Control law design for elastic aircraft based on intelligent variable structure

Abstract

With the continuous expansion of aircraft flying envelope and the extensive application of new structural materials, the flexibility of the aircraft increases. Under the actual flight conditions, the elastic vehicle will be elastically deformed under aerodynamic force which will produce elastic vibration. In addition to the rigid body movement of the inertia vehicle, the inertial measurement element of the control system will introduce the structural elastic vibration noise into the control system. Under certain unfavorable conditions, the control system will increase the amplitude of the vibration, resulting in the final destruction of the structure of the aircraft. In the engineering application, the elastic noise in the control signal is generally suppressed by the structural notch filter. However, when the structural bending frequency is low and the bandwidth is close to the control system, the suppression effect of the structural notch filter to the elastic noise will be reduced. By combining the fuzzy sliding mode control method with the classical attitude control structure, the longitudinal attitude controller is designed for the static unstable elastic aircraft, and the equilibrium state of the system at the origin is proved to be consistent and stable by using the Lyapunov stability analysis method. The time-domain response characteristics of the designed controller and the stability of the program signal under disturbing conditions are verified by the fixed-point simulation. The simulation results show the effectiveness and robustness of the controller.

Keywords

Fuzzy sliding mode control robustness active control hypersonic aircraft

1 Introduction

With the continuous expansion of aircraft flying envelope and the wide application of new structural materials, aircraft flexibility increases greatly. Under the actual flight conditions, the elastic vehicle will be elastically deformed under aerodynamic force and produce elastic vibration [1 –7]. In addition to the rigid body movement of the inertia vehicle, the inertial measurement element of the control system will introduce the structural elastic vibration noise into the control system [8 –14]. Under certain unfavorable conditions, the control system will increase the amplitude of the vibration, resulting in the final destruction of the structure of the aircraft. Thus, it is particularly important to consider the effects of aerodynamic elasticity and find a viable solution in the design of the control system [15 –19].

The study on the elastic aircraft is widely concerned by researchers [20 –26]. Zeng et al. analyzed the adverse coupling of structural moduli in the structural modal coupling test of an aircraft and reduced the adverse coupling of the aircraft structure modal by adjusting parameters of the structural trap filter of the flight control system, which improved the stability margin of the control loop in 2009 [1]. Choi et al. applied the adaptive notch technique to filter out the structural elastic vibration noise in the feedback channel for the aerodynamic servo elasticity of the KSR (Korea Sounding Rocket)-II two-stage sound rocket, thus ensuring the stability of the system in 2000 [2]. It should be noted that when use the structural notch filter to filter the elastic noise, it need to know the vibrating frequency of the elastic noise in advance. If the elastic vibration frequency changes greatly, the filtering effect in literature [1] will be greatly reduced, and the adaptive method used in literature [2] to identify the elastic vibration information will increase the complexity of the control system.

As a kind of nonlinear control method, variable structure control has been widely concerned by scholars of various countries due to its strong robustness to system parameters perturbation and external disturbance. In literature [3], the longitudinal controller for missile is designed by using the variable structure theory and the simulation results show that the controller has strong robustness to the uncertain disturbance. However, the nonlinear switch of traditional variable structure is an ideal switch, which requires the system to switch between different subsystems at infinite speed. But the actual systems have inertia, hysteresis and other phenomena, and do not have the ideal switching characteristics, which led to the vibration. The current method to weaken the vibration is the saturation function method and the approach law. However, there are some defects in the two methods. The robustness of sliding mode control with saturation function method decreases with the increase of the thickness of the boundary layer, while the effect to suppress vibration will be reduced when decreases the thickness of the boundary layer [4]. The approach law method reduces the speed of the system through the sliding surface to weaken the vibration, and the control coefficients are concerned with the interference values, and the vibration problem is still present when the interference is large [5].

In this paper, the fuzzy control is used to suppress the inherent vibration problem of variable structure control. The longitudinal controller is designed for static unstable elastic vehicle. The robustness of the designed controller and the vibration suppression effect are tested by mathematical simulations.

The paper was organized as following five sections. The first section was the introduction of this paper. The second section was used to describe the main model of elastic aircraft. The third section was the design of fuzzy sliding mode controller and the proof of stability of controller. The forth part showed the detailed simulation with different assumptions. And the fifth section was the main conclusion of this paper.

2 Mathematical model

The Longitudinal small disturbed motion equation considering first order elasticity is given in Equation 1. ${\begin{matrix} \ddot{ϑ} + a_{1} \dot{ϑ} + a_{11} \dot{α} + a_{2} α + a_{3} δ_{z} = 0 \\ \dot{θ} - a_{4} α - a_{5} δ_{z} = 0 \\ ϑ - θ - α = 0 \\ \ddot{q} + 2 ξ ω \dot{q} + ω^{2} q \\ = D_{1} \dot{ϑ} + D_{2} α + D_{3} δ_{z} + D_{4} \dot{q} + D_{5} q \\ {\dot{ϑ}}_{Σ} = \dot{ϑ} - \frac{\partial Φ (x_{g})}{\partial x} \dot{q} (t) \\ ϑ_{Σ} = ϑ - \frac{\partial Φ (x_{g})}{\partial x} q (t) \end{matrix}$ (1)

Where ϑ is the pitching angle, θ is the angle of the ballistic, α is the angle of attack, β is the side slip angle, δ_z is the pitching rudder angle, a₁, a₁₁, a₂, a₃, a₄ and a₅ are the pitching dynamic coefficient and q is the first-order mode generalized coordinate, ξ is the first-order mode structure damping ratio, ω is the first-order mode frequency, D₁, D₂, D₃, D₄ and D₅ are the elastic dynamic coefficients, Φ is the first-order mode, x_g is the angular velocity measurement element mounting position, ${\dot{ϑ}}_{Σ}$ is the pitching angle velocity signal with elastic noise, ϑ_Σ is the pitching angle signal containing elastic noise.

Over the years, the classic two-way attitude control method has been widely used in aircraft control system design, and its control signals directly come from the strap-down inertial navigation system, which is easy to achieve. However, this structure is used with PID control. This paper attempts to combine it with the sliding mode variable structure to design the longitudinal controller with the pitching angle and pitching angle velocity as the feedback signals. The block diagram is shown in Fig. 1.

Fig. 1

Block diagram of two-way variable structure control principle.

Fig. 2

The influence of vibration on the pitching angle tracking traces.

In Fig. 1, S is the sliding mode, u_N is the non-linear control quantity obtained by non-linear transform, and the dotted line is used to calculate the equivalent control u_eq. u_k0, u_k1, u_k2, c₁ and c are the control parameters to be designed.

3 Design of fuzzy sliding mode controller

3.1 Variable structure controller design

According to Fig. 1, the sliding mode is depicted as the Equation 2. $S = \int (ϑ_{Σ} - ϑ_{cx}) dt + c_{1} (ϑ_{Σ} - ϑ_{cx}) + c_{2} {\dot{ϑ}}_{Σ}$ (2)

When the system slides on the sliding surface, Equation 2 is equal to zero. In the given feature point, Equation 1 can be regarded as a constant system, take Laplace transformation, the Equation 3 can be obtained with Equations 1 and 2. $ϑ_{Σ} = W (s) ϑ_{cx}$ (3)

Where s is Laplace operator, and $W (s) = \frac{c_{1} s + 1}{c_{2} s^{2} + c_{1} s + 1}$ (4)

According to the characteristic equation of the transfer function, change of c1 and c2 values can arbitrarily configure the closed-loop poles and adjust the dynamic characteristics of the sliding phase of the system. In the case of the eigenvalues λ₁ and λ₂ with negative real parts, the sliding mode coefficients c1 and c2 are given as Equation 5. ${\begin{matrix} c_{1} = - \frac{1}{λ_{1}} - \frac{1}{λ_{2}} \\ c_{2} = \frac{1}{λ_{1} λ_{2}} \end{matrix}$ (5)

When the program pitching angle is a step instruction, according to the final value theorem, Equation 6 is obtained. $lim_{t \to \infty} ϑ_{Σ} (t) = lim_{s \to 0} W (s) ϑ_{cx} = ϑ_{cx}$ (6)

Therefore, when the system slides on the sliding surface, the pitching angle can track the program signal without net error.

According to the existence of sliding mode, derivation of Equation 2 for the time t is calculated as Equation 7. $\dot{S} = (ϑ_{Σ} - ϑ_{cx}) + c_{1} ({\dot{ϑ}}_{Σ} - {\dot{ϑ}}_{cx}) + c_{2} {\ddot{ϑ}}_{Σ}$ (7)

Take Equation 1 into consideration, Equation 8 can be got.

$\begin{matrix} {\ddot{ϑ}}_{Σ} & = & \ddot{ϑ} - \frac{ϑ Φ (x_{g})}{\partial x} \ddot{q} (t) \\ = & - a_{1} \dot{ϑ} - a_{11} \dot{α} - a_{2} α - a_{3} δ_{z} \\ - \frac{\partial Φ (x_{g})}{\partial x} \ddot{q} (t) \\ = & - a_{1} {\dot{ϑ}}_{Σ} - a_{2} α - a_{3} δ_{z} \\ - (a_{1} \frac{\partial Φ (x_{g})}{\partial x} \dot{q} (t) + \frac{\partial Φ (x_{g})}{\partial x} \ddot{q} (t) + a_{11} \dot{α}) \end{matrix}$ (8)

The contents in the brackets include the effects of elastic vibration angular velocity, angular acceleration, angle of attack, and so on, which are treated as equivalent rudder partial interference, and Equation 9 is derived from Equations 7 and 8.

$\begin{matrix} \dot{S} & = & (ϑ_{Σ} - ϑ_{cx}) + c_{1} {\dot{ϑ}}_{Σ} - c_{2} (a_{1} {\dot{ϑ}}_{Σ} \\ + a_{2} α + a_{3} δ_{z}) + δ_{zf} \end{matrix}$ (9)

Where δ_zf is the equivalent interference. Let the above Equation 9 be zero, sliding mode control signal is obtained as Equation 10. $δ_{z} = u_{ko} (ϑ_{Σ} - ϑ_{cx}) + u_{k 1} {\dot{ϑ}}_{Σ} + u_{k 2} α + u_{N}$ (10)

Where $u_{k 0} = \frac{1}{c_{2} a_{3}}, u_{k 1} = \frac{c_{1} - c_{2} a_{1}}{c_{2} a_{3}}, u_{k 2} = - \frac{a_{2}}{a_{3}}$

In Equation 10, u_N is used to resist external interference, and it is taken as a symbolic function, and mathematical simulation is carried out without interference.

According to the simulation results, the symbol function is an ideal switch, which requires the system to switch between different structures at an infinite rate, but there is a time delay for an actual system, and it is impossible to switch the control structure at an infinite rate that is limited by the servo power, therefore there is vibration. In the missile control system, the vibration not only affects the accuracy of the control system, increases fuel consumption, but also and the high frequency un-modeled dynamic characteristics of the system can easily be excited, which destroys the performance of the system, and even leads the system to oscillate or lose stability.

3.2 Vibration suppression method based on fuzzy control

There are three ways for combination use of fuzzy control and sliding mode control [6].

First, the amplitude of the symbolic function is adjusted adaptively by the fuzzy control rule. The method can keep the robustness of the system better under the premise of ensuring the approaching speed and reducing the vibration. However, due to the existence of the precision problem of the conventional fuzzy logic itself, the control parameters derived from it are not very accurate.

Secondly, the fuzzy control rule is directly used to determine the fuzzy control quantity, that is, take the switch function S and its derivative as direct input, and u_N is obtained by fuzzy reasoning. This method is simple and direct, and has strong robustness to parameter change and external disturbance.

Thirdly, the composite control strategy of variable structure control and fuzzy control is adopted. In the large deviation, variable structure control is adopted. Fuzzy control is adopted in small deviation. The method avoids the vibration phenomenon because variable structure control is not used in small deviation.

Base on the simple and practical principle, select the second method. The two-dimensional fuzzy controller is constructed by using the switching function S and its change rate SC as input and the nonlinear control quantity u_N as output. When the control system is running, S and SC are calculated according Equations 2 and 7, then quantized into fuzzy variables S₁ and SC₁ by quantization factors K_s and K_SC, and fuzzy variable U is obtained according to fuzzy control rules and fuzzy logic reasoning, and the exact control amount u_N is obtained by multiplying U by the output scale factor K_U. Define the fuzzy language words of S₁, SC₁ and U are the following. ${NB, NM, NS, O, PS, PM, PB}$

Where NB is negative maximum, NM is negative medium, NS is negative minimum, O is zero, PS is positive minimum, PM is positive medium, and PB is positive maximum.

Define the fuzzy sets of S₁ and SC₁ as the following. $\begin{matrix} {- 6, - 5, - 4, - 3, - 2, - 1, \\ 0, + 1 + 2 + 3, + 4 + 5, + 6} \end{matrix}$

Define the fuzzy set of U $\begin{matrix} {- 7, - 6, - 5, - 4, - 3, - 2, - 1, \\ 0, + 1, + 2, + 3, + 4, + 5, + 6, + 7} \end{matrix}$

Each fuzzy language word set contains seven variables, and the fuzzy set domain consists of 13 or 15 quantization levels, so that the number of elements of the domain is twice the number of fuzzy language lexeme elements, ensuring that each fuzzy language variable better covering the domain.

In order to establish a connection between the fuzzy set theory and the fuzzy language words, we need to determine the membership degree of the fuzzy language variable in the fuzzy set domain, that is, assign the fuzzy language variable, select the triangular membership function, and determine the membership function of fuzzy language words for S₁, SC₁ and U, as shown in Figs. 3 to 5.

Fig. 3

Membership function of fuzzy language words for S₁.

Fig. 4

Membership function of fuzzy language words for SC1.

Fig. 5

Membership function of fuzzy language words for U.

The two input signals S₁ and SC₁ of the two-dimensional fuzzy controller include seven fuzzy language variables respectively, corresponding to 49 fuzzy control rules under different input conditions. According to the existence of sliding mode conditions, $S_{1} \times {SC}_{1} < 0$ (11)

Define the fuzzy control rule as the Table 1 [7].

Table 1

Fuzzy control rule

S₁\SC₁	NB	NM	NS	O	PS	PM	PB
NB	NB	NB	NB	NB	NM	NS	O
NM	NB	NB	NB	NM	NS	O	PS
NS	NB	NB	NM	NS	O	PS	PM
O	NB	NM	NS	O	PS	PM	PB
PS	NM	NS	O	PS	PM	PB	PB
PM	NS	O	PS	PM	PB	PB	PB
PB	O	PS	PM	PB	PB	PB	PB

According to the fuzzy control rule table and fuzzy logic reasoning algorithm, we can determine the fuzzy output U, and the exact nonlinear control quantity u_N is obtained multiplying U by the output scale factor k_U, and the fuzzy nonlinear term is expressed as Equation 12. $u_{N} = k_{U} \times U$ (12) where k_U is a fuzzy output quantization factor greater than zero. Substitute Equation 12 is into Equation 10, and the fuzzy sliding mode control law is obtained as Equation 13. $δ_{z} = u_{k 0} (ϑ_{Σ} - ϑ_{cx}) + u_{k 1} {\dot{ϑ}}_{Σ} + u_{k 2} α + k_{U} \times U$ (13)

When the sliding mode non-linear switching control u_N is generated by the symbolic function, the control signal u_N is switched back and forth between the positive and negative outputs whenever the sign of the sliding mode S changes, but the actual drive cannot achieve such an ideal switching effect, and resulting in vibration. In fact, the fuzzy control output u_N is the two-dimensional function of S and its derivative, the real output shown in Fig. 6.

Fig. 6

Nonlinear output u_N of fuzzy sliding mode variable structure.

According to Fig. 6, when S is the same, the output u_N becomes smaller as S becomes smaller. Considering the influence of the sliding mode change trend, the fuzzy controller can adjust the control effect according to the change trend of the sliding mode S. Assuming that the system deviates the sliding mode from the switching plane due to disturbance, if the trend of the sliding mode is zero at this time, the fuzzy controller applies a smaller force. If the sliding mode has the tendency to continue to deviate, the greater control force is applied. If the sliding mode has a tendency to converge to the switching plane, and no control force is applied, which is similar to the human brain decision process. However, in the symbolic function, the controller exerts the maximum force in the negative direction whenever S is negative, and if S is positive, the maximum force is applied in the positive direction. Compared with the symbol function, the fuzzy controller takes into account the trend of sliding mode, the control effect is smoother, and it reduces the velocity of the system through the switching plane, and is more conducive to weaken the vibration.

3.3 Stability proof of fuzzy sliding mode controller

The stability of fuzzy sliding mode controller is proved based on Lyapunov stability analysis. Take the Lyapunov function as Equation 14. $V = \frac{1}{2} S^{2}$ (14)

Where S is the sliding surface. According to the characteristics of sliding mode variable structure control, the two states are used to prove the stability of the controller

when the system is moving or not on the sliding surface.

Case 1: when the system moves on the sliding surface.

When the system moves on the sliding surface, $S = 0$ (15)

The aircraft pitching angle and the program signal satisfy the transfer function relationship given by Equation 4, and the transfer function characteristic roots λ₁ and λ₂ are both negative and negative, so the system is stable.

Case 2: when the system is not moving on the sliding surface.

When the system does not reach the sliding mode switching surface, that is, S ≠ 0, V is a positive scalar function, and when ||S||→ ∞, V→ ∞.

Derive V to time and get: $\dot{V} = S \times \dot{S}$ (16)

According to Lyapunov’s large-scale progressive stability theorem, when the Equation 14 is a semi-negative scalar function, the equilibrium state of the system at the origin is consistent and stable. Substituting Equations 7 and 13 into 14, we get Equation 17.

$\begin{matrix} \dot{V} & = & S \times \dot{S} = S \times SC \\ = & - c_{2} a_{3} k_{U} \times U \times S \end{matrix}$ (17)

In Equation 14, a₃ represents the efficiency of the elevator, which is greater than zero; c2 is greater than zero from Equation 5.

When the sliding mode function S < 0, the corresponding fuzzy language variable is NB, NM or NS.

According to Table 1, if the initial state point sliding mode change rate SC ⩾ 0, then the corresponding fuzzy language variable is NB, NM, NS or O, the fuzzy nonlinear output variable U is NB, NM or NS, that is, $u_{N} = k_{U} \times U < 0$ (18)k_U is a fuzzy output quantization factor greater than zero, then Equation 14 is less than zero.

If the fuzzy state variable corresponding to the initial state point sliding mode change rate SC is PS, PM or PB, then the system is moving towards the sliding surface. To reduce the vibration, the fuzzy control is controlled according to the sliding mode and its rate of change. There are six fuzzy control rules as the following under the control in Table 1. if S₁ = NB and SC₁ = PS, if S₁ = NB and SC₁ = PM if S₁ = NB and SC₁ = PB, if S₁ = NM and SC₁ = PS if S₁ = NM and SC₁ = PM, if S₁ = NS and SC₁ = PS

Output U is NM, NS or O, that is $u_{N} = k_{U} \times U ⩽ 0$ (19)

For the following three fuzzy control rules:

if S₁ = NM and SC₁ = PB,

if S₁ = NS and SC₁ = PM,

if S₁ = NS and SC₁ = PB,

Output U is PM or PS, that is $u_{N} = k_{U} \times U > 0$ (20)

Equation 14 is greater than zero, and these three fuzzy control rules are designed to reduce the speed of the system through the sliding surface and to suppress vibration. If the cross-slip surface leads to S > 0 and SC > 0, the stability is shown in the next section.

When the sliding mode function S > 0, the corresponding fuzzy language variable is PB, PM or PS.

According to Table 1, if the initial state point sliding mode change rate SC ⩾ 0, then the corresponding fuzzy language variable is PB, PM, PS or O, the fuzzy nonlinear output variable U is PB, PM or PS, that is, $u_{N} = k_{U} \times U > 0$ (21)

Then Equation 14 is less than zero.

If sliding modulus change rate SC > 0 at the initial state point, then the corresponding fuzzy language variable is NS, NM or NB, then the system is moving to the sliding surface. To reduce the vibration, the fuzzy control is controlled according to the sliding mode and its rate of change. There are six fuzzy control rules as the following under the control in Table 1.

if S₁ = PB and SC₁ = NB, if S₁ = PB and SC₁ = NM

if S₁ = PB and SC₁ = NS, if S₁ = PM and SC₁ = NS

if S₁ = PM and SC₁ = NM, if S₁ = PS and SC₁ = NS

Output U is PM, PS or O, that is $u_{N} = k_{U} \times U ⩾ 0$ (22)

Then Equation 14 is less than zero.

For the following three fuzzy control rules:

if S₁ = PM and SC₁ = NB,

if S₁ = PS and SC₁ = NM,

if S₁ = PS and SC₁ = NB,

Output U is NM or NS, that is $u_{N} = k_{U} \times U ⩾ 0$ (23)

Equation 14 is greater than zero, and these three fuzzy control rules are designed to reduce the speed of the system through the sliding surface and to suppress vibration. If the cross-slip surface leads to S < 0 and SC < 0, the stability is shown in the previous section.

Thus, when the system is not moving on the sliding surface, the equilibrium state of the system at the origin is consistent and stable under the fuzzy control.

In summary, the fuzzy controller designed in this paper can guarantee the stability of the stage, and the stability of the desired pole can guarantee the stability of the sliding stage. Therefore, the fuzzy sliding mode controller has stability.

4 Simulation verification

4.1 The effect of structural elastic vibration

An elongated body of the aircraft, the first-order vibration frequency is 8 Hz, structural damping ratio is 1%, the rigidity coefficient a₁ = 1.54, a₂ = −2.07, a₃ = 12, a₄ = 0.45, a₅ = 0.02. The steering gear is simplified to the first inertia, the transition time is about 60 ms. The PID control law is designed according to the rigid body dynamics model, and the result is kp = 2.4, ki = 1.6, kd = 0.8. The simulation model is established by using Matlab / Simulink, as shown in Fig. 7.

Fig. 7

Simulink simulation model.

In Fig. 7, Transfer Fcn1 is a rigid body model, and Transfer Fcn2 is a structural elastic vibration model, and the sum of the two is an elastomeric object model given in Chapter 2 considering the first-order elasticity. Keeping the control parameters unchanged, the controlled objects were rigid body and elastomer, and the unit step response curve as shown below.

According to Fig. 5, considering the influence of structural elastic vibration, the sensor-sensitive structural elastic vibration signal enters the PID control law through the feedback channel, which eventually leads to the divergence of the attitude of the control system. This is the typical aerodynamic servo elasticity problem.

Figures 8 to 11 show the effect of structural elastic vibration on the stability of the control system from the perspective of time domain. In order to further study the reason of posture divergence, Fig. 12 shows the bode diagram in the case of controlled objects with rigid body and elastomer, respectively.

Fig. 8

Effect of structural elastic vibration on pitching angle.

Fig. 9

Effect of structural elastic vibration on control voltage.

Fig. 10

Effect of structural elastic vibration on pitching angle velocity.

Fig. 11

Effect of structural elastic vibration on angle of attack.

According to Fig. 12, when the controlled object is a rigid body model, the stability margin of designed attitude control system is 20.1 dB, 75.7 deg, with sufficient stability. However, without changing the control law, when the controlled object model is replaced by the elastomer model, the control system in the vicinity of 8 Hz is in the situation of amplitude response spike, phase frequency characteristics of the jump, which results in control system instability.

Fig. 12

The bode diagram of rigid and elastomeric objects.

4.2 Solving the problem of pneumatic servo elasticity by structural notch filter

In normal case, the use of structural filter is to suppress the elastic vibration noise in control signal, and the filter equation is as follows: $F (s) = \frac{\frac{1}{ω_{N}^{2}} s^{2} + \frac{2 ζ_{N}}{ω_{N}} s + 1}{\frac{1}{ω_{D}^{2}} s^{2} + \frac{2 ζ_{D}}{ω_{D}} s + 1}$ (24)

Since the first-order vibration frequency of the controlled object in Section 4.1 is 8 Hz, the filter notch frequency ω_N=ω_D= 25.12rad/s, damping ratio ς_N= 0.01, ς_D= 0.8. The bode diagram of the notch filter is as Fig. 13.

Fig. 13

The bode diagram of structured notch filter.

According to Fig. 13, the amplitude of the structural notch filter is attenuated by –38.1 dB at the notch frequency, that is, the frequency signal is attenuated to 1.24%. When the notch filter is added to Fig. 7, the Simulink simulation block diagram becomes the followed Fig. 14.

Fig. 14

Simulink simulation model with structural notch filters.

Keeping the control parameters unchanged, the controlled objects are rigid body and elastomer, respectively, and unit step response curves are shown in Figs. 15 to 19.

Fig. 15

Effect of structural notch on pitching angle.

Fig. 16

Effect of structural notch on pitching angle velocity.

Fig. 17

Effect of structural notch on angle of attack.

Fig. 18

Effect of structural notch on servo control voltage.

Fig. 19

comparisons of servo control voltage before filtering for elastomer.

According to Figs. 15 to 19, after the PID control signal is filtered by the structural notch filter, the sensor-sensitive structural elastic vibration signal is suppressed and the control system is stabilized. In order to further study the reasons for the stability of the attitude, Fig. 20 shows the bode diagram that the controlled objects are rigid and elastomer, respectively.

Fig. 20

The bode diagram of rigid and elastomeric objects.

According to Fig. 20, when the object is a rigid body model, due to the addition of the notch filter, the stability margin of the attitude control system is slightly reduced to 19.8 dB, 59.1 deg, but it is still sufficiently stable. However, without changing the control law and the controlled object model is replaced by the elastomeric model, amplitude response peak was greatly attenuated for the control system in the vicinity of 8 Hz, and the control system is stable.

4.3 The problem of structural notch

During the flight of the aircraft, with the fuel consumption, weight reduction, aerodynamic heating, structural temperature changes and other effects, the first order mode structural frequency will not always remain at 8 Hz, and it will inevitably perturb. According to Fig. 20, the structural notch filter can only attenuate the sensor-sensitive structural elastic vibration signal at the designed notch frequency, and the first-order mode frequency perturbation can cause the notch filter to fail.

In this example, assuming that the first-order mode frequency perturbation range is 6 Hz∼10 Hz, then the PID control law designed in Section 4.1 and the notch filter designed in Section 4.2 are used to simulate the impact of the first-order frequency variation on control system stability. The results are shown in the Figs. 21 to 24.

Fig. 21

Effect of first-order mode frequency perturbation on control.

Fig. 22

Effect of first-order mode frequency perturbation on pitching angle velocity.

Fig. 23

Effect of first-order mode frequency perturbation on angle of attack.

Fig. 24

Effect of first-order mode frequency perturbation on control voltage.

According to Figs. 21 to 24, when the first-order mode frequency perturbation is in the range of 6 Hz∼10 Hz, the notch filter cannot inhibit the structural elastic vibration, and the control system is still in the risk of instability. Figure 25 shows the control system bode diagram of the first-order mode frequency perturbation. It can be seen that the notch filter cannot effectively attenuate the sensor noise after the first-order mode frequency perturbation, and lead to the instability of the control system.

Fig. 25

The bode diagram of the first-order mode frequency perturbation.

4.4 Control law design based on intelligent variable structure control method

Considering the frequency pull-off of the first-order mode, the frequency of the reference curve is 8 Hz, the negative pull-off (6.0 Hz) and the positive pull-off (10.0 Hz), structural damping ratio and aerodynamic parameters do not perturb, the robustness of intelligent variable structure controller to the elastic vibration frequency perturbation is analyzed, and the results are shown in Figs. 26 to 28.

Fig. 26

Effect of a frequency perturbation on the controller.

Fig. 27

Effect of a frequency perturbation on the servo control voltage.

Fig. 28

Effect of a frequency perturbation on angle of attack.

The first-order vibration mode damping ratio exists perturbation for different aircraft affected by production, manufacture and other effects, and the reference damping ratio of 1%, negative pull 0.5%, positive pull 2% can be selected for the three cases. The first-order vibration frequency and the aerodynamic parameters are not perturbed, and the robustness of the intelligent variable structure controller to the structural damping ratio perturbation is analyzed. The results are shown in Figs. 29 to 31.

Fig. 29

Effect of structural damping ratio perturbation on the controller.

Fig. 30

Effect of structural damping ratio perturbation on the servo control voltage.

Fig. 31

Effect of structural damping ratio perturbation on angle of attack.

During the flight of the aircraft, due to the inaccurate calculation of the pneumatic data, it will lead to the change of the missile model, 30% for the aircraft rudder and 30% for the static stability. The simulation results are shown in Figs. 32 to 34.

Fig. 32

Effect of aerodynamic parameter perturbation on the pitch angle.

Fig. 33

Effect of aerodynamic parameter perturbation on servo control voltage.

Fig. 34

Effect of aerodynamic parameter perturbation on angle of attack.

The simulation results show that the designed controller can stably track the instruction program signal under the condition of adding the first order elastic bending frequency pull and the aerodynamic parameter disturbance. Compared with Fig. 2, the fuzzy sliding mode controller proposed in this paper can not only effectively suppress the vibration but also realize stable signal tracking under the condition of interference. Therefore the fuzzy sliding mode controller designed in this paper can effectively suppress the vibration and ensure the robustness of the system at same time, thus having the advantages of both fuzzy control and sliding mode control.

5 Conclusion

In this paper, the traditional variable structure and fuzzy control are combined to form fuzzy sliding mode variable structure control, and combined with the classical attitude control structure, the static instability elastic vehicle controller is designed and proved its stability by using the Lyapunov stability analysis method. The simulation results show that the controller has good dynamic performance and no overshoot, and has strong robustness to the first order elastic bending frequency deviation and aerodynamic parameter disturbance. This shows that the fuzzy variable structure controller inherits the advantages of the two controllers, while maintaining the strong robustness of the traditional variable structure and eliminating the vibration, and has a good application prospect in the flight control. And future work will focus on the intelligent control of elastic aircraft with neural network method.

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