The local structure of affine systems

The local structure of affine systems César Aguilar and Andrew D. Lewis Department of Mathematics and Statistics, Queen s University 05/06/2008 C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 1 / 22

Algebra in geometry D TM a distribution, x 0 M. π Dx0 : T x0 M T x0 M/D x0 the canonical projection. Define B D (x 0 ) 2 (D x 0 ) T x0 M/D x0 by B D (x 0 )(u, v) = π Dx0 ([U, V](x 0 )), where U and V are vector fields extending u, v D x0. Frobenius: If D is analytic or smooth with constant rank, it is integrable if and only if B D (x) = 0 for all x M. C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 2 / 22

Feedback invariance Many controllability theorems are not feedback invariant. Example On R m R n m consider a system with governing equations ẋ 1 = u, ẋ 2 = Q(x 1 ), where Q is a R n m -valued homogeneous polynomial of degree 2. Write Q(x 1 ) = (B 1 (x 1, x 1 ),..., B n m (x 1, x 1 )) for quadratic forms B 1,..., B n m. C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 3 / 22

Feedback invariance Example (cont d) By the generalised Hermes condition, (Sussmann 1 ) the system is STLC from (0, 0) if the diagonal entries in the matrices B 1,..., B n m are zero. Is this condition necessary? No. Is this condition invariant under feedback transformations of the form u Pu for P GL(m; R)? No. However... the system is STLC from (0, 0) if and only if there exists P GL(m; R) such that the diagonal entries of the matrices P T B 1 P,..., P T B n m P are zero. 1 SIAM J. Control Optim., 25(1), 158 194 C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 4 / 22

Feedback invariance Control-affine systems ẋ(t) = f 0 (x(t)) + ẋ(t) = g 0 (x(t)) + m u a (t)f a (x(t)), a=1 n v b (t)g b (x(t)) on a manifold M are feedback equivalent when they have the same trajectories (essentially). Have a relation like g 0 (x) = m λ a (x)f a (x), g b (x) = a=0 b=1 m Λ a b (x)f a(x), a=1 b {1,..., n}. C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 5 / 22

Feedback invariance Since trajectories are of most interest, one s approach to control theory should be feedback invariant, i.e., identify systems that are feedback equivalent. Working with a specific f 0, f 1,..., f m is like working with coordinates in differential geometry: Sometimes it is necessary, but it is best avoided if comprehension is what is sought. Two possible approaches: 1 Choose f 0, f 1,..., f m, then show results are independent of this choice. 2 Develop an approach that does not involve this choice. C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 6 / 22

Feedback invariance Example (cont d) Consider again: ẋ 1 = u, ẋ 2 = Q(x 1 ). TFAE: 1 STLC from (0, 0); 2 there exists P GL(m; R) such that the diagonal entries of the matrices P T B 1 P,..., P T B n m P are zero; 3 0 int(conv(image(q))). C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 7 / 22

Feedback invariance Condition 2 represents the choose generators and show independence on these approach. Condition 3 represents generator independent approach. Generalisation of the first approach seems hopeless. Generalisation of the second approach... how does one do it? Let us choose ignorance over hopelessness. C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 8 / 22

Affine systems Replace the data {f 0, f 1,..., f m } with a subset A TM such that, in a neighbourhood of any point x 0 M, there exist vector fields X 0, X 1,..., X k such that A x A T x M = { X 0 (x) + k u a X a (x) u R k}. The object A is a locally finitely generated affine distribution on M. The (not uniquely defined) vector fields X 0, X 1,..., X k are local generators. a=1 C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 9 / 22

Affine systems Typically controls for a control-affine system ẋ(t) = f 0 (x(t)) + m u a (t)f a (x(t)) are restricted to lie in a subset U R m. We generalise this by... An affine system in an affine distribution A assigns to each point x M a subset A(x) A x. Require the nondegeneracy condition that aff(a(x)) = Ax and require some fussy smoothness conditions that I will not state here. a=1 A trajectory of A is a locally absolutely continuous curve γ : I M such that γ (t) A(γ(t)) for a.e. t I. An A-vector field is a map ξ : M TM such that ξ(x) A(x). C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 10 / 22

Problems for affine systems We aim to study the local properties of affine systems. To what end? 1 Understand controllability. 2 Understand stabilisability. 3 Understand relationships between controllability and stabilisability. What geometric properties of affine distributions come into play to address these problems? We sort of think Lie brackets are involved, but how? Today: Understand this through controllability. C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 11 / 22

Controllability of affine systems Have the expected notions of controllability for an affine system A. Denote R A (x 0, T) = {γ(t) γ is a trajectory on [0, T] such that γ(0) = x 0 } and R A (x 0, T) = t [0,T] R A (x 0, t). Two flavours of controllability from x 0 M: 1 Accessibility: int(r A (x 0, T)) ; 2 Small-time local controllability (STLC): x 0 int(r A (x 0, T)). C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 12 / 22

Controllability of affine systems Bang bang type theorems = it suffices to consider piecewise constant controls. Thus need to consider trajectories of the form Φ ξ 1 t 1 Φ ξp t p (x 0 ) for A-vector fields ξ 1,..., ξ p and t 1,..., t p R 0. Baker Campbell Hausdorff: For k Z >0 there exists BCH k (η 1,..., η p ) such that Φ ξ 1 t 1 Have Φ ξp t p (x 0 ) = Φ BCH k(t 1 ξ 1,...,t pξ p) 1 (x 0 ) + O((t 1 + + t p ) k+1 ). BCH k (η 1,..., η p ) = η 1 + + η p + 1 2 j,k {1,...,p} j<k [η j, η k ] + C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 13 / 22

Controllability of affine systems Standard technique in controllability: Find curves in the reachable set so the (possibly higher-order) tangent vector of this curve at x 0 is a reachable direction. Construction of such tangent vectors: 1 Abbreviate Φ ξ1 t 1 Φ ξp t p (x 0 ) = Φ ξ x 0 (t 1,..., t p ). 2 Consider the map R p 0 t Φξ x 0 (t) M. 3 Consider a curve R 0 s τ (s) R p 0 with τ (0) = 0. 4 The composition Φ ξ x 0 τ is a curve in the reachable set. 5 Suppose the first k 1 derivatives of Φ ξ x 0 τ vanish. 6 Then dk ds k s=0 Φ ξ x 0 τ (s) (assume nonzero) is a kth-order tangent vector to the reachable set. 7 Denote ord x0 (ξ, τ ) = k. C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 14 / 22

Controllability of affine systems Example Take ξ = ( Y, X, Y, X) and τ (s) = (s, s, s, s). Then ord x0 (ξ, τ ) = 2 and d 2 ds 2 Φ ξ x 0 τ (s) = [X, Y](x 0 ). s=0 This is nothing new. C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 15 / 22

Algebraic constructions using jets For analytic systems data is contained in the Taylor expansions at x 0, i.e., the infinite jets at x 0. Jet bundle notation and properties. 1 For manifolds M and N, J k (x,y)(m; N) denotes the k-jets of maps for which x y. J k (x,0)(m; R) is a R-algebra. Elements of J k (x,y)(m; N) are homomorphisms of the R-algebras J k (y,0) (N; R) and Jk (x,0)(m; R). 2 For a vector bundle π : E M, J k xπ denotes the k-jets of sections of E at x. J k xπ is a R-vector space. C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 16 / 22

Algebraic constructions using jets If ord x0 (ξ, τ ) = k we are interested in the k-jet of s Φ ξ x 0 τ (s). We have j k (Φ ξ x 0 τ )(0) = j k τ (0) j k Φ ξ x 0 (0) (composition of algebra homomorphisms). j k τ (0) J k (0,0) (R; Rp ) is a rather canonical object. What about j k Φ ξ x 0 (0)? C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 17 / 22

Algebraic constructions using jets Some notation: 1 Denote T k x M = J k (x,0) (M; R) and (Rp ) k = J k (0,0) (Rp ; R). 2 π p TM : TMp M denotes the p-fold Whitney sum. 3 For a R-vector space V, S j (V) denotes the symmetric tensors of degree j. 4 S k (V) = k j=1 Sj (V). 5 Define k : V S k (V) v v (v v) (v v). 6 L(U; V) denotes the linear maps between R-vector spaces U and V. 7 Hom(A; B) denotes the homomorphisms of R-algebras A and B. C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 18 / 22

Algebraic constructions using jets Theorem For each k, p Z >0 there exists a unique map such that T k p (x 0 ) L(S k (J k 1 x 0 π p TM ); L(T k x 0 M; (R p ) k )) T k p (x 0 )( k (j k 1 ξ(x 0 ))) = j k Φ ξ x 0 (0) for every family ξ = (ξ 1,..., ξ p ) of C -vector fields. Moreover, the diagram 1 (J 0 x 0 π p TM ) 2 (J 1 x 0 π p TM ) 3 (J 2 x 0 π p TM ) Tp 1 (x 0 ) Hom(T 1 x M; (R p ) 1 ) 0 T 2 p (x 0 ) Hom(T 2 x 0 M; (R p ) 2 ) T 3 p (x 0 ) Hom(T 3 x M; (R p ) 3 ) 0 commutes, where the horizontal arrows are the canonical projections. C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 19 / 22

Algebraic constructions using jets Important points: 1 The diagram implies proj lim k exists. 2 The map Tp k (x 0 ) can be computed (and the proof of its existence is made) using Baker Campbell Hausdorff. 3 The map Tp k (x 0 ) is system independent: It depends only on k, p, and dim(m). How is T k p (x 0 ) used for controllability? C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 20 / 22

Controllability of affine systems (again) Fix k 2. Neutralisation: Find A-vector fields ξ 1,..., ξ p and τ : R 0 R p 0 such that j k 1 τ (0) T k 1 p ( k 1 (j k 2 ξ(x 0 ))) = 0. Bad news: This is an intractable nonlinear equation for j k 1 τ (0) and j k 2 ξ(x 0 ). Good news: The equation is algebraic and it can be written down! The point: If j k 1 τ (0) and j k 2 ξ(x 0 ) satisfy the neutralisability condition, then j k τ (0) T k p ( k (j k 1 ξ(x 0 ))) is a kth-order tangent vector to the reachable set. Note: The system only comes in through its jets. C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 21 / 22

Conclusions and future work Conclusion: The local structure of an analytic affine system about x 0 is captured by the canonical algebraic objects Tp k (x 0 ) restricted to system data. Future work: 1 Must do examples = need symbolic software to do computations. 2 See what kind of controllability theorems come from this framework. 3 Think about stabilisability = connection with Control Lyapunov Functions? 4. C. Aguilar, A. Lewis (Queen s University) The local structure of affine systems 05/06/2008 22 / 22