Properties of Non-Linear Systems

Task: Stabilization of pendulum at the origin
x&
1
x&
2
= x
2
= a
[ sin ( x + δ)
−sin
( δ)
] −bx
+ cu
1. Choose u as follows to cancel nonlinear term:
2. Stabilizing fb controller
1
a
u =
1 +
c
[ sin ( x + δ)
− sin ( δ)
]
c
3. Re-substituting to or fb control law
⎛a⎞
1
u = ⎜ ⎟ 1
1 1 +
⎝c
⎠
c
Pendulum Example
[ sin ( x + δ)
−sin
( δ)
] − ( k x k x )
2
v
Closed loop
2
2
CAS-DSP, Sigtuna 2007-Control Theory-S.Simrock
Linear system
x&
1
x&
2
= x
2
= −b⋅
x
1 2
= −k1x1
k2x
2
x&
2 = k1x1
− ( k2
+ b)
x2
v −
x&
=
x
2
+ v
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