On Modern Guidance Laws
C.A. Rabbath
Defence R&D Canada Valcartier
Third
Thi
dM
Meeting
ti off th
the STANAG 4618 W
Working
ki G
Group
DRDC Valcartier, Quebec City, Canada
4-6 October 2011
Outline*
1. Context
2 Proportional-Derivative
2.
Proportional Derivative Navigation Guidance Law
3. Near-Optimal Trajectory Shaping of Guided
Projectiles with Constrained Energy Consumption
*Two
2
recently proposed schemes part of a group of guidance laws labeled as “modern”
1. Context
Guidance
“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
law, estimation/filtering)
3
1. Context
One-on-one engagement
Weapon & target in vicinity of collision course
vt
θt
nt
β
Target
nt : target
g normal
acceleration
r
Line of sight
vm
y
α
nm
λ
Missile
Inertial frame
x
Fig. 2D Point-mass Geometry
4
nm : weapon
acceleration ⊥ LOS
(typical of TPN)
vcl = -rr
1. Context
Simplified block diagram
Di it l guidance
Digital
id
law
l
acceleration commands
LOS rate, range, …
Estimator
Digital autopilot
ADC
ADC
IMUs
Seeker
T
Target
t
5
DAC
Actuators
Missile dynamics
1. Context
More elaborate block diagram…
diagram for simulations
6
Classical Guidance: PNG
PNG [[zarchan]:
h ] nmc = N vcl λ
0
Guidance
λ
nmc
Flight
g control
Kinematics
nm
Feedback
control
interpretation
[White et al.]
Goal of PNG:
To make LOS rate zero when near or on collision course
PNG optimal [Kreindler] when 1) nt= 0 and 2) nm assumed
equal to nmc
However, 1) target typically maneuvers (nt ≠ 0), and 2) nm ≠ nmc
i.e. missile flight control has finite time constant
7
Issues, challenges, limitations
 Uncertainties in missile control
 Flight control dynamics always approximately known
 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 p
post-synthesis
y
approximations
pp
(nonlinear guidance includes terms related to nonlinear kinematics )
 Highly maneuverable targets (w.r.t. missile) may prohibit use of
small-angle approximations
8
Issues, challenges, limitations
 Maneuverable targets
 Optimality typically guaranteed under stringent (unrealistic)
assumptions
p
 Target behavior not known, but can be estimated/predicted?
 Monte Carlo simulations necessary for effectiveness demo
 Realistic assumptions
p
 Errors/noise in guidance input signals
 Constrained processing for guidance algorithms
 Delays in transmission of information
 Guidance command bounded/saturated in magnitude
9
Issues, challenges, limitations
 High
High-Precision
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.)
10
Outline
1. Context
2 Proportional-Derivative
2.
Proportional Derivative Navigation Guidance
Law
3 Near-Optimal
3.
Near Optimal Trajectory Shaping of Guided
Projectiles with Constrained Energy Consumption
11
3. Proportional-Derivative Navigation Guidance
Law
Homing guidance (terminal phase of an air-to-air or an air-to-surface engagement)
Single-missile, single-target 2D engagement
Flight
Control
Pursuer (Missile or rocket)
Guidance
Law
Evader (Target)
Goal
To minimize pursuer/evader miss distance by generating
appropriate acceleration commands.
12
3.1 Three issues
3.1.1 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 model.
low-order
model
Some solutions: Neoclassical PNG, Adaptive NL guidance with SMC.
3 1 2 Constrained digital implementation
3.1.2
Digital guidance law
T
2
A digital implementation of guidance
and control laws may adversely
T1
T2
affect the performance of a weapon
due to inherent computational delays,
T1 > T2
quantization effects and constrained
control
t l update
d t andd sampling
li rates.
t Converted to discrete-time and implemented on digital hardware
Especially important for small
Reference acceleration
δ
weapons: munitions, rockets.
+
Missile
+
Actuators
Estimator
Digital autopilot
ADC
ADC
DAC
Actuators
Missile dynamics
IMUs
Seeker
Target
Kp
+
Ki
13
1
s
dynamics
+
Kr
Normal acceleration in yaw/pitch channel
Yaw/pitch rate
Sensors
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:
- robustness to uncertain weapon dynamics,
- target maneuvering
i (to
( some extent…).
)
14
3.2 Concept of passivity
Fundamental to the proposed guidance law.
B i conceptt
Basic
Recall: Lyapunov function V is some energy function (relates to state of system)
“Stability”: dV/dt ≤ 0 along trajectories x
Passive system:
System which cannot store more energy than is supplied by
some source, with difference between stored and supplied
energy namedd dissipated
di i t d energy.
Energy in
15
System
Stored energy
Energy out
I/O concept
Passivity and stability
u
y
Σ
2
y T u ≥ V ( x) + β u + δ y
2
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.
16
Passivity and stability
Target
acceleration
at
t
Missile-target
System
To damp out through guidance
Weapon-target
Weapon
target law,
law then expect small miss
separation
y
Strictly Passive
∈ L2
∈ L2
t
Strict passivity implies L2 stability (input-output)
A passivity-based
passivity based guidance law renders
the closed-loop system strictly passive
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
17
3.3 Design of PDNG
Steps to go from synthesis to digital implementation
Continuous-time plant model
Passivity-based synthesis
Continuous-time controllers
Closed-loop discretization
Discrete-time controllers
Implementation
p
on
digital electronics
Digital controllers
18
Passivity-based guidance synthesis
Obj ti T
Objective:
To make
k the
th uncertain
t i missile-target
i il t
t system
t passive.
i
1. Model the simplified missile target engagement
Assuming small flight-path angles or small deviations from collision course
vt
nt
θt
β
Target
nt
β
at
am
λ
r
nm
Line of sight
vm
y
α
nm
λ
Missile
Inertial frame
19
x
y
Second-order linear uncertain dynamic model
Actual
Uncertain parameters
Known bounds
20
Commanded
2. Decompose missile-target model into feedback system
at
Kinematics
am
at
Guidance
ag
Missile - autopilot
Kinematics
am
y, y
y, y
Missile - autopilot
G1
z
G2
ag
Time
varying
gains
21
at
Kinematics
y, y
G1
z
1
ρ
am
22
Missile & autopilot
ag
2
G2
3. Calculate G1 and G2
G1 { kp(t),
(t) kd (t) }
Use an extension of KYP Lemma to time-varying
systems,
y
, and apply
pp y Sylvester’s
y
theorem:
Sufficient passivity conditions for kp(t) and kd(t)
23
3. Calculate G1 and G2
G2 { k1, k2 }
2 is made strictly input passive by solving for K,
K P2 in LMI
at the vertices of the polytopic form of the system.
E ample
Example
Si = (Ai,Bi,Ci,Di)
β
S1
S2
Fourtuple at each vertex, 16 sets of
LMIs solved simultaneously.
To account ffor entire realm off parameter
p
values, put all possible cases within a
polytope (convexity) and solve at the
vertices.
24
β S3
α
S4
α
α
Closed-loop discretization
Objective:
Obj
i To
T place
l
into
i
a format
f
suitable
i bl for
f a discrete-time
di
i
implementation and that results in a satisfactory performance for
a wide range
g of sampling
p g rates.
Calculate the gains of a discrete-time control law Gd that minimizes
the L2
L2-gain
gain of an error system
Σcδ
Target
at
acceleration
acce
ea o
zΣ
+
−
z
Pδ
H
Hold
a g ,hg
Gd
ε
Error between reference
system
sys
e aand
d DT sys
system
e
y
S
Sampler
DT guidance law (to be calculated)
Minimax problem is solved to obtain time-invariant DT guidance law
25
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,
- controller computations with limited number of bits,
- fixedfixed and floating
floating-point
point arithmetic.
arithmetic
5. Generate code, load to target processor, and compile.
26
3.4 Results – Model & parameters
Missile-target
g kinematics
Initial conditions
Uncertain missile parameters
Missile flight control dynamics
Recall
Maneuvering target
∈ L2
27
Note: parameter values inspired
from [Gurfil] who simplified a
real system
Simulink model for simplified engagement simulations
G2
G1
28
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 environment:
Simulation
i
Matlab
M l b - Simulink
Si li k
To solve the LMIs: Matlab’s LMI Toolbox
29
Results – PDNG
Stable poles placed within circle centered at –22 with radius 11.2
2
kp(t) = N / t2go, kd (t) = N / tgo t = tf - t
Calculated the minimum k1, k2
solving the LMIs (via Matlab)
k1 = 170, k2 = 1.3
at
Kinematics
am
30
y, y
Missile - autopilot
G1
z
G2
ag
Results – OGL
Optimal Guidance Law
Guidance yielding zero miss and minimizing integral of
square of commanded acceleration
nmc = f(N, tgo, y, y, nt, L), L is time lag of missile control
31
 Assume nt , tgo known exactly
 No uncertainties considered
 First-order approximation to missile control dynamics
Results - Miss
Unconstrained digital implementation
300
250
ωOGL
o , ξo
ωG12
,ξ
ω ,ξ
OGL
Range (m)
R
200
150
PDNG
100
50
0
0
0.5
1
1.5
2
2.5
Time (s)
3
3.5
4
4.5
5
Miss distances in meters
Range
→ Taking into account the uncertain flight dynamics in the design reduces miss
32
Results - Miss
Unconstrained digital implementation
Target maneuvering
Unconstrained digital implementation
Various initial separations
33
Results - Miss
Constrained digital implementation
TPDNG: Discrete-time PDNG obtained with classical local discretization method
RO2PDNG: Discrete-time
Discrete time PDNG obtained with
ith closed-loop
closed loop discreti
discretization
ation and
order reduction (2nd order G2)
RO4PDNG: Discrete-time PDNG obtained with closed-loop discretization and
order reduction (4th order G2)
P i off parameters:
Pairs
t
Sampling/control update rate = 10 Hz
34
Results - Accelerations
Unconstrained digital implementation
Saturation
35
Outline
1. Context
2 Proportional-Derivative
2.
Proportional Derivative Navigation Guidance Law
3. Near-Optimal Trajectory Shaping of Guided
Projectiles with Constrained Energy
Consumption
36
4. Trajectory Shaping
Objective
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
speed,
d andd tterminal
i l fli
flight-path
ht th angle.
l
Projectile dynamics
Characterized by control constraints (actuator saturation), model
nonlinearities, wind turbulence.
37
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 in
range and with an orientation tailored to the delivery of an optimal
effect.
Bias
Classical PNG law:
Navigation constant
- No terminal impact angle requirement enforced,
- May result in high acceleration demands (saturation → miss).
38
Closing velocity
LOS rate
BPNG
Guidance law that simultaneously nullifies (1) the LOS rate,
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
difference
bbetween a ddesired
i d LOS angle
l and
d the
h actuall LOS angle
l is
i as small
ll as
required.
Precision conditions allow us to conclude that the angle
g between the velocityy
vector of the missile and the ground plane complies to a desired angle at some
time instant. NOT CONSTRUCTIVE.
Desired impact angle
39
BPNG
Desired LOS angle
LOS angle
40
22. Constrained nonlinear optimization by application of level set
theory and viability theory
q
solvingg Hamilton-Jacobi-Isaacs ppartial differential
- requires
equations (Bayen, Mitchell, Oishi, and Tomlin)
- computationally expensive (parallel processing feasible?)
- Complex, impractical (based on our own experience)
41
4.2 Proposed trajectory shaping
11. Enables satisfying tight terminal constraints while taking into
account nonlinear flight dynamics.
p
non-gradient-based
g
iterative searches to qquicklyy determine
2. Exploits
the 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 accelerationtime pairs generated by TS by means of an appropriate
guidance law and autopilot
autopilot.
42
Block diagram
Initial conditions
Trajectory
Shaping
Altitude
G Target set
Latax
∼u
Datalink
t
u
R
Range
Guidance
Law *
Y(t)
Wind Turbulence
w
Autopilot
Actuators
Flight Dynamics
of the Projectile
INS
Closed-Loop Dynamics : X = f ( X , u, w)
* Biased PNG
43
[Rabbath & Lestage]
x
y
v
γ
4.3 Formulation of the problem
4000
Ballistic
trajectory
3500
Target set
3000
0] / xmin≤ x ≤ xmax,
v min ≤ v ≤ v max , γmin≤ γ ≤ γmax, y=0}.
y (m)
G {(
G={(x,y,
v, γ)
2500
∈3×[-π,
[
2000
Despinned
projectile
u≡0
1500
1000
500
0
Finite-state command generator
Provides a piecewise constant signal
-500
v
2000
4000
6000
x (m)
8000
10000
γ
12000
S it hi ti
Switching
time iinstants
t t
u(t)=uk∈U, for all t∈[tsw,k, tsw,k+1),
where U ={-U, -(n-1)U /n, …,-U/n,0,
u
0
Trajectory of
the guided
projectile
u≠0
Elevation
U/n,…, (n-1)U /, U},
t
z denotes the sequence {uk, 1≤k≤p}.
44 Control input sequence for entire flight
t sw,1 t sw, 2 t *f
t f > t *f
t sw, p −1 t sw, p = ηt *f Flight
Time
Block diagram
Augmented Closed-Loop Dynamics
Trajectory Shaper
u
Smoothing Filter
u
tsw,i
sw i
∼
u
t
Closed-Loop Dynamics
X = f ( X , ∼
u, w)
x
v
γ
Trajectory shaper
• Off
Off-line
line finite
finite-state
state command generator
• Smoothing filter: to comply with the bandwidth
projectile’s
j
controllers and
of the p
u
X u = F filter ( X u , u ),
u~ = G
( X , u ).
)
filter
to reduce the energy consumption
u
Wind Turbulence
w
Guidance
Law *
Autopilot
Actuators
Flight Dynamics
of the
o
e Projectile
ojec e
INS
45
Closed-Loop Dynamics : X = f ( X , u , w)
x
y
v
γ
Control problem
Find the sequence of control signals z that minimizes dG (xf,vvf,γf; z)
where dG (xf,vf,γf; z) = dG (xf; z)+ dG (vf; z)+ dG (γf; z)
position
iti
dG (χf; z) =
χf
velocity
l it
angle
l
0 if χ f ∈ [χ min χ max ],

i (| χ min − χ f | / χ min , | χ max − χ f | / χ max ),
) otherwise,
th
i
min(|
denotes either xf, vf, or γf
z : specifies that the terminal state is obtained by applying the
sequence of control signals z
46
To obtain sequence Zi :
- Exhaustive search (intensive)
- Cross-Entropy-Minimization-Based Search
(CEMBS): a stochastic search method to quickly
find a near
near-optimal
optimal solution to the TS problem
- speed is a function of model complexity, sample
47 size
Sample set
control in
nputs
N
Z1 = {Z1,…,Z
ZN}
For i=1 to N
Zi={uk,1≤… ≤p}
S
Simulations
s
Iterative simulation-based approach
• Obtain identically distributed random sequence Zi
• Conduct
C d t bbatch
t h off simulations
i l ti
ffor every sequence
Zi
• Use results of simulations to select,, at the next
iteration, a new sample set that tends to decrease
dG (xf,vf,γf; z)
X = f ( X , u~, w)
X u = F filter
fil ( X u , u )
u~ = G
( X , u)
filter
ta
Stochasttic
e”
search
h “distance
4.4 Trajectory shaping solution
u
tf <η tf*
dG (x,v,γ)
CEMBS
CEMBS (Rubinstein & Kroese)
Developed originally to conduct rare
rare-event
event simulations and solve complex
combinatorial optimization problems such as TSP.
Cross Entropy (CE) minimization – to estimate probabilities of rare events,
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 term
u~( s) =
ωn2
s 2 + 2ωnξs + ωn2
u (s)
Augmented Closed-Loop Dynamics
Trajectory Shaper
u
Smoothing Filter
u
tsw,i
49
t
∼
u
Closed-Loop Dynamics
X = f ( X , ∼
u , w)
x
v
γ
4.5 Results - model
3 DOF model
3-DOF
d l off dde-spinned
i d projectile
j til
v y =
−Cd* (v,
y )vˆ(v y − w y ) +
x = v x ,
y = v y ,
v=
v x2
uv y
mv
uv y
,
mv
0.3
0.25
− g,
Cd
v x = −Cd* (v, y )vˆ(v x − wx ) −
0.35
0.2
w wind turbulence
+ v 2y
Cd* (v, y ) =
vˆ = (v x − wx ) + (v y − wy )
ρSCd ( M )
2m
2
2
0.15
0.1
0
0.5
1
1.5
Mach Number
M
2
2.5
Smoothing filter
v
M=
vs ( y )
u~ ( s) =
ωn2
s
2
+ 2ωnξs + ωn2
u ( s)
Additional dynamics
50
ω 2 eτ d s
u (s) =
sat10 N (u~ ( s ))
2ξ
s2 +
s + ω2
ω
∼
u
es
s2+1.4s+1
u
x, y
Simplified
vx, vy
Model
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
2
Target set C2
Same as Target set C1 but with v max=1.1v*(tf)
u~ ( s) =
TS algorithm
n=1, p=10, U=92.5 N, η=1.1, α=0.08, ρ=0.1, κ=5, and N=p2
51
ωn2
s
2
+ 2ωnξs + ωn2
09
0.9
u ( s)
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
5
95
4.5
90
4
Average Terminal Error (%)
A
Empirical Frequency of Zero
o Terminal Error (%)
100
85
80
75
70
65
3.5
3
2.5
2
15
1.5
60
1
55
0.5
50
10
12
14
16
18
Number of Switches
20
22
Empirical frequency of dG (xf,vf,γf; z)=0
# switches ↑ → probability of zero-error ↑
Plateau at ≈ 90%
53
0
10
12
14
16
18
Number of Switches
20
Average terminal (relative) error when
the target set is not reached
# switches ↑ → average terminal error ↓
22
Proposed
P
d TS algorithm
l ith
Average 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.
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
Energy
4
18
x 10
Unfiltered command
Actual command
16
14
12
10
8
6
4
2
0
0
5
10
15
Time (s)
20
25
30
Energy expenditure u2dt for unfiltered (---) and filtered (⎯) commands with
smoothing filter having parameters ωn= 1 rad/s and ξ=0.9
55
Example trajectories
One guided projectile trajectory, and sequence of control inputs
B lli ti
Ballistic
With TS
56