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I have a system with two inputs (throttle and brake) and one output (speed). How does one design a controller in such a way that the two outputs of the controller (throttle and brake) are never both greater than zero (so that it doesn't accelerate and brake simultaneously)?
Thanks
The system is nonlinear for acceleration. If the desired acceleration value is bigger than 0, use the throttle, if smaller than 0, use the break.
Other than that, you should be fine using a cascaded control loop with an inner acceleration loop and an outer speed control loop. Despite the fact that it has a discontinuity in the acceleration control loop, you should be able to treat it as a linear one if both break and throttle can be set to arbitrary small values, creating a smooth transition between the use of throttle and break at 0 acceleration.
This should be straightforward, because the throttle can never provide negative acceleration and the brake can never provide positive acceleration.
It sounds like you want to create a linear mapping from an input acceleration $(a)$ to a throttle and brake output. If the response of the throttle system for a given velocity $v$ is
$$ f(x_{throttle}) = a $$
(where $a > 0$), and the response of the braking system is
$$ g(y_{brake}) = a $$
(where ($a < 0$ if $v > 0$ and is 0 otherwise), you will just define the controller in a piecewise fashion.
$$ control_{throttle}(a) = \begin{cases} f^{-1}(a) & a \gt 0 \\ 0 & a \le 0 \end{cases} $$ and $$ control_{brake}(a) = \begin{cases}g^{-1}(a) & a \lt 0 \\ 0 & a > 0 \\ C_{parkingbrake} & a = 0 \end{cases} $$
