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Specifically, constraining the rate at which the dynamics of the switching function A(t) can evolve on the equilibrium manifold, it is possible to address input rate saturation constraints. 24) that a constraint on the rate at which the dynamics of the parameter A(t) can evolve on the equilibrium manifold can be enforced effectively placing a rate constraint on the control u(t). 4, we present an algorithm that outlines the key steps in constructing the hierarchical switching feedback controller.
4a. If no solution exists, )~s(x(t)) is unchanged. 4b. If one solution ~1 exists and p(~l) < p(,~) then switch )~s(x(t)) to )~1. 4c. ). Note that multiple solutions can be avoided by modifying the ca 's. Step 5. 11) holds. ) such that Step 3 is satisfied, can be guaranteed by modifying the first part Step 4 as follows: Step 4'. , search for the solutions of V~(x(tk)) < c~, )~ 9 As. In this case, the switching set S C_ As need not be explicitly defined and is computed online. Furthermore, the case where AT ~ 0 recovers the continuous framework described in this section.
34), ~ x x x=Xs(x) = q(z)vxscz)(z)' it follows that ~'(x) - dp(A)dA As(x) = q(x)vxs(x)~s(x). 41) is immediate. 33). 5. 43) where c : De --~ R l• is an arbitrary row vector such that c(x)vT(x) # O, x E De, A E S. 33). Proof. 43). 33). 43), with c(x) = Vxs(z)(x)M;sl(z)(x, q(x)), x E 7)c, implicitly characterizes the Lagrange multiplier q(x), x E De, since q(x) appears in M;ls(x)(x, q(x)). The next result provides an explicit characterization for the Lagrange multiplier q(x), x E I)c. 6. 45) where ck (x, q(x) ), x E 7)c, k = O, 1,...