Assessment and Future Directions of Nonlinear Model Predictive ...

318 D. Limon et al.The system is subject to constraints on both the state and the control input.These constraints are given byu ∈ U, x ∈ X (5)where X is a closed set and U a compact set, both of them containing the origin.The objective of this paper is to present some robust MPC controllers whichare able to robustly stabilize this system. These controllers deal with the uncertaintyby means of interval arithmetic with an affordable increment of thecomputational burden with respect to the nominal problem. In the followingsection, a brief introduction to interval arithmetic is presented.2 A Brief Introduction to Interval ArithmeticAn interval X =[a, b] istheset{ x : a ≤ x ≤ b }. The unitary interval isB =[−1, 1]. The set of real compact intervals [a, b], where a, b ∈ R and a ≤ b,is denoted as I. A box is an interval vector. A unitary box, denoted as B m ,isabox composed by m unitary intervals. Given a box Q =([a 1 ,b 1 ],...,[a n ,b n ]) ⊤ :mid(Q) denotes its center and diam (Q) =(b 1 −a 1 ,...,b n −a n ) ⊤ . The Minkowskisum of two sets X and Y is defined by X ⊕ Y = { x + y : x ∈ X, y ∈ Y }.Given a vector p ∈ R n and a matrix H ∈ R n×m ,theset:p ⊕ HB m = { p + Hz : z ∈ B m }is called a zonotope of order m. Given a continuous function f(·, ·) andsetsX ⊂ R n and W ⊂ R nw , f(X, W) denotes the set { f(x, w) : x ∈ X, w ∈ W }.Interval arithmetic and Kühn’s method provides two approaches to obtain outerbounds of set f(X, W).The interval arithmetic is based on operations applied to intervals. An operationop can be extended from real numbers to intervals as: for a given A, B ∈ I,AopB= { aopb : a ∈ A, b ∈ B }. The four basic interval operationsare defined in [1], where the sum is [a, b]+[c, d] =[a + c, b + d], and the productis [a, b] ∗ [c, d] = [min(ac,ad,bc,bd), max(ac,ad,bc,bd)], for instance. Theinterval extension of standard functions {sin, cos, tan, arctan, exp, ln, abs, sqr,sqrt} is possible too. A guaranteed bound of the range of a non-linear functionf : R n → R can be obtained by means of the natural interval extension, that is,replacing each occurrence of each variable by the corresponding interval variable,by executing all operations according to interval operations and by computingranges of the standard functions.Theorem 4. [1] A natural interval extension □(f) of a continuous functionf : R n → R over a box X ⊆ R n satisfies that f(X) ⊆ □(f(X)).The natural interval extension is a particular and efficient way to compute aninterval enclosure. However, natural interval extension may lead to unnecessaryoverestimation when a variable appears several times in the same expression(multi-occurrence). To reduce this overestimation, Kühn’s method can beused [2].