On Modern Guidance LawsOn Modern Guidance...
Transcript of On Modern Guidance LawsOn Modern Guidance...
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
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
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*Two recently proposed schemes part of a group of guidance laws labeled as “modern”
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
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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
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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
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However, 1) target typically maneuvers (nt ≠ 0), and 2) nm ≠ nmi.e. missile flight control has finite time constant
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
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
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Guidance command bounded/saturated in magnitude
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
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To minimize pursuer/evader miss distance by generating appropriate acceleration commands.
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
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KiKr
1s
- ++
Normal acceleration in yaw/pitch channel
Yaw/pitch rateKi
Kr
1s1s
- ++
Normal acceleration in yaw/pitch channel
Yaw/pitch rate
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
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SystemEnergy in Energy out
Stored energy
I/O concept
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
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
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
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x
Missile
Inertial framex
Missile
Inertial frame
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
1ρ
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.
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αβ
α α
S3 S4
f f pvalues, put all possible cases within a polytope (convexity) and solve at the vertices.
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
zΣ
+
−
Σcδ
εat
zΣ
+
−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)
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DT guidance law (to be calculated)
Minimax problem is solved to obtain time-invariant DT guidance law
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
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- fixed- and floating-point arithmetic.
5. Generate code, load to target processor, and compile.
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
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∈ L2from [Gurfil] who simplified a real system
Simulink model for simplified engagement simulations
G2
G1
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G1
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
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am agMissile - autopilot
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
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Assume nt , tgo known exactly No uncertainties considered First-order approximation to missile control dynamics
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
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
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Pairs of parameters:
Sampling/control update rate = 10 Hz
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
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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.
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Desired impact angle
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
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guidance law and autopilot.
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 :
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* Biased PNG
[Rabbath & Lestage]
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
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*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
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
vγ
GuidanceLaw *
y
Wind Turbulencew
u
Flight Dynamicsof the ProjectileActuatorsAutopilot
x
vγ
GuidanceLaw *
y
Wind Turbulencew
u
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o e ojec eINS
γ
),,( wuXfX =Closed-Loop Dynamics :
Law o e ojec eINS
γ
),,( wuXfX =Closed-Loop Dynamics :
Law
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
fχ
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
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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
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/
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
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t γtsw,i t γtsw,i
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
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
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n=1, p=10, U=92.5 N, η=1.1, α=0.08, ρ=0.1, κ=5, and N=p2 0.9
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}
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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 ↓
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Plateau at ≈ 90%
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
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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
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Example trajectories
One guided projectile trajectory, and sequence of control inputs
B lli tiBallistic
With TS
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