Closed loop stability

Question

Regarding the Lyapunov stability, we check if a nonlinear system stays near the equilibrium point or approaches to e.p. as time goes to infinity, when it is disturbed.

Let's assume that we have a nonlinear system (an automobile) and designed an optimal controller. The controller executes the given driver input for safely driving by controlling the braking forces and steering angle. The case might be a lane change at high velocities.

In this case, or in similar cases like control of an airplane/AUV/ship/etc., there is a moving (not at rest) system; the controller takes the system from a system of states to another system of states. How can we talk about the stability in this case? If I designed a controller by using an unknown method, say, my method, how can I check that the closed loop system is stable?

Edit: It is an autonomous system. My point is how I can prove that the stability is guaranteed for that controlled system. How will I know that my controller will not make the dynamic system unstable?

Answer 1

Let me recollect the above pieces of comments to an answer.

There is a difference between stability of a control, the stability of an autonomous system and the stability of a non-autonomous system. If you have a controller coupled to an autonomous system, then you have as result a non-autonomous system and the stability you look for will be about the whole non-autonmous system.

In general this is about an own stack of theory see here >>>

And on StackExchange here >>>

Answer 2

Firstly, you have to define what you mean with stability.

Nonlinear systems are different to linear systems since the former can have more than one equilibrium point, whereas for the linear system, with a change of coordinates which always exists, is always the origin.

In the literature you can find such definitions of stability such as globally, almost-globally, semi-globally, locally stability. Input to state stability, Input to output stability, asymptotically or exponentially stability, etc, etc.

Also note that with Lyapunov you can not conclude stability, you need the help of LaSalle's principle if your equilibria is time-invariant, or Barballat's lemma if the equilibria is time-variant. In addition, there are more techniques to show the stability of a system such as the Centre Manifold theorem.

One starting point to show if one equilibrium point for some equilibrium set is stable in a non-linear system is to look at the linearized dynamics of the system at this equilibrium point. Then apply linear techniques to conclude that your system is locally stable.