Nonlinear Control Sy.. - Free

10.6. THE ZERO DYNAMICS 281
i=Ax+Bu (10.31)
y = Cx.
To simplify matters, we can assume without loss of generality that the system is of third
order and has a relative degree r = 2. In companion form, equation (10.31) can be expressed
as follows:
0
±1
±2
x3
0 1 0
0 0 1 x2 +
- 'I,
4o -ql 42 X3 1
y = [ Po P1 0]
The transfer function associated with this system is
+ Pls
H(s) =
qo + q1S + g2S2 + Q3S3
Suppose that our problem is to design u so that y tracks a desired output yd. Ignoring
the fact that the system is linear time-invariant, we proceed with our design using the
input-output linearization technique. We have:
Thus, the control law
Y = Pox1 +P1x2
y = POxl +P1±2
= POx2 + Plx3
= P0X2 +P1X3
produces the simple double integrator
= POx3 + pl(-40x1 - 41x2 - 42x3) + Plu
1
u = 40x1 + q, X2 + Q2x3 - P0 X3 + - v
P1 P1
y=v.
Since we are interested in a tracking control problem, we can define the tracking error
e = y - yd and choose v = -kle - k2e + yd. With this input v we have that
u = Igoxl + 41x2 + 42x3 - x3J + 1[- kle - k2e + yd]
U.
0
U