MPC Distributed MPC MPC for water networks Multi-level MPC Road networks Summary Control of large-scale systems with applications to water distribution and road traffic networks Bart De Schutter, Andreas Hegyi, Rudy Negenborn, Solomon Zegeye, Lakshmi Baskar, Anna Sadowska Delft Center for Systems and Control Delft University of Technology www.dcsc.tudelft.nl Lucca, July 5, 2013 Control of large-scale transportation systems 1 / 85
MPC Distributed MPC MPC for water networks Multi-level MPC Road networks Summary Outline 1 Recapitulation: Model predictive control 2 Distributed MPC 3 MPC for water networks 4 Multi-level MPC 5 MPC for road traffic networks 6 Summary Lucca, July 5, 2013 Control of large-scale transportation systems 2 / 85
MPC Distributed MPC MPC for water networks Multi-level MPC Road networks Summary Model predictive control (MPC) Features Very popular in process industry Model-based Easy to tune Multi-input multi-output (MIMO) Allows constraints on inputs and outputs Adaptive / receding horizon Uses off-line or on-line optimization Lucca, July 5, 2013 Control of large-scale transportation systems 3 / 85
MPC Distributed MPC MPC for water networks Multi-level MPC Road networks Summary MPC: Principle of operation Performance/objective control measurements function (e.g., reference system inputs tracking versus input energy) MPC controller Prediction model control optimization actions Constraints objective, model (On-line) optimization prediction constraints Receding horizon J MPC Nonlinear optimization problem: min k , N p ( u k ) u k subject to system dynamics, operational constraints where u k = [ u T ( k ) u T ( k + 1) · · · u T ( k + N p − 1)] T Lucca, July 5, 2013 Control of large-scale transportation systems 4 / 85
MPC Distributed MPC MPC for water networks Multi-level MPC Road networks Summary MPC: Receding horizon approach future past setpoint predicted outputs computed control inputs k + 1 k + N c k + N p k control horizon prediction horizon Lucca, July 5, 2013 Control of large-scale transportation systems 5 / 85
MPC Distributed MPC MPC for water networks Multi-level MPC Road networks Summary Challenges in control of large-scale networks Large-scale nature of the system Distributed vs centralized control Optimality ↔ computational efficiency/tractability Global ↔ local Scalability Communication requirements (bandwidth) Robustness against failures Lucca, July 5, 2013 Control of large-scale transportation systems 6 / 85
MPC Distributed MPC MPC for water networks Multi-level MPC Road networks Summary Challenges in MPC of large-scale networks Major problem for MPC in practice : In general: nonlinear, nonconvex optimization problem → huge computation time, in particular for large-scale systems Solutions : Choice of the prediction model: accuracy versus computational complexity Use parametrized control laws Use distributed and/or multi-level approach Right optimization approach parallel and/or distributed optimization approximate original MPC optimization problem by another optimization problem that can be solved efficiently Include application-specific knowledge Lucca, July 5, 2013 Control of large-scale transportation systems 7 / 85
MPC Distributed MPC MPC for water networks Multi-level MPC Road networks Summary Distributed MPC Subsystems instead of overall system Single agent/controller for each subsystem limited action capabilities limited information gathering Challenge : agents should choose local inputs that are globally optimal control agent control agent Ag2 Ag1 Ag3 Ag4 Ag5 optimizer optimizer control agent optimizer Lucca, July 5, 2013 Control of large-scale transportation systems 8 / 85
MPC Distributed MPC MPC for water networks Multi-level MPC Road networks Summary Distributed MPC Interconnection between control agents v i v j w in , ji w out , ij d i d j x i x j y i w out , ji y j w in , ij u i u j x i ( k + 1) = f i ( x i ( k ) , u i ( k ) , d i ( k ) , v i ( k )) Lucca, July 5, 2013 Control of large-scale transportation systems 9 / 85
MPC Distributed MPC MPC for water networks Multi-level MPC Road networks Summary Distributed MPC Interconnection between control agents v i v j w in , ji w out , ij d i d j x i x j y i w out , ji y j w in , ij u i u j x i ( k + 1) = f i ( x i ( k ) , u i ( k ) , d i ( k ) , w in , j 1 i ( k ) , . . . , w in , j mi i ( k )) w out , ji ( k + 1) = h ji out ( u i ( k ) , y i ( k ) , x i ( k + 1)) for each neighbor j of i Lucca, July 5, 2013 Control of large-scale transportation systems 9 / 85
MPC Distributed MPC MPC for water networks Multi-level MPC Road networks Summary Local MPC control problem of agent i at decision step k x i ( k +1) J local , i (˜ u i ( k ) , ˜ min x i ( k + 1)) u i ( k ) , ˜ ˜ subject to subsystem dynamics: prediction model x i ( k + 1) = f i ( x i ( k ) , u i ( k ) , d i ( k ) , . . . ) w in , j 1 i ( k ) , . . . , w in , j mi i ( k ) w out , ji ( k + 1)= h ji out ( u i k , y i k , x i k +1 ) for each neighbor j of i . . . x i ( k + N ) = f i ( x i ( k + N − 1) , u i ( k + N − 1) , d i ( k + N − 1) , . . . ) j mi i w j 1 i in , k + N − 1 , . . . , w in , k + N − 1 w out , ji ( k + N )= h ji out ( u i k + N − 1 , y i k + N − 1 , x i k + N ) initial local state, disturbances, and additional constraints Lucca, July 5, 2013 Control of large-scale transportation systems 10 / 85
MPC Distributed MPC MPC for water networks Multi-level MPC Road networks Summary Local MPC control problem of agent i at decision step k min x i ( k +1) J local , i (˜ u i ( k ) , ˜ x i ( k + 1)) ˜ u i ( k ) , ˜ subject to subsystem dynamics: prediction model x i ( k + 1) = f i ( x i ( k ) , u i ( k ) , d i ( k ) , w in , j 1 i ( k ) , . . . , w in , j mi i ( k )) w out , ji ( k + 1) = h out , ji ( u i ( k ) , y i ( k ) , x i ( k + 1)) for each neighbor j of i . . . x i ( k + N ) = f i ( x i ( k + N − 1) , u i ( k + N − 1) , d i ( k + N − 1) , w in , j 1 i ( k + N − 1) , . . . , w in , j mi i ( k + N − 1)) w out , ji ( k + N ) = h out , ji ( u i ( k + N − 1) , y i ( k + N − 1) , x i ( k + N )) initial local state, disturbances and additional constraints Lucca, July 5, 2013 Control of large-scale transportation systems 10 / 85
MPC Distributed MPC MPC for water networks Multi-level MPC Road networks Summary Interconnecting constraints Constraints on interconnecting variables subnetwork j 1 Imposed by dynamics of overall network What goes in into i equals subnetwork j 2 what goes out from j subnetwork i Satisfaction necessary for accurate predictions w in , ji ( k ) = w out , ij ( k ) For agent controlling subsystem i w out , ji ( k ) = w in , ij ( k ) w in , ij and w out , ij of neighbor . . . . . . j unknown w in , ji ( k + N − 1) = w out , ij ( k + N − 1) How to make accurate w out , ji ( k + N − 1) = w in , ij ( k + N − 1) predictions? → via negotiations Lucca, July 5, 2013 Control of large-scale transportation systems 11 / 85
MPC Distributed MPC MPC for water networks Multi-level MPC Road networks Summary Multiple-iterations scheme to agree on values of interconnecting variables Each agent computes optimal local and interconnecting variables communicates interconnecting variables to neighbors updates parameters ˜ λ ji in , ˜ λ ji out of additional cost term J i inter Iterations continue until stopping criterion satisfied Scheme converges to overall optimal solution under convexity assumptions � min J local , i (˜ u i ( k ) , ˜ x i ( k + 1)) + J inter , i (˜ w in , ji ( k ) , ˜ w out , ji ( k )) u i , ˜ ˜ x i , ˜ w in , li , ˜ w out , li j ∈ neighbors i subject to dynamics of subsystem i over the horizon initial local state, disturbances, additional constraints Lucca, July 5, 2013 Control of large-scale transportation systems 12 / 85
MPC Distributed MPC MPC for water networks Multi-level MPC Road networks Summary Scheme based on augmented Lagrangian and block coordinate descent + serial implementation Additional objective function J ( s ) inter , i (˜ w in , ji ( k ) , ˜ w out , ji ( k )) = � � T � ˜ � �� � � ˜ ( s ) 2 ˜ � � in , ji ( k ) w in , ji ( k ) + γ w in,prev , ij ( k ) − ˜ w out , ji ( k ) λ � � , � � ( s ) w out , ji ( k ) ˜ w out,prev , ij ( k ) − ˜ ˜ w in , ji ( k ) − ˜ 2 out , ij ( k ) λ 2 where for each j that is a neighbor that solved its problem before i in iteration s : w ( s ) w ( s ) w in,prev , ij ( k ) = ˜ ˜ and w out,prev , ij ( k ) = ˜ ˜ in , ij out , ij and where for each j that has not solved its problem in iteration s yet w ( s − 1) w ( s − 1) w in,prev , ij ( k ) = ˜ ˜ and w out,prev , ij ( k ) = ˜ ˜ in , ij out , ij Lucca, July 5, 2013 Control of large-scale transportation systems 13 / 85
MPC Distributed MPC MPC for water networks Multi-level MPC Road networks Summary Multiple-iterations scheme (continued) Update of ˜ λ in , ji : � � ( s +1) ( s ) w ( s ) w ( s ) ˜ ( k ) = ˜ λ λ in , ji + γ ˜ in , ji ( k ) − ˜ out , ij ( k ) in , ji Alternative: auxiliary problem principle with parallel implementation Lucca, July 5, 2013 Control of large-scale transportation systems 14 / 85
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