computed torque control or feedback linearization of a 1DOF system

Question

I have a nonlinear system of

$f = m \ddot{x} + b \dot{x} + c \sin{x} , \tag{1}$

for which I want to design a model based PD controller or as I have seen on other references using computed torque control or feedback linearization.

I can define error as:

$e = x - x_d , \tag{2}$

where $x_d$ is the desired value. Following this presentation the computed torque control law is:

$f = m \left( \ddot{x}_d - u \right) + N , \tag{3}$

where $u$ is

$u = -k_d \dot{e} -k_p e , \tag{4}$

and $N$ is

$N = b \dot{x} + c \sin{x} . \tag{5}$

Combining the above equations I get

$\ddot{e} - k_d \dot{e} - k_p e = 0 , \tag{6}$

which in addition to contradicting the above reference, is very confusing. I would appreciate if you could help me know if my calculations are correct or I'm making a mistake here? Thanks for your support in advance.

P.S. reading this paper it seems that $u$ should actually be defined as

$u = k_d \dot{e} + k_p e , \tag{7}$

which leads to

$\ddot{e} + k_d \dot{e} + k_p e = 0 , \tag{6}$

Answer 1

We want the error dynamics $\ddot{e}+k_\text{D}\dot{e}+k_\text{P}e=0$. The error is given by $e=x_\text{d}-x$, which results in

$$\ddot{x}_\text{d}+k_\text{D}\dot{x}_\text{d}+k_\text{P}x_\text{d}=\ddot{x}+k_\text{D}\dot{x}+k_\text{P}x.$$

But we have $\ddot{x}=-b/m\dot{x}-c/m\sin x - f/m$. This results in

$$\ddot{x}_\text{d}+k_\text{D}\dot{x}_\text{d}+k_\text{P}x_\text{d}=-b/m\dot{x}-c/m\sin x - f/m+k_\text{D}\dot{x}+k_\text{P}x.$$

Solving for $f$ will give us the desired input $f$.

$$f/m=-\ddot{x}_\text{d}-k_\text{D}\dot{x}_\text{d}-k_\text{P}x_\text{d}-b/m\dot{x}-c/m\sin x +k_\text{D}\dot{x}+k_\text{P}x.$$ $$f =-m\ddot{x}_\text{d}-mk_\text{D}\dot{x}_\text{d}-mk_\text{P}x_\text{d}-b\dot{x}-c\sin x +mk_\text{D}\dot{x}+mk_\text{P}x.$$

This only works if we can measure $x$, $\dot{x}$ and if your desired trajectory $x_\text{d}$ is at least twice continuously differentiable.

Answer 2

There is a difference between PD computed-torque controller which requires a precise model of the system and a pure PD which is nothing but using the error and its derivative with some positive gains. PD computed-torque controller is a nonlinear approach that maps the nonlinear system into a linear model. That is why it requires the model of the system to explicitly cancel any nonlinear terms and hence applies linear techniques. The PD controller usually is used when the system is already linear and one only needs to place the poles of the system at desired poles which meet some requirements such as overshoot, steady state error, etc.