on perspective functions, vanishing constraints, and
TRANSCRIPT
On Perspective Functions, VanishingConstraints, and Complementarity Programming
Fast Mixed-Integer Nonlinear Feedback Control
Christian Kirches1, Sebastian Sager2
1Interdisciplinary Center for Scientific Computing (IWR)Heidelberg University
2Institute for Mathematical OptimizationUniversity of Magdeburg
17th International Workshop on Combinatorial Optimization
Aussois, France
January 9, 2013
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Cyclic adsorption chillers
[Gräber, K., Bock, Schlöder, Tegethoff, Köhler, 2011]
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Cyclic adsorption chillers
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Cooling plants
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Automotive control
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Automotive control
courtesy Lewis Hamilton via twitter
[Kehrle 2010]
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Predictive cruise control for heavy duty trucks
Aim: Time/Energy optimal driving with automatic gear choice
500
1000
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2000
0
1000
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30000
0.01
0.02
0.03
0.04
Realization: Online computation of mixed-integer feedback controls ona moving horizon
8 available gears, 20 possible shifts = more than 1018 continuousproblems! [K., 2010]
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Mixed-integer feedback controls on the Autobahn
slope profile
velocity
effective torque
engine speed
gear choice
[K., 2010]
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
A mixed integer feedback control loop
(Simulated) process
Feedback Observer
Evaluate process model
Solve model-predictive control problem
observables
stateestimate
new continuous, integerfeedback control
most recent continuous,integer feedback control
state
state andcontrol trajectories
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Mixed integer optimal control problems (MIOCPs)Dynamic & switched process control problem on the prediction horizon [0, T]:
minx(·), z(t), u(·), v(·)
∫ T
0
l(x(t), z(t), u(t), v(t), p) dt+m(x(T), z(t), p)
s.t. x(t) = f(x(t), z(t), u(t), v(t), p) t ∈ [0, T]
0= g(x(t), z(t), u(t), v(t), p) t ∈ [0, T]
0= x(0)− x0
0≤ c(x(t), z(t), u(t), v(t), p) t ∈ [0, T]
0≤ d(x(t), z(t), u(t), p) t ∈ [0, T]
0µ r(x(ti), z(t)0≤i≤N, p) ti0≤i≤N ⊂ [0, T]
v(t) ∈ Ω t ∈ [0, T]
Objective: typically economic/tracking part l and terminal weight part m
Constraints: Initial value, path constraints c, d, point constraints r on a time grid
Dynamic process (x(·), z(·)) modeled by an ODE/DAE system f
Continuous controls u(·) from set U ⊂ Rnu ,
Controls v(·) from discrete set Ω := v1, . . . , vnΩ ⊂ Rnv holding finitely many choices vj formode-specific parameters
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Nonlinear model-predictive control (NMPC) scheme
v(t)v
v
v
v
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Classic NMPC benchmark problem: CSTR
[Klatt & Engell, 1993]
Worst-case runtimes for one iteration of the NMPC loop:
1997 [Chen] 60 seconds Pentium 166 MHz2001 [Diehl] 500 milliseconds Celeron 800 MHz2011 [Houska, Ferreau, Diehl] 400 microseconds Intel i7 3.6 GHz
100.000x times faster than 15 years ago!
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Computational approaches in MIOC
Known fixed sequence of mode switches
Solve a single multi-stage continuous OCP =⇒ easy
Relax first, then discretize and solve a single OCP
Direct relaxation of the integer controls
then solve a single continuous OCP
Build on NMPC technology available for continuous OCPs
Model functions must be evaluated in fractional points
Integer feasibility? Bounds on the loss of optimality?
Optimal control problem based branch & bound
First treat combinatorics in a branch & bound framework
then solve continuous OCPs in the tree nodes
Affordable for small trees only, per-node cost is prohibitive
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Example: branch & bound for MIOCP
Solve MIOCP to find time optimal gear shift sequence:
N t∗f [sec] CPU time20 6.779751 000:23:5240 6.786781 232:25:3180 ? ?
[Gerdts, 2005]
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Computational approaches in MIOC
Discretize first, then treat combinatoricsFirst obtain a discretized problem, e.g. using a direct andsimultaneous method (collocation, multiple shooting)then solve a structured possibly nonconvex MINLPSophisticated methods: outer approximation, cut generation, divingBonami, Wächter, . . . (Bonmin), Leyffer, Linderoth, . . . (FilMint, MINOTAUR),
Belotti, Biegler, Floudas, Fügenschuh, Grossmann, Helmberg, Koch, Lee, Liberti, Lodi, Luedtke,
Marquardt, Martin, Michaels, Nannicini, Oldenburg, Rendl, Sahinidis, Wächter, Weismantel, . . .
But: Extremely expensive for optimal control problemsLong horizons, fine discretization in time, little opportunity for earlypruning
Exploit control theory knowledge properlyyI ∈ 0, 1nI comes from a time discretization, nI likely is very largeBang-bang arcs of an optimal solution of a relaxation are integerfeasibleInteger variables only enter inside an integral
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Partial outer convexification for MIOCP
Introduction of convex multipliers ωj(·) ∈ 0,1 for choices v(·) = vj ∈ Ω,j= 1, . . . , nΩ:
bijection: v(t) = vj ∈ Ω ⇐⇒ ωj(t) = 1,nΩ∑
k=1
ωk(t) = 1
Modeling of MIOCP as a partially convexified optimal control problem:
minx(·), u(·), ω(·)
∫ T
0
nΩ∑
j=1
ωj(t) · l(x(t), u(t), vj, p) dt+m(x(T), p)
s.t. x(t) =∑nΩ
j=1ωj(t) · f(x(t), u(t), vj, p) t ∈ [0, T]
0= x(0)− x0(τ)
0≤ωj(t) · c(x(t), u(t), vj, p), j= 1, . . . , nΩ, t ∈ [0, T]
0≤ d(x(t), u(t), p), t ∈ [0, T]
ω(t) ∈ 0,1nΩ , 1=∑nΩ
j=1ωj(t) t ∈ [0, T]
[Sager, 2005, K., 2010]
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Partial outer convexification for MIOCP
Introduction of convex multipliers ωj(·) ∈ 0,1 for choices v(·) = vj ∈ Ω,j= 1, . . . , nΩ:
bijection: v(t) = vj ∈ Ω ⇐⇒ ωj(t) = 1,nΩ∑
k=1
ωk(t) = 1
Relaxation then yields a continuous, larger optimal control problem:
minx(·), u(·), α(·)
∫ T
0
nΩ∑
j=1
αj(t) · l(x(t), u(t), vj, p) dt+m(x(T), p)
s.t. x(t) =∑nΩ
j=1αj(t) · f(x(t), u(t), vj, p) t ∈ [0, T]
0= x(0)− x0(τ)
0≤ αj(t) · c(x(t), u(t), vj, p), j= 1, . . . , nΩ, t ∈ [0, T]
0≤ d(x(t), u(t), p) t ∈ [0, T]
α(t) ∈ [0, 1]nΩ , 1=∑nΩ
j=1αj(t) t ∈ [0, T]
[Sager, 2005, K., 2010]
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Approximation theoremsTheorem (MIOCP, function space)Let (x∗(·), u∗(·),α∗(·)) be the optimal solution of the convexified relaxed MIOCP withobjective ΦCR.∀ ε > 0 ∃ ωε binary feasible and xε(·) such that (xε(·), u∗(·),ωε(·)) is a feasible solution ofthe (convexified) MIOCP with objective ΦCB, and
(ΦCR ≤ ) ΦCB ≤ ΦCR + ε.
[Sager, Reinelt, Bock, 2009]
Theorem (NLP, discretized control)Consider for t ∈ [0, T] the two affine-linear systems
x(t) = A(t, x(t)) α∗(t), x(0) = x0, y(t) = A(t, y(t))ω(t), y(0) = y0,
for α∗, ω measurable, A ∈ C1 essentially bounded by M, Lipschitz in x with constant L, and
with total t-derivative bounded by C. Assume ω satisfies
∫ T
0ω(t)−α∗(t) dt
≤ ε.(bang-bang arcs, or sum-up rounding)Then for all t ∈ [0, T]:
||x(t)− y(t)|| ≤
||x0 − y0||+ (M+ C(t− t0))ε
eL(t−t0).
[Sager, Bock, Diehl, 2011]
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Example: b & b vs. outer convexification for MIOCP
Solve MIOCP to find time optimal gear shift sequence:
N t∗f [sec] CPU time
20 6.779751 000:23:5240 6.786781 232:25:3180 ? ?
N t∗f [sec] CPU time10 6.798389 00:00:0720 6.779035 00:00:2440 6.786730 00:00:4680 6.789513 00:04:19
[K., Bock, Schlöder, Sager, 2010]
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Example: b & b vs. outer convexification for MIOCP
Solve MIOCP to find time optimal gear shift sequence:
[K., Bock, Schlöder, Sager, 2010]
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
The mixed-integer NMPC loop
(Simulated) process
Feedback Observer
Evaluate dynamic process model(ODE/DAE) and compute sensitivities
Sum-Up Rounding
One iteration= solve a QPVC
yk
xk0
∆uk(0), ∆vk(0)
xk−1(0), uk−1(0), vk−1(0)
xk(0)
(xkα(·), uk(·),αk(·))
(xkω(·), uk(·),ωk(·))
[Diehl, 2001, K., 2010]
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Complementarity/Vanishing Constraint FormulationConstraints 0≤ c(x(t), u(t), v(t), p) depend on v(·)
Approximation theorem does not address feasibility of c(·) afterrounding
Tightest formulation: Complementarity and vanishing constraints(MPCCs, MPVCs)
0≤ αj(t) · c(x(t), u(t), vj, p), j= 1, . . . , nΩ, t ∈ [0, T]
Violates constraint qualifications LICQ, MFCQ, ACQ in αj(t) = 0,c(·) = 0, but GCQ and hence KKT-based optimality holds
Numerical methods: Solve a sequence of NLPs obtained byregularization, smoothing, or a combination thereof
MPCC: Fletcher, Leyffer, Munson, Ralph, Stein, ...
MPVC: Achtziger, Hoheisel, Kanzow, ...
Best convergence properties for sequential linear-quadratic methodsfor MPCC/MPVC [Leyffer, Munson, 2004]
Open: Actual implementation?
Tailored active set quadratic MPVC solver [K., Potschka, Bock, Sager, 2012]
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Predictive cruise control for a heavy-duty truck
Partial outer convexification and relaxation for gear shift
Vanishing constraint formulation for gear-dependent engine speedlimits
Direct multiple shooting discretization in time
Sequential QPVC active set solver for the truck model MPVC
Exploitation of block structures in linear algebra
Sampling times of 10 to 100 ms on my desktop system
Save 3%-5% fuel when compared to experienced driver’sperformance (105 km/year, 30-40 l/100km)
Methodology is extensibleto future hybrid technologies
Patent [Bock, K., Sager, Schlöder]
jointly with Mercedes Trucks, Stuttgart
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
(I) One-Row Relaxation Formulation
Constraints 0≤ c(x(t), u(t), v(t), p) depend on v(·)One-row relaxation formulation
0≤nΩ∑
j=1
αj(t) · c(x(t), u(t), vj, p), t ∈ [0, T]
Is obtained as the convex combinstion of residuals for theconstraints on the choices vj
Satisfies LICQ, but often suffers from compensatory effects
Open: Can we efficiently add a few cuts (in MIOCP, in an MI-NMPCscheme) and effectively (reducing the integrality gap)?
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
(II) Generalized Disjunctive Programming
Constraints 0≤ c(x(t), u(t), v(t), p) depend on v(·)Generalized disjunctive programming Balas, Grossmann, ...
minx(·),u(·),Y(·)
e(x(T))
s.t. ∨i∈1,...,nω
Yi(t)x(t) = f(x(t), u(t), vi)0≤ c(x(t), u(t), vi)
, ∀t ∈ [0, T]
x(0) = x0
0≤ d(x(t), u(t)), ∀t ∈ [0, T]Y(t) ∈ false, true, ∀t ∈ [0, T]
Obtain convex hull description using big-M or perspective(MILP procedure: Ceria, Soares, 1999, Stubbs, Mehrotra, 1999)
Requires time discretization of the disjunction literal Y(·)Involves lifting the ODE system and the initial value constraints
[Jung, K., Sager, 2012]
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
(III) Liftings of the Differential Equations
minx(·),u(·),α(·)
e(x(T))
s.t. xi(t) = αkif(xi(t)/αki, ui(t)/αki, vi) t ∈ [tk, tk+1]
xi(tk) = αkisk
sk+1 =nω∑
i=1
xi(tk+1; tk, sk, ui(·)/αki, vi)
0≤ αkic(xi(t)/αki, ui(t)/αki, vi) t ∈ [tk, tk+1]
0≤ d
nω∑
i=1
xi(t),nω∑
i=1
ui(t)
!
t ∈ [tk, tk+1]
nω∑
i=1
αki = 1, 0≤ xi(t)≤ αkiMs, 0≤ ui(t)≤ αkiM
u t ∈ [tk, tk+1]
Several numerical difficulties:
ODE system has significantly grown in size
Positivity of states and controls
Perspective curvature ill-defined near zero
Vanishing constraint structure still present
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
An Example for the Constraint Formulations
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[Jung, K., Sager, 2012]C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Key points and future work
Mixed integer optimal control problems
Partial outer convexification for MIOCP
Solve a large, continuous OCP – typically no exponential runtime
Sum-up-rounding or MILP to reconstruct the integer control
has optimality certificate in function space and after discretization
has feasibility certificate for nonconvex MPCC/MPVC formulation
Mixed integer nonlinear model predictive control
Advanced SQP and QP techniques for NMPC available
Partial outer convexification allows transfer to mixed–integer NMPC
Future developments for constraints on integer controls
An SLP-EQP solver for the MPCC/MPVC formulation?
Use tight convex relaxations from GDP, instead of MPCC/MPVC?
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Acknowledgements
Hans Georg BockAlexander Buchner
Michael JungFlorian KehrleSven Leyffer
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control
Thank you very much!
Questions?
C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control