Integrated design of lateral-directional control law and vertical tail load for civil aircraft

According to new requirements of airworthiness regulation CS25.353, the advanced lateral-directional P-Beta control law design is studied in this paper. The LQR method based on the tracking command model is used to design the lateral-directional control law parameters. At the same time, the vertical tail load model of aircraft for yaw maneuver is introduced into the lateral-directional control law design model. The aim is to build the integrated design of control law and flight load. Finally, the simulation results show that the P-Beta control law can not only improve Dutch roll damping and coordination turn characteristics, but it can also reduce the yaw maneuver load by 18.5%. Thus, the structure weight of the aircraft’s vertical tail can be reduced.


Introduction
Today, with the rapid development of civil aircraft transport, Boeing and Airbus consistently adopt more advanced control laws in lateral-directional control to further improve the economy of civil aircraft.This can improve directional static stability, Dutch roll damping, and coordination turn performance; it can also reduce the workload of pilots and improve ride comfort [1] .Moreover, it can also reduce the vertical tail area.More importantly, it can reduce the yaw maneuver load of the vertical tail.
The accident of American Airlines Flight 587 in 2001 showed that even when airspeed is below maneuver speed V A , the vertical tail load may also exceed the limit load if pilots control the pedal repeatedly [2] .As a result, the National Transportation Safety Board requested the FAA to amend FAR25.1583(a)(3).At the same time, a requirement in AFM (Airplane Flight Manual) is added.It is illustrated that even at speed below V A , controlling the pedal sharply and repeatedly or multi-axis fullstroke cross maneuvering is not permitted.However, the incident of the Canada AC190 flight in 2008 showed that even well-trained pilots may also encounter the problem of controlling the pedal repeatedly [3][4] .Therefore, in airworthiness regulation CS25 (Amendment No. 22) released by EASA on November 5, 2018, CS25.353, "Rudder control reversal conditions", was added, and the reversal of the rudder was taken as the limit maneuver load condition [5] .The FAA released the "Yaw Maneuver Conditions-Rudder Reversals" proposed terms for public comment on 16 July 2018 [6] .In summary, two key problems must be solved in the design of lateral-directional control law for civil aircraft.First, relaxed directional static stability shall be adopted to reduce the vertical tail area and improve the economy based on satisfying lateral-directional control quality.Second, requirements of the yaw limit maneuver load of airworthiness regulation CS25.353 should be met, which requires control law design and flight load to be integrated design.
Berger et al. [7] proposed an improved lateral-directional control law instead of a traditional yaw damper.Aerodynamic data and inertial navigation signal synthesis were used to obtain the estimated side-slip angle for feedback control.The advantage is that side-slip angle sensor installation is not required.However, it depends on the high precision side-slip angle estimation method and accurate aerodynamic data.Gangsaas and Hodgkinson [8] carried out the design of control law parameters using the LQR method and successfully applied it to civil aircraft.Many domestic scholars have also studied the control law design based on the LQR method [9] and the airworthiness requirements of the yaw maneuver load [10] .Nevertheless, there is a lack of research on the integrated design of control law and flight load according to the airworthiness regulation CS25.353.
Focusing on the airworthiness requirements of CS25.353, this paper studies advanced lateraldirectional P-Beta control law (side-stick commands roll rate, pedal commands side-slip angle).It carries out the control law parameter design using the LQR method based on the command model tracking.Meanwhile, the aircraft yaw maneuver load model is introduced into the control law design model to build an integrated design for control law and flight load.Finally, the simulation results show that, compared with the traditional yaw damper method, the P-Beta control law enhances the directional static stability and improves the Dutch roll damping and coordination turn characteristics.It also reduces vertical tail load by 18.5%, according to CS25.353.

Airworthiness requirements for lateral-directional control law
The design of lateral-directional control law for civil aircraft should meet the relevant requirements of flight quality, flight load, and flutter in airworthiness regulations.Among them, flight quality regulations include CS25.147 lateral-directional control, CS25.161(b) lateral-directional trim, CS25.177 lateral-directional static stability, CS25.181(b) lateral-directional dynamic stability, and CS25.233 ground directional stability and controllability.Flight load regulations include CS25.341 gust and turbulence load, CS25.349 roll maneuver load, CS25.351 yaw maneuver load, and the CS25.353rudder control reversal conditions.Flutter regulations include CS25.629(b) aeroelastic stability envelope.Studies on airworthiness requirements of flight quality and aeroelastic stability are mature, which can be referred in [5] [11] [12] .This chapter mainly focuses on the study of CS25.353.
The yaw maneuver load of CS25.351 refers to a fast full-stroke maneuver of the pedal or rudder to maintain a steady and balanced side-slip angle and then restore to a neutral position quickly (Figure 1  The new airworthiness requirement of CS25.353 causes the example aircraft (Figure 2) that only adopts a yaw damper to encounter great challenges.
where q represents dynamic pressure, CAS V represents calibrated airspeed, rud K represents feed-forward gain, cmd r _  represents rudder command, and other symbols are referred in Equation (1). Figure 3 shows the example of an aircraft yaw maneuver at 170 kN in accordance with the requirements of CS25.353.It is obvious that the vertical tail load will increase when the pedal is operated repeatedly.For example, the yaw damper design value of the aircraft is 0.7.The vertical tail load will increase from 41700 N by the first pedal input with a 20.9° side-slip angle to -47000 N by the second pedal input with a -23.7° side-slip angle (an increase of 12%).If the yaw damper is insufficient, the vertical tail load will be larger.The main reason is that the adverse side-slip angle and rudder deflection increase vertical tail load during the control pedal or rudder rapidly and repeatedly.Therefore, in [6] , researchers pointed out that such civil aircraft in service can meet the requirements of CS25.353 by changing the rudder authority.

P-Beta control law design
According to the requirements of CS25.353, the yaw maneuver simulation in Chapter 1.2 conducted by an example aircraft only equipped with a yaw damper shows that if side-slip angle feedback control is adopted, the yaw maneuver load can be reduced.Therefore, this chapter focuses on the design of the P-Beta control law considering vertical tail load.

Side-slip angle synthesis
One of the difficulties in P-Beta control law is obtaining an accurate and reliable side-slip angle signal.The side-slip angle sensor can accurately measure the steady-state side-slip angle.However, its dynamic characteristics are poor, and it is easy to be affected by gusts and turbulence.Inertial navigation can effectively predict the rate of inertial side-slip angle (as shown in Equation ( 1)), but it is difficult to accurately obtain the steady-state side-slip angle.Therefore, the complementary filtering method was adopted in this paper to integrate the signal measured by the side-slip angle sensor with the estimated rate of side-slip angle (Figure 6).The obtained side-slip angle signal is used for feedback control.The frequency n  and damping ratio of the complementary filter should be determined by combining the Dutch roll mode characteristics of the bare aircraft and the control law parameters.The estimation method for the rate of inertia side-slip angle is where ys n represents lateral acceleration, TAS V represents true airspeed, r represents yaw rate, p represents roll rate, represents roll angle, represents pitch angle, represents angle of attack.
Figure 7 shows the scaled model flight test, which indicates that beta vanes exist in small oscillations due to turbulence in the air.Synthesis side-slip angle eliminates the small oscillations caused by turbulence and tracks dynamic response quickly during rudder reversal control input.

P-Beta control law parameters design
P-Beta control law parameters were designed using the LQR method with an integrator.Hence, the response of roll rate and side-slip angle will track the command model of the side-stick and pedal command model (Figure 8).When the vertical tail load exceeds the limit value, the side-slip angle command model can be adjusted appropriately to reduce the vertical tail load.By referring to the LQR design method in [13] , the expected response cost function of roll rate and side-slip angle and the output extended model in Equation ( 2) are constructed.Therefore, the cost function is minimized to realize the command model in which the response of roll rate and side-slip angle will track the side-stick and pedal command model completely.

Simulation results
The example aircraft selected takeoff configuration and corrected the flight state of 170 kN airspeed and 2000 ft altitude.Considering the requirements of yaw maneuver ability and crosswind ability, the maximum side-slip angle of pedal command under this condition is set as 18°.
Figure 9 shows the comparison of the yaw damper and P-Beta control law for the yaw maneuver in accordance with the requirements of CS25.353.The results show that the yaw maneuver load can be reduced from -47000 N to -38300 N (a decrease of about 18.5%) by using the P-Beta control law Figure 10 shows the comparison simulation results of coordination turn performance.The results show that the P-Beta control law has better coordination turn performance, and the side-slip angle is reduced from 1.2° to 0.2°.

Conclusion
This paper studies the new requirements of CS25.353 airworthiness regulation issued by EASA.The problem of which vertical tail load will increase due to the traditional yaw damper control law is analyzed.The P-Beta control law is proposed.The control law parameters are designed by the LQR method to track side-stick and pedal command models.At the same time, the vertical tail load model is introduced into the control law design.Finally, simulation results show that the P-Beta control law . a).At the same time, CS25.353 requires the pedal or rudder to restore to a neutral position quickly at least 3 times (Figure 1. b).

Figure 2 .
Figure 2. Yaw damper control law for the example aircraft.

Figure 3 .
Figure 3. Yaw maneuver load for the example aircraft only with yaw damper.

Figure 5 Figure 5 .
Figure 5 shows the command model in which the pedal commands the side-slip angle.The pedal input signal is shaped to cmd  , and the output side-slip angle command m  is generated by the second- order module.The parameters m   and m  of the second-order module are determined by flight quality

Figure 7 .
Figure 7. Flight test of synthesis side-slip on a scaled model.

Figure 8 .
Figure 8. P-Beta control law design structure.
function of roll rate response and side-slip angle response can be minimized.Thus, the LQR feedback matrix is obtained: , and R are weighted matrices.

Figure 9 .
Figure 9.Comparison of yaw maneuver load for an aircraft.

Figure 10 .
Figure 10.Comparison of coordination turn performance.
the vertical tail load by 18.5%.This can improve the Dutch roll and coordination turn performance.
2023 Journal of Physics: Conference Series 2691 (2024) 012030 compared with the yaw damper only.Considering overshoot, the maximum side-slip angle is -19.2° by P-Beta control law, which is less than -23.7° achieved by yaw damper.