Local nested transverse feedback linearization

Authors: Alireza Doosthoseini Christopher Nielsen

Publish Date: 2015/08/29

Volume: 27, Issue: 4, Pages: 493-522

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Abstract

We study two local feedback equivalence problems for a nonlinear controlaffine system with two nested controlledinvariant embedded submanifolds in its state space The first less restrictive result gives necessary and sufficient conditions for the dynamics of the system restricted to the larger submanifold and transversal to the smaller submanifold to be linear and controllable This normal form facilitates designing controllers that locally stabilize the smaller set relative to the larger set The second more restrictive result additionally imposes that the transversal dynamics to the larger set be linear and controllable This result can simplify designing controllers to locally stabilize the larger submanifold This is illustrated by sufficient conditions under which these normal forms can be used to locally solve a nested set stabilization problemLet Usubseteq mathbb Rn be an open set containing barx and set V 1 = S 1cap U If dim T xS 1cap G 0x is constant on V 1 then since dim T xS 1 and dim G 0x are constant on V 1 the function sigma x in 11 is constant on V 1 If both dim T xS 2cap G 0x and dim T xS 1cap G 0x are constant on V 2 = S 2cap U then the functions nu and rho in 11 are constant on V 2Conversely if the function sigma is constant on an open set V 1 subset S 1 with barxin V 1 then since T xS 1 and G 0x are constant dimensional and from the definition of sigma it follows that dim T xS 1cap G 0x is constant on V 1 If nu rho are constant on an open set V 2 subset S 2 with barxin V 2 then from their definitions it follows that dim T xS 2cap G 0x is constant on V 2 square Let barx in S 2 be a regular point of the distributions 10 Then by Proposition 2 and Definition 4 P is nonsingular in a neighbourhood V 2=V 1cap S 2 with V 1subseteq S 1 and containing barx Lemma 3 proves that P is also smooth in a neighbourhood of barx without loss of generality V 2 Proposition 1 shows that Q is nonsingular on V 2 and R is nonsingular on V 1 Furthermore by Proposition 2 the assumed nonsingularity of G 0 and Lemma 3 we have by possibly shrinking V 1 and hence V 2 that G 0 cap TS 1 and left G 0cap TS 1right perp are smooth on V 1 and left G 0xcap TS 2right perp is smooth on V 2 Therefore Q and R are the nonsingular intersection of smooth nonsingular distributions and by 14 Lemma 135 they are smooth themselvesConversely suppose that the distribution R in 10 is smooth and nonsingular in a neighbourhood V 1 subseteq S 1 containing barx and distributions P and Q in 10 are smooth and nonsingular in V 2=V 1cap S 2 By Proposition 1 and Definition 4 barx is a regular point of 11 square Let mathcal NM be a tubular neighbourhood of M By 20 Proposition 1020 there exists a smooth retraction r mathcal NM rightarrow M Let Usubseteq mathcal NM be an open set containing x Then restriction left r right U is a smooth retraction of U to Mcap U square Apply Lemma 2 to obtain an open set U subseteq mathbb Rn containing barx and maps varPhi 1 and barvarPhi 2 such that V 1=varPhi 110 and V 2=barvarPhi 2varPhi 110 where V 1=S 1cap U and V 2=S 2cap U24 Let N subset M be an ndimensional submanifold of the mdimensional manifold M Let p in N be a regular point of a ddimensional distribution D on M Suppose there exists an open neighbourhood V of p in N such that k = dim T qN cap Dq is constant for all q in V Then there exists a neighbourhood U of p in V such that TNcap D is smooth on ULet W psi be a coordinate chart of M adapted to N that is such that psi N cap W = x in psi W x n+1 = cdots = x m = 0 and let f 1 ldots f d be a set of local generators of D around p Let pi x 1 ldots x m mapsto x 1 ldots x n be the projection onto the first n factors By making W smaller we can assume that f 1 dots f d are linearly independent on W

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