The Exact Solution of Min-Time Optimal Control Problem in Constrained LTI Systems: A State Transition Matrix Approach

In this paper, the min-time optimal control problem is mainly investigated in the linear time invariant (LTI) continuous-time control system with a constrained input. A high order dynamical LTI system is firstly considered for this purpose. Then the Pontryagin principle and some necessary optimality conditions have been simultaneously used to solve the optimal control problem. These optimality conditions would usually lead to some complicated equations while some integral terms may be presented. Then a systematic procedure based on state transition matrix will be addressed to overcome and simplify the mentioned complexities. Therefore the state transition matrix would be used to determine the exact solution of the min-time control problem in a typical LTI system. The min-time problem would be converted to some algebraic nonlinear equations by using of the state transition matrix. These algebraic equations are depended on some definite parameters. Hence the required design parameters as well as switching times and the possible minimum time would be analytically determined in the minimum-time optimal control problem. Thus the min-time control signal would be explicitly determined by computing of the switching times and also some other constants. The proposed control scheme is applied in some typical dynamical examples to show the effectiveness of the suggested control method.