On Perspective Functions, Vanishing Constraints, and Complementarity Programming

Transcription

1 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 2 Institute for Mathematical Optimization University of Magdeburg 17 th International Workshop on Combinatorial Optimization Aussois, France January 9, 2013

2 Cyclic adsorption chillers [Gräber, K., Bock, Schlöder, Tegethoff, Köhler, 2011] C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control

3 Cyclic adsorption chillers C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control

4 Cooling plants

5 Automotive control C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control

6 Automotive control courtesy Lewis Hamilton via twitter [Kehrle 20]

7 Predictive cruise control for heavy duty trucks Aim: Time/Energy optimal driving with automatic gear choice x y Realization: Online computation of mixed-integer feedback controls on a moving horizon 8 available gears, 20 possible shifts ˆ= more than 18 continuous problems! [K., 20]

8 Mixed-integer feedback controls on the Autobahn slope profile velocity effective torque engine speed gear choice [K., 20]

9 A mixed integer feedback control loop Feedback new continuous, integer feedback control (Simulated) process most recent continuous, integer feedback control Observer observables state Evaluate process model Solve model-predictive control problem state and control trajectories state estimate

10 Mixed integer optimal control problems (MIOCPs) Dynamic & switched process control problem on the prediction horizon [0, T]: min x( ), 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. ẋ(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) ˆx 0 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(t i ), z(t)} 0 i N, p) {t i } 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 n u, Controls v( ) from discrete set Ω := {v 1,..., v n Ω } n v holding finitely many choices v j for mode-specific parameters

11 Nonlinear model-predictive control (NMPC) scheme v v(t) v v v

12 Classic NMPC benchmark problem: CSTR Worst-case runtimes for one iteration of the NMPC loop: [Klatt & Engell, 1993] 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 0.000x times faster than 15 years ago!

13 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

14 Example: branch & bound for MIOCP Solve MIOCP to find time optimal gear shift sequence: N t [sec] f CPU time :23: :25:31 80?? [Gerdts, 2005]

15 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 y I {0, 1} n I comes from a time discretization, n I likely is very large Bang-bang arcs of an optimal solution of a relaxation are integer feasible Integer variables only enter inside an integral

16 Partial outer convexification for MIOCP Introduction of convex multipliers ω j ( ) {0, 1} for choices v( ) = v j Ω, j = 1,..., n Ω : bijection: v(t) = v j Ω ω j (t) = 1, n Ω ω k (t) = 1 k=1 Modeling of MIOCP as a partially convexified optimal control problem: min x( ), u( ), ω( ) T 0 s.t. ẋ(t) = n Ω ω j (t) l(x(t), u(t), v j, p) dt + m(x(t), p) j=1 nω j=1 ω j(t) f(x(t), u(t), v j, p) t [0, T] 0 = x(0) ˆx 0 (τ) 0 ω j (t) c(x(t), u(t), v j, p), j = 1,..., n Ω, t [0, T] 0 d(x(t), u(t), p), t [0, T] ω(t) {0, 1} n Ω, 1 = nω j=1 ω j(t) t [0, T] [Sager, 2005, K., 20]

17 Partial outer convexification for MIOCP Introduction of convex multipliers ω j ( ) {0, 1} for choices v( ) = v j Ω, j = 1,..., n Ω : bijection: v(t) = v j Ω ω j (t) = 1, n Ω ω k (t) = 1 k=1 Relaxation then yields a continuous, larger optimal control problem: min x( ), u( ), α( ) T 0 s.t. ẋ(t) = n Ω α j (t) l(x(t), u(t), v j, p) dt + m(x(t), p) j=1 nω j=1 α j(t) f(x(t), u(t), v j, p) t [0, T] 0 = x(0) ˆx 0 (τ) 0 α j (t) c(x(t), u(t), v j, p), j = 1,..., n Ω, t [0, T] 0 d(x(t), u(t), p) t [0, T] α(t) [0, 1] n Ω, 1 = nω j=1 α j(t) t [0, T] [Sager, 2005, K., 20]

18 Approximation theorems Theorem (MIOCP, function space) Let (x ( ), u ( ), α ( )) be the optimal solution of the convexified relaxed MIOCP with objective Φ CR. ɛ > 0 ω ɛ binary feasible and x ɛ ( ) such that (x ɛ ( ), u ( ), ω ɛ ( )) is a feasible solution of the (convexified) MIOCP with objective Φ CB, and (Φ CR ) Φ CB Φ CR + ɛ. Theorem (NLP, discretized control) Consider for t [0, T] the two affine-linear systems [Sager, Reinelt, Bock, 2009] ẋ(t) = A(t, x(t)) α (t), x(0) = x 0, ẏ(t) = A(t, y(t)) ω(t), y(0) = y 0, for α, ω measurable, A C 1 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 ɛ. (bang-bang arcs, or sum-up rounding) Then for all t [0, T]: x(t) y(t) x 0 y 0 + (M + C(t t 0 ))ɛ e L(t t 0). [Sager, Bock, Diehl, 2011]

19 Example: b & b vs. outer convexification for MIOCP Solve MIOCP to find time optimal gear shift sequence: N t f [sec] CPU time :23: :25:31 80?? N t [sec] f CPU time :00: :00: :00: :04:19 [K., Bock, Schlöder, Sager, 20]

20 Example: b & b vs. outer convexification for MIOCP Solve MIOCP to find time optimal gear shift sequence: [K., Bock, Schlöder, Sager, 20]

21 The mixed-integer NMPC loop u k (0), v k (0) (Simulated) process y k Feedback x k 1 (0), u k 1 (0), v k 1 (0) Observer x k (0) Evaluate dynamic process model (ODE/DAE) and compute sensitivities Sum-Up Rounding (x k ω ( ), uk ( ), ω k ( )) (x k α ( ), uk ( ), α k ( )) One iteration ˆ= solve a QPVC ˆx k 0 [Diehl, 2001, K., 20]

22 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 α j (t) c(x(t), u(t), v j, p), j = 1,..., n Ω, t [0, T] Violates constraint qualifications LICQ, MFCQ, ACQ in α j (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]

23 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 to 0 ms on my desktop system Save 3%-5% fuel when compared to experienced driver s performance ( 5 km/year, l/0km) Methodology is extensible to future hybrid technologies Patent [Bock, K., Sager, Schlöder] jointly with Mercedes Trucks, Stuttgart

24 (I) One-Row Relaxation Formulation Constraints 0 c(x(t), u(t), v(t), p) depend on v( ) One-row relaxation formulation 0 n Ω j=1 α j (t) c(x(t), u(t), v j, p), t [0, T] Is obtained as the convex combinstion of residuals for the constraints on the choices v j 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)?

25 (II) Generalized Disjunctive Programming Constraints 0 c(x(t), u(t), v(t), p) depend on v( ) Generalized disjunctive programming Balas, Grossmann,... min e(x(t)) x( ),u( ),Y( ) s.t. i {1,...,n ω } Y i (t) ẋ(t) = f(x(t), u(t), v i ) 0 c(x(t), u(t), v i ) x(0) = x 0 0 d(x(t), u(t)), t [0, T] Y(t) {false, true}, t [0, T], 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]

26 (III) Liftings of the Differential Equations min e(x(t)) x( ),u( ),α( ) s.t. ẋ i (t) = α ki f(x i (t)/α ki, u i (t)/α ki, v i ) t [t k, t k+1 ] x i (t k ) = α ki s k n ω s k+1 = x i (t k+1 ; t k, s k, u i ( )/α ki, v i ) i=1 0 α ki c(x i (t)/α ki, u i (t)/α ki, v i ) t [t k, t k+1 ] nω n ω 0 d x i (t), u i (t) t [t k, t k+1 ] i=1 i=1 n ω α ki = 1, 0 x i (t) α ki M s, 0 u i (t) α ki M u t [t k, t k+1 ] i=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

27 An Example for the Constraint Formulations Track elevation [m] Velocity v [m/s] Interval root relaxation Velocity v [m/s] outer convex., one-row relaxation Velocity v [m/s] Dynamic Programming Velocity v [m/s] Multipliers α Velocity v [m/s] Multipliers α Velocity v [m/s] Multipliers α Velocity v [m/s] vanishing constraints (IPOPT stuck) relaxed VC smoothed VC GDP, Big-M Velocity v [m/s] GDP, VC Multipliers α Multipliers α Multipliers α Multipliers α Multipliers α

28 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?

29 Acknowledgements Hans Georg Bock Alexander Buchner Michael Jung Florian Kehrle Sven Leyffer

30 Thank you very much! Questions?