# How are stability, performance, and robustness different? They seem very similar

"How are stability, performance, and robustness different? They seem very similar."

1. For dynamical systems, stability is perhaps the most important thing we want to make sure on top of all other requirements (performances). There are lots of different definitions of stability depending on different contexts as you go through the course. Roughly speaking, it means that the state of a dynamical system is always near some point (the equilibrium point). Performances are usually extra requirements on top of the stability which are in general design and application specific. Robustness comes to the analysis when we have to consider uncertain factors in designing the controller. Let's take the inverted pendulum for an example.

Stability: the controller you shall design has to make sure that it never falls down.

Performance 1: the controller you shall design has to move the pendulum to the vertical position from another position in minimum time. Or Performance 2: the controller you shall design has to move the pendulum to the vertical position from another position with minimum energy.

(notice that performance 1 and 2 are in general contradicting objectives, that is you cannot in general design a controller with minimum time and using minimum energy)

Robustness 1: suppose the pendulum is put outside your lab where there's some wind. the controller you shall design has to take the wind into consideration and has to make sure it never falls down.

Robustness 2: there's some error in your measurement about the length of the pendulum (but you know that the error should be within 0.01cm). the controller you shall design has to take this error term into consideration and has to make sure it never falls down.