Switching surfaces and null-controllable region of a class of LTI systems using Gröbner basis

The problem of time optimal feedback control of a single input, continuous time, linear time invariant (LTI) system is considered. The control input is constrained to obey |u(t)| ≤ 1. It is known that the solution to this problem is bang-bang with the input switching between the extreme values ±1 according to some “switching surfaces” in the state space. It is shown that for a class of LTI systems, these switching surfaces can be represented as semi-algebraic sets and a method to construct these switching surfaces using Gröbner basis, is proposed. Numerical simulations demonstrate the effectiveness of this feedback control strategy.