Stochastic Model Predictive Control
•stochastic finite horizon control
•stochastic dynamic programming
•certainty equivalent model predictive control
Prof. S. Boyd, EE364b, Stanford University
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Stochastic Model Predictive Control-Software and Convex Optimization-Lecture Slides, Slides of Convex Optimization
Dr. Hanumant Chawd delivered this lecture at Alagappa University for Convex Optimization course. Its main points are: Stochastic, Model, Predictive, Control, Dynamic, Programming, Equivalent, Model, Causal, State, Feedback
Typology: Slides
2011/2012
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Stochastic Model Predictive Control
stochastic finite horizon control
stochastic dynamic programming
certainty equivalent model predictive control
Prof. S. Boyd, EE364b, Stanford University
docsity.com
Causal state-feedback control
linear dynamical system, over finite time horizon:
x
t
Ax
t
Bu
t
w
t
t
,... , T
x
t
R
n
is state,
u
t
R
m
is the input at time
t
w
t
is the process noise (or exogeneous input) at time
t
X
t
x
0
,... , x
t
is the state history up to time
t
causal state-feedback control:
u
t
φ
t
X
t
ψ
t
x
0
, w
0
,... , w
t
−
1
t
,... , T
φ
t
R
(
t
+1)
n
R
m
called the control
policy
at time
t
1 docsity.com
Stochastic finite horizon control
an infinite dimensional problem: variables are
functions
φ
0
,... , φ
T
−
1
can restrict policies to finite dimensional subspace,
e.g.
φ
t
all affine
key idea: we have
recourse
(a.k.a. feedback, closed-loop control)
we can change
u
t
based on the observed state history
x
0
,... , x
t
cf standard (‘open loop’) optimal control problem, where we committo
u
0
,... , u
T
−
1
ahead of time
in general case, need to evaluate
J
(for given control policies) via
Monte Carlo simulation
3 docsity.com
‘Solution’ via dynamic programming
let
V
t
X
t
be optimal value of objective, from
t
on, starting from initial
state history
X
t
V
T
X
T
T
x
T
J
⋆
E
V
0
x
0
V
t
can be found by backward recursion: for
t
T
V
t
X
t
) = inf
v
∈U
t
x
t
, v
E
V
t
X
t
, Ax
t
Bv
w
t
X
t
V
t
t
,... , T
are convex functions
optimal policy is causal state feedback
φ
⋆t
X
t
) = argmin
v
∈U
t
x
t
, v
E
V
t
X
t
, Ax
t
Bv
w
t
X
t
4 docsity.com
Linear quadratic stochastic control
special case of linear stochastic control
• U
t
R
m
x
0
, w
0
,... , w
T
−
1
are independent, with
E
x
0
E
w
t
E
x
0
x
T 0
E
w
t
w
Tt
W
t
t
x
t
, u
t
x
Tt
Q
t
x
t
u
Tt
R
t
u
t
, with
Q
t
R
t
T
x
T
x
TT
Q
T
x
T
, with
Q
T
6 docsity.com
can show value functions are quadratic,
i.e.
V
t
x
t
x
Tt
P
t
x
t
q
t
t
,... , T
Bellman recursion:
P
T
Q
T
q
T
; for
t
T
V
t
z
inf
v
z
T
Q
t
z
v
T
R
t
v
E
Az
Bv
w
t
T
P
t
Az
Bv
w
t
q
t
works out to
P
t
A
T
P
t
A
A
T
P
t
B
B
T
P
t
B
R
t
−
1
B
T
P
t
A
Q
t
q
t
q
t
Tr
W
t
P
t
7 docsity.com
Certainty equivalent model predictive control
at every time
t
we solve the certainty equivalent problem
minimize
T
−
1
τ
=
t
t
x
τ
, u
τ
T
x
T
subject to
u
τ
∈ U
τ
τ
t,... , T
x
τ
Ax
τ
Bu
τ
w
τ
|
t
τ
t,... , T
with variables
x
t
,... , x
T
u
t
,... , u
T
−
1
and data
x
t
w
t
|
t
w
T
−
1
|
t
w
t
|
t
w
T
−
1
|
t
are predicted values of
w
t
,... , w
T
−
1
based on
X
t
e.g.
, conditional expectations)
call solution
˜x
t
x
T
u
t
˜u
T
−
1
we take
φ
mpc
X
t
u
t
φ
mpc
is a function of
X
t
since
w
t
|
t
w
T
−
1
|
t
are functions of
X
t
9 docsity.com
Certainty equivalent model predictive control
widely used,
e.g.
, in ‘revenue management’
based on (bad) approximations: –
future values of disturbance are exactly as predicted; there is nofuture uncertainty
in future, no recourse is available
yet, often works very well
10 docsity.com
Stochastic MPC: Sample trajectory
sample trace of
x
1
and
u
1
0
10
20
30
40
50
−1 −
2 1 0
0
10
20
30
40
50
−0.
0
x 1 )t( u
1 )t(
t
Prof. S. Boyd, EE364b, Stanford University
12 docsity.com
Cost histogram
0
200
400
600
800
1000
0 50 150 100
0
200
400
600
800
1000
0 50 150 100
J
mpc
J
relax
J
relax
J
sat
13 docsity.com