## 3 Method to Obtain the Rolling Information
### Analysis of Performance and Parameters Optimized  
Obviously, analysis of performance and parameters optimization starts from the premise that tracking system is capable of tracking signal originated from spinning projectile rotating in the form of acceleration function. Thus, analysis of steady-state error will be presented at first. According to Figure 5, and (10), the difference between \(u_{\rm o}(s)\) and \(u_{\rm i}(s)\) is described as  
\[u_{\rm e}(s)=\frac{s}{s+KF(s)}u_{\rm i}(s) \tag{16}\]  
when the object rotates in the form of acceleration function, \(u_{\rm i}(s)=\frac{k_{\rm r}}{s! }(k_{\rm r}\) is the change of rate in speed) and \(F(s)=F_{\rm P}(s)\). Referring to final-value theorem [30], the steady-state error \(e_{\rm ss}(\infty)\) of \(H_{\rm pe}(s)\) can be described as  
\[e_{\rm ss}(\infty)=\lim_{s\to 0}su_{\rm e}(s) \tag{17}\]  
thus,  
\[e_{\rm ss}(\infty)=\lim_{s\to 0}\frac{k_{\rm r}s}{s^{3}+b_{3}\omega_{\rm np}s^{2 }+a_{3}\omega_{\rm np}^{2}s+\omega_{\rm np}^{3}}=0 \tag{18}\]  
From analysis above, the designed third-order PLL is capable of tracking the phase signal changing in the form of acceleration function. Similarly, the designed second-order FLL is able to track the frequency signal which changes in the form of ramp function. According to (14), the noise bandwidth of PLL, \(B_{\rm PLL}\), is given by  
\[B_{\rm PLL}=\int\limits_{0}^{\infty}\bigl{|}H_{\rm P}(j2\pi f)\bigr{|}^{2}df =\frac{b_{3}^{2}a_{3}+a_{3}^{2}-b_{3}}{4(a_{3}b_{3}-1)}\omega_{\rm np} \tag{19}\]  
where, \(f\) is the frequency and \(H_{\rm p}\) is the frequency response function of the third-order PLL. Referring to [29],  
\[b_{3}=2.4,\ a_{3}=1.1 \tag{20}\]  
As shown in Figure 8, the amplitude-frequency characteristic and step response of PLL vary as \(B_{\rm PLL}\) changes:  
Figure 8 shows how noise bandwidth \(B_{\rm PLL}\) influence amplitude-frequency and transient characteristics. Larger the bandwidth is, the better the transient performance is (shorter settling time). However, with a lager \(B_{\rm PLL}\), it will have a worse amplitude-frequency characteristic and a greater cut-off frequency, which leads to a reduction of tracking accuracy. Figure 8: (**a**) Descriptions of amplitude–frequency characteristic. (**b**) Descriptions of step response of third-order PLL. According to (15), the noise bandwidth of FLL, \(B_{\text{FLL}}\), is given by  
\[B_{\text{FLL}}=\int\limits_{0}^{\infty}|H_{\text{F}}(j2\pi f)|^{2}df=\frac{ \omega_{\text{nf}}}{2}\bigg{(}\xi+\frac{1}{4\xi}\bigg{)} \tag{21}\]  
where, \(H_{\text{F}}\) is the frequency response function of second-order FLL. The damping ratio \(\xi\) and noise bandwidth \(B_{\text{FLL}}\) both determine the frequency at the \(-3\) dB point, settling time, and overshoot, which is shown in Figure 9.  
Similar to PLL, the results of the amplitude-frequency and step response show that with the damping ratio \(\xi\) and noise bandwidth \(B_{\text{FLL}}\) increased, the cut-off frequency becomes larger, and the overshoot and settling time become shorter. Influence of parameters on tracking system is shown in Table 1.  
From the analysis above, one can make a conclusion that for the tracking system designed, the performance of amplitude-frequency contradicts the transient performance. For PLL, which ensures the tracking precision, amplitude-frequency performance plays a more important role. FLL in the combined tracking loop is designed to lock the frequency and pull the loop into phase-locking state as quickly as possible, therefore transient performance should be given priority. Furthermore, choosing optimum damping \(\xi=0.707\) would optimize the two-order FLL to the greatest extent. Meanwhile, the selected noise bandwidths \(B_{\text{PLL}}\) and \(B_{\text{FLL}}\) should fit to different rotational speed which ranges from 3 Hz to 300 Hz. For the low speed, the selected noise bandwidth must enable the frequency at \(-3\) dB point to be less than 3 Hz and for a higher rotational speed the noise bandwidth should be improved depending on the practice.