A new form of the Proportional Navigation (PN) guidance law for short-range homing missiles is proposed. Named the Switched Bias Proportional Navigation (SBPN) law, it is derived by invoking sliding-mode control theory and is structured around the basic PN, with an additive switched bias term. This additional term depends only on the polarity of the line-of-sight rate, which is readily available with a seeker. It is shown that the bias term acts as an estimate of the target acceleration and other unmodeled dynamics. An adaptive procedure is suggested to select the gain of this term, which results in improved performance. The SBPN is almost as simple to implement as the PN law itself, as it does not require any additional information related to the engagement, in the form of either measurements or estimates. Simulation results show that the acceleration profiles of SBPN closely follow those of augmented PN guidance law, after a short initial transient. They further demonstrate the robustness of the proposed SBPN in the presence of missile velocity variation.

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The effectiveness of the redesigned control law is illustrated by a simple numerical example. Let us consider a CT-LTI system described by Eq. (1) withThe feedbackgain matrix F D [¡1 ¡1:4] gives the closed-loopsystem a second-orderdynamics with a damping ratio 0.7 and a natural frequency1 rad/s. For a sampling time T D 0:5 s and the feedforward gain matrix G D 1, the DT gain matrices are computed by Eqs. (10) and (11) as Figure 1 shows time histories of the states and control input of the CT and DT closed-loop systems for a step input r.t / D 1. The DT state responses agree with the CT ones very well, and it can be seen that the DT control input satis es the PEA in each sampling period. ConclusionsA digital redesign method for the linear state-feedback control law is derived based on the PEA. The effectiveness of the DT control law is illustrated through a numerical example. The redesigned control law is the same as the one derived by other approaches. As mentioned in the Introduction, however, the PEA also plays an important role in redesign of one-degree-of-freedom controllers and conversion of PAM inputs into PWN inputs. These characteristics of the PEA imply that the PEA is an underlying idea generally applicable to control-input redesign, and the analysis using the mean value theorem explains why it does work.

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