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    • Some autopilots have been designed

    2018-10-25

    Some autopilots have been designed for blended controlled missiles based on nonlinear models [7–10]. In Ref. [7], the zero dynamics of missile were analyzed, and a feedback linearization approach was employed to design an optimal controller for the missile. In Ref. [8], pitch and yaw motions were decoupled by feedback linearization, and RBF neural network sliding mode control was applied to design a controller. The lateral thrust was treated as a continuous variable [7,8]. In Ref. [9], the predictive control was applied to design a lateral thrust control law, and the active disturbance rejection technique was used in aerodynamic control. Although the pulse-like dynamics of pulse thrusters were considered in the simulation, the rock inhibitor of the closed-loop system was not analyzed. In Ref. [10], the nonlinear model has been analyzed and simplified, and then the sliding mode control was employed to design a controller for tracking an angle of attack command. The pulse-like property of lateral thrust was discussed only in the implementation of autopilot rather than in the stability analysis. In practice, a pulse thruster works in on-off mode with a fixed period. Therefore, the lateral thrust is a pulse-like variable and can be approximated by a zero-order holder. The aerodynamic force is a continuous variable. The mixed control inputs, continuous and discontinuous control inputs, bring a challenge to the autopilot design and stability analysis of agile missile control system. To handle this problem, two strategies have been proposed. In Ref. [11], a two-step strategy was proposed and the autopilot was designed by state-dependent Riccati equation (SDRE) approach. Based on the nonlinear model of agile missile, a continuous SDRE feedback controller was designed. Accounting for the pulse-like lateral thrust, the nonlinear tolerance of the feedback system was analyzed when the control cannot be implemented ideally. Some sufficient conditions were given to guarantee the asymptotic stability of the closed-loop system. In Ref. [12], an indirect robust control strategy via SDRE approach was presented for the missile. It was supposed that there exists a continuous feedback controller for the acceleration tracking problem. There were errors between actual inputs from actuators and computed inputs from the controller. A virtual robust control problem was formed by treating the errors as input uncertainties. Then, the robust control problem was translated into a nonlinear quadratic optimal control problem, and the performance index was modified with uncertainty bound. SDRE approach is applied to solve the problem to obtain a continuous feedback controller. However, to track the acceleration command quickly enough, the fin deflections usually reach their upper or lower limits in the simulation. In sensor paper, the indirect robust control strategy is integrated with Theta-D technique [13,14] to solve the problem. Theta-D solution to the Hamilton-Jacobi-Bellman (HJB) equation is obtained by adding some small perturbations to the performance index and using the power series expansion to the co-state. A closed-form solution is obtained by taking finite terms in the series expansion. Design flexibility is brought from the parameters in the additional perturbations, and an appropriate tracking accuracy can be achieved easily. The saturation of fin deflections can be avoided by the free split of the state weighted matrix in the performance index.
    Nonlinear model of agile missile The nonlinear dynamics of the agile missile [11] in pitch and yaw planes is given in terms of angle of attack α, sideslip angle β, yaw rate ω, and pitch rate ω aswhere δ and δ are the deflections of rudder and elevator, respectively; n and n are the equivalent ignition numbers of pulse thrusters in the pitch and yaw planes, respectively; m, V, S and L are the mass, velocity, reference area and reference length of the missile, respectively; J and J are the moments of inertia of the missile; q is the aerodynamic pressure; C, c, c, , , m, m, , , and are aerodynamic coefficients; d is the moment arm of the lateral thrust; T0 is the equivalent thrust produced by one thruster.