High-order Iterative Learning Identification Of Projectile's ... - IEEE Xplore

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 6, NO. 4, JULY 1998

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High-Order Iterative Learning Identification of Projectile’s Aerodynamic Drag Coefficient Curve from Radar Measured Velocity Data Yangquan Chen, Changyun Wen, Jian-Xin Xu, and Mingxuan Sun

Abstract— Extracting projectile’s optimal fitting drag coefficient curve Cdf from radar measured velocity data is considered as an optimal tracking control problem (OTCP) where Cdf is regarded as a virtual control function while the radar measured velocity data are taken as the desired output trajectory to be optimally tracked. With a three-degree of freedom (DOF) point mass trajectory prediction model, a high-order iterative learning identification scheme with time varying learning gains is proposed to solve this OTCP with a minimax performance index and an arbitrarily chosen initial control function. The convergence of the high-order iterative learning identification is analyzed and a guideline to choose the time varying learning gains is given. The curve identification results from a set of actual flight testing data are compared and discussed for different learning gains. These results demonstrate that the high-order iterative learning identification is effective and applicable to practical curve identification problems. Index Terms— Aerodynamic drag coefficient, curve identification, data reduction, iterative learning control, minimax tracking, optimal tracking control.

I. INTRODUCTION

A

N effective artillery attack depends on the accuracy of firing tables of guns while the accuracy of a firing table is related to many aspects such as various fitting-factor corrections, nominal trajectories calculations, the physical and aerodynamic data determinations, etc. The firing table is essentially related to the obtained aerodynamic drag coefficient which is of crucial importance in determining the projectile’s ballistic behaviors. How to obtain the effectively and convincingly has been a constant objective since last century. The wind tunnel test and the numerical code for aerodynamic property prediction are the two common and other aerodynamic coefficients. ways to obtain the However, accuracy of the results is highly subject to certain factors such as the impractical assumptions in numerical modeling, the wind tunnel wall interferences, etc. Therefore, identifying the aerodynamic properties of real or full-scale flying objects from flight testing data has been an important Manuscript received February 22, 1996; revised August 27, 1997. Recommended by Associate Editor, R. Ravi. The work of M. Sun was supported in part by the National Science Foundation of China (NSFC) under Project 69404004. Y. Chen was with the School of Electronic and Electrical Engineering, Nanyang Technological University, Singapore 639798. He is now with the Department of Electrical Engineering, National University of Signapore, Singapore 119260. C. Wen is with the School of Electronic and Electrical Engineering, Nanyang Technological University, Singapore 639798. J.-X. Xu is with the Department of Electrical Engineering, National University of Singapore, Singapore 119260. M. Sun is with the Department of Electrical Engineering, Xi’an Institute of Technology, Xi’an 710032, China. Publisher Item Identifier S 1063-6536(98)04579-5.

research area of the modern flight dynamics which is in essence a combination of modern control theory and flight dynamics. Clearly, the identification results using flight testing data are more convincing, hence are suitable for verifying and improving design objectives, computational aerodynamic property prediction codes, etc. The Doppler tracking radar such as TERMA’s OPOS DR582 has become a common equipment in proving grounds. accuMany efforts have been made to extract the rately and rapidly based on a parameter identification [1], [2] framework [3]–[9]. By combining the ballistic model into a Newton–Raphson-like iteration scheme, the differential correction method is proposed in [3] for searching aerodynamic parameters, which was applied to determine the aerodynamic drag from radar data in [4]. A maximum likelihood (ML) estimation was applied in [6] for aerodynamic parameter identification. From the characteristics of aerodynamics, we believe that an aerodynamic property curve can be regarded as a deterministic virtual control function. Therefore, the aerodynamic property curve identification can be considered as an optimal tracking control problem (OTCP) where the collected testing data are the desired output trajectories to be optimally tracked. Based on this observation, an optimal control solution method was proposed in [10], the result of which is a continuous but nonsmooth time function. This result cannot be applied directly because it should be postprocessed by some kind of data fitting software, e.g., the , which is a one introduced in [11]. The resultant fit to the identification model used, may be a multivalued function in certain situations. To simplify the procedure for the application of the identified result, an optimal dynamic fitting directly. method was proposed in [12] to get a smooth Several schemes for the simplification were also introduced in [12] which made the optimal dynamic fitting method more attractive than that of [10]. All the existing methods are more or less restrictive in the choice of an initial control function or initial values of some parameters. The results obtained are not in a global sense either. Furthermore, the performance index is in the least squares sense. The above debit sides can be overcome by using the iterative learning control (ILC) method formulated by Arimoto et. al. [13]. It is important to note that, in the iterative learning concept, the performance index is in the minimax sense and the initial control function can be selected arbitrarily using the iterative learning control. The control effort of the current iteration is updated by using the control effort and the tracking error information of the previous iteration. In other words, it is learning from the past practice and experience. The more past iterations are used, the more improved learning

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 6, NO. 4, JULY 1998

Fig. 1. Illustration of Doppler radar tracking.

performance can be expected. This is called a high-order learning updating law, which was first proposed in [14] and further investigated in [15]. In this paper, we propose a highorder learnig updating law with time-varying learning gains. The learning convergence properties are established and a guideline to choose the time varying learning parameters is suggested. The curve identification results from a real flight testing data are compared and discussed for different choices of learning gains. It is observed that the results of this paper are consistent with those of optimal dynamic fitting method [12]. The improved learning performance for a real flight testing data has been achieved. These results demonstrate that the high-order iterative learning identification is effective and applicable for practical curve identification problems. It should also be pointed out that, from the aerodynamics curve cannot be taken as point of view, the identified the zero lift drag coefficient curve or incidence induced drag is just the fitting drag coefcoefficient curve. The ficient curve with respect to the trajectory model employed, which comprehensively reflects the effects of zero lift drag and the drag induced by the angular motion around the center of mass. Because the firing table generation is mainly based on a single ballistic coefficient, a drag law and some fitting factors, more accurate firing table can be produced when we of this paper directly. utilize the identified The remainder parts of this paper are organized as follows. The problem is formulated in Section II and the curve identification method using a high-order iterative learning law is given in Section III. The curve identification results for a set of real flight testing data with different high-order learning parameters are presented in detail in Section IV. The conclusion is drawn in Section V. A set of flight testing data together with some

other flight testing information, which are necessary to make the results of this paper repeatable, are listed in the Appendix. II. PROBLEM STATEMENT For brevity of our discussion, a 3-degree of freedom (DOF) point mass ballistic model will be used. A more complete ballistic model [16] may be used if other aerodynamic coefficients are available and the method presented here can still be applied. Suppose at time the position of the projectile in earth coordinate system (ECS) is and its relative velocity vector with respect to ECS is . The position of the radar in ECS is which is known as illustrated in Fig. 1. The 3-DOF point mass trajectory model can be described by nonlinear state space equations as follows:

(1) and is known; is the gravitational where are the wind components in ECS known acceleration; is projectile’s relative from meteorological measurements; velocity with respect to the wind and (2) is the air density; is the reference area of the is the mass of projectile and is the projectile’s diameter; is the fitting drag coefficient curve the projectile and

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 6, NO. 4, JULY 1998

with respect to the trajectory model (1) which is regarded as the unconstrained control function. as Define a state vector of system (1) . The initial state is given. Also let denote the Mach number, i.e., , can be composed where is the local sonic speed. and which is more interesting from the from aerodynamics point of view. represent the distance Referring to Fig. 1, let and the projectile . Then we between the tracking radar have

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D-type ILC updating law [13] in a high-order form with time-varying learning parameters, i.e., (7) where is the ILC iteration number, i.e., the cycle number, and are the time varying learning parameter, is the . To study the order of ILC updating law, and learning convergence condition, the the following norms are used:

(3) To formulate our problem, we must transform the projectile’s which is in the radial tangential velocity into a velocity direction of the doppler radar as follows. Let

(4)

where is a vector, is a is a real function. matrix, and For the convergence analysis in a general sense, consider a class of repetitive uncertain nonlinear time-varying system

One obtains that

(8) (5)

the Doppler radar measured velocity data. Hence, Denote . Then the control the tracking error is that problem is to achieve the following objective: (6) It can be noted that (1), (5), and (6) formulate an OTCP, where is the desired the doppler radar measured velocity data and the fitting drag trajectory to be tracked by “output” is the “control” function. coefficient curve However, a lot of difficulties will arise in numerical computation for this OTCP with minimax performance index in (6). Even when a quadratical performance index is used, the OTCP is obviously singular as pointed out in [12] and thus the mathematical programming-based techniques [10] are impractical both in computing cost and in convergence property. We shall show that, in the next section, this OTCP can be easily solved by iterative learning concept. III. CURVE IDENTIFICATION BY HIGH-ORDER ITERATIVE LEARNING We now turn to discuss the curve identification from the control point of view. It is assumed in ILC that the system to be controlled must be repetitive, i.e., it performs an identical desired task in a finite fixed time interval repeatedly. To the curve identification problem of this paper, the operation of the sysem (1) in one repetition or one cycle is simply a numerical to with initial condition integration of (1) from under the current control . The current control is and tracking obtained by using the information from of previous cycles by applying the conventional error

is the time and is given, denotes the where th iteration of the system’s repetitive operation, , , and are the state, control input, and output of the system, respectively, the functions , and are piecewise continuous and satisfy the Lipschitz continuity conditions, i.e.,

where are the Lipschitz constants; are uncertainty or disturbance to the system with bounds defined as

Denote Assume with

and

. also satisfy Lipschitz continuity conditions their Lipschitz constants, respectively; are bounded as

The proposed high-order D-type ILC updating law is (9)

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 6, NO. 4, JULY 1998

where, and in the remainder part of this paper, the order of ILC updating law; are bounded timevarying learning parameter matrices of proper dimensions; is the tracking error and is the means realizable desired system output. The realizable such that that there exists a unique bounded

where

(14) ,

where , Similarly, we have

(10) As the initial state at each ILC operation may not be the same, , we assume the bounded initialization i.e., and for brevity, we denote errors as . We will show that under certain conditions on the timethe varying learning matrices ILC will converge in such a way that when starting from , through an arbitrary continuous initial control input as the ILC updating law, where the tracking error bound is related to the uncertainty, disturbance and initialization error. A sufficient condition for ILC convergence is given in the following theorem. Theorem 3.1: Consider the ILC updating law (9) applied to the repetitive nonlinear uncertain system (8) with bounded uncertainty, disturbance, and initialization error. For a given on the fixed time interval realizable desired trajectory , under the conditions ; C1) and C2)

, .

, and

(15) where Investigating and referring to (C1), (9), (14), and (15), we observe that

(16) Estimating the norms of (16), we obtain (17)

(11) where

where and Integrating (15) and then taking the norm yield

then, as

, the bounds of the tracking errors and converge asymptotically to a residual ball centered at the origin. Moreover, when the bounds of the initialization error, disturbance and uncertainty tend to zero, all the bounds of the tracking errors tend to zero. Before presenting a proof of Theorem 3.1, the following lemma [15] is needed. satisfies Lemma 3.1: Suppose a real positive series

d d

(18)

Noticing the fact that

(19) , where , then the following holds:

and (12)

where operation for (18), we can obtain

, performing the -norm (20)

Proof: Refer to [15]. We now proceed to present a proof of Theorem 3.1: For in the equations when brevity, we will drop the time necessary. From (8), it is easy to see that (13)

and , if we choose

where a sufficiently large

such that (21)

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Fig. 2. Convergence history comparison of tracking error bound eb (Case 1).

Taking -norm of (17) with the substitution of (20), we simply have (22)

Therefore, according to (C1) and (C2) (27) , one can get

Set

(28) where Based on (C2), we can make

and

Because

(28) can be simplified as (29)

(23) by using a sufficiently large . Applying Lemma 3.1, we have (24) From (13) and (20) we get (25)

are varying with Notice that in the above ideal case, respect to the iteration number and cannot be conveniently applied. Nevertheless, (29) gives a guideline to select . For the simplicity of the curve identification algorithm, choose an by substituting with and approximation of with , i.e., (30)

(26) It is clear that if the bounds of the uncertainty, disturbance, and initialization error tend to zero, the final tracking error bound will tend to zero, too, which completes the proof of Theorem 3.1. Now the above general convergence result can be applied to the OTCP. Note that the OTCP here is a single input single output system without uncertainty, disturbance and initialization errors. To find a suitable selection of the practical time varying learning parameters, we first study the first-order , and by referring to learning updating law. In this case (1) and Theorem 3.1, it is easy to see that

is the standard air density at the sea level which can where be easily determined. Based on Theorem 3.1 and with some practical considerations, the following high-order ILC updating law with time-varying learning parameters is proposed for the OTCP: (31) is a time-varying learning curve given in (30). where Its role is to speed up the convergence of iterative learning identification procedures compared to the constant learning parameters. The use of (30) is to give a guideline for the choice of a time-varying learning gain. The effectiveness of this scheme over constant learning gains will be verified by the flight testing results given in the next section. IV. IDENTIFICATION RESULTS FROM FLIGHT TESTING The results presented here are based on a set of real flight testing data which are listed in the Appendix. To

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 6, NO. 4, JULY 1998

Fig. 3. Convergence history comparison of tracking error bound eb (Case 2).

Fig. 4. Convergence history comparison of tracking error bound eb (Case 3).

investigate and compare the curve identification results under different situations, various choices of learning parameters , as listed in the right-upper corner of each , the figure, are considered. Clearly, if choosing . If learning updating law is of the first-order, i.e., or , a second-order learning scheme is employed. The identification is studied and the results are presented in three different cases. In each case, the RK-4 method is used to numerically integrate the trajectory model with step size second and a maximal allowable absolute tracking is set to be 0.05 m/s for the termination of the error . identification program. For the first two cases, Case 1: Identification Results with Constant Learning PaFig. 2 shows the rameters: In this situation, in the iterative diconvergence of tracking error bound and . From rection (locally zoomed) for both

the results, it can be noted that the convergence is significantly improved by adding a second-order term . Case 2: Identification Results with Time-Varying Learning Parameters: The time-varying scheme (30), (31) is used and different choices of . From with the results, it is clear that the second-order updating law with time varying learning parameters gives a much better convergence. By comparing Figs. 2 and 3, it can be noted that the convergence of the first-order with time varying learning is better than that of the second-order scheme with only constant learning parameters. Case 3: Identification Results with Different Selections of Initial Controls: To demonstrate that the iterative learning identification method converges in a global sense, results from are compared different choices of initial control in Fig. 4. It is clear that the ILC-based curve identification

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 6, NO. 4, JULY 1998

Fig. 5. Comparison of the identified optimal fitting drag coefficient curves

scheme is much more insensitive to the choice of initial control in comparison with the other methods in [10] and [12]. It should be noted that the converged drag coefficient curve must be a positive function. A comparison of the converged between the optimal dynamic fitting method curves in [12] and the iterative learning identification is given in Fig. 5. From Fig. 5, the average value of the drag coefficient is about 0.28 even when there are some negative initial controls . Furthermore, the final converged results of this paper with various learning parameter selections are all the same and they are consistent with the result of optimal dynamic fitting method [12]. V. CONCLUSION A high-order iterative learning identification method with time varying learning parameters is proposed for effectively extracting projectile’s optimal fitting drag coefficient curve from radar measured velocity data. The performance can be index is a minimax function and the initial arbitrarily set. A general convergence analysis is given and a guideline to choose time varying learning parameters is given. Practical curve identification results by high-order learning are presented. From the comparison of several schemes for learning gain determination, it is shown that the proposed highorder iterative learning identification scheme is effective and applicable to practical curve identification problems. APPENDIX FLIGHT TESTING DATA The complete flight testing data used in this paper are given as follows: m, 1) Projectile’s physical parameters: kg; m/s; 2) Atmosphere: ICAO Standard, ; 3) Initial state:

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Cdf (M ). 4) Radar position: 5) Radar measured data time interval of s). 652.9757 610.3392 572.0804 538.0422 507.3897 479.6691 454.1041 430.4870 408.7190 388.3574 369.5753 351.8651 335.2063 319.5674 304.9160 291.0726 278.1360 267.2413 257.4787 248.6151 240.3281

644.2016 602.2492 565.0996 531.5992 501.6705 474.3199 449.2432 426.0405 404.4716 384.5102 365.9011 348.4608 331.9882 316.5649 302.0498 288.3348 275.9500 265.2183 255.6622 246.9184

m; series: (row by row in equal s over time period

635.4464 594.5349 557.9661 525.3740 495.9301 469.1093 444.4515 421.5356 404.4716 380.7192 362.3417 345.0592 328.8130 313.5905 299.2961 285.7450 273.6631 263.2231 253.8853 245.2147

627.0872 586.8960 551.2083 519.2748 490.4462 464.0607 439.6902 417.1813 396.3070 376.9328 358.8414 341.7309 325.6838 310.6943 296.4955 283.1665 271.5456 261.2731 252.1319 243.5340

618.8109 579.4882 544.5161 513.2719 484.9349 459.0376 435.0269 412.8641 392.3452 373.1922 355.2994 338.4507 322.6044 307.7371 293.7404 280.6173 269.4207 259.3871 250.3103 241.9693

REFERENCES [1] L. Ljung and S. T. Soderstrom, Theory and Practice of Recursive Identification. Cambridge, MA: MIT Press, 1983. [2] L. Ljung, System Identification. Englewood Cliffs, NJ: Prentice-Hall, 1987. [3] G. T. Chapman and D. B. Kirk, “A method for extracting aerodynamic coefficients from free flight data,” AIAA J., vol. 18, no. 4, 1970. [4] R. F. Lieske and A. M. Mackenzie, “Determination of aerodynamic drag from radar data,” Aberdeen Proving Ground Tech. Rep., 1972. [5] W. E. Williamson, Jr., “Instrument modeling for aerodynamic coefficient identification from flight test data,” AIAA J. Guidance and Contr., vol. 3, no. 3, pp. 225–279, 1980. [6] C. Fratter and R. F. Stengel, “Identification of aerodynamic coefficients using flight testing data,” AIAA Paper 83-2099, 1983. [7] K. W. Illif, “Parameter estimation for flight vehicle,” AIAA J. Guidance and Contr., vol. 12, no. 5, pp. 609–622, 1989.

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[8] Y. Chen, “Nonlinear yaw card data reduction,” System Engineering and Electronic Techniques, vol. 1991, no. 3, pp. 53–58, 1991 (in Chinese). [9] D. J. Linse and R. F. Stengel, “Identification of aerodynamic coefficients using computational neural networks,” J. Guidance, Contr. and Dynamics, vol. 16, no. 6, pp. 1018–1025, 1994. [10] Y. Chen and H. Dou, “Researches on the optimal control solution of identifying fitting drag coefficient curve from radar measured velocity data,” Aerodynamic Experiment and Measurement Contr., vol. 1993, no. 2, pp. 81–89, 1993 (in Chinese). , “Data fitting algorithm and general purpose software by con[11] strained multistaged polynomials with different orders,” Aerodynamic Experiment and Measurement Contr., vol. 5, no. 3, pp. 78–86, 1991 (in Chinese). [12] Y. Chen, C. Wen, Z. Gong, and M. Sun, “Drag coefficient curve identification of projectiles from flight tests via optimal dynamic fitting,”

J. Contr. Eng. Practice, vol. 5, no. 5, pp. 627–636, May 1997. [13] S. Arimoto, S. Kawamura, and F. Miyazaki, “Bettering operation of robots by learning,” J. Robotic Syst., vol. 1, no. 2, pp. 123–140, 1984. [14] Z. Bien and K. M. Huh, “High-order iterative learning control algorithm,” Proc. Inst. Elect. Eng., vol. 136, pt. D, no. 3, pp. 105–112, 1989. [15] Y. Chen, M. Sun, B. Huang, and H. Dou, “Robust higher order repetitive learning control algorithm for tracking control of delayed repetitive systems,” in Proc. 31st IEEE Conf. Decision and Contr., Tucson, AZ, Dec. 1992, pp. 2504–2510. [16] Y. Chen, “Researches on trajectory prediction models and software for spin-stabilized projectiles (I), (II), (III),” Projectile and Rocket Fascicule of Acta Armamentarii, vol. 1990, no. 3, pp. 1–7; vol. 1990, no. 4, pp. 1–11, 1990; vol. 1991, no. 1, pp. 1–10, 1991 (in Chinese).