Proceedings of the International Congress of Mathematicians August 16-24, 1983, "Warszawa

E. W. BBOOKBTT

Nonlinear Control Theory and Differential Geometry

This report concerns recent developments in the use of differential geo­metric methods to study nonlinear problems in automatic control. This has been an active subject for more than a decade now with contributions, coming from researchers in many countries. Eather than focusing here on a particular subarea of this discipline we have allowed ourselves to range rather broadly over the field using the discussion of a few unsolved problems as the main thread. In this way we hope to give some indication of the scope of the current activity and to touch on a representative sample of the geometrical ideas which play a role.

Feedback

Finite-dimensional, continuous time control systems have as their de­scription in local coordinates (x = dxjdt)

x(t) =r(x(t),u(t))

with x(t) being a point in Bn and u(t) being a point in Bm. Without loosing too much generality, we may describe a corresponding global object as follows. Let X be a finite-dimensional manifold and let ut: E-+X be a rank m vector bundle over X. Let TX denote the tangent bundle of X and let ut*TX denote the pullback of TX over B. Begarding (x, u) as a point in B, a section of the pullback of TX over B can be given locally by a function y(x, u) which, for each (x, u), specifies a velocity x = y(x, u). We denote the set of all such sections by r(B, ut*TX) and call the elements of this set control systems. Notice that associated with any y e r(B, ut*TX) there is a vector field y0 which is obtained by setting u equal to 0 ; we call this vector field the drift.

This definition includes a great many situations which are of techno­logical and mathematical interest such as mechanics problems with

[13Ö7]

1358 Section 14: E. W. Brockett

u representing exogeneous forces or torques, electrical networks with u representing voltage or current sources, etc. In order to help fix ideas we consider a specific example which illustrate the main points of the definitions. Let X be the unit sphere bundle over JB?3. (Think of {(x9 x) | x e B3

f

\\x\\ = 1}.) Construct B by taking the tangent bundle of the two-sphere and pulling it back over X. (Think of B = {(x9 x9 u)\x eB*9 \\cb\\ = 1, 0 = (u9 x}}.) The second order equation

tè = u9 <u9 x) = 0, \\x(t)\\ = 1

then defines a control system, i.e. an element of T(B9 ut*TX). This control system has a number of possible interpretations. On one hand, it describes the motion of a newtonian particle of unit mass and unit speed being acted on by a controllable force which is constrained to do no work. On the other hand, it can be thought of as describing the end point of a curve in JE73 whose curvature is \\u\\. In this latter guise y is a substitute for the more familiar Frenet-Serret system; x is the unit tangent vector, uj\\u\\ the unit normal, etc.

We return now to generalities. Given a control system y we can replace u by u + a{x) and get a modified system which we denote by ya. Since a section oiut: E->X is specified locally by a function a we see that what has just been given is the local coordinate description of a mapping

v: r(E,ut*TX)xr(X,B)~>r(B9ut*TX)9

v: (y9 «)->/ ,

where ya has the local coordinate description x = y(x, u + a(x)). The section a e r(X9 B) is called a feedback control law. Notice that we can also think of v as defining an action of the additive group r(X9 B) on the set of control systems r(B9 ut*TX). Because it is usually easy to implement a feedback control law, as opposed to making other modifica­tions in the system which y describes, it is important to have a good under­standing of this group action. A specific question which arises in this way is the

FEEDBACK STABILIZATION PROBLEM. Given y e T(B, ut*TX) and given a subset X1a X containing a distinguished point x0, does there exist a feedback control law a e r(X, B) such that xQ is an asymptotically stable critical point for (ya)0 with a domain of attraction which includes all of Xx ?

We will discuss some partial results on this problem below, after we have has a chance to make further definitions and have introduced one more problem.

Control Theory and Differential Geometry 1359

Beturning now to the mapping v9 notice that for a fixed y it de­scribes a mapping of the vector space F(X, B) into the vector space r(B9 ut*TX). If this mapping is affine then we see that y must admit a local description of the form

x =f(x)+G(x)u.

In the literature such systems are said to be input-linear control systems or affine control systems. As a further specialization, we will say that an input-linear control system y e F(B9 ut*TX) defines a linear control system if there exists a connection V on X with respect to which (i) X a complete flat affine space, (ii) the image of y~y0: E-+TX a flat subbundle of TX and (iii) in a neighborhood of each x we can describe y by equations of the form

x = Ax + Bu + £

with x19 x2,..., xn satisfying V^djdx^ = 0. Linear control systems can be described quite concretely. Let (Ni} n{)

denote the affine transformation x-^Nix+ni. Let {(^,%)} denote a group of affine transformations which act freely and properly discontinu-ously on Bn. As is well-known, the quotient space X == Bnl{Ni9ni)} then admits the structure of a complete flat affine space and all complete flat affine spaces arise in this way. In order to construct a linear system on Bnj{Ni, n4} we need to find A and B and a homomorphism

r . pr.,%)}->Gl(m)

such that A = H^Nj1, NtB = Bcp(Ni, n{) and J.% = 0. Under these circumstances the range of B defines a flat subbundle of TX (which we take to be B) and the local description

x = Ax + Bu

defines a linear control system on B.

FEEDBACK LINEARIZATION PROBLEM. Given y e r(B, n*TX) under what circumstances does there exist a e T(X, B) such that ya is a linear control système

In describing some results on stabilization and linearization we re­strict discussion to the input-linear case. Obviously linearization can not be achieved without this assumption and rather little can be said about stabilization in the more general situation. With this assumption in force we define a subset of TXX, the tangent space at x, by

&M = {œ\œ = ya(x, 0)-y0(oO, « e T(X, B)}.

1360 Section 14: R, W. Brockett

If we write x =f(x)+G(x)u this is just the range of G(x). Denote the resulting distribution by «^v Proceeding inductively, we define &h 3 «^"A-1

ZD . . . => #*0 by taking the distributions which correspond to ( [ , ] denotes Lie bracket) {

I t is not too hard to deduce from this definition that ^k(x) is, for x = f(x) + + G(x)u, simply

&rk(x) = span(E, AB,..., AkB),

where A = (df\dx)x and B =G(x). The linear system x ~Ax + Bu is said to be controllable if rank(B, AB,...) = dima?. For the purpose of this paper we want to call a y G T(B, ut* TX) quasi-linear if the dimensions of the ^h(x) are, for fixed Jc, independent of x and the controllability condition ^k(x) = TXX is satisfied for some Jc.

As a local question feedback linearization is understood, i.e. necessary and sufficient condition for there to exist a such that ya is linear are

(i) y should be quasi-linear; (ii) The distributions ^^,^x, ...,^h should be integrable. Furthermore if x0 belongs to the set of possible rest points

So = {x\ f(x) G Bange G(x)}

then for suitable a, ya has the local description x = Ax +Bu. This has some implications for stabilizability. It is well known that

a linear system x = Ax +Bu which is controllable can be stabilized to any point in 80 using control laws of the forms u = Cx+i-. Since asympto­tic stability of equilibrium point of a nonlinear system is determined by the linearization if the linearization does not yield eigenvalues on the imaginary axis, this means that any quasi-linear system can be stabilized to any point in S0 having a domain of attraction which includes some neighborhood of x0.

We now turn to a result on the nonexistence of stabilizing control laws. Consider the class of systems for which yQ = 0, d i m ^ is constant and equal to the dimension of the fibers of B, and for &k+1 = \ßk, &k] + + &k with 0O = ^ 0 have the property that the <Si are constant dimension with ^ r = TX for some r. For reasons which will be explained in the next section we call such systems guasi-Biemannian control systems. The con­dition 9r = TX insures that it is possible to find a control which steers any initial state to any final state. The condition that yQ is zero insures that the set 80 of potential equilibrium states is all of X. The quasi-Eieman-

Control Theory and Differential Geometry 1361

nian systems are quasi-linear only in the very special case ^"0 = TX œ B and, except for this case, represent an opposite extreme. As an example we have the following nonstabilizability result.

If y is feedback equivalent to a quasi-Biemannian control system with J^o rjk TX then SQ = X but ya has no asymptotically stable critical points regardless of the choice of a.

We emphasize that this rules out even local asymptotic stability, regardless of the choice of x0. The proof of this theorem is based on the fact that if x0 were asymptotically stable there would be a Liapunov function V whose derivative V = (dVldx9 (ya)0> would be negative on B = {x\ V(x) = e} but that by a degree argument dV/dx and J ^ must be perpendicular at some point on B unless ^"0 = TX.

In this brief account we were unable to mention the interesting work on invariant distributions, decoupling, etc. (see the references to Hirschorn and Isidori et al.). In connection with this kind of work it would be useful to know the answer to the following

DIMENSIONALITY PROBLEM FOR INTEGRABLE SUBDISTRIBUTIONS. Given integers n>m> 0 find the largest integer <p(n, m) sucJi tliat we may assert that every locally defined distribution of dimension m in Bn Jvas a locally defined subdistribution of dimension cp(n,m) wJiicJi is integrable.

There are obvious global versions of this problem as well but the local version seems to be what is needed the most in control theory.

We close this section with some remarks on the literature. Beferences [1 ,3 , 9-12] pertain to this section, references [3] and [9] contain papers by many authors working in this field and can be used to trace the litera­ture; a complete set references would be several pages long.

Hamilton-Jacobi theory

The qualitative issues raised in the previous section are reflected in the solutions of concrete optimizations problems. The problem of minimizing

t

rj = j (u(o), u(o)}da 0

for x =f(x)+G(x)u, subject to the constant x(0) = 0 , x(t) = xx will now be used to illustrate this and to give some indication about solved and unsolved problems. In addressing these questions we will asume that x =f(x)+G(x)u is controllable in the sense that for any given t > 0 and

1362 Section 14: E. W. Brockett

any xt near 0 there is a control such that x(t) = x±. The most interesting behaviour corresponds to the case /(0) — 0 and we limit ourselves to this case.

Introduce the value function S(t, x) defined to be the minimum value of rj expressed as a function of * and the end point x. Assuming that the indicated partial derivatives exist, 8 satisfies the well known Hamilton-Jacobi equation

Since there may be many points where these derivatives do not exist we must interpret this equation in some weak sense.

DEGENERATE HAMILTON-JACOBI PROBLEM. Give a complete theory of the solution of the above Eamilton-Jacobi equation in a neighborhood of an equilibrium point of f assuming only the controllability of x =» f(x) + +G(x)u.

H / is identically zero and G(x) is of rank dimX then this Hamilton-Jacobi equation appears in Eiemannian geometry. More precisely, if the metric tensor is expressed as (G(x)GT(x))~1 then 8(19 x) is the square of the distance between 0 and x. In this sense, the special case / .= 0 is a generalization of Eiemannian geometry. This also explains why we called this situation quasi-Eiemannian in the last section.

To begin with we solve the above equation in the linear case. This solution is well known, easy to verify, and will be qualitatively correct for the quasi-linear problems of the previous section. Using the notation x = Ax +Bu9 introduce

t W(t) = feÄ{i^BBTeAT(-i^da.

0

Assuming controllability this matrix will be invertible for all t > 0 and 8(t9x) = <#, W^tyx} will satisfy the corresponding Hamilton-Jacobi equation. W"1^) has a pole at t = 0 which we now describe. Let r be the least integer such that (B9 AB,..., ArB) is of rank n. Then for t small and positive

W~l(t) = £ Ei*+M{t)

with M(t) analytic near 0, M(0) = 0 and the E{ having range (B, AB9... ..., Ai~2B) in their kernel. TSote that 8(t9x) is homogeneous in t only in the uninteresting case corresponding to rank B =n9A = 0 .

Control Theory and Differential Geometry 1363

We now ton to a quasi-Biemannian situation. We coordinatile space X by an m-vector y and an m by m skew-symmetric matrix Z. The equa­tions are

y ~u,

Z = yuT —uyT.

It is easy to see this system is controllable on its m(m+1)/2-dimensional state space. Variational arguments show that an optimal y satisfies y + Qy = 0 for same skew symmetric matrix Û. If y(0) = 0 and Z(0) = 0 then y(t) = emX — X and Z(t) is given by

t

Z(t) = y(t)yT(t)-2 jen°'lXTeQTadoQ. 0

The corresponding 8(t,y,Z) satisfies the following identities:

B{t,y,o) =» •%

8(t,y,Z) =8(l,y,Z)lt9

8(t,y,Z) =S(t,ay,a*Z)la*,

8(t, y, Z) = 8(t, by, 6ZdT), 6dT = I .

From the last of these we see that 8(t, 0,Z) must be expressible in terms of the eigenvalues of Z and it has been shown that

8(t,0,Z) - 2TT(/11+2;12+ ... +rAr)/t,

where /Lx > A2 > ... > A,. are the positive eigenvalues of iZ. The derivation of this result shows, moreover, a rather remarkable "exclusion principle" which accounts for the different weights. Briefly stated, the optimal controls which steer this system from (0, 0) to (0, Z) have the form emb with the eigenvalues of Q being multiples of 2izi. However, it turns out that the nonzero eigenvalues of Q cannot be repeated and that the number of distinct eigenvalues must equal rank Z.

For the special case m = 2 it is illuminating to write down the first few terms of the Newton-Puiseux expansion for 8. From the scaling properties we see that 8(t,y,z) = (\&\lt)'s(y*l\2\). In fact it has an ex­pansion in y\}/z which we may express as

8(h y, «) - (|*|/t)(2w-V87u 112/11 *\z + \\yf+ . . . ) , this asymptotic expansion being valid off the plane z = 0. It is possible to interpret VB(19 y,z) as a distance function and this has been persued in some detail in our paper on singular Eiemannian geometry cited below.

1364 Section 14: E. W. Brockett

Although it may not be apparent immediately, this second case is a paradigm for the general class of problems whose Hamiltion-Jacobi equation is

88 11 d8 - asr\ - r - = ~--r(#1(x)-—-, # (0)-T—>

under the hypothesis that rank G(0) = m and dima? = dim([0, 0 ] + ^ ) = m (m+l)/2, provided that we restrict our attention to a neighborhood of x = 0 . This is explained in more detail in the paper just refered to.

I t is clear that one can raise many questions in this area. The following is obviously one of interest.

HAMILTON-JACOBI ASYMPTOTICS PROBLEM. Bind the correct gener­alization of the given Newton-Puiseux expansion for cm arbitrary quasi-Biemannian problem, assuming analyticity of &.

Beferences [2, 4, 5, 8] deal with aspects of this material. We should point out that there is a discrepancy between our formula for 8(1, 0, Z) (which is taken from [2]) and the claims of [4], Section 5.3. I t seems that in [4] the possibility of optimal trajectories corresponding to "higher harmonics" has been overlooked.

Stochastic phenomena

ÎSTo discussion of the relationships between control theory and differential geometry should fail to touch on how these ideas illuminate problems related to Ito models of the form

dx =-f(x)dt + ^gi(x)dwi. i=i

We sketch out how this goes and later on describe its connection with nonlinear filtering.

The first idea is that it is, for some purposes at least, worthwhile to study the above stochastic equation with the help of the control model

& = /(as) - - JT l~\ gt{x) + JT ft (») »«

= f(œ)+6(œ)u.

Control Theory and Differential Geometry 1365

The explanation of w h y / got traded fo r / when we replaced "dw/dt" by u is to be found in the asymmetrical definition of the Itô integral — a point that has been discussed many times. Actually one gets more than just a control system x = f(x)+G(x)u out of this. Because Wiener pro­cesses in Bm are defined with respect to the quadratic form defining the variance, we get, automatically, an innerproduct on u as well.

Beturning to the stochastic equation again, it has associated to it a second order differential operator L with the property that the prob­ability density ç(t, x) evolves according to

If m is less then dima?, L will be degenerate; in fact it is of the form LQ + +1/1+ ... + 1 4 where the Li are first-order differential operators. The first point of contact between the control system and the stochastic equa­tions is that x =f(x)+G(x)u has the property that at any time t > 0 the reachable set of states from a given x0 has nonempty interior if and only if djdt — L is hypoelliptic. Thus the support and the smoothness of the probability density is what one would guess it to be by looking at the sample path behavior of the related control system. The following problem can serve as a focus for our remaining remarks.

HAMILTON-JACOBI/FOKKER-PLANCK PROBLEM. Belate tJie small time behavior of tJie solution of Bolcher-PlancJc equation to the solution of the Eamilton-Jacobi equation associated witlv the corresponding control problem.

Of course in the case of full ellipticity this is an absolutely standard idea.

As we have done previously, we recall the situation for the linear case. Consider

dx = Axdt+Bdw.

There is a beautiful formula for the probability density corresponding to a>(0) = 0

p(t, x) = 1 = e-****, \/(àet8xx)(27z)n

where 8 is the solution of the Hamilton-Jacobi equation discussed above. What this equation says is that the probability density at (t, x), given (c(0) = 0, is inversely proportional to the exponential of the cost of getting

1366 Section 14: E. W. Brockett

to x from zero in t units of time. The expansion of 8(t, x) given in connection with Hamilton-Jacobi theory gives a rather detailed picture of the small time asymptotics of p. I t is apparent, for example, that the rate of growth of p(t, x) depends strongly on the particular subspace in which x lies.

We now turn to the quasi-Eiemannian prototype

dy = dn,

dZ =y dwT — dwyT.

P. Levy studied a special case of these equation corresponding to àìmy = 2 and recently there has been a great deal of interest in the general situation by probabilists and analysts alike, due in part to the many ex­plicit formulae which describe the relevant probability distributions.

Of principle interest for our purpose is the geometrical interpretation of the right-hand side of the diffusion equation as an analog of the Lap-lace-Beltrami operator. This goes hand in hand with the interpretation of VS(1, y, Z) as distance and leads to a generalization of the well known formula of Varadhan

lim2Ünp(t,x) = -d*(x)

relating the distance from x0 to x and the probability density at (*, x) given that x(0) = x0.

Oui1 final problem concerns the area of estimation theory and illus­trates again the value of sample path considerations in those situations which are sufficiently robust. The key idea is to try to capture the sym­metries which make finite-dimensional estimatimation possible in a Lie algebra setting.

We are given a stochastic differential equation

dx =f(x)dt + g(x)dw

together with an observation

dy = h(x)dt + dv9 y eB1

and wish to find Q(1,X\ y[Qtt]), the probability density at (t,x) condi­tioned on the observations over the whole interval [0, *]. We assume Q(09X)

is given. The equation for Q is nonlinear but a certain path dependent multiple of Q, which we denote by Q, satisfies the stochastic partial differen­tial equation (here L is the Fokker-Planck operator associated with dx =f(x)dt + g(x)dw)

ÔQ = Lçdt + dyh(x)Q.

Control Theory and Differential Geometry 1367

If we associate to this equation a control system using the procedure outlined above we obtain

^-^(L-^(x))e + uh(x)e.

Abstractly this is an equation of the form x = Ax + uBx.

If it happens that the Lie algebra generated by the operators A and B is finite-dimensional we would be lead to postulate a solution of the form

x(t) = e^me^n . . . eH^r o?(0),

where Ex, E2 9... 9 Er is a suitably chosen ordered basis for the Lie algebra generated by A and B and {%} is a set of scalar functions dependent on u. In this context the Lie algebra

{L-\li(x),Ji(x)}LA

is called the estimation algebra. Bather remarkably, in the standard Gauss-Markov cases it is the same as the oscillator algebra of quantum mech­anics. Our final problem is the

CLASSIFICATION OF FINITE-DIMENSIONAL ESTIMATION ALGEBRAS.

Bind all tlie finite-dimensional Lie algebras which can occur as the estimation algebras for diffusion processes.

The literature on the connections between the Hamilton-Jacobi equa­tions and degenerate diffusions is very large. We mention Gaveau [4] and the more recent works [7,13]. The book [6] contains a number of papers on the connection between Lie algebasr and filtering. The literature can be traced from these.

Acknowledgements

This work was supported in part by the U.S. Army Besearch Office under Grant No. DAAG29-79-0-0147, Air Force Grant Mb. AFOSB-81-7401, the Office of Naval Eesearch under JSBP Contract No. N00014-76-C-0648, and ^he National Science Foundation under Grant No. EOS-81-21428.

References

[1] Brockett R. W., Feedback Invariants for Nonlinear Systems. In: Proc. IFAO Congress, Helsinki, 1978.

[2] Brockett R. W., Control Theory and Singular Riemannian Geometry. In: P. Hilton and G. Young (eds.), New Direction in Applied Mathematics, Sprin­ger-Verlag, New York, 1981.

34 — Proceedings..., t. II

1368 Section 14: R. W. Brockett

[3] Broçkefct R. W., et al. (eds.), Differential Geometric Control Theory, Birkhäuser, Boston, Ma., 1983.

[4] Gaveau B., Principle de Moindre action propagation de la chaleur et estimées sons elliptiques sur certains groupes nilpotents, Acta Mathematica 139 (1977), pp. 95-153.

[5] Günther N., Hamiltonian Mechanics and Optimal Control, Ph.D. thesis, Harvard University, 1982.

[6] Hazewinkel M. and Willems J. C. (eds.), Stochastic Systems: The Mathematics of Filtering and Identification and Applications, Reidel, 1981.

[7] Helmes K. and Schwane A.., Levy's Stochastic Area Formula in Higher Dimensions, Springer Lecture Notes in Control and Information Sciences 42, Springer-Verlag, New York, 1982.

[8] Hermann E. , Geodesies of Singular Riemannian Metrics, Bull. AMS 79 (1973), pp. 780-872.

[9] Hinrichsen D. and Isidori A. (eds.), Feedback Control of Linear and Nonlinear Systems, Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin, 1982.

[10] Hirschorn R., {A, JB)-Invariant Distributions and the Disturbance Decoupling of Nonlinear Systems, SIAM J. Control and Optimization 17 (1981), pp. 1-19.

[11] Isidori A., Kroner A^J., Gori-Gorgi C , and Monaco S., Nonlinear Decoupling via Feedback : A Differential Geometric Approach, TEFF Trans. Aut. Control 16 (1981), pp . 331-345.

[12] Jakubczyk B. and Respondek W., On the Linearization of Control Systems, x Bulletin de VAcadémie Polonaise des Sciences 28 (1980), pp. 517-522.

[13] Taylor Th., Hypoelliptic Diffusions and Nonlinear Control Theory, Ph.D. thesis, Harvard University, 1983.