br Numerical analysis In this section numerical analysis

Numerical analysis In this section, numerical analysis is carried out to show the effectiveness of the proposed optimal technique for autopilot gain design. Data of a typical missile system [14], listed in Table 1, are used for this purpose. Moreover, the actuator is considered as a second-order dynamic system with natural frequency and damping factor . Let be used as the upper limitation to the open-loop crossover frequency, which guarantees the open-loop system to have about 30° of the phase margin after considering the phase lags caused by autopilot\'s hardware systems [14]. In the following, the numerical analysis is introduced through two steps. In the first step, the proposed technique is applied to introduce some analysis for the considered optimization problem, and the nature of its cost function also emphasizes the achievement of required crossover frequency for practical optimal design of autopilot system. Moreover, the lower bound of damping factor is examined for proper setting. In the second step, the introduced technique is compared to another design strategy with the same crossover frequency requirement. First, the objective function (Eq.  14) of the optimization problem (Eq. 15) is the performance evaluation scale and it xanthine oxidase inhibitors could be described as three-dimensional search space defined by the design parameters τ, ζ and ω. This space is formed by multi-surfaces for different values of ζ. Specifically, each particular point in this space corresponds to particular autopilot gain combination with particular response based on the analytical equation (7). Fig. 2(a) exhibits the smooth objective space for the specified region of the three design parameters without applying the crossover frequency constraint. The objective space after applying the crossover frequency constraint can be observed in Fig. 2(b), where the unfitted portion is removed. In both cases the optimization technique easily converges to the minimum point. The optimal design of both cases, with and without the crossover frequency constraint, is introduced in Table 2. Clearly, the constraint design achieves the open-loop crossover frequency within the constraint limit, while the unconstrained autopilot design derives very high open-loop crossover frequency which is compatible with the conclusion in Ref. [1]. Furthermore, as listed in Table 3, for the constraint case, the smallest value of objective ISE mostly belongs to the surface of smaller value of ζ. As a result, the optimum performances express degradation in phase margin and overshoot value even with good tracking performance. In order to achieve sufficient system damping and phase margin, a prescribed is set as the lower bound of damping factor ζ. The results show that the bound achieves a good compromise between system response and robustness with sufficient phase margin, appropriate overshoot and settling time. Second, the proposed technique is compared to the design strategy of Ref. [14] for the three-loop autopilot design. This strategy is keen on adjustment of the design parameters for single objective of minimizing . Specifically, this strategy is pole adjustment technique where the pole position is described by τ, ζ and ω. Both τ and ζ are prescribed, while ω is tuned to minimize the crossover frequency objective with unclear relation between ω and . Likewise, the strategies in Refs. [4–7] are LQR approach with weight adjustment procedure for minimizing the same objective. On the other hand, the proposed approach freely optimizes the whole three parameters for minimum tracking objective that satisfies the crossover frequency constraint. To illustrate, the tracking performances of acceleration command 5g are displayed in Fig. 3(a), and the numerical comparison results are stated in Table 4. The simulation results show that the proposed optimal design method derives much better tracking performance than that in Ref. [14], even if the open-loop crossover frequencies and phase margins in both cases are almost the same. Moreover, the related fin deflections are demonstrated in Fig. 3(b), where the exhibited elevator deflections reach the same steady value since petals is totally determined by the aerodynamic parameters and the flight velocity of the missile. Besides, the slight larger elevator deflection and deflection rate introduced by the proposed technique during transient time serves as a payment for the faster response. Since the role of the autopilot system is to drive the missile to track the acceleration commands, so its tracking performance should be the main point to evaluate the design quality within the applicability dynamic constraints.