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On Perspective Functions, Vanishing Constraints, and Complementarity - - PowerPoint PPT Presentation

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On Perspective Functions, Vanishing Constraints, and Complementarity Programming Fast Mixed-Integer Nonlinear Feedback Control Christian Kirches 1 , Sebastian Sager 2 1 Interdisciplinary Center for Scientific Computing (IWR) Heidelberg University

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On Perspective Functions, Vanishing Constraints, and Complementarity Programming

Fast Mixed-Integer Nonlinear Feedback Control

Christian Kirches1, Sebastian Sager2

1Interdisciplinary Center for Scientific Computing (IWR)

Heidelberg University

2Institute for Mathematical Optimization

University 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

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Cyclic adsorption chillers

[Gräber, K., Bock, Schlöder, Tegethoff, Köhler, 2011]

  • C. Kirches (Heidelberg), S. Sager (Magdeburg)

Fast Mixed-Integer Nonlinear Feedback Control

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Cyclic adsorption chillers

  • C. Kirches (Heidelberg), S. Sager (Magdeburg)

Fast Mixed-Integer Nonlinear Feedback Control

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Cooling plants

  • C. Kirches (Heidelberg), S. Sager (Magdeburg)

Fast Mixed-Integer Nonlinear Feedback Control

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Automotive control

  • C. Kirches (Heidelberg), S. Sager (Magdeburg)

Fast Mixed-Integer Nonlinear Feedback Control

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Automotive control

courtesy Lewis Hamilton via twitter

[Kehrle 2010]

  • C. Kirches (Heidelberg), S. Sager (Magdeburg)

Fast Mixed-Integer Nonlinear Feedback Control

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Predictive cruise control for heavy duty trucks

Aim: Time/Energy optimal driving with automatic gear choice

x y z

  • 0.1
  • 0.05

0.05 0.1

  • 1
  • 1

1 ×104

  • 50

50 100 150 200

500 1000 1500 2000 1000 2000 3000 0.01 0.02 0.03 0.04

Realization: Online computation of mixed-integer feedback controls on a moving horizon 8 available gears, 20 possible shifts ˆ = more than 1018 continuous problems!

[K., 2010]

  • C. Kirches (Heidelberg), S. Sager (Magdeburg)

Fast Mixed-Integer Nonlinear Feedback Control

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SLIDE 8

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

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A mixed integer feedback control loop

(Simulated) process Feedback Observer Evaluate process model Solve model-predictive control problem

  • bservables

state estimate new continuous, integer feedback control most recent continuous, integer feedback control state state and control trajectories

  • C. Kirches (Heidelberg), S. Sager (Magdeburg)

Fast Mixed-Integer Nonlinear Feedback Control

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Mixed integer optimal control problems (MIOCPs)

Dynamic & switched process control problem on the prediction horizon [0,T]: min

x(·), z(t), u(·), v(·)

T 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) − ˆ 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] 0 ≦ r({x(ti),z(t)}0≤i≤N,p) {ti}0≤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 ⊂ nu, Controls v(·) from discrete set Ω := {v1,...,vnΩ} ⊂ nv holding finitely many choices vj for mode-specific parameters

  • C. Kirches (Heidelberg), S. Sager (Magdeburg)

Fast Mixed-Integer Nonlinear Feedback Control

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Nonlinear model-predictive control (NMPC) scheme

v(t) v v v v

  • C. Kirches (Heidelberg), S. Sager (Magdeburg)

Fast Mixed-Integer Nonlinear Feedback Control

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Classic NMPC benchmark problem: CSTR

[Klatt & Engell, 1993]

Worst-case runtimes for one iteration of the NMPC loop: 1997 [Chen] 60 seconds Pentium 166 MHz 2001 [Diehl] 500 milliseconds Celeron 800 MHz 2011 [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

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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

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Example: branch & bound for MIOCP

Solve MIOCP to find time optimal gear shift sequence: N t∗

f [sec]

CPU time 20 6.779751 000:23:52 40 6.786781 232:25:31 80 ? ?

[Gerdts, 2005]

  • C. Kirches (Heidelberg), S. Sager (Magdeburg)

Fast Mixed-Integer Nonlinear Feedback Control

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Computational approaches in MIOC

Discretize first, then treat combinatorics First obtain a discretized problem, e.g. using a direct and simultaneous method (collocation, multiple shooting) then solve a structured possibly nonconvex MINLP Sophisticated methods: outer approximation, cut generation, diving

Bonami, 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 problems

Long horizons, fine discretization in time, little opportunity for early pruning

Exploit control theory knowledge properly yI ∈ {0,1}nI comes from a time discretization, nI likely is very large Bang-bang arcs of an optimal solution of a relaxation are integer feasible Integer variables only enter inside an integral

  • C. Kirches (Heidelberg), S. Sager (Magdeburg)

Fast Mixed-Integer Nonlinear Feedback Control

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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: min

x(·), u(·), ω(·)

T

nΩ

  • j=1

ωj(t) · l(x(t),u(t),vj,p) dt + m(x(T),p) s.t. ˙ x(t) = nΩ

j=1 ωj(t) · f(x(t),u(t),vj,p)

t ∈ [0,T] 0 = x(0) − ˆ 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) ∈ {0,1}nΩ, 1 = nΩ

j=1 ωj(t)

t ∈ [0,T]

[Sager, 2005, K., 2010]

  • C. Kirches (Heidelberg), S. Sager (Magdeburg)

Fast Mixed-Integer Nonlinear Feedback Control

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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: min

x(·), u(·), α(·)

T

nΩ

  • j=1

αj(t) · l(x(t),u(t),vj,p) dt + m(x(T),p) s.t. ˙ x(t) = nΩ

j=1 αj(t) · f(x(t),u(t),vj,p)

t ∈ [0,T] 0 = x(0) − ˆ 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=1 αj(t)

t ∈ [0,T]

[Sager, 2005, K., 2010]

  • C. Kirches (Heidelberg), S. Sager (Magdeburg)

Fast Mixed-Integer Nonlinear Feedback Control

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Approximation theorems

Theorem (MIOCP , function space) Let (x∗(·),u∗(·),α∗(·)) be the optimal solution of the convexified relaxed MIOCP with

  • bjective ΦCR.

∀ ǫ > 0 ∃ ωǫ binary feasible and xǫ(·) such that (xǫ(·),u∗(·),ωǫ(·)) is a feasible solution of the (convexified) MIOCP with objective ΦCB, and (ΦCR ≤ ) ΦCB ≤ ΦCR + ǫ.

[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

0 ω(t) − α∗(t) dt

  • ≤ ǫ.

(bang-bang arcs, or sum-up rounding) Then for all t ∈ [0,T]: ||x(t) − y(t)|| ≤ ||x0 − y0|| + (M + C(t − t0))ǫeL(t−t0).

[Sager, Bock, Diehl, 2011]

  • C. Kirches (Heidelberg), S. Sager (Magdeburg)

Fast Mixed-Integer Nonlinear Feedback Control

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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 000:23:52 40 6.786781 232:25:31 80 ? ? N t∗

f [sec]

CPU time 10 6.798389 00:00:07 20 6.779035 00:00:24 40 6.786730 00:00:46 80 6.789513 00:04:19

[K., Bock, Schlöder, Sager, 2010]

  • C. Kirches (Heidelberg), S. Sager (Magdeburg)

Fast Mixed-Integer Nonlinear Feedback Control

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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

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The mixed-integer NMPC loop

(Simulated) process Feedback Observer Evaluate dynamic process model (ODE/DAE) and compute sensitivities Sum-Up Rounding One iteration ˆ = solve a QPVC yk ˆ xk ∆uk(0), ∆vk(0) xk−1(0), uk−1(0), vk−1(0) xk(0) (xk

α(·),uk(·),αk(·))

(xk

ω(·),uk(·),ωk(·))

[Diehl, 2001, K., 2010]

  • C. Kirches (Heidelberg), S. Sager (Magdeburg)

Fast Mixed-Integer Nonlinear Feedback Control

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Complementarity/Vanishing Constraint Formulation

Constraints 0 ≤ c(x(t),u(t),v(t),p) depend on v(·) Approximation theorem does not address feasibility of c(·) after rounding 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 by regularization, smoothing, or a combination thereof

MPCC: Fletcher, Leyffer, Munson, Ralph, Stein, ... MPVC: Achtziger, Hoheisel, Kanzow, ...

Best convergence properties for sequential linear-quadratic methods for 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

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Predictive cruise control for a heavy-duty truck

Partial outer convexification and relaxation for gear shift Vanishing constraint formulation for gear-dependent engine speed limits 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’s performance (105 km/year, 30-40 l/100km) Methodology is extensible to 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

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(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 the constraints 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-NMPC scheme) and effectively (reducing the integrality gap)?

  • C. Kirches (Heidelberg), S. Sager (Magdeburg)

Fast Mixed-Integer Nonlinear Feedback Control

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(II) Generalized Disjunctive Programming

Constraints 0 ≤ c(x(t),u(t),v(t),p) depend on v(·) Generalized disjunctive programming

Balas, Grossmann, ...

min

x(·),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

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(III) Liftings of the Differential Equations

min

x(·),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) ≤ αkiMu 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

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An Example for the Constraint Formulations

  • 10

10 20 30 40 50 60 70 5 10 15 20 25 30 35 40 Track elevation [m] Interval

Multipliers α

2 4 6 8 10 12 14 16

Multipliers α

2 4 6 8 10 12 14 16 5 10 15 20 25 30 5 10 15 20 25 30 35 40

Velocity v [m/s]

root relaxation

Multipliers α

2 4 6 8 10 12 14 16

Multipliers α

2 4 6 8 10 12 14 16 5 10 15 20 25 30 5 10 15 20 25 30 35 40

Velocity v [m/s]

  • uter convex., one-row relaxation

Multipliers α

2 4 6 8 10 12 14 16

Multipliers α

2 4 6 8 10 12 14 16 5 10 15 20 25 30 5 10 15 20 25 30 35 40

Velocity v [m/s]

Dynamic Programming

Multipliers α

2 4 6 8 10 12 14 16

Multipliers α

2 4 6 8 10 12 14 16 5 10 15 20 25 30 5 10 15 20 25 30 35 40

Velocity v [m/s]

vanishing constraints (IPOPT stuck)

Multipliers α

2 4 6 8 10 12 14 16

Multipliers α

2 4 6 8 10 12 14 16 5 10 15 20 25 30 5 10 15 20 25 30 35 40

Velocity v [m/s]

relaxed VC

Multipliers α

2 4 6 8 10 12 14 16

Multipliers α

2 4 6 8 10 12 14 16 5 10 15 20 25 30 5 10 15 20 25 30 35 40

Velocity v [m/s]

smoothed VC

Multipliers α

2 4 6 8 10 12 14 16

Multipliers α

2 4 6 8 10 12 14 16 5 10 15 20 25 30 5 10 15 20 25 30 35 40

Velocity v [m/s]

GDP , Big-M

Multipliers α

2 4 6 8 10 12 14 16

Multipliers α

2 4 6 8 10 12 14 16 5 10 15 20 25 30 5 10 15 20 25 30 35 40

Velocity v [m/s]

GDP , VC

Jung, K., Sager, 2012

  • C. Kirches (Heidelberg), S. Sager (Magdeburg)

Fast Mixed-Integer Nonlinear Feedback Control

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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

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Acknowledgements

Hans Georg Bock Alexander Buchner Michael Jung Florian Kehrle Sven Leyffer

  • C. Kirches (Heidelberg), S. Sager (Magdeburg)

Fast Mixed-Integer Nonlinear Feedback Control

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Thank you very much! Questions?

  • C. Kirches (Heidelberg), S. Sager (Magdeburg)

Fast Mixed-Integer Nonlinear Feedback Control