Optimal control for third degree polynomial systems and its automotive application

Optimal control for linear systems with time delay in control input

“…For linear systems without delay, if the optimal control exists, then the optimal filter exists for the dual linear system with Gaussian disturbances and can be found from the optimal control problem solution, using simple algebraic transformations (duality between the gain matrices and between the gain matrix and variance equations), and vice versa (see [1]). Taking into account the physical duality of the filtering and control problems, the last conjecture should be valid for all cases where the optimal control (or, vice versa, the optimal filter) exists in a closed finite-dimensional form [18]. This proposition is now applied to the optimal filtering problem for linear system states over observations with delay, which is dual to the stated optimal control problem (1), (2), and where the optimal filter has already been obtained (see [19][20][21]).…”

“…1, which presents the graphs of the optimally controlled state (18) xðtÞ in the interval ½0; T; the shifted ahead by 0.1 cost function (12) Jðt À 0:1Þ in the interval ½0:1; T þ 0:1; and the shifted ahead by 0.1 optimal control (16) u à ðt À 0:1Þ in the interval ½0; T: The values of the state (18) and the cost function (12) at the final moment T ¼ 0:25 are xð0:25Þ ¼ 1:668 and Jð0:25Þ ¼ 35:3248: There is a definitive improvement in the values of the controlled state to be maximized and the cost function to be minimized, in comparison to the preceding case, due to the optimality of regulator (16), (17) for the linear system (11) with time delay in control input.…”

“…In doing so, the optimal regulator gain matrix is constructed as dual transpose to the optimal filter gain one and the optimal regulator gain equation is obtained as dual to the variance equation in the optimal filter. The results obtained by virtue of the duality principle can be rigorously verified through the general equations of the maximum principle [22,8] or the dynamic programming method [23,9] applied to a specific time-delay case, although the physical duality seems obvious: if the optimal filter exists in a closed form, the optimal closed-form regulator should also exist, and vice versa [18]. In this paper, the obtained results are proved using the maximum principle [22,8].…”

“…This paper concentrates on the solution of the optimal control problem for a linear system with delay in control input and a quadratic cost function, which is based on the duality principle in a closed-form situation [18] applied to the optimal filter for linear systems with delay in observations obtained in [19][20][21]. Taking into account that the optimal control problem can be solved in the linear case applying the duality principle to the solution of the optimal filtering problem [1], this paper exploits the same idea for designing the optimal control in a linear system with time delay in control input, using the optimal filter for linear systems with delay in observations.…”

“…Section 2 states the optimal control problem for a linear system with time delay in control input and describes the duality principle for a closed-form situation [18]. For reference purposes, the optimal filtering equations for a linear state and linear observations with delay [19][20][21] are briefly reviewed in Section 3.…”