Robust MPC using min-max differential
inequalities
Boris Houska, Mario Villanueva, Benoît Chachuat
ShanghaiTech, Texas A&M, Imperial College London
1
Overview
Introduction to Robust MPC
Min-Max Differential Inequalities
2
Model Predictive Control (MPC)
Certainty equivalent MPC:
minimize distance to dotted line
subject to: system dynamics and constraints
3
Model Predictive Control (MPC)
Certainty equivalent MPC:
minimize distance to dotted line
subject to: system dynamics and constraints
4
Model Predictive Control (MPC)
Repeat:
wait for new measurement
re-optimize the trajectory
5
Model Predictive Control (MPC)
Problem:
certainty equivalent prediction is optimistic
infeasible (worst-case) scenarios possible
6
What is Robust MPC?
Main idea:
take all possible uncertainty scenarios into account
important: we can react to uncertainties
7
What is Robust MPC?
Main idea:
take all possible uncertainty scenarios into account
important: we can react to uncertainties
8
What is Robust MPC?
Main idea:
take all possible uncertainty scenarios into account
important: we can react to uncertainties
9
What is Robust MPC?
Main idea:
take all possible uncertainty scenarios into account
important: we can react to uncertainties
10
What is Robust MPC?
Problem:
exponentially exploding amount of scenarios possible
much more expensive than certainty equivalent MPC
11
Tube-based Robust MPC [Langson’04, Rakovic’05,. . .]
Idea:
optimize set-valued tube that encloses all possible scenarios
no exponential scenario tree, but set enclosures needed
12
Tube-based Robust MPC [Langson’04, Rakovic’05,. . .]
Idea:
optimize set-valued tube that encloses all possible scenarios
no exponential scenario tree, but set enclosures needed
13
Notation 1
Closed-loop system dynamics:
ẋ(t) = f (x(t), µ(t, x(t)), w(t))
14
Notation 2
Constraints:
µ(t, x(t)) ∈ U , x(t) ∈ X , w(t) ∈ W (all compact sets)
15
Notation 3
Set-valued tubes:
Xµ (t) ⊆ Rnx denotes robust forward invariant tube: all solutions of
ẋ(t) = f (x(t), µ(t, x(t)), w(t))
with x(t 0 ) ∈ Xµ (t)
satisfy x(t) ∈ Xµ (t) for all t ≥ t 0 and all w with w(t) ∈ W.
16
Mathematical Formulation of Robust MPC
Optimize over future feedback policy µ:
Z
inf
µ:R×X→U
s.t.
t+T
`(Xµ (τ )) dτ
t


X (t) = {x̂t } ,

 µ
Xµ (τ ) ⊆ X



 optional terminal constraints
Function ` denotes scalar performance criterion
x̂t denotes current measurement
T denotes finite prediction horizon
17
Overview
Introduction to Robust MPC
Min-Max Differential Inequalities
18
Differential Inequalities
Scalar case:
uncertain scalar ODE without controls:
ẋ(t) = f (x(t), w(t))
with x(0) = x0
19
Differential Inequalities
Scalar case:
Interval X (t) = x L (t), x U (t) is robust forward invariant if
ẋ L (t) ≤
minw∈W f (x L (t), w)
ẋ U (t) ≥
maxw∈W f (x U (t), w)
(Differential Inequalities)
20
Min-Max Differential Inequalities
Scalar case with controls:
Interval X (t) = x L (t), x U (t) is robust forward invariant if
ẋ L (t) ≤ maxu∈U minw∈W f (x L (t), u, w)
ẋ U (t) ≥ minu∈U maxw∈W f (x U (t), u, w)
x L (t) ≤ x U (t)
21
Generalized Differential Inequalities
General case:
The state vector x(t) may have more than one component,
ẋ(t) = f (x(t), u(t), w(t))
with x(0) = x0
22
Generalized Differential Inequalities
Definition:
The support function of a compact set X is denoted by
V [X ](c) = max c T x
x∈X
23
Generalized Differential Inequalities
Theorem [Villanueva et al., 2016]:
If f Lipschitz, X (t) ⊆ X convex and compact, and



x ∈ X (t)


T
V̇ [X (t)](c) ≥ min max c f (x, u, w) c T x = V [X (t)](c)
u∈U x,w 



w∈W









for a.e. (t, c), then X (t) is a robust forward invariant tube.
24
Application to Robust MPC
Conservative reformulation:
Z t+T
inf
`(Y (τ )) dτ
Y
t


X (t) = {x̂t } ,






X (τ ) ⊆ X










s.t.

V̇
[X
(t)](c)
≥
min
max
c T f (x, u, w)


x,w 
u∈U












 optional terminal constraints
x ∈ X (t)
T
c x = V [X (t)](c)
w∈W









Parameterize set X (t); not the feedback law µ!
25
Affine Set Parameterizations
Affine Parameterization:
X (t) = {A(t)ξ + b(t) | ξ ∈ Em }
Basis set: Em , domain constraint: [A(t), b(t)] ∈ Dm,n
26
Numerical Example
Spring-mass-damper system:



ẋ1 (t)
x2 (t) + w1 (t)

 = 
1 )x1 (t)
ẋ2 (t)
− k0 exp (−x
− hd xM2 (t) +
M

u(t)
M
+
w2 (t)
M

27
Numerical Example
Spring-mass-damper system:



ẋ1 (t)
x2 (t) + w1 (t)

 = 
1 )x1 (t)
ẋ2 (t)
− k0 exp (−x
− hd xM2 (t) +
M

u(t)
M
+
w2 (t)
M

28
Numerical Example
Spring-mass-damper system:



ẋ1 (t)
x2 (t) + w1 (t)

 = 
1 )x1 (t)
ẋ2 (t)
− k0 exp (−x
− hd xM2 (t) +
M

u(t)
M
+
w2 (t)
M

29
Conclusions
Introduction to Robust MPC
Needs accurate model of the system and uncertainties
Useful whenever certainty equivalent MPC is too optimistic
Apply if primary objective is safety (rather than CPU-time),
example: human-robot cooperation
Min-Max Differential Inequalities
Min-Max DI leads to conservative reformulation, but
parameterizes sets rather than feedback laws
Boundary feedback law is minimizer of the RHS of min-max DI
30
Conclusions
Introduction to Robust MPC
Needs accurate model of the system and uncertainties
Useful whenever certainty equivalent MPC is too optimistic
Apply if primary objective is safety (rather than CPU-time),
example: human-robot cooperation
Min-Max Differential Inequalities
Min-Max DI leads to conservative reformulation, but
parameterizes sets rather than feedback laws
Boundary feedback law is minimizer of the RHS of min-max DI
31
References
M.E. Villanueva, B. Houska, B. Chachuat.
Unified Framework for the Propagation of Continuous-Time Enclosures for
Parametric Nonlinear ODEs.
JOGO, 2015.
B. Houska, M.E. Villanueva, B. Chachuat.
Stable Set-Valued Integration of Nonlinear Dynamic Systems using Affine Set
Parameterizations.
SINUM, 2015.
M.E. Villanueva, R. Quirynen, M. Diehl, B. Chachuat, B. Houska.
Robust MPC via Min-Max Differential Inequalities.
AUTOMATICA, (provisionally accepted).
32