Previous work on nonlinear MPC with feedback linearization assumed the state x(t) is perfectly known to the controller at any moment in time Simon et al. [2013]. In practice, only a state estimate ˆx(t) is available, and ˆx(t)6=x(t). Thus a controller de- signed to work optimally when operating on the true statexis in general sub-optimal when operating on ˆx (and may even lead to instability). Robust MPC (RMPC) Bemporad and Morari [1999a] has been investigated as a way of handling state esti- mation errors for linear systems Richards and How [2005a], Kouvaritakis and Cannon [2015] and nonlinear systems Mayne and Kerrigan [2007], Streif et al. [2014], Oliveira and Morari [1994], but not via feedback linearization. In particular, for non-linear systems, Mayne and Kerrigan [2007] develops a non-linear MPC with tube like con- straints for robust feasibility, but involves solving two (non-convex) optimal control problems. In Streif et al. [2014], the authors solve a non-linear Robust MPC through a bi-level optimization that involves solving a non-linear, non-smooth optimization which is challenging. Streif et al. [2014] also guarantees a weaker form of recursive feasibility than Richards and How [2005a] and what we guarantee in this work. In Zhao and Go [2014] the authors approximate the non-linear dynamics of a quadrotor by linearizing it around hover and apply the RMPC of Richards and How [2005a] to the linearized dynamics. This differs significantly from our approach, where we formulate the RMPC on the feedback linearized dynamics directly, and not on the dynamics obtained via Jacobian linearization of the non-linear system. Existing work on MPC via feedback linearization and input/state constraints has also assumed that either T is the identity, which simplifies the subsequent stability and performance analysis Simon et al. [2013], or, in the case of uncertainties in the parameters, that there are no state constraints Son et al. [2001]. A non-identity T is problematic when the state is not perfectly known, since the state estimation errore= ˆx−xmaps to the linearized dynamics via T in non-trivial ways greatly complicating the analysis. In particular, the error bounds for the state estimate inz-space now depend on the cur- rent nonlinear statex. One of the complications introduced by feedback linearization is that the bounds on the input (u∈ U) may become a non-convex state-dependent constraint on the input v to Sf l: V = {v(x, U) ≤ v ≤ v(x, U)}. In Simon et al.
[2013] forward reachability is used to provide inner convex approximations to the in- put set V. A non-identity T increases the computational burden since the non-linear reach set must be computed (with an identity T, the feedback linearized reach set is sufficient).
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