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On Modern Guidance Laws On Modern Guidance Laws C.A. Rabbath Defence R&D Canada Valcartier Thi d M ti f th STANAG 4618 W ki G Third Meeting of the STANAG 4618 Working Group DRDC Valcartier, Quebec City, Canada 4-6 October 2011

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Page 1: On Modern Guidance LawsOn Modern Guidance Lawscradpdf.drdc-rddc.gc.ca/PDFS/unc118/p536605_A1b.pdf · On Modern Guidance LawsOn Modern Guidance Laws C.A. Rabbath Defence R&D Canada

On Modern Guidance LawsOn Modern Guidance Laws

C.A. RabbathDefence R&D Canada Valcartier

Thi d M ti f th STANAG 4618 W ki GThird Meeting of the STANAG 4618 Working GroupDRDC Valcartier, Quebec City, Canada4-6 October 2011

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Outline*

1. Context2 Proportional Derivative Navigation Guidance Law2. Proportional-Derivative Navigation Guidance Law3. Near-Optimal Trajectory Shaping of Guided

Projectiles with Constrained Energy ConsumptionProjectiles with Constrained Energy Consumption

2

*Two recently proposed schemes part of a group of guidance laws labeled as “modern”

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1. ContextGuidance

“To bring weapon on or near target”

Missile, munition, rocket, …

The guidance system relies on:

– Hardware (seeker, other sensors, datalinks, digital processor)

Software (guidance law estimation/filtering)– Software (guidance law, estimation/filtering)

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1. Context One-on-one engagement

v ntv nt

Weapon & target in vicinity of collision course

vt

Target

tθt

β

vt

Target

tθt

β

nt : target normal

yvm

Line of sight

r

yvm

Line of sight

r

t gaccelerationnm : weapon acceleration ⊥ LOSy

Missile

λ

αnm

y

Missile

λ

αnm

v = r

acceleration ⊥ LOS(typical of TPN)

Fig. 2D Point-mass Geometry

xInertial frame

xInertial frame

vcl = -r

4

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1. ContextSimplified block diagramSimplified block diagram

Di it l id l acceleration commandsDigital guidance law

Digital autopilot Missile dynamicsActuatorsDACEstimatorLOS rate, range, …

acceleration commands

ADC ADC

IMUs

Seeker

T tTarget

5

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1. ContextMore elaborate block diagram for simulationsMore elaborate block diagram… for simulations

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cPNG [ h ] N λ

Classical Guidance: PNGcPNG [zarchan]: nm = N vcl λ

Feedback Guidance Flight control0 nmc nm

control interpretation[White et al.]

g

Kinematicsλ

Goal of PNG: To make LOS rate zero when near or on collision course

PNG optimal [Kreindler] when 1) n = 0 and 2) n assumedPNG optimal [Kreindler] when 1) nt= 0 and 2) nm assumed equal to nm

However, 1) target typically maneuvers (nt ≠ 0), and 2) n ≠ n

c

c

7

However, 1) target typically maneuvers (nt ≠ 0), and 2) nm ≠ nmi.e. missile flight control has finite time constant

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Issues, challenges, limitations Uncertainties in missile control

Flight control dynamics always approximately known

Inherent dynamic variations over flight path Inherent dynamic variations over flight path

System lags adversely affect performance ( ↑ miss)

Deviations in subsystems performance: expected vs. actual

Nonlinear kinematics Usually, linearization is involved

Intuition: Recover PNG terms with post-synthesis approximations p y pp(nonlinear guidance includes terms related to nonlinear kinematics )

Highly maneuverable targets (w.r.t. missile) may prohibit use of

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small-angle approximations

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Maneuverable targets

Issues, challenges, limitations

Maneuverable targets

Optimality typically guaranteed under stringent (unrealistic) assumptionsp

Target behavior not known, but can be estimated/predicted?

Monte Carlo simulations necessary for effectiveness demo

Realistic assumptionsp

Errors/noise in guidance input signals

Constrained processing for guidance algorithms

Delays in transmission of information

9

Guidance command bounded/saturated in magnitude

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High-Precision Requirements

Issues, challenges, limitations

High Precision Requirements

Terminal impact angle, speed

Bound on acceptable miss distance

Constraints on energy

To connect guidance law design steps with required terminal effects (e.g. lethality statistics, etc.)

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Outline

1. Context2 Proportional Derivative Navigation Guidance2. Proportional-Derivative Navigation Guidance

Law3 Near Optimal Trajectory Shaping of Guided3. Near-Optimal Trajectory Shaping of Guided

Projectiles with Constrained Energy Consumption

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3. Proportional-Derivative Navigation Guidance LawHoming guidance (terminal phase of an air-to-air or an air-to-surface engagement)Single-missile, single-target 2D engagement

FlightControl

Pursuer (Missile or rocket)

Control

GuidanceLaw

Evader (Target)Goal

12

To minimize pursuer/evader miss distance by generating appropriate acceleration commands.

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3.1 Three issues3.1.1 Robustness

Flight control system dynamics typically used in the design of missile guidance laws are considered to be either ideal (as a unitary gain) or an exact low order modellow-order model.

Some solutions: Neoclassical PNG, Adaptive NL guidance with SMC.

3 1 2 Constrained digital implementation3.1.2 Constrained digital implementation

A digital implementation of guidance and control laws may adversely

Digital guidance law

Digital autopilot Missile dynamicsActuatorsDAC

ADC ADC

Estimator

T1 T2

T2

Digital guidance law

Digital autopilot Missile dynamicsActuatorsDAC

ADC ADC

Estimator

T1 T2

T2

affect the performance of a weapon due to inherent computational delays, quantization effects and constrained

t l d t d li t

Seeker

Target

IMUs

T1 > T2

Seeker

Target

IMUs

T1 > T2

control update and sampling rates. Especially important for small weapons: munitions, rockets. δ

Kp+ +

+

Reference acceleration

MissiledynamicsActuators Sensors

Converted to discrete-time and implemented on digital hardware

δKp

+ ++

Reference acceleration

MissiledynamicsActuators Sensors

Converted to discrete-time and implemented on digital hardware

13

KiKr

1s

- ++

Normal acceleration in yaw/pitch channel

Yaw/pitch rateKi

Kr

1s1s

- ++

Normal acceleration in yaw/pitch channel

Yaw/pitch rate

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3.1.3 Target maneuvers

Potentially significant in air-to-air engagements.

If information (e.g. an estimate) on target acceleration is available during an engagement, a guidance law that exploits such signal is preferred.

PDNG addresses:PDNG addresses:

- robustness to uncertain weapon dynamics,

i ( )- target maneuvering (to some extent…).

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3.2 Concept of passivity

Fundamental to the proposed guidance law.

B i tRecall: Lyapunov function V is some energy function (relates to state of system)Basic concept

“Stability”: dV/dt ≤ 0 along trajectories x

Passive system:Passive system: System which cannot store more energy than is supplied by some source, with difference between stored and supplied

d di i t denergy named dissipated energy.

Energy out

15

SystemEnergy in Energy out

Stored energy

I/O concept

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Passivity and stability

Σu y

22)( yuxVuyT δβ ++≥ V(x) ≥ 0

Strictly Input Passive: β > 0

Strictly Output Passive : δ > 0

Key results: [Khalil] A feedback connection of SIP systems results in a strictly passive closed-loop system.

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Passivity and stabilityTo damp out through guidance law then expect small missTarget Weapon-target

Missile-target System

at ylaw, then expect small miss

accelerationWeapon target

separation

tt

Strictly Passive

∈ L2 ∈ L2

Strict passivity implies L2 stability (input-output)

A passivity-based guidance law renders Small miss expectedA passivity based guidance law renders the closed-loop system strictly passive

Notes:

Small miss expected

Stability guaranteed

Notes: • Internal stability (guidance+autopilot+seeker) could be obtained provided system is zero-state detectable• y(tf) is miss distance

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y(tf) is miss distance

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3.3 Design of PDNGSteps to go from synthesis to digital implementation

Continuous-time plant model

Continuous-time controllers

Passivity-based synthesis

Closed-loop discretization

Discrete-time controllers

Implementation on pdigital electronics

Digital controllers

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Digital controllers

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Passivity-based guidance synthesis

Obj ti T k th t i i il t t t iObjective: To make the uncertain missile-target system passive.

1. Model the simplified missile target engagement

Assuming small flight-path angles or small deviations from collision course

vt

Target

tθnt

β

vt

Target

tθnt

β

yvm

Line of sight

r

λ

αnm

yvm

Line of sight

r

λ

αnm

ynmam λntatβ

19

x

Missile

Inertial framex

Missile

Inertial frame

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Second-order linear uncertain dynamic model Commanded

Actual

Uncertain parametersUncertain parameters

Known bounds

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2. Decompose missile-target model into feedback system

GuidanceKinematicsat

am

y, y

aMissile - autopilot

ag

G1Kinematicsat y, y

G2z

Missile - autopilotam ag

Time varying gains

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G1Kinematicsat y, y z

am ag G22

Missile & autopilot

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3. Calculate G1 and G2

G1 { k (t) k (t) }G1 { kp(t), kd (t) }

Use an extension of KYP Lemma to time-varying systems, and apply Sylvester’s theorem:y , pp y y

Sufficient passivity conditions for kp(t) and kd(t)

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3. Calculate G1 and G2G2 { k1, k2 }

is made strictly input passive by solving for K P in LMI2 is made strictly input passive by solving for K, P2 in LMI

at the vertices of the polytopic form of the system.

E ampleExample

βS1 S2

Si = (Ai,Bi,Ci,Di)

β S3To account for entire realm of parameter

Fourtuple at each vertex, 16 sets of LMIs solved simultaneously.

24

αβ

α α

S3 S4

f f pvalues, put all possible cases within a polytope (convexity) and solve at the vertices.

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Closed-loop discretization

Obj i T l i f i bl f di iObjective: To place into a format suitable for a discrete-time implementation and that results in a satisfactory performance for a wide range of sampling rates.g p g

Calculate the gains of a discrete-time control law Gd that minimizes the L2-gain of an error systemthe L2 gain of an error system

Σcδ

εat

+

Σcδ

εat

+

−Target

accelerationError between reference system and DT system

P δ

G S

zy

ghga ,H

P δ

G S

zy

ghga ,H

acce e a o sys e a d sys e

Gd SH Gd SH

SamplerHold

DT guidance law (to be calculated)

25

DT guidance law (to be calculated)

Minimax problem is solved to obtain time-invariant DT guidance law

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Implementation on digital electronics

1. Reduce the order of the controllers obtained (if high).

2. Select sampling/control update rates.

3. Choose an implementation structure (direct form, canonical form, etc.)

4. Simulate finite wordlength effects. These effects include:

- finite resolution of ADC,

- representation of control parameters with limited number of

bits,bits,

- controller computations with limited number of bits,

fixed and floating point arithmetic

26

- fixed- and floating-point arithmetic.

5. Generate code, load to target processor, and compile.

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3.4 Results – Model & parametersMissile-target kinematics Initial conditionsg Initial conditions

Missile flight control dynamicsUncertain missile parameters

Recall

Maneuvering target

∈ LNote: parameter values inspired from [Gurfil] who simplified a

27

∈ L2from [Gurfil] who simplified a real system

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Simulink model for simplified engagement simulations

G2

G1

28

G1

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Time of flight: tf ≥ 5 sec. (From triggering of TG until intercept)

Some parameters

Time of flight: tf ≥ 5 sec. (From triggering of TG until intercept)

Desired miss distance: y(tf) < 1 m

Acceleration saturation: ± 20g

Si l i i M l b Si li kSimulation environment: Matlab - Simulink

To solve the LMIs: Matlab’s LMI Toolbox

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Results – PDNG

Stable poles placed within circle centered at 2 with radius 1 2Stable poles placed within circle centered at –2 with radius 1.2

kp(t) = N / t2go, kd (t) = N / tgo t = tf - t

k1 = 170, k2 = 1.3 Calculated the minimum k1, k2solving the LMIs (via Matlab)

G1at

a

y, yG2zKinematics

30

am agMissile - autopilot

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Results – OGL Optimal Guidance Law

Guidance yielding zero miss and minimizing integral ofGuidance yielding zero miss and minimizing integral of square of commanded acceleration

nm = f(N, tgo, y, y, nt, L), L is time lag of missile controlc

31

Assume nt , tgo known exactly No uncertainties considered First-order approximation to missile control dynamics

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Results - Miss300

OGLξ

Unconstrained digital implementation

250

OGLG12

ωo , ξo

ξξω ,

ω ,OGL

150

200

Ran

ge (m

)

PDNG

50

100

R PDNG

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

Time (s)

Range

Miss distances in meters

32→ Taking into account the uncertain flight dynamics in the design reduces miss

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Results - MissUnconstrained digital implementation

Target maneuvering

Unconstrained digital implementation

Various initial separations

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Results - Miss

Constrained digital implementation

TPDNG: Discrete-time PDNG obtained with classical local discretization method

RO2PDNG: Discrete time PDNG obtained ith closed loop discreti ation andRO2PDNG: Discrete-time PDNG obtained with closed-loop discretization and order reduction (2nd order G2)

RO4PDNG: Discrete-time PDNG obtained with closed-loop discretization and order reduction (4th order G2)

P i f t

34

Pairs of parameters:

Sampling/control update rate = 10 Hz

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Results - AccelerationsUnconstrained digital implementation

Saturation

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Outline

1. Context2 Proportional Derivative Navigation Guidance Law2. Proportional-Derivative Navigation Guidance Law3. Near-Optimal Trajectory Shaping of Guided

Projectiles with Constrained EnergyProjectiles with Constrained Energy Consumption

36

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4. Trajectory ShapingObjective

Given up-to-date information prior to firing, find the lateral acceleration profile to steer a precision-guided munition to a prescribed target set.

Target set

Defined by a number of constraints - projectile’s range, terminal d d t i l fli ht th lspeed, and terminal flight-path angle.

Projectile dynamics

Characterized by control constraints (actuator saturation), model nonlinearities, wind turbulence.

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4.1 Approaches to satisfy terminal constraints

1. Biased Proportional Navigation Guidance (BPNG)

To steer the rocket to the target with as high a precision as possible inTo steer the rocket to the target with as high a precision as possible in range and with an orientation tailored to the delivery of an optimal effect.

Bias

Classical PNG law: Navigation constant

Bias

- No terminal impact angle requirement enforced,

- May result in high acceleration demands (saturation → miss).

Navigation constant LOS rate

Closing velocity

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Guidance law that simultaneously nullifies (1) the LOS rate and (2) the differenceBPNG

Guidance law that simultaneously nullifies (1) the LOS rate, and (2) the difference between a desired LOS angle and the actual LOS angle at or near impact.

BPNG theory allows us to state whether there exists a time instant at which the diff b d i d LOS l d h l LOS l i lldifference between a desired LOS angle and the actual LOS angle is as small as required.

Precision conditions allow us to conclude that the angle between the velocity g yvector of the missile and the ground plane complies to a desired angle at some time instant. NOT CONSTRUCTIVE.

39

Desired impact angle

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Desired LOS angleBPNG

LOS angleLOS angle

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2 Constrained nonlinear optimization by application of level set2. Constrained nonlinear optimization by application of level set theory and viability theory

- requires solving Hamilton-Jacobi-Isaacs partial differential q g pequations (Bayen, Mitchell, Oishi, and Tomlin)

- computationally expensive (parallel processing feasible?)

- Complex, impractical (based on our own experience)

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4.2 Proposed trajectory shaping1 Enables satisfying tight terminal constraints while taking into1. Enables satisfying tight terminal constraints while taking into account nonlinear flight dynamics.

2. Exploits non-gradient-based iterative searches to quickly determine p g q ythe sequence of lateral accelerations.

3. Simulations indicate that energy expenditure can be limited by simply including a smoothing filter rather than trying to minimize a more complex objective function.

4. We assume

- TS calculations are done prior to launch,

- the controlled projectile can robustly track the acceleration-time pairs generated by TS by means of an appropriate guidance law and autopilot

42

guidance law and autopilot.

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Block diagram

TrajectoryShaping

Y(t)Initial conditions

Wind Turbulencet

G

Datalink

Altitude

R

u∼Latax

Target set

Flight DynamicsActuatorsAutopilot

Wind Turbulence

x

v

tw

Guidance

Range

y

u

of the ProjectileActuatorsAutopilot

INS

vγLaw *

),,( wuXfX =Closed-Loop Dynamics :

43

* Biased PNG

[Rabbath & Lestage]

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4.3 Formulation of the problem4000

G {( ) 3 [ 0] / ≤ ≤

Target set2500

3000

3500

4000Ballistictrajectory

G={(x,y, v, γ) ∈3×[-π, 0] / xmin≤ x ≤ xmax,

v min ≤ v ≤ v max , γmin≤ γ ≤ γmax, y=0}.

500

1000

1500

2000

y (m

)

Despinnedprojectileu ≡ 0

Trajectory ofthe guidedprojectile

0 2000 4000 6000 8000 10000 12000-500

0

x (m)

v γ

projectile u ≠ 0

Finite-state command generatorProvides a piecewise constant signal S it hi ti i t t

Elevation

Provides a piecewise constant signal

u(t)=uk∈U, for all t∈[tsw,k, tsw,k+1),

where U ={-U, -(n-1)U /n, …,-U/n,0,

Switching time instants

U/n,…, (n-1)U /, U},u

t

44

*ft *

ff tt > Flight Time

*, fpsw tt η=1,swt 2,swt 1, −pswt

t

z denotes the sequence {uk, 1≤k≤p}.Control input sequence for entire flight

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Trajectory Shaper

Augmented Closed-Loop DynamicsBlock diagram

u

t

Smoothing Filter

),,( wuXfX = vγ

xu u ∼

tsw i

Closed-Loop Dynamics

tsw,i

Trajectory shaper

• Off-line finite-state command generator

)(~),,(

uXGu

uXFX ufilteru

=

=

Off line finite state command generator

• Smoothing filter: to comply with the bandwidth

of the projectile’s controllers and ).,( uXGu ufilter=p j

to reduce the energy consumptionFlight Dynamicsof the ProjectileActuatorsAutopilot

x

GuidanceLaw *

y

Wind Turbulencew

u

Flight Dynamicsof the ProjectileActuatorsAutopilot

x

GuidanceLaw *

y

Wind Turbulencew

u

45

o e ojec eINS

γ

),,( wuXfX =Closed-Loop Dynamics :

Law o e ojec eINS

γ

),,( wuXfX =Closed-Loop Dynamics :

Law

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Find the sequence of control signals z that minimizes dG (xf vf γf; z)

Control problemFind the sequence of control signals z that minimizes dG (xf,vf,γf; z)

where dG (xf,vf,γf; z) = dG (xf; z)+ dG (vf; z)+ dG (γf; z)

iti l it l

[ ] ∈

th i)/||/|i (|, if 0 maxmin χχχ fdG (χf; z) =

position velocity angle

−− otherwise,),/||,/|min(| maxmaxminmin χχχχχχ ff

dG (χf; z)

denotes either xf vf or γff

z : specifies that the terminal state is obtained by applying the

denotes either xf, vf, or γf

z : specifies that the terminal state is obtained by applying thesequence of control signals z

46

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4.4 Trajectory shaping solutionZ1 = {Z1 ZN}Nse

t np

uts

Iterative simulation-based approach• Obtain identically distributed random sequence Zi

C d t b t h f i l ti f

Z1 {Z1,…,ZN}

For i=1 to N

Sam

ple

cont

rol i

n

• Conduct batch of simulations for every sequence Zi

• Use results of simulations to select, at the next )(

),~,(

uXFX

wuXfX

fil=

=

Zi={uk,1≤… ≤p}

s,iteration, a new sample set that tends to decrease dG (xf,vf,γf; z)

),(~),(

uXGu

uXFX

ufilter

ufilteru

=

=

Sim

ulat

ions

To obtain sequence Zi : - Exhaustive search (intensive)

ta tf <η tf*

Se”

- Cross-Entropy-Minimization-Based Search (CEMBS): a stochastic search method to quickly find a near-optimal solution to the TS problem

dG (x,v,γ)

“dis

tanc

etic

h

47

find a near optimal solution to the TS problem- speed is a function of model complexity, sample size

CEMBS

Sto

chas

tse

arch

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CEMBS (Rubinstein & Kroese)

Developed originally to conduct rare event simulations and solve complexDeveloped originally to conduct rare-event simulations and solve complex combinatorial optimization problems such as TSP.

Cross Entropy (CE) minimization – to estimate probabilities of rare eventsCross Entropy (CE) minimization to estimate probabilities of rare events, and to solve difficult combinatorial optimizations, derived from the CE measure of information.

Here, the algorithm updates the sequences ZiN with parameter values that minimize the CE between two probability distributions on control input sequences. The algorithm stops when a condition on the “distance” to the target set is fulfilled.

48 See: http://iew3.technion.ac.il/CE/

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Second-order smoothing filter

• Filters out high-frequencies in the control input signal (inherent to switching)

• Results in reduced energy consumption (wrt not having a filter) → no need to complexify the optimization with an energy termcomplexify the optimization with an energy term

)(2

)(~22

2susu n

ξω

++=

Augmented Closed-Loop DynamicsAugmented Closed-Loop Dynamics

2 22 ss nn ωξω ++

u

Trajectory Shaper

Smoothing Filter

),,( wuXfX = vγ

xu u ∼

t

Closed-Loop Dynamicsu

Trajectory Shaper

Smoothing Filter

),,( wuXfX = vγ

xu u ∼

t

Closed-Loop Dynamics

49

t γtsw,i t γtsw,i

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4.5 Results - model

3 DOF d l f d i d j til

,)(ˆ),(* yxxdx mv

uvwvvyvCv −−−=

3-DOF model of de-spinned projectile

0.3

0.35

,

,)(ˆ),(*

x

yyydy

vx

gmvuv

wvvyvCv

=

−+−−=

0.2

0.25

Cd

w wind turbulence,yvy =

22 vvv += 22 )()(ˆ yyxx wvwvv −+−=0 0.5 1 1.5 2 2.50.1

0.15

Mach Number M

w wind turbulence

yx vvv += )()( yyxx wvwvv +

mMSCyvC d

d 2)(),(* ρ= )(yv

vMs

=Smoothing filter

)()(~2

susu nω=m2 )(ys

Additional dynamics

)(2

)( 22 suss

sunn ωξω ++

=

50

))(~(sat2)( N1022

2su

ss

esusd

ωωξ

ω τ

++= es

s2+1.4s+1Simplified

Model

x, y

vx, vy

u∼ u

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Parameters

m=18.5 kg, S=18.8⋅10-3 m2, g=9.81 m/s2

x(0) =0 m, y(0) =0 m, v(0) = 625 m/s, and γ(0)=π/4 rad

Target set C1

xmin=10599 m, xmax=10712 m, v min = 241.6 m/s, v max= v*(tf), γmin=-1.15 rad, and γmax=-0.94 v*(tf)=268 m/s

Target set C2

Same as Target set C1 but with v max=1.1v*(tf) 2

2

TS algorithm

)(2

)(~22

2su

sssu

nn

n

ωξωω

++=

0 9

51

n=1, p=10, U=92.5 N, η=1.1, α=0.08, ρ=0.1, κ=5, and N=p2 0.9

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TS algorithm (continued)TS algorithm (continued)

150 simulation runs

1 GB, 2.4-GHz Xeon computer

Matlab/Simulink-compiled code

CEMBS tested with p∈{12,13,14,15,18,20}

52

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100

%) 5

80

85

90

95

o Te

rmin

al E

rror

(

3

3.5

4

4.5

Err

or (%

)

65

70

75

80

Freq

uenc

y of

Zer

o

1 5

2

2.5

3

Aver

age

Term

inal

10 12 14 16 18 20 2250

55

60

Em

piric

al

10 12 14 16 18 20 220

0.5

1

1.5A

10 12 14 16 18 20 22Number of Switches

10 12 14 16 18 20 22Number of Switches

Empirical frequency of dG (xf,vf,γf; z)=0 Average terminal (relative) error when

the target set is not reachedthe target set is not reached

# switches ↑ → probability of zero-error ↑

Plateau at ≈ 90%

# switches ↑ → average terminal error ↓

53

Plateau at ≈ 90%

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P d TS l ithAverage computation time is about 2 min 30 s when p∈{12,13,14,15} and increases to 3 min 48 s and to 5 min 30 s when p=18 and p=20, respectively.

Proposed TS algorithm

Exhaustive search

With p=10 → 310 = 59049 executions, a total computation time of 4 h 30 min.

For this case, exhaustive search has not resulted in dG (xf,vf, γf;z)=0.

# switches ↑ → processing time ↑

Processing time: proposed TS << Exhaustive search

54

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18x 104

Unfiltered command

Energy

12

14

16 Actual command

6

8

10

0

2

4

0 5 10 15 20 25 30Time (s)

Energy expenditure u2dt for unfiltered (---) and filtered (⎯) commands with smoothing filter having parameters ωn= 1 rad/s and ξ=0.9

55

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Example trajectories

One guided projectile trajectory, and sequence of control inputs

B lli tiBallistic

With TS

56

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