# How to prove the stability of trajectory tracking with PD control using Lyapunov method?

I'm having a problem to prove the stability of PD control in trajectory tracking. Let say we have a system with double integrator dynamics and we what to minimalize the tracking error $$\tilde{x}$$. The problem is stated as follows:

$$$$\ddot{x} = u,$$$$ $$$$\tilde{x} = x - x_d, \quad \dot{\tilde{x}} = \dot{x} - \dot{x}_d, \quad \dot{x}_{d} \neq 0, \quad x_d(0) = 0, \quad \dot{x}_{d}(0) = 0,$$$$ $$$$u = -K_P\tilde{x} -K_D\dot{\tilde{x}}, \quad where \quad K_P,K_D \in R^+$$$$

Finally my question is:

What Lyapunov function could be chosen ($$V = ?$$) and what assumptions should be made to prove the stability of the proposed controler?

Is it even possible to prove the stability without assuming that $$\dot{x}_d \approx 0 ?$$

• If you add the desired acceleration to your control input, then it would behave just as a regulation problem. If not, it's probably possible to compute bounds on tracking error if you assume some bounds on the magnitude of the desired velocity and the desired acceleration. Is that what you are interested in? Without bounds on the reference, the tracking error can be arbitrarily high.
– Alex
Commented Feb 16, 2021 at 6:11
• In my opinion adding the desired acceleration to the control signal would be the easiest approach. Unfortunately the problem is to somehow prove the stability of the proposed control, not to change it. The desired velocity and the desired accelerations are bounded but I still have trouble to prove the stability using Lyapunov method. Any ideas how to tackle this problem ? Commented Feb 16, 2021 at 15:43