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Modern Homing Missile Guidance Theory and Techniques

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Modern Homing Missile Guidance Theory and Techniques Modern Homing Missile Guidance Theory and Techniques Neil F. Palumbo, Ross A. Blauwkamp, and Justin M. Lloyd lassically derived homing guidance laws, such as proportional navigation, can be highly effective when the homing mis- sile has significantly more maneuver capability than the threat. As threats become more capable, however, higher performance is required from the missile guidance law to achieve intercept. To address this challenge, most modern guid- ance laws are derived using linear-quadratic optimal control theory to obtain analytic feedback solutions. Generally, optimal control strategies use a cost function to explicitly optimize the missile performance criteria. In addition, it is typical for these guidance laws to employ more sophisticated models of the target and missile maneuver capabil- ity in an effort to improve overall performance. In this article, we will present a review of optimal control theory and derive a number of optimal guidance laws of increasing complexity. We also will explore the advantages that such guidance laws have over proportional navigation when engaging more stressing threats INTRODUCTION Classical guidance laws, with proportional navigation optimal control theory to missile guidance problems had (PN) being the most prominent example, had proven to sufficiently matured, offering new and potentially prom- be effective homing guidance strategies up through the ising alternative guidance law designs.1–3 Around this 1960s and early 1970s. By the mid-1970s, however, the time, the computer power required to mechanize such predicted capability of future airborne threats (highly advanced algorithms also was sufficient to make their maneuverable aircraft, supersonic cruise missiles, tacti- application practical. cal and strategic ballistic missile reentry vehicles, etc.) Most modern guidance laws are derived using linear- indicated that PN-guided weapons might be ineffective quadratic (LQ) optimal control theory to obtain against them. However, by that time, the application of analytic feedback solutions.1, 4, 5 Many of the modern 42 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) $ formulations take target maneuver into account to deal x()t= f((x t), u()t,) t . (1) with highly maneuvering target scenarios (particularly true for terminal homing guidance). The availability In Eq. 1, x is the n-dimensional state vector of real of target acceleration information for the guidance law elements ()x!!RRn, u m is the control vector, and varies, depending on targeting sensor capability and t represents time (later we will provide more detail as type and the specific guidance law formulation. Typi- to the structure of the state and control vectors for the cally, there also is an explicit assumption made about homing guidance problem). With this general system, we the missile airframe/autopilot dynamic response charac- associate the following scalar performance index: teristics in the modern formulations. We will show later that PN is an optimal guidance law in the absence of J((x t0), u()t 0 ,) t0 = ((x tf), t f) airframe/autopilot lag (and under certain other assumed t f (2) conditions). To some extent, the feedback nature of the + L((x t), u()t,) t dt . #t homing guidance law allows the missile to correct for 0 inaccurate predictions of target maneuver and other unmodeled dynamics (see Fig. 1). However, the require- In this equation, [t0, tf] is the time interval of inter- ment for better performance continues to push optimal est. The performance index comprises two parts: (i) guidance law development, in part by forcing the con- a scalar algebraic function of the final state and time sideration (inclusion) of more detailed dynamics of the (final state penalty), ((x tf), t f), and (ii) a scalar inte- interceptor and its target. It is interesting that most if gral function of the state and control (Lagrangian), t not all modern guidance laws derived using optimal con- f # L((x t), u()t,) t dt . The choice of ((x tf), t f) t0 t trol theory can be shown to be supersets of PN. and # f L((x t), u()t,) t dt (this choice is a significant part In the following sections, we first provide a cursory t0 review of dynamic optimization techniques with a focus of the design problem) will dictate the nature of the on LQ optimal control theory. Using this as background, optimizing solution. Thus, the performance index we then develop a number of terminal homing guid- is selected to make the plant in Eq. 1 exhibit desired ance laws based on various assumptions, in the order of characteristics and behavior (transient response, stabil- assumption complexity. In our companion article in this ity, etc.). issue, “Guidance Filter Fundamentals,” we introduce For our purposes, the optimal control problem is * guidance filtering, with a focus on Kalman guidance to find a control, u ()t , on the time interval [t0, tf] that filter techniques. drives the plant in Eq. 1 along a trajectory, x*()t , such that the scalar performance index in Eq. 2 is minimized. It is difficult to find analytic guidance law expressions REVIEW OF LQ OPTIMAL CONTROL for such general nonlinear systems. Therefore, we will Here, we start by considering a general nonlinear turn our attention to a subset of optimal control that dynamics model of the system to be controlled (i.e., the can yield tractable analytic solutions, known as the LQ “plant”). This dynamics model can be expressed as optimal control problem. Kalman Filter Optimal Control Servomechanism Theory Theory Theory (1960s) (1960s) (1930s–1950s) Target Target state Acceleration Actuation direction estimates commands commands Target motion Target Guidance Guidance Autopilot Airframe/ sensors filter law propulsion Inertial navigation Vehicle motion Figure 1. The traditional guidance, navigation, and control topology for a guided missile comprises guidance filter, guidance law, autopilot, and inertial navigation components. Each component may be synthesized by using a variety of techniques, the most popular of which are indicated here in blue text. JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) 43­­­­ N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYD The LQ Optimal Control Problem Then, from the calculus of variations, four necessary 5 Here, we assume that the nonlinear model in Eq. 1 conditions for optimality must be satisfied to solve our problem: state, costate, boundary, and stationarity con- can be linearized about an equilibrium point (,x0 u0) and represented by the time-varying linear dynamic ditions must all hold. These four conditions are listed in Table 1. system described in Eq. 31, 4: From Table 1 (and based on our previous description $ of the dimension of the plant state vector), it can be seen x()t= A() t x()t + B() t u()t . (3) that there is a system of 2n dynamic equations that must be solved; n equations must be solved forward in time In this model (as in the nonlinear case), x ! Rn is the over [t , t ] (state equations), and n equations must be state vector, u ! Rmis the control vector, and t repre- 0 f solved backward in time over [t , t ] (costate equations). sents time. Here, however, A(t) and B(t) are the time- f 0 We further note that the equations are coupled. Apply- varying Jacobian matrices ing the stationarity condition to Eqs. 3–6 yields the fol- lowing result for the control: 2 2 x= x and 2 2 x= x , respectively) ((f x,,ut)/ x 0 f(, x u,)t / u 0 u= u u= u 0 0 u()t= –( R–1 t)(BT t)( t). (7) of appropriate dimension. Using the “shorthand” notation z()t2 _ zT () t Sz()t , S Using Eq. 7 in the state and costate necessary conditions, we next define the quadratic performance index (a and taking into account the boundary condition, leads special case of the performance index shown in Eq. 2) to the following two-point boundary value problem: shown in Eq. 4: $ –1 T x()t 1 2 x()t A()t –(BRt) B $ = , J((x t0), u()t 0 ,) t0 = 2 x()tf T ()t Qf =()t G =–(Q t) –(A t) G; E (8) tf (4) x()t given + 1 x()t2 + u() t 2 dt . 0 . 2 #t Q()t R()t ()t= Q x() t 0 8 B ) f f f 3 In Eq. 4, the following assumptions are made: the ter- Here, we are concerned with solution methods that minal penalty weighting matrix, Qf, is positive semi- can yield analytic (closed-form) solutions rather than definite (the eigenvalues of Qf are 0, expressed as iterative numerical or gain scheduling techniques. We Qf 0); the state penalty weighting matrix, Q(t), is note, however, that the sparseness/structure of the con- positive semi-definite Q( (t) 0); and the control pen- stituent plant and cost matrices—e.g., A(t), B(t), Q(t), alty matrix, R(t), is positive definite R( (t) > 0). Thus, the and R(t)—will dictate the ease by which this can be LQ optimal control problem is to find a control, u*()t , accomplished. Qualitatively speaking, the level of dif- such that the quadratic cost in Eq. 4 is minimized sub- ficulty involved in obtaining analytic solutions is related ject to the constraint imposed by the linear dynamic primarily to the complexity of state equation coupling system in Eq. 3. into the costate equations, most notably by the structure To solve this continuous-time optimal control prob- of Q(t) and, to a lesser extent, by the structure of A(t). As lem, one can use Lagrange multipliers, ()t , to adjoin we will see later, however, this fact does not negate our the (dynamic) constraints (Eq. 3) to the performance ability to derive effective guidance laws.
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