Controllability, Observability & Local Decompositions

Transcription

1 ontrollability, Observability & Local Decompositions Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University

2 Outline Lie Bracket Distributions ontrollability ontrollability Distributions ontrollability Rank ondition Examples Observability Observability odistributions Observability Rank ondition Examples Local Decompositions

3 Lie Bracket Definition: If v,w are vector fields on M, then their Lie bracket [v,w] is the unique vector field defined in local coordinates by the formula Property: [ v, dw w] = ( t, x) w x ( Ψ ) dt v t= = v x [ vw, ] w x The rate of change of w along the flow of v

4 Lie Bracket Interpretation Let us consider the Lie bracket as a commutator of flows. Beginning at point x in M follow the flow generated by v for an infinitesimal time which we take as for convenience. This takes us to point ε y = exp( ε v)x Then follow w for the same length of time, then -v, then -w. This brings us to a point ψ given by εw εv εw εv ψε (, x) = e e e e x

5 Lie Bracket Interpretation ontinued u -w -v w z ψ(ε,x) [v,w] v y x d Ψ = dε ( +, x) [ vw, ] x

6 Iterated Lie Bracket We recursively define higher order Lie Brackets: k v ad ( w) = w k 1 v ad = v, ad ( w) v

7 Distributions v,, v is a set of vector fields on M 1 r { } = span v( p),, v ( p) is a subspace of TM p 1 r p Definition: A smooth distribution on M is a map which assigns to each point p M, a subspace of the tangent space to M at p, p TMp such that p is the span of a set of smooth vector fields v 1,..,v r evaluated at p. We write =span{v 1,..,v r }. Definition: An integral submanifold of a set of vector fields v 1,..,v r is a submanifold N M whose tangent space TNp is spanned by {v 1 (p),..,v r (p)} for each p N. The set of vector fields is (completely) integrable if through every point p M there passes an integral submanifold.

8 Involutive Distributions Definition: A system of smooth vector fields {v 1,..,v r } on M is in involution if there exist smooth real valued functions c ij k (p), p M and i,j,k = 1,..,r such that for each i,j r ij i, j cv k k k = 1 v v = Proposition: (Froebenius) Let {v 1,..,v r } be an involutive system of vector fields with dim [span{v 1,..,v r }]=k on M. Then the system is integrable with all integral manifolds of dimension k. Proposition: (Hermann) Let {v 1,..,v r } be a system of smooth vector fields on M. Then the system is integrable if and only if it is in involution.

9 Example 3 M = R = span{ v, w} v = y x w = z 2 2zx 2yz + 1 x 2 y 2 [, vw] so the distribution is completely integrable. The distribution is singular because dim = 2 everywhere except on the z-axis x =, y = 2 2 and on the circle x + y = 1, z = where dim = 1 The z-axis and the circle are one-dimensional integral manifolds. All others are the tori: T c = {(,, ) ( ) ( 1) 2} / x y z R x + y x + y + z + = c >

10 Example

11 Invariant Distributions Definition: A distribution = { v 1,, v r } on M is invariant with respect to a vector field f on M if the Lie bracket [f,v i ], for each i = 1,,r is a vector field of. Notation: [ f, ] = span{[ f, vi ], i = 1,, r} that is invariant with respect to f may be stated [f, ]. In general +[f, ] = +span{[f,v i ], i=1,..,r } = span{v 1,..,v r,[f,v 1 ],..,[f,v r ]}

12 Involutive losure ~ 1 Problem 1: find the smallest distribution with the following properties It is nonsingular It contains a given distribution It is involutive It is invariant w.r.t. a given set of vector fields, τ 1,,τ q τ1, τ q

13 Algorithm Algorithm for Problem 1: = [ τ, ] = + k k 1 i= 1 i k 1 q stop when = k k 1

14 Affine Systems x = f( x) + Gxu ( ) = f( x) + g( xu ) m i= 1 i i y = hx ( ) n p m x R, y R, u R

15 ontrollability x f is U-reachable from x if given a neighborhood U of x containing x f, there exists t f > and u(t) on [,t f ] such that x goes to x f along a trajectory contained entirely in U. The system is locally reachable from x if for each neighborhood U of x the set of states U-reachable from x contains a neighborhood of x. If the reachable set contains merely an open set the system is locally weakly reachable from x. The system is locally (weakly) controllable if it is locally (weakly) reachable from every initial state. R. Hermann and A. J. Krener, "Nonlinear ontrollability and Observability," IEEE Transactions on Automatic ontrol, vol. 22, pp , 1977

16 ontrollability Distributions = { } f, g,, g span f, g,, g 1 m 1 m = { } f, g,, g span g,, g 1 m 1 m O, satisfy O { f } + span O { f } { } { } xa regular point of + span f ( x) + span f( x) = ( x) if and + span are of constant dim, then O dim dim 1 O O O O

17 ontrollability Rank ondition Proposition: A necessary and sufficient condition for the system to be locally weakly controllable is ( ) dim x = n, x R A necessary and sufficient condition for the system to be locally controllable is ( ) dim x = n, x R n n

18 Example: Linear System ontrollability n x = Ax + Bu, x R, u R m f ( x) = Ax, g ( x) = b, i = 1,, m [ Ax, b ] i b Ax = Ax b = x x i i i i b,, [, ] i bj = Ax Ax = i Ab

19 Example: Linear System ontinued = k = 1 span span { B} { B AB} span { k 1 B AB A B} { n 1 } H Thm = span B AB A B = = = { Ax B} span, { n 1 Ax B AB A B} span,,,

20 Example: Bilinear System x = x 2 + xu = span 1 weakly locally controllable 1 1 = span 2x 2 not locally controllable 3x 3

21 Example: Extended Wheeled Robot The extended system is: θ ω x vx cosθ d y vx sinθ u = + dt vx z T ω 1 z 1

22 Example: Extended Wheeled Robot = span,,,, rank 6 controllable at generic state tanθ = span,,, x,, rank 5 uncontrollable at x = = 1 1 Actually, this is true for any point 1 with vx and ω both zero. For generic points with only v =, the system is controllable. x

23 Example: Parking body fixed frame y b x b v θ u= vu, = ω 1 2 φ (x,y) steering & drive wheel y d drive x cos( φ+ θ) y sin( φ+ θ) u = θ 1 steer space frame 1 x dt φ sinθ u2

24 Parking, ontinued = ( ) ( ) ( θ) { } f, g,, g span f, g,, g 1 m 1 m = { } f, g,, g span g,, g 1 m 1 m O ( ) ( ) ( θ) cos φ+ θ cos φ+ θ sin φ θ sin φ θ + + = =, span, O sin sin 1 1

25 Example: Parking, new directions from Lie bracket sin( θ + φ) cos( θ + φ) wriggle = [ steer, drive] = cosθ sinφ cosφ slide = [ wriggle, drive] =

26 Parking, ontinued cos( φ+ θ) sin( θ + φ) sinφ sin ( ) cos( θ φ) cosφ φ+ θ + span,,, sin ( θ ) cosθ 1 1 1,,, 1 1

27 Recall Lie Bracket Interpretation as ommutator of Flows Let us consider the Lie bracket as a commutator of flows. Beginning at point x in M follow the flow generated by v for an infinitesimal time which we take as for convenience. This takes us to point ε y = exp( ε v)x Then follow w for the same length of time, then -v, then -w. This brings us to a point ψ given by εw εv εw εv ψε (, x) = e e e e x

28 Example: Parking, implementation y b v θ wriggle=steer+drive-steer-drive x b φ (x,y) slide=wriggle+drive-wriggle-drive

29 More ontrollability Distributions { k f ad g i m k n } = span,,1, 1 L f i { k ad g 1 i m, k n 1} = span, L f i k k 1 adv( w) = w, adv = v, adv ( w) weak local controllability = n = n local controllability = n = n L L

30 Example: Linear Systems Revisited f ( x) = Ax, G( x) = B = = L { n 1 Ax B AB A B} span,,,, = = L { n 1 B AB A B} span,,,

31 ontrollability Hierarchy ( ) dim ( ) weak local controllability dim x = n x = n L ( ) dim ( ) local controllability dim x = n x = n = { } f, g,, g span f, g,, g 1 m 1 m = { } f, g,, g span g,, g 1 m 1 m O k { f ad g i m k n } k { ad g i m k n } = span,,1, 1 L f i = span,1, 1 L f i L dim n 1 linear controllability B AB A B = n

32 Observability onsider an open set U in R n. x 1,x 2 are U-distinguishable if there exists a control u(t) whose trajectories from both x 1,x 2 remain in U such that y(t;x 1,u) y(t;x 2,u). Otherwise they are U- indistinguishable. The system is strongly locally observable at x if for every nbhd U of every state in U other than x is U-distinguishable from x. It is locally observable at x if there exists a nbhd W of x such that for every nbhd U of x contained in W every state in U other than x is U-distinguishable from x. The system is (strongly) locally observable if it is (strongly) locally observable at x for every x in R n.

33 Observability odistributions Ω = ( ) { } f, g,, g span dh,, dh O 1 m 1 p L f i =Ω O { k,1, 1} Ω = span L dh i p k n The distribution O is invariant wrt f,g 1,,g m and and it is contained in the kernel of span{dh 1,,dh p }. If it is nonsingular, it is also involutive. O

34 Observability Rank ondition Proposition: If Ω O (equivalently, O ) is of constant dimension on some open set U, then the system is locally observable on U if and only if dim Ω O = n, or equivalently, dim O =.

35 Example: Linear System Observability x = Ax + Bu, y = x f ( x) = Ax, g ( x) = b, dh = c L c = c A, L c = i i j j Ax j j b j A Ω = span { }, Ω 1 = span,, Ω k = span A k 1 A A H Thm rank = n n 1 A i

36 Example: Role of Input x 1 x2 x x x = + u, y = x x The linearized system is not observable, The system with g(x)=, yields [ 1] [ ] Ω O = span 1 On the other hand, for this system [ 1 ] Ω O = span [ 1 ] [ 1]

37 Observability Hierarchy Ω = ( ) { } f, g,, g span dh,, dh locally observable dimω x = n O 1 m 1 p O L f i { k,1, 1} Ω = span L dh i p k n ( ) zero input observable dimω x = n ( ) linearly observable dim = n n 1 A L

38 Local Decompositions, ΩO, +ΩO, all of constant dimension on U ζ = Ψ( x) such that ζ 1 f1( ζ1, ζ2, ζ3, ζ4) G1( ζ1, ζ2, ζ3, ζ4) f 2 2( ζ2, ζ4) ζ G2( ζ2, ζ4) = + u ζ f ( ζ, ζ ) ζ f 4 4( ζ 4) restricted to ζ 3 = y = h( ζ, ζ ) , ζ = locally controllable restricted to ζ =, ζ = locally observable