Assessment and Future Directions of Nonlinear Model Predictive ...

Interval Arithmetic in Robust Nonlinear MPC 319Kühn’s method is a procedure to bound the orbits of discrete dynamical systems.In Kühn’s method, the evolution of the system is approximated by azonotope and sub-exponential overestimation is proven [2]. First the zonotopeinclusion operator is introduced in the following theorem:Theorem 1 (Zonotope inclusion). Consider a family of zonotopes representedby Z = p ⊕ MB m where p ∈ R n is a real vector and M ∈ I n×m isan interval matrix. A zonotope inclusion, denoted by ⋄(Z), is defined by:[ ]Bm⋄(Z) =p ⊕ [mid(M) G]B n = p ⊕ JB m+nwhere G ∈ R n×n is a diagonal matrix that satisfies:m∑ diam (M ij )G ii =, i =1,...,n.2j=1Under these definitions it results that: Z ⊆⋄(Z).The following theorem is a generalization of Kühn’s method for a function whichdepends on an unknown but bounded vector of parameters [3].Theorem 5. Given a function f(x, w) : R n × R nw → R n ,azonotopeX =p ⊕ HB m and a zonotope W = c w ⊕ C w B sw , consider the following intervalextensions:• A zonotope q ⊕ SB d such that f(p, W ) ⊆ q ⊕ SB d .• An interval matrix M = □(∇ x f(X, W))H.• A zonotope Ψ(X, W) =q ⊕ SB d ⊕⋄(MB m )=q ⊕ H q B l with l = d + n + mUnder the previous assumptions it results that f(X, W) ⊆ Ψ(X, W)Note that the zonotope q ⊕ SB d of theorem 5 can be obtained by means of anatural interval extension of f(p, W ). It is worth remarking that this methodincreases the dimension of the obtained zonotope, and hence its complexity. Inorder to reduce this effect, a lower order zonotope which bounds the obtainedzonotope can be used instead. This bound can be obtained by means of theprocedure proposed in [3].In the following section it is shown how an outer estimation of the reachableset can be obtained by means of the presented interval arithmetic methods.3 Guaranteed Approximation of the Reachable SetFor a system in absence of uncertainties and for a given sequence of control inputs,the prediction of the evolution of the system is a trajectory. However, whenuncertainties or noises are present, there exists a trajectory for each realizationof the uncertainties and noise. The set of these trajectories forms a tube that willbe denoted as the sequence of reachable sets. a precise definition is the following: