This paper develops the double generating function method for the discrete-time linear quadratic optimal control problem. This method can give generators for optimal solutions only in terms of pre-computed coefficients and boundary conditions, which is useful for the on-line repetitive computation for different boundary conditions. Moreover, since each generator contains inverse terms, the invertibility analysis is also performed to conclude that the terms in the generators constructed by double generating functions with opposite time directions are invertible under some mild conditions, while the terms with the same time directions will become singular when the time goes infinity which may cause instability in numerical computations. Examples demonstrate the effectiveness of the developed method.

To achieve precise measurement without landing, the high-mobility vehicle-mounted theodolite needs to be leveled quickly with high precision and ensure sufficient support stability before work. After the measurement, it is also necessary to ensure that the high-mobility vehicle-mounted theodolite can be quickly withdrawn. Therefore, this paper proposes a hierarchical automatic leveling strategy and establishes a two-stage electromechanical automatic leveling mechanism model. Using coarse leveling of the first-stage automatic leveling mechanism and fine leveling of the second-stage automatic leveling mechanism, the model realizes high-precision and fast leveling of the vehicle-mounted theodolites. Then, the leveling control method based on repeated positioning is proposed for the first-stage automatic leveling mechanism. To realize the rapid withdrawal for high-mobility vehicle-mounted theodolites, the method ensures the coincidence of spatial movement paths when the structural parts are unfolded and withdrawn. Next, the leg static balance equation is constructed in the leveling state, and the support force detection method is discussed in realizing the stable support for vehicle-mounted theodolites. Furthermore, a mathematical model for “false leg” detection is established furtherly, and a “false leg” detection scheme based on the support force detection method is analyzed to significantly improve the support stability of vehicle-mounted theodolites. Finally, an experimental platform is constructed to perform the performance test for automatic leveling mechanisms. The experimental results show that the leveling accuracy of established two-stage electromechanical automatic leveling mechanism can reach 3.6″, and the leveling time is no more than 2 mins. The maximum support force error of the support force detection method is less than 15%, and the average support force error is less than 10%. In contrast, the maximum support force error of the drive motor torque detection method reaches 80.12%, and its leg support stability is much less than the support force detection method. The model and analysis method proposed in this paper can also be used for vehicle-mounted radar, vehicle-mounted laser measurement devices, vehicle-mounted artillery launchers and other types of vehicle-mounted equipment with high-precision and high-mobility working requirements.