UTILIZATION OF AIR-TO-AIR MISSILE SEEKER CONSTRAINTS IN THE MISSILE EVASION. September 13, 2004

Transcription

1 Mat Independent Research Project in Applied Mathematics UTILIZATION OF AIR-TO-AIR MISSILE SEEKER CONSTRAINTS IN THE MISSILE EVASION September 13, 2004 Helsinki University of Technology Department of Engineering Physics and Mathematics Systems Analysis Laboratory Henri Hytönen 54394U

2 Contents 1 Introduction 2 2 Problem Formulation Aircraft Model Missile Model Optimal Control Problem Solution Methods Direct Multiple Shooting Moving Horizon Control Numerical examples Optimization Examples Comparison of Solution Methods Analysis of the Missile Launch Point Conclusions 32 References 36

3 1 Introduction The primary weapon used in modern air combat is a guided air-to-air missile. In order to determine guidance commands needed to reach the target, most guided missiles incorporate some sort of a seeker system to receive position and velocity information from the target. To be able to track the target, the target has to be in the field of view of the missile s seeker head. The seeker head can be rotated to track the target when the target is not directly in front of the missile. The seeker head cannot rotate arbitrarily, however. The physical limit for the gimbal angle, i.e. the angle between the tracking boresight axis and the missile centerline, is typically about ±40 to ±60 degrees [9]. Also, the rate at which the seeker head can turn is also limited. The seeker can only follow its target if the relative position and movement of the target aircraft does not violate these constraints. Thus, the missile will lose its lock on the target if (i) the target maneuvers outside the gimbal limit of the seeker head or (ii) the rate at which the angle between the missile centerline and the line of sight vector is changing is too high. In this paper, we examine the abilities for a modern fighter aircraft to utilize the aforementioned seeker limits to evade a medium-range air-to-air missile launched towards the aircraft. The goal of the study is to investigate whether either the gimbal limit or the line of sight rate limits of the missile can be exceeded early enough for the aircraft to avoid the missile. We consider only situations where the missile is launched relatively close to the aircraft so that the aircraft cannot outrun the missile. A dynamic, extended point mass model is used to describe both the aircraft and the missile. The missile is assumed to be guided by proportional navigation (see, for example, [14]). The aircraft trajectory is obtained as a solution of an optimal control problem. Due to the complexity of the dynamic model of the aircraft and missile, closed-form solutions of the problem cannot be formulated. However, optimal, numerical open-loop solutions can be obtained when the state and control variables of the problem are discretized, and the problem can be treated as a nonlinear programming problem. In the numerical solutions, table data containing performance and limitation statistics for a generic fighter aircraft and a generic 2

4 1 INTRODUCTION medium-range air-to-air missile are used in the aircraft and the missile model of this study. As mentioned, the numerical optimization approach yields only open-loop solutions, i.e., solutions where the optimal control u is determined as a function of time for a specified initial state, or u (t) = e(x(t 0 ), t) [7]. However, feedback solutions where the optimal control could be derived from the current state, that is, u (t) = f(x(t), t), would be more preferable due to their generality and possibly less demanding computational effort. Therefore the problem is also analyzed with control strategies involving moving horizon control methods, where the control and state trajectories are selected so that they are optimal over some fixed time horizon with respect to the current state. The area of optimal control in air combat has been studied quite widely. Maximization of the miss distance of the missile has been studied in [11]. Some aspects of the numerical methods applied to trajectory optimization in the air combat has been discussed in [10]. Previous studies also include research on usage of genetic algorithms in the generation of initial solution for the nonlinear solver [12], a method that is also utilized in this research project. The report is organized as follows. Chapter 2 describes the model of the aircraftmissile system and presents the principles of proportional navigation guidance used by the missile. The optimal control problem is also formulated in this chapter. Chapter 3 gives an overview of the numerical solution methods that are used in the problem solving. In Chapter 4, results of numerical examples are presented. Comparison of the results as well as concluding remarks are presented in Chapter 5. Some qualitative comparisons are also made with real-world evasion strategies. 3

5 2 Problem Formulation The purpose of this chapter is to formulate a problem that allows the analysis of breaking the lock-on of the missile. To perform this, the basic properties of the dynamics of the aircraft-missile system have to be correctly modeled. The engagement between the aircraft and the missile lasts only a few seconds. Still, the differential equations describing the kinematics of the system are fairly complex and highly nonlinear. The model needs to be accurate enough to capture the essential aspects of the problem. Yet too complex system would result in a problem that would be too challenging to solve with numerical methods. The dynamic model of the aircraft and the missile as well as its guidance dynamics are presented in Section 2.1 and 2.2, respectively. Using these, a dynamic optimal control problem is formulated in Section 2.3. The aircraft model is described thoroughly for example in [8]. More detailed explanation of the proportional navigation used by the missile can be found in [14]. 2.1 Aircraft Model The aircraft model is a three-dimensional, three degrees-of-freedom point mass model with state equations ẋ a = v a cos γ a cos χ a (2.1) ẏ a = v a cos γ a sin χ a (2.2) ḣ a = v a sin γ a (2.3) γ a = 1 {{L(α, h a, v a, M(h a, v a )) + m a v a ηt max (h a, M(h a, v a )) sin α} cos µ m a g cos γ a } (2.4) χ a = 1 {L(α, h a, v a, M(h a, v a )) + m a v a cos γ a ηt max (h a, M(h a, v a )) sin α} sin µ (2.5) v a = 1 m a {ηt max (h a, M(h a, v a )) cos α D(α, h a, v a, M(h a, v a ))} g sin γ a. (2.6) 4

6 2 PROBLEM FORMULATION h µ α η γ a v a,α v a v h = v a sin γ a y χ a v x = v a cos γ a cos χ a v y = v a cos γ a sin χ a x Figure 2.1: Coordinate system and control variables of the aircraft The state variables of the model, presented also in Fig 2.1, are x a, y a, h a, γ a, χ a, and v a. Variables x a, y a and h a represent the position of the aircraft with x and y-coordinates and altitude, respectively. γ a is the flight path angle of the aircraft, i.e. the angle between the velocity vector of the aircraft and the (x, y)-plane. χ a is the heading angle, which is the angle between the projection of the velocity vector on the (x, y)-plane and the x-axis. Finally, v a is the velocity of the aircraft, which, together with the altitude, defines the Mach number, M(h a, v a ), i.e. the velocity relative to the speed of sound at the current altitude. The aircraft is controlled with its angle of attack α, bank angle µ, and throttle setting η. The angle of attack is the angle between the aircraft centerline vector v a,α and the velocity vector v a. Bank angle µ is the angle of roll around the centerline vector. The throttle setting η [0, 1] is the relative thrust of the aircraft engine with 0 corresponding to idle and 1 to full power. The thrust of the aircraft engine is therefore ηt max (h a, M(h a, v a )), where T max (h a, M(h a, v a )) is the maximum available thrust at the current altitude and velocity. The thrust force is directed parallel to the centerline vector of the aircraft. 5

7 2 PROBLEM FORMULATION Being a point mass model the moment of inertia and exact rotation dynamics of the aircraft are not included in the model. Therefore the aircraft can be controlled directly with the angle of attack and the bank angle. The limitations imposed by rotation dynamics are considered in the model as constraints to the magnitude of the derivatives of the pitch and the bank angle. Yaw rotation, or rotation around the h-axis is assumed to be negligible in the control of the jet aircraft and is therefore omitted. Due to relatively short distances and time intervals considered in this study, the Earth can be regarded as ideally flat and non-rotating. Acceleration due to gravity, g, is assumed to be constant. Also, the mass of the aircraft m a, is assumed to be constant during the flight time interval. No wind or turbulences are assumed to exist. L(α, h a, v a, M(h a, v a )) in the state equations is the lift force generated by the air flow deflected by the aircraft wing. The lift force is directed upwards and perpendicular to the plane of the aircraft body axes. The magnitude of the lift force can be calculated from L(α, h a, v a, M(h a, v a )) = C L (α, M(h a, v a ))Sq(h a, v a ), where C L (α, M(h a, v a )) is the aircraft-specific lift coefficient corresponding to the angle of attack α, S the reference wing area of the aircraft and q(h, v) = 1/2ρ(h)v 2 the dynamic pressure of the air flow. ρ(h) is the density of air at the altitude h. The greater the angle of attack α, the bigger the lift force generated. On the other hand, increasing the angle of attack also increases the drag force D(α, h a, v a, M(h a, v a )). The magnitude of the drag force is defined as D(α, h a, v a, M(h a, v a )) = C D (C L (α, M(h a, v a )))Sq(h a, v a ), where C D (C L ( ), M( )) is the aircraft-specific drag polar, defining the drag coefficient as a function of the lift coefficient C L and the Mach number M. Due to the fact that the state equations (2.1) (2.6) do not explicitly include rotation dynamics, additional constraints are introduced to keep the bank and pitch angle rates in acceptable limits. Also, the structural strength of the aircraft and the ability of the aircraft wing to withstand different lift coefficients without stalling 6

8 2 PROBLEM FORMULATION impose limitations to the achievable maneuvers. The state and control variables are constrained by inequalities α 0 (2.7) n(α, h a, v a ) N max 0 (2.8) C L (α, M(h a, v a )) C L,max (M(h a, v a )) 0 (2.9) µ χ a sin γ a K max (h a, M(h a, v a ), α) 0 (2.10) α + γ a cos µ + χ a cos γ a sin µ P max 0 (2.11) h min h a 0 (2.12) q(h a, v a ) q max 0. (2.13) Equation (2.7) is the angle of attack limit, limiting the angle of attack only to positive values due to lack of data for negative angles. The next two are the load factor and lift coefficient limits. The load factor is defined as n(α, h a, v a ) = L(α, h a, v a, M(h a, v a )) m a g and represents the structural stress of the aircraft, which cannot exceed the aircraft structural limit, N max. The maximum lift coefficient defines the maximum angle of attack in the current situation that the aircraft can withstand without stalling. If the angle of attack is too great and the lift coefficient exceeds its maximum allowed value, the aircraft stalls, i.e., the air flow on the aircraft wing becomes turbulent. This results in loss of lift force of the aircraft wing. Equations (2.10) and (2.11) are the pitch and bank angle rate constraints, which define the maximum pitch and roll rates for the aircraft. The last two inequalities are the minimum altitude and dynamic pressure constraints. The dynamic pressure constraint defines the maximum allowable speed for the aircraft in the current altitude. Diagram of the angle of attack, load factor, lift coefficient, and dynamic pressure constraints is presented in Fig These limitation are also discussed in more detail in [13]. All of these limits except the altitude limit are aircraft-specific. They are generated with interpolation from table data representing a generic fighter aircraft. 7

9 2 PROBLEM FORMULATION n 10 8 N max 6 4 C L,max Feasible region q max 2 0 α = stall speed v a max speed Figure 2.2: The angle of attack, load factor, lift coefficient, and dynamic pressure limits. 2.2 Missile Model The state equations of the missile system are ẋ m = v m cos γ m cos χ m (2.14) ẏ m = v m cos γ m sin χ m (2.15) ḣ m = v m sin γ m (2.16) γ m = a 1 g cos γ m (2.17) v m χ m = v m = a 2 v m cos γ m (2.18) 1 m m (t) {T (t) D(a, h m, v m )} g sin γ m (2.19) ȧ 1 = a 1c a 1 τ (2.20) ȧ 2 = a 2c a 2. (2.21) τ The basic properties of the missile model are similar to those of the aircraft. The missile is a point-mass object with state variables x m, y m, h m, γ m, χ m, and v m. 8

10 2 PROBLEM FORMULATION In addition to those, the state variables of the missile include current vertical and horizontal accelerations with respect to the current missile centerline vector, (a 1 and a 2 ). These accelerations correspond to some vertical and horizontal angles of attack, α 1 and α 2. Coordinate system of the missile is depicted in Fig It is assumed that the rotation dynamics of the missile are faster than those of the aircraft and the missile does not need to bank to turn and thus the bank angle of the missile is irrelevant. The missile can be directly controlled with commanded pitch and yaw accelerations a 1c and a 2c. Nonetheless, there exists a dynamic lag between the commanded accelerations and the actual accelerations due to the missile seeker, guidance system and actuator dynamics. The rocket motor of the modeled missile is boost-sustain type, i.e., it provides large thrust in the beginning of the flight and small sustain thrust after the initial acceleration. The propulsive force of the missile engine at time t is therefore defined as a step function T Boost t < t B T (t) = T Sust t B t t S, (2.22) 0 t > t S where t B is the length of the boost stage and t S t B the length of the sustain stage. The fuel of the missile s engine, which constitutes a significant proportion of the mass of the missile, is used during the two stages during early phase of the missile s flight time. Thus, the mass of the missile m m (t), is also a function of time. The commanded accelerations a 1c and a 2c are related to current accelerations with first-order dynamics (equations (2.20) and (2.21)) and with time constant τ. a = a a 2 2 is the total transversal acceleration. The commanded accelerations are determined with proportional navigation (PN). The purpose of the proportional navigation is to guide the missile to a direction that will result in zero angular velocity between the missile centerline and the line of sight vector and thus will result in the missile roughly trying to follow the lead collision course for a non-maneuvering aircraft. According to PN, required transversal acceleration for the missile is directly proportional to the closing speed of the target and to the relative angular velocity of the target (hence the name). Acceleration commands 9

11 2 PROBLEM FORMULATION h v m,α v m α 1 x y γ m α 2 χ m Figure 2.3: Coordinate system of the missile. v m,α is the missile centerline vector and v m is the velocity vector of the missile. suggested by PN are divided to vertical and horizontal components. The vertical acceleration component is directed perpendicular to the velocity vector of the missile and upwards, and the horizontal component is perpendicular to both the velocity vector and the vertical acceleration component. The suggested vertical and horizontal accelerations by PN guidance are then a 1P N = N e v c ω 1 λ + g cos γm (2.23) a 2P N = N e v c ω 2 λ (2.24) and the total acceleration is a P N = a 2 1P N + a2 2P N. Here, N e is the navigation constant, a design parameter of the missile, usually between 3 and 5 [14]. v c is the closing speed which is the negative of the time derivative of the distance between the aircraft and the missile. λ is the line-of-sight angle rate or the time derivative of the angle between missile centerline and the line-of-sight vector. ω 1 λ and ω2 λ are the vertical and horizontal LOS rates, respectively. The term g cos γ m in the vertical component compensates the acceleration due to gravitation. In practice, v c and λ are estimated from measurements of the missile s optical or radar seeker, 10

12 2 PROBLEM FORMULATION if possible. Here, perfect measurement is assumed and exact current data is used with the proportional navigation guidance. The acceleration commands given by PN guidance cannot exceed the maximum achievable accelerations of the missile imposed by the lift coefficient limit and structural limits. These missile limits cannot be incorporated as constraints in the dynamic model as that would prohibit any solution in which the maneuvering capability of the missile would be exceeded. Thus the commanded accelerations a 1c and a 2c presented in state equations (2.20) and (2.21) are defined as a ic = { min{a n,max,a CL,max} aip N a P N a ip N if a P N > min{a n,max, a CL,max} if a P N min{a n,max, a CL,max}, where a n,max and a CL,max are the missile s load factor and lift coefficient limits, defined as a n,max = g n max a CL,max = v2 mc L,max Sρ(h) 2m m. 2.3 Optimal Control Problem The purpose of the constructed dynamic model is to find the aircraft control trajectories that maximize the gimbal angle λ or the gimbal angle rate λ. Because of computational difficulties and additional complexity in the calculation of the LOS rate, the performance criterion in the rest of the study is the line-of-sight angle. All the techniques used in the study also apply to the analysis of the LOS rate, however. Velocity vector of the missile, v m and missile centerline vector v m,α as well as the angle λ are represented in the Fig According to section 2.2, the missile velocity vector v m can written with its x, y 11

13 2 PROBLEM FORMULATION v m,α v m α m λ Missile LOS v a r Aircraft Figure 2.4: The geometry of the missile-target system and h -components as v m = v m cos γ m cos χ m v m cos γ m sin χ m. (2.25) v m sin γ m The missile is controlled with vertical and horizontal accelerations a 1 and a 2, which correspond to the angle of attacks α 1 and α 2. These are the horizontal and vertical angles between between the velocity vector and the missile centerline, and can be approximated with a sufficient accuracy with the current acceleration components as a 1 = L m (α 1, M(h m, v m ))/m m a 2 = L m (α 2, M(h m, v m ))/m m, when the angles α 1 and α 2 are small. Here, L m ( ) = C L,m (α, M(h m, v m )S m q(h m, v m ) is the lift force of the missile. The missile centerline vector v m,α, presented also in 12

14 2 PROBLEM FORMULATION Fig. 2.3, can then be defined similarly to (2.25) as v m,α = cos(γ m + α 1 ) cos(χ m + α 2 ) cos(γ m + α 1 ) sin(χ m + α 2 ) sin(γ m + α 1 ) (2.26) if the length of the centerline vector is set to unity. Now the cosine of the gimbal angle λ can be calculated as cos λ = = v m,α r v m,α r ( cos(γ m + α 1 ) cos(χ m + α 2 )(x a x m ) + cos(γ m + α 1 ) sin(χ m + α 2 )(y a y m ) + ) sin(γ m + α 1 )(h a h m ) / (xa x m ) 2 + (y a y m ) 2 + (h a h m ) 2. (2.27) This is a suitable performance measure since it is minimized when the gimbal angle λ is maximized. It also ignores the sign of λ. In this problem, only the final value of the gimbal angle λ is important, denoted here with λ(x, t f ). Thus, the optimal control problem can now be stated as min cos λ(x, t f ) s.t. ẋ = f(x, u, t) (2.28) g(x, u, t) 0 (2.29) r(x, t f ) 2 = d, (2.30) where x = [ x a, y a, h a, γ a, χ a, v a, x m, y m, h m, γ m, χ m, v m, a 1, a 2 ] T is the state vector and u = [ α, µ, η ] T is the control vector of the aircraft. Constraint (2.28) denotes the state equations of the aircraft (2.1) (2.6) and the missile (2.14) (2.21). Inequalities (2.29) are the control constraints (2.7) (2.13). (2.30) is the final distance constraint between the aircraft and the missile. Unlike the altitudes h a and h m, the absolute (x, y) -positions of the aircraft and the missile are irrelevant for the dynamics of the model. Therefore, the origin can be fixed to the aircraft in the 13

15 2 PROBLEM FORMULATION (x, y)-plane and the two variables, x a and y a, can be eliminated. 14

16 3 Solution Methods This chapter presents the numerical methods used to solve the problem formulated in the previous chapter. The optimal control problem under study is numerically fairly challenging, consisting of twelve state variables and three control variables. To obtain both optimal open-loop and approximate feedback solutions to the problem, two different approaches are adopted. First, a method of direct multiple shooting is used, yielding optimal open-loop solutions to the problem. The direct multiple shooting is described in Section 3.1. The approach results in a constrained nonlinear program, which is solved with a hybrid method of a genetic algorithm and sequential quadratic programming, or SQP. Relying solely on the SQP method could result in a poor convergence or convergence only to a local optimum. Thus, the approximate initial solution to the problem is generated with a genetic algorithm. The implementation of the genetic algorithm is introduced and analyzed in [12]. Secondly, to obtain comparative results, and more importantly, to analyze what kind of feedback results could be achievable with the studied problem, a moving horizon control approach is also used. The approach is presented in Section 3.2. The resulting trajectories with the method are suboptimal but feedback in nature. The algorithms of the both methods as well as the problem dynamics are implemented with Fortran 77/90 programming language. The nonlinear programming problem generated with multiple shooting method is solved with NPSOL optimization package [3], which uses the SQP algorithm to solve the problem. More on theory and applications of the SQP algorithm itself can be found, e.g., in [1] and [3]. Other problems of same type have also been solved with direct multiple shooting and hybrid methods, see [11] and [5]. 3.1 Direct Multiple Shooting Most methods which are currently used to solve optimal control problems can be classified either as direct or indirect methods. Direct methods optimize the 15

17 3 SOLUTION METHODS objective criterion directly by discretizing the state and control variables to finitedimensional approximation and then solving the resulting problem with regular nonlinear programming algorithms, such as sequential quadratic programming. Indirect methods try to find the solution by solving the two-point boundary-value problem resulting from the necessary conditions derived from Pontryagin s maximum principle [7], [6]. A general simple shooting method can be used to solve two-point boundary-value problems, where we want to find an initial value x(t i ) such that with dynamics ẋ(t) = f(x(t), t)) the value at the final point x(t f ) is some specified x f (see, e.g., [2], [4]). The simple shooting method is therefore a suitable solution approach for indirect methods. First the initial conditions x(t i ) are guessed. Then, the differential equations are integrated forward from the initial point t i to the final point t f with some numerical integration method, referred to as shooting. Finally, the difference between the obtained value and the desired value at the final point x(t f ) x f, is measured. By adjusting the initial values with the methods of nonlinear programming and repeating the shooting step, the difference in the endpoint can be reduced to zero. The problem with the simple shooting method is that even small alteration in the initial values could result in very large changes in the variables at the endpoint and therefore the problem can be very difficult to solve numerically or show poor convergence [4]. To remedy this problem, the time interval [t i, t f ] can be divided into shorter steps and each interval can be processed separately. In these multiple shooting methods, the values of the function x(t) are guessed in the beginning of each step x 0, x 1,..., x k 1, and the integration is performed to the beginning of the next step. Of course, since the state trajectories need to be continuous, the endpoint of the previous step must coincide with the starting point of the next step. Thus, by dividing the problem into smaller steps, additional variables and continuity constraints need to be introduced. Similar approach can also be used with direct methods, which do not require the formulation of the necessary conditions of the optimal control problem. With a given initial state x(t i ), the state and control variables are guessed at the beginning 16

18 3 SOLUTION METHODS of each interval and the state variables are then integrated using the guessed control variables over the interval. Since the controls do not need to be constant during each step, they can be approximated with greater precision than the state variables. In order to achieve continuity or differentiability in the control trajectories, Hermite interpolation or cubic polynomial interpolation of the control trajectories can be used. n {}}{ s t 0 1 t2 0 t 0 n t 1 c 1 t 1 2 t 1 n c t 2 1 t 2 2 t n d 2 2 t n d 1 t f n c t n d 1 1 t n d 1 Figure 3.1: Discretization of states (n d ), controls (n c ), and interpolation points of control trajectories (n s ) n c The discretization points for the evasion problem are presented in Fig The state trajectories x(t), t [t i, t f ], are discretized on n d points t 1, t 2,..., t nd 1 in the time interval [t i, t f ]. The controls are discretized with greater accuracy, each interval [t k, t k+1 ] is further divided to n c subintervals t k 1, t k 2,..., t k n c. Finally, to increase the accuracy of the numerical integration, the controls are linearly interpolated on n s points between the control trajectory discretization points. Since the final time t f is initially unknown, it must be treated as a variable. Combining the discretized state and control variables to a single column vector of decision variables y, we get y = [ h a 1 h a 2 h a n d γ a 1 γ a 2 γ a n d. a 1 1 a 1 2 a 1 n d a 2 1 a 2 2 a 2 n d α1 0 αn 0 c α1 1 αn 1 c α n d 1 1 α n d 1 n c µ 0 1 µ 0 n c µ 1 1 µ 1 n c µ n d 1 1 µ n d 1 n c η1 0 ηn 0 c η1 1 ηn 1 c η n d 1 1 η n d 1 n ] c T t f. α n d 1 µ n d 1 η n d 1 (3.1) The vector y consists of the 12 n d discretized state variables followed by 3(n d n c +1) 17

19 3 SOLUTION METHODS control variables and the final time t f. The controls α n d 1, µ n d 1, and η n d 1 at the final state are included because they are required by the interpolation. The state trajectories need to be continuous as noted above and in addition need to satisfy the state equations in every discretization point. In every discretization interval t [t k, t k+1 ], an initial value problem ẋ(t) = f(x(t), ũ(t), t), x(t k ) = x k (3.2) can be formulated from the state equations (2.1) (2.6) and (2.14) (2.21). Here, ũ(t) is the approximation of the control trajectory, generated with interpolation and x k is the vector of state variables at the time t k. Specifically, x k = [ h a k, γa k,..., a2 k ]T. We define ˆx k = x k (t) + tk+1 t k f(x k (t), ũ(t), t) dt (3.3) to be the solution to the initial value problem (3.2) in the interval [t k, t k+1 ] at time t k+1 with ˆx k. Then, to satisfy the state equations, the values of the discretized state variables have to satisfy x k ˆx k 1 = 0 in every discretization point t k. By introducing a column vector h(y) as h(y) = [ h a 1 ĥa 0 h a 2 ĥa 1 h a n d ĥa n d 1 γ a 1 ˆγ a 0 γ a 2 ˆγ a 1 γ a n d ˆγ a n d 1 a 1 1 â 1 0 a 1 2 â 1 1 a 1 n d â 1 n d 1. (3.4) a 2 1 â 2 0 a 2 2 â 2 1 a 2 n d â 2 n d 1 ] T r nd 2 d, the equality constraints of the discretized problem can be stated in the general form h(y) = 0. The last term represents the final distance constraint (2.30). The inequality constraints of the original problem can be written in a similar fashion 18

20 3 SOLUTION METHODS as g(y) 0 by defining g(y) = [ c 1 (1, 1, 1) c 1 (n d, n c, n s ) c 1 (n d + 1, 1, 1) c 2 (1, 1, 1) c 2 (n d, n c, n s ) c 2 (n d + 1, 1, 1) c 3 (1, 1, 1) c 3 (n d, n c, n s ) c 4 (1, 1, 1) c 4 (n d, n c, n s ) c 5 (1, 1, 1) c 5 (n d, n c, n s ) c 6 (1, 1, 1) c 6 (n d, n c, n s ) v c ] T, (3.5) where c 1 (i, j, k) = n(α i,j,k, ĥa i,j,k, ˆv a i,j,k) N max c 2 (i, j, k) = C L (α i,j,k, M(ĥa i,j,k, ˆv a i,j,k)) C L,max (α i,j,k, M(ĥa i,j,k, ˆv a i,j,k)) c 3 (i, j, k) = α i,j,k+1 α i,j,k δt c 4 (i, j, k) = α i,j,k+1 α i,j,k δt c 5 (i, j, k) = µ i,j,k+1 µ i,j,k δt c 6 (i, j, k) = µ i,j,k+1 µ i,j,k δt ˆ γ a i,j,k cos µ i,j,k ˆ χ a i,j,k cos ˆγ a i,j,k sin µ i,j,k P max + ˆ γ a i,j,k cos µ i,j,k + ˆ χ a i,j,k cos ˆγ a i,j,k sin µ i,j,k P max K max (ĥa i,j,k, M(ĥi,j,k, ˆv i,j,k ), α i,j,k ) K max (ĥa i,j,k, M(ĥi,j,k, ˆv i,j,k ), α i,j,k ), and v c is the closing velocity. The constraints c m (i, j, k) correspond to inequalities (2.7)-(2.13). δt is the time interval between the interpolated control trajectory points, i.e., δt = (t f t i )/(n d n c n s ), which is used in the difference approximation of the derivatives of the control variables. The term v c in (3.5) is the requirement that the closing velocity is nonnegative at the endpoint, i.e., the missile has not yet passed the aircraft. By combining the inequality and equality constraints with the performance measure (2.27) introduced in the previous chapter, the problem can now be written in the standard NLP form as min cos λ(y, t f ) s.t. g(y) 0 h(y) = 0. The problem consists of 12 n d + 3 n d n c + 4 variables, 6 n d n c n s + 3 inequality 19

21 3 SOLUTION METHODS constraints and 12 n d + 1 equality constraints. 3.2 Moving Horizon Control The purpose of the moving horizon control method in this study is to compare the optimal open-loop solution with results obtained with a technique producing feedback solutions and to analyze whether similar results to open-loop optimization could be obtained with computationally simpler and faster methods. The trajectories resulting from the moving horizon control method are suboptimal, but of feedback type. Thus the suggested controls in every point are function of the current state. In the moving horizon control method the performance measure is not optimized over the whole time interval but up to some specified time ahead. Possible control trajectories are then considered up to the specified point in time and the problem is to choose the best control trajectory within the time interval examined. By discretizing the possible control trajectories at each point, the problem at each point can be reduced to selection from a set of discrete alternatives. Naturally, when the discretization is performed with high precision, the number of alternatives becomes very large. The examination horizon can be extended several steps ahead with recursion. However, an increase in the recursion depth causes an exponential increase in the number of alternatives and thus in the computational workload. To reduce the number of alternative control sequences from which to choose, the aircraft throttle setting η is fixed to η 1. This was done because in all the numerical open-loop solutions of the problem the throttle setting was constantly at unity, and therefore it seems reasonable to assume that it would also be the case in the corresponding feedback strategies. At each step, n α different values for the angle of attack and n µ different values for the bank angle are generated between the allowed minimum and maximum values. Thus, with n α angle of attack and n µ bank angle alternatives and recursion depth of p steps, the number of alternative control combinations on each step is (n α n µ ) p. 20

22 3 SOLUTION METHODS The possible controls for the following stage are generated as α i (t + t) = α(t) + α i t i 1,..., n α (3.6) µ j (t + t) = µ(t) + µ j t j 1,..., n µ, (3.7) where the alternatives for the derivatives of the control variables, α i and µ j are uniformly distributed between the minimum and maximum values in the current state P max = α 1 < α 2 <... < α nα = P max (3.8) K max = µ 1 < µ 2 <... < µ nµ = K max. (3.9) We denote a single control combination with a length of p steps with u k, u k+1,..., u k+p 1, where u k = [ α k, µ k ] T. After the alternatives have been generated, each of them are integrated the step length t forward, and after that the ones not satisfying the model constrains are dropped. The performance criterion cos λ is calculated for each feasible state and the one minimizing the criterion is selected as the next step. More specifically, given a set of n α n µ possible control sequences of length p u k i, u k+1 i,..., u k+p 1 i, i 1,..., n α n µ at time t k in state x k, the state x k is integrated p steps forward with each of the control sequences, tk+1 x k+1 i = x k + x k+2 i = x k+1 i +. x k+p i = x k+p 1 i + t k tk+2 f(x k, u k i, t) dt t k+1 f(x k+1 i, u k+1 i, t) dt (3.10) tk+p f(x k+p 1 i t k+p 1, u k+p 1, t) dt i for each i 1,..., n α n µ. Next, any possible control sequences leading to states x n i that are infeasible with respect to constraints (2.7) (2.13), are dropped. Then, the performance criterion cos λ(x k+p i, t k+p ) is calculated for each feasible i 1,..., n α n µ 21

23 3 SOLUTION METHODS and the one minimizing the criterion is chosen, denoted by ū k, ū k+1,..., ū k+p 1. The next state x k+1 is then found by integrating the system one step forward with the first control in the best control sequence, i.e, x k+1 = x k + tk+1 t k f(x k, ū k, t) dt. (3.11) As a conclusion, one step of the algorithm can be outlined as follows: 1. Generate the possible controls α i and µ j for next step and do the following for each combination of α i and µ j : (a) Integrate the system t forward with some numerical integration technique. (b) Check if the new state satisfies the model inequality constraints. If not, abandon this alternative and go to step 1a with the next alternative. (c) Calculate the performance criterion for this alternative, or, if recursion depth is greater than one, call step 1 with one smaller recursion depth. 2. Return the best alternative. If recursion depth is greater than 1, return the first controls in the control sequence. After this, we integrate the system t forward with the best controls found. The algorithm is continued this way until the distance between the aircraft and the missile is the same as the final distance d or the closing velocity v c is less than zero. 22

24 4 Numerical examples Examples of the numerical results obtained with both methods are presented in this chapter. The computations were carried out on a computer with 1.4 GHz processor and 256 MB of memory. In the numerical examples solved with direct multiple shooting, values of n d, n c and n s (see Section 3.1) are 8, 3, and 3, respectively, and thus the resulting problem has 172 variables, 435 inequality, and 97 equality constraints. The initial velocity for both the missile and the aircraft is 250 m/s as it is assumed that the missile is launched by the adversary. The final distance d between the missile and the aircraft is 200 meters. 4.1 Optimization Examples This section presents some numerical examples to examine the qualitative nature of the results and to find out how much the final nonlinear optimization can improve the results of the genetic algorithm. Two example trajectories are presented in Fig The figure shows the optimal trajectory to evade a missile launched in of the aircraft at the distance of 20 km. The initial altitude for both the aircraft and the missile is 10 km. Also, the initial solution generated by the genetic algorithm is shown, which was used as an initial guess for the nonlinear solver. The initial state for the example is presented in Table 4.1. Table 4.1: Initial state and the final distance for the example in Fig. 4.1 x a y a h a γ a χ a v a x m y m h m γ m χ m v m d π It can be seen that both the genetic algorithm and the final optimal trajectory of the aircraft are tightly downward-sloping, with most noticeable maneuvers performed in the beginning and end of the flight. Although both the trajectories look qualitatively similar, there are considerable differences in the final value of the gimbal angle λ(t f ). The value of λ(t f ) is 31.1 degrees with the genetic algorithm and 41.4 degrees after solving the problem using the multiple shooting method. 23

25 4 NUMERICAL EXAMPLES (x, h) plane h [m] Aircraft Aircraft (initial solution) Missile Missile (initial solution) x [m] 4000 (x, y) plane 2000 y [m] Aircraft Aircraft (initial solution) Missile Missile (initial solution) x [m] Figure 4.1: Example initial guess from the genetic algorithm and the final optimal solution Fig. 4.2 shows the velocities of the aircraft and missile in the example. The effect of the missile thrust stages, T Boost and T Sust, can be seen in the figure. The missile acceleration is highest during the boost stage, t [0, 3]. The velocity of the missile v m is highest at the end of the sustain stage, t S = 8s. After that, the missile velocity starts to drop because of the drag force. Then, the longer it takes for the missile to reach its target, the lower its speed advantage it has, giving the target better changes for evading the missile or breaking the lock-on. The controls as a function of time corresponding to the trajectories of Fig. 4.1 are shown in Fig Some interesting remarks can be made about the control 24

26 4 NUMERICAL EXAMPLES v a (t), v m (t) Aircraft Missile v [m/s] t [s] Figure 4.2: Velocity of the aircraft v a (t) and the missile v m (t) in the initial guess and the final solution α, µ [deg] α(t), µ(t) 300 α µ η η(t) a 1, a 2 [m/s 2 ] a 1 (t), a 2 (t) 80 a a λ [deg] λ(t) Figure 4.3: Control trajectories α(t), µ(t) and η(t) of the example in Fig. 4.1 as well as true vertical and horizontal accelerations of the missile, a 1 and a 2, and the angle λ(t). 25

27 4 NUMERICAL EXAMPLES h [m] y [m] x [m] Missile Aircraft Figure 4.4: Example trajectories for the missile and the aircraft. The missile is launched from a direction 45 degrees relative to the heading angle of the aircraft at the range of 15 km and altitude equal to that of the aircraft. The color in the aircraft trajectory represents the aircraft speed. Dark blue is roughly 250 m/s and cyan 360 m/s. trajectories. The throttle setting is constantly at η = 1 as it is in all the optimization runs performed. The biggest angle-of-attack values are in the beginning of the flight as the aircraft starts turning rapidly downwards, and in the end, when the aircraft turns away from the missile. According to almost constant value of 180 degrees in the bank angle µ, the aircraft turns upside down in the early phases of the flight. The vertical and horizontal accelerations of the missile, a 1 and a 2, initially are relatively small, but increase rapidly near the final point. The performance criterion λ seems to exhibit similar kind of behavior, as it increases almost linearly in the middle phase of the flight up to the final seconds where it rises sharply. It seems that breaking the missile s lock-on, i.e., achieving the highest possible values of the gimbal angle λ requires letting the missile relatively close before the final break turn, which can be also very risky. 26

28 4 NUMERICAL EXAMPLES Fig. 4.4 presents aircraft trajectory with a slightly different initial state. Here, the missile is launched from a direction 45 degrees off the nose of the aircraft. The qualitative aspects of the trajectory are somewhat similar to Fig The aircraft first turns upside down and dives towards the ground to gather airspeed. Near the final point the aircraft again breaks hard to increase the angle between the aircraft and the missile centerline. 4.2 Comparison of Solution Methods In this section, the problems of the previous section are solved using the moving horizon control approach are the obtained solutions are compared to those of the previous section. The initial state for both methods was identical. The moving horizon control trajectories are calculated in the example with n α = n µ = 5 and recursion depth of 3 steps. The time step length is t = 0.2 seconds. The selected parameters would result in (5 5) 3 = alternatives on each step if all the trajectories were feasible. In practice, many of the extreme control alternatives are infeasible, decreasing the number of true alternatives substantially. The resulting trajectory and the trajectory from the direct multiple shooting method are depicted in Fig Both trajectories show a downward-sloping flight path, but the controls suggested by the moving horizon control approach are considerably smoother than those of the open-loop solution. Also, the moving horizon control approach turns the aircraft rightward and away from the missile, whereas in the optimal trajectory, the aircraft flies directly towards the missile in the (x, y)-plane almost for the duration of the flight. The quantitative differences in the trajectories with respect to the gimbal angle λ can be seen in Fig The values of λ(t f ) for direct multiple shooting and moving horizon control approach are 41.4 and 32.9 degrees, respectively. Although the suboptimal solution given by the moving horizon control is perfectly sensible, the difference of almost ten degrees in the evasion could be critical in some situations. The values of the gimbal angle λ as a function of time is quite similar in both solutions, but the final time t f is approximately two seconds less in the moving 27

29 4 NUMERICAL EXAMPLES (x, h) plane h [m] Aircraft (optimization) Aircraft (moving horizon control) Missile (optimization) Missile (moving horizon control) x [m] (x, y) plane Aircraft (optimization) Aircraft (moving horizon control) Missile (optimization) Missile (moving horizon control) y [m] x [m] Figure 4.5: Comparison of optimal open-loop and approximate moving horizon control trajectories 28

30 4 NUMERICAL EXAMPLES 45 λ(t) λ [deg] Optimization Moving Horizon Control t [s] Figure 4.6: The gimbal angle λ(t) for the open-loop and moving horizon control solutions of Fig. 4.5 horizon control solution than in the optimization. The aircraft is thus able to avoid the missile for a longer time in the open-loop solution, thus resulting in a greater loss of the missile s speed. 4.3 Analysis of the Missile Launch Point In this example, several problems with different initial state are solved in order to examine the effect of the missile launch point to the maximum obtained gimbal angle at the final moment. Presumably, closer launch distances would result in smaller values of the gimbal angle λ. The problems with eight initial distances ranging from 2.5 km to 20 with 2.5 km interval are solved. Also, the launch point direction relative to the aircraft is varied from 0 to 345 degrees with 15 degree interval, giving a total of 192 launch points. The initial states are presented in Fig The final distance d is 200 meters, as in the previous examples. A little less than half of the initial states can be omitted due to symmetry, however, leaving 104 problems to be solved. Using two-dimensional interpolation, the results can be used to approximate the value of the gimbal angle λ that could be obtained in the missile evasion when the missile is launched at a given point in the vicinity of the aircraft. 29

31 4 NUMERICAL EXAMPLES Figure 4.7: The initial states in the example. Single dot presents one initial state for the missile. The aircraft is in the middle, the direction of the aircraft is upwards in the graph. The dotted range rings are 5, 10, 15, and 20 km. The solutions of the problems can be seen in Fig Single point in the Figure represents a single launch point of the missile and the level curves indicate the corresponding optimal value of the gimbal angle λ when the missile is launched from that point. Under the range of 15km the resulting values of the gimbal angle λ are quite small. According to the results, breaking the lock-on becomes considerably easier with longer initial distances. This is likely due to reduced velocity of the missile and increased velocity of the aircraft (see Fig. 4.2), as the aircraft has more time to accelerate. In the rear hemisphere of the aircraft, the relative direction of the launch point does not seem to have significant impact on the success of breaking the lock-on. On the other hand, head-on shots seem to be harder to evade because of larger closing velocities. 30

32 4 NUMERICAL EXAMPLES Figure 4.8: The optimal value of the gimbal angle λ(t f ) (in degrees) with respect to the missile launch point relative to the aircraft in the middle of the graph. 31

33 5 Conclusions In this study, the missile seeker constraints and their utilization in the missile evasion were analyzed with the help of dynamic optimization. A three degrees-offreedom aircraft model representing a generic modern fighter aircraft was used as well as a model for a modern medium-range air-to-air missile using proportional navigation as its guidance law. Aerodynamic performance data for a generic aircraft and missile has been used in the models. Using the dynamic models, an optimal control problem was presented with the missile s gimbal angle as the performance criterion. The optimal control problem was discretized and formulated as a nonlinear programming problem. A nonlinear solver was used to find a solve to the problem, yielding optimal, open-loop solutions. A suboptimal feedback control strategy based on the moving horizon control approach was also presented. The numerical solutions given by the open-loop and the feedback methods were compared. Also, numerical open-loop solutions were generated for several different initial states. There are several limitations and simplification in the model used in order to keep it simple enough for the numerical optimization software. In this study, it is assumed that the missile s seeker obtains perfect tracking information. In reality, all radar and optical detection equipment are subject to tracking errors arising from inaccuracy of the seeker as well as environment and weather conditions. Thus, different kind of noise filters and error correction methods (see for example [14]) are needed with all missile guidance systems. In this study, the target information of the missile was also constantly updated, while the target detection can be different during different phases of the missile flight for the real missiles. For example, the missile can first be guided by the target information given by the radar of the launching aircraft. Thereafter, it may be guided by the inertial guidance for a moment until it locks on the target at some locking distance and transits to active mode. The flight paths of the actual missiles might therefore be somewhat different from the trajectory of the nearly ideal missile model used in this paper. As a representation of the capabilities of a modern air combat missile, the constructed model should, however, work reasonably well. 32

34 5 CONCLUSIONS The aircraft has also other defensive capabilities than evasive maneuvers. These include countermeasures of the aircraft, noise and deception jamming (explained for example in [13]), for example. On the other hand, in this study a perfect information about the states of the aircraft and the missile are assumed. In reality, it would be difficult to spot the incoming missile, a warning given by the radar warning receiver of the aircraft might be the only indication that a missile is homing towards the aircraft. Fairly accurate knowledge of the position of the incoming missile is required to successfully execute the most maneuvers suggested by this study. Results seem to suggest that under some circumstances, the missile lock-on can be broken by exceeding the gimbal limit of the missile. This, however, requires a suitable initial state and the probability of success depends on several aspects. To achieve the largest value for the gimbal angle, the final break turn need to be executed as late as possible. This introduces another risk, if the final distance between the missile and its target gets too small. The aircraft might not be able to escape the missile blast even if the lock-on of the missile were broken. According to the results, the velocities of both the aircraft and the missile also seem to be of critical importance in trying to break the lock-on. In optimal trajectories, the aircraft almost always pulls down to obtain more velocity. This naturally requires that the initial altitude of the aircraft is large enough so that a deep dive is possible. Also other factors, such as other threats nearby might limit the execution of the maneuver. As stated by R. L. Shaw in [13] For nearly head-on or tail-on threats, the break direction is the pilot s choice, with vertically nose-down usually preferable if that option is available. Another critical factor is naturally the launch distance of the missile. As the missile rocket motor is used only during the early phases of the flight, the missile launched from large distance slows significantly down before reaching the aircraft. And as the results show, much larger gimbal angles can be obtained if the velocity difference between the aircraft and the missile is small. When comparing the optimal open-loop solution with the feedback solution given 33