Performance and H INFINITY 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 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 optimality.