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Although I know that a linear systems is the one that follows principle of superposition but I want to know what topics/contents do we study in each of both categories linear control and non linear control?Please kindly give some examples of both categories

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It is not entirely clear what you are asking exactly, so I’ll try to answer as best as I can. In linear control we consider systems described by linear differential equations subject to holonomic constraints (based on position). An example for such a system is an RC network.

The benefits of having a linear system is that the principle of superposition applies. So you can add the inputs together to get the output. In a non-linear system this is no longer true. In particular in becomes difficult to take derivatives while remaining on your constraint surface, which can now be dependent on the velocities (non-holonomic). This is prevalent in mechanical systems with inertia. After all, you cannot stop a car instantaneously.

In non-linear control we construct manifolds (Riemannian or Symplectic) and use the tools of differential geometry to steer the system to the desired state. A typical example is a vehicle. The problem with the non-holonomic constraints is that the constraint surface moves with changes to the velocity. So it is difficult to get back to it after you do numerical differentiation in Euclidean space. If you calculate the constraint force needed to get back to the constraint surface, the velocity will change and the constraint surface will move such that you are no longer on it. Hence the need to ensure that you do not leave it through the tools of differential geometry.

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