Copyright © by SIAM. Unauthorized reproduction of this article is ...

HOMOGENEOUS OBSERVER DESIGN 1829bi-limit stabilizing control law for the origin of the system (4.1), then there is one forthe origin of the system with state X i+1 =(X 1,...,X i+1) in R i+1 defined by(4.2) Ẋ 1 = X 2,...,Ẋ i+1 = u, i.e., Ẋ i+1 = S i+1 X i+1 + B i+1 u.1Let d 0 and d ∞ be in (−1,n−1) and consider the weights and degrees defined in (3.2).Theorem 4.1 (homogeneous in the bi-limit backstepping). Suppose there existsa homogeneous in the bi-limit function φ i : R i → R with associated triples (r 0 , d 0 +r 0,i ,φ i,0 ) and (r ∞ , d ∞ + r ∞,i ,φ i,∞ ) such that the following hold:1. There exists α i ≥ 1 such that the function ψ i (X i ) = φ i (X i ) αi is C 1 andfor each j in {1,...,i} the function ∂ψi is homogeneous in the bi-limit with∂X jweights (r 0,1 ,...,r 0,i ), (r ∞,1 ,...,r ∞,i ), degrees α i (r 0,i + d 0 ) − r 0,j , α i (r ∞,i +d ∞ ) − r ∞,j , and approximating functions ∂ψi0, ∂ψi∞.∂X j ∂X j2. The origin is a globally asymptotically stable equilibrium of the systems(4.3)Ẋ i = S i X i + B i φ i (X i ) , Ẋ i = S i X i + B i φ i,0 (X i ) , Ẋ i = S i X i + B i φ i,∞ (X i ) .Then there exists a homogeneous in the bi-limit function φ i+1 : R i+1 → R withassociated triples (r 0 , d 0 + r 0,i+1 ,φ i+1,0 ) and (r ∞ , d ∞ + r ∞,i+1 ,φ i+1,∞ ) such that thesame properties hold, i.e.,1. there exists a real number α i+1 > 1 such that the function ψ i+1 (X i+1 ) =φ i+1 (X i+1 ) αi+1 is C 1 and for each j in {1,...,i+1} the function ∂ψi+1is∂X jhomogeneous in the bi-limit with weights (r 0,1 ,...,r 0,i+1 ), (r ∞,1 ,...,r ∞,i+1 ),degrees α i+1 (r 0,i+1 + d 0 ) − r 0,j , α i+1 (r ∞,i+1 + d ∞ ) − r ∞,j , and approximatingfunctions ∂ψi+1,0∂X j, ∂ψi+1,∞;∂X j2. the origin is a globally asymptotically stable equilibrium of the systems(4.4)X i+1 = S i+1 X i+1 + B i+1 φ i+1 (X i+1 ) ,X i+1 = S i+1 X i+1 + B i+1 φ i+1,0 (X i+1 ) ,X i+1 = S i+1 X i+1 + B i+1 φ i+1,∞ (X i+1 ) .Proof. We prove this result in three steps. First, we construct a homogeneous inthe bi-limit Lyapunov function; then we define a control law parametrized by a realnumber k. Finally, we show that there exists k such that the time derivative, alongthe trajectories of systems (4.4), of the Lyapunov function and of its approximatingfunctions is negative definite.Step 1. Construction of the Lyapunov function. Let d V0 and d V∞ be positive realnumbers satisfying(4.5) d V0 > maxj∈{1,...,n} {r 0,j}, d V∞ > maxj∈{1,...,n} {r ∞,j} ,and(4.6)d V∞r ∞,i+1≥ d V 0r 0,i+1> 1+α i .With this selection, Theorem 2.20 gives the existence of a C 1 , proper, and positivedefinite function V i : R i → R + such that, for each j in {1,...,n}, the function ∂Vi∂X jCopyright © by SIAM. Unauthorized reproduction of this article is prohibited.