory ( x r ( t ) , y , ( t ) , z r ( t ) , Q , - ( t ) , Q r ( t ) , $ , ( t ) ) is followed.

1.10.4 (Spacecraft) The dynamic model of a spacecraft driven by gas jet actuators is given by

O = w,cosq3 - w,sin$ w, sin @ + w, cos Q

$2 = cos s

in which: ($ ,Q, d l ) are t,he roll, pit,ch and yaw angles; (d,, w,. J,) are the co~nponent~s of t,he angular velocity of t,he spacecraft expressed in a frame attached tlo t,he spacecraft; J is the symmet,ric posit,ive definite inertia matrix of the spacecraft; ( T ~ , r Y r 7,) are the torques supplied by the gas jet actuators. The control input,s (r,, r,, r,) are to be designed so t'hat the reference attitude ( d l ( t ) , O , ( t ) , ( t ) ) is tracked. The entries of t,he inertia mat,rix J may be unkrlown

26 Introduction

1.10.5 ( I n v e r t e d p e n d u l u m ) The dynamic equat,ions of an invert,ed pendulum on a moving cart are given by

( ~ L I + m )i + m l cos p p - ml slncpcp2 = 711 r n l c o s c p x + m 1 2 ~ - m g l s i n p = 112

in which cp is the angle displacement of the pendulum from the vertical con- figurat,ion, x is the position of t,he cart,, 1 is t he length of the pendulum, h.1 is t,he mass of the cart , m is the point mass at,t,ached at the end of the pendulum, g is t,he gravity constbnt,. The input t ~ ] is t,he force applied t,o t,he cart while t,he input ( 1 2 is the t,orque applied a t t,he base of t,he pendulum. Several cont,rol problems arise:

( a ) Assuming t,he st,at,es ( x , x , cp, +) are measured: ( i) ( u l , r r 2 ) are to be designed in order t.o track x , ( t ) and cp,.(t); (ii) is t,o be designed in order t,o track cp,(t), in particular ~ , ( t ) = 0,

assuming 712 = 0; (iii) 7 ~ 2 is t,o be designed in order t,o t,rack ~ , - ( t ) ( i in particular cp,.(t) = 0,

when vl is const,ant and unknown; (iv) the above cont,rol problems ( i ) , jii), (iii) a re 60 be solved when t,he

parameters are unknown. ( b ) Assuming t,hat only p is measured:

( i) 111 is t,o be designed in order to track cp,.(t) when 712 = 0; (ii) 7r2 is to be designed in order to t,rack cp,(t) when 711 is constant and

unknown.

1.10.6 ( B a l l a n d b e a m ) A hall of point mass m is in a tube of infinite length and of moment of inertia J around t,he middle point a t which a t.orque r is esert,ed: neglecting any friction, t,he rnodel is given by

I.' = - g sin p + rd"

-

-2m7 ~ ,u , ri lgr cos 9 * - - + 7 J + m r L J + ~ n ~ l

The posit,ion of t,he ball with respect t,o the origin is given by r while cp denot,es the angle by which the tube is rotat,ed with respect to the horizontal line. Assuming t,hat, the s ta te ( r , v , cp. d) is measured a control T is t,o be designed so that r tracks any const,ant reference value r,. including r , = 0.

1.10.7 ( P o i n t rrlass sa t e l l i t e ) A point mass rn in a plane. subject t,o an inverse square law force field wit,h potent,ial energy k / r and bot,li radial and t>angential thrusts t11 and 7 1 2 , respect,ively, is governed by t,he equations

Physical control problems

in which ( r , (F) are t,he polar coordinat,es of the mass, v is t he radial speed and w is t,he angular speed. Assuming t,hat the stat,e ( r , v , cp, w ) is measured, or t,hat only the variables r and q are measured, the following problems can be formulat,ed bot,h when t2he t,wo parameters ( k , m,) are known and when t,lley are unknown:

(a) (1r1, are t,o be designed so t,hat constant r,. and w, are tracked; (b) 11,2 is t,o be designed in order to track a const,ant r,. when 711 = 0

1.10.8 (Robot with flexible joint) T h e dynamic equations of a single link robot a rm wit,h a revolute elastic hint rotfating in a vertical plane are given by

in which ql and q2 are the link displacement and the rotor displacement, respect,ively. The link inert,ia J 1 , the motor rotor inertia J,, t,he elastic const,ant. k , t,he link mass M , the gravity constant g , t he center of mass 1 and t,he viscous friction coefficients Fl , F, are positive constant parameters. The control o is t,he t,orque delivered by t,he motor. The control problems are:

(a) assuming that t,he whole st,at,e (q l , ql, qz, i z ) is measured, 11 is t,o be designed so t,hat q1 tracks a desired reference qr l ( t ) in two situations:

(i) t,he parameters are assumed t,o be known, (ii) t,he parameters are unknown;

jb) assuming that only ql is measured, 71 is t30 be designed so t,hat ql t,racks a desired reference qrl( t) in two situations:

( i) the parameters are assumed to be known, (ii) all parameters are unknown.

1.10.9 (Synchronous generator) A synchronous generator connected through purely react,ive transmission lines to the rest of a net,work which is represent,ed by an infinite bus (i.e. a machine rot,at,ing a t synchronous speed w, and capable of absorbing or deliveriilg any amount of energy) is modelled by

6 = J - J , ; = c, - 02w.Gfsin 6 - f93~'4A sin 6 - O ~ W $ B cos 6 + O5 sin 6 cos 6

Introduction

$B = - e 1 4 w ~ B - O I 5 sin 6

in which: 6 is the generator rotor angle referred to the infinit,e bus, w is t,he rot,or angular speed, , y ~ f is the field flux linkage, W A is the direct axis damper winding flux linkage, 4~ is the q ~ a d r a t ~ u r e axis damper winding flux linkage, uf is the field excitat,ion voltage, c, is the acceleration provided by the turbine; Bi are constant parameters depending on the machine (in particular O2 > O ) , the transmission line electrical parameters, rotor inertia, and infinite bus const,ant volt,age. Let (he, w,, qfe, dAe, be the stable operat,ing condition corresponding to t,he inputs (c,,, v f e ) A short circuit, in the transmission line may cause an abrupt variation of most parameters and impose a different operating condit,ion. The control problem is to design feedback cont,rols c, - c,,, vf- vf, to drive t,he generator to a stable operating condition bot,h in the case in which t,he short, circuit, is not cleared and in the case in which it is promptly cleared. I\ieglect,ing tke dynamics of the damper windings, the reduced order model (8: are new pa;ameters) is

ij = c, - 19iw$~ sin 6 + 8; sin 6 cos 6

Since the input vf may be varied more quickly than the input c,, c, may be viewed as a const,ant parameter (c, = 0;) and vf is to be designed as a feedback control to stabilize the generator after t,he occurrence of a short circuit or a turbine failure. Flux linkage (,hf, 4,4, ,$B) measurements are usually not available.

1.10.10 (Synchronous motor) The dynamic equations of a permanent magnet synchronous motor with sinusoidal flux distribution are given by

d ? , -

- -

R K m ' 1 a - - I , + -d sin(p6) + -

t f t L L L dl h

-

R h', -

-

7' b - -16 - -L,, cos(p6) + -

d t L L L

in which b and w are the rotor position and speed, and (i,, 7,b) and ( v , , 11.b) are st,ator currents and stator voltages expressed in a fixed stator frame. The parameters of the motor are the stator winding resistance R and self- inductance L , the motor torque constant K,, the rotor inertia J , the viscous friction F and the number of pole pairs p. All parameters, except,ing TL,

Exercises 29

are positive. Assuming that. (6, w, i,, I+) are measured, 71, and 7/,b are to be designed so that 6 tracks a smoot,h reference signal 6,(t) in two situations:

(i) the parameters are assumed to be known; (ii) t,he parameters are unknown

1.10.11 ( U n k n o w n p o i n t m a s s ) Consider an unknown point mass m. subject to: a control input, force u , an unknown constant disturbance force d , unknown viscous damping force k,x and unknown elastic force kPx. According to Newt,onls law the model is

wit,h m,, kp, k, unknown positlive constant,^ and d unknown. The control task is to design a feedback control algorithm so that x ( t ) asymptotically tracks a reference signal x , ( t ) in two cases:

(a) x and x are measured, (b) only x is measured.

t 1.10.12 (Sa te l l i t e w i t h so la r a r r a y s ) A simplified model of a satellite with flexible appendages (solar arrays) along each principal axis of inertia when couplings are neglected is given by

1 S' + 2(,Ld,s + UJ; 4 ( s ) = - C

JsZ ,=, y s 2 + 2(,w,s + w: in which N is the number of elastic modes with w, cantilever frequencies, C, damping ratios and pi coupling fact,ors ( J > 2p,2), J is the satellite moment of inertia about the considered axis, r$ is the angle of rotation, u is the control given by the torque delivered by act,uators (gas jets or reaction wheels), dT is the const,ant disturbance torque. Since the solar arrays may rotate, the parameters J , p,, J,, C,, dT are assumed to be constant but unknown: the control problem is to design a stabilizing feedback control u on the basis of Q measurenlent only.

1.11 EXERCISES

The following examples have been discussed in this chapter and are collected for easy reference and co~nparison. The reader may try to design observers, s ta te and output feedback stabilizing or tracking controls using his or her background and then compare with the solutions sketched in this chapter. All the exercises are globally solvable on the basis of the techniques introduced in this book.

1.11.1 Design a stabilizing feedback control for the system (compare with (1 .2 ) ) r. = 0 : C 3 + , 1 L

30 Introduction

1.11.2 Design a stabilizing s tate feedback control for the system (compare with (1.4))

1.11.3 Design a stabilizing s tate feedback control for the system (compare with (1.6))

1.11.4 Design an adaptive stabilizing s tate feedback control for the system (compare wi th (1.20), (1.21))

t 1.11.5 Design a stabilizing s ta te feedback control for the system (compare with (1.10), (1.11))

1.11.6 Design an adaptive stabilizing s tate feedback control for the system (compare with (1.16))

1.11.7 1)esigrr a stabilizing s tate feedhack control for the systern (cornpare with (1.15)) r , =

2 1'2 - r2 "

1.2 = I ,

I / = (' 1

1.11.8 Design a stabilizing s tate feedback control for t h e system (1.14) 1 1 = I.: + ,r:,, rL' = I ,

w111ch rrlnkes the origin a globally asy~nptotically stable equilibrium point Htr i f : use the, func,tion V = ( r ? t . r : ) / 2 .

Exercises 31

1.11.9 Design a stabilizing s tate feedback control for the linear system (compare with (1.18))

1 .11 .10 Design an observer for the system (compare with (1.26))

1.1 1.11 Design an adaptive observer for the system (compare with (1.28)-(1.29))

1 .11.12 Design an output feedback adaptive tracking control for the system (compare with (1.23) and (1.24))

1.11.13 Design both s tate feedback and output feedback stabilizing controls for the system (compare with (1.31), (1.32), (1.33), (1.35), (1.36), (1.37))

1 .11.14 Design an output feedback adaptive tracking control for the system (1.34)

1 .11 .15 Desigri an output feedback stabilizing control for the systern (compare with (1.39))

Part I

STATE FEEDBACK

Feedback linearization

In this chapt,er we consider single input syst,ems witah no ~nce r t~a in t i e s

in a neighborhood U,, c R n of a n equilibrium point x, corresponding to 71 = 0, i.e. j (x , ) = 0; j and g are assuhed t80 be smooth vector fields defined on Rn with g(x,) # 0. We present two results given in Sections 2.2 and 2.3, respectively, stating necessary and sufficient condit,ions under which system (2.1) is transformable into a linear cont,rollable system by nonlinear change of coordinates only (Section 2.3) or by nonlinear feedback and change of c ~ o r d i n a t ~ e s ( S e ~ t ~ i o n 2.2) . These t,wo problems are called st,at,e linearizat,ion and feedback linearization, respect,ively. Feedback linearizat,ion is viewed as a generalization of pole placement for linear systems, which is recalled in Section 2.1. More general problems such as partial feedback linearizat,ioil (Section 2.4) and its application to st,abilizat,ion (Section 2 .5) , global issues (Sect,ion 2.6) , and ext,ensions to multivariable systems (Section 2.7) are also addressed. Physical examples are discussed in Section 2.8.

2.1 P O L E P L A C E M E N T FOR L INEAR S Y S T E M S

We first recall a well known result from linear system theory, t he pole placement theorem, and problde a proof whlch generalizes to a class of nonlinear systems (2 1 ) Theorem 2.1.1 (Pole Placement) Consider the single input linear system

.i = FX + grr. I. E Rn, I / E R . (2.2) Given v < n / 2 arbitrary pairs of complex conjugate nllnlbers 1, & I & , , 1 5 I < v , and 11 - 2v arbitrary real numbers A,, 2v t 1 5 J 5 7 ~ , there exists a linear state feedback transfornlation; i .e . a linear change of coordinat,es (T nonsingular)

Feed back ljnearjzation

and a s ta te feedback (v E R )

such t,hat t he closed-loop system

and

if. and only if, the Kalman controllability condition

span{g, F g , . . . , Fn--lg) = Rn ( 2 . 3 ) holds

Proof. Suficzency. By virt,ue of t,he cont~rollabilit~y assumption ( 2 . 3 ) t he n x n, controllability matrix

is nonsingular and therefore there exist,s a unique row vector h which solves t,he linear equat,ion

h R = h [ g , F g . . . , F n - ' g ] = [hg, h ,Fg, . . . , h , ~ " - ' ~ ] = [ O , . . . , 0 , 11 ( 2 . 4 ) that 1s

Pole placement for lineal. systems

Let

be the characteristic polynomial of the matrix F Recall tha t by the Cayley- Hamilton Theorem

Define the n x n nonsingular matrix T

To show that T is a nonsingular matrix, observe that the n x n Toeplitz matrix

is nonsingular since, according to definition (2 4 ) .

Slncr R is nonsingular by assumption T = L ~ ~ - ' is also nonsingular and z = T x is a lirlrar change of coordinates In z-coordinates the linear system x = Fx t g r ~ becomes z = T F T ~ ' Z + T ~ I I hIore precisely. b y virtue of ( 2 8 ) , we directly compute

Applying ( 2 . 6 ) . we have


which, substituted in (2.9), gives

i.e. a linear system in con t ro l l e r f o r m . Define now t,he stat,e feedback

which, s ~ b s t i t u t ~ e d in (2.10), gives

and concludes the sufficiency part . Necesszty. We proceed by cont,radict,ion. Assume t-hat condition (2.3) does not, hold, that, is

rank[g, F g , . . . , Fn-lg] = 1. < TI . It follows that g , F g , . . . , FT-'g are linearly independent and that. the subspace

R = span{g, F g , . . . , Fr- 'g) is F-invariant,, i.e. FR c R. Let e,, r + 1 5 j < n be vectors which, along wit,h g . . . . , F ' - ' ~ , form a basis in Rn. In the new basis {FT-'g, . . . , g , e , + l , . . . , e n ) the pair (F , g) becomes

with

T - I = [ ~ ' - l g . . . . , g, e r + l , . . . , e,l. For any choice of the vector k t,he st,ate feedback

Pole placement for linear systems 39

does not. modify the eigenvalues of F 2 2 , which contradicts the hypothesis. The necessity of t,he controllabilit,y condition (2.3) can be alternatively proved by show- ing that, (2.3) is invariant under linear change of ~oordinat~es and state feedback and t,hen observing that (2.3) is satisfied for the pair ( A , b) for any choice of the eigenvalues of the matrix A .

R e m a r k 2.1.1 As is well known from linear system theory, the condition (2.3) is necessary and sufficient for t,he existence of a piecewise continuous input driving t,he syst,em from any arbitrary stat,e X I int,o any arbitrary state x2 in any (arbitrarily small) positive time. Condition (2.3) is equivalent to the Popov-Belevich- H a u t u s control labi l i ty condi t ion :

A linear system (F, g ) satisfying these condit,ions is said to be control lable . R e m a r k 2.1.2 Provided that A is asympt,otically stable, a Lyapunov function for the closed loop system in z-coordinates V = z T p z , which becomes in the original X-coordinates V = xTTTpTx, c n be computed by solving the Lyapunov matrix equation for P 4

R e m a r k 2.1.3 The sufficiency part of t,he proof is c~nst~ructive. To place the poles of the closed loop system a t desired locations in the complex plane one has t,o apply t,o t,he system (2.2) the state feedback given by (2.11), that is

11 = (cr - a ) T ~ z + v where a and a are the vect.ors ~ n a d e by t,he coefficients of t,he characteristic polyno- rnials of the given mat,rix F and of t,he closed loop mat,rix A (which has the desired eigenvalues) , respect,ively.

If we consider the family of all ~ont~rollable pairs (F, g) , and the action of l inear s t a t e feedback t r a n s f o r m a t i o n s defined as

with T a nonsingular n x 77 matrix and k. row vector, Theorem 2 1 1 can be restated as follows

T h e o r e m 2.1.2 A linear system ( F , g ) is transformable by a linear state feedback transformation int,o a B r u n o v s k y control ler fo rm

40 Feedback linearization

i.e. there exist a nonsingular mat,rix T and a row vector k such that

if, and only if, the pair ( F , g ) is controllable. We say t,hat t,wo single input linear syst,ems are feedback equivalent if they are

related by a linear state feedback t,ransformation: ail syst,ems are then divided int,o feedback equivalence classes. According to Theorem 2.1.2, all single input linear ~ont~rollable syst,ems belong t,o the same feedback equivalence class represent,ed by the Brunovsky controller form (2.12).

If we only consider a linear change of coordinates acting on the pair ( F , g ) as (F , g) --, (TFT-', Tg)

we can stat,e a result which was obtained as an intermediate st,ep in the proof of Pole Placement Theorem 2.1.1.

Theorem 2.1.3 Any controllable pair ( F , g ) with de t ( s I - F) = sn + a,-IS"-' + . . + a l s + a 0 is transformable by a linear change of coordinates into:

rank[g. F g . . . ~ " - ' g ] = n

(a) the controller form t

we can choose t h t n linearly independent vectors F " - ' ~ , . . . , g as a new basis so t!iat in the new coordinates 2 = T2x defined by

T~ FT;' =

tlie system is expressed in controllable form. This is easily seen since, by the Caj.ley-Ham~lton Theorem,

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

. .

, T l g = 111. . 0 0 0 . . . 1

- a 0 -a1 - 0 2 . . . -ffn-1

,

(b) the controllable form 1;-

T~FTF' =

Proof. (a) Stat,ement (a ) is proved in the first part of the proof of Theorem 2.1.1 (see (2 .10) ) . ( h ) Since

-a,-1 1 0 . . . 0 -0,-2 0 1 . . . 0

. . . . . . ' . . I -a1 0 0 . . . 1 -a,, 0 0 . . . 0

, Tzg =

0 0

I 0 1

,


2.2 FEEDBACK LINEARIZATION

Consider now single input nonlinear systems (2.1) which generalize linear ones (2.2) since t,he vector field f (x) is not restricted to being linear and g(x) is not re~trict~ed t,o being constant. We restrict ourselves to a neighborhood U,, of an equilibrium point x, (f (x,) = 0) which is assumed to be the origin since we can always perform a change of coordinates r = x - x, so that 2 = 0 is an equilibrium point. However, the same considerations apply in a neighborhood of any point xo, not necessarily an equilibrium point for t,he vector field f , provided that f (so) and g(xo) are linearly dependent, i.e. f (xo) = yg(xo) with g(xo) # 0. Indeed, we can define the change of ~oordinat~es z = x - xo so that in new coordinates

Let v be the new control variable defined as v = 11 + y which, s u b s t i t ~ t ~ e d in (2.13), gives

witah fz(0) = 0 and g,(O) # 0. We generalize the notion of linear state feedback transformations by allowing:

(a) nonlinear local change of coordinates

wit,h T : Uo - Rn a local diffeomorphism in Uo, a neighborhood of the origin; (b) nonlinear st,at,e feedback

with k(0) = 0 , P(0) # 0, k : Co - R, 3 : Uo - R both smooth functions The above transformation is called nonlinear state feedback transformation.

The act,ion of nonlinear state feedback (2.14) on the vector fields f and y is to transform them int,o the vector fields

The action of nonlinear state feedback t,ransformation on the single input system (2.1) is to t,ransform (2.1) into

'4 natural problem is to determine the nonlinear feedback equivalence class con- taining the Brunovsky cont,roller form. As we have already seen, it contains all conirollable single input, linear syst,ems: we would like to determine all nonlinear syst,ems (2.1) belonging to such an equivalence class.


Definition 2.2.1 Two systems

where fl, f2, g l , g2 are smooth vector fields In Rn wlth fl(0) = 0 f2(0) = 0, gl(0) # 0, g2(0) # 0, are said to he locally feedback equivalent ~f there exist two smooth functions k(xl), P(xl), wlth k(0) = 0, O(0) # 0 definlng a s ta te feedback

and a local diffeomorphism in a neighborhood of the origin in Rn.

such that tJhe closed loop syst,em

in x2 coordinates is t

Definition 2.2.2 T h e nonlinear single input system (2 .1) is said to be locally state feedback linearizable if it is locally feedback equivalent t,o a linear syst,em in Brurlovsky controller form

bVe wol~ld like to determine those nonlinear systems which are locally feedback lincarizablc. The following theorem identifies t,liose syst,ems by means of necessary and sufficierlt conditions.

Theorern 2 .2 .1 (Feedback Linearization) The single input system (2 .1 ) is locally s ta te feedback lirlearizable if, and only if, in Croj a neighborhood of the origin:

( i ) sp;~n{g. . udnf-lg) = R n ,

Feedback linearization 43

(ii) the dist,ribut,ion GnP2 = span{g, . . . , is involutive and of constant rank rl. - 1.

or, eq~ivalent~ly , if and only if:

(iii) t,he disth-ibutions

are involutive and of constant rank I + 1.

Proof: Suficzency. By virt,ue of assumptions ( i ) and (ii) we can apply Frobenius's Theorem 14.4.3 which guarantees the existence of a srnooth funct,ion h : Rn -t R such t,hat in a neighborhood of t,he origin

arid h (0) = 0. In ot,her words (i) and (ii) guarant,ee t,he existence of a solut,ion h for t,he set of linear part,ial different,ial equations

( (dh , g) . (dh, ad(-~) ,g) . . . . . (dh. ad;:&)) -- (0 , . . . : 0 ,?(X)) (2.17)

for sorne snlooth function y ( x ) , y ( 0 ) # 0 Defining the 71 funct~oris h ( r ) , L f h ( x ) , L;-'h(x), we can show that

is a local d~ffeomorphisrn In a ileighborhood of the orlgin we just need to shorn that its 7, t~obian d T / d x 1s nons~ngular and then apply Inverse Function Theorem .4 1 1 Drhlie the T I x n rnatrlx

By defin~tiori (2.17). t he first row of the rnatrix ;l:(x) is given by [O. . . , 0, -r (.~.)j. By v ~ r t u e of Lcibniz's formula and of (2.17), we have


so that the second row of mat,rix N ( x ) is [O, . . . , 0 , ( x ) , ( d ( L f h , ) , Ap- plying repeatedly Leibniz's formula, the remaining rows of the matrix N ( x ) can be computed, yielding

i.e. N ( x ) is nonsingular in a neighborhood of t,he origin. This fact along with assumption (i) imply that d T / d x is nonsingular: by Inverse Function Theorem A . l . l it follows that z = T ( x ) is a local diffeomorphism. The system

is expressed in z-coordinates as

i, = ~ ) h , ( x ) + L , L > - ' ~ , ( X ) ~ ~ , , 1 5 i 5 n . From (2 .17) , by applying Leibniz's formula 4

and equation (2 .19) becomes

Define the state feedback

and not,e t,hat k ( 0 ) = 0 since f ( 0 ) = 0 and 0 ( 0 ) # 0 since L , L ~ - ' h ( 0 ) = ~ ( 0 ) # 0 . Substituting (2 .21 ) in (2 .20 ) we obt,airi t,he Brunovsky controller form (2 .15 ) . While condition (i) implies that rank G, = z t 1 for 0 1 < 11 - 1 in Uo, in~pect~ing the matrix .V (L) we observe that

d h . . d ( ~ ; - ~ h ) E G,'


and that dh,, . . . , d ( ~ ) h , ) are linearly independent for 0 _< i. < n - 2, which proves, according to Frobenius's Theorem A.4.3, that the distributions Gi are involutive for I; = 0,. . . , n. - 2. Since is involut,ive by assumption (i) , this shows that conditions (i) and (ii) imply condition (iii). On the other hand, condition (iii) implies (i) and (ii): t,herefore (i) and (ii) are equivalent to (iii). Necessity. By assumption there exists a nonlinear state feedback taransformation from syst,em (2.1) to system (2.15) and vice versa. Consider the linear system (2.15) in Brunovsky controller form for which we can compute the distributions

which are involut,ive and of dimension 1: + 1. We now show that they are invariant under nonlinear st,at,e feedback tJransformations. The di~t~ributions Gi are invariant under st,at,e space change of coordinates. Hence we need only consider the action of a nonlinear state feedback transformation on a system ( f , g)

Clearly

We proceed by induction. Assume that

we claim that

Hence

S~nce G, are irivolutive, in particular adgG, C G,, it follows that

Contiltions (iii) are then satisfied for any feedback linearizable system (2.15).

Feed back linearization

Remark 2.2 .1 In z-coordinat,es t,he vect,or field g is

so that

G~ = span

Since in z-coordinates the vect,or field f is

w e compute

d = - L,LT;-'~,- - L

azn-l dz, j

Since L , L ; - ' ~ # 0, we have d

= span { L dz, -) dz,.-] Analogous comput,ations lead t,o

Reniark 2 . 2 . 2 Polv placement. Throrern 2.1.1, stat,ed as Theorem 2.1.2, is a corollary of Fred1)ack Linearization Theorem 2.2.1. In fact for linear systems (2.2), condition (ii) is always satisfied while contlit,ion (i) becomes t,he Kalman controllabilit,y c,ur~ciit ion

r a r ~ k [ ~ . F g . . . Fn-' 91 = T I

Remark 2 . 2 . 3 S o t e t,!~at the proofs of t,he sufficiency parts of the pole placement and frt,dbac,k Ilnt~arizatlon tlieorerns conceptually follow tile same steps. While thtl c,ontrollat)~lit,v assurriptiori (2.3) is sufficient t,o solve the linear equation (2 .4) . co~~ciitions ( i ~ allti ( i ~ ) are sufficient t,o solve the linear partial different,ial equation (2 17) . this I.: the niain difference ket,ween the t,wo proofs and it is a technical orlc'. CI


Remark 2.2.4 T h e linearizing t,ransformat,ion for a feedback linearizable system is not unique since tlhe solut,ion of (2.16) is not unique.

We now st,at,e three consequences of Feedback Linearization Theorem 2.2.1.

Corollary 2.2.1 A planar nonlinear system (2.1)

is locally st,at,e feedback linearizable in a neighborhood of t,he origin, if and only if it,s linear approximation about t,he origin

Proof. Suf ic iency . Since g(0) # 0 and g is a smooth vect.or field g does not vanish in a neighborhood of t>he origin qnd 50 = span{g) is involutive of constant rank 1. Since

and f (0) = 0, a t t he origin

By virt,ue of the cont,rollability a s s ~ m p t ~ i o n on the linear approximation. g(0) and ad /g(O! are linearly independent). Since g and a d f g are smooth vector fields, they also are linearly independent in U o , a neighborhood of the origin, i.e.

so t,hat Feedback Linearization Theorem 2.2.1 applies. Necess~ty . Feedback Linearization Theorem 2.2.1 implies (2.24) which, by virtue of (2.22) and (2.23) implies tha t g and F g are linearly independent,. 0

Sirnple examples show t,hat controllabilit,y of the linear approximation about the origin is no longer a sufficient condit,ion for local s ta te feedback linearizat,ion for syst,ems (2.1) wit,h n > 3; it, is, however, a necessary condit,ion which is easy to check.

Corollary 2.2.2 If the nonlinear system (2.1) is locally st,ate feedback linearizable then its linear approxirriatior~ about t,he origin

is controllable, i.e. rank[g, F g . . . . , Fn-'g] = n

Feed back linearization

Proof. This is left t,o t,he reader as an exercise.

C o r o l l a r y 2.2.3 The system in t r i a n g u l a r f o r m

in which . . . , 4, are smooth functions such that @,(O) = 0. 1 i 5 n,; is locally feedback linearizable.

Proof. A proof consists in comput,ing the distributions GnW1 and G,-z and verifying that conditions ( i ) and (ii) of Theorem 2.2.1 apply. .4lternatively, the feedback linearizing transformation may be directly computed as follows:

R e m a r k 2 . 2 . 5 More generally a system

x = I x ) 1 1 7 L n - 1 X n = @n(xlr xn) O(x1. , xn)71

with o,, 1 < 7 < n , and 3 smooth functions such that o,(O) = 0, 1 < 7 5 n , d ~ , / d x ~ + l ( O ) # 0 , 1 5 I n - 1, and 3 (0 ) + 0 is said to be in t r i a n g u l a r f o r m and 1s locally feedback llnearlzable

R e r n a r k 2 .2 .6 \Vhen the control 11 enters nonilnearly in the s ta te equations, i e

L = f (x 1 1 ) f (0. 11) = 0. V I I E R t h e feedback l ~ r i e a r ~ ~ a t l o n problem may be posed for the e x t e n d e d s y s t e m

s = f ( x 11) I1 = IL (2 27)

111 uhlch ti 1s the nev, control and (x , 11) 1s the extended s ta te If the cond~t ions of ihe fveciback li~lea~izatiorl theorem apply to the extended system (2 27), t he resulting control

is a d y n a m i c s t a t e f eedback l inear iz ing c o n t r o l for t,he original system.


Example 2.2.1 Consider the syst,em

It is not feedback linearizable in any neighborhood of the origin by virtue of Corol- lary 2.2.1, since the linear approximation about the origin is not controllable.

Example 2.2.2 Consider t,he system

Note tha t t,he origin cannot be made globally asymptotically stable by a linear s ta te feedback 1 1 = -k lx l - kzxz. The system is feedback linearizable by virtue of either Corollary 2.2.1 or Corollary 2.2.3. Denoting

so that

= span{g, adfg) = R'. Vx E R~

The linearizing transformations are computed by determining a function h such that h(0) = 0 and (see (2.16))

In this case any localiy invert,ible function of zl, h,(xl), is such tha t

If we choose h l ( x l ) = ~ 1 , the feedback linearizing transformation is (see (2.18), ( 2 21))


while if we choose h,%(xl) = xl + x:, the feedback linearizing t,ransformat,ion is

This shows that the feedback linearizing t,ransformat,ion is not unique. It,s choice act,ually affects the complexit,y of t,he resulting linearizing st.ate feedback.

Exarnple 2 . 2 . 3 Consider t,he system

where cu is a smooth function. Denote f = x2d /dx l , g = cu(x2)d/dxl + d / d x 2 . LLTe comput,e a d f g = - a / d z l . Hence rank G1 = 2 in R'. The system is locally feedback linearizable. We look for a function h ( x l , x2) such t,hat.

The funct,ion

solves t,hr partial differential equation (2.30). In the global coordinat,es

I he >ystt>rn becomes

Yotc that onlv a change of c oordlnatcs 15 neec-lcd no feedback 1s rpqliired to make the s~ stern I~nea r slid controllable

Exarrlple 2.2.4 Cons~der tlic \?stem ( o , dl , 8 2 srnooth functions)

with J2 f 0. V.1. E R ' ~ which coincides wit,h Example 2.2.3 whet 01 = 0, ,02 = 1. This systcl~n 1s fveclback linearizable by the diffeornorphisrn


and the s ta te feedback

E x a m p l e 2.2.5 Corls~drr t,he systrn!

The linear approximat,ion about the origin is controllable: ~ievertlieless, a s we shall see, this system is not feedback linearizable. Denote

We compute

Since span{g, ad fg , ad2fg) = R3 in the whole of R3, c ~ n d i t ~ i o n (i) of Theorem 2.2.1 is satisfied. Since [y, adfg] $! span{g, a d f y ) , the d i~ t r ibu t~ ion G I = span ig , adfg) is riot involut.ive, so t,hat condit,ion (ii) of Theorem 2.2.1 is violated. T h e system is not feedback linearizable. This example shows tha t controllability of t,he l i i~ear approx~niat,ion about. the origin is riot sufficient for feedback linearizahilit,~.

Exarrlple 2.2.6 Consider the system

which is not, according to Corollary 2.2.3 or Remark 2.2.5, in triangular form. Denote


We compute a a

adfg = -zl- - - ax1 ax2

and we have

rank{g, adfg, ad:g) = 3, Vx E {x E R~ : x2 # 1) On the other hand since [g, a d f g ] = 0, t,he distxibution

Gi = span{gl adrg) is involutive of constant rank 2 in R ~ . Hence conditions (i) and (ii) of Theorem 2.2.1 are satisfied and t,he syst,em is state feedback linearizable. To compute t.he linearizing diffeomorphism and state feedback we need to determine a function h, such that h.(O) = 0 and, in Uol

The function h,(x) = xlePz2 satisfies the syst,em of partial differential equations (2.32). The diffeomorphism is given by

z = (21, 22, 23) = (h.(x), Lrh,(x), L ;~ . (x ) ) = T ( z ) with

The linearizing state feedback is

with

This example shows that a feedback linearizable system need not be in triangular form ( 2 . 2 6 ) .

Linearization by change of coordinates

Example 2.2.7 Consider the system

which is not in triangular form (2.26). It is linearizable by the global change of coordinates

No state feedback is needed. + i

2.3 LINEARIZATION BY CHANGE OF COORDINATES

As we have seen in Examples 2 2 3 and 2 2 7, there are nonlinear systems transformable into linear and controllable ones, by using nonlinesr change of coordinates only In this section such systems are identified in terms of necessary and sufficient conditions A procedure to construct the l~nearizlng coordinates is also presented

Definition 2.3.1 Two systems

where f l , f2, gl, gz are smooth vector fields in R7' with fl(0) = 0, f2(0) = 0, gl(0) # 0, g2(0) # 0, are said to be locally state equivalent if there exists a local diffeomorphism in a neighborhood of the origin in Rn,

such that


Definition 2.3.2 T h e nonlinear single input system (2 .1) is said to be locally state linearizable if it is locally s ta te equivalent to a linear controllable system, tha t is

with ( A , b) a controllable pair.

Our goal is t,o characterize all locally s ta te linearizable syst,ems. We consider a nonlinear local change of coordinates

and would like to determine those nonlinear single input systems

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