Assessment and Future Directions of Nonlinear Model Predictive ...

Coordination of Networked Dynamical Systems 5874ˆα(t)4α(t)3.53.5332.52.5221.51.5110.50.500 100 200 300 400 500 600 700 800 900# step00 100 200 300 400 500 600 700 800 900# stepFig. 4. (Left) (Middle) ˆα(t) computed, r(ˆα(t)) is the corresponding reference. (Right)α(t) applied, r(α(t)) is the corresponding reference.0.5u(t)0.05ɛ(t)Input [N]0Error0−0.50 100 200 300 400 500 600 700 800 9000.5−0.050 100 200 300 400 500 600 700 800 900 10000.05Input [N]0Error0−0.50 100 200 300 400 500 600 700 800 9000.5−0.050 100 200 300 400 500 600 700 800 900 10000.05Input [N]0Error0−0.50 100 200 300 400 500 600 700 800 9000.5−0.050 100 200 300 400 500 600 700 800 900 10000.05Input [N]0Error0−0.50 100 200 300 400 500 600 700 800 900# step−0.050 100 200 300 400 500 600 700 800 900 1000# stepFig. 5. (Left) Forces F x,yi,jthe prescribed limits.within the saturation limits. (Right) Tracking errors withinThe LDR strategy will be used and a direct link between the two vehicles isassumed.The state-space representation of the overall system is given by the followingequationsm 1 ẍ 1 = −k(x 1 − x 2 ) − β(ẋ 1 − ẋ 2 )+F1 x,m 1ÿ 1 = −k(y 1 − y 2 ) − β(ẏ 1 − ẏ 2 )+F y 1m 2 ẍ 2 = −k(x 2 − x 1 ) − β(ẋ 2 − ẋ 1 )+F2 x ,m 2 ÿ 2 = −k(y 2 − y 1 ) − β(ẏ 2 − ẏ 1 )+F y 2(9)where m 1 and m 2 are the two masses, k the spring constant, β the viscouscoefficient of the damper and Fix and F yi, i =1, 2, the forces acting as inputs.Each subsystem is locally pre-compensated by a suitable controller ensuringoffset-free tracking error to constant set-points on positions.The constraint set C is described by ∣ ∣F x,yi,j (t)∣ ∣ < 1[N], |y i (t) − r i (α(t))|