Assessment and Future Directions of Nonlinear Model Predictive ...

A Two-Time-Scale Control Scheme for Fast Unconstrained Systems 553Predictive control of (1) is based on repeatedly solving the following optimizationproblem:t k∫+T pL(x(τ),u(τ))dτ (2)min k + T p )) +u[t k ,t k +T c)t ks.t. ẋ = F (x, u), x(t k )=x m (t k ) (3)S(x, u) ≤ 0 T (x(t + T p )) = 0 (4)where Φ is an arbitrary scalar function of the states and L an arbitrary scalarfunction of the states and inputs. x m (t) represents the measured or estimatedvalue of x(t). S is a vector function of the states and inputs that representsinequality constraints and T is a vector function of the final states that representsequality constraints. The prediction horizon is noted T p and the control horizonis noted T c . In the following, T c = T p = T will be used. The solution to problem(2)-(4) will be noted (x ∗ ,u ∗ ). A lower bound for the re-optimization intervalδ = t k+1 − t k is determined by the performance of the available optimizationtools.2.2 System Inversion for Flat SystemsIf the system (1) is flat in the sense of [7, 8], with y = h(x) being named the flatoutput, then, for a given sufficiently smooth trajectory y(t) and a finite numberσ of its derivatives, it is possible to compute the corresponding inputs and states:u = u(y, ẏ, ..., y (σ) ) x = x(y, ẏ, ..., y (σ) ). (5)u(y, ẏ, ..., y (σ) ) is a nonlinear function that inverts the system.2.3 Neighboring-Extremal ControlUpon including the dynamic constraints of the optimization problem in the costfunction, the augmented cost function, ¯J, reads:¯J = Φ(x(t k + T )) +t∫k +Tt k(H − λT ẋ ) dt (6)where H = L + λ T F (x, u), and λ(t) isthen-dimensional vector of adjoint statesor Lagrange multipliers for the system equations. The first-order variation of¯J is zero at the optimum. For a variation ∆x(t) =x(t) − x ⋆ (t) ofthestates,minimizing the second-order variation of ¯J, ∆ 2 ¯J, with respect to ∆u(t) =u(t)−u ∗ (t) represents a time-varying Linear Quadratic Regulator (LQR) problem, forwhich a closed-form solution is available [3]:∆u(t) =−K(t)∆x(t) (7)K = Huu−1 (Hux + Fu T S) (8)