Suboptimal Guidance for Passive Homing Missile With Angle-Only-Measurement

SUBOPTIMAL GUIDANCE FOR PASSIVE HOMING MISSILES WITH ANGLE-ONLY-MEASUREMENT Ho-Il Lee· Hungu Lee" Min-Jea Tahk . ,1

• Division of Aerospace Engineering, Korea Advanced Institute of Science and Technology, 979-1 Kusong-Dong, Yusong-Gu, Taejon, Korea 905-701 . .. Satellite Technology Research Center Initiative, 18F Sahak Bldg., 929 Dunsan-Dong, Seo-Gu, Taejon, Korea 902-120.

Abstract: The optimal guidance law obtained from a nonlinear stochastic problem may have a dual-control effect. The guidance law with dual-control effect is addressed by a trade-off study by maintaining good guidance performance and small estimation errors. The direct stochastic optimization is solved by the co-evolutionary augmented Lagrangian method. Because the solution is an open-loop type for various engagement scenarios, a neural-network is implemented to derive a feedback guidance law. Simulation results show that the proposed guidance law is superior to the conventional proportional navigation. Keywords: Estimation, Missiles, Guidance systems, Neural networks, Stochastic systems, Optimization

1. INTRODUCTION

Generally, the guidance law of a missile is designed from a deterministic model by assuming of perfect measurements and tested by a stochastic model including the target state estimator. For passive homing missiles with angleonly-measurement, however, the intercept performance of the guidance law based on the deterministic model is not guaranteed since the missile has an inherent observability problem (Hammel & Aidala, 1985; Nardone & Aidala, 1981). As t he missile approaches the target, the line-of-sight (LOS) rate quickly goes to zero, resulting in reduced target observability, and consequently, poor intercept performance. The guidance kinematics with angle measurements are expressed as nonlinear stochastic equations. It is extremely hard to obtain the solution 1 All correspondence should be forwarded to Professor Min-Jea Tahk. (E-mail: [email protected]. Phone: +82-42-869-3718, Fax: +82-42-869-3710)

because the existence conditions of an optimal controller for the system are unknown and the separation theorem does not hold. One known thing is that the optimal solution has a dual effect: The guidance command attempts to drive the reference output to the desired value, but it may need to introduce intentional perturbations of trajectory when the target states are poorly estimated, which improves the estimation performance. The optimal guidance law considering the dual control effect achieves a correct balance between maintaining good control and small estimation errors (Astrom & Wittenmark, 1989). Although the dual control problems have not been solved rigorously, several studies have been performed under the assumption of the existence of the solution (e.g. Casler Jr., 1978; Hull, Speyer & Burris, 1990). However, these results have complex forms so that they do not have much practical use. Trajectory modulation techniques are also developed to improve target observability under the assumption that better target observability

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would result in smaller miss distances (e.g. Hull, Speyer & Tseng, 1985; Speyer, Hull, Tseng & Larson, 1984; Tahk & Speyer, 1989). In this paper, a direct stochastic optimization is performed in order to minimize the expectation of the miss distance and the control effort of the missile. The solutions for the selected scenarios are obtained by Monte Carlo simulations using co-evolutionary augmented Lagrangian method (CEALM) (Tahk & Sun, 2000) . Additionally, because these are open-loop solutions, a feedback guidance law implemented a neural network (NN) for the case of application. It is motivated by the fact that the NN for deriving an feedback guidance law which is suitable for real-time implementation (Song, 1999). The NN guidance law is trained using the optimal solutions obtained by various situations and verified by applying to other scenarios. The results show that the NN guidance law is superior intercept performances to the conventional proportional navigation (PN) guidance law. 2. STOCHASTIC OPTIMAL SOLUTIONS The guidance kinematics are derived as 2-D linear equations. It is assumed that the process noise and time delay do not exist except state estimation processing time.

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where v is the white Gausian measurement noise. The Kalman filter to estimate states can not directly applied to this problem because the system is expressed as a set of linear dynamics but the measurement as a nonlinear equation. In this paper, the relative range is estimated by a modifiedgain pseudo-measurement filter (MGPMF). It is known that the MGPMF is faster than an extended Kalman filter (Song, Ahn & Park, 1988). Generally, the optimal guidance law is derived by solving the optimal problem to minimize the control efforts specified final condition. The final constraints can be augmented into the performance index. In the stochastic optimization problem, the 444

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Fig. 1. Parameterization and interpolation of guidance law. performance index is defined as the sum of the expected miss distance and the control effort.

where r = y'r; + r; is the relative range, B is the weighting factor and a c is the guidance command of which direction is normal to the missile velocity vector. The desired probability of kill PD is introduced as an inequality constraint.

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where, R is the lethal radius(or range). On the other hand, the missile with angle-only measurement seeker is limited to the look-angle between the missile flight path angle "Im and the LOS angle to track the target, which is also an inequality condition. E [max I"Im(t) - a(t)1l

where (rz, r z ) is the relative position vector, and (v z , vz ) is the relative velocity vector. The target and missile acceleration are denoted by (atz, atz) and (a mz , amz), respectively. For the angle-onlymeasurement missile, the measurement y is the LOS angle expressed as y=a=tan-

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where Bo is a maximum allowable look-angle. The optimization of the guidance law is performed by the CEALM because it does not require any gradient information or the initial guesses for both states and co-states and can obtain the global minimum within a given search space. The optimal guidance law is parameterized by n nodes dependent on the relative range. The last node is fixed to zero to minimize the zero effort miss distance. The commands a c between each node are interpolated by a cubic spline method (see figure 1). The CEALM generates a set of the guidance law and performs a number of Monte Carlo simulations. From simulations, the statistical information is computed and next generation parameter sets are evolved to optimize the performance index. Figure 2 simply shows the proposed optimization technique. Since the numerical solutions are different with respect to various engagement scenarios, it is necessary to obtain solutions for as many scenarios

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Fig. 2. Flow chart of stochastic optimization using CEALM. as possible. The engagement scenario determined by the initial heading of the missile and the target, and the initial range. If the initial range is fixed , the engagement scenarios are considered as figure 3(a). Considering actually possible initial look angle conditions, it can be chosen that 9 possible independent scenarios for the angle-onlymeasurement missiles depicted in the figure 3(b).

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It is well known that the target is not observable in the absence of a maneuver. From this view point, the target is considered to have constant velocity. The initial relative range is fixed as 3000 m, the missile velocity is 500 m/s and the target velocity is 200 m/s o For each selected scenario, the deterministic and stochastic optimal solutions (denoted DOS and SOS) are obtained. For the stochastic model, target states are estimated using the MGPMF presented in Song et al. (1988) and Tahk et al. (1989). The filter uses the constantvelocity target model and the filter initial states are set to the true initial states for each engagement scenarios. The initial covariance P(O) is set diag(P(O)) = [10 6 ,106 ,10 6 ,106 ]. LOS angle measurements are corrupted by the white Gaussian noise sequence with zero-mean and variance 0.03 2 • When the observability is relative bad, the stochastic solutions are different from the deterministic solutions because the guidance tries to enhance the target observability for better intercept performance. Especially, the case that the initial velocity vectors of the missile and the target are co-linear shows the effect most significant as shown in figure 4.

(a) General engagement scenarios. 3. FEEDBACK GUIDANCE LAW USING NEURAL NETWORK 4

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(b) Possible independent engagement scenarios. Fig. 3. Possible independent engagement scenarios for angle-only-measurement missiles.

In this paper, a multilayer neural network guidance law is proposed to obtain the feedback solution from the open-loop solution obtained by the stochastic optimization. The network inputs are the state vectors and the output is the guidance command. The NN is trained by a standard back propagation algorithm (Haykin, 1994). The NN has 2 hidden layers and the inputs are the estimated relative range f, the estimated LOS angle rate &, the LOS angle er, and the flight path angle of the missile "(m, respectively. The numbers of hidden neurons are 10 and 5 for the first and second hidden layer, respectively. Figure 5 shows the structure of the proposed network. SESSION 8

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(7) where W 1 , W 2 and W3 are weighting matrices and f 1,2( ') are activation functions. The activation functions of the hidden neurons are hyperbolic tangent functions. The learning sets are composed of 9 stochastic optimal solutions as shown in figure 6. However, the data near the final phase (r = 0) are not used for train because of the divergence of the LOS rate. It is also observed in the figure that when for the training sets with the same initial "fm ' S, the states except LOS rate are similar but the commands are very different near the initial phase (r = ro) . Hence, the data near the initial phase are excluded for reliable training. During the initial and final few seconds when the NN is not trained, a conventional guidance law is applied. Figure 7 shows the guidance commands and the estimated relative ranges out of the learning sets. Here, 'case if means the case of i-th missile and jth target condition shown in figure 3(b). Figure 8 is the results from the trained weighting matrices, which shows that the network is well trained.

Fig. 6. Learning sets.

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In order to verify the training result of the proposed guidance law, two test scenarios without measurement noise are selected. Next, another test scenario is selected to evaluate the intercept performance of the proposed guidance law with measurement noise. The test scenarios to evaluate the NN training performance are different from the learning scenarios. A missile is fired 500 m/s of speed and a target moves at 300 m/s of a constant speed.

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lateral target acceleration of -15 m/s 2 near the final time for about 1 second is also considered in these cases. The proposed guidance is compared with the PN guidance. Target states are also estimated using the MGPMF. The target model is applied to the constant-acceleration model and the filter state veCtors is XT = [rz r z V z V z atz atzV. The initial covariance P(O) is set diag(P(O)) = [106,106,106,106,104,104]. Monte Carlo simulations are performed with respect to four different initial filter states.

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Fig. 10. Trajectories of the proposed and the PN guidance for Case2. • CaseI: the flight path angle of the missile 30° and that of the target 60°. The target located at (2000 m, Om). • Case2: the flight path angle of the missile 0° and that of the target 100°. The target located at (3500 m, Om).

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The proposed guidance is compared with the a conventional PN guidance .. Figures 9 and 10 show the trajectories for each case. It is observed that the trajectories of the proposed guidance are detached from those of the PN. In the PN guidance, as the missile approaches the target, the intercept trajectory forms into the collision triangle and the LOS rate becomes zero. But the missile does not make the triangle in the proposed guidance, so that the missile enhances the target observability. Another test scenario is selected to evaluate the intercept performance with measurement noise of the proposed guidance law. In view of the target observability, a constant velocity target is rather important. The missile is fired in x direction with a velocity of 550 m/so The target is initially located at (2000 m, 100 m) from the missile, and moves at a constant speed of 250 m/so A constant

• Case 1: Xo = [2,1, -0.3,0,0, O]T, true initial states. • Case 2: Xo = [3,0, -0.3,0,0, oV, initial states of learning sets. • Case 3: Xo = [2,1, -0.2,0,0, oV • Case 4: Xo = [1,O,O .I,O,O,OV Figure 11 is a sample trajectory of 1000 simulations using the proposed guidance. Similar to figures 9 and 10, the trajectory of missile is detached from those of the PN in order to enhance the target observability. Figure 12 shows that the LOS rate of the PN maintains itself small, while the proposed guidance increases to enhance the observability and decreases to intercept the target. The LOS rate is an important measure of the target observability for the angle-only-measurement filter. The superior estimation performance of the proposed guidance law implies that it has the dual control effect. It is shown in figure 13 that the relative range is well estimated by the proposed guidance. Table 1 shows the probability of destruction with respect to the lethal range for each case, which shows that the proposed guidance is superior to the conventional PN guidance though the presented intercept performance depends on various factors such as filter parameters. On the other hand, the performance of the proposed guidance is somewhat inferior to the PN in the Case 4. It is because the filter initial states are too different from the learning sets.

Table 1. Comparison of probability of destruction w.r.t. the lethal range. case2

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6. REFERENCES

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5. CONCLUSION The stochastic optimal solutions are directly obtained by an evolutionary computation method for a passive homing missile. Using the neural network, a feedback guidance law is obtained from the optimal open-loop solutions. The proposed neural-network guidance shows the dual control effect which enhances the target observability and 448

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the intercept performance. Though only constantvelocity target and 2-D kinematics are considered in this paper, this method can be easily extended to the maneuvering target and 3-D case .

Astrom, K.J., and B . Wittenmark (1989) . Adaptive Control, Addison-Wesley Publishing Company, New York. Casler Jr., R.J. (1978). Dual-control guidance strategy for homing interceptors taking angleonly measurements. Journal of Guidance, Control, and Dynamics, 1,63-70. Hammel, S.E. and V.J. Aidala (1985) . Observability requirements for three-dimensional tracking via angle measurements. IEEE 1hlnsactions on Aerospace and Electronic Systems, 21, 200-207. Haykin, S (1994). Neural Networks, Prentice-Hall International Inc., London. Hull, D. G., J. L. Speyer, and D. B. Burris (1990). Linear-quadratic guidance law for dual control of homing missiles. Journal of Guidance, Control, and Dynamics, 13, 137-144. Hull, D.G., J .L. Speyer and C.Y. Tseng (1985) . Maximum-information guidance for homing missiles. Journal of Guidance, Control, and Dynamics, 8,494-497. Nardone, S.C. and V.J. Aidala (1981). Observability criteria for bearings-only target motion analysis. IEEE Transactions on Aerospace and Electronic Systems, 17, 162-166. Song, E.J. and M.J .Tahk (1999). Real-time midcourse missile guidance robust against launch conditions. Control Engineering Practice, 7, 507-515. Song, T .L., J.Y. Ahn and C. Park (1988) . Suboptimal Filter Design with Pseudo-measurements for Target 'fracking. IEEE Transactions on Aerospace and Electronic Systems, 24, 28-39. Speyer, J.L., D.G. Hull, C.Y. Tseng and S.W Larson (1984) . Estimation enhancement by trajectory modulation for homing missiles . Journal of Guidance, Control, and Dynamics, 7, 167-174. Tahk, M. J., and J . L. Speyer (1989). Use of Intermittent Maneuvers for Miss Distance Reduction in Exoatmospheric Engagements, Proceedings of 1989 AIAA Guidance and Control Conference, Boston, Mass., 1041-1047. Tahk, M.J. and B.C. Sun (2000). Co-evolutionary augmented Lagrangian methods for constrained optimization. IEEE Transactions on Evolutionary Computation, 4, 114-124.