Assessment and Future Directions of Nonlinear Model Predictive ...

Towards the Design of Parametric Model Predictive Controllers 203and ψ g (x(t f )) = −x(t f ) + 1. The last expression represents that at final timethe beam should be positioned at a point where x(t f ) ≥ 1. We also assume thatt 0 = 0. Although, the problem has already been solved for a fixed initial pointx 0 =[0 0] T (as for example in [3]), here the optimal solution is derived for thefirst time as a function of the initial point coordinates.The problem is first dealt in discrete-time as described in Section 2. Thecontinuous-time system is turn into a discrete-time system assuming a samplingtime ∆t such that [t 0 ,t f ] is divided in three equally spaced time intervals of ∆ti.e. [t 0 ,t f ]=3∆t. The discrete-time problemmin ∆tγ kx k+1 = x k +(2gy k ) −1/2 cosγ k ∆t , k =0, 1, 2y k+1 = y k +(2gy k ) −1/2 sinγ k ∆t , k =0, 1, 2y k − 0.5x k − 1 ≤ 0 ,k=0, 1, 2, 3x 3 ≥ 1is then solved by transforming the above problem in (5) and solving the mp-LP problem to acquire the PWA solution. The continuous-time is solved nextfollowing Algorithm 3.1. The results for both the discrete-time case and thecontinuous-time case together with a simulation for x 0 =[0 0] T are shownin Figure 1 and 2. The straight line in both diagrams represents the boundaryof the linear constraint y − 0.5 − 1 ≤ 0. There are three control laws for thecontinuous-time case, depending in which region the initial state is. The controllaw in the unconstrained region (Figure 2.) is obtained by solving the followingsystem of algebraic equalities with respect to c 1 and γ0=c 2 1 − arccos(c 1√ y) − xc2 √ √1 + c 1 y sin arccos c1 yγ = −1/2(2g) 1/2 √c 1 t + arccos c 1 yIn the constrained region the control law is obtained as following. First, thefollowing system of equalities is solvedx(τ ′′ ) − (2g) 1 22c 1τ ′′ + 12c 2 1sin (2g) 1 2 c1 (t f − τ ′′ )=1− (2g) 1 22c 1t f = arccos c √ 11 y +0.5(2g) 2 c 1 τ ′′,x(τ ′′ )=0.1989gτ ′′2 +(2g 1 √0.5(2g) 1 2 )0.896 y0 τ ′′ + x 02 c 1y(τ ′′ )=(0.222(2g) 1 2 τ ′′ + √ y 0) 2, 0.46 = 0.5(2g)12 c1 (t f − τ ′′ )which is a system of five equations with five unknowns t f ,τ ′′ ,c 1 ,x(τ ′′ ),y(τ ′′ ).Then, the control to be applied is given asIf t ≤ τ ′′ then γ =0.46 = arctan(0.5) Else If t ≥ τ ′′ then γ =0.5(2g) 1 2 c 1 (t f − t)As it can be observed from Fig. 1 the approximating, discrete-time, PWAsolution is not the optimal one comparing to the optimal solution as it is givent f