Time Optimal Feedback Control using Chebyshev Polynomials and Gröbner Basis∗

The problem of synthesizing time optimal “feedback” control for a single input linear time invariant system with constrained control input is considered. It is well known that the optimal open loop control is bang-bang and the switching takes place according to some switching surfaces in the state space. A parametric representation of the switching surfaces using Pontryagins’ minimum principle (PMP) is first computed. This parametric representation is in general not semi-algebraic. For such situations, Chebyshev polynomials are used to approximate the switching surfaces. Then, a Gröbner basis based elimination algorithm is used to find an implicit representation for approximated switching surfaces. The computed representation is used to construct a state feedback based nested switching logic which will achieve a near time optimal state transfer of system states to the origin.