Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile
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Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile
Presented By Mr. Partha Saradhi Vankara st 1 Semester, M.Tech, Roll No:10GNC08 Department of Electrical Engineering College of Engineering, Trivandrum Thiruvananthapuram16 2010 Abstract In this topic, an interceptor (missile) with bounded acceleration, pursues a target which makes a sudden step maneuver. A multiplemodel adaptive estimatorguidance law is pre sented for state estimation of the target. The estimators proposed for each target model is placed in a filter bank. These estimators differ in the expected timing of the target ma neuver jump and feature a mechanism to efficiently identify such a jump. The certainty equivalence principle seems to be invalid in the case of bounded missile acceleration. There fore the guidance gain for each target model, depends on the measurement noise level, target maneuver statistics, and saturation limit. These gains can be computed a priori and stored in lookup table to be used online, which are different for each target model in the bank. The expressions for the zeroeffort miss for all models are identical; however, since their value is independently evaluated by each estimator in the bank, they are in general different. Simulation results show significant improvement over the deterministic optimal guidance law when jump times vary from 10 to 3 missile time constants before intercept. A sensitivity analysis to various noise levels and expected target maneuvers emphasizes the robustness of this scheme. Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile.pdf (Size: 791.39 KB / Downloads: 53) Contents 1 Introduction 1 2 Problem Formulation 3 3 Single Element EstimatorController Formulation 6 3.1 Single Element Temporal MultipleModel Estimator Equations . . . . . . . . 6 3.2 Describing Function Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.3 Saturated Optimal Guidance Law built for a Single Temporal MultipleModel Element Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.4 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.4.1 Homing Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.4.2 Gains Sensitivity to Noise Level . . . . . . . . . . . . . . . . . . . . . . 12 3.4.3 Gains Sensitivity to Expected Target Maneuver Magnitude . . . . . . 13 4 MultipleModel Adaptive Control Approach Solution 15 4.1 MultipleModel Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 MultipleModel Adaptive Control Solution . . . . . . . . . . . . . . . . . . . . 16 4.3 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.3.1 Comparison to the Bound . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.3.2 Measurement Noise Level Impact . . . . . . . . . . . . . . . . . . . . . 19 4.3.3 Expected Target Maneuver Magnitude Impact . . . . . . . . . . . . . 20 5 Conclusions 22 iii List of Figures 2.1 Linearized endgame geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.1 Error function(erf) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Sample SOGL guidance gains . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Sample CDF for target maneuver at θjump=7 , σjump = 60[m/s2 ] 3.3 . . . . . . . . 11 The rms miss distance vs θjump , σjump = 60[m/s2 ] . . . . . . . 3.4 . . . . . . . . 12 3.5 N’ sensitivity to angular measurement noise level σφd . . . . . . . . . . . . . 13 3.6 N’ sensitivity to expected target maneuver magnitude σjump . . . . . . . . . 14 4.1 MMAC11F approach solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 The rms miss distance vs θjump ,σjump = 60 [m/s2 ]; Comparision of MMAC11F 4.2 with the performance bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 The rms miss distance vs θjump , σjump = 60 [m/s2 ].Target jumps ±σjump 4.3 Comparision of MMAC11F with the performance bound . . . . . . . . . . . . 19 MMAC11F:The rms miss distance vs θjump ,σjump = 60 [m/s2 ]; Target behaves 4.4 as its model. Comparision of noise levels. . . . . . . . . . . . . . . . . . . . . 19 The rms miss distance vs θjump , σjump = 60 [m/s2 ].Target jumps ±σjump ; 4.5 Comparision of noise levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 MMAC11F:The rms miss distance vs θjump , σjump = 40,60,90 [m/s2 ]. Target 4.6 jumps with σjump . Comparison of expected target maneuver effect. . . . . . . 20 The rms miss distance vs θjump , σjump = 40,60,90 [m/s2 ]; Target jumps 4.7 ±σjump .Comparison of expected target maneuver effect . . . . . . . . . . . . 21 iv Chapter 1 Introduction The most popular and widely used guidance law is the PN (Proportional Navigation) guid ance law. PN issues a guidance acceleration command, which is jointly proportional to the inertial angular rate of the LOS between the missile and target, and the missiletotarget closing velocity. PN results in a zero miss distance, if ideal dynamics are assumed and if there are no constraints on the acceleration. If missile and target speeds are constant, as suming ideal dynamics, for a nonmaneuvering target, both the target and the missile move close to straight lines that meet each other at the intercept point. These lines together with the LOS of missile and target are referred to as the Collision Triangle which is the basis for a linearized version of the general guidance problem. PN also has its drawbacks. For example, in realistic interception scenarios between a pursuer with limited maneuver ability and a target with large maneuver capability, it results in a significant miss distance. To reduce the miss distance and to relax the large acceleration requirements, Modern guidance laws are formulated, which required additional information such as the predicted time to intercept (or) the time to go tgo and the missile and target accelerations. An important concept of modern guidance laws is the Zeroeffort miss (ZEM). ZEM is the miss distance that would result if the missile made no further corrective com mands, and the target followed the assumed acceleration model. The PN guidance law can be expressed using the ZEM concept. The ZEM is constructed assuming ideal dynamics and taking target’s acceleration to be zero. The Augmented Proportional Navigation guidance law (APN) suggests an augmentation term compensating for constant target acceleration by including its effect in the ZEM term. The direct effect of APN compared to PN is to reduce missile acceleration at the end of the conflict at the expense of larger acceleration at the initial time, and to reduce the total required maneuver. For a maneuvering target, assuming ideal dynamics for the missile, using the optimal control theory, it was proven that APN with navigation coefficient N’=3 is in fact an opti mal guidance law. This guidance law minimizes, a quadratic cost function, of the integral of the square of the missile acceleration subject to ZEM constraint. Removing the ideal missile dynamics assumption and replacing it with firstorder dynamics, and using the same optimal control tools, resulted in the birth of the Optimal Guidance Law (OGL). In realworld prob lems, sensor and system noise are the major influences on miss distance in a pursuerevader scenario. In realistic scenario, it can be assumed that the pursuer’s state is available for use in the guidance law. However, the evader’s state is generally not known to the pursuer. A welldesigned evader guidance law will make the evading target as difficult to predict as possible. Therefore, target state estimation is essential. The common linear state estimator is the Kalman filter. For a nonlinear system or nonlinear observations, the extended Kalman filter may be used. The EKF updates a linearization around the current state estimate. When the assumed target maneuver and the observation noise statistics differ from the ac tual ones, the performance of the Kalman filter is degraded. 1 Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile One way of overcoming this disadvantage is a method known as the MultipleModel Adaptive Estimation (MMAE)method [1]. Here, the target acceleration is assumed to follow one of the several target models. Then processing the observations simultaneously, using all possible models and finally using a weighted average of the resulting estimates, is in fact an optimal estimator structure.The MMAE assumes that the system is in one of a finite number of modes. Each filter’s estimate is weighted through the filter’s weighting coefficient to form the full multiplemodel filter’s estimate. The weighting coefficients are a function of the measurements available up to the current time. The disadvantage of the MMAE scheme is that, the estimator is required to contain a large number of filters, describing all the possible system modes. An approximation of the MMAE scheme is the Temporal MultipleModel (TMM) esti mator, with the important addition of input estimation. For the assumed target model, the target acceleration is represented as a white noise process. Pursuer (missile) acceleration saturation is another major source of increased miss distance. Saturation brings another important issue for the estimatorguidance scheme. The generally assumed certainty equiv alence principle (CEP) is not valid for a limited acceleration control missile.The certainty equivalence principle states that the optimal control law for a stochastic control problem is the optimal control law for the associated deterministic problem for which the known states are replaced by the estimated ones. When the CEP holds, it is possible to design an estimator and a control law independently of each other. Here the control law may be designed on the basis of deterministic states. When the CEP is not valid, it was proved that the estimator may be designed independently of the control problem, but the optimal controller depends on the statistics of the estimated state. This principle is known as the oneway separation (or) General Separation Theorem (GST). It can be used to suggest an approach for saturation inclusion in the guidance law. This new linear guidance law named Saturated OGL (SOGL) describes saturation using the random input describing function (RIDF) method and solves the stochastic controller problem. This unique formulation results in a guidance law gain that depends on the estimated state statistics. [2] provides the basis for the guidance law used in the current work. A later work [3] proposed the optimal nonlinear solution for this problem by numerically solving the stochastic optimization problem. When certainty equivalence is not valid, a separate guidance law should be constructed for each estimator, wherein the initial part of the scenario is devoted to the model identifica tion after which one filterguidance law branch is selected. The suboptimal multiplemodel adaptive control (MMAC) scheme [4] can be used, which degenerates to an MMAE estimator and the APN guidance law.The derivation of the guidance law proceeds in two stages. In the first stage, it is assumed that the instance that the target issues a step command in its acceleration is known. But the size of the target maneuver jump must be estimated. In the second stage, it is assumed that the target performs a single step maneuver at an unknown time near intercept. A suboptimal guidance law is derived using the MMAC formulation, making this estimationdependent guidance law adaptive to the target’s maneuver with a gain, varying according to the current estimated state error statistics.The remainder of this paper [5] is organized as follows. The next section depicts the problem formulation, the model used, the assumptions, and the available measurements. Next is the solution for a single estimator element matched to the new guidance law and a performance evaluation. This is followed by an MMAC type solution, results, analysis and conclusions. Department of Electrical Engineering, College of Engineering, Trivandrum 2 Chapter 2 Problem Formulation The dynamic model of the problem is constructed using the following assumptions. 1) the missiletarget engagement occurs in one plane, 2) missile target are represented by a point mass model with linear dynamics, 3) the relative endgame trajectory is linearized about the initial LOS direction, 4) the missile and target speeds remain constant, 5)missile acceleration is bounded, 6) missile and the target are modeled as firstorder dynamic systems and 7) missile acquires noisy measurement of some statevariables. Figure 2.1 shows the linearized endgame geometry. Here X axis is aligned with the initial LOS direction, r is the range between the missile and the target, φ is the angle between current LOS and initial LOS, y is the relative displacement perpendicular to the initial LOS, aT and aM are the accelerations of the target and missile normal to the initial LOS respectively. Figure 2.1: Linearized endgame geometry The proposed target model for the estimator is a firstorder system, represented by a time constant τ T and noise. The deterministic part of the target model is given as: 1 aT = (2.1) aTC 1 + s τT where aTc is the target’s acceleration command. For a constant acceleration command, the general target dynamics are described by: y ̇T = υT (2.2) υ ̇T = aT (2.3) −1 1 1 a ̇T = aT + aTc + wT (2.4) τT τT τT aTc = 0 ̇ (2.5) 3 Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile where yT ,υ T , aT are the target’s position, velocity and acceleration respectively and wT is the white noise process, representing the target acceleration uncertainties( small, possibly random, changes as would result from wind gusts, turbulence, target flight control effects, etc.). All are perpendicular to the initial LOS direction. The acceleration command satisfies the following assumptions: 1) acceleration command is a single step function. 2) magnitude of the step command is taken from a Gaussian distribution. The state used in this problem consists of the relative position, velocity, missile acceleration, target acceleration and target acceleration command, all being perpendicular to initial LOS direction. aTc ]T X = [y y aM ̇ aT (2.6) The missile is modeled as a firstorder dynamic system with a time constant τ M having the following acceleration transfer function: aM 1 (2.7) = u 1 + sτM where ‘u’ is the maneuver command. Since constant closing speed uC is assumed, and the initial range between the missile and target is ‘r0 ’ , the final intercept time tf is approximated by: tf = r0 /υc (2.8) The time to go is defined as tgo ∆ tf − t (2.9) = and the normalized time to go is θ∆ tgo /τM (2.10) = The bounded control is described as a standard saturation function. = UM if u > UM (2.11) sat(u) = u if − UM ≤ u ≤ UM (2.12) = −UM if u < −UM (2.13) here UM = aMmax is the missile’s maximal maneuver capability. The equation of motion is given as: ̇ X = Ax + bsat(u) + Gw; x(0) = x0 (2.14) where 0 1 0 0 0 −1 0 0 1 0 −1/τM 0 0 0 0 A = (2.15) −1/τT 0 0 0 1/τT 0 0 0 0 0 0 0 b = 1/τM (2.16) 0 0 0 0 0 0 G = 1/τM 0 (2.17) 0 1/τT 0 0 Department of Electrical Engineering, College of Engineering, Trivandrum 4 Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile and w = [wM ,wT ]T is a two dimensional white noise, and wM and wT are the white Gaussian process representing the uncertainty in missile and target acceleration. The standard devia tion of the missile’s acceleration is taken to be small (0.11g) and represents the uncertainties of the missile’s sensors. The continuoustime measurement taken by the pursuing missile is given as: z = φ + υ ≈ cx + υ (2.18) where the measurement matrix ‘c’ is c = [1/r 0 0 0 0] (2.19) and ‘u’ is the angle measurement error modeled as a continuoustime white Gaussian noise with spectral density V. 2 V = σφ (2.20) 2 2 2 (For discretetime measurements taken every Ts seconds with variance σφd , σφ = σφd Ts ). This range ‘r’ is assumed to be measured accurately. The cost function to be minimized is: tf T (X T QX + Ru2 )dt] J = E[Xf Sf Xf + (2.21) 0 where the subscript ‘f’ denotes the value of the final time. For this interception problem, the following matrices are chosen sf 0 0 0 0 0 0 0 0 0 Sf = 0 0 0 0 0 (2.22) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Q = 0 0 0 0 0 (2.23) 0 0 0 0 0 0 0 0 0 0 therefore the weighting coefficients ‘sf ’ and ‘R’ become the design parameters of the problem. Department of Electrical Engineering, College of Engineering, Trivandrum 5 Chapter 3 Single Element EstimatorController Formulation The problem of estimating a sudden step maneuver during the intercept of timetogo values (equal to several missile time constants) is a very practical and difficult one. This is the time span in which a large portion of the miss distance is formed. The problem becomes more complicated when the missile’s acceleration is bounded, causing another growth in the miss distance. Knowing these two elements’ contribution to the guidance performance degradation, we make use of the special attributes of the TMM estimator, and the saturation inclusion in the guidance law to deal with these two major problems in the guidance system. In this section, a single TMM  type element estimator matched to a target acceleration step, of known magnitude distribution, at a known time is used. Later, this estimator will be combined with the SOGL algorithm to create the guidance law (using the oneway sepa ration). The single element TMM estimator is the optimal estimator for the assumed target model, and the SOGL will be matched exactly to this optimal estimator. This combination will yield the best available performance such a pair can achieve. 3.1 Single Element Temporal MultipleModel Estimator Equa tions The initial statistics of the estimated state vector x are ˆ E[x(0)] = x0 ˆ (3.1) E[ˆ (0) xT (0)] = Pxx 0 (3.2) x ˆˆ the estimation error, denoted as (e), is defined as: e=x−x ˆ (3.3) Assuming that x0 , w(t) and υ(t) are independent and using Eqn.(2.14), we obtain the ˆ following continuoustime filter equations: ̇ x = Aˆ + bsat(u) + kf (z − cˆ) ˆ x x (3.4) kf = Pee cT V −1 (3.5) Pee = APee + Pee AT − Pee cT V −1 cPee + GGT ̇ (3.6) 0 Pee (0) = Pee (3.7) The special feature of the TMM estimator, that allows it to identify sudden target accelera tion jumps, lies in the estimator initialization. The initialization includes two operations: 1) 6 Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile initializing the estimator’s state vector, 2) initializing the estimator’s error covariance matrix. The state vector is initialized using the best available data at the time of the initialization, and setting the fifth state member to aTc = 0. The error covariance matrix is initialized by ˆ taking the previous value of the 4 by 4 matrix, forcing p55 to the jump uncertainty value σ2jump and setting remaining elements in the fifth row and column to zero. The setting of the remaining elements in the fifth row and column to zero, is due to the fact that the (assumed) new acceleration command is history independent, and hence uncorrelated to the other state variables. It is shown schematically below p− p− p− p− 0 11 12 13 14 p− p− p− p− 0 21 22 23 24 p− p− p− p− + 0 Pee = (3.8) 31 32 33 34 p− p− p− − p44 0 41 42 43 2 0 0 0 0 σjump where ( )− represents the values before the initialization, and ( )+ the values after ini tialization. This is, in fact, the optimal estimator when (1) the time of the jump in the target’s acceleration command is known, (2) and the target’s acceleration step command is uncorrelated with the previous commands. 3.2 Describing Function Approach The missile acceleration saturation will be approximated by an RIDF as shown. Let φ (ζ) be an odd singleinputsingleoutput nonlinearity driven by a zero mean Gaussian process(ζ). The approximation of φ (ζ) by a linear gain leads to the following minimization problem: min − Lζ]2 L E[φ (ζ) (3.9) where ‘L’ is a RIDF (random input describing function). Let φ (ζ) be the saturation function defined by Eq. (2.11),(2.12)and (2.13) where ζ = u. The associated RIDF in this case is given by: √ L = erf (Um / 2σu ) (3.10) where η 1 2 e−ψ dψ erf (η) = √ (3.11) π −η 2 u ≈ N (0, σu ) (3.12) L and (1L) represent the probability of not reaching and reaching saturation respectively. √ Examine figure 3.1, the xaxis is the error function’s argument Um / 2σu as in Eq.(3.10) and the yaxis is the RIDF’s equivalent gain L. σu represents the range of values of the control signal in a closedloop system (due to state uncertainties). The following is observed: 1) if the standard deviation of the acceleration command approaches zero, the argument increases and the gain ‘L’ approaches 1, meaning the saturation is inactive. 2) If the standard deviation of the acceleration command rises as in stochastic case, the argument decreases and the gain ‘L’ drops to values lower than 1, inducing an attenuation factor, simulating the effect of saturation in the closedloop system. Using the RIDF concept, we will obtain the optimal linear controller such that u = kc xˆ (3.13) Calculating the control signal variance yields 2 T σu = kc Pxx kc (3.14) ˆˆ the calculation of the covariance matrix Pxx is based on the approximation of the saturation ˆˆ nonlinearity by the RIDF. Using Eqns. (2.14), (3.10) and (3.13) ̇ x = (A + Lbkc )ˆ + kf (z − cˆ) ˆ x x (3.15) Department of Electrical Engineering, College of Engineering, Trivandrum 7 Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile Figure 3.1: Error function(erf) is obtained. Because υ = z − cˆ is the innovation sequence, x E[υ(t)υ T (τ )] = V δ(t − τ ) (3.16) we can write the state covariance differential equation as: ̇ Pxx= (A + Lbkc )Pxx + Pxx (A + Lbkc )T + kf V kf T (3.17) ˆˆ ˆˆ ˆˆ 0 Pxx (0) = Pxx (3.18) ˆˆ ˆˆ 3.3 Saturated Optimal Guidance Law built for a Single Tem poral MultipleModel Element Estimator After the estimator has been formulated and the error covariance matrix is known for the entire scenario duration, the optimal guidance law may be calculated using the oneway separation principle. The cost function to be minimized is given by tf tf f J = E[xT Sf xf + (xT Qx+Ru2 )dt] = tr[(Pxx +Pee )Sf ]+ f T (tr[(Pxx +Pee )Q]+Rkc Pxx kc )dt ˆˆ ˆˆ f ˆˆ 0 0 (3.19) where the minimization is performed with respect to kc , subject to Eq. (3.17) and (3.18). Pee does not depend on kc (the oneway separation principle) and the problem to be solved is tf min f T tr(Pxx Sf ) + [tr(Pxx Q) + Rkc Pxx kc ]dt (3.20) kc ˆˆ ˆˆ ˆˆ 0 subject to Eq. (3.17) and (3.18), where Um L = erf (3.21) T 2kc Pxx kc ˆˆ The Hamiltonian associated with this problem is H = [tr(Pxx Q) + Rkc Pxx kc ] + tr [(A + Lbkc )Pxx + Pxx (A + Lbkc )T + kf V kf ]S + (3.22) T T ˆˆ ˆˆ ˆˆ ˆˆ 2 Um T λ kc Pxx kc − ˆˆ 2[erf −1 (L)]2 ,where the symmetric matrix S and the scalar λ are Lagrange multipliers. The optimal solution is obtained by solving the following equations: ∂H =0 (3.23) ∂kc Department of Electrical Engineering, College of Engineering, Trivandrum 8 Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile ∂H ̇ S=− (3.24) ∂Pxxˆˆ and the transversality condition S(tf ) = Sf (3.25) L is also an optimization variable under the constraint of Eq.(3.21); thus, ∂H =0 (3.26) ∂L Equation (3.23) leads to the relation L T kc = − b S (3.27) R+λ and substituting Eq. (3.27) into Eq. (3.21) yields 2 2 L Um bT SPxx S T b = (3.28) ˆˆ 2[erf −1 (L)]2 R+λ Equations (3.24) and (3.25) are equivalent to the following differential equation: L2 ̇ S = −Q − AT S − SA + SbbT S (3.29) R+λ S(tf ) = Sf (3.30) Equation (3.26) is used to derive an expression of λ in terms of L: R λ= (3.31) √ π exp{[erf −1 (L)]2 } −1 2 L erf −1 (L) Equation (3.28) can be rewritten as R+λ Um bT SPxx Sb = √ (3.32) ˆˆ 2erf −1 (L) L Substituting Eq. (3.31) into Eq. (3.32), we obtain R R+ √ π exp{[erf −1 (L)]2 } −1 L Um erf −1 (L) 2 bT SPxx Sb = √ (3.33) ˆˆ 2erf −1 (L) L Equations (3.27),(3.29) and (3.30) induce the following guidance law structure: NSOGL (tgo ) u = ac = ZEMSOGL (3.34) t2 go with the navigation gain N’ SOGL taken from Eq. (3.27) NSOGL (tgo ) = kc (tgo )t2 [1 0 0 0 0]T (3.35) go and 1 2 ZEMSOGL = y + y tgo − aM τM (e−θ +θ −1)+ aT τT (e−θT +θT −1)− aˆ c τT (e−θT +θT −1− θT ) ̇ ˆ 2 ˆ 2 2 ˆ ˆ T 2 (3.36) θT = tgo /τT (3.37) Department of Electrical Engineering, College of Engineering, Trivandrum 9 Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile 3.4 Performance Evaluation To evaluate the potential of the newly proposed estimatorguidancelaw pair a performance evaluation was carried out using a simulation of the onedimensional engagement (perpen dicular to the initial LOS). A discretetime implementation was chosen, sampled over the sampling period Ts . The scenario’s parameters are described in Table 3.1. In the scenario described, the target produces an acceleration step at a predefined normalized time (denoted as θjump ) which is varied in steps of 0.1s during the last second of the scenario. The tar get generates commands as assumed by the estimator’s model, that is, commands a step in acceleration for which the amplitude is chosen from a Gaussian distribution with the known σ jump . For each target jump time, a matched single TMM element estimator was constructed, inducing the covariance change as in Eq. (3.8) exactly at the known jump time. Then, the covariance of this single element TMM estimator is fed into the SOGL algorithm, resulting in a matched guidance law gain for this specific element estimator. This estimatorguidance law pair represents the best available match and leads to a performance bound. Typical guidance gains are shown in Figure 3.2 as a function of the normalized time to go θ for different values of target maneuver times. As expected, the effective navigation gain N’ is influenced by the covariance increase of the filter. All N’ gains, including that of OGL, converge asymptotically to 3, the constant gain of PN. When the filter initialization occurs a long time before the intercept (θjump = 10, 7) the gain resembles that of the original formulated SOGL. When the initialization is expected to happen exactly at the engagement Figure 3.2: Sample SOGL guidance gains termination (θjump = 0) the gain is identical to that of the classical OGL, as for this cases a nonmaneuvering target is expected. If the target jump is expected to happen close to the terminal time (θjump = 5, 2) the covariance increase causes a substantial growth in the N’ Department of Electrical Engineering, College of Engineering, Trivandrum 10 Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile values for θ ∈ [0.5, 2]. This illustrates the inherent feature of the SOGL to increase, early in the engagement, the gain when uncertainties rise. In a full functional estimator, this increase in the covariance represents the chance of a target maneuver, which means that an increase in N’ would occur when the target performs a jump. This agrees with the common guidance engineering practice of increasing the gain when the target maneuvers. 3.4.1 Homing Performance , OGL and SOGL guidance laws were compared, all using the same target state estimates from a single TMM estimator. Hence, the achieved improvement is due solely to the dif ferent guidance laws used. For each jump time 100 Monte Carlo runs were simulated and the rms of the miss distance was calculated. Figure 3.3 shows a sample missdistance (MD) cumulative distribution function (CDF) for a target maneuvering at θ = 7. The superiority of SOGL over OGL and PN is evident. For example, when using SOGL, in 90 percent of the cases the miss distance was smaller than about 0.25m, whereas for OGL and PN it was lower than about 0.5 and 2.5 m, respectively. We will use the RMS of the MD from the 100 runs, when comparing the different guidance laws. Here it is 0.29, 0.54, 1.39 m when using SOGL, OGL, and PN respectively. Figure 3.4 shows the rms MD obtained form 100 Monte Carlo runs for each target ma Figure 3.3: Sample CDF for target maneuver at θjump=7 , σjump = 60[m/s2 ] neuver time. In Figure 3.4 a line representing PN performance is included to emphasize the substantial difference between PN, OGL, and SOGL. Figure 3.4: The rms miss distance vs θjump , σjump = 60[m/s2 ] In all comparisons similar PN behavior was observed, and so, to show the comparable Department of Electrical Engineering, College of Engineering, Trivandrum 11 Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile difference between OGL and SOGL, only these two will be presented in the figures to follow. An examination of Figure 3.4 shows typical behavior: little to no improvement occurs in the vicinity of θjump = 10 Most of the improvement appears around θjump = 5, 6, 7 ( about 30 percent), and from θjump = 3 and on, OGL and SOGL performance coincide. θjump = 0 represents the case of a nonmaneuvering target. The behavior at θjump = 10 are explained by the fact that both OGL and SOGL have enough time to close the ZEM and obtain a similar miss distance. Also, by examining Figure 3.2, it is obvious that SOGL’s N’ for target jumps in this region resembles that of the ordinary OGL. The region of θjump = 5, 6, 7 is where most improvement is observed. This is expected because this is the region when N’ of SOGL is significantly different from that of the ordinary OGL. This is the region in which, if a target maneuvered and the maneuver were properly identified, it would be wise for the pursuer to apply as much acceleration command as possible to null the ZEM. SOGL’s higher gain does just that. The region of θjump ∈ [0, 4] is characterized by the fact that both OGL and SOGL (and from θjump = 2, PN) gains are high enough to saturate the acceleration command (as N’ is divided by tgo), so that the performance of SOGL and OGL are similar. It is important to understand that the extra information used in SOGL (the covariance enlargement) is of no use to the conventional OGL and does not change its structure or behavior; thus, OGL is the reasonable comparison basis for SOGL. 3.4.2 Gains Sensitivity to Noise Level For a matched estimator SOGL, the only difference between OGL and SOGL lies in the dif ferent behavior of the guidance gain N’. This fact implies that, by exploring the behavior of N’ with some varying parameters, we will gain better understanding of this guidance system and its expected behavior. In this section, we explore the varying behavior of the guidance gain N’ when the angular measurement noise level changes. The scenario parameters are de fined in Table 3.1 with the varying angular measurement noise as defined in table 3.2. For the sake of clarity, and to deliver only the important details, the labels of the axes were removed from the figure and are elaborated as follows: 1) the horizontal x axis is the normalized time to go θ, 2) the vertical y axis is the guidance gain N’ and 3) each graph’s title represents the corresponding θ in which a jump had occurred and for which the estimatorguidancelaw pair was constructed. The gains shown in Figure 3.5 , which corresponds to target jump times of θ = 9,8,7,....,1, are the gains for which most of the important phenomena may be observed: (1) As observed before, a short time after a filter has been initialised, the gain increases. This increase in N’ grows larger with the increase in noise level. The phenomenon is best observed in θjump = 4, 5. This is the SOGL’s reaction to an increase in the estimation statistics uncertainties due to larger measurement noise. (2) The increase in measurement noise causes the gains to rise earlier in the last few missile time constants of the intercept. Again, this represents the SOGL’s feature of increasing the gain when uncertainties rise and saturating the missile’s acceleration command earlier in the scenario. The phenomenon Department of Electrical Engineering, College of Engineering, Trivandrum 12 Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile Figure 3.5: N’ sensitivity to angular measurement noise level σφd is best observed in θjump = 7, 8, and 9. Both effects represent the physical and common engineering practice of increasing he gain when uncertainties in the expected ZEM values increase to react fast to a possible target maneuver. The high values of the navigation gain serve to saturate the guidance command for relatively small values of ZEM. The value of the guidance gain primarily influences the specific value of ZEM required to saturate the guidance command. 3.4.3 Gains Sensitivity to Expected Target Maneuver Magnitude In this section, we explore the varying behavior of the guidance gain N’ when the expected target maneuver magnitude changes. The scenario parameters are defined in Table 3.1 with the varying expected maneuver magnitude σjump as defined in Table 3.3. For the sake of clarity, and to deliver only the important details, the labels of the axes were removed from the figure and are elaborated as follows: 1) the horizontal x axis is the normalized time to go θ, 2) the vertical y axis is the guidance gain N’ and 3) each graph’s title represents the corresponding θ in which a jump had occurred and for which the estimatorguidancelaw pair was constructed. The gains shown in Figure 3.6 which corresponds to target jump times of θ = 9, 8, 7,...,1, are the gains with which most of the important phenomena may be observed. (1) A short time after a filter has been initialized, the gain increases. This increase in N’ grows larger with the increase in expected target maneuver magnitude. The phenomenon is best observed in θjump = 6, 7, and 8. Again this is the SOGL’s reaction to an increase in the estimation statistics uncertainties due to larger expected target maneuver magnitude. (2) The increase in expected target maneuver magnitude causes the gains to rise earlier in the last few missile time constants of the intercept. This represents the SOGL’s feature of increasing the gain when uncertainties rise, and saturating the missile earlier in the scenario. Department of Electrical Engineering, College of Engineering, Trivandrum 13 Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile Figure 3.6: N’ sensitivity to expected target maneuver magnitude σjump The phenomenon is best observed in θjump = 3, 4, and 5. (3) It is observed that the increase in expected target maneuver magnitude does not increase the gain proportionally, but rather causes an abrupt and powerful increase in N’. this is due to the fact that SOGL identifies the expected target maneuver magnitude as being a very large expected ZEM uncertainty, causing a large increase in N’ and inducing a saturated pursuer acceleration command as early as possible. These effects, again explain the common engineering practice of increasing the gain when uncertainties in the expected ZEM values increase, to react fast to a possible target maneuver. The high gains, in the last few missile time constants of the intercept yield a saturating pursuer acceleration command. Department of Electrical Engineering, College of Engineering, Trivandrum 14 Chapter 4 MultipleModel Adaptive Control Approach Solution In this section, an MMAC scheme is used to construct a feasible implementation of an estimatorguidance law scheme. In this scheme, the time of the jump in the target’s ac celeration is no longer assumed to be known. To derive this solution, a brief review of the multiplemodel approach is presented, followed by an overview of the MMAC approach principles and the specific suggested MMAC solution for the problem at hand. 4.1 MultipleModel Approach Our approach is based on the use of the MM approach, which is now introduced. In the basic MM approach, it is assumed that the system obeys one of a finite number of models. The filter is constructed using a Bayesian framework. Starting with previous probabilities of each model being correct, the corresponding posterior probabilities are obtained. It is assumed that the model the system obeys is fixed (referred to as the system mode being fixed) for the entire estimation process and that it is one of the ‘r’ possible models (the system is in one of ‘r’ possible modes). Mathematically formulating the above, the system mode M is M ∈ {Mj }r j=1 (4.1) The prior probability that Mj is correct (the system is in mode j) given Z0 is p Mj Z 0 = μj (0) ; j = 1, ..., r (4.2) where Z0 is the prior information. It can be noted that Σr μj (0) = 1 (4.3) j=1 because the correct model is among the assumed r possible models. It is assumed that all models are linear Gaussian. Using Bayes formula, the posterior probability of model j being correct, given the mea surement data up to time k ( referred to as the weighting coefficients), is obtained recursively as p z (k) Z k−1 , Mj μj (k − 1) μj (k) = p Mj Z k = r ; j = 1, ...., r (4.4) Σj=1 p [z (k) Z k−1 , Mi ] μi (k − 1) starting the recursive calculation with Eq. (4.2). The probability appearing in the nominator of Eq. (4.4) is referred to as the likelihood function of mode j at time k, which, under the linear Gaussian assumptions, is given by the expression 15 Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile Λj (k) ≡ p z (k) Z k−1 , Mj = p[vj (k)] ≈ N [0, Sj (k)] (4.5) where vj and Sj are the innovation and its covariance from the mode matched filter cor responding to mode j. A filter matched to each mode is constructed, yielding a mode conditioned state estimate and a modeconditioned error covariance. The probability of each mode being correct is obtained according to Eq. (4.4) based on its likelihood function (4.5) relative to the other filters’ likelihood functions. This filter structure consists of r linear filters. After the filters are initialized, they run recursively on their own estimates. Their likelihood functions are used to update the mode probabilities. The latest mode probabili ties are used to combine the modeconditioned estimate and covariance for that time. Under these assumptions, the probability density function of the state is a Gaussian mixture with r terms: p[x(k)Z k ] = Σr μj (k) N [ˆj (kk), Pj (kk)] x (4.6) j=1 The combination of the modeconditioned estimates and covariance is carried out as follows: x(kk) = Σr μj (k) xj (kk) ˆ (4.7) j=1 and P (kk) = Σr μj (k) P j (kk) + [xj (kk) − x(kk)][xj (kk) − x(kk)]T (4.8) j=1 Equations (4.7) and (4.8) are exact, under the following assumptions: 1) the correct model is among the set of models considered, and 2) the same model has been in effect from the initial time. This multiplemodel formulation is usually referred to as the multiplemodel adaptive estimation scheme. 4.2 MultipleModel Adaptive Control Solution Multiplemodel adaptive control is an approximate (suboptimal) scheme for combining an MMAE type estimator with an estimatordependent controller. When the controller is independent of the estimator, it is possible to match the combined output estimate of the MMAE estimator with a single controller, but this is not the case for out matched estimator controller pair. In the MMAC scheme, each MMAE type filter is combined with its matched controller and the controllers outputs are weighted using the same weighting coefficients of the MMAE estimator, to form an equivalent control signal for the system’s plant. Most realworld targets will rarely switch their maneuver direction, type, or magnitude in the last 10 time constants of the interception. At most, a single new maneuver will start at these last instances of the engagements. And any maneuver starting at that time, with a target time constant as assumed here (0.3s), will appear mostly as a step or some part of it. Therefore, the assumption of a single step occurring in these last 10 time constants is per fectly reasonable and will be used here.The combined TMMSOGL algorithm does not yield a significant improvement if initialized earlier than 10 missile time constants before the inter cept time. This is also evident from the almost unchanged N’ resulting from initializations before the 10 missile time constants limit. Understanding that the important time span to be dealt with is the last 10 missile time constants, a MMAC type guidance system structure is formed.The MMAC system (figure 4.1) is constructed from 11 filterguidance law branches (this system will be identified here as “MMAC11F”) matched to the 10 possible target jump times of the span (θjump = 10, 9, 8,1) and another filter representing a jump at θjump = 0, which is equivalent to a filter representing no target jump at all. Each of the component models of the multiplemodel architecture assumes a jump in the target acceleration com mand at a known time, but with an unknown magnitude. The jump magnitude is assumed to be zero mean, Gaussian distributed, with a known σ jump . In this case, until a jump is iden tified, the expected value of the target acceleration command remains unchanged, regardless Department of Electrical Engineering, College of Engineering, Trivandrum 16 Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile Figure 4.1: MMAC11F approach solution of the known different jump times that are assumed in each of the component models of the multiplemodel architecture. That is, a constant acceleration command is assumed for all models. Therefore, all the ZEMSOGLi , have identical expressions: 1 2 ZEMSOGLi = yi +yi tgo −aMi τM (e−θ +θ−1)+aˆ i τT (e−θT +θT −1)−aTci τT (e−θT +θT −1− θT ) ˆ ˆ ˆ 2 2 ˆ 2 ̇ T 2 (4.9) Although for all MMAC based guidance laws the ZEM structure is identical, for each dif ˆ ˆ ˆ ferent value of i (different model), appropriate values of yi , yi , aMi , aˆ i , and aTci are used. ̇ ˆ T The MMAC  based control law is 11 Ni (tgo ) ac = μi zi (tgo ) ˆ (4.10) t2 go i=1 Where zi (tgo ) is the expectation if the zeroeffort miss conditioned on the measurements ˆ until the current time and on the assumed time ti of the step in the target acceleration, μi is the conditional probability of the target acceleration step occurring at time ti , and Ni (tgo ) is the optimal guidance gain associated with a step in the target acceleration at time ti of unknown (random) amplitude. Remark1: The stochastic target model used in the current work specifies the random behavior of the target acceleration from the beginning of the engagement to intercept time. Consequently, the assumed model dictates the zero effortmiss calculation. The model assumes that the target acceleration is the sum of the two components: a firstorder GaussMarkov process and a single step in the target acceleration command, occurring at an unknown (random) time. In particular, the ZEM calculation in Eq.(4.9)shows that, at any instant, the future values of the target acceleration are the sum of the present estimate of the target acceleration (aˆ i ) decaying with the assumed target time T constant for the Gauss  Markov component and a constant acceleration command (aTci ) ˆ acting through the assumed target time constant. Remark 2: If the SOGL is replaced by OGL, then Ni (tgo ) = N (tgo ) for all i. In this case, Eq. (4.10) can be rewritten as 11 N (tgo ) N (tgo ) ac = μi zi (tgo ) = ˆ ze (tgo ) ˆ (4.11) 2 t2 tgo go i=1 Department of Electrical Engineering, College of Engineering, Trivandrum 17 Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile where 11 ze (tgo ) = ˆ μi zi (tgo ) ˆ (4.12) i=1 Remark 3: The guidance gains Ni (tgo ) of the different models in Eq. (4.10) can be computed a priori. As is common in the missile guidance community, these gains can then be stored in a lookup table to be used online. 4.3 Performance Evaluation 4.3.1 Comparison to the Bound To compare the performance with that of the previous section , the same scenario will be addressed and two types of behavior will be evaluated: 1) A target behaving according to the assumed model, for example, commands a step in acceleration for which the amplitude is chosen form a Gaussian distribution with the known σjump . 2) A target performing a step with a magni 


