Hello,
As far as I know, PID controllers are most widely used in the industry although all real-life systems are nonlinear.
Is there any mathematical proof that PID controllers (with sufficiently high gains) can guarantee robustness and stability (to some extent) for uncertain nonlinear systems?
Although PID is very strong controller. but it is difficult to tune its parameters in order to well regulate some nonlinear systems
So it depends on your system
Your PID controller will work well when it is operating near to the working point, i.e., the linearization of your non-linear system is valid inside a particular region. At the time when the real system is working at a different point from the working one, the PID controller will show some difficulties.
In addition to the marvellous example of Mr. Polat, let we don't forget, that by default PID algorithm is particularly suitable predominantly for sluggish processes with time delays and low-frequency disturbances. If the process is too fast as the case of inverted pendulum, or the disturbances are mid- or high-frequency ranged with regard to the process dynamics, PID fails to reach good efficiency. Moreover, in the case of that fancy pendulum benchamark, which is disturbed maximally to the bottom position, the disturbance amplitude would be non-admissibly large and the system will jump on a steady-state point with probably maximum non-linear effect. Then the PID control system with the pendulum will probably start to behave itself as a singular stiff point system. In that case, the process model (and the controller tuning params) would be too far from the truth, that provokes immediate instability.
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