Assessment and Future Directions of Nonlinear Model Predictive ...

A Two-Time-Scale Control Scheme for Fast Unconstrained Systems 557the simplified model can be written as:⎡ ⎤ ⎡⎤ ⎡ ⎤ẍsin(η 2 )0⎢ ⎥ ⎢⎥ ⎢ ⎥⎣ ÿ ⎦ = C xyz ⎣ −cos(η 2 )sin(η 1 ) ⎦ v 1 − ⎣ 0 ⎦ (20)¨zcos(η 2 )cos(η 1 ) g¨φ = C φ v 2 (21)1¨η 1 =Cη 1 1cos(η 2 ) 2 + Cη 2 (C d sin(η 2 )v 2 + C s dcos(η 2 )(sin(φ)v 3 + cos(φ)v 4 )1− IA M cos(η 2)˙η 2 v 5 ) (22)¨η 2 = C η2 (−cos(φ)v 3 + sin(φ)v 4 + IM A cos(η 2 )˙η 1 v 5 ) (23)˙v k = u k k =1, ..., 4 (24)with the constantsC xyz =C sM 5 +4mC dC φ =4IM A +4md2 + IM5C 1 η 1= −I 5 M + I5 S − 4IA M +4IA S − 2md2 +4mh 2 C 2 η 1= I M 5 +4IA M +4md2C s dC η2 =4IS A +2md2 +4mh 2 + IS5The inputs u k , k =1...4, do not represent the physical inputs (the motortorques M k ), but are related to them by invertible algebraic relationships. Thenumerical values of the parameters used in the simulations are given in Table 1.Parameters IM 5 , I5 S , IA M , IA S are the inertias of the main body and the propellers,respectively. Parameters C s and C d are the aerodynamical parameters of thepropellers. Parameters M 5 , m are the masses of the main body and propellers,respectively.The simplified model is flat. The flat outputs are: Y = [ xyzφ ] .Table 1. Model parametersC s 3.64 × 10 −6 Ns 2 d 0.3 mC d 1.26 × 10 −6 Nms 2 I 5 M 181 × 10 −4 Nms 2M 5 0.5 kg I 5 S 96 × 10 −4 Nms 2m 2.5 × 10 −2 kg I A M 6.26 × 10 −6 Nms 2h 0.03 m I A S 1.25 × 10 −6 Nms 24.2 Control ProblemThe control problem is of the tracking type: the VTOL structure must be drivensmoothly from some initial configuration to another predefined configuration. A