Transcript Semi Plenary 3 (ppt, 361kb)

Colloquium on Optimisation and Control
University of Sheffield
Monday April 24th 2006
Sensitivity Analysis and Optimal Control
Richard Vinter
Imperial College
Sensitivity Analysis
Sensitivity analysis: the effects of parameter changes on the solution of an
optimisation problem:
Practical Relevance:
• Resource economics (economic viability
of optimal resource extraction in changing environment)
• Design (buildings to withstand earthquakes, . .)
Theoretical Relevance:
• Intimate links with theory of constrained optimization
(Lagrange multipliers, etc.)
• Intermediate step in mini-max optimisation
• ‘parametric’ approaches to MPC
The Value Function
m vector parameter
Minimize f ( x,  ) over
P( )
Data:
s.t.
0
x  Rn
g ( x,  )  0
and
h ( x,  )  0
f : R n  R m  R, g : R n  R m  R k , h : R n  R m  R k
Value function:
V ( ) : inf{ f ( x,  ) : x s. t. g ( x,  )  0, h( x,  )  0}
(describes how minimum cost changes with
f (., 1 )
V (1 )
)
f (.,  2 )
V ( 2 )
(no constraints case)

Links With Lagrange Multipliers
Special case:
P( )
Minimize
f (x )
over
x  Rn
h(x)  
s.t.
(m vector parameter

is value of equality constraint function )
Lagrange multiplier rule:
Fix
*
   * . Suppose x
is a minimiser for P( ) .
Then for some m vector ‘Lagrange multiplier’
*

f ( x* ,  * )  T h( x* ,  * )  0
Value function:
V ( ) : inf{ f ( x) : x such that h( x)   }
Fact: The Lagrange multiplier

has interpretation:
T   V ( *)
(   is the gradient of the value function associated with
perturbations of the constraint )
Show this:
For any
f (x )  
x,
for
  h(x)
, so
f ( x)  V (h( x))  0
By ‘minimality’:
Hence
f ( x*)  V (h( x*))  0 (since
f ( x* )  T h( x* )  0
h( x*)   * )
, where
(Caution: analysis not valid unless V is differentiable.)
T   V ( * )
Consider now the optimal control problem:
T
Minimize
 L( x(t ), u(t ))dt
0
s.t.
(data:
x (t )  f ( x(t ), u (t ))
u (t )  
x(0)  x0 and x(T )  C
a
 R n , L : Rnn  R m  R, sets , C and x0  R n
d
f : Rn  Rm
)
Most significant value function is associated with perturbation of initial data:
T
Minimize
P( ,  )
s.t.
 L( x(t ), u(t ))dt
x (t )  f ( x(t ), u (t ))
u (t )  
x( )  
x(T )  C
V ( , x)  inf{ P( , x)}
x
Domain of P( , x)
,x
t
Pontryagin Maximum Principle
Take a minimizer ( x * (.), u * (.))
Define ‘Hamiltonian’:H ( x, p, u ) 
pT f ( x, u )  L( x, u )
Then, for some ‘co-state arc’ p (.) ,
 p (t )  f xT ( x * (t ), u * (t )) p (t )  L( x * (t ), u * (t ))
(adjoint equation)
h(t )  H ( x * (t ), p (t ), u * (t ))
(max. of Hamiltonian cond.)
 p (T )  N C ( x * (T ))
(transversality cond.)
where
h(t ) : max u H ( x * (t ), p (t ), u * (t ))
NC (x) is the normal cone at x
(maximised Hamiltonian)
C
x
‘normal vector’ at x
Sensitivity Relations in Optimal Control
Gradients of value function w.r.t. ‘initial data’ are related to co-state variable
( p(0),h(0))  V (0, x0 )
What if V is not differentiable?
Interpret sensitivity relation in terms of set valued ‘generalized gradients’:
V ( z )  co{lim i V ( zi ) : zi  z}
(definition for ‘Lipshitz functions’, these are ‘almost everywhere’ differentiable)
f
For some choice of co-state p(.)
+1
x
f
-1
x
( p(t ), h(t ))  V (t , x * (t )) , 0  t  T
(Valid for non-differentiable value functions)
Generalizations
T
Minimize
 L( x(t ), u(t ),  )dt
Dynamics and cost depend on par.

0
P( )
s.t.
x (t )  f ( x(t ), u (t ),  )
u (t )  
x(0)  x0 and x(T )  C
V ( )  inf{ P( )}
(  * , nominal value )
Obtain sensitivity relations (gradients of V’ ) by ‘state augmentation’.
~
Introduce P ( , ( ,  )) with extra state equation
  0
~
V ( )  V (  0, ( ,  )  ( x0 ,  *))
expressible in terms of co-state arcs for state augmented problem
Application to ‘robust’ selection of feedback controls
Classical tracker design:
Step 1: Determine nominal trajectory using optimal control
Step 2: Design f/b to track the nominal trajectory
(widely used in space vehicle design)
• Can fail to address adequately conflicts between performance and robustness
Alternatively,
Integrate design steps 1 and 2
• Append ‘sensitivity term’ in the optimal control cost to reduce effect of
model inaccuracies
This is ‘robust optimal control’
Robust Optimal Control
1) Model dynamics:
x (t )  f ( x(t ), u(t ),  ), x(0)  x0
T
2) Model cost:
J ( x(.), u (.))   L( x(t ), u (t )) dt
0
3) Model variables requiring de-sensitisation:
T
( x(.))    ( x(t )) dt  d ( x(T ))
0
Example: magnitude of deviation from desired terminal location :
 ( x(.)) || x(T )  xd (T ) ||2
4) Feedback control law:
u  u (t )  K ( x  x (t ))
Objective: find sub-optimal control which reduces sensitivity of  (x (.))
to deviation of

from
 *.
Sensitivity Relations
*

x

x
)
u
(.)
(Write
x
(.)
For control
, let
be state trajectory for  .
The `sensitivity function’
T
u ( )   ( x (.))   L( x(t ), u (t )) dt

0
has gradient:
T
 u ( *)   pT (t ) f ( x(t )) dt  d ( x(T ))
0
where the arc p (.) solves
 p (t )  (( f xT  K T f uT )( x (t ), u (t ),  *)) p(t )   x ( x (t ))
p(T )  d ( x (T ))
Optimal Control Problem with Sensitivity Term
T
 L( x(t ), u (t ))dt
Minimize
 
0
s.t.
T
T
2
p
(
t
)
f
(
x
(
t
),
u
(
t
),

*)
dt
|


0
x (t )  f ( x(t ), u (t ),  *)
u (t )  
x(0)  x0 and x(T )  C
 p (t )  (( f xT  K T f uT )( x (t ), u (t ),  *)) p (t )   x ( x (t ))
p (T )  d ( x (T ))
Pink blocks indicate extra terms to reduce sensitivity

is sensitivity tuning parameter
sensitive
0

values
insensitive

Trajectory Optimization for Air-to-Surface Missiles with Imaging Radars
Researchers: Farooq, Limebeer and Vinter. Sponsors: MBDA, EPSRC
‘Terminal guidance strategies for air-to-surface missile using DBS radar seeker’.
Specifications include:
• Stealthy terrain phase, followed by climb and dive phase (‘bunt’ trjectory)
• Sharpening radars impose azimuthal plane constraints on trajectory
Bunt phase
Stealth phase
• Six degree of freedom model of skid-to-turn missile
(two controls: normal acceleration demains
Select cost function to achieve motion, within constraints.
References:
R B VINTER, Mini-Max Optimal Control, SIAM J. Control and Optim., 2004
V Papakos and R B Vinter, A Structured Robust Control Technique, CDC 2004
A Farooq and D J N Limebeer, Trajectory Optimization for Air-to-Surface
Missiles with Imaging Radars, AIAA J., to appear