Take a look at this system $$ \begin{align} \dot{x}_1 &= \cos x_2 + (x_2+1)x_3 \tag{1}\\ \dot{x}_2 &= x^3_1+x_3 \tag{2}\\ \dot{x}_3 &= x^2_1+u \tag{3}\\ y&=x_1 \tag{4} \end{align} $$ The objective is to design tracking reference signal $y_d(t)$. Since there is no direct relation between $y$ and $u$, we need to ascertain this by taking the derivative of Eq(4) twice, hence: $$ \ddot{y} = (x_2+1)u + (x^3_1+x_3)(x_3-\sin x_2) + (x_2+1)x^2_1 \tag{5} $$ Designing the input $u$ to explicitly cancel the linearities, we get $$ u = \frac{1}{x_2+1}\Big(v-(x^3_1+x_3)(x_3-\sin x_2) + (x_2+1)x^2_1\Big) \tag{6} $$ Substituting Eq(6) in Eq(5), we $$ \ddot{y} = v $$
where $$ v = y_d -k_1e -k_2\dot{e} \tag{7} $$ and $e=y-y_d$ which yields
$$ \ddot{e} + k_2\dot{e} + k_1e = 0 \tag{8} $$ This is what has been shown in the book but I'm expecting an error in this equation in Eq(7) which I believe should be $$ v = \ddot{y}_d -k_1e -k_2\dot{e} $$ In order to get Eq(8). Is this a printed error? The book has no errata list. Block Backstepping Design of Nonlinear State Feedback Control Law for Underactuated Mechanical Systems
I've simulated the system based on my correction to Eq(7) and I got this results. It seems to me this is a horrible printed mistake.
