Performance and H∞ optimality of PID trajectory tracking controller for Lagrangian systems

This paper suggests an inverse optimal proportional-integral-derivative (PID) control design method to track trajectories in Lagrangian systems. The inverse optimal PID controller exists if and only if the Lagrangian system is extended disturbance input-to-state stable. First, we find the Lyapunov function and the control law that satisfy the extended disturbance input-to-state stability by using the characteristics of the Lagrangian system. The control law has a PID control form and satisfies the Hamilton-Jacobi-Isaacs equation. Hence, the H-infinity inverse optimality of the closed-loop system dynamics is acquired through the PID controller if the conditions for the control law are satisfied. Also, simple coarse/fine performance tuning laws are suggested based on a performance limitation analysis of the inverse optimal PID controller. Selection conditions for gains are proposed as functions of the tuning variable. Experimental results for a typical Lagrangian system show that our analysis provides performance and H-infinity optimality.