Emendation unit analysis for the original system
Autopilot is important to improve the stability and dynamic performance of the original control system, but it needs completely redesign of the missile. Therefore, autopilot is not suitable in the original control system. In this paper, we proposed a method to improve the stability and dynamic performance of the original control system, in which the framework of the original control system is unchanged. The content of the instructions generated by control box is modified. In fact, the instructions are digital and easy to change. The best way to modify the instructions of the control box is to modify the emendation unit.
Specifically, the improvement to the missile control system includes three steps [2].

(1)
Build the transfer function of the control loop;

(2)
Design the emendation unit in frequency domain;

(3)
Validate the design via trajectory simulation.
The transfer function of the over shoot network unit in the original control system can be represented as
$$ {W}_D(s)= K\cdotp \frac{0.455 s+1}{\left(0.045 s+1\right)\left(0.022 s+1\right)} $$
(4)
Due to Eq. (5), the emendation unit belongs to lead emendation, which can provide with positive phase angle and higher cutoff frequency. Therefore, the system dynamic performance can be enhanced. But the promotion of the phase stability margin γ is not obvious, resulting in the smaller phase stability margin for the emendation unit and lower stability [3].
$$ \arctan \frac{0.455}{1} \arctan \frac{0.045}{1} \arctan \frac{0.022}{1}>0 $$
(5)
The openloop phase stability margin γ is merely −160.15°. Larger phase stability margin γ is needed and the cutoff frequency is not too small.
According to the empirical data of the classical control theory, lead emendation can provide positive phase angle and increase the cutoff frequency. But it can only provide phase stability margin about 40° to 60°; the lag emendation decrease the cutoff frequency for a larger phase angle. Therefore, we need to combine the lead and lag emendation together to increase the system phase stability angle and enhance the stability.
Bode diagram design of the laglead emendation
Design requirement for emendation network
The transfer function of the laglead emendation network can be written as
$$ {G}_c(s)= K{G}_{c1}(s)\times {G}_{c2}(s)= K\frac{1+{T}_1 s}{1+\beta {T}_1 s}\cdotp \frac{1+{T}_2 s}{1+\alpha {T}_2 s} $$
(6)
In the design process, adjustment parameters of the characteristic points are designed separately. Then, the emendation parameters at each time points in the flying process are calculated with linear interpolation. The design requirements are shown in below.

(1)
To keep the gain stability margin h, the phase stability margin γ is not less than 30°;

(2)
To avoid resonance of the missile body, the closeloop cutoff frequency must be less than 0.3 times of the inherent frequency of the missile body, ω
_{
c
} ≤ 14.13 rad/s;

(3)
To ensure the missile not falling on the ground in the uncontrolled flying stage, the system overshoot needs as small as possible.
Procedure for emendation network design
According to classical automatic control theory, the procedure for design phase lag emendation unit using bode diagram is shown as follows [4, 5].

(1)
Calculate the openloop gain K satisfying the performance requirement of given error coefficients.

(2)
Plotting the system bode diagram before emendation, while calculate the amplitude stability margin L
_{
m
} (G
_{
m
}), crossover frequency ω
_{
g
} of –π, phase stability margin γ (P
_{
m
}), and cutoff frequency ω
_{
c1} (ω
_{
cp
}). Check whether the requirements are satisfied. If not, continue to the next step.

(3)
Determine the parameters of transfer function for the lag emendation unit:
$$ {G}_{c1}(s)=\frac{1+{T}_1 s}{1+\beta {T}_1 s} $$
(7)
Where these parameters can be chosen in engineering as follows:
$$ \frac{1}{T_1}=0.1{\omega}_{c1}\kern1.5em \beta =8\sim 10 $$

(4)
Choose a new system cutoff frequency ω
_{
c2} to make the phase provided by lead emendation unit satisfying the requirement of system phase stability margin γ. Meanwhile, the total amplitude frequency attenuation of the original system add the lag emendation unit reaching 0 dB, which means the L curve crosses the horizontal coordinate axis at ω
_{
c2}.

(5)
Determine the parameters of transfer function for the lead emendation unit:
$$ {G}_{c2}(s)=\frac{1+{T}_2 s}{1+\alpha {T}_2 s} $$
(8)
Where α < 1 and can be expressed by the following equation.
$$ 20 \log \alpha = L\left({\omega}_{c2}\right) $$
(9)
L(ω
_{
c2)} (expressed by dB) is the amplitude frequency characteristic of the original system plus lag emendation unit. Where ω
_{
c2} and T
_{2} are shown in below.
$$ {\omega}_{c2}={\omega}_m=\frac{1}{\sqrt{\alpha}{T}_2}\kern1.5em {T}_2=\frac{1}{\sqrt{\alpha}{\omega}_m} $$
(10)

(6)
Plot the system bode diagram after emendation and check system frequency domain performance.

(7)
Build closeloop system and validate the performance of the system.
Calculate the parameters of the emendation network
We will analyze the openloop gain G(s) for the original guidance loop at 3 s of missile flying. For the emendation unit is not added to this moment, therefore, it is not reasonable to design the emendation network when the gravity compensation is added to the transfer function [6]. The gravitycompensating unit is deleted when designing emendation network, for gravitycompensating instruction is disturbing signal in the guidance loop and does not affect the properties of the system itself. The gravitycompensating instruction is added to the guidance loop in system performance simulation. The openloop transfer function can be represented as
$$ \begin{array}{l}{G}_k^{\prime }(s)={W}_{\varepsilon}(s)\cdotp {W}_K^{\overset{.}{\varTheta}}(s)\cdotp {W}_{\mathrm{Y}}(s)\\ {}\kern2.25em =\frac{204}{0.05 s+1}\cdotp \frac{23.11}{s^2+1.93 s+121.13}\cdotp \frac{1}{s^2}\\ {}\kern2.25em =\frac{4714.4}{s^2\left(0.05 s+1\right)\left({s}^2+1.93 s+121.13\right)}\end{array} $$
(11)
According to the method above, the laglead emendation procedure is realized in Matlab environment. First, design the lag unit:
% Openloop gain
K = 1/1.65;
% Lag emendation unit
wc = 12; beta = 8.5;
T = 1/(0.1*wc);
betat = beta*T;
Gc1 = tf([T 1],[betat 1])

According to the code above, the transfer function of the lag emendation network can be obtained:
$$ {G}_{c1}(s)=\frac{0.8333 s+1}{7.083 s+1} $$
(12)
Substitute the transfer function above into the transfer function of the original system to modify the original transfer function. The Matlab code of design lead emendation network is below.
% Lead emendation unit
den1 = conv([1 0 0 0],[1 1.93 121.13]);
den2 = conv([7.083 1],[0.05 1]);
den = conv(den1,den2);
num = conv([0 4714.7],[0.8333 1]);
sope = tf(num,den);
gama = 50;wc = 12;
[Gc] = leadc(1,sope,[gama])

According to the code above, the transfer function for the lead emendation network is
$$ {G}_{c2}(s)=\frac{1.232 s+1}{0.1225 s+1} $$
(13)
Therefore, the transfer function of the emendation unit is
$$ {G}_c(s)= K{G}_{c1}(s)\times {G}_{c2}(s)=\frac{1}{1.65}\cdotp \frac{0.8333 s+1}{7.083 s+1}\cdotp \frac{1.232 s+1}{0.1225 s+1} $$
(14)
Bode diagram check in frequency domain
Substitute the modified transfer function into the original guidance loop and the transfer function of the whole system can be obtained:
$$ \begin{array}{l}{G}_k(s)=\left[{W}_{\varepsilon}(s)\cdotp {G}_c(s)+ G(s)\right]\cdotp {W}_K^{\overset{.}{\varTheta}}(s)\cdotp {W}_{\mathrm{Y}}(s)\\ {}=\left[\frac{204}{0.05 s+1}\cdotp \frac{1}{1.65}\cdotp \frac{0.8333 s+1}{7.083 s+1}\cdotp \frac{1.232 s+1}{0.1225 s+1}+\frac{3.31 s+0.0678}{s^2}\right]\cdotp \frac{23.11}{s^2+1.93 s+121.13}\cdotp \frac{1}{s^2}\\ {}\approx \left[\frac{204}{0.05 s+1}\cdotp \frac{1}{1.65}\cdotp \frac{0.8333 s+1}{7.083 s+1}\cdotp \frac{1.232 s+1}{0.1225 s+1}+\frac{3.31}{s}\right]\cdotp \frac{23.11}{s^2+1.93 s+121.13}\cdotp \frac{1}{s^2}\\ {}=23.11\cdotp \frac{127.98{s}^3+84.66{s}^2+150.91 s+3.31}{s^3\left(0.05 s+1\right)\left(7.083 s+1\right)\left(0.1225 s+1\right)\left({s}^2+1.93 s+121.13\right)}\end{array} $$
(15)
Analyze the modified transfer function in frequency domain.
den1 = conv([1 0 0 0],[1 1.93 121.13]);
den2 = conv([0.05 1],[7.083 1]);
den3 = conv(den2,[0.1225 1]);
den = conv(den1,den3);
num = 23.11.*[127.98 84.66 150.91 3.31];
G = tf(num,den);
bode(G)
margin(G)
[Gm,pm,Wg,Wp] = margin(G)

Output the system openloop bode diagram after emendation, as shown in Fig. 2. Where M is openloop amplitude margin (unit, dB) and P is phase margin (unit, deg)
According to the figure above, the amplitude crossover frequency is ω
_{
c
} = 3.1191 rad/s, the phase stability margin γ = 46.2027°, the phase crossover frequency ω
_{
g
} = 8.4006 rad/s, and the amplitude stability margin is h = 1.6856.
According to the results above, γ = 46.2027° satisfies the commonly used 30°–70°criterion in engineering design and the crossover frequency is indeed less than 14.13 rad/s satisfying the requirement above.
When the missile flied 7 s, only the second order oscillation block changed and the code modifies work that is omitted here. After analysis, it can be concluded that the transfer function of the emendation unit satisfies the stability condition in frequency domain.
In short, the laglead emendation unit based on bode diagram can satisfy the control system stability of the missile in highaltitude environment.