Sum-of-Squares Applications in Nonlinear Controller Synthesis

4.2 Controller Tuning Illustrative Examplesthe origin is used. This of course implies, that the global stability for the approximatesystem does not hold in general for the full model. Still our method provides a very straightforward way to come up with a control law for a system that is particularly difficult tocontrol.Table 2: Parameters for the simulation.m 1 m 2 l g2.5 1.0 0.5 9.81We use the parameters from table 2 and the polynomial approximationwith⎡ ⎤ ⎡ ⎤ ⎡ ⎤ẋ 1 x 30ẋ 2x 40=+u . (72)⎢⎣ẋ 3⎥ ⎢⎦ ⎣f 3 (x 2 , x 4 ) ⎥ ⎢⎦ ⎣g 3 (x 2 ) ⎥⎦ẋ 4 f 4 (x 2 , x 4 ) g 4 (x 2 )f 3 (x 2 , x 4 ) = 0.11707 x 5 2 + 0.035908 x 3 2 x 2 4 − 1.6032 x 3 2 − 0.17201 x 2 x 2 4 + 3.0313 x 2 ,f 4 (x 2 , x 4 ) = 0.24902 x 5 2 + 0.13049 x 3 2 x 2 4 − 5.6188 x 3 2 − 0.29147 x 2 x 2 4 + 23.9892 x 2 ,g 3 (x 2 ) = 0.02905 x 4 2 − 0.11289 x 2 2 + 0.3955 ,g 4 (x 2 ) = 0.096371 x 4 2 − 0.54277 x 2 2 + 0.7831 ,which is obtained as a least-square approximation of the full equations for x 2 × x 4(− π 2 , π 2 ) × (− 3 π ). The linearized system then is2 , 3 π2⎡ ⎤ ⎡⎤ ⎡ ⎤ ⎡ ⎤ẋ 1 0 0 1 0 x 1 0ẋ 20 0 0 1x 20=+u (73)⎢⎣ẋ 3⎥ ⎢⎦ ⎣0 3.0313 0 0⎥⎢⎦ ⎣x 3⎥ ⎢⎦ ⎣0.3955⎥⎦ẋ 4 0 23.9892 0 0 x 4 0.7831and solving the associated algebraic Riccati equations for Q =[ 1 0 0 0]0 1 0 00 0 1 00 0 0 1and R = 1 yields⎡⎤3.1990 −15.2287 4.6168 −3.6086−15.2287 541.5121 −45.1076 115.4561P =. (74)⎢⎣ 4.6168 −45.1076 13.0926 −10.6973⎥⎦−3.6086 115.4561 −10.6973 24.8493∈28