Robust counterpart optimization techniques for linear optimization and mixed integer linear optimization problems are studied in this paper. Different uncertainty sets, including those studied in literature (i.e., interval set; combined interval and ellipsoidal set; combined interval and polyhedral set) and new ones (i.e., adjustable box; pure ellipsoidal; pure polyhedral; combined interval, ellipsoidal, and polyhedral set) are studied in this work and their geometric relationship is discussed. For uncertainty in the left hand side, right hand side, and objective function of the optimization problems, robust counterpart optimization formulations induced by those different uncertainty sets are derived. Numerical studies are performed to compare the solutions of the robust counterpart optimization models and applications in refinery production planning and batch process scheduling problem are presented.

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To improve the stability of the autonomous vehicle for high speed tracking, a vehicle estimator scheme integrated into a path-tracking system has been proposed in this paper. Vehicle stability is related to road condition (low road adhesion, high road adhesion, and changing road adhesion) and vehicle state, thus a state observer has been preferred in this paper to estimate vehicle state and tire-road friction as a means of judging vehicle stabilization. For the approach to the estimation, an unscented Kalman filter (UKF) employing a three degrees-of-freedom vehicle model combined with a Magic Formula (MF) tire model was designed. As a widely used model control method, the multi-constraints model predictive control (MMPC) was proposed and that was then used to calculate the desired front steering angle for tracking the planned path. The performance of the MMPC controller, with the estimator, was evaluated by the vehicle simulation software CARSIM and Matlab/Simulink. The simulation results show that the designed MMPC controller with the estimator successfully performs path-tracking at high speed for the intelligent vehicle.

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