Guidance and Navigation Systems CHAPTER 2: GUIDANCE

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1 Guidance and Navigation Systems CHAPTER : GUIDANCE Guidance as closed loop system Guidance Classification Proportional Navigation Guidance (PNG) Optimal Guidance Other Guidance Schemes Advanced Guidance Waypoint Guidance Noise and Uncertainties Examples and Applications.1. Guidance as closed loop system The most important feature of guidance, at least in this context, is the tight association with the autopilot and the inner control loop. Therefore, unless otherwise specified, we will address only those topics that involve some form of feedback. As described in the introduction, the guidance process is primarily a kinematic process and has several theoretical and computational components. Measurement of vehicle position and velocity, Assessment of target dynamic behavior (final point or intermediate points), Correction and adjustment to external noise and disturbances, Computation of control actions necessary to properly adjust position and velocity, Delivery of suitable adjustment commands to the vehicle's control system. Let us review the guidance block diagram from chapter 1. 1

2 The first block that we can identify is the inner control loop, which includes the complete dynamic model of the vehicle, actuators, external disturbances, and flight control system (inclusive of autopilot). The resulting command of interest, from a guidance stand point, is the acceleration, here a lateral acceleration A my. This loop, as we can see from the input going into the flight control system, is responsible for disturbance rejection, rotational motion control, and translational motion control in the direction of the vehicle s acceleration vector. The dynamic behavior that relates the motion of the vehicle with the motion of the target point is shown above. The objective in this particular case is the management of the relative motion of the two systems. The target dynamics may be known or unknown, and the output of this process is related to the relative instantaneous distance between the two, indicated by the term 1/R. The next block shows how the information relative to distance and direction is measured by the vehicle (through a component called seeker ), processed with noise models based on the component, target, and environment knowledge, and finally sent to the guidance process. In this particular application, the main variable used by the guidance law is the so-called line of sight rate or LOS-rate ( σ& ). Of course this is only one possible implementation (applicable to high performance, manoeuvrable vehicles), and others are possible.

3 The next block diagram describes the details of the main interconnections among the guidance processor, seeker, and angular measurement unit (navigation system). The actual guidance system is shown below, from a process point of view, meaning that that guidance method and structure is not detailed here. As we can see, the input LOS rate is smoothed and filtered. In addition vehicle s attitude rate is fed in and command to the flight control system is given. An interesting element is the saturation block, which is present due to limitations in the actuators, and introduces stability problems. Another example of the closed loop nature of guidance is shown next, taken from a marine rather than aerospace application. The guidance process is defined using the concept of waypoints (more on this later). In very general terms, a surface ship block diagram is shown below: 3

4 As we can see, the guidance system is responsible for providing desired position and rate variables [ ν d, η d] to be implemented by the control system (or autopilot). The loop is closed by the navigation system responsible for locating the ship, reference command and information about disturbances. A different look at the same problem is shown next. For a ship or an underwater vehicle, the guidance and control system usually consists of: an attitude control system a path control system In its simplest form the attitude control system is a course autopilot. The main function of the attitude feedback control system is to maintain the vessel in the desired attitude on the ordered path by controlling the vessel in roll, pitch, and yaw. The task of the path controller is to keep the vessel on the predefined path with some predefined dynamics (e.g. forward speed) by generating orders to the attitude control system. The way-points are stored in a way-point database and used for generation of a trajectory or a path for the moving vessel to follow. Both trajectory and maneuvering control systems can be designed for this purpose. Sophisticated features like weather routing, obstacle avoidance and mission planning can be incorporated in the design of way-point guidance systems... Guidance Classification There are many implementations of guidance, according to the type of vehicle, type of application, required performance, sensors and actuators, etc. Therefore many different classifications are possible. 4

5 ..1. Trajectory Phases Classification One way of separating guidance design and requirements is as a function of the part of the mission. In the case of a vehicle or a rocket, for instance, there are three phases: initial, midcourse, and terminal. The initial phase is characterized by a series of tasks, some of them extremely demanding from the standpoint of the control system, since the dynamic behavior is highly nonlinear and uncertain. In terms of guidance we can recognize: Launcher clearance Initial acceleration Deployment of flight surfaces Inertial navigation initialization Alignment Error In the mid-course phase, guidance solutions may accommodate open loop strategies with preset corrections: Cruise course/speed Closure on the target Flight Control System & Navigation Control System corrections Onboard Sensors External Tracking The third and last phase is the terminal part of the task. In this phase, the dynamics depend heavily on the relative motion with the target, if the target is not stationary. The guidance system may rely on onboard processing of imagery and radar information, in order to achieve the objective. This phase is the most challenging from the point of view of control system theory and application. Target sensing Vehicle performance Violent, high G-force maneuvers 5

6 Target Intercept The three phases of flight can be also identified in a completely different mission. Let us consider the mission of traveling to Mars (courtesy of NASA-JPL). It is obviously a complex mission with different engineering areas of interest. We consider here the mission of the Global Surveyor spacecraft (second from the top). The main scientific tasks of the spacecraft are relative to mapping of the surface, analysis of the atmosphere, and relay for future missions. The general timeline is shown above. The initial phase is given by the launch, parking orbit, preparation for Mars transfer; the midcourse phase, which is the longest; the final phase that includes all operations about Mars. The initial phase starts with the launch and all preset operations to reach Earth orbit. Based on the current astronomical configuration of Earth, Sun and Mars, and the timing of arrival, the launch is set, the rocket rolls to the Earth orbit osculating plane and ends its operation, once reached a given altitude and velocity capable of inserting the spacecraft in a parking orbit. The guidance system in this phase is essentially coincident with the trajectory control system of the launcher. The mid-course phase consists of three parts. The first is the departure from Earth s gravity field, the second is the insertion into the transfer trajectory to Mars, and the third is the deceleration into Mars orbit. The approximate technique used for this type of motion is based on the method of patched-conics, which connects segments of 6

7 Keplerian orbits, to come up with the mid-course phase. The guidance system is strongly connected to the navigation system since precise position and timing are crucial. The implementation of guidance commands is usually pre-set, unless major position errors are found along the flight. Earth departure is obtained by performing a + V change, which puts the spacecraft from the parking orbit into a hyperbolic escape trajectory. Then another impulsive change is required to enter the transfer orbit about the Sun (Hohmann transfer). Mars approach can be achieved in different ways, always via impulsive decelerations. In this case, the initial burn transfers the spacecraft from heliocentric orbit, into an elongated elliptical orbit about Mars. Successive decelerations bring the Surveyor to a circular orbit. In this course, we will concentrate on the mid-course and terminal phases of guidance.... Guidance Classification In addition to phases of travel, guidance methods can be classified according to the type of information, the amount of closed loop interaction, and methods for actually deriving the guidance law itself. We can start from the guidance algorithms described in the first flowchart. In a preset guidance algorithm, all information is stored on the vehicle at launch. In particular, 7

8 target information is not updated, although the guidance system is capable of maintaining the course and reducing course error with the help of the autopilot. Direct guidance methods are characterized by the fact that information about the target is received at launch, as well as during the flight. This feature allows interception of moving targets and the implementation makes the difference between one guidance law and another. Direct guidance methods are well established, and further improvement will depend only on hardware development. Command guidance is the guidance of a vehicle by means of electronic commands generated outside. The basic concept is to study the relative position and velocity of vehicle-target, and then uplink the information so that intercept at some position is achieved. The information can be given to the vehicle in various ways (radar station, radio frequency, laser beam, uncoiled wire, etc.). The main advantage of this solution is that there is no need of on-board seeker; the main limitation is the influence of noise on the data transmission to the vehicle. CLOS is a derivative of command guidance, where the line-of-sight information is in some way given to the vehicle, thus improving the capability of intercept. The vehicle approaches the target along a line joining a control point and the target itself. At the control point, the target is tracked by a radar (or other method), while the vehicle is tracked by the same radar or by some other point. If the vehicle can maintain the straight line between tracker and target (LOS), it will intercept. The input to the guidance algorithm is therefore the relative error vector and its rate, producing an appropriate compensating acceleration. This can be obtained by applied different control methods, from classical control, to optimal control, from feedback linearization, to neural and fuzzy systems. Beam-rider guidance is a three-point guidance where the guidance algorithm maintains the vehicle inside a beam directed to the target. The beam is usually either radar or laser generated. Homing guidance is typically applied to the terminal phase of the guidance trajectory. In this case, the vehicle is guided by information coming directly from the target, in three 8

9 main solutions: active, semi-active, and passive. In active homing, the vehicle has a transmitter and a receiver and illuminates the target, thus obtaining information about its position and velocity. In semi-active homing, the illumination comes from a source outside the vehicle. In passive homing, the target itself is known due to some special radiating information that is unique to the target itself (heat, electromagnetic waves, etc.). In all three cases, however, the vehicle must have a seeker (system that is capable of performing the homing information measurement and processing it to the guidance law. In order to intercept a target or to reach a specified point precisely, the vehicle must constantly travel in the appropriate direction, which is achieved by a guidance law built into the global guidance processor. The synthesis of a guidance law is obtained using different control methods, and to be complete it needs to be combined with appropriate filtering and estimation, which reduce the influence of noise sources. A schematic of currently used guidance laws is shown below. Although similar in principle (LOS information is crucial to the success of the guidance), there are differences with respect to a particular aspect of intercept: the capability of achieving intercept in the presence of a maneuverable target and/or a target having speed higher than that of the vehicle. For a brief explanation of the differences, we refer to the figure below, which illustrates the variables of interest. As we said before, the main purpose of the guidance law is to provide an appropriate commanded acceleration history, which will guarantee intercept. Based on the engagement geometry, we can compute the vehicle acceleration as: 9

10 R σ&& A = σ & V + + V& tanσ m m m m lead cos σlead After some manipulations (the details will be given later), the kinematic transfer function becomes θm(s) 1 1 = A m(s) s ( Vm V& mr m / R& m) + ( VmR m / R& m) s Rms The LOS loop becomes unstable under small perturbations, and requires a stabilizing lead compensator as shown in the block diagram: The homing guidance system, which contrasts with LOS guidance, is designated as a two-point guidance system and it is implemented mostly as LOS rate guidance. The next block diagram shows a block diagram implementation of the latter solution. Since there is only a pole at the origin, instability issues are less critical, at least until near intercept. Since however it is desirable to keep the guidance loop bandwidth as small as possible, a solution called Proportional Navigation is usually adopted. The 10

11 implementation of LOS rate guidance in terms of commanded flight path angle is given by: γ & = λσ& Where λ is the so-called navigation gain. c And the proportional navigation implementation loop is shown here: Pursuit guidance was the first implemented time wise. It resembles the classical houndhare course. The vehicle aims directly at the target during the entire engagement. While it permits a simple implementation and is less sensitive to noise, pursuit guidance is somewhat impractical against highly moving targets, resulting in energy consuming tail chase. This is NOT the case for instance for ships and underwater vehicles, where the speed advantage is much greater. A less demanding course is the one that predicts the target path and aims the vehicle to the predicted intercept point. This works well if the predictor scheme is accurate, and 11

12 the autopilot is commanded to keep the direction (LOS) to the target constant, resulting in the so-called constant bearing guidance shown below. Proportional Navigation Guidance is a derivation of constant bearing guidance, and will be studied in detail in the next section. The basic concept is to make the vehicle s heading rate proportional to the LOS rate from vehicle to target, yielding a trajectory like in the figure. The term Advanced guidance identifies all new developments in the field of homing guidance. Here the problem is generalized to include prediction, and optimality is taken into account in many possible ways (calculus of variations optimization, constrained nonlinear optimization and on-off control, differential games, fuzzy logic, robust multivariable control methods, etc.)..3. Proportional Navigation Guidance (PNG) Because of the widespread use, and connection to the optimal solution of a two point boundary value problem in certain cases, PNG is examined in detail in this section Development The base principle common to all types of PNG is that of controlling the direction of the vehicle s velocity vector, with a maneuver speed proportional to the LOS rate. It appears that the principle was known and used by pirates in order to achieve collision course with merchant ships starting from More recently, the method was studied by Germany at the end of WW II, and by the USA in the same period. Due to its physical simplicity, the method was implemented in the early 50, and first published in 1948 in the Journal of Applied Physics. Mathematical derivation of its optimality under certain conditions followed only 0 years later with the work by Arthur Bryson. PNG gives the possibility of defining the acceleration command to give the vehicle for intercept. While the magnitude of the acceleration is common to most of the variants and proportional to LOS rate, the commanded acceleration direction changes according to different solutions. In addition, augmented PNG has a term that takes into account target motion capabilities. 1

13 Pure PNG has the acceleration command normal to the velocity of the vehicle; True PNG has the direction normal to the LOS, In modified True PNG, the acceleration direction is still normal to the LOS, but the magnitude is proportional to the product between LOS rate and closing speed (relative speed), Ideal PNG has the direction of the acceleration normal to the closing speed, Generalized True PNG has the direction of the acceleration at a specified angle with respect to the LOS..3.. Optimality The optimality of PNG was proved in 1969 by Bryson, by casting the scenario as a differential game between two players: pursuer and evader. Differential games are a particular set of optimization problems involving players with conflicting interest. As imagined, they involve continuous differential equations and functional performance criteria. Game theory and optimization are very close in mathematical terms, but they were established independently. In guidance problems requiring intercept, the two players are the vehicle and target (pursuer and evader), and their conflicting interest is the solution of a MinMax optimization problem. In mathematical form we have the following problem: Given the dynamic system x& = f(x,u,v,t) x(t ) = x 0 0 (1) Terminal constraints And the performance criterion ( x(t ),t ) Ψ =0 () f f 0 0 Find the couple ( u,v ) t f ( ) [ ] J x(t ),t L(x,u,v,t) dt =Φ f f + t0 such that (3) 13

14 ( 0 ) ( 0 0 ) ( 0 ) J u,v J u,v J u,v (4) The first order necessary conditions are the same as calculus of variations; therefore we have for this minmax problem: =λ + T H f L λ & T = T H, λ x (t) =Φ f x(t f ) H H u v = 0 = 0 (5) Or H 0 = maxminh v u (6) In order to obtain sufficiency of the solution, we also require the existence of a gametheoretic saddle point, which not necessarily exists unless H is separable. Fortunately in our applications this is not the case since both f and L are in general separable. Note that for the function L(u,v ) we have: Game theoretic saddle point Lu = 0 Lv = 0 L L uu Calculus saddle point L = 0 L = 0 u 0 vv 0 LuuLvv ( Luv ) Another interesting point is the evaluation of optimization (4) as an open loop or closed loop strategy. In fact from (4) it is not clear if one player strategy is optimal based on the other s operating in an open loop or closed loop fashion. Although this is not part of the course, it suffices to notice that optimality of one player s behavior is much more stringent when the other operates in feedback fashion. Consider now the standard intercept planar problem based on the next figure, reformulated as a differential game with limits on the vehicles acceleration. 0 v 14

15 In a pursuit-evasion game, the pursuer s control is its acceleration a(t) p normal to the initial LOS, and the evader does the same with its acceleration a(t). t In the following, vector and scalar quantities will be understood from the context. If v(t) and y(t) are the relative velocity and displacement normal to the initial LOS, the equations of motion are: v& = ap at v(t ) = v y& = v y(t f ) = (7) The pursuer wishes to minimize the terminal miss y(t f ), while the evader tends to maximize it, therefore a feasible performance index is: 1 J = y(t ) [ ] f (8) The controls are bounded as: at ap a pm,a pm > a tm a tm (9) The Hamiltonian for the problem is: H = λv(ap a t ) +λ yv. The adjoint equations and optimality conditions are readily obtained as: λ & v = λy λ v(t f ) = 0 λ & y λ y(t f ) = y(t f ) a(t) = a sgnλ a(t) t = atmsgnλv p pm v Solving the Euler-Lagrange equations, we can determine an expression for y(t f ) as 1 v 0 ( tf t0 ) ( apm atm) tf t0 y(t f ) = 1 v ( tf 0 t0 ) ( apm atm) tf t0 However no solution exists for, v 0 if > 1 ( tf t0 )( apm atm) v 0 if < 1 ( tf t0 )( apm atm) v 0 1< if < 1 ( tf t0 )( apm atm) In fact, for this range of initial conditions, the pursuer can always achieve zero miss distance, for instance choosing its acceleration such that: 15

16 v(t ) a p(t) = a(t) t + t t f. The term t f t is also called time to go or t go. Let us consider now a linear quadratic differential game, with a quadratic performance index that contains the acceleration limitations, in a 3-D environment. The equations of motion are: v& p = fp + ap v& t = ft + at, & rp = vp r & t = vt (10) Assuming the gravity contribution f the same, the only variable of interest is the relative position vector r(t) r(t). Consider the performance index: p t t { p t p t } b J = r (t) r (t) r (t) r (t) + 1 f 1 1 { p ( p p) t ( t t) } 0 c a a c a a dt (11) Using (5) yields: c (t t) r (t) r (t) + (v (t) v (t))(t t) p f p t p t f ap = 3 1 (tf t) + (cp c t ) b 3 (1) ct at = ap cp In (1) we note that: If cp > ct the feedback control gain has always the same sign, If cp < ct the feedback control gain changes sign when and t f sufficiently large. 1 b (t t) 3 3 f + (cp c t ) = 0 (13) Equation (13) is actually the conjugate point condition for the optimization problem (10)- (11), that is the relationship for which sufficient conditions for global minimum are not satisfied according to the second variation properties. Therefore case cp < ct is not longer optimal, which means that intercept is not possible when cp < ct and evident for b=. At this point we can select the optimal solution corresponding to cp > ct. With b=, the pursuer s control strategy becomes: 16

17 3 a = r (t) r(t) (v (t) v (t))(t t) c + (t t) p p t p t f p 1 f c t (14) Now let the pursuer and the follower be on a nominal collision course with range and closing velocity respectively given by R Vc R = t t f Let the lateral deviation from collision be given by r(t) p r(t) t as shown below From the engagement geometry, the lateral acceleration to be applied by the pursuer from (14) is given by: a LAT 3 = Vcσ& (15) c t 1 cp Which is simply Proportional Navigation with an effective navigation gain of 3 N = ct 1 cp Based on experimental results, the best value of N has been found to be between 3 and 5. Note that N=3 corresponds to non maneuverable evader (c t =0), whereas N=5 corresponds to a weighting ratio c t /c p =/5. 17

18 .3.3. D-Scenario In this section, we will limit ourselves to a two-dimensional scenario, which is the most common in guidance problems. Nomenclature: A,A m A,A m t t Vehicle and target accelerations Vehicle and target velocities γm, γ t Vehicle and target flight path angles σ LOS angle R Range Vc = R & Closing velocity (relative velocity along the inst. LOS) The inertial reference system has the abscissa in the down range direction and the ordinate either altitude (depth) or cross range. The angle α = L+ He = γm σ has two components: L is the lead angle that is the theoretical value for a collision course (no additional acceleration commands are required for intercept). H e is known as heading error, or the initial deviation from the collision triangle. The equations of motion of the geometry above can be written in Cartesian form: x&& = A sinγ z&& m = Amcosγm x&& t = At sinγt z&& t = At cosγt m m m γ m =tan z& x & z& -1 m m γ t=tan R= ( x x ) +( z - z ) -1 zo zm σ=tan x o x m -1 t x & t (16) o m o m 18

19 ( xt x m)( x t x m)+(zt-z m )(z o - z m) V= c ˆ - R & & & & & = - R ( xt x m)(z & o - z & m)-( zt-z m )( x & t x & m) σ & = - R (17) Or in Polar form: R& = Vc = Vt cos( γt σ) Vmcos( γm σ) Vt sin( γt σ) Vmsin( γm σ) σ & = R R && = R σ & + Amsin( γm σ) At sin( γt σ) σ & R& + At cos( γt σ) Amcos( γm σ ) σ && = R Am γ & m = V m A γ o & t = V t (18) (19) In the previous section, we have seen the mathematical proof of optimality from the standpoint of a differential game approach. Consider now the collision triangle ( R & < 0 ) below: If the pursuer velocity is constant, the evader is not maneuvering, and the LOS does not rotate in the inertial space, collision will occur. Consider in fact the second of (18), differentiating and substituting back into the first yields: Rσ+ && R NVmcos( m) & + σ γ = V& sin( σ γ ) +γ& Vcos( σ γ ) Vsin( & σ γ ) m m t t t t t (0) From (15), for non maneuvering target, γ & m = Nσ& and pursuer constant speed, the RHS of (0) becomes equal to zero and (0) is a homogeneous differential equation in LOS rate. 19

20 Set the navigation ratio N as: Λ R& = Λ Λ > n N n, Vmcos( σ γm) n (1) And substituted in (0), yields: [ ] Rσ&& R& Λ σ & = 0 () n This has an analytical solution R σ=σ & & 0 R0 Λn Λn R γ & m = γ & m0 R0 (3) The solution tends to zero (intercept) for R & < 0 Λ n =3, and asymptotic decrease for Λn.. Linear decrease is achieved for NOTE: From the PNG, it is possible to go back to the expressions for pursuit and constant bearing guidance laws. Recall that in pursuit guidance, the vehicle aims directly at the target during the entire engagement. Assuming the usual constraints on pursuer and evader speed ( Vt = const. γ t = const. = 0 ;Vm = const. ), the guidance law assumes the form: γ m = σ (4) Am = Vmσ& From (18) R& = Vt cos γt Vm (5) R σ & = -Vt sin γ t But σ& =0 if σ=0 or σ=π (head-on, or tail-on), which means that the pursuer must always be turning. The case of constant bearing guidance is even closer to PNG. It derives directly from the intercept triangle conditions, and it requires: Example σ& 0 (6) Consider the following engagement; the pursuer has a heading angle of -0 degrees. 0

21 The results in terms of PNG guidance are shown next Two-dimensional tactical missile-target engagement simulation 15 Two-dimensional tactical missile-target engagement simulation red -> N=4 green-> N=5 Altitude (Ft) Acceleration of missle (G) Downrange (Ft) x Time (sec) Consider the second engagement, with the evader having a sudden acceleration (maneuverable target) The next two figures show the vehicles trajectories, as well as pursuer acceleration profiles for the two navigation ratios of 4 and 5. Even if the optimality of PNG is not guaranteed for maneuvering targets, in this example intercept occurs. 1.5 x 104 Two-dimensional tactical missile-target engagement simulation 4.5 Two-dimensional tactical missile-target engagement simulation Altitude (Ft) Red-> N=4 Acceleration of missle (G) Downrange (Ft) x Time (sec).3.4. Linear Approximation 1

22 As usual, many important feature of dynamic systems behaviour can be inferred by a linear analysis. In the previous sections we introduced PNG and evaluated its optimality, given some engagement assumptions. In simulation, however, the intercept was successful even in the case of maneuverable target (at least for the case shown). A linear view of the problem will help understanding why. Recall the D scenario described in section.3.3.: Based on the above geometry, consider the relative acceleration normal to the reference axis 1, A = y&& = A cosβ A cos σ R R t m Using small angle approximation (small flight path angles) we have: AR At Am yr σ R Treating the closing velocity as a positive constant, since R 0 at and of the flight, we can use the linear expression for the range: R = V c(tf t ) = Vct go,where tf = const.

23 The linearized miss distance, or relative separation ad the end of the engagement, is y (t ) R f If we consider the previous two examples, we have: From the time histories we can see that in the case of non maneuvering target the linear approximation is excellent, while an overestimate is present when the target is maneuvering because linearized guidance assumes the target maintaining the same acceleration vector all throughout the flight. Since the problem is linearized, a closed form solution can be obtained (for the two examples above). Recall that Then: HE yr = miss dis tan ce initial heading error At Am = y && R, MAN yr = miss dis tan ce t arg et maneuverable y&& = NV σ& y&& NV A HE R c MAN R = cσ+ & t Solving for the transition matrix and using the initial conditions, yields: N HE HE Vm N t Amc = 1 tf tf A N t = 1 1 N MAN mc N t f A t (*) An important aspect of the linearized approach is the physically insight gained by examining the solution, and the definition of Zero Effort Miss (ZEM), a parameter important also in other types of guidance laws. Definition: Define ZEM as the distance the pursuer would miss the evader, if the evader continued along its present course, and the pursuer made no further corrective maneuvers. 3

24 From the above equations, and the standard form of PNG (15) we have: Amc = NVc σ& R = V (t t) = V t yr σ R c f c go (7) Substituting and differentiating yields: From which: d y =σ & R Amc NVc dt V c (t f t ) N Amc = yr + y& Rt go t go,zem = yr + y& Rtgo Amc = NVc σ& (8) In (8) we can see that the term in brackets is nothing but the ZEM, therefore PNG, when linearization applies, can be thought as a guidance law which is also proportional to the ZEM and inversely proportional to the square of the time-to-go. The Adjoint Method The adjoint method is a powerful optimization technique dating to Volterra (1870), and used originally for artillery studies. It combines gradient based optimization with a linearized guidance loop dynamics. The above diagram, based on the preceding discussions, is the simplest form of a linearized homing loop. We can see that noise processing and autopilot are considered perfect, and the LOS rate information is computed perfectly by the seeker; this absence 4

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