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    • Based on Eqs and the autopilot design is

    2018-11-12

    Based on Eqs. (5) and (7), the autopilot design is handled by the parameters τ, ζ and ω to achieve the desired performance requirements. For optimal gain design, these parameters are tuned to the optimum of certain cost function. The tuning technique should be carried out under the system limitations, so the crossover frequency constraint will be highlighted in the next section.
    Crossover frequency constraint function For robust and adequate control properties, frequency domain specifications are specified using the crossover frequency and protein phosphatase margins of the open-loop system. The crossover frequency must be chosen to be high enough to ensure a wide autopilot bandwidth but low enough to prevent stability problems due to actuator, rate gyro and other un-modeled dynamics. The limit of crossover frequency can be determined by system stability requirements and fin actuator performance. According to the classical “rule of thumb”, the crossover frequency should be less than one-third of the actuator bandwidth [1,3]. The open-loop transfer function of the three-loop autopilot with loop broken right before the fin actuator is expressed as The open loop magnitude ratio is written as For practical autopilot, the crossover frequency is beyond the airframe fundamental dynamic frequency, i.e. . Moreover, the magnitude trajectory of the open-loop transfer function will cross the 0 dB line only once. At the crossover frequency, the magnitude ratio is equal to one, i.e. . Therefore crossover frequency constraint should satisfy the following inequality:where is the prescribed limitation of , that is, for any gain combination K1, K2 and K3 satisfying Eq. (10), then the corresponding will be . Referring to Eqs. (6) and (10), the crossover frequency constraint function is equivalently expressed in terms of τ, ζ, ω and aswhere
    Optimal autopilot gain design The main objective of the autopilot system is to force the missile to follow the steering commands developed by the guidance system. According to Eqs. (5), (6) and (9), the positive parameters τ, ζ and ω are tuned to minimize the integral error criteria of the input–output of the autopilot closed-loop transfer function: Integral of various functions of error between the reference input and the controlled plant output is a powerful quantitative measurement of the system performance. There are several kinds of integral error criteria to describe the system performance [8]. Among them, the integral of the square of the error (ISE) is chosen as the command tracking performance index of the autopilot with the consideration of obtaining its analytical expression. Referring to Eq. (12), with partial fraction expansion, the unit step response is expressed aswhere By inverse Laplace transform, the unit step output response in the time-domain is given asand the error for unity command tracking is expressed as Therefore, the analytical objective function of ISE performance index can be written as The value of the above ISE analytical formula is always a real positive number due to complex conjugated relation in Eq. (14). Then optimal autopilot gain design is stated in the form of constrained optimization problem as This optimization problem is the core of the introduced optimal technique, where the objective function is the performance evaluation scale and the constraint g(τ,ζ,ω) is the performance limitation for practical design. The bounds of design parameters are set to positive values to ensure the system stability for the whole space of the cost function. Since the crossover frequency cannot satisfy the total performance requirements, it is needed to set a lower limit for the closed loop damping parameter , usually it is about 0.7. Referring to Table 3, this limit shall guarantee a sufficient system damping and phase margin. Particularly, the optimization problem (Eq. 15) is a nonlinear constrained multi-variable optimization problem. As expected, the function fmincon of the MATLAB Optimization Toolbox can solve such kind of smooth objective optimization problem well effectively with feasible initial design parameters. Moreover, it converges to the same minimum point even starting from different initial guesses. In this line of thought, the optimal autopilot gains , and Kg are easily calculated by substituting the optimum parameters τ, ζ and ω into Eqs. (7) and (8) without the need for design weight adjustment.