This paper addresses the optimal controller problem for a polynomial system over linear observations with respect to different Bolza-Meyer criteria, where: 1) the integral control and state energy terms are quadratic and the nonintegral term is of the first degree or 2) the control energy term is quadratic and the state energy terms are of the first degree. The optimal solutions are obtained as sliding mode controllers, each consisting of a sliding mode filter and a sliding mode regulator, whereas the conventional feedback polynomial-quadratic controller fails to provide a causal solution. Performance of the obtained optimal controllers is verified in the illustrative example against the conventional LQG controller that is optimal for the quadratic Bolza-Meyer criterion. The simulation results confirm an advantage in favor of the designed sliding mode controllers.

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This paper designs a predefined-time convergent continuous control algorithm to stabilize a permanent-magnet synchronous motor (PMSM) system. Three cases have been considered: disturbance-free, in presence of a deterministic disturbance satisfying a Lipschitz condition, and in presence of both a stochastic white noise and a deterministic disturbance satisfying a Lipschitz condition. The designed control law is free from the restrictions of exponential control growth and exact initial conditions knowledge. This is the first predefined-time convergent continuous control algorithm applied to stabilizing a PMSM system with both deterministic and stochastic disturbances, which enables one to a priori set the predefined convergence time even in presence of various disturbances of different nature. Numerical simulations are provided for a PMSM system to validate the obtained theoretical results in each of the three considered cases. The simulation results demonstrate that the employed values of the predefined-time convergent control inputs are applicable in practice.

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