Multiple-input m ultiple-output longitudinal robust control for aircraft

Nội dung text: Multiple-input m ultiple-output longitudinal robust control for aircraft

1. TNU Journal of Science and Technology 226(11): 38 - 46 MULTIPLE-INPUT M ULTIPLE-OUTPUT LONGITUDINAL ROBUST CONTROL FOR AIRCRAFT Nguyen Tien Hung* TNU - University of Technology ARTICLE INFO ABSTRACT Received: 19/6/2021 This paper is dealt with the design of a multiple-input multiple-output robust controller for the longitudinal flight dynamics of an aircraft Revised: 29/6/2021 control system. The design objective is to achieve robust stability and Published: 30/6/2021 good dynamic performance against the variation of aircraft parameters in which the aircraft forward speed is considered to be a real KEYWORDS uncertainty. The controller synthesis is aimed at maintaining robust performance for frozen values of the aircraft forward speed in a Flight control specified operating range. The proposed robust controller is Robust controller implemented using the robust control toolbox in Matlab. The obtained Mixed sensitivity approach results verify the performance of the proposed controller for aircraft control system with respect to different values of the aircraft forward Loop-shaping design speeds. Robust performance ĐIỀU KHIỂN BỀN VỮNG ĐA BIẾN CHUYỂN ĐỘNG DỌC CỦA CÁC THIẾT BỊ BAY Nguyễn Tiến Hưng Trường Đại học Kỹ thuật Công nghiệp – ĐH Thái Nguyên THÔNG TIN BÀI BÁO TÓM TẮT Ngày nhận bài: 19/6/2021 Bài báo này trình bày việc thiết kế bộ điều khiển bền vững đa biến cho mô hình động học chuyển động dọc của hệ thống điều khiển bay. Ngày hoàn thiện: 29/6/2021 Mục tiêu của bài toán thiết kế là nhằm đạt được tín ổn định bền vững Ngày đăng: 30/6/2021 với chất lượng động học mong muốn để chống lại các biến đổi tham số trong mô hình máy bay với vận tốc của máy bay được coi là một TỪ KHÓA bất định. Việc tổng hợp bộ điều khiển là nhằm duy trì chất lượng động học bền vững của hệ thống khi vận tốc của máy bay nằm trong Điều khiển bay một khoảng xác định trước. Bộ điều khiển bền vững được thực hiện Bộ điều khiển bền vững nhờ hộp công cụ Matlab. Các kết quả nghiên cứu đã kiểm chứng chất lượng của bộ điều khiển đối với các giá trị khác nhau của vận tốc Phương pháp độ nhạy hỗn hợp máy bay. Thiết kế Loop-shaping Chất lượng bền vững DOI: Email: h.nguyentien@tnut.edu.vn 38 Email: jst@tnu.edu.vn
2. TNU Journal of Science and Technology 226(11): 38 - 46 1. Introduction In flight control systems, the performance requirements should be maintained over the entire range of aircraft speeds and altitudes. For improving the performance of the closed-loop longitudinal control, multi-input multi-output (MIMO) controller designs can be employed for both elevator and throttle servos with the linear quadratic reg- ulator (LQR) control [1, 2]. On the other hand, it is well-known in flight control that the actuator dynamics depend on angle of attack regions [3]. In order to improve the system robustness against changes in the machine parameters and exogenous inputs, several controller designs have been proposed for such aircraft in literature. In [2, 4], the H∞ controller design is proposed for multivariable vertical short take-off and landing (VSTOL) aircraft system. In [1], the same approach is also applied to generic VSTOL aircraft model. H∞ control provides a very powerful tool for controller synthesis of multivariable linear time-invariant systems in the presence of uncertainty. However, the design techniques become more complicated if we consider uncertain linear time- varying systems. At the expense of conservativeness and possibly poor performance, the varying parameters can be treated as uncertainties and a single robust parameter- independent LTI controller can be designed for the entire operating range. On the other hand, if the parameter value is measurable online, one might instead try to design a parameter-dependent controller in order to improve performance. Recently, the linear parameter-varying (LPV) control approach, which takes the parameter variations into account directly in the control design, is applied for flight control systems of several types of aircraft and flight conditions [3, 57]. A common feature of the above publications is that the mathematical models of aircraft are not provided explicitly for some reasons. Furthermore, the robustness of the controlled system with respect to the changes of parameters is not also given clearly. Therefore, a robust H∞ controller design to improve the performance of the elevator deflection control loop with respect to machine parameter variations and in view of the throttle servos disturbance is presented in [8]. The obtained results show that the designed H∞ controller achieves the required performance specifications over the operating range of the aircraft. In addition, robustness of the controlled system against parameter changes as well as the impact of throttle servos on the robustness of the control system are also considered by means of some substantial simulation results. Since the throttle servo is considered to be a disturbance input, this design falls into the category of the single-input single-output (SISO) configuration. However, as it will be shown in the next section, the longitudinal dynamics model of aircraft is described by a MIMO system. Therefore, in order to improve the performance of the closed-loop control system, the effect of the crossing-term in the aircraft model should be taken into account. In this work, we present a MIMO robust H∞ controller design that guarantees the tracking performance for both channels from the references to their corresponding outputs over the specified range of the aircraft forward speed. In addition, robustness of the controlled system against changes of aircraft parameter is also evaluated. The study results will be given to demonstrate the obtain performance of the proposed controller design. In the next section, we will present the longitudinal dynamics model of aircraft. This model can be found in [8] but it is reproduced here for the reader's convenience. The multiple-objective H∞ controller synthesis for a class of linear time-invariant systems 39 Email: jst@tnu.edu.vn
3. TNU Journal of Science and Technology 226(11): 38 - 46 will be given as the content of the design section. Similarly to the previous work in [8], the method presented in this section is especially focused on affine parameter-dependent systems. The synthesis is based on the linear matrix inequality (LMI) approach and the bounded real lemma as a powerful tool for turning H∞-constraints into LMIs. More detail of the approach can be found in the literature, for instance in [911]. Finally, some simulation results and conclusions will be presented in the last sections. 2. Longitudinal dynamics model of aircraft Consider the aircraft body axes, (i, j, k) and the north, east, down (NED) local horizon frame, (I, J, K) as shown in Figure 1. Note that longitudinal motion is normally repre- sented by a small displacement from an equilibrium (unaccelerated) flight condition in the longitudinal plane. The flight variables in such an equilibrium are denoted with a subscript e. In this fashion, the pitch angle can be represented as Θ = θe + θ. Similarly, the forward speed U = Ue + u, the downward (or plunge) velocity W = v, the pitch rate Q = q, the forward force X = Xe + X, the downward force Z = Ze + Z, and the pitching moment M = M where θ, u, w, q, X, Z, M are the perturbation quantities. Let J
   be the inertia tensor, where , , are the moments J = ik i,k={x,y,z} Jxx Jyy Jzz of inertia, and Jxy, Jyz, Jxz are the products of inertia. Note that, for a symmetrical plane, Jxy = Jyz = 0. Let α be the angle of attack, m be the aircraft's mass, X be the forward force, Z be the downward force, M be the pitching moment, U be the forward speed. Denote ∂F , where , as the first-order of a Taylor series Fx = ∂x e F ∈ {X, Z, M} expansion at the equilibrium point. Figure 1. The aircraft body axes [1] By neglecting products of small perturbation quantities, we obtain the longitudinal dynamics model of the aircraft in the state-space form as [1] x˙ l = Alxl + Blwl (1) yl = Clxl (2) 40 Email: jst@tnu.edu.vn
4. TNU Journal of Science and Technology 226(11): 38 - 46 where  Xu Xα  m m −g cos θe 0  Zu Zα − g sin θe 1 + Zq   mU mU U mU  (3) Al =  0 0 0 1  ,   Mu + Mα˙ Zu Mα + Mα˙ Zα − Mα˙ g sin θe Mq + Mα˙ (mU+Zq) Jyy mUJyy Jyy mUJyy UJyy Jyy mUJyy  Xδ XT  m m Zδ ZT    mU mU  1 0 0 0 Bl =   ,Cl = , (4)  0 0  0 0 0 1 Mδ + Mα˙ Zδ MT + Mα˙ ZT Jyy mUJyy Jyy mUJyy
   T
   T xl = u α θ q is the state variable, wl = δE βT is the input of the system, δE is the elevator deflection, βT is the throttle servos. Let 1 and express as an uncertainty element , where is va = U va va = vn(1 + pvδv) vn the nominal value of , indicates the variation of around its nominal value, va pv ∈ R va , , we can write δv ∈ R −1 ≤ δv ≤ 1 Al = Aln + δvAlv (5) Bl = Bln + δvBlv (6) in which  Xu Xα  m m −g cos θe 0 Zu Zα Zq  m vn m vn −g sin θevn 1 + m vn  Aln =   (7)  0 0 0 1  Mu + Mα˙ Zu v Mα + Mα˙ Zα v − Mα˙ g sin θe v Mq + Mα˙ + Mα˙ Zq v Jyy mJyy n Jyy mJyy n Jyy n Jyy Jyy mJyy n  0 0 0 0  Zu Zα Zq  m vnpv m vnpv −g sin θevnpv m vnpv  Alv =   (8)  0 0 0 0  Mα˙ Zu v p Mα˙ Zα v p − Mα˙ mg sin θe v p Mα˙ Zq v p mJyy n v mJyy n v mJyy n v mJyy n v  Xδ XT    m m 0 0 Zδ ZT Zδ ZT  m vn m vn   m vnpv m vnpv  Bln =   Blv =   (9)  0 0   0 0  Mδ Mα˙ Zδ MT Mα˙ ZT Mα˙ Zδ Mα˙ ZT + v + v vnpv vnpv Jyy mJyy n Jyy mJyy n mJyy mJyy Equation (1) can now be expressed as x˙ l = (Aln + δvAlv)xl + (Bln + δvBlv)wl      
   xl
   xl
   xl = Aln Bln + δv Alv Blv = Aln Bln + wv (10) wl wl wl where    
   xl
   xl wv = δv Alv Blv = δvzv, zv = Alv Blv . (11) wl wl 41 Email: jst@tnu.edu.vn
5. TNU Journal of Science and Technology 226(11): 38 - 46 In (11), wv and zv represent the input and output signals of the disturbance channel corresponding to the time-varying parameter va. Rewrite equations (10), (11), and (2) in a matrix form as       x˙ l Aln Blw Bln xl zv  =  Alv Blz Blv  wv  , (12) yl Cln Dlw Dlu wl wv = ∆vzv, ∆v = δvI4 (13) where Blw = I4 is an 4×4 unity matrix, Blz = Z4 is an 4×4 zero matrix, Dlw = Dlu = 0. ∆v is also called the perturbation block. Let Gla be the transfer function with the state- space realization (12), i.e.     Aln Blw Bln ∆ Ala Bla (14) Gla = =  Alv Blz Blv  . Cla Dla Cln Dlw Dlu The system can then be generally described by         zv wv Gzw Gzu wv = Ga = , (15) yl wl Gyw Gyu wl where Gyu is the transfer function mapping wl to yl. 3. H∞ control design In this section, we start with H∞-synthesis for the above mentioned frozen values of the aircraft forward speed. Then the performance of the linear time-invariant (LTI) controller designed for a fixed value of va is evaluated with other constant values of its. The content of this section is similarly to one in [8] but this design is for MIMO systems in stead of SISO ones. 3.1. H∞ loop shaping design A standard control structure for the synthesis of an H∞-controller is depicted in Figure 2. Here, ∆v is the uncertainty block as given in (13), Kle is the H∞ controller that is
   T
   T to be designed. In this configuration, the reference input is rl = ud αd , δE βT
   T is the controller output, yl = u α is the controlled output, and ele = rl − yl is the controller input which is equal to the tracking error. The interconnection of the system used for the controller synthesis is shown in Figure 3 where Gln is the LTI part of the plant as given in (14). The external control
   T input wle consists of the throttle servo and the angle of attack wle = ud αd . The
   T controlled variable is zle = zt zs . Note that the component βT of the external control inputs are considered as disturbances and their influences on the controlled outputs must be reduced as much as possible. The weighting function Ws is used to shape the transfer function from the external control input wle to the tracking error ele. Ws is kept large over the low frequency range for tracking. The weighting function Wt is used to shape the transfer function from the external control input wle to the controlled output yr. The selection of the weighting 42 Email: jst@tnu.edu.vn
6. TNU Journal of Science and Technology 226(11): 38 - 46 + - Figure 2. Structure of the closed-loop system in H∞ design function Wt is not only intended to keep the closed loop bandwidth at a desired value, but also to reject the effects of the component βT on the controlled outputs as discussed above. Note that a large bandwidth corresponds to a faster rise time but the system is more sensitive to noise and to parameter variations [12]. + - + - Figure 3. The interconnection of the system The standard H∞ control problem is to find a stabilizing LTI controller Kle at fixed frozen values of va such that the H∞-norm of the channel wle → zle is smaller than a given number γ: W S  s le ≤ γ. W T t le ∞ 3.2. Simulation results with the H∞ current controller The set of the aircraft parameters that is given as follows [1]: , Zu , θe = 0 m = −0.36 /s Xα = 1.96 m/s2, Zα = 108 m/s2, Mα = −8.6 /s2, Mα˙ = −0.9 /s, Mq = −2 /s, Xδ ≈ 0, m m Jyy Jyy Jyy m Zδ = 0.3 m/s2/rad, Mδ = 0.1243 s−2, XT = 0.2452 m/s2/rad, ZT ≈ 0, and MT ≈ 0. m Jyy m m Jyy During the controller design stage, a trial-and-error-repetition technique is used in order to achieve the desired performance specifications by adjusting the weighting functions. The design steps were repeated until we are able to meet the required performance specifications. Finally, the following weighting functions were obtained: 43 Email: jst@tnu.edu.vn
7. TNU Journal of Science and Technology 226(11): 38 - 46 0.5 0.5 W = W W
   T ,W = ,W = , (16) t tu ta tu 1.15s + 1.98 ta 1.15s + 1.98 1 0.99 W = W W
   T ,W = ,W = . (17) s su sa su 5s + 1.05 sa 5s + 1.05 For the chosen frozen value of U = 55m/s, the controlled system with the H∞ current controller for the above given weighting functions achieves a norm of 0.943. Reference airspeed to output 1 Reference airspeed to error 1 From: uref To: [+Gm(1)] From: uref To: [+uref-Gm(1)] 80 80 60 60 40 40 20 20 0 0 -20 Magnitude (dB) Magnitude (dB) -20 -40 -60 -40 -80 -60 10-2 100 102 10-2 100 102 Frequency (rad/s) Frequency (rad/s) (a) (b) Reference angle of attack to output 1 Reference angle of attack to output 2 From: aref To: [+Gm(1)] From: aref To: [+Gm(2)] 20 0 0 -20 -20 -40 -40 -60 -60 -80 -80 Magnitude (dB) Magnitude (dB) -100 -100 -120 -140 -120 10-2 100 102 10-2 100 102 Frequency (rad/s) Frequency (rad/s) (c) (d) Figure 4. The performance of the controlled system with H∞ current controller in the frequency domain for the variation of va from 0.5vn to 1.5vn. Figure 4 shows the frequency responses of the controlled system with the H∞ current controller and the inverse of the weighting functions Wtu (see equations (16) and (17)) with 11 frozen values of the aircraft forward speed from 50% up to 150% of its nominal magnitude. In this figure, the thick solid lines show the responses of the closed-loop system with respect to the normal value of the aircraft speed. Figures 4a,b show the relevant magnitude plots of the complementary sensitivity and sensitivity functions of the closed-loop system with the performance requirements achieved by Wt and Ws. Figure 4a shows the response of the output u with respect to the reference inputs ud. 44 Email: jst@tnu.edu.vn
8. TNU Journal of Science and Technology 226(11): 38 - 46 Reference airspeed to output 1 10-3 Reference airspeed to output 2 1.2 2 1 1.5 0.8 1 0.6 0.5 Magnitude Magnitude 0.4 0 0.2 -0.5 0 -1 0 5 10 15 20 0 5 10 15 20 Time (s) Time (s) (a) (b) 10Reference-3 angle of attack to output 1 Reference angle of attack to output 2 2 1 1.5 0.8 1 0.6 0.5 Magnitude 0.4 Magnitude 0 0.2 -0.5 -1 0 0 5 10 15 20 0 5 10 15 20 Time (s) Time (s) (c) (d) Figure 5. The performance of the controlled system with H∞ current controller in the time domain for the variation of va from 0.5vn to 1.5vn. The performance of the reference input ud to the control error ud −u is shown in Figures 4b. The inverse of the weighting function Wtu (see Figure 3) is depicted by the dashed line in Figure 4a and the inverse of the weighting function Ws is depicted by the dashed lines in Figure 4b, respectively. Figures 4c,d show the relevant magnitude plots of the transfer functions from the reference input αd to output u and α, respectively. It is clear from Figure 4 that the sensitivity and complementary sensitivity functions are below the inverse of the performance weighting functions. The gains of the frequency responses of the reference angle of attack for some values of aircraft speeds are bigger than zero. This indicates that the influence of crossing-terms into the channel from reference airspeed to its output is not small. Note that these performance curves are obtained for 11 values of the aircraft forward speeds as mentioned above. Figure 5 shows the time responses of the controlled system for a step input in the consistence to the curves in the frequency domain as shown in Figure 4 with 11 values of the aircraft forward speed as shown above. 4. Conclusion This paper has briefly presented an LMI-based loop-shaping design of the multiple- input multiple-output robust H∞ controller for the linear simplified longitudinal model of a aircraft, in which the aircraft forward speed is considered as an uncertain param- 45 Email: jst@tnu.edu.vn
9. TNU Journal of Science and Technology 226(11): 38 - 46 eter. The robust H∞ controller is then synthesized to guarantee that the H∞-norm of the closed-loop system is smaller than some given number for different frozen values of the aircraft forward speed. Next, the robust performance of the robust controller with respect to the other the aircraft forward speeds is investigated in the range from 50% up to 150% of its nominal values. Some simulation results are given to demon- strate the performance and robustness of the control algorithm. Since the effect of the crossing-term in the aircraft model was not small, it is difficult to obtain good tracking performance for both channels from the reference airspeed to its output as well as from the reference angle of attack to its output. Therefore, this problem should be take into account for improving the tracking performance of the closed-loop control system in future works. REFERENCES [1] A. Tewari, Advanced control of aircraft, spacecraft and rockets. Wiley, 2011. [2] J. Zarei, A. Montazeri, M. R. J. Motlagh, and J. Poshtan, Design and comparison of LQG/LTR and H∞ controllers for a VSTOL flight control system, Journal of the Franklin Institute, vol. 344, no. 5, pp. 577594, 2007. [3] B. Lu, F. Wu, and S. W. Kim, Switching LPV control of an F-16 aircraft via controller state reset, IEEE Transactions on Control Systems Technology, vol. 14, no. 2, pp. 267277, 2006. [4] R. A. Hyde, The application of robust control to VSTOL aircraft, Ph.D. disser- tation, Girton College Cambridge, 1991. [5] N. Aouf and B. Boulet, Linear fractional transformation gain-scheduling flight control preserving robust performance, Journal of Aerospace Engineering, vol. 226, no. 7, pp. 763773, 2011. [6] N. Wen, Z. Liu, Y. Sun, and L. Zhu, Design of LPV-based sliding mode controller with finite time convergence for a Morphing aircraft, International Journal of Aerospace Engineering, vol. 11, pp. 955969, 2017. [7] C. Weiser, D. Ossmann, and G. Looye, Design and flight test of a linear parameter varying flight controller, CEAS Aeronautical Journal, vol. 11, pp. 955969, 2020. [8] N. T. Hung and N. T. M. Huong, A robust controller design for aircraft, The conference of applying high technology into practice, Thainguyen University of Technology, 2021. [9] N. T. Hung and N. D. Minh, Performance of robust controller for DFIM when the rotor angular speed is treated as a time-varying parameter, Vietnam Conference on Control and Automation, Hanoi, 2011. [10] H. N. Tien, C. W. Scherer, J. M. A. Scherpen, and V. Muller, Linear param- eter varying control of doubly fed induction machines, IEEE Transactions on Industrial Electronics, vol. 63, no. 1, pp. 216224, 2016. [11] N. T. M. Huong and N. T. Hung, Robustness stability analysis for H-infinity control of separately excited DC motors, TNU Journal of Science and Technology, vol. 188, pp. 2937, 2018. [12] S. Skogestad and I. Postlethwaite, Multivariable feedback control - Analysis and design. John Wiley & Sons, 1996. 46 Email: jst@tnu.edu.vn