# Design on a lag-lead emendation network for some missile control system

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## Abstract

Aiming at solving the problem that the adaptability of some missiles in highland environment is bad, the stability of some missile control system in highland environment is analyzed by the method of frequency domain. For the problem existed, the lag-lead emendation network for the missile control system is designed and the conclusion analyzed proves that the emendation network improves the stability of the control system. At last, the circuit for this emendation network is designed and analyzed, and also the parameters for components and integrate circuits are also given.

## Keywords

Control system Lag-lead Emendation network## 1 Introduction

A certain anti-tank missile can only be used in plain area, and it cannot adapt to high-altitude environment in which it cannot fly stably even to drop to the ground. To analyze the above problem in low-cost condition, this paper analyzes the stability of the control system under high-altitude environment with frequency domain analysis method. To improve the adaptability of the missile in highland environment, emendation network for the control system is designed. The feasibility of the emendation network is validated and it can improve the adaptability of the control system under high-altitude environment, for the phase stability and the amplitude stability are both eligible. The work in this paper is based on [1], in which the control system of this missile is analyzed and the model of control system is constructed.

## 2 Control system analysis at high altitude

*α*can be obtained. The transfer function from the input instructions to output of the inclination angle variation rate can be represented as [1]

*α*is shown in Eq. (2).

Where inherent frequency is \( {\omega}_n=\sqrt{a_2+{a}_1{a}_4} \); amplification coefficients are \( {k}_K^{\overset{.}{\varTheta}}\approx \frac{a_3{a}_4}{a_2+{a}_1{a}_4} \) and \( {k}_K^{\alpha}\approx \frac{a_3}{a_2+{a}_1{a}_4} \), respectively; and the relative damping coefficient is \( \xi =\frac{a_1+{a}_4}{2\sqrt{a_2+{a}_1{a}_4}} \). Parameters from *a* _{1} to *a* _{5} are dynamic coefficients of the pitching movement, with unit s^{−1}.

In order to analyze the influence of atmosphere density to the stability of the control system theoretically, stability criterion of minimum phase systems is used to analyze the stability of the control system under highland environment. In contrast, the parameter *ρ* depicting atmosphere density under high-altitude environment is set to 0.6 times of the plain area atmosphere density. The other parameters are kept unchanged.

Where the amplitude crossover frequency is *ω* _{ c } = 13.685 rad/s, the phase stability margin is *γ* = −160.144°, the phase crossover frequency is *ω* _{ g } = 8.683 rad/s, and the gain stability margin is *h* = 0.342, respectively.

According to the stability criterion of minimum phase system, the simulated amplitude crossover frequency *ω* _{ c } is larger than phase crossover frequency *ω* _{ g }. It indicates that the original control system is unstable at high altitudes. Meanwhile, the oscillation amplitude and frequency is large, which also means the control system is unstable. The adjusting time of the system is too long for the crossover frequency is too low, which results in the response of the system to instructions being slow [1].

## 3 Design of the lag-lead emendation network

### 3.1 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.

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

- (2)
Design the emendation unit in frequency domain;

- (3)
Validate the design via trajectory simulation.

*γ*is not obvious, resulting in the smaller phase stability margin for the emendation unit and lower stability [3].

The open-loop 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.

### 3.2 Bode diagram design of the lag-lead emendation

#### 3.2.1 Design requirement for emendation network

- (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 close-loop 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.

#### 3.2.2 Procedure for emendation network design

- (1)
Calculate the open-loop 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 close-loop system and validate the performance of the system.

#### 3.2.3 Calculate the parameters of the emendation network

*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 gravity-compensating unit is deleted when designing emendation network, for gravity-compensating instruction is disturbing signal in the guidance loop and does not affect the properties of the system itself. The gravity-compensating instruction is added to the guidance loop in system performance simulation. The open-loop transfer function can be represented as

% Open-loop gain % Lag emendation unit beta Gc1 = tf([T 1],[betat 1]) |

% 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]) |

#### 3.2.4 Bode diagram check 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) |

*M*is open-loop 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 lag-lead emendation unit based on bode diagram can satisfy the control system stability of the missile in high-altitude environment.

## 4 Circuit design of the emendation network

For emendation network design, circuit design is needed to achieve the final goal. In engineering, the circuit is usually emendation equipment composed by operational amplifier and resistor-capacitor network [7, 8, 9, 10]. We will design active emendation device for the lag-lead emendation network.

The value of each parameter can be obtained: *G* _{0} = 0.606, *T* _{1} = 7.083, *T* _{2} = 1.232, *T* _{3} = 0.8333, and *T* _{4} = 0.1225

Substitute *C* _{1} = 1μF, *C* _{2} = 10μF, *R* _{4} = 10*Ω*, and *R* _{7} = 1KΩ into the above equation, and the value of the resistance and the capacitance is *R* _{1} = 14.3MΩ, *R* _{2} = 7MΩ, *R* _{3} = 1.5KΩ, *R* _{5} = 83KΩ, and *R* _{6} = 58*Ω*.The magnification of the amplifier is *K* = 50.

From the above analysis, the value of each element is determined. According to the current and voltage of the circuit, the components of the circuit can be obtained.

## 5 Conclusions

In order to improve the stability of a certain missile in highland environment, the lag-lead emendation network is designed. The stability of the control system is checked using bode diagram based on frequency domain analysis. The simulation results show that the designed emendation network can improve the stability of the control system. The circuit implementation of the emendation network is analyzed and the values of the components in the circuit are given. Electromagnetic shielding, disturbance of different wire, or the voltage matching situation needs to be considered in the implementation process of the circuit. These factors will cause the circuit occurring minor change. Therefore, in the process from emendation network to actual circuit, there are still many specific problems which need to be further studied.

## Notes

### Acknowledgements

This research was supported the by National Natural Science Foundation of China under Grant No. 1494, 61602346.

### Competing interests

The authors declare that they have no competing interests.

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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