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Unformatted text preview: A SYMPTOTICS TABILITY A ND F EEDBACKS TABILIZATION R. W. Brockett Abstract. We consider the loca1 behavior of control problems described by (* = dx/dr) i=f(x,u) ; f(x,0)=0
o and more specifically, smooth function asymPtotically and 2 be1ow. would insure u(x) stable. Whereas it the question such that of determining when there point in exists which is a x = xo is an equilibrium are formulated that Our main results night Theorems 1 have been suspected control is far controllability Theorem I uses the case. an the exlstence of a stabilizing law, a degreetheoretic The positive application result of high argument to show this of from being of Theorem 2 can be thought a nonlinear as providing gain feedback in setting. 1. Introduction In this paper we establish among other general theorems which are strong enough to irnply, a) there is things, that law (u,v) = (u(xry, z) rv(x,y,z)) stable for a continuous control which makes the origin asympEoticatly x=u y=v z=xy
and t hat b) there exists v(x,y,z)) no continuous control law (urv) = (u(xryrz), stable for which makes the origin asymptotically 181 182 x=u y=v z = y uxv l + :
*
t g. i
1 that the null solutlon of Eulerrs stable wlth two The flrst angular control velocity of these irnplies equations can be made asymptotlcally principle axes. torques aligned but not this wlth (see [1,2)for a general a discusslon, counter particurar repeated result.) conjecture The second provldes asserting that a of example to form of control theory. the oft local law. reasonable stabillzing in control controllability Section 2 gives iurplles certain the existence a background rnaterlal result In section 3 we formulate for our nonexistence of and ln control section 1aws. 4 we glve a criterion the existence stabilizing Sussmann [3] abie trol in a strong gives an example of a system in fails R2which is controrrcon sense and yet global to have a continuous stability. feedback law yielding asymptotic His example involves ours involves both bounds on the controls nei ther. 2. Control we intend lrorthwhile detailed Systems to work locally and nonlocal considerations, in this a bout paper, but even so it is perhaps t o m ake a f ew r emarks a g 1oba1 f ormulation. [3]r A m ore and systeuratic account can be found in familiar with control theory but in any case to section be a vector T*TX 3. the reader can go directly Let X be a differentiable bundle over x. Let manifold and 1et n:E+X bundle of of rx denote the tangent of TX over E. vector in X and let denote the pullback assignment of A section TX for n*TX is in then an E. rf we a velocity each point choose a 1oca1 trivializati,on Y € f (E,tl*TX) ls equivalent (in of E and pick coordjnates to then a section an obvlous notation) x = f (x,u)
such a y is ca11ed a control o€f(X,f) is it system. an assignment of a pair is given by a function in coordinates (u,o) o(x). by corresponding we denote bv A section to each x and so locally Yo the section of f(E,n*TX) defined l B3 i = t(x,u*cr(x))
and say that control field yo ls obtained fron with Y by the appllcatlon every control of the feedback ls will a vector be law o. which is Associated obtalned system Y there field by settlng u=0. This vector We now state ydenoted b Yo and is Glven Y€f(Err*TX) o€f(X,E) such that called the drift. a preclse problen. exist and xo€X xo ls under what clrcumstances point does there ls an equillbriusr thls of Y: whlch locally asymptotlcally problem. stabLe. We call the loca1 feedback stabilizatlon Now the fibers and so it f (E,r*TX) we call of E and the fibers of r*TX are both veccor xf(E,n*TX;* to o. sPaces makes sense to ask if defined by (c,y) l* Ehe napping of f(X,E) Vo is afflne with respect If so the systeu lnputlinear. inputlinear systems have a local description of the Observe that form i{.; = f(x)+rurer(x)
By a standard y€f belng (E,I*TX) wlth g iven by linear x=lRo, control svstem r.re understand (the trivlal a sectlon E=IRmx Rn bundle) wlth .1 x=Ax* m Ib*u*
I+I
I I 3. Nonexistence of Stabilizlng Control Laws There is been completely here in one situation understood 1n which for the stabilization We formulate to the laEter problen has some time. a guide the results developments such a way as to provide Renark: Consider the standard linear control system x = Ax*Bu x(t) € nn (*) A necessary and sufficient conditlon for there to exist a control Point ls 1aw that ls (*) o which nakes x = 0 an asyflptotically there exists of a neighborhood time ua(') N of stable equilibrium for x = 0 such that on [0,) each E € N there the solution of a function defined such that 1 Bl .r
wlth lnitial condltton x(0) =f and control u(') =us'(') goes to zero as t goes to lnflnlty. Proof:
startlng It as t T hls c ondltlon l s c learly a n ecessary o ne s lnce s olutlons
stable equtllbrlum thls ls polnt ls must approach sufficlent subspace for near an asymptotlcally goes to lnflnlty. To see thet condltton one observes that Range (BrA3r...rAtlB) (*) an invarlant as A and by change of basis can be ltritten ^ "] . [';] [;;][ ^l [ :;] "
wlth Rang" (B.rrAl1B.r,...,a,lrlnul r eal p arts lt = dLm x,r. if x* is Clearly to the eigenvalues of A22 t rlust h ave n egatlve lnflnlty. if 0n the other is g o t o z ero a s t g oes t o Proven that K hand, of is well known and easlly (n,A3,...,AtlB) (A+BK) rank n then there exists eigenvalues ln an m by n matrix haLfplane. such that has lts the open left The remark then follows. The rank condition r?=Ax*Bu pair x. just mentioned is controllability necessary Property. and sufficlent If for for to have a certain any glven a n d x ,/ a n d a n y g i v e n T > 0 , t h e r e 1 s a c o n t r o l u ( ' ) d e f i n e d o n [0rT] such that u(') steers (*) from x, at t=0 to x, at t=T we say system (*) for ls controllable. The rank condition in this sense. of is thaE the control necessary surilnary of stabllizable with and sufficient all if this controllabtlity A short (*) is reads as fo11ows. The nu1l solution with anll.qnlv if, all real parts this the nodes associated eiFenvalues nonnegative It is, in for are controllable. background, such as not completely unreasonable to view of nonlinear hope that systens f (x) + Iu. g. (x) something similar might happen. A speciflc question in this direction state 1n a neighborhood of xo can be steered Eo is, t'If every initial x by a control defined on [0r*) does there necessarily stable?" exist In a feedthis paper back control we show that law which makes xo asymptotically the anst/er is no provided that we want a control law with a some smoothness. In general we need someEhing more than just r Rc condltlon since, as we will see, the controls whlch 1n a controllabiltty steer the trajectors to zero cannot always be Patched together smooth way. We n entlon o ne m ore w ell k nown f act. If w e h ave x = f (xru) f(0,0) = 0
respect to both arguments' systen wlth and if f(.,.) continuously eifferentlable with we deflne a= (3f/0x)o and B=Ef/Eu, then the control x=Ax+Bu system a is called the lineatized the condition lar,r u=Kx system at (0,0)' If Ehe llnearized =n satisfies linear Rank (BrAB,.",AtlB) such that A*BK then there exists eigenvalues ;=?(*) in the control has its open lefr halfplane. Moreover, if stable we reca1l equilibrium that Point witn ?(O) =O the has 0 as an asymPtotically eigenvalues that of (Ef/0x) provided have real is Parts which are negative' for i= f(x,u) then we see the feedback stabillzation possible provided llnearized this system 1s control1able. still In view of the previous discusslon can be stated law for more precisely. with f(0,0) There exists a stabilizing nodes of control x = f(x,u) = 0 Provided the gnstable lhe llnear control l afr i f t he l inearized s vstem h asan result that u nglable m ode w hich i s knom if here depends on the well point is Part uncontrollable. result (Ef/Ox) positive. of The negative asserting Liapunov an equllibrium with unstable which is at lhat point has any eigenvalue a real FromtheseremarksweSeethatinsofaraSlocalasymPtotlc problems involve cases where is concerned, the only difflcult stability (3f/3x) has eigenvalues modes of on the iuraginary the associated axis whlch correspond to systems and all to
I uncontrollabte other linearized uncontrollable stable modes of behavlor. the linearized In the study system correspond of stability' part asymptotically where (8f/Ex) vanishes study of at the cases which The the has one or more eigenvalues the equilibrium point with a real critlcal are called far cases. the critieal in cases is [ 5]' still from complete' in this (See, e'8' PaPer' remarks of Arnold priurarily page 59). understanding We are then, certain concerned with features of the critical' 185
cases. The followlng theoren glves a necessary raw which, (ili) eonditlon (i) for the exissumrnprizes tence of a stabllizing our previous is declslve dlscussion as far control and ln under and (ii) introduces a new element which lncruding the second as large crass of problens, are concerned. example of the introductlon, Theorem 1: Let x=f(x,u) be glven with f(xorO) =0 and f(.,.) (xor0). contl_ nuously dlfferentiabre condltion for in a nelghborhood of of a continuously stable A necessary contror the exlstence differentiable is that: 1aw which makes (xo,0) (i) asyuptorically the linearlzed associated with system shourd. have no uncontrollable eigenvalues whose real (xo,0) part is modes positive. for each (ii) there exists E€N that x=f a neighborhood N of a control steers ur(.) such that there exisrs this contror defined on [0,) of *=f(x,ur) such frour the solution at t=0 to *=*o at t=o. (ili) the rnapping Y : AxIRm r Rn defined by Y : (x,u) F+ f(x,u) should be onto an open ser containing Proof: n
of rf (iii), xo is the necessity an equilibrium (i) we prove the necessity above. stable of point and (ii) having been explained which is asymptotically of i=a(x) [6] that we know fron v such that the work of wilson v is positive for xlxo, vanishes "a *o, is continuously differentiable, and has level sets '(o) v which are homotopy spheres. The compacrness of ,1(o) inplies that there exisrs c and € > 0 such rhar on .,r1(o) , llAv/E*l I .1/€ and <Dv/Dx,a(x)><€. This inplies rhar if sna11 the llqll is sufficiently vector f ield associated with x = a(x) * 1 { p o i n t s i n w a r d o . , . , , r 1 o ; . By evaluating at time t=1 the solution of x=a(x)+{ which passes through x at t=0, Applying fixed we get a conrinuous map of fixed point formula {xlr(*) <oi this into itself. the Lefschetz there exists a Liapunov function we see that urap has a point which musr be an equilibrium we could use a vers'on with of poinr Alternatively, which applies off the proof . of i = a(x) + 6. the poincar5Hopf Theoren page 41) to finish a(x) = 6 to manifolds This, in boundary (see [7], irylies turn, that we can solve l8T
for all 6 sufflclently srnall. Now l f a(x) = f(x,u(x)) is t o h ave x 1t
1S as an equllibrium clearly necessary polnt, that and if E = f (x,u) xo ls to be asymPtoticallY for f srnall . stable, be solvable Remark: If the control system is of the form x = f (x) + Iu. g, (x)
(iii) there x(t) €Ncnn then condition a solution if implies is that the stabilization problem cannot have f(') and a smooth distribution One further D which contains special case: If g1(.),...,8r(') with dim D<n. we have i = 1 "L.e . ( x ) 1
with exists In this the vectors a solutlon x(t) €NcRn Bi (x) being linearly independent problern if at xo then there m=n. to the stabilization and only if case we must have as manv control Of course the matter is parameters conpletely In as we have different in Ehe dimensions of X. set {gr(xo)} with drops dimension precisely singularitles at xo. this sense' distri butions are the only interesting kind. Remark: There is no stabilizing conditions (iii) control law for i=t, (1) and (ii) of Y=V, z=xvYu. This system satisfies fails to satisfy the theorem, but it condition Existence of Stabilizing Control L As mentioned above, 1n the study of asymptotic null solution of *=t("); cases, i.e. part. stability of the f(0) =0 one singles out for those for which (3f/3x) special attention the critical a zero real there has an eigenvalue with situation is negative Liapunov showed that a quadratic function zero. In in a noncritical whose derivative fact, it is always exists definite that in a neighborhood of may be chosen so a negative or unstable, definite there the derivative form. has a leading term which quadratic exlsts in In such cases, be they stable 0 a function v(x) a neighborhood of such that <Ev/Ex,f(x)> 1BB
is negatLve for x I 0 and which has the further for some reasonable class property that it remains negatlve definite of perturbations. with f(xo,0) = 0. We will a function is 0, v(x) >o Conslder the control say that v(') for this system x= f(x,u) system has finite gain at xo lf there exists v(xo) mapping a neighborhood of xo into xI x a nd f or
U rR sueh that s ome k x I
{ ; < 3 v / 3 x , f( x , u ) > <  0 ( x ) * k ( u , u ) with 0(x) > 0 a n d 0 o n l y w h e n X = x ^ . Remark: N ote t h a t i f fx ( 0,0) a nd f ( 0 , 0 ) ,, L * S is all conrinuously differentiable with f(0r0), zero then x = Ax*Bu*f (x,u) has finite real parts. generally, gain at zero provided the eigenvalues of A have negative More equilibrium there (lrlecan take v(x) =xrQx with QA+ArQ=_I.) if ;=f(x) has zero as an asymptbticalry has finite v(x) for gain at x=f(x) stable poinE then i=f(x)+ug(x) exists a Liapunov function zero provided that whose rate of decay satisfies of (0,0). ,] < urrP*2, M > 0, r^rith <Vr,g>/vP/2 bo,rrrd.d in a neighborhood setting To establish this last = <Vv,g>/v al1ows us to write 28 assertion we note that Iv,u] < V v , f ( x ) + u g ( x ) > k u 2 I u  J"J [
definite by taking k to study stability in some critical u [0,"0 ll"1 and this large guadratic form can be made negative enough. We now use this definition cases. Lennna: Consider the coupled differential equations x = Ax*By+g(x,y) h(x,y) 189
wlth g(','), (0,0). h(.r.), s*(.,.) and gy(.,.) all contlnuous tn a neighborof A have negatlve (0,0). real hood of parts Suppose that g, h, the eigenvalues g* and g, all and suppose that solution palr point vanlsh at Let rp be the contlnuous y=0. stable of Arf(y) +By*g(0(V),y) of equations provided i has (0,0) = 0 whlch vanishes at as an asymptotically has finlte gain ar y = 9. Then this crltlcal = fr("rr<y),y) Proof: Changevariables according to i=x0(y), i=y. Then ji. l ; = ni + eU i) + Bi + s (i+rp !y,;, ( 1 CIE = e i+ e (i+rp1;y  g (i',i) , i)
= Ax+ g(x,y; where !not only vanishes together with its first derivatives y = 0 but, in fact, vanlshes when?= 0. Wehave, then at i=0. diclE x = A x + g ( x , y ) d* t v = r rtxrl (i ),i ) ' d Let n be the Liapunov for the i functlon which establlshes Using the Liapunov the finite function gain property equation. v(x) = oi'qi+n(il
where qi+itQ=1, we compuEe v =  c<;,;> + <Q;,E(i,i)t + < vn ,h (;v(;), ;)
Since B(x,y) dominates
,N t is second order and vanishes of for wt^en x does o<xrx> gain hypothesis, large. arguments. of the second term and by virtue side is negative definite the finite the lefthand Asymptotic Notice the stability I equatlon) a sufficiently Liapunov stability that then follows of this from standard lernma is the effect to to reduce the study problen (the the study of a lower diuensional by elirni.nation of the "uninteresting" part. noncritical problem I
I * 190 This ttr" i can also part be i.nterpreted to ln in Eerms of time scales. represents The solution of goes quickly zero whereas the ! the manifold motlon whlch occurs much more slowly defined by ?=0. Remark: cally If there exists a control point for law w hlch m akes x = 0 a n a symptoti' stable equllibrium x = Ax*Bu then there stable fact, if exists a control point, law which makes xo an asymptotically ko *Buo = 0 can be solved stable for uo. point point. a whole In critical provided u = Kx grakes x = 0 an asynptotically equllibrium then u=Kx+u^ Thus if makes x^ an asymptotically stable equilibriurn o o x = 0 can be made asynptotically stable then there is points subspace U = {x IRx e Range B} of stable. Incidently, which can be made asymptotieally (iii) of theorem 1 we see that for ic=f(x,u). we in view of part can apply Sardfs Namely, if there theorem to conclude something sirnilar exists a feedback control then for for all 1 a r ^ rw h i c h m a k e s x = 0 N., of i = 0 and some set of measure space. makes sense to law into is two asymptotically stable some neighborhood 6€Nr, neighborhood N of x=0 zero, {(x,u)lf(x,n) except a possible in (x,u) it =6}nN defines a nanifold theorem is a stabilizing The idea behind divide parts, totically linear up the question one being stable control the followlng of finding of that control the choice a slow mode behavior and the other to. the which asympof a on a submanifold law to drlve being the choice the systen slow mode regime. Theorem 2: hood of their Let f and g be continuously differentiable (0,0,0) for in a neighbor (0,0,0), and suppose they vanish at A sufficient together wirh first derivatlves. condition i = e"*Fy*Bu*f(x,y,u) y = Cy*g(x,y,u)
to be stabilizable that A * BK has its at (0r0,0) is in rhat there exist a pair (K,uo(.)) and for such \ s
F I
l" eigenvalues solution of the open lefthalfplane , {
T
l 0 the continuous (A+Bx)rl(y) * Fy * nuo (l) + 0(V(v) ,y, uo (Y) + KQ(y)) = 0 which vanishes ar 0 I 191 y = c y + e ( x  ! , ( y ), y , u o ( y ) + K U ( y))
I tras a flnlte gain at (0,0) wlth x regarded as the lnput. Proof: This ls an lrnnedlate appllcatlon of the lenma wlth u belng taken to be Kx * u (v). example of the lntroductLon' Remark: To apply this to the flrst . . . ? v=yz The s10wmode equation is ,*yrwe x=u; y=v; let u=xlz, . ? 7 2 and r1(z)=z shows that this equation has then z =z'*x2'y2xy flnite galn at 0. 4. A cknowledgements It is a pleasure helpful to thank Chris Byrnes, John Balllieul and Peter Crouch for discussions in on this part material. This work ltas supported by the U.S. Arny Research Office under Grant No. DMG2979C0147, the Office ALr Force Grant No. AFOSR817401' No. N0001475C0648' of Naval Research under JSEP Contract Science Founcation and the Natlonal under Grant No. ECS8L2I428. 5. tf] References "The Geometry of Homogeneous Polynomial Dynanical J. Ball1ieu1, Theory Methods and Applications, Systems," Nonlinear Analvsis, Vo1. 4 (1980) pp. 879900. "Spacecraft AtEltude P.E. Crouch, (suboitted . for publication) Control and Stabilization" J. of Diff. and 12) t3] t4] 15] t6] l7l H.J. Sussmann, "Subanalytic Sets and Feedback Control, Eqs. Vol. 31, No. 1, Jan. L979, pp. 3I52. Fraurework for Nonlinear R.W. Brockett, A Geonetrical N o t e s f o r C B M SC o n f e r e n e e ( t o a p p e a r ) . Estimation, Control Probleurs (Felix frorn Hilbertrs Matheuratical Developments Arising Providence, RI , L976. Browder, ed.), A.elil4n_Xggh.._Sog., F.W. Wilson, Jr.n "The Structure of the Level Surfaces of a Eqs. Vol. 4 (1967) 323329. Lyapunov Function," J. of Diff. Viewpoint, J.W. Milnor, Topology, A Differentiable VA, 1965. Press of Virginia, Charlottesville, University ...
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This note was uploaded on 12/29/2011 for the course ECE 594d taught by Professor Teel,a during the Fall '08 term at UCSB.
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