Implementing control system to track position and velocity references simultaneously

Question

I've been trying to implement a PI (proportional-integral) controller to allow a launch vehicle to track 2 reference states simultaneous, namely position and velocity, so that the launch vehicle will end its ascent burn in a stable circular orbit at a 600km orbital altitude. I tried doing this by first solving an optimal guidance problem which provides the required reference position, reference velocity and reference control input (which is used as feedforward control) for the rocket to follow along its ascent. The PI controller is then used primarily to reject disturbances, with the feedforward control providing the bulk of the control input.

What I've noticed is that the PI controller does a good job at tracking 1 reference at a time (i.e. tracking a position reference or a velocity reference), but if I try and track both position and velocity at the same time, it gives disappointing results, with only 1 of the reference states being tracked accurately.

Attached is an example of implementing the controller to track only a position reference. Radial position is given in the left plot, and total velocity is given in the right plot, where the blue plots are the references to follow, and the orange plots are the actual trajectories taken by the rocket.

As can be seen, the controller tracks the position reference very well, but isn't able to track the velocity reference with good accuracy.

If we then try to track just the velocity reference, we have the following plots, where the velocity is tracked well, but the position deviates from our desired trajectory.

As such, I was wondering if there is a way to accurately track both position and velocity for the launch vehicle's ascent to orbit. How would tracking multiple reference targets be achieved during a real launch event?

Answer 1

As such, I was wondering if there is a way to accurately track both position and velocity for the launch vehicle's ascent to orbit.

If possible, have more accurate actuators and reference models. Improves the tracking of states easier.

But since there is a relationship between position and velocity you cannot properly control both. The reason is simple: If you have a reference trajectory position and velocity, that are both aligned with time. If at any point you are lagging behind in position, you can only compensate by accelerating and thus increasing velocity which would make you deviate from the reference velocity. If you tried only to control the reference velocity, then, over time, your position would increasingly deviate from its reference.

What I'd suggest you consider for such a simplified problem is that, while you generate $p(t)$ and $v(t)$ as references based on time, you should try to control the velocity as a function of the position, i.e. $v(p)$. This makes sense because it is better to be at the relevant height with the relevant speed than at the right time.

How would tracking multiple reference targets be achieved during a real launch event?

That is pretty much proprietary information for each rocket and launching site. But keep in mind a few things: 1. Not only height and speed are monitored, there is also latitude, longitude, absolute speed, vertical speed and horizontal speed. 2. The attitude of the rocket matters a lot. 3. It is not simple to modulate the trust (I don't even think any rocket can control inner pressure with a reference). And turning on and off the thrusters takes long if possible. 4. Any modern rocket controls the pointing of the thrust, but in a very limited manner, withing a small angle, and the direction can only (and should only) change slowly.

Answer 2

As such, I was wondering if there is a way to accurately track both position and velocity for the launch vehicle's ascent to orbit. How would tracking multiple reference targets be achieved during a real launch event?

You cannot. Position and velocity are dependent variables (velocity is by definition the time derivative of the position), so controlling one will affect the other and vice versa (see Mefitico's answer).

As a theoretical exercise you could parameterize the trajectory to get velocity as function of position as suggested by Mefitico, but this approach does not make any physical sense: physical systems either allow you to apply a force, yielding an acceleration (anything that behaves like a mass, $a = \frac{1}{m}F$), or displace it yielding a force (spring-like things, $F = Kx$).

So I suggest you model your rocket first as a point mass with an omnidirectional thruster and see if you can track your trajectory. Then gradually refine your model from there.