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FLIGHT TEST OF IN-FLIGHT SIMULATOR CONTROLLER FOR SIMULTANEOUS SIMULATION OF GUST RESPONSE AND HANDLING RESPONSE
FLIGHT TEST OF IN-FLIGHT SIMULATOR CONTROLLER FOR SIMULTANEOUS SIMULATION OF GUST RESPONSE AND HANDLING RESPONSE Masayuki Sato ∗ Atsushi Satoh ∗∗ ∗
Institute of Space Technology and Aeronautics, Japan Aerospace Exploration Agency, Mitaka, Tokyo 181-0015, Japan ∗∗ Department of Mechanical Engineering, Iwate University, Morioka, Iwate 020-8551, Japan
Abstract: This paper presents flight test result of In-Flight Simulator (IFS) controller which simulates gust response as well as handling response. To simulate these two responses simultaneously, we first design a robust feedback controller to simulate gust response via scaled H∞ control, we then design a feedforward controller to simulate handling response via H∞ control. We design two controllers for the lateral-directional motions of a research aircraft for two different target aircraft. Hardware-In-the-Loop (HIL) simulations and flight tests with the designed controllers confirm that the controllers achieve our objective; that is, the controllers simultaneously simulate gust response as well as handling response. c Copyright IFAC 2007. Keywords: Flight test; In-flight simulator; H∞ control; Inverse system.
1. INTRODUCTION In-Flight Simulators (IFSs) have been recognized as one of the useful tools for the research of flying qualities, real aircraft development, etc. (Weingarten, 2005). For this reason, much research for developing IFSs has been conducted for a long time, e.g. (Motyka et al., 1972; Komoda et al., 1998; Hanke and Lange, 1988; Weingarten, 2005; Sato, 2006b). As pointed out in (Weingarten, 2005), to realize real-world cues is the most important design specification when designing controllers of IFSs; that is, the controllers of IFSs are to be designed to simulate the response of target aircraft. However, to authors’ knowledge, there is no research to realize the simultaneous simulation of two responses, handling response and gust response; that is, the existing papers have realized only one of these two responses. To
simulate handling response of the target aircraft is very important for IFSs, however, it is almost impossible to avoid wind gust in flight tests. Further, gust response simulation is also very important for the research of handling qualities in special environment, e.g. approaching and landing phases. Therefore, to simulate these two responses simultaneously has been desired for IFSs. In this paper, we apply the method proposed in (Sato, 2006a), which has been proposed for realizing arbitrary maneuverability and gust rejection, to IFS controller design for the simultaneous simulation of handling response and gust response. We design two IFS controllers for the LateralDirectional (L/D) motions of a research aircraft of Japan Aerospace Exploration Agency (JAXA), MuPAL-α, which is shown in Fig. 1, for two different target aircraft, and their Hardware-In-the-
troller, the robustness against the modeling uncertainties is not considered. This is because it is very difficult to choose appropriate weighting functions for the feedforward controller design under some uncertainties. Therefore, when designing the feedforward controller, we do not take into account of the uncertainties, however, after designing it, we check the robust performance against the uncertainties. Similar to (Sato, 2006a), the designed controllers in this paper are also confirmed to have robust performance against the uncertainties. Fig. 1. Research aircraft MuPAL-α Loop (HIL) simulations and flight tests confirm that our method is very effective to achieve our objective; that is, our controllers realize the simultaneous simulation of gust response and handling response. The contribution of this paper is to demonstrate that MuPAL-α can be used as an IFS for the L/D motions of several target aircraft with the simultaneous simulation of gust response as well as handling response. This paper is organized as follows: First we briefly describe our design method with showing two design examples for the L/D motions of MuPALα. The method is based on the method proposed in (Sato, 2006a), therefore, the details including plant model data, actuator model data, etc. are omitted because they are all included in (Sato, 2006a). We next show experimental results of HIL simulations and flight tests of the controllers. Finally, we summarize this paper. In the following, diag(X1 , · · · , Xk ) denotes a block diagonal matrix composed of X1 , · · · and Xk .
2. CONTROLLER DESIGN FOR L/D MOTIONS OF MUPAL-α We apply the method proposed in (Sato, 2006a), which has been proposed to realize arbitrary maneuverability and gust rejection, to the simultaneous simulation of handling response and gust response. The method is based on model-matching controller design via H∞ control. We design two L/D IFS controllers for the linearized motions of MuPAL-α at an altitude 1520 [m] and a true airspeed 66.5 [m/s] for two different target aircraft; Boeing 747 at an altitude 0 [m] and a true airspeed 67.4 [m/s] and Convair 880M at an altitude 0 [m] and true airspeed 68.9 [m/s] (Heffley and Jewell, 1972). Our controller design method is as follows: First, a feedback controller to simulate gust response, which is robust against modeling uncertainties existing in high frequency range, is designed via scaled H∞ control, then a feedforward controller to simulate handling response is designed via H∞ control. When designing the feedforward con-
2.1 Feedback Controller Design We first design feedback controllers, which are robust against plant uncertainties existing in high frequency range, via H∞ model matching problem. The uncertainties are modeled as delay models, in which each delay time belongs to an interval [0.08, 0.16] [s], at plant inputs with their first-order Pad´e approximations. Further, they are represented as LFT (Δd, Pd ), where LFT denotes linear fractional transformation, Δd is a 2 × 2dimensional diagonal matrix and its entries are normalized as unities, and Pd is an appropriately defined system. Fig. 2 shows the block diagram for designing feedback controller K. In this figure, uF B denotes T the feedback input vector, w1T wdT denotes the disturbance input vector for the generalized plant T Pg , z1T zdT denotes the performance output vector for the generalized plant Pg , y denotes the measurement output vector. In the generalized plant Pg , P denotes plant model, Pa denotes firstorder actuator model, M denotes the model of target aircraft to simulate, W1 denotes a weighting function to specify the model-matching performance of K and its frequency range. In the ˜ Δp denotes a augmented uncertainty block Δ, 2×2-dimensional full block fictitious uncertainties for model-matching. Plant model P has four state T variables x = [vi p φ r] ; inertial side velocity (vi ), roll rate (p), roll angle (φ) and yaw rate (r), one disturbance input w1 = vg ; sideway gust input T (vg ), two inputs uF B = [δa δr ] ; aileron input (δa ) and rudder input (δr ), two controlled outputs z1 = [va φ]; side air speed (va ) and roll angle, and four measurement outputs y = [va p φ r]T . We set W1 as follows. ⎧ 2s2 + s + 1.1 ⎪ ⎨γ diag (1, 0.81) , B747 s2 + 5.3s + 1.1 W1 : , 2 ⎪ ⎩ γ 2s + s + 2.2 diag (1, 0.67) , C880M s2 + 3.9s + 2.2 (1) where γ is a design parameter to be maximized because the larger γ means the better modelmatching performance. These weighting functions
~ Δ
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wd
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Fig. 2. Block diagram for designing feedback controller K
6.3040 × 10−3 0.23147 0.10042 0.81281 9.3628 × 10−4 −0.31419 0.94285 2.4858
0.016274 0.537941 0.24957 0.92461 −3.6195 × 10−3 0.099165 1.7458 3.5892
, B747,
, C880M,
diag 6.5821 × 10−4 I2 , 0.10534, 0.029202 , B747,
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Fig. 4. Bode plots from vg to va and φ with Kopt for C880M model 2.2 Feedforward Controller Design
(2)
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Fig. 3. Bode plots from vg to va and φ with Kopt for B747 model
φ
Similar to (Sato, 2006a), we design (local) optimal static output feedback controller (Kopt ) for the generalized plant Pg because the on-board computers have limited memories. However, note that our design approach is directly applicable for designing dynamical controllers, which are worth considering for better performance with more computing resources. The obtained maximum γ is as follows: 1.7698 (B747 model) and 1.4273 (C880M model). The corresponding controller Kopt and scaling matrix are respectively shown in (2) and (3):
closed-loop system (four lines) P B747
-40 -60 180
Frequency [rad/s]
Phase [deg]
respectively have their lowest gain at the frequency of the Dutch-roll mode of the each target aircraft.
0 -20
diag 9.4684 × 10−4 I2 , 0.022297, 8.1526 × 10−3 , C880M.
(3) Figs. 3 and 4 show the Bode plots of the closedloop system, plant, and corresponding models from vg to va and φ. In these figures, we draw four lines for the closed-loop system (we hardly distinguish them in this scale); the combinations of maximum delay time and minimum delay time of aileron input and rudder input. The designed controllers confirm that they have good modelmatching performance in [0.2, 2] [rad/s] against the prescribed uncertainties. Figs. 5 and 6 show the responses of step-type gust input for the closed-loop system and the corresponding models. Similar to Figs. 3 and 4, we draw four lines for the closed-loop system (we hardly distinguish them in this scale). The controllers almost simulate gust responses of their target aircraft.
We next design feedforward controllers to simulate handling responses without considering the plant uncertainties. We design a pseudo right inverse system based on H∞ norm used in (Sato, 2006b), in which controllers designed using the method confirmed to have good model-matching performance in flight tests. Fig. 7 shows the block diagram for designing the feedforward controller F . In this figure, w2 denotes model outputs, z2 denotes the fictitious performance output, which is weighted by weighting function W2 , to design a pseudo right inverse system of the closed-loop system Pcl . Weighting function W2 specifies the right inverse performance and its frequency range. In the closedloop system Pcl , Pdn denotes the nominal delay model (nominal delay time is set as 0.12 [s] for both aileron and rudder inputs), Kopt denotes the designed feedback controller. For both cases of B747 simulation controller and C880M simulation controller, we set W2 as follows. 10 (4) I2 s + 0.10 This function is set for F to satisfy the same design specifications used in (Sato, 2006a). We design full-order (tenth-order) controller F . Due W2 :
1
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Fig. 5. Step responses with Kopt for B747 model (numerical simulation)
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Fig. 6. Step responses with Kopt for C880M model (numerical simulation) w2 F Pcl
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Fig. 7. Block diagram for designing feedforward controller F to lack of space, we omit the state-space representation of F . Figs. 8 and 9 respectively show the Bode plots of the closed-loop system Pcl in Fig. 7, the augmented system composed of the closed-loop system Pcl , F and the corresponding model M , and corresponding models from model inputs to va and φ. In these figures, we draw five lines for the augmented system, which is composed of Pcl , F and M , with setting the nominal delay model Pdn as follows; the nominal delay time, and four combinations of maximum delay time and minimum delay time of aileron input and rudder input. The designed controllers confirm that they have good model-matching performance under the prescribed uncertainties. Next, we exactly check their robust performance via μ analysis. Fig. 10 shows the block diagram for checking the robust performance of F . In this figure, wd and zd are the same signals used in Fig.
Pcl FM (five lines) Pcl B747
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Fig. 9. Bode plots of Pcl F M for C880M model Δp
F
wd
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Fig. 10. Block diagram for checking the robust performance of F 2, w2 and z2 are the same signals used in Fig. 7, and W3 denotes the weighting function to evaluate the robust performance of F and its frequency range. We first set W2 used in designing F as W3 . In Figs. 11 and 12, the upper bounds of μ using W2 are respectively shown for B747 simulation controller and C880M simulation controller. However, both maxima of the upper bounds exceed unity; that is, we cannot conclude that the designed controllers have robust performance against the prescribed uncertainties. After much trial and error, we use (5), which is shown at the top of the next page, as W3 . In Figs. 11 and 12, the upper bounds of μ using W3 in (5) are respectively shown for B747 simulation controller and C880M simulation controller. Both maxima of the upper bounds are
10
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Fig. 11. upper bound of Fig. 12. upper bound of μ for B747 simulaμ for C880M simution controller lation controller less than unity; that is, the controller have robust right inverse performance which is defined by the weighting function in (5). The use of W3 concludes that the feedforward controller F has robust performance defined as follows. • B747 · ω ∈ [0, 0.13], 1/ζ(ω) = 0.02, · ω ∈ (0.13, 0.39], 1/ζ(ω) = 0.1, · ω ∈ (0.39, 0.59], 1/ζ(ω) = 0.2, and · ω ∈ (0.59, 2], 1/ζ(ω) = 0.28 • C880M · ω ∈ [0, 0.15], 1/ζ(ω) = 0.02, · ω ∈ (0.15, 0.60], 1/ζ(ω) = 0.1, · ω ∈ (0.60, 1.09], 1/ζ(ω) = 0.2, and · ω ∈ (1.09, 2], 1/ζ(ω) = 0.28, where 1/ζ(ω), roughly speaking, denotes the maximum error from ideal inverse system (for the precise definition of 1/ζ(ω), see (Sato, 2006a)). These results do not satisfy the specifications when designing F , however, the degradation is not so big; that is, the designed controller F are both confirmed to have robust right inverse performance under the prescribed uncertainties.
3. EXPERIMENT
10 5 0 -5 10 5 0 -5 5 0 -5 -10 5 0 -5 -10 2 1 0 -1 -2 6 4 2 0 -2 2 1 0 -1 5 0
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Fig. 13. Step gust responses in HIL simulation (B747 model) v g [m/s]
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2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 -2 10
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14.945s 5 + 412.82s 4 + 2890.9s 3 + 4435.0s 2 + 2828.1s + 1183.0 I2 , B747 + 30.096s 6 + 257.16s 5 + 675.13s 4 + 1058.3s 3 + 853.30s 2 + 187.06s + 11.841 4 3 2 14.837s + 449.27s + 3213.4s + 3277.2s + 3093.4 I2 , C880M s6 + 32.056s 5 + 276.49s 4 + 550.72s 3 + 832.33s 2 + 403.06s + 31.435 s7
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Fig. 14. Step gust responses in HIL simulation (C880M model) Figs. 15 and 16 respectively show handling responses of B747 simulation controller and C880M simulation controller. Discrepancies are hardly shown in both HIL simulations; that is, our designed controllers simulate handling responses very faithfully.
3.1 HIL Simulations 3.2 Flight Tests We first conducted HIL simulations, in which the motions of MuPAL-α is calculated with six-degree nonlinear equations. Figs. 13 and 14 respectively show gust responses of B747 simulation controller and C880M simulation controller. Although some discrepancies are shown in both HIL simulations, the gains and frequencies of the Dutch-roll mode in va and φ are almost the same as the models; that is, our designed controllers almost simulate gust responses of the target aircraft.
We next conducted flight tests. Figs. 17 and 18 respectively show time histories of B747 simulation flight and C880M simulation flight. In these figures, model data are recalculated with recorded gust input and model inputs which are driven by pilot, because model data in flight are calculated only with model inputs; that is, gust input is not considered. In these flight, the pilot were demanded to make aircraft steadily turn. Although some discrepancies are shown in
φ [deg] p [deg/s] va [m/s] 20
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Fig. 16. Pilot handling responses in HIL simulation (C880M model) both flight tests, va and φ are almost the same as the models; that is, our designed controllers almost simulate gust responses as well as handling responses of the target aircraft in real flight.
4. SUMMARY This paper presents a design method of IFS controllers to simulate gust response as well as handling response. The controller is designed in the following sequence: Feedback controller is first designed to simulate gust response, feedforward controller is then designed to simulate handling response. Feedforward controller is designed without considering plant uncertainties, however, it is confirmed that designed feedforward controller has robust performance against the uncertainties via μ analysis. Our designed controllers are confirmed to have good model-matching performance in hardware-in-the-loop simulations and flight tests.
REFERENCES Hanke, D. and H. H. Lange (1988). Flight evaluation of the ATTAS digital fly-by-wire/light
B747
MuPAL-α
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80
t [s]
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140
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Fig. 17. Time histories of flight test (B747 model) 4 2 0 -2 -4 4 2 0 -2 -4 5
v g [m/s]
C880M
φ [deg] p [deg/s] va [m/s]
ψ [deg]
r [deg/s]
φ [deg]
p [deg/s] va [m/s]
Fig. 15. Pilot handling responses in HIL simulation (B747 model)
0 -5 5 0
δ r [deg] δ a [deg] ψ [deg] r [deg/s]
10
5
v g [m/s]
MuPAL-α
B747
0 -5 20 10 0 -10 -20 4 2 0 -2 -4 100 50 0 -50 4 2 0 -2 -4 2 1 0 -1 -2 0
δ r [deg] δ a [deg] ψ [deg] r [deg/s]
p [deg/s] va [m/s] φ [deg] r [deg/s] ψ [deg] δ r [deg] δ a [deg]
10 5 0 -5 -10 10 5 0 -5 -10 20 10 0 -10 4 2 0 -2 -4 10 5 0 -5 10 5 0 -5 -10 10 5 0 -5 -10 0
C880M
20
40
60
t [s]
80
100
MuPAL-α
120
140
Fig. 18. Time histories of flight test (C880M model) flight control system. Proceedings of ICAS 16th Congress pp. 866–876. Heffley, R. K. and W. F. Jewell (1972). Aircraft handling qualities data. NASA CR-2144 pp. 193–209 and 210–242. Komoda, M., N. Kawahata, Y. Tsukano and T. Ono (1998). VSRA in-flight simulator — its evaluation and applications. AIAA Flight Simulation Technologies Conference, AIAA88-4605-CP. Motyka, P. R., E. G. Rynaski and P. A. Reynolds (1972). Theory and flight verification of the TIFS model-following system. Journal of Aircraft 9, 347–353. Sato, M. (2006a). Flight test of flight controller for arbitrary maneuverability and wind gust rejection. Proc. CCA pp. 2534–2540. Sato, M. (2006b). Flight test of model-matching controller for in-flight simulator MuPAL-α. Journal of Guidance, Control, and Dynamics 29, 1476–1481. Weingarten, N. C. (2005). History of in-flight simulation at general dynamics. Journal of Aircraft 42, 290–298.
Institute of Space Technology and Aeronautics, Japan Aerospace Exploration Agency, Mitaka, Tokyo 181-0015, Japan ∗∗ Department of Mechanical Engineering, Iwate University, Morioka, Iwate 020-8551, Japan
Abstract: This paper presents flight test result of In-Flight Simulator (IFS) controller which simulates gust response as well as handling response. To simulate these two responses simultaneously, we first design a robust feedback controller to simulate gust response via scaled H∞ control, we then design a feedforward controller to simulate handling response via H∞ control. We design two controllers for the lateral-directional motions of a research aircraft for two different target aircraft. Hardware-In-the-Loop (HIL) simulations and flight tests with the designed controllers confirm that the controllers achieve our objective; that is, the controllers simultaneously simulate gust response as well as handling response. c Copyright IFAC 2007. Keywords: Flight test; In-flight simulator; H∞ control; Inverse system.
1. INTRODUCTION In-Flight Simulators (IFSs) have been recognized as one of the useful tools for the research of flying qualities, real aircraft development, etc. (Weingarten, 2005). For this reason, much research for developing IFSs has been conducted for a long time, e.g. (Motyka et al., 1972; Komoda et al., 1998; Hanke and Lange, 1988; Weingarten, 2005; Sato, 2006b). As pointed out in (Weingarten, 2005), to realize real-world cues is the most important design specification when designing controllers of IFSs; that is, the controllers of IFSs are to be designed to simulate the response of target aircraft. However, to authors’ knowledge, there is no research to realize the simultaneous simulation of two responses, handling response and gust response; that is, the existing papers have realized only one of these two responses. To
simulate handling response of the target aircraft is very important for IFSs, however, it is almost impossible to avoid wind gust in flight tests. Further, gust response simulation is also very important for the research of handling qualities in special environment, e.g. approaching and landing phases. Therefore, to simulate these two responses simultaneously has been desired for IFSs. In this paper, we apply the method proposed in (Sato, 2006a), which has been proposed for realizing arbitrary maneuverability and gust rejection, to IFS controller design for the simultaneous simulation of handling response and gust response. We design two IFS controllers for the LateralDirectional (L/D) motions of a research aircraft of Japan Aerospace Exploration Agency (JAXA), MuPAL-α, which is shown in Fig. 1, for two different target aircraft, and their Hardware-In-the-
troller, the robustness against the modeling uncertainties is not considered. This is because it is very difficult to choose appropriate weighting functions for the feedforward controller design under some uncertainties. Therefore, when designing the feedforward controller, we do not take into account of the uncertainties, however, after designing it, we check the robust performance against the uncertainties. Similar to (Sato, 2006a), the designed controllers in this paper are also confirmed to have robust performance against the uncertainties. Fig. 1. Research aircraft MuPAL-α Loop (HIL) simulations and flight tests confirm that our method is very effective to achieve our objective; that is, our controllers realize the simultaneous simulation of gust response and handling response. The contribution of this paper is to demonstrate that MuPAL-α can be used as an IFS for the L/D motions of several target aircraft with the simultaneous simulation of gust response as well as handling response. This paper is organized as follows: First we briefly describe our design method with showing two design examples for the L/D motions of MuPALα. The method is based on the method proposed in (Sato, 2006a), therefore, the details including plant model data, actuator model data, etc. are omitted because they are all included in (Sato, 2006a). We next show experimental results of HIL simulations and flight tests of the controllers. Finally, we summarize this paper. In the following, diag(X1 , · · · , Xk ) denotes a block diagonal matrix composed of X1 , · · · and Xk .
2. CONTROLLER DESIGN FOR L/D MOTIONS OF MUPAL-α We apply the method proposed in (Sato, 2006a), which has been proposed to realize arbitrary maneuverability and gust rejection, to the simultaneous simulation of handling response and gust response. The method is based on model-matching controller design via H∞ control. We design two L/D IFS controllers for the linearized motions of MuPAL-α at an altitude 1520 [m] and a true airspeed 66.5 [m/s] for two different target aircraft; Boeing 747 at an altitude 0 [m] and a true airspeed 67.4 [m/s] and Convair 880M at an altitude 0 [m] and true airspeed 68.9 [m/s] (Heffley and Jewell, 1972). Our controller design method is as follows: First, a feedback controller to simulate gust response, which is robust against modeling uncertainties existing in high frequency range, is designed via scaled H∞ control, then a feedforward controller to simulate handling response is designed via H∞ control. When designing the feedforward con-
2.1 Feedback Controller Design We first design feedback controllers, which are robust against plant uncertainties existing in high frequency range, via H∞ model matching problem. The uncertainties are modeled as delay models, in which each delay time belongs to an interval [0.08, 0.16] [s], at plant inputs with their first-order Pad´e approximations. Further, they are represented as LFT (Δd, Pd ), where LFT denotes linear fractional transformation, Δd is a 2 × 2dimensional diagonal matrix and its entries are normalized as unities, and Pd is an appropriately defined system. Fig. 2 shows the block diagram for designing feedback controller K. In this figure, uF B denotes T the feedback input vector, w1T wdT denotes the disturbance input vector for the generalized plant T Pg , z1T zdT denotes the performance output vector for the generalized plant Pg , y denotes the measurement output vector. In the generalized plant Pg , P denotes plant model, Pa denotes firstorder actuator model, M denotes the model of target aircraft to simulate, W1 denotes a weighting function to specify the model-matching performance of K and its frequency range. In the ˜ Δp denotes a augmented uncertainty block Δ, 2×2-dimensional full block fictitious uncertainties for model-matching. Plant model P has four state T variables x = [vi p φ r] ; inertial side velocity (vi ), roll rate (p), roll angle (φ) and yaw rate (r), one disturbance input w1 = vg ; sideway gust input T (vg ), two inputs uF B = [δa δr ] ; aileron input (δa ) and rudder input (δr ), two controlled outputs z1 = [va φ]; side air speed (va ) and roll angle, and four measurement outputs y = [va p φ r]T . We set W1 as follows. ⎧ 2s2 + s + 1.1 ⎪ ⎨γ diag (1, 0.81) , B747 s2 + 5.3s + 1.1 W1 : , 2 ⎪ ⎩ γ 2s + s + 2.2 diag (1, 0.67) , C880M s2 + 3.9s + 2.2 (1) where γ is a design parameter to be maximized because the larger γ means the better modelmatching performance. These weighting functions
~ Δ
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P
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M
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wd
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20
Fig. 2. Block diagram for designing feedback controller K
6.3040 × 10−3 0.23147 0.10042 0.81281 9.3628 × 10−4 −0.31419 0.94285 2.4858
0.016274 0.537941 0.24957 0.92461 −3.6195 × 10−3 0.099165 1.7458 3.5892
, B747,
, C880M,
diag 6.5821 × 10−4 I2 , 0.10534, 0.029202 , B747,
-360 20 0 -20 -40 -60 0 -90 -180 -270 -360 -450 -3 10
10
-2
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-1
10
0
1
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10
2
Gain [dB]
Phase [deg]
va
Gain [dB]
20 0
closed-loop system (four lines) P C880M
-20 -40 -60 180 0 -180 -360 20 0 -20 -40 -60 0 -90 -180 -270 -360 -450 -3 10
10
-2
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-1
10
0
1
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10
2
Frequency [rad/s]
Fig. 4. Bode plots from vg to va and φ with Kopt for C880M model 2.2 Feedforward Controller Design
(2)
0 -180
Fig. 3. Bode plots from vg to va and φ with Kopt for B747 model
φ
Similar to (Sato, 2006a), we design (local) optimal static output feedback controller (Kopt ) for the generalized plant Pg because the on-board computers have limited memories. However, note that our design approach is directly applicable for designing dynamical controllers, which are worth considering for better performance with more computing resources. The obtained maximum γ is as follows: 1.7698 (B747 model) and 1.4273 (C880M model). The corresponding controller Kopt and scaling matrix are respectively shown in (2) and (3):
closed-loop system (four lines) P B747
-40 -60 180
Frequency [rad/s]
Phase [deg]
respectively have their lowest gain at the frequency of the Dutch-roll mode of the each target aircraft.
0 -20
diag 9.4684 × 10−4 I2 , 0.022297, 8.1526 × 10−3 , C880M.
(3) Figs. 3 and 4 show the Bode plots of the closedloop system, plant, and corresponding models from vg to va and φ. In these figures, we draw four lines for the closed-loop system (we hardly distinguish them in this scale); the combinations of maximum delay time and minimum delay time of aileron input and rudder input. The designed controllers confirm that they have good modelmatching performance in [0.2, 2] [rad/s] against the prescribed uncertainties. Figs. 5 and 6 show the responses of step-type gust input for the closed-loop system and the corresponding models. Similar to Figs. 3 and 4, we draw four lines for the closed-loop system (we hardly distinguish them in this scale). The controllers almost simulate gust responses of their target aircraft.
We next design feedforward controllers to simulate handling responses without considering the plant uncertainties. We design a pseudo right inverse system based on H∞ norm used in (Sato, 2006b), in which controllers designed using the method confirmed to have good model-matching performance in flight tests. Fig. 7 shows the block diagram for designing the feedforward controller F . In this figure, w2 denotes model outputs, z2 denotes the fictitious performance output, which is weighted by weighting function W2 , to design a pseudo right inverse system of the closed-loop system Pcl . Weighting function W2 specifies the right inverse performance and its frequency range. In the closedloop system Pcl , Pdn denotes the nominal delay model (nominal delay time is set as 0.12 [s] for both aileron and rudder inputs), Kopt denotes the designed feedback controller. For both cases of B747 simulation controller and C880M simulation controller, we set W2 as follows. 10 (4) I2 s + 0.10 This function is set for F to satisfy the same design specifications used in (Sato, 2006a). We design full-order (tenth-order) controller F . Due W2 :
1
va [m/s]
0.5
Gain [dB]
closed-loop system (four lines) B747
Phase [deg]
0 -0.5 -1 1
Gain [dB]
φ [deg]
0.5 0
Phase [deg]
-0.5 -1 -1.5
0
5
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t [s]
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35
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Fig. 5. Step responses with Kopt for B747 model (numerical simulation)
Gain [dB]
va [m/s]
Phase [deg]
-0.5 -1 1
Gain [dB]
0 -0.5
Phase [deg]
φ [deg]
0.5
-1 0
5
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15
20
t [s]
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Fig. 6. Step responses with Kopt for C880M model (numerical simulation) w2 F Pcl
uFB
Pd n
Pa Kopt
P
+
-
W2
z2
y
Fig. 7. Block diagram for designing feedforward controller F to lack of space, we omit the state-space representation of F . Figs. 8 and 9 respectively show the Bode plots of the closed-loop system Pcl in Fig. 7, the augmented system composed of the closed-loop system Pcl , F and the corresponding model M , and corresponding models from model inputs to va and φ. In these figures, we draw five lines for the augmented system, which is composed of Pcl , F and M , with setting the nominal delay model Pdn as follows; the nominal delay time, and four combinations of maximum delay time and minimum delay time of aileron input and rudder input. The designed controllers confirm that they have good model-matching performance under the prescribed uncertainties. Next, we exactly check their robust performance via μ analysis. Fig. 10 shows the block diagram for checking the robust performance of F . In this figure, wd and zd are the same signals used in Fig.
Pcl FM (five lines) Pcl B747
(2,2) element
(2,1) element
-2
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Frequency [rad/s]
Fig. 8. Bode plots of Pcl F M for B747 model
closed-loop system (four lines) C880M
0
-1.5
(1,2) element
(1,1) element
Frequency [rad/s]
1 0.5
60 40 20 0 -20 -40 -60 180 90 0 -90 -180 -270 -360 60 40 20 0 -20 -40 -60 180 90 0 -90 -180 -270 -360 -3 10
60 40 20 0 -20 -40
(1,2) element
(1,1) element
Pcl FM (five lines) Pcl C880M
180 90 0 -90 -180 -270 -360 100
(2,2) element
(2,1) element
50 0 -50 180 90 0 -90 -180 -270 -360 -3 10
-2
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-3
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Frequency [rad/s]
Frequency [rad/s]
Fig. 9. Bode plots of Pcl F M for C880M model Δp
F
wd
Δd
w2
zd F
uFB
Pd
Pa Kopt
P
+
-
W3
z2
y
Fig. 10. Block diagram for checking the robust performance of F 2, w2 and z2 are the same signals used in Fig. 7, and W3 denotes the weighting function to evaluate the robust performance of F and its frequency range. We first set W2 used in designing F as W3 . In Figs. 11 and 12, the upper bounds of μ using W2 are respectively shown for B747 simulation controller and C880M simulation controller. However, both maxima of the upper bounds exceed unity; that is, we cannot conclude that the designed controllers have robust performance against the prescribed uncertainties. After much trial and error, we use (5), which is shown at the top of the next page, as W3 . In Figs. 11 and 12, the upper bounds of μ using W3 in (5) are respectively shown for B747 simulation controller and C880M simulation controller. Both maxima of the upper bounds are
10
0
1
10 10 Frequency [rad/s]
2
10
0 -2 10
-1
10
0
1
10 10 Frequency [rad/s]
2
10
Fig. 11. upper bound of Fig. 12. upper bound of μ for B747 simulaμ for C880M simution controller lation controller less than unity; that is, the controller have robust right inverse performance which is defined by the weighting function in (5). The use of W3 concludes that the feedforward controller F has robust performance defined as follows. • B747 · ω ∈ [0, 0.13], 1/ζ(ω) = 0.02, · ω ∈ (0.13, 0.39], 1/ζ(ω) = 0.1, · ω ∈ (0.39, 0.59], 1/ζ(ω) = 0.2, and · ω ∈ (0.59, 2], 1/ζ(ω) = 0.28 • C880M · ω ∈ [0, 0.15], 1/ζ(ω) = 0.02, · ω ∈ (0.15, 0.60], 1/ζ(ω) = 0.1, · ω ∈ (0.60, 1.09], 1/ζ(ω) = 0.2, and · ω ∈ (1.09, 2], 1/ζ(ω) = 0.28, where 1/ζ(ω), roughly speaking, denotes the maximum error from ideal inverse system (for the precise definition of 1/ζ(ω), see (Sato, 2006a)). These results do not satisfy the specifications when designing F , however, the degradation is not so big; that is, the designed controller F are both confirmed to have robust right inverse performance under the prescribed uncertainties.
3. EXPERIMENT
10 5 0 -5 10 5 0 -5 5 0 -5 -10 5 0 -5 -10 2 1 0 -1 -2 6 4 2 0 -2 2 1 0 -1 5 0
δ r [deg] δ a [deg] ψ [deg] r [deg/s]
W2 W3 -1
-5 0
MuPAL-α
B747
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t [s]
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Fig. 13. Step gust responses in HIL simulation (B747 model) v g [m/s]
0.5
φ [deg] p [deg/s] va [m/s]
1
φ [deg] p [deg/s] va [m/s]
v g [m/s]
1.5
W2 W3
(5)
10 5 0 -5 10 5 0 -5 10 5 0 -5 -10 5 0 -5 -10 4 2 0 -2 10 5 0 -5 4 2 0 -2 4 2 0 -2 0
MuPAL-α
C880M
δ r [deg] δ a [deg] ψ [deg] r [deg/s]
2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 -2 10
⎩
14.945s 5 + 412.82s 4 + 2890.9s 3 + 4435.0s 2 + 2828.1s + 1183.0 I2 , B747 + 30.096s 6 + 257.16s 5 + 675.13s 4 + 1058.3s 3 + 853.30s 2 + 187.06s + 11.841 4 3 2 14.837s + 449.27s + 3213.4s + 3277.2s + 3093.4 I2 , C880M s6 + 32.056s 5 + 276.49s 4 + 550.72s 3 + 832.33s 2 + 403.06s + 31.435 s7
upper μ
upper μ
W3 :
⎧ ⎨
5
10
15 t [s]
20
25
30
Fig. 14. Step gust responses in HIL simulation (C880M model) Figs. 15 and 16 respectively show handling responses of B747 simulation controller and C880M simulation controller. Discrepancies are hardly shown in both HIL simulations; that is, our designed controllers simulate handling responses very faithfully.
3.1 HIL Simulations 3.2 Flight Tests We first conducted HIL simulations, in which the motions of MuPAL-α is calculated with six-degree nonlinear equations. Figs. 13 and 14 respectively show gust responses of B747 simulation controller and C880M simulation controller. Although some discrepancies are shown in both HIL simulations, the gains and frequencies of the Dutch-roll mode in va and φ are almost the same as the models; that is, our designed controllers almost simulate gust responses of the target aircraft.
We next conducted flight tests. Figs. 17 and 18 respectively show time histories of B747 simulation flight and C880M simulation flight. In these figures, model data are recalculated with recorded gust input and model inputs which are driven by pilot, because model data in flight are calculated only with model inputs; that is, gust input is not considered. In these flight, the pilot were demanded to make aircraft steadily turn. Although some discrepancies are shown in
φ [deg] p [deg/s] va [m/s] 20
30 t [s]
40
50
60
δ r [deg] δ a [deg]
10 5 0 -5 -10 20 10 0 -10 -20 20 10 0 -10 -20 5 0 -5 15 10 5 0 -5 4 2 0 -2 -4 10 5 0 -5 -10 0
MuPAL-α
-5 5 0 -5 20 10 0 -10 -20 4 2 0 -2 -4 100 50 0 -50 20 10 0 -10 -20 2 1 0 -1 0
5
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t [s]
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Fig. 16. Pilot handling responses in HIL simulation (C880M model) both flight tests, va and φ are almost the same as the models; that is, our designed controllers almost simulate gust responses as well as handling responses of the target aircraft in real flight.
4. SUMMARY This paper presents a design method of IFS controllers to simulate gust response as well as handling response. The controller is designed in the following sequence: Feedback controller is first designed to simulate gust response, feedforward controller is then designed to simulate handling response. Feedforward controller is designed without considering plant uncertainties, however, it is confirmed that designed feedforward controller has robust performance against the uncertainties via μ analysis. Our designed controllers are confirmed to have good model-matching performance in hardware-in-the-loop simulations and flight tests.
REFERENCES Hanke, D. and H. H. Lange (1988). Flight evaluation of the ATTAS digital fly-by-wire/light
B747
MuPAL-α
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t [s]
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Fig. 17. Time histories of flight test (B747 model) 4 2 0 -2 -4 4 2 0 -2 -4 5
v g [m/s]
C880M
φ [deg] p [deg/s] va [m/s]
ψ [deg]
r [deg/s]
φ [deg]
p [deg/s] va [m/s]
Fig. 15. Pilot handling responses in HIL simulation (B747 model)
0 -5 5 0
δ r [deg] δ a [deg] ψ [deg] r [deg/s]
10
5
v g [m/s]
MuPAL-α
B747
0 -5 20 10 0 -10 -20 4 2 0 -2 -4 100 50 0 -50 4 2 0 -2 -4 2 1 0 -1 -2 0
δ r [deg] δ a [deg] ψ [deg] r [deg/s]
p [deg/s] va [m/s] φ [deg] r [deg/s] ψ [deg] δ r [deg] δ a [deg]
10 5 0 -5 -10 10 5 0 -5 -10 20 10 0 -10 4 2 0 -2 -4 10 5 0 -5 10 5 0 -5 -10 10 5 0 -5 -10 0
C880M
20
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60
t [s]
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Fig. 18. Time histories of flight test (C880M model) flight control system. Proceedings of ICAS 16th Congress pp. 866–876. Heffley, R. K. and W. F. Jewell (1972). Aircraft handling qualities data. NASA CR-2144 pp. 193–209 and 210–242. Komoda, M., N. Kawahata, Y. Tsukano and T. Ono (1998). VSRA in-flight simulator — its evaluation and applications. AIAA Flight Simulation Technologies Conference, AIAA88-4605-CP. Motyka, P. R., E. G. Rynaski and P. A. Reynolds (1972). Theory and flight verification of the TIFS model-following system. Journal of Aircraft 9, 347–353. Sato, M. (2006a). Flight test of flight controller for arbitrary maneuverability and wind gust rejection. Proc. CCA pp. 2534–2540. Sato, M. (2006b). Flight test of model-matching controller for in-flight simulator MuPAL-α. Journal of Guidance, Control, and Dynamics 29, 1476–1481. Weingarten, N. C. (2005). History of in-flight simulation at general dynamics. Journal of Aircraft 42, 290–298.