state transition matrix

Employing discrete-time techniques, the min-time control of continuous-time dynamical systems is mainly studied through an analytical framework. To this aim, the exact discrete-time model of the linear time-invariant systems is specified through a zero-order hold. The optimal solution could be directly determined from some necessary conditions. However, the structure of the optimum control sequences is derived by utilizing the well-known Pontryagin principle. Employing the state transition matrix, the states of the control system are computed at the switching times. The switching times of the control signal would be found from a set of nonlinear algebraic equations. Accordingly, the transformation of the system’s states, from a known initial point to a specific value, would be accomplished in the minimum possible time. Applying the proposed scheme, the exact (integer) values of the switching times and the final time are numerically determined from the solution of an algebraic equation. Several discrete-time and continuous-time examples are discussed and simulated to show the feasibility and effectiveness of the suggested procedure in the dynamical systems. The simulation results confirm the method’s advantages over the existing ones.

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.