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

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 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 -0.1 -0.05 0 200 0.04 150 0.03 100 0.02 z 50 -1 0.01 0 2000 -50 0 3000 0 1500 2000 -1 1000 1000 0.1 1 500 0 0.05 y ×10 4 x Realization: Online computation of mixed-integer feedback controls on a moving horizon = more than 10 18 continuous 8 available gears, 20 possible shifts ˆ problems! [ 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 new continuous , integer feedback control (Simulated) process observables most recent continuous , integer feedback control Feedback Observer state Evaluate process model Solve model-predictive control problem state and state control trajectories estimate 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 ] : � T min l ( x ( t ) , z ( t ) , u ( t ) , v ( t ) , p ) d t + m ( x ( T ) , z ( t ) , p ) x ( · ) , z ( t ) , u ( · ) , v ( · ) 0 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 x 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 C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control

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

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: t ∗ N 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

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

Partial outer convexification for MIOCP Introduction of convex multipliers ω j ( · ) ∈ { 0,1 } for choices v ( · ) = v j ∈ Ω , j = 1,..., n Ω : n Ω v ( t ) = v j ∈ Ω � bijection : ⇐⇒ ω j ( t ) = 1, ω k ( t ) = 1 k = 1 Modeling of MIOCP as a partially convexified optimal control problem: � T n Ω � ω j ( t ) · l ( x ( t ) , u ( t ) , v j , p ) d t + m ( x ( T ) , p ) min x ( · ) , u ( · ) , ω ( · ) 0 j = 1 � n Ω j = 1 ω j ( t ) · f ( x ( t ) , u ( t ) , v j , p ) s.t. x ( t ) = ˙ t ∈ [ 0, T ] 0 = x ( 0 ) − ˆ x 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 ] � n Ω ω ( t ) ∈ { 0,1 } n Ω , 1 = j = 1 ω j ( 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 ω j ( · ) ∈ { 0,1 } for choices v ( · ) = v j ∈ Ω , j = 1,..., n Ω : n Ω v ( t ) = v j ∈ Ω � bijection : ⇐⇒ ω j ( t ) = 1, ω k ( t ) = 1 k = 1 Relaxation then yields a continuous, larger optimal control problem: � T n Ω � α j ( t ) · l ( x ( t ) , u ( t ) , v j , p ) d t + m ( x ( T ) , p ) min x ( · ) , u ( · ) , α ( · ) 0 j = 1 � n Ω j = 1 α j ( t ) · f ( x ( t ) , u ( t ) , v j , p ) s.t. x ( t ) = ˙ t ∈ [ 0, T ] 0 = x ( 0 ) − ˆ x 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 ] � n Ω α ( t ) ∈ [ 0,1 ] n Ω , 1 = j = 1 α j ( t ) t ∈ [ 0, T ] [ Sager, 2005, K., 2010 ] C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control

Approximation theorems Theorem (MIOCP , function space) Let ( x ∗ ( · ) , u ∗ ( · ) , α ∗ ( · )) be the optimal solution of the convexified relaxed MIOCP with objective Φ CR . ∀ ǫ > 0 ∃ ω ǫ binary feasible and x ǫ ( · ) such that ( x ǫ ( · ) , u ∗ ( · ) , ω ǫ ( · )) is a feasible solution of the (convexified) MIOCP with objective Φ CB , and (Φ CR ≤ ) Φ CB ≤ Φ CR + 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 ) = x 0 , ˙ ˙ y ( 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 � T � � � � 0 ω ( t ) − α ∗ ( t ) d t with total t-derivative bounded by C . Assume ω satisfies � ≤ ǫ . � � � � � � � (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 )) ǫ � e L ( t − t 0 ) . [ Sager, Bock, Diehl, 2011 ] C. Kirches (Heidelberg), S. Sager (Magdeburg) Fast Mixed-Integer Nonlinear Feedback Control