Nonlinear Control Sy.. - Free

286 CHAPTER 10. FEEDBACK LINEARIZATION
First we find the relative order. Differentiating y, we obtain
y=i2=-x2+x1u.
Therefore, r = 1. To determine the zero dynamics, we proceed as follows:
Then, the zero dynamics is given by
y=0 x2=0 t-- u=0.
which is exponentially stable (globally) if k > 0, and unstable if k < 0.
Example 10.19 Consider the system
Differentiating y, we obtain
Therefore
I
xl = x2 + xi
i2=x2+u
x3 = x1 +X 2 + ax3y = X1
y = LEI =x2+xi
2x111 + i2 = 2x1 (x2 + xl) + x2 + u
r=2
To find the zero dynamics, we proceed as follows:
Therefore the zero dynamics is given by
#-.i1=0=x2+2i =>' X2=0
i2=x2+u=0 u=-x2.
x3 = ax3.
Moreover, the zero dynamics is asymptotically stable if a = 0, and unstable if a > 0.