NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

Now we regard as the control variable and construct a control law y 1( ) that
will make the translational coordinates (z 1 and z 2) globally asymptotically stable.
Jankovic et.al. [1] choose y 1( )=; arctan(c 0z 2). However, y 1 is not the control vari-
able, and it follows its own dynamic equations. Therefore we de ne new angular
variables to implement the desired trajectories of and _ : 1 = + arctan(c 0z 2) and
2 = _ 1. Substituting into the previous system equation yields the following modi ed
system equation:
z_ 1 = z2 z_ 2 = ;z1 + sin( 1 ; arctan(c0z2)) _ 1 = 2
_ 2 = w
w = v ; 2c3 0z2 1+c2 0z2 (;z1 + sin( ))
2
2 +
c 0
1+c 2
0z2 2
(;z 2 + _ cos( )):
(3.20)
Now all that is required is to stabilize the -subsystem which can be done with the
simple feedback w = ;K . By following the approach outlined in [1], but with the
system modi ed for voltage as the input instead of torque, the following control law
is obtained:
u = k(I + mc2 )
m + M
= ;k1(y1 + arctan(c0z2)) ; k _
k2c0(;z1 + sin( )
2 ;
1+c2 0z2 2
+ 2c0z2(;z1 + sin( )) 2
(1 + c2 0z2 2) 2 + c0(;z2 + _ cos( )
1+c2 0z2 2
mc
= p
2 (I + mc )(m + M)
(3.21)
The primary di culty with implementing this control is understanding the purposes
of the various transformations and substitutions. Additionally, this last expression for
the control input is quite complicated { this makes it di cult to assign physical mean-
ing to the control parameters, c 0, k 1, k 2. The parameters used in the commissioning
of this design are as follows: c 0 =1,k 1 = 10, and k 2 =1.
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