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Variable phase control of wing rock
Aerospace Science and Technology 10 (2006) 27–35 www.elsevier.com/locate/aescte
Variable phase control of wing rock Zeng Lian Liu ∗ , Chun-Yi Su, Jaroslav Svoboda Department of Mechanical Engineering, Concordia University, 1455, de Maisonneuve Blvd. W., Montreal, Quebec, Canada, H3G IMS Received 12 February 2004; accepted 24 November 2004 Available online 9 June 2005
Abstract This paper addresses a variable phase control issue for suppressing wing rock with hysteresis. In free-to-roll tests, as the angle of attack (AOA) is increased, the roll angle versus the rolling moment indicates hysteresis and provides clues about where wing rock motion is being driven and where the motion is being damped. We present the analysis method of wing rock energy to explain the mechanism of wing rock and the formation of hysteresis, and then develop a variable phase control (VPC) scheme to compensate the phase and magnitude distortions. The effectiveness and robustness of the proposed scheme are demonstrated by suppressing wing rock phenomenon at various AOA and any initial conditions. © 2005 Elsevier SAS. All rights reserved. Keywords: Hysteresis; Nonlinear system; Variable phase control; Wing rock
1. Introduction Many modern combat aircraft often operate at subsonic speeds and high angles of attack. At a sufficiently high angle of attack (AOA), these aircraft become unstable and enter into a limit-cycle oscillation (LCO), mainly rolling motion known as wing rock [3,5,7,13]. In practice, high-speed civil transport and combat aircraft can fly in conditions where this self-induced oscillatory rolling motion is observed; moreover, wing rock phenomenon can be highly annoying to the pilot and may pose serious limitations to the combat effectiveness of the aircraft. Therefore, the control of wing rock phenomenon is of significant importance. Considerable research has been conducted on the motion of 80◦ swept delta wing to help understand the fundamental mechanisms causing wing rock [1,3–5,7–9,12]. Free-to-roll tests are usually used to determine build-up and limit-cycle characteristics of wing rock. These results reveal the magnitude of limit cycles of wing rock varying with the AOA. In addition, the tracking tests [1,3,4,8,9,12] of the primary vor* Corresponding author.
E-mail address: [email protected] (Z.L. Liu). Deceased.
1270-9638/$ – see front matter © 2005 Elsevier SAS. All rights reserved. doi:10.1016/j.ast.2004.11.002
tex positions in the cross-flow plane provide the data to understand the driving mechanism of wing rock phenomenon. That hysteresis exists between the roll angle and the rolling moment provides clues for explaining wing rock phenomenon. This hysteresis shows three loops during one cycle. The researchers [1,3,7,9] have observed that the work done by the rolling motion is driving the oscillation during the central loop since the aerodynamic motion acts in the direction of wing rolling motion whereas during the two reverse outer loops the oscillation is being damped. In this paper, we will theoretically analyze this hysteresis mechanism instead of physical insight. In this work, inspired by the hysteresis mechanism of driving wing-rock motion, we will directly apply hysteresis compensation or reduction methods to suppress wing rock. The conception of the phaser proposed by Cruz and Hernandez [2] is adopted to design variable phase control (VPC) schemes. The main objective of this paper is to study the VPC to suppress wing rock with hysteresis. The rate of energy change of wing rock is derived to analyze the hysteresis mechanism of driving wing rock. The hysteresis loops, based on a defined critical angle, are divided into two parts: a center loop and two reverse outer loops. We develop a VPC scheme to compensate the hysteresis effects of wing rock
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for each part. To verify the effectiveness and robustness of the proposed method, we will demonstrate the several cases of wing rock suppression. Simulation results show that the proposed control scheme can quickly suppress wing rock at various AOAs and any initial conditions.
2. Wing rock model The phase plane representation of wing rock shows that wing rock phenomenon is dominated by nonlinear damping and a relationship can be established with one-degree-offreedom analytical models [3]. The differential equation of describing wing rock is given by [6,11,14] 2 Sb/2Ixx )Cl + u (1) φ¨ = (ρU∞ where φ(t) is the roll angle, ρ is the density of air, U∞ is the freestream velocity, S is the wing reference area, b is the chord, Ixx is the mass moment of inertia, u(f ) is the control input, and Cl is the rolling moment coefficients written as ˙ φ˙ + b4 φ 3 + b5 φ 2 φ. ˙ Cl = b0 + b1 φ + b2 φ˙ + b3 |φ| (2) The aerodynamic parameters bi (i = 0, 1, 2, 3, 4) are the time-varying functions of AOA. Substituting (2) into (1), we have [10] ˙ φ˙ + a3 φ 3 + a4 φ 2 φ˙ = u (3) φ¨ + a0 φ + a1 φ˙ + a2 |φ| where ai (i = 0, 1, 2, 3, 4) are the parameters relative to free-to-roll experiment conditions [3,4]. A typical set of coefficients ai (at Reynolds number = 636 000) is depicted in Fig. 1. To illustrate the behaviors of wing rock, the uncontrolled wing-rock motion at AOA = 32.5◦ with the initial condition ˙ = 0 is demonstrated in Fig. 2, which φ(0) = 0.1◦ and φ(0) shows a small initial disturbance is enough to cause wing rock.
Fig. 1. Coefficients ai in Eq. (3).
3. Hysteresis analysis of wing rock The uncontrolled wing-rock model in (3) can be written in the following form: ˙ + a4 φ 2 φ˙ + (a0 φ + a3 φ 3 ) = 0. φ¨ + a1 + a2 |φ| (4) ˙ Eq. (4) is then expressed in a stateLet x1 = φ and x2 = φ; variable form: x˙1 = x2 , (5) x˙2 = −(a1 + a2 |x2 | + a4 x12 )x2 − (a0 x1 + a3 x13 ). Eq. (5) is further expressed in the phase-trajectory equation: dx2 −(a1 + a2 |x2 | + a4 x12 x2 − (a0 x1 + a3 x13 ) = . (6) dx1 x2 By integrating Eq. (6) and substituting x2 = x˙1 we have t 1 2 x + U (x1 ) = C − (7) a1 + a2 |x2 | + a4 x12 x22 dt 2 2 0
˙ Fig. 2. Time history of wing rock at φ(0) = 0.1◦ and φ(0) = 0.
where U (x1 ) = (a0 x1 + a3 x13 ) dx1 is the potential energy and C is an integral constant. Define E as the total mechanical energy of the system given by [15] 1 E = x22 + U (x1 ). 2
(8)
Z.L. Liu et al. / Aerospace Science and Technology 10 (2006) 27–35
Fig. 3. Build up phase of the hysteresis at AOA = 32.5◦ .
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1) If x2 = 0 and dE/dt = 0, then x1 = ±Xc . We define this roll angle Xc as a critical angle, marking B and C in Fig. 4. It lies on the boundary between increasing energy and decreasing energy and is an important parameter for designing the VPC in Section 4. If x2 = 0 and dE/dt = 0, then x1 = ±φmax ; we mark them as A and D in Fig. 4. 2) If x2 = 0 and |x1 | < Xc , then dE/dt > 0; this means that the wing-rock motion is absorbing energy and developing a single-loop hysteresis in a clockwise direction, as shown in Fig. 3. If x2 = 0 and |x1 | > Xc , then dE/dt < 0; this implies that the motion dissipates energy and develops two additional outer-loop hysteresis as in Fig. 4 where the direction of the motion is anticlockwise. 3) Similarly, the role of x2 can be analyzed because the relationship of x1 and x2 is constrained by wing rock to limit cycles. If |x2 | > −(a1 + a4 x12 )/a2 or |x2 | < −(a1 + a4 x12 )/a2 , then dE/dt > 0 or dE/dt < 0, respectively. With Eq. (4), it is easy to see that in this case the roll damping has a variable sign. Based on the above analysis, we conclude that the condition of wing rock developing into LCO is the hysteresis that has one center loop with two reverse outer loops.
4. Variable phase control 4.1. Phase control
Fig. 4. The hysteresis of wing rock at AOA = 32.5◦ .
By differentiating Eq. (7), we obtain the rate of change of total energy E: dE = − a1 + a2 |x2 | + a4 x12 X22 . (9) dt Eq. (9) shows that the total energy E is not constant and increases or decreases according to the sign of a1 + a2 |x2 | + a4 x12 x22 . We now turn our attention to the hysteresis of wing rock. Substituting (3) into (1), we obtain a Cl − φ relationship to illustrate the hysteresis ˙ φ˙ + a3 φ 3 + a4 φ 2 φ˙ /k (10) Cl = − a0 φ + a1 φ˙ + a2 |φ| 2 Sb/2I . where k = ρU∞ xx We simulate the build-up phase of the hysteresis in Fig. 3 and the full hysteresis in Fig. 4, and draw arrows indicating the developing direction of hysteresis over time, assuming ˙ = 0, AOA = 32.5◦ , initial conditions φ(0) = 0.1◦ and φ(0) and k ≈ 1/30. Finally, we analyze the change of dE/dt as well as the relationship between dE/dt and the hysteresis. Note that a1 < 0, a2 > 0, and a4 > 0 (except AOA = 45◦ ) in Fig. 1.
In general, the effects of hysteresis can be seen as a phase lag between a periodic input and the corresponding output. The natural way of correcting this lag is to reverse its effects. When a periodic input signal with period T is applied to a system with hysteresis, the output signal generally has the same period T as the input signal does, but is shifted in some phase angle [2]. In the wing-rock case, the output signal of wing-rock motion is complex and can be approximated by y(ωt − ϕ) = a0 +
∞ ak cos(kωt) + bk sin(kωt) . k=1
If this signal is referenced to its input u(t) (see Fig. 5), all the components of y(t) are shifted by some phase angle. It is possible to speak of phase shift between the input and output. Here we introduce the concept of the phaser. Definition [2]. A phaser Lph is an operator that shifts a periodic input signal by a constant angle ϕ 0 and has unity gain. In the frequency domain, the phaser can be expressed as
Fig. 5. Phase control scheme.
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Lph (j ω) = p + j q,
Lph (j ω) = 1,
and Lph (j ω) = φ.
(11)
The basic idea of phase control can be explained with a diagram as drawn in Fig. 5, where a phaser Lph is connected in series with a plant. If the plant is with hysteresis, the phase of the output signal y(t) will be shifted by −φs < 0 with respect to the control input signal u(t); if the phaser is used as a controller, then the phase of the control input signal u(t) will be shifted by ϕ > 0 with respect to the input signal r(t). Suppose we design this phaser with ϕ ≈ ϕ, the compensation can be obtained because one block cancels the other. 4.2. VPC When wing rock develops into the LCOs, it will exhibit three loops of hysteresis (see Fig. 4): a center loop and two reverse outer loops. The loop in the center resembles singleloop hysteresis with saturation. The two outer loops are symmetric and have a reverse phase angle. Therefore, the overall hysteresis compensation involves two kinds of phasers to provide a positive angle ϕ1 > 0 for the center loop and a negative angle ϕ2 < 0 for the outer loops. These angles can be obtained by considering each loop independently from the other. A VPC, based on the input signal |r| to switch from one phaser to the other, is given by Lph1 (j ω)r(j ω) if |r| Xc or dE/dt 0, (12) u(j ω) = Lph2 (j ω)r(j ω) if |r| > Xc or dE/dt < 0 where Lph1 (j ω) = ϕ1 > 0, Lph2 (j ω) = ϕ2 < 0, and Xc is the critical angle defined in Section 3. It should be noted that when |r| > Xc , the phaser Lph2 (j ω) is introduced in order to speed the oscillation attenuation even if in this situation dE/dt < 0. Obviously, the single-loop hysteresis during the build-up phase of wing rock like in Fig. 3 is a special case of the VPC. 4.3. VPC implementation
ω = [ω1 ω2 ]. In other words, if we know the phase angles ϕ1 and ϕ2 we can calculate all of the coefficients of Eq. (13), which is described in detail in the Appendix. To design the phaser, the angle ϕ is the sole parameter we need to know. In a close-loop nonlinear system, it is difficult to find this angle. In general, the bigger the angle ϕ is, the faster will the system response be, but a too large value of ϕ can cause instability. If the smaller ϕ is chosen, wing rock motion will develop into smaller LCOs. One simple method is to choose a big ϕ and then to adjust this angle by evaluating control performance. As for the frequency range ω = [ω1 ω2 ] of compensation, because lower harmonic components usually have bigger coefficients, a low frequency range should be selected in order to obtain a better phase compensation. In this study, we choose a six-order linear approximation for phaser ϕ1 = 36◦ in the frequency range [ω1 ω2 ] = [0.01 1] and design it as Lph1 (s) = (s 6 + 1.652s 5 + 0.7709s 4 + 0.1259s 3 + 0.00745s 2 + 0.00015s + 0.00000075) × (s 6 + 3.017s 5 + 2.32s 4 + 0.5477s 3 + 0.05108s 2 + 0.0015s + 0.00001579)−1 . (14) The Bode Diagrams of phaser ϕ1 = 36◦ are plotted in Fig. 6. As to the phaser ϕ2 , for simplicity, let the phaser ϕ2 = −ϕ1 = −36◦ ; similarly, we have Lph2 (s) = (s 6 + 1.652s 5 + 0.7709s 4 + 0.1259s 3 + 0.00745s 2 + 0.00015s + 0.00000075) × (s 6 + 0.7665s 5 + 0.2658s 4 + 0.02618s 3 + 0.001122s 2 + 0.000012s + 0.00000004)−1 . (15) Note that the different phasers have only the different values of the qi ’s in the denominator. We should point out the order of linear approximation usually depending on the value of phaser ϕ. If ϕ is big, we
The definition of the phaser in Eq. (11) is an ideal one. We can approximate implementation by applying two important properties of the phaser [2]: superposition and noncausality. Superposition follows from linearity. Noncausality prevents direct online implementation. Thus, two linear filters are adopted to approximate the two phasers as follows: ⎧ s n + p1,n−1 s n−1 + · · · + p1,1 s + p1,0 ⎪ ⎪ (s) = L ⎪ ph1 ⎪ ⎪ s n + q1,n−1 s n−1 + · · · + q1,1 s + q1,0 ⎪ ⎪ ⎨ for ϕ1 > 0, (13) ⎪ s n + p2,n−1 s n−1 + · · · + p2,1 s + p2,0 ⎪ ⎪ ⎪ Lph1 (s) = n ⎪ ⎪ s + q2,n−1 s n−1 + · · · + q2,1 s + q2,0 ⎪ ⎩ for ϕ2 < 0 where the coefficients pi,n and qi,n are determined such that the phase of the filters is almost constant around the design parameters ϕ1 and ϕ2 within the input signal frequency range
Fig. 6. Bode diagrams of phaser ϕ1 = 36◦ and [ω1
ω2 ] = [0.01
1].
Z.L. Liu et al. / Aerospace Science and Technology 10 (2006) 27–35
need a high order linear approximation; otherwise, it will affect the compensation effect. Besides, the VPC method we consider above focuses on the phase shift or compensation of the output signal, whereas ignoring the magnitude compensation of the signal. The magnitude compensation will be discussed in Section 5.
5. The implementation of the VPC scheme of wing rock Before implementing the VPC scheme to suppress wing rock phenomenon, we need to address two issues: the input signal of the phaser and the magnitude compensation of the phaser. Input signal of the phaser: The hysteresis in Fig. 4 is seen as the phase shift between the rolling moment and the roll angle. However, it is not the input-output relationship of the wing-rock model given in Eq. (3); we cannot directly use the VPC technique without the rolling moment information. Fortunately, we could obtain the term kCl from Eq. (10) as the input signal of the phaser.
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6. Simulation results The control objective in this paper is to suppress wing rock. This means to achieve the roll angle φ = 0 and the roll rate φ˙ = 0 when wing rock phenomenon occurs and commanded signal is φ = 0 and φ˙ = 0. The following six cases are used to verify the performances of the VPC schemes. Case 1: AOA = 32.5◦ , kp = 1, ϕ1 = −ϕ2 = 36◦ . To compare the performances achieved with the linearizing control law (u = −kCl ) and with the overall control law (u = −kCl + uph ), the uncontrolled wing-rock motion is allowed to exhibit some LCOs, and then the controllers are activated to suppress wing rock, as shown in Fig. 8. From this simulation, we have observed that if the system is with u = −kCl , it is unstable; if the system is with u = −kCl + uph , it is stable and can suppress wing rock quickly.
The magnitude compensation of the phaser: As mentioned earlier, the phaser only corrects the phase distortion of the system whereas ignoring the magnitude distortion. To suppress wing rock completely, both the phase and the magnitude need to be compensated. For this purpose, a phaser gain, kp 1, is applied in the control system. In general, the bigger the phaser gain kp is, the faster will the system response be, but a too large value of kp can cause overshoot, or even instability. In addition, we have observed that the system with a big kp is more sensitive to the output measurement noise than the system with a small kp . Control structure: In Fig. 5, the phaser is connected in series with the plant, which forms an open-loop control system. To suppress wing rock, we need the closed-loop control system and then propose the VPC scheme of wing rock, as shown in Fig. 7. In this control structure, phasers Lph1 (s) and Lph2 (s) are switched depending on the sign of dE/dt, kp is the phaser gain, and kCl is the input signal of the phaser. The overall control law in Fig. 7 can be expressed by the sum of two terms: the linearizing control law and the phaser control law with magnitude compensation, i.e., u = −kCl + uph .
(a)
(b)
Fig. 7. Structure of wing-rock control system.
Fig. 8. The comparison of performances with different control laws. (a) Responses with the control law u = −kCl . (b) Responses with the control law u = −kCl + uph .
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(a)
(b)
(c)
(d)
Fig. 9. Variable phase control for Case 2. (a) Responses in hysteresis. (b) Responses on phase plane. (c) The rate of change of energy. (d) Control input.
Case 2: The same as Case 1 except kp = 6. As Fig. 8(b) in Case 1 was simulated, we observed that the control system is unstable in some initial conditions where they imply dE/dt > 0, for example, φ = 0 and φ˙ = φ˙ − max . If we modify the phaser gain from kp = 1 to kp = 6, then the VPC can guarantee a stable control system and the system states tend towards zero over time, as shown in Figs. 9(a) and 9(b). Fig. 9(c) shows the corresponding dE/dt decreasing to zero over time, which reveals that the VPC essentially makes the system energy dissipative. Fig. 9(d) displays the corresponding control input u(t). If we compare this u(t) with the umax (t) ≈ −0.9 of Ref. [15], we have observed that the VPC scheme needs only a very small control input, which is important for aircraft control at a high AOA. Case 3: The same as in Case 2 except AOAs. This case will verify the robustness of the proposed method. Assume the initial condition of wing rock to be φ(0) = 5◦ ˙ and φ(0) = 0. The controller can be activated at any time, say at t = 800 time steps. Although the controller parameters remain the same as Case 2, the VPC can suppress the different wing rock motions at other fixed AOAs, i.e.,
AOA = 25◦ , 27.5◦ , 30◦ , 35◦ , 37.5◦ , 40◦ , 42.5◦ , or 45◦ , as shown in Fig. 10. Case 4: Single-phaser control (AOA = 32.5◦ , k1 = 6, ϕ1 = 36◦ ). This simulation illustrates a special case. We hope to suppress wing rock at the beginning of wing rock motion. In fact, single phaser ϕ1 = 36◦ can easily achieve this aim because the wing rock at this time exhibits a singleloop hysteresis like in Fig. 3. If the phaser is activated at t = 600 time steps, Fig. 11(a) shows the comparison outputs between the control system and uncontrolled system, and Fig. 11(b) indicates the corresponding change of dE/dt. From Fig. 11(c), we have observed that once the phaser is activated, the direction of hysteresis loops will change from clockwise (the system absorbing energy) to anticlockwise (the system dissipating energy). We mark S as a starting point and draw arrows indicating its direction. Similarly, in Figs. 8 and 9(a), regardless of the starting point at the center loop or the outer loops of hysteresis, the hysteresis development in all cases is forced to move in anticlockwise direction until the wing rock is completely suppressed.
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(a)
(b)
Fig. 10. Output responses (φ (deg) vs. time steps) at different AOAs.
Case 5: Comparison of different order phasers (AOA = 32.5◦ , kp = 1, ϕ1 = −ϕ2 = 36◦ ). This case illustrates the performance comparison between the six-order phaser approximation in (14) and (15) and the one-order phaser approximation which is designed as Lph1 (s) = (s + 0.0075)/ (s + 0.0285) for phaser ϕ1 = 36◦ and Lph2 (s) = (s + 0.0075)/(s + 0.00195) for phaser ϕ1 = −36◦ . Assume the ˙ = 0. We have siminitial condition is φ(0) = 42◦ and φ(0) ulated the wing rock suppression with above two kinds of phaser approximations and observed that the response of the system with the one-order phaser approximation is too much slower than the response of the system with the six-order phaser approximation, as shown in Figs. 12(a) and 12(b), respectively. Case 6: System output with the measurement noise. Lastly, we illustrate a case of the VPC with the output measurement
(c) Fig. 11. Phase control for Case 4. (a) Comparison of output responses. (b) The rate of change of energy. (c) Responses in hysteresis.
noise. Assume a band limited white noise disturbance to be added to the system output variables, for example, φ with the noise (its power is 10−4 , sample time is 0.05, and seed is 23341) and φ˙ with the noise (its power is 10−6 , sample time is 0.05, and seed is 23 341). The other parameters are the same as we simulate Fig. 12(b). Fig. 13 shows that the response of the control system is almost the same as Fig. 12(b)
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(a)
Fig. 13. Wing rock suppression with the measurement noise.
small control power need, and good robustness and suppression effect. Therefore, the proposed VPC scheme may be the effective approach for suppressing wing rock. For future work, we will consider wing rock dynamics integrating with a lateral flight dynamics and deal with the suppressing and tracking problems of time-varying wing rock with disturbances.
Appendix A. A six-order approximate phaser The form of the six-order linear filter is given by Lpa (s) = (b) Fig. 12. Response comparison of different order phaser approximations. (a) Responses with the first-order phaser approximations. (b) Responses with the six-order phaser approximations.
except the small error at the equilibrium because the rate of signal and noise is lower. This result implies that the control system can work well with the output measurement noise.
s 6 + p 5 s 5 + · · · + p 1 s + p0 . s 6 + q5 s 5 + · · · + q1 s + q0
(A.1)
For a given frequency ω = [ω1 ω2 ] and a phase angle ϕ, the location for the pole and the zero can be found. The location of the first zero z1 is located somewhere behind the location of ω1 ; for example, z1 = ω1 − ω1 /4. The remainder of the zeros will be located between ω1 and ω2 . They are logarithmically equally spaced between ω1 and ω2 , as shown in Fig. A.1. The values of pi ’s are obtained using the values of the zeros num(s) = (s + z1 )(s + z2 )(s + z3 )(s + z4 )(s + z5 )(s + z6 )
7. Conclusions The variable phase control (VPC) scheme has been developed to suppress wing rock with hysteresis at various AOAs and any initial conditions. After analyzing the energy mechanism of wingrock hysteresis, and we have designed the VPC scheme to compensate the phase and magnitude distortions for each part of the hysteresis. Simulation studies show that the proposed control scheme, in nature, can guarantee wing-rock motion always to dissipate energy such that the wing-rock phenomenon can be suppressed. The main advantages of proposed method are simple design and calculation,
= s 6 + p 5 s 5 + p4 s 4 + p3 s 3 + p2 s 2 + p1 s + p 0 . (A.2) Substituting s = j ω into (A.1), we have
Fig. A.1. Six zeros logarithmical equal space.
Z.L. Liu et al. / Aerospace Science and Technology 10 (2006) 27–35
Lph (j ω) = (j ω)6 + p5 (j ω)5 + p4 (j ω)4 + p3 (j ω)3 + p2 (j ω)2 + p1 j ω + p0 × (j ω)6 + q5 (j ω)5 + q4 (j ω)4 + q3 (j ω)3 −1 + q2 (j ω)2 + q1 j ω + q0 = (−ω6 + p4 ω4 − p2 ω2 + p0 ) + j (p5 ω5 − p3 ω3 + p1 ω) × (−ω6 + q4 ω4 − q2 ω2 + q0 ) −1 + j (q5 ω5 − q3 ω3 + q1 ω) = (K1 K3 + K2 K4 ) + j (K2 K3 − K1 K4 ) × (K32 + K42 )−1
(A.3)
where K1 = −ω6 + p4 ω4 − p2 ω2 + p0 , K2 = p5 ω5 − p3 ω3 + p1 ω, K3 = −ω6 + q4 ω4 − q2 ω2 + q0 , and K4 = q5 ω5 − q3 ω3 + q1 ω. The phase angle of the transfer function is K2 K3 − K1 K4 Lph (j ω) = Φ(ω) = tan−1 . (A.4) K1 K3 + K2 K4 Substituting the values of K3 and K4 , into (A.4) we obtain (−ω6 + q4 ω4 − q2 ω2 + q0 ) tan Φ(ω) K1 − K2 + (q5 ω5 − q3 ω3 + q1 ω) tan Φ(ω) K2 + K1 = 0. Let K5 = tan(Φ(ω))K1 − K2 and K6 = tan(Φ(ω))K2 + K1 ; we have −K5 ω6 + K5 q4 ω4 − K5 q2 ω2 + K5 q0 + K6 q5 ω5 − K6 q3 ω3 + K6 q1 ω = 0.
(A.5)
The unknown variables in (A.5) are the qi ’s and ω. We can make the value of the phase angle equal to ϕ at the values where the zeros are located. If ω = [ω1 ω2 ], then we have Lpa (j ω) = Φ(ω) = ϕ and tan(Φ(ω)) = tan(ϕ). The following set of equations are then obtained ⎧ ⎪ −K7 z16 + K7 z14 q4 − K7 z12 q2 + K7 q0 + K8 z15 q5 ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ − K8 z1 q3 + K8 z1 q1 = 0, ⎪ ⎪ ⎪ ⎪ −K7 z26 + K7 z24 q4 − K7 z22 q2 + K7 q0 + K8 z25 q5 ⎪ ⎪ ⎪ ⎪ ⎪ − K8 z23 q3 + K8 z2 q1 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ −K7 z36 + K7 z34 q4 − K7 z32 q2 + K7 q0 + K8 z35 q5 ⎪ ⎪ ⎪ ⎪ ⎨ − K8 z3 q3 + K8 z3 q1 = 0, 3 (A.6) 6 ⎪ −K7 z4 + K7 z44 q4 − K7 z42 q2 + K7 q0 + K8 z45 q5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − K8 z43 q3 + K8 z4 q1 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ −K7 z56 + K7 z54 q4 − K7 z52 q2 + K7 q0 + K8 z55 q5 ⎪ ⎪ ⎪ ⎪ ⎪ − K8 z53 q3 + K8 z5 q1 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −K7 z66 + K7 z64 q4 − K7 z62 q2 + K7 q0 + K8 z65 q5 ⎪ ⎪ ⎪ ⎩ − K8 z63 q3 + K8 z6 q1 = 0
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where K7 = tan(ϕ)K1 − K2 and K8 = tan(ϕ)K2 + K1 . Eq. (A.6) can be re-ordered in the form Ax = C ⎤ ⎡ K7 K8 z1 −K7 z12 −K8 z13 K7 z14 K8 z15 ⎧ q0 ⎫ ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎢ ⎪ ⎪ q1 ⎪ ⎢ K7 K8 z2 −K7 z22 −K8 z23 K7 z24 K8 z25 ⎥ ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎢ ⎪ ⎨q ⎪ ⎬ ⎢ K7 K8 z3 −K7 z2 −K8 z3 K7 z4 K8 z5 ⎥ ⎪ 2 ⎢ 3 3 3 3⎥ ⎥ ⎢ ⎢ K7 K8 z4 −K7 z2 −K8 z3 K7 z4 K8 z5 ⎥ ⎪ q3 ⎪ ⎪ 4 4 4 4 ⎥⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎥ ⎢ ⎪ 3 5 2 4 ⎪ ⎪ q 4 ⎪ ⎣ K7 K8 z5 −K7 z5 −K8 z5 K7 z5 K8 z5 ⎦ ⎪ ⎪ ⎩ ⎪ ⎭ 3 5 2 4 q 5 K7 K8 z6 −K7 z6 −K8 z6 K7 z6 K8 z6 ⎧ ⎫ K7 z16 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 6⎪ ⎪ ⎪ K z ⎪ ⎪ 7 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ K7 z6 ⎪ ⎬ 3 . (A.7) = 6 ⎪ ⎪ K7 z4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ K7 z56 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 6 K7 z6 Now, the values qi ’s can be obtained. References [1] A.S. Arena Jr., R.C. Nelson, Experimental investigation on limit cycle wing rock of slender wings, J. Aircraft 31 (1994) 1148–1155. [2] J.M. Cruz-Hernandez, V. Hayward, Phase control approach to hysteresis reduction, IEEE Trans. on Control Systems Technology 9 (2001) 17–26. [3] G. Guglieri, F. Quagliotti, Experimental observation and discussion of the wing rock phenomenon, Aerospace Sci. Technol. 2 (1997) 111– 123. [4] G. Guglieri, F. Quagliotti, Analytical and experimental analysis of wing rock, Nonlinear Dynamics 24 (2001) 129–146. [5] C. Hsu, C.E. Lan, Theory of wing rock, J. Aircraft 22 (1985) 920–924. [6] S.V. Joshi, A.G. Sreenatha, Chandrasekhar, Suppression of wing rock of slender delta wings using a single neuron controller, IEEE Trans. on Control System Technology 6 (1998) 671–677. [7] J. Katz, Wing/vortex interactions and wing rock, Progr. Aerospace Sci. 35 (1999) 727–750. [8] J. Katz, D. Levin, Self-induced roll oscillations measured on a delta wing/canard configuration, J. Aircraft 23 (1986) 814–819. [9] P. Konstadinopoulos, D.T. Mook, A.H. Nayfeh, Subsonic wing rock of slender delta wings, J. Aircraft 22 (1985) 223–228. [10] Z.L. Liu, C.-Y. Su, J. Svoboda, Control of wing rock phenomenon using fuzzy PD controller, in: IEEE Int. Conf. on Fuzzy System, MO, USA, May 25–28, 200. [11] J. Luo, C.E. Lan, Control of wing rock motion of slender delta wings, J. Guidance, Control, and Dynamics 16 (1993) 225–231. [12] A.H. Nayfeh, J.M. Elzebda, D.T. Mook, Analytic study of the subsonic wing rock phenomenon for slender delta wings, J. Aircraft 26 (1989) 805–809. [13] L.V. Schmidt, Wing rock due to aerodynamic hysteresis, J. Aircraft 16 (1979) 129–133. [14] S.N. Singh, W. Yim, W.R. Wells, Control of wing rock motion of slender delta wings, J. Guidance, Control, and Dynamics 18 (1995) 25–30. [15] S.-S. Wanda, The Behaviour of Nonlinear Vibrating Systems, Kluwer Academic, 1990.
Variable phase control of wing rock Zeng Lian Liu ∗ , Chun-Yi Su, Jaroslav Svoboda Department of Mechanical Engineering, Concordia University, 1455, de Maisonneuve Blvd. W., Montreal, Quebec, Canada, H3G IMS Received 12 February 2004; accepted 24 November 2004 Available online 9 June 2005
Abstract This paper addresses a variable phase control issue for suppressing wing rock with hysteresis. In free-to-roll tests, as the angle of attack (AOA) is increased, the roll angle versus the rolling moment indicates hysteresis and provides clues about where wing rock motion is being driven and where the motion is being damped. We present the analysis method of wing rock energy to explain the mechanism of wing rock and the formation of hysteresis, and then develop a variable phase control (VPC) scheme to compensate the phase and magnitude distortions. The effectiveness and robustness of the proposed scheme are demonstrated by suppressing wing rock phenomenon at various AOA and any initial conditions. © 2005 Elsevier SAS. All rights reserved. Keywords: Hysteresis; Nonlinear system; Variable phase control; Wing rock
1. Introduction Many modern combat aircraft often operate at subsonic speeds and high angles of attack. At a sufficiently high angle of attack (AOA), these aircraft become unstable and enter into a limit-cycle oscillation (LCO), mainly rolling motion known as wing rock [3,5,7,13]. In practice, high-speed civil transport and combat aircraft can fly in conditions where this self-induced oscillatory rolling motion is observed; moreover, wing rock phenomenon can be highly annoying to the pilot and may pose serious limitations to the combat effectiveness of the aircraft. Therefore, the control of wing rock phenomenon is of significant importance. Considerable research has been conducted on the motion of 80◦ swept delta wing to help understand the fundamental mechanisms causing wing rock [1,3–5,7–9,12]. Free-to-roll tests are usually used to determine build-up and limit-cycle characteristics of wing rock. These results reveal the magnitude of limit cycles of wing rock varying with the AOA. In addition, the tracking tests [1,3,4,8,9,12] of the primary vor* Corresponding author.
E-mail address: [email protected] (Z.L. Liu). Deceased.
1270-9638/$ – see front matter © 2005 Elsevier SAS. All rights reserved. doi:10.1016/j.ast.2004.11.002
tex positions in the cross-flow plane provide the data to understand the driving mechanism of wing rock phenomenon. That hysteresis exists between the roll angle and the rolling moment provides clues for explaining wing rock phenomenon. This hysteresis shows three loops during one cycle. The researchers [1,3,7,9] have observed that the work done by the rolling motion is driving the oscillation during the central loop since the aerodynamic motion acts in the direction of wing rolling motion whereas during the two reverse outer loops the oscillation is being damped. In this paper, we will theoretically analyze this hysteresis mechanism instead of physical insight. In this work, inspired by the hysteresis mechanism of driving wing-rock motion, we will directly apply hysteresis compensation or reduction methods to suppress wing rock. The conception of the phaser proposed by Cruz and Hernandez [2] is adopted to design variable phase control (VPC) schemes. The main objective of this paper is to study the VPC to suppress wing rock with hysteresis. The rate of energy change of wing rock is derived to analyze the hysteresis mechanism of driving wing rock. The hysteresis loops, based on a defined critical angle, are divided into two parts: a center loop and two reverse outer loops. We develop a VPC scheme to compensate the hysteresis effects of wing rock
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for each part. To verify the effectiveness and robustness of the proposed method, we will demonstrate the several cases of wing rock suppression. Simulation results show that the proposed control scheme can quickly suppress wing rock at various AOAs and any initial conditions.
2. Wing rock model The phase plane representation of wing rock shows that wing rock phenomenon is dominated by nonlinear damping and a relationship can be established with one-degree-offreedom analytical models [3]. The differential equation of describing wing rock is given by [6,11,14] 2 Sb/2Ixx )Cl + u (1) φ¨ = (ρU∞ where φ(t) is the roll angle, ρ is the density of air, U∞ is the freestream velocity, S is the wing reference area, b is the chord, Ixx is the mass moment of inertia, u(f ) is the control input, and Cl is the rolling moment coefficients written as ˙ φ˙ + b4 φ 3 + b5 φ 2 φ. ˙ Cl = b0 + b1 φ + b2 φ˙ + b3 |φ| (2) The aerodynamic parameters bi (i = 0, 1, 2, 3, 4) are the time-varying functions of AOA. Substituting (2) into (1), we have [10] ˙ φ˙ + a3 φ 3 + a4 φ 2 φ˙ = u (3) φ¨ + a0 φ + a1 φ˙ + a2 |φ| where ai (i = 0, 1, 2, 3, 4) are the parameters relative to free-to-roll experiment conditions [3,4]. A typical set of coefficients ai (at Reynolds number = 636 000) is depicted in Fig. 1. To illustrate the behaviors of wing rock, the uncontrolled wing-rock motion at AOA = 32.5◦ with the initial condition ˙ = 0 is demonstrated in Fig. 2, which φ(0) = 0.1◦ and φ(0) shows a small initial disturbance is enough to cause wing rock.
Fig. 1. Coefficients ai in Eq. (3).
3. Hysteresis analysis of wing rock The uncontrolled wing-rock model in (3) can be written in the following form: ˙ + a4 φ 2 φ˙ + (a0 φ + a3 φ 3 ) = 0. φ¨ + a1 + a2 |φ| (4) ˙ Eq. (4) is then expressed in a stateLet x1 = φ and x2 = φ; variable form: x˙1 = x2 , (5) x˙2 = −(a1 + a2 |x2 | + a4 x12 )x2 − (a0 x1 + a3 x13 ). Eq. (5) is further expressed in the phase-trajectory equation: dx2 −(a1 + a2 |x2 | + a4 x12 x2 − (a0 x1 + a3 x13 ) = . (6) dx1 x2 By integrating Eq. (6) and substituting x2 = x˙1 we have t 1 2 x + U (x1 ) = C − (7) a1 + a2 |x2 | + a4 x12 x22 dt 2 2 0
˙ Fig. 2. Time history of wing rock at φ(0) = 0.1◦ and φ(0) = 0.
where U (x1 ) = (a0 x1 + a3 x13 ) dx1 is the potential energy and C is an integral constant. Define E as the total mechanical energy of the system given by [15] 1 E = x22 + U (x1 ). 2
(8)
Z.L. Liu et al. / Aerospace Science and Technology 10 (2006) 27–35
Fig. 3. Build up phase of the hysteresis at AOA = 32.5◦ .
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1) If x2 = 0 and dE/dt = 0, then x1 = ±Xc . We define this roll angle Xc as a critical angle, marking B and C in Fig. 4. It lies on the boundary between increasing energy and decreasing energy and is an important parameter for designing the VPC in Section 4. If x2 = 0 and dE/dt = 0, then x1 = ±φmax ; we mark them as A and D in Fig. 4. 2) If x2 = 0 and |x1 | < Xc , then dE/dt > 0; this means that the wing-rock motion is absorbing energy and developing a single-loop hysteresis in a clockwise direction, as shown in Fig. 3. If x2 = 0 and |x1 | > Xc , then dE/dt < 0; this implies that the motion dissipates energy and develops two additional outer-loop hysteresis as in Fig. 4 where the direction of the motion is anticlockwise. 3) Similarly, the role of x2 can be analyzed because the relationship of x1 and x2 is constrained by wing rock to limit cycles. If |x2 | > −(a1 + a4 x12 )/a2 or |x2 | < −(a1 + a4 x12 )/a2 , then dE/dt > 0 or dE/dt < 0, respectively. With Eq. (4), it is easy to see that in this case the roll damping has a variable sign. Based on the above analysis, we conclude that the condition of wing rock developing into LCO is the hysteresis that has one center loop with two reverse outer loops.
4. Variable phase control 4.1. Phase control
Fig. 4. The hysteresis of wing rock at AOA = 32.5◦ .
By differentiating Eq. (7), we obtain the rate of change of total energy E: dE = − a1 + a2 |x2 | + a4 x12 X22 . (9) dt Eq. (9) shows that the total energy E is not constant and increases or decreases according to the sign of a1 + a2 |x2 | + a4 x12 x22 . We now turn our attention to the hysteresis of wing rock. Substituting (3) into (1), we obtain a Cl − φ relationship to illustrate the hysteresis ˙ φ˙ + a3 φ 3 + a4 φ 2 φ˙ /k (10) Cl = − a0 φ + a1 φ˙ + a2 |φ| 2 Sb/2I . where k = ρU∞ xx We simulate the build-up phase of the hysteresis in Fig. 3 and the full hysteresis in Fig. 4, and draw arrows indicating the developing direction of hysteresis over time, assuming ˙ = 0, AOA = 32.5◦ , initial conditions φ(0) = 0.1◦ and φ(0) and k ≈ 1/30. Finally, we analyze the change of dE/dt as well as the relationship between dE/dt and the hysteresis. Note that a1 < 0, a2 > 0, and a4 > 0 (except AOA = 45◦ ) in Fig. 1.
In general, the effects of hysteresis can be seen as a phase lag between a periodic input and the corresponding output. The natural way of correcting this lag is to reverse its effects. When a periodic input signal with period T is applied to a system with hysteresis, the output signal generally has the same period T as the input signal does, but is shifted in some phase angle [2]. In the wing-rock case, the output signal of wing-rock motion is complex and can be approximated by y(ωt − ϕ) = a0 +
∞ ak cos(kωt) + bk sin(kωt) . k=1
If this signal is referenced to its input u(t) (see Fig. 5), all the components of y(t) are shifted by some phase angle. It is possible to speak of phase shift between the input and output. Here we introduce the concept of the phaser. Definition [2]. A phaser Lph is an operator that shifts a periodic input signal by a constant angle ϕ 0 and has unity gain. In the frequency domain, the phaser can be expressed as
Fig. 5. Phase control scheme.
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Lph (j ω) = p + j q,
Lph (j ω) = 1,
and Lph (j ω) = φ.
(11)
The basic idea of phase control can be explained with a diagram as drawn in Fig. 5, where a phaser Lph is connected in series with a plant. If the plant is with hysteresis, the phase of the output signal y(t) will be shifted by −φs < 0 with respect to the control input signal u(t); if the phaser is used as a controller, then the phase of the control input signal u(t) will be shifted by ϕ > 0 with respect to the input signal r(t). Suppose we design this phaser with ϕ ≈ ϕ, the compensation can be obtained because one block cancels the other. 4.2. VPC When wing rock develops into the LCOs, it will exhibit three loops of hysteresis (see Fig. 4): a center loop and two reverse outer loops. The loop in the center resembles singleloop hysteresis with saturation. The two outer loops are symmetric and have a reverse phase angle. Therefore, the overall hysteresis compensation involves two kinds of phasers to provide a positive angle ϕ1 > 0 for the center loop and a negative angle ϕ2 < 0 for the outer loops. These angles can be obtained by considering each loop independently from the other. A VPC, based on the input signal |r| to switch from one phaser to the other, is given by Lph1 (j ω)r(j ω) if |r| Xc or dE/dt 0, (12) u(j ω) = Lph2 (j ω)r(j ω) if |r| > Xc or dE/dt < 0 where Lph1 (j ω) = ϕ1 > 0, Lph2 (j ω) = ϕ2 < 0, and Xc is the critical angle defined in Section 3. It should be noted that when |r| > Xc , the phaser Lph2 (j ω) is introduced in order to speed the oscillation attenuation even if in this situation dE/dt < 0. Obviously, the single-loop hysteresis during the build-up phase of wing rock like in Fig. 3 is a special case of the VPC. 4.3. VPC implementation
ω = [ω1 ω2 ]. In other words, if we know the phase angles ϕ1 and ϕ2 we can calculate all of the coefficients of Eq. (13), which is described in detail in the Appendix. To design the phaser, the angle ϕ is the sole parameter we need to know. In a close-loop nonlinear system, it is difficult to find this angle. In general, the bigger the angle ϕ is, the faster will the system response be, but a too large value of ϕ can cause instability. If the smaller ϕ is chosen, wing rock motion will develop into smaller LCOs. One simple method is to choose a big ϕ and then to adjust this angle by evaluating control performance. As for the frequency range ω = [ω1 ω2 ] of compensation, because lower harmonic components usually have bigger coefficients, a low frequency range should be selected in order to obtain a better phase compensation. In this study, we choose a six-order linear approximation for phaser ϕ1 = 36◦ in the frequency range [ω1 ω2 ] = [0.01 1] and design it as Lph1 (s) = (s 6 + 1.652s 5 + 0.7709s 4 + 0.1259s 3 + 0.00745s 2 + 0.00015s + 0.00000075) × (s 6 + 3.017s 5 + 2.32s 4 + 0.5477s 3 + 0.05108s 2 + 0.0015s + 0.00001579)−1 . (14) The Bode Diagrams of phaser ϕ1 = 36◦ are plotted in Fig. 6. As to the phaser ϕ2 , for simplicity, let the phaser ϕ2 = −ϕ1 = −36◦ ; similarly, we have Lph2 (s) = (s 6 + 1.652s 5 + 0.7709s 4 + 0.1259s 3 + 0.00745s 2 + 0.00015s + 0.00000075) × (s 6 + 0.7665s 5 + 0.2658s 4 + 0.02618s 3 + 0.001122s 2 + 0.000012s + 0.00000004)−1 . (15) Note that the different phasers have only the different values of the qi ’s in the denominator. We should point out the order of linear approximation usually depending on the value of phaser ϕ. If ϕ is big, we
The definition of the phaser in Eq. (11) is an ideal one. We can approximate implementation by applying two important properties of the phaser [2]: superposition and noncausality. Superposition follows from linearity. Noncausality prevents direct online implementation. Thus, two linear filters are adopted to approximate the two phasers as follows: ⎧ s n + p1,n−1 s n−1 + · · · + p1,1 s + p1,0 ⎪ ⎪ (s) = L ⎪ ph1 ⎪ ⎪ s n + q1,n−1 s n−1 + · · · + q1,1 s + q1,0 ⎪ ⎪ ⎨ for ϕ1 > 0, (13) ⎪ s n + p2,n−1 s n−1 + · · · + p2,1 s + p2,0 ⎪ ⎪ ⎪ Lph1 (s) = n ⎪ ⎪ s + q2,n−1 s n−1 + · · · + q2,1 s + q2,0 ⎪ ⎩ for ϕ2 < 0 where the coefficients pi,n and qi,n are determined such that the phase of the filters is almost constant around the design parameters ϕ1 and ϕ2 within the input signal frequency range
Fig. 6. Bode diagrams of phaser ϕ1 = 36◦ and [ω1
ω2 ] = [0.01
1].
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need a high order linear approximation; otherwise, it will affect the compensation effect. Besides, the VPC method we consider above focuses on the phase shift or compensation of the output signal, whereas ignoring the magnitude compensation of the signal. The magnitude compensation will be discussed in Section 5.
5. The implementation of the VPC scheme of wing rock Before implementing the VPC scheme to suppress wing rock phenomenon, we need to address two issues: the input signal of the phaser and the magnitude compensation of the phaser. Input signal of the phaser: The hysteresis in Fig. 4 is seen as the phase shift between the rolling moment and the roll angle. However, it is not the input-output relationship of the wing-rock model given in Eq. (3); we cannot directly use the VPC technique without the rolling moment information. Fortunately, we could obtain the term kCl from Eq. (10) as the input signal of the phaser.
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6. Simulation results The control objective in this paper is to suppress wing rock. This means to achieve the roll angle φ = 0 and the roll rate φ˙ = 0 when wing rock phenomenon occurs and commanded signal is φ = 0 and φ˙ = 0. The following six cases are used to verify the performances of the VPC schemes. Case 1: AOA = 32.5◦ , kp = 1, ϕ1 = −ϕ2 = 36◦ . To compare the performances achieved with the linearizing control law (u = −kCl ) and with the overall control law (u = −kCl + uph ), the uncontrolled wing-rock motion is allowed to exhibit some LCOs, and then the controllers are activated to suppress wing rock, as shown in Fig. 8. From this simulation, we have observed that if the system is with u = −kCl , it is unstable; if the system is with u = −kCl + uph , it is stable and can suppress wing rock quickly.
The magnitude compensation of the phaser: As mentioned earlier, the phaser only corrects the phase distortion of the system whereas ignoring the magnitude distortion. To suppress wing rock completely, both the phase and the magnitude need to be compensated. For this purpose, a phaser gain, kp 1, is applied in the control system. In general, the bigger the phaser gain kp is, the faster will the system response be, but a too large value of kp can cause overshoot, or even instability. In addition, we have observed that the system with a big kp is more sensitive to the output measurement noise than the system with a small kp . Control structure: In Fig. 5, the phaser is connected in series with the plant, which forms an open-loop control system. To suppress wing rock, we need the closed-loop control system and then propose the VPC scheme of wing rock, as shown in Fig. 7. In this control structure, phasers Lph1 (s) and Lph2 (s) are switched depending on the sign of dE/dt, kp is the phaser gain, and kCl is the input signal of the phaser. The overall control law in Fig. 7 can be expressed by the sum of two terms: the linearizing control law and the phaser control law with magnitude compensation, i.e., u = −kCl + uph .
(a)
(b)
Fig. 7. Structure of wing-rock control system.
Fig. 8. The comparison of performances with different control laws. (a) Responses with the control law u = −kCl . (b) Responses with the control law u = −kCl + uph .
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(a)
(b)
(c)
(d)
Fig. 9. Variable phase control for Case 2. (a) Responses in hysteresis. (b) Responses on phase plane. (c) The rate of change of energy. (d) Control input.
Case 2: The same as Case 1 except kp = 6. As Fig. 8(b) in Case 1 was simulated, we observed that the control system is unstable in some initial conditions where they imply dE/dt > 0, for example, φ = 0 and φ˙ = φ˙ − max . If we modify the phaser gain from kp = 1 to kp = 6, then the VPC can guarantee a stable control system and the system states tend towards zero over time, as shown in Figs. 9(a) and 9(b). Fig. 9(c) shows the corresponding dE/dt decreasing to zero over time, which reveals that the VPC essentially makes the system energy dissipative. Fig. 9(d) displays the corresponding control input u(t). If we compare this u(t) with the umax (t) ≈ −0.9 of Ref. [15], we have observed that the VPC scheme needs only a very small control input, which is important for aircraft control at a high AOA. Case 3: The same as in Case 2 except AOAs. This case will verify the robustness of the proposed method. Assume the initial condition of wing rock to be φ(0) = 5◦ ˙ and φ(0) = 0. The controller can be activated at any time, say at t = 800 time steps. Although the controller parameters remain the same as Case 2, the VPC can suppress the different wing rock motions at other fixed AOAs, i.e.,
AOA = 25◦ , 27.5◦ , 30◦ , 35◦ , 37.5◦ , 40◦ , 42.5◦ , or 45◦ , as shown in Fig. 10. Case 4: Single-phaser control (AOA = 32.5◦ , k1 = 6, ϕ1 = 36◦ ). This simulation illustrates a special case. We hope to suppress wing rock at the beginning of wing rock motion. In fact, single phaser ϕ1 = 36◦ can easily achieve this aim because the wing rock at this time exhibits a singleloop hysteresis like in Fig. 3. If the phaser is activated at t = 600 time steps, Fig. 11(a) shows the comparison outputs between the control system and uncontrolled system, and Fig. 11(b) indicates the corresponding change of dE/dt. From Fig. 11(c), we have observed that once the phaser is activated, the direction of hysteresis loops will change from clockwise (the system absorbing energy) to anticlockwise (the system dissipating energy). We mark S as a starting point and draw arrows indicating its direction. Similarly, in Figs. 8 and 9(a), regardless of the starting point at the center loop or the outer loops of hysteresis, the hysteresis development in all cases is forced to move in anticlockwise direction until the wing rock is completely suppressed.
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(a)
(b)
Fig. 10. Output responses (φ (deg) vs. time steps) at different AOAs.
Case 5: Comparison of different order phasers (AOA = 32.5◦ , kp = 1, ϕ1 = −ϕ2 = 36◦ ). This case illustrates the performance comparison between the six-order phaser approximation in (14) and (15) and the one-order phaser approximation which is designed as Lph1 (s) = (s + 0.0075)/ (s + 0.0285) for phaser ϕ1 = 36◦ and Lph2 (s) = (s + 0.0075)/(s + 0.00195) for phaser ϕ1 = −36◦ . Assume the ˙ = 0. We have siminitial condition is φ(0) = 42◦ and φ(0) ulated the wing rock suppression with above two kinds of phaser approximations and observed that the response of the system with the one-order phaser approximation is too much slower than the response of the system with the six-order phaser approximation, as shown in Figs. 12(a) and 12(b), respectively. Case 6: System output with the measurement noise. Lastly, we illustrate a case of the VPC with the output measurement
(c) Fig. 11. Phase control for Case 4. (a) Comparison of output responses. (b) The rate of change of energy. (c) Responses in hysteresis.
noise. Assume a band limited white noise disturbance to be added to the system output variables, for example, φ with the noise (its power is 10−4 , sample time is 0.05, and seed is 23341) and φ˙ with the noise (its power is 10−6 , sample time is 0.05, and seed is 23 341). The other parameters are the same as we simulate Fig. 12(b). Fig. 13 shows that the response of the control system is almost the same as Fig. 12(b)
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(a)
Fig. 13. Wing rock suppression with the measurement noise.
small control power need, and good robustness and suppression effect. Therefore, the proposed VPC scheme may be the effective approach for suppressing wing rock. For future work, we will consider wing rock dynamics integrating with a lateral flight dynamics and deal with the suppressing and tracking problems of time-varying wing rock with disturbances.
Appendix A. A six-order approximate phaser The form of the six-order linear filter is given by Lpa (s) = (b) Fig. 12. Response comparison of different order phaser approximations. (a) Responses with the first-order phaser approximations. (b) Responses with the six-order phaser approximations.
except the small error at the equilibrium because the rate of signal and noise is lower. This result implies that the control system can work well with the output measurement noise.
s 6 + p 5 s 5 + · · · + p 1 s + p0 . s 6 + q5 s 5 + · · · + q1 s + q0
(A.1)
For a given frequency ω = [ω1 ω2 ] and a phase angle ϕ, the location for the pole and the zero can be found. The location of the first zero z1 is located somewhere behind the location of ω1 ; for example, z1 = ω1 − ω1 /4. The remainder of the zeros will be located between ω1 and ω2 . They are logarithmically equally spaced between ω1 and ω2 , as shown in Fig. A.1. The values of pi ’s are obtained using the values of the zeros num(s) = (s + z1 )(s + z2 )(s + z3 )(s + z4 )(s + z5 )(s + z6 )
7. Conclusions The variable phase control (VPC) scheme has been developed to suppress wing rock with hysteresis at various AOAs and any initial conditions. After analyzing the energy mechanism of wingrock hysteresis, and we have designed the VPC scheme to compensate the phase and magnitude distortions for each part of the hysteresis. Simulation studies show that the proposed control scheme, in nature, can guarantee wing-rock motion always to dissipate energy such that the wing-rock phenomenon can be suppressed. The main advantages of proposed method are simple design and calculation,
= s 6 + p 5 s 5 + p4 s 4 + p3 s 3 + p2 s 2 + p1 s + p 0 . (A.2) Substituting s = j ω into (A.1), we have
Fig. A.1. Six zeros logarithmical equal space.
Z.L. Liu et al. / Aerospace Science and Technology 10 (2006) 27–35
Lph (j ω) = (j ω)6 + p5 (j ω)5 + p4 (j ω)4 + p3 (j ω)3 + p2 (j ω)2 + p1 j ω + p0 × (j ω)6 + q5 (j ω)5 + q4 (j ω)4 + q3 (j ω)3 −1 + q2 (j ω)2 + q1 j ω + q0 = (−ω6 + p4 ω4 − p2 ω2 + p0 ) + j (p5 ω5 − p3 ω3 + p1 ω) × (−ω6 + q4 ω4 − q2 ω2 + q0 ) −1 + j (q5 ω5 − q3 ω3 + q1 ω) = (K1 K3 + K2 K4 ) + j (K2 K3 − K1 K4 ) × (K32 + K42 )−1
(A.3)
where K1 = −ω6 + p4 ω4 − p2 ω2 + p0 , K2 = p5 ω5 − p3 ω3 + p1 ω, K3 = −ω6 + q4 ω4 − q2 ω2 + q0 , and K4 = q5 ω5 − q3 ω3 + q1 ω. The phase angle of the transfer function is K2 K3 − K1 K4 Lph (j ω) = Φ(ω) = tan−1 . (A.4) K1 K3 + K2 K4 Substituting the values of K3 and K4 , into (A.4) we obtain (−ω6 + q4 ω4 − q2 ω2 + q0 ) tan Φ(ω) K1 − K2 + (q5 ω5 − q3 ω3 + q1 ω) tan Φ(ω) K2 + K1 = 0. Let K5 = tan(Φ(ω))K1 − K2 and K6 = tan(Φ(ω))K2 + K1 ; we have −K5 ω6 + K5 q4 ω4 − K5 q2 ω2 + K5 q0 + K6 q5 ω5 − K6 q3 ω3 + K6 q1 ω = 0.
(A.5)
The unknown variables in (A.5) are the qi ’s and ω. We can make the value of the phase angle equal to ϕ at the values where the zeros are located. If ω = [ω1 ω2 ], then we have Lpa (j ω) = Φ(ω) = ϕ and tan(Φ(ω)) = tan(ϕ). The following set of equations are then obtained ⎧ ⎪ −K7 z16 + K7 z14 q4 − K7 z12 q2 + K7 q0 + K8 z15 q5 ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ − K8 z1 q3 + K8 z1 q1 = 0, ⎪ ⎪ ⎪ ⎪ −K7 z26 + K7 z24 q4 − K7 z22 q2 + K7 q0 + K8 z25 q5 ⎪ ⎪ ⎪ ⎪ ⎪ − K8 z23 q3 + K8 z2 q1 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ −K7 z36 + K7 z34 q4 − K7 z32 q2 + K7 q0 + K8 z35 q5 ⎪ ⎪ ⎪ ⎪ ⎨ − K8 z3 q3 + K8 z3 q1 = 0, 3 (A.6) 6 ⎪ −K7 z4 + K7 z44 q4 − K7 z42 q2 + K7 q0 + K8 z45 q5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − K8 z43 q3 + K8 z4 q1 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ −K7 z56 + K7 z54 q4 − K7 z52 q2 + K7 q0 + K8 z55 q5 ⎪ ⎪ ⎪ ⎪ ⎪ − K8 z53 q3 + K8 z5 q1 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −K7 z66 + K7 z64 q4 − K7 z62 q2 + K7 q0 + K8 z65 q5 ⎪ ⎪ ⎪ ⎩ − K8 z63 q3 + K8 z6 q1 = 0
35
where K7 = tan(ϕ)K1 − K2 and K8 = tan(ϕ)K2 + K1 . Eq. (A.6) can be re-ordered in the form Ax = C ⎤ ⎡ K7 K8 z1 −K7 z12 −K8 z13 K7 z14 K8 z15 ⎧ q0 ⎫ ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎢ ⎪ ⎪ q1 ⎪ ⎢ K7 K8 z2 −K7 z22 −K8 z23 K7 z24 K8 z25 ⎥ ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎢ ⎪ ⎨q ⎪ ⎬ ⎢ K7 K8 z3 −K7 z2 −K8 z3 K7 z4 K8 z5 ⎥ ⎪ 2 ⎢ 3 3 3 3⎥ ⎥ ⎢ ⎢ K7 K8 z4 −K7 z2 −K8 z3 K7 z4 K8 z5 ⎥ ⎪ q3 ⎪ ⎪ 4 4 4 4 ⎥⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎥ ⎢ ⎪ 3 5 2 4 ⎪ ⎪ q 4 ⎪ ⎣ K7 K8 z5 −K7 z5 −K8 z5 K7 z5 K8 z5 ⎦ ⎪ ⎪ ⎩ ⎪ ⎭ 3 5 2 4 q 5 K7 K8 z6 −K7 z6 −K8 z6 K7 z6 K8 z6 ⎧ ⎫ K7 z16 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 6⎪ ⎪ ⎪ K z ⎪ ⎪ 7 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ K7 z6 ⎪ ⎬ 3 . (A.7) = 6 ⎪ ⎪ K7 z4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ K7 z56 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 6 K7 z6 Now, the values qi ’s can be obtained. References [1] A.S. Arena Jr., R.C. Nelson, Experimental investigation on limit cycle wing rock of slender wings, J. Aircraft 31 (1994) 1148–1155. [2] J.M. Cruz-Hernandez, V. Hayward, Phase control approach to hysteresis reduction, IEEE Trans. on Control Systems Technology 9 (2001) 17–26. [3] G. Guglieri, F. Quagliotti, Experimental observation and discussion of the wing rock phenomenon, Aerospace Sci. Technol. 2 (1997) 111– 123. [4] G. Guglieri, F. Quagliotti, Analytical and experimental analysis of wing rock, Nonlinear Dynamics 24 (2001) 129–146. [5] C. Hsu, C.E. Lan, Theory of wing rock, J. Aircraft 22 (1985) 920–924. [6] S.V. Joshi, A.G. Sreenatha, Chandrasekhar, Suppression of wing rock of slender delta wings using a single neuron controller, IEEE Trans. on Control System Technology 6 (1998) 671–677. [7] J. Katz, Wing/vortex interactions and wing rock, Progr. Aerospace Sci. 35 (1999) 727–750. [8] J. Katz, D. Levin, Self-induced roll oscillations measured on a delta wing/canard configuration, J. Aircraft 23 (1986) 814–819. [9] P. Konstadinopoulos, D.T. Mook, A.H. Nayfeh, Subsonic wing rock of slender delta wings, J. Aircraft 22 (1985) 223–228. [10] Z.L. Liu, C.-Y. Su, J. Svoboda, Control of wing rock phenomenon using fuzzy PD controller, in: IEEE Int. Conf. on Fuzzy System, MO, USA, May 25–28, 200. [11] J. Luo, C.E. Lan, Control of wing rock motion of slender delta wings, J. Guidance, Control, and Dynamics 16 (1993) 225–231. [12] A.H. Nayfeh, J.M. Elzebda, D.T. Mook, Analytic study of the subsonic wing rock phenomenon for slender delta wings, J. Aircraft 26 (1989) 805–809. [13] L.V. Schmidt, Wing rock due to aerodynamic hysteresis, J. Aircraft 16 (1979) 129–133. [14] S.N. Singh, W. Yim, W.R. Wells, Control of wing rock motion of slender delta wings, J. Guidance, Control, and Dynamics 18 (1995) 25–30. [15] S.-S. Wanda, The Behaviour of Nonlinear Vibrating Systems, Kluwer Academic, 1990.