Assessment and Future Directions of Nonlinear Model Predictive ...

Optimal Online Control of Dynamical Systems Under Uncertainty 333In the control process under consideration the optimal value of the objectivefunction was equal to 0.1046290478. Figure 1 presents the estimates ˆα i (τ), ˆβ i (τ),i = 1, 4; t ∈ [0, 2]. Figure 2 shows the optimal control function u ∗ (t), t ∈ T ,and the projections on the phase planes xẋ and ϕ ˙ϕ of the corresponding optimaltrajectories.Fig. 1. Linear estimates ˆα i(τ), ˆβ i(τ), τ ∈ T h , i = 1, 4Fig. 2. Optimal control and projections of the optimal trajectoriesFig. 3. Two-mode algorithm: estimate ρ(t), t ∈ T h , and optimal trajectoriesTo estimate the delay s(τ) assume that σ is maximal time required to integrateprimal or adjoint equations on the whole interval T . We used methods from [9, 11]to perform online corrections of solutions of optimal control and observationproblems. The maximal length of the intervals, where differential equations wereintegrated, was l max =0.043t ∗ (4.3% of the control interval). If σ is such thatσl max