NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

weighting matrix R, and P : K = R ;1 B T P , where P is the solution to the following
algebraic Riccati equation:
0=A T P + PA+ Q ; PBR ;1 B T P (3.2)
As the name implies, a linearization of the system equations is necessary to implement
this control strategy. The linearized equations for system (2.4) were derived as follows:
_~x = rF (x)jx0 ~x + rG(x)jx0 ~u = A~x + B~u (3.3)
where F (x) and G(x) are as de ned in (2.4). A and B are calculated to be:
0
0
B (mc
B
A = B
@
1 0 0
2 +I)k
Mmc2 +I(m+M ) 0 0
mcK2 mK2 g
Rm(Mmc2 1 0
0 0 0
0
C B
C B ;mcKmKg
C B
+I(m+M )) C B = B Rm(Mmc
B
1 C B
A @
0 0
2 1
C
+I(m+M )) C
0 C
A
mck
Mmc 2 +I(m+M )
;K2 mK2 g (m+M )
Rm(Mmc2 +I(m+M ))
(m+M )KmKg
Rm(Mmc 2 +I(m+M ))
(3.4)
After the constants are substituted, the linearized model becomes:
0
1 0 1
0
B
B;48
_x = B 0
@
1
0
0
0
0
0
0
:44
1
0
C B C
C B C
C B;:69C
C x + B C
B C u(x):
C B 0 C
A @ A
(3.5)
86 0 0 ;19:9 31:7
Next, simulations were run to determine which matrices Q and R would result in a
strongly damping control while not running the actuator past it's physical limitations.
Once suitable weighting matrices were obtained, the control was implemented on the
hardware, and the weights were tuned again to give good performance on the physical
plant. Tuning is always a subjective endeavor, and there is no guarantee that the
values we found were, in fact, the absolute best choices, nevertheless the parameters
that we found to give the best performance were the following:
0
1
2000
B 0
Q = B 0
@
0
100
0
0
0
70
0
C
0 C
0C
A
0 0 0 0
R = :01:
16