Controlling Complex Plants: Perspectives from Network Theory

Transcription

1 Controlling Complex Plants: Perspectives from Network Theory Prodromos Daoutidis University of Minnesota Wentao Tang, Davood Babaei Pourkargar University of Minnesota Sujit S. Jogwar Indian Institute of Technology, Bombay

2 Chemical Plant: Integrated Process Network Vinyl acetate monomer process Naphtha reforming process Tight integration - key to sustainability/energy efficiency Feedback interconnections: network level behavior Disturbance propagation, multi-time-scale dynamics 2

3 Control Paradigms Decentralized control: inherent limitations Centralized control: impractical as size increases Slow controller Slow dynamics Controller 2 Subsystem 2 Fast controller Fast dynamics Controller 1 Subsystem 1 Controllers Plant Controllers Plant Hierarchical/multi-layer control Scattolini, R. (2009). J. Process Contr., 19(5), Christofides, P. D. et al. (2013). Comput. Chem. Eng., 51, Negenborn, R. R., & Maestre, J. M. (2014). IEEE Contr. Syst., 34(4), Distributed/quasidecentralized control 3

4 Distributed MPC - Distributed optimization Primal-dual approach Acceleration - Stability and optimality Nash equilibrium/pareto optimum Multi-agent games - Cooperation Iterative/sequential configurations Non-cooperative counterparts - Communication Neighbor-to-neighbor communication Event-triggered Communication limitations handling Wakasa, Y. et al. (2008). Proc. 47th IEEE Conf. CDC Giselsson, P. et al. (2013). Automatica, 49(3), Teixeira, A. et al. (2016). IEEE Trans. Signal Process., 64(2), Venkat, A. N. et al. (2005). Proc. 44th IEEE Conf. CDC Giselsson, P., & Rantzer, A. (2010). Proc. 49th IEEE Conf. CDC Maestre, J. M. et al. (2011). Optim. Control Appl. Methods, 32(2), Stewart, B. T. et al. (2010). Syst. Contr. Lett., 59(8), Liu, J. et al. (2010). AIChE J., 56(8), Farina, M., & Scattolini, R. (2012). Automatica, 48(6), Hu Y, El-Farra N.H. (2013) Amer. Contr. Conf., Liu, J., Chen, X., de la Peña, D. M., & Christofides, P. D. (2012). IEEE Trans. Autom. Contr., 57(2), Li, H., & Shi, Y. (2014). Automatica, 50(4), Tippett, M. J., & Bao, J. (2014). AIChE J., 60(5),

5 Plant Decomposition Computational tractability Robustness to faults Σ Decomposition Simpler algorithms Reduced communication Σ 1 Σ 2 Σ 3 Σ 4 Σ 5 Σ 6 Optimal plant decomposition: an open problem, related to the underlying network structure 5

6 Structural Analysis and Control Synthesis Multi-layer/Multi-echelon control decomposition (Morari, Arkun and Stephanopoulos, 1980) Structural controllability / observability (Lin, 1974, Glover and Silverman, 1976, Davison, 1977) Structural interactions and synthesis of control systems (Morari and Stephanopoulos, 1980, Johnston and Barton, 1984, Russel and Perkins, 1987, Georgiou and Floudas, 1989, Daoutidis and Kravaris, 1992, Schne and Hangos, 2010) Graph-theory tools used extensively 6

7 Large-scale System Decomposition u 1 Subsystem I u 2 II u 3 III u 4 Graph-theoretic algorithms for decomposing coefficient matrix - strict hierarchical/acyclic decomposition, nested epsilon decomposition, etc. (Siljak et al., 1980s) x 1 x 2 x 3 x 4 x 5 x 6 x 7 Decentralized / blockdecentralized control Require special, restrictive structure y 1 y 2 y 3 y 4 Optimization-based approaches (Hangos and Tuza, 2001, Barcelli et al. 2010, Boem et al., 2015, Anderson and Papachristodoulou, 2012) 7

8 Control Structure Selection u 1 u 2 u 3 u 1 u 2 u 3 K K -T y 1 y 2 y 3 y 1 y 2 y 3 Input-output partitioning by RGA-based interaction analysis Network topology? Plant-wide control (Buckley, Foss, Stephanopoulos, Luyben, Georgakis, Skogestad, ) Decentralized 8

9 Hierarchical Control for Plants with Inventory Flow Segregation Singular perturbation analysis for time-scale separation HOLDUPS STABILIZATION DISTRIBUTED CONTROL FAST TIME SCALE REGULATORY LAYER HOLDUPS STABILIZATION SUPERVISORY CONTROL INTERMEDIATE T.S. PRODUCT PURITY PRODUCTION RATE IMPURITY LEVELS SUPERVISORY CONTROL INTERMEDIATE T.S. SUPERVISORY CONTROL SLOW TIME SCALE SUPERVISORY LAYERS FASTER TIME SCALE OPTIMIZATION NETWORK LEVEL OPTIMIZATION LAYER Baldea, M., & Daoutidis, P. (2012). Dynamics and nonlinear control of integrated process systems. Cambridge University Press. 9

10 This Work: Seeking Optimal Decompositions for Distributed/Hierarchical Control Hierarchical/ distributed control Complex systems Network theory for plant decomposition 10

11 Communities in Real-World Networks Communities: dense subnetworks with sparse interactions Proteins and their interactions in a rat cell Pages of a website and their hyperlinks Identification of communities: Community detection Hierarchical clustering, modularity maximization, etc. Fortunato, S. (2010). Phys. Rep., 486(3), Newman, M. E. J. (2012). Nat. Phys., 8(1),

12 Graph Representations of Control Systems Nonlinear control system mm xx = ff xx + jj=1 gg jj (xx)uu jj yy ii = h ii xx, ii = 1,, ll Input/output bipartite graph System digraph u 1 y 1 u 1 x 1 y 1 x 2 u 2 y 2 u 2 y 2 u 3 y 3 u 3 x 3 x 4 y 3 u 4 y 4 u 4 x 5 y 4 u 5 y 5 u 5 y 5 x6 x 7 Inputs Outputs Inputs Outputs States Šiljak, D. D. (1978). Large-scale dynamic systems: stability and structure. North Holland. Reinschke, K. J. (1988). Multivariable control: A graph theoretic approach. Springer. 12

13 Input-Output Connectivity Input-output path in system digraph - Length of the shortest path: measure of physical closeness / directness of effect u 1 u 2 x 1 x 2 y 1 y 2 Relative degree u 3 x 3 x 4 y 3 - The smallest integer r ij such that u 4 x 5 y 4 L ggii L ff rr iiii 1 hjj (xx) 0 - Sluggishness of output response u 5 y 5 x6 x 7 (uu jj = SS tt, uu jj = 0) - Shortest path length on the digraph rr iiii = ll iiii 1 dd rr iiiiyy ii ddtt rr iiii tt=0 0 Daoutidis, P. & Kravaris, C. (1992). Chem. Eng. Sci., 47(5),

14 Optimal Fully Decentralized Configuration Distribute inputs and outputs to maximize structural decoupling ll mm ss(uu jj, yy ii ) = max rr ii jj rr iiii + rr iiii rr iiii, 0 ii =1 jj =1 u j y i Integer programming formulation Objective: Sum of structural decoupling indices Integer variables: Assignment matrix (0-1) Heo, S. et al. (2015). Chem. Eng. Sci., 136,

15 Agglomerative Hierarchical Clustering Distance between input-output pairs dd uu jj1, yy ii1, uu jj2, yy ii2 = 2 max rr ii1 jj 1, rr ii2 jj 2 rr ii1 jj 2 + rr ii2 jj 1 Distance between input-output clusters dd CC 1, CC 2 = min uu jj 1,yy ii1 CC 1, uu jj 2,yy ii2 CC 2 dd uu jj1, yy ii1, uu jj2, yy ii2 Heo, S. et al. (2015). Chem. Eng. Sci., 136,

16 Optimal Fully Decentralized Configuration Graph-theoretic formulation - Bipartite input/output graph weighted by s ij - Maximum matching problem - Hungarian algorithm u 1 u 2 u 3 u 4 u 5 y 1 y 2 y 3 y 4 y 5 Kang, L. et al. (2016). J. Process Contr., 46,

17 Agglomerative Hierarchical Clustering Graph-theoretic formulation - Complete graph of input-output pairs weighted by distances - A clustering pattern = A tree in complete graph - Minimum spanning tree problem - Kruskal s algorithm (u 5, y 5 ) 3 (u 1, y 1 ) 3 0 (u 2, y 2 ) d (u 4, y 4 ) 5 (u 3, y 3 ) (u 1, y 1 ) (u 2, y 2 ) (u 3, y 3 ) (u 4, y 4 ) (u 5, y 5 ) 17

18 Divisive Hierarchical Clustering Divisive Clustering - Recursive bisection - Quality of bisection: Decentrality measure For a bisection of C into C and C =C\C Compactness : connectivity inside C and C Distance : (Inverse of) connectivity between C and C Decentrality maximized by integer programming Heo, S., & Daoutidis, P. (2016). AIChE J., 62(9),

19 Hierarchical Clustering Remarks - Hierarchy of control configurations generated - A-posteriori evaluation necessary - Agglomerative clustering can be formulated graphtheoretically - Divisive clustering more general but computationally more expensive Next step community detection by modularity maximization - Avoid complete enumeration and a-posteriori evaluation - Exploit efficient modularity maximization algorithms - Incorporate quantitative response information 19

20 Modularity-based Community Detection Newman-Girvan Type Modularity For a partition P of the nodes in the network into communities Modularity Q(P) = Fraction of intracommunity edges (or edge weights) observed in the existing graph - Fraction of intracommunity edges (or edge weights) expected in a random graph Captures the statistical significance of the communities in the network Community detection : max PP QQ PP Newman, M. E., & Girvan, M. (2004). Phys. Rev. E, 69(2),

21 Modularity on Weighted Bipartite Graph Nodes {u 1,, u m } {y 1,, y l } Edges captured by weight matrix a ij - weight of edge y i u j Degree of u j ll kk jj uu = ii =1 aa ii jj Degree of y i mm kk ii yy = jj =1 aa iijj Total number of edges ll mm mm = kk yy ii = ii=1 jj=1 kk jj uu Barber, M. J. (2007). Phys. Rev. E, 76(6),

22 Modularity on Weighted Bipartite Graph Pairwise Modularity Measure bb iiii = Pairwise modularity measure aa iiii /mm Fraction of edges between y i and u j kk yy ii /mm Fraction of edges incident to y j kk jj uu /mm Fraction of edges incident to u j Expected fraction of edges between y i to u j 22

23 Modularity of Partition on Bipartite Graph For partition P of nodes into communities C 1, C 2, C 1 C 2 C 3 C 1 C 2 C 3 QQ(PP) = CC PP bb iiii II yy ii CC II uu jj CC ii,jj Sum of all intra-community modularity measures Global maximization of modularity: computationally intractable Newman s spectral algorithm Recursive bisection, approximately maximal modularity increase Louvain fast unfolding algorithm Recursive aggregation, local maxima obtained by moving nodes 23

24 Bipartite Community Detection Input-output affinity: Weight of edges in bipartite graph aa iiii = 1 LL iiii LL min + 1 Shortest path length from u j to y i (sum of edge weights w in digraph) Shortest path in the network u j = εs(t) L ij L min a ij 1 0 ww ee = 1 log 10 SS(ee) Sensitivity obtained by linearizing at operating point u j = 0 y i L ij : accounts for connectivity and response sensitivity dominates the short time response when the product of S(e) 1 on the shortest path, L ij r ij Tang, W. & Daoutidis, P. (2016). Submitted to IEEE Trans. Autom. Contr. 24

25 Modularity Maximization on Bipartite Graph Spectral Algorithm - Recursive bisection by maximizing modularity increase For a bisection of C into C and C =C \ C max ss ΔQQ = 1 2 sstt BB CC tt 1 TT BB CC 1 t i = 1 if y i C t i = -1 if y i C s j = 1 if u j C s j = -1 if u j C - Approximation by SVD T BB CC = σσ rr uu rr vv rr - Stop when max Q < 0 rr ss = sign uu 1, tt = sign vv 1 B C the modularity matrix block corresponding to the nodes in C C C C C Newman, M. E. J. (2006). Proc. Nat. Acad. Sci., 103(23),

26 Modularity on System Digraph Nodes {v 1,, v N }= {u 1,, u m, x 1,, x n, y 1,, y l } Edges captured by adjacency matrix - a ij = 1 if there is an edge from v i to v j - a ij = 0 otherwise Out-degree of v i NN kk ii = jj =1 aa iijj In-degree of v j NN kk jj + = ii =1 aa ii jj Total number of edges NN NN mm = kk ii = ii=1 jj=1 kk jj + Leicht, E. A., & Newman, M. E. (2008). Phys. Rev. Lett., 100(11),

27 Modularity on System Digraph Digraph Modularity Expected fraction of edges from v i to v j bb iiii = aa iiii /mm kk ii /mm kk jj + /mm Pairwise modularity measure Fraction of edges from v i to v j Fraction of edges starting at v i Fraction of edges ending at v j QQ(PP) = CC PP ii,jj=1 Modularity Maximization Spectral Algorithm - Recursive bisection by maximizing modularity increase max ss NN bb iiii II vv ii CC II vv jj CC = CC PP ii,jj=1 - Approximation by spectral decomposition NN bb iiii II vv ii CC II vv jj CC bb iiii = bb jjjj = bb iiii +bb jjjj 2 ΔQQ = ss TT BB CC ss 1 TT BB CC 1 /2 s j = 1 if v j C s j = -1 if v j C BB CC = λλ rr uu rr uu T rr, ss = sign uu 1 rr 27

28 System Digraph Community Detection Subsystem 1 u 1 x 1 y 1 Controller 1 U 1 u 2 x 2 y 2 Y 1 u 3 x 3 x 4 y 3 Subsystem 2 Controller 2 U 2 u 4 x 5 u 5 y 5 x 6 x 7 y 4 Y 2 Modularity maximization in system digraph: minimization of cross-community edges Jogwar, S. S. & Daoutidis, P. (2016). Submitted to Chem. Eng. Sci.

29 Remarks on Community Detection Approaches Bipartite community detection accounts for input/output interactions state variables are implicitly distributed Guarantees input/output reachability State variables overlap Digraph community detection explicitly distributes state variables Allows local model-based control strategies reduced complexity Controllability check necessary - verifiable Disturbances can be incorporated in both decompositions

30 Example Reactor-Separator Process Model Cold Feed Cold Feed A CSTR 1 CSTR 2 Flash A B C A B C B 9 Inputs Feed rates Flow rates Heating power Liu, J. et al. (2009). AIChE J., 55(5), Stewart, B. T. et al.. (2010). Syst. Contr. Lett., 59(8), States Temperatures Concentration of A Concentration of B Liquid levels 9 Outputs Temperatures Concentration of B Liquid Levels 30

31 Example Reactor-Separator Process System Digraph 30 nodes, 60 edges F R F f1 V 1 F f2 V 2 V 3 x V 1 V V 3 A1 x x A2 A3 2 F 1 F 2 F 3 x B1 x B1 x B2 x B2 T 1 T 2 T 3 x B3 x B3 Q 1 T 1 Q 2 T 2 Q 3 T 3 31

32 Example Reactor-Separator Process Community detection in system digraph F R F f1 V 1 F f2 V 2 V 3 x V 1 V V 3 A1 x x A2 A3 2 F 1 F 2 F 3 x B1 x B1 x B2 x B2 T 1 T 2 T 3 x B3 x B3 Q 1 T 1 T 2 Q 2 Q 3 T 3 Communities are distributed around the units 32

33 Example Reactor-Separator Process Community detection in bipartite graph cold feed F R F f1 V 1 F f2 V 2 V 3 x V 1 V V 3 A1 x x A2 A3 2 F 1 F 2 F 3 x B1 x B1 x B2 x B2 T 1 T 2 T 3 x B3 x B3 Q 1 T 1 T 2 Q 2 Q 3 T 3 Communities are distributed along a combination of mass and energy balances and units 33

34 Example Reactor-Separator Process Community detection in bipartite graph hot feed F R F f1 V 1 F f2 V 2 V 3 x V 1 V V 3 A1 x x A2 A3 2 F 1 F 2 F 3 x B1 x B1 x B2 x B2 T 1 T 2 T 3 x B3 x B3 Q 1 T 1 T 2 Q 2 Q 3 T 3 Communities are distributed around units Different interaction intensity between feed and temperatures 34

35 Simulation Case Study Different Decompositions - Suboptimal 1: non-maximum modularity observed in digraph community detection - Suboptimal 2: intuitive decomposition by mass balance, energy balance and kinetics Decomposition Subsystem Inputs Outputs Centralized - All All Optimal Digraph Communities Optimal Bipartite Communities I F f1, F R, Q 1 V 1, T 1, x B1 II F f2, F 1, Q 2 V 2, T 2, x B2 III F 2, F 3, Q 3 V 3, T 3, x B3 I F f1, F R, F 1 V 1, x B1, x B3 II F f2, Q 2 T 2, x B2 III F 2, F 3 V 2, V 3 IV Q 1, Q 3 T 1, T 3 Suboptimal 1 I F f1, Q 1 V 1, T 1, x B1 II F f2, F 1, Q 2 V 2, T 2, x B2 III F 2, F 3, F R V 3, x B3 IV Q 3 T 3 Suboptimal 2 I F f1, F f2, F R x B1, x B2, x B3 II F 1, F 2, F 3 V 1, V 2, V 3 III Q 1, Q 2, Q 3 T 1, T 2, T 3 35

36 Simulation Case Study Distributed MPC simulation (for 120 min) - Non-cooperative, iterative algorithm - Control/state profile communicated between controllers - Receding horizon N = 15 - Sampling time Δt = 1.5 min Babaei Pourkargar, D., Almansoori, A., & Daoutidis, P. (2017). Submitted to IFAC 20 th World Congress. Tang, W., Babaei Pourkargar, D. & Daoutidis, P. (2017). Unpublished. 36

37 Simulation Case Study Distributed MPC simulation (for 120 min) - MATLAB 2015b, SQP GHz Intel Core i processor Decomposition Perf. index Comp. time Centralized min Digraph Communities min Bipartite Communities min Suboptimal min Suboptimal min 37

38 Simulation Case Study Distributed MPC simulation (for 120 min) Outputs (x B3 for example) Inputs (F 3 for example) 38

39 Example - Hydrodealkylation (HDA) Plant 23 inputs, 23 outputs System digraph: 175 nodes, 848 edges Luyben, W. L., Tyréus, B. D., & Luyben, M. L. (1999). Plantwide process control. McGraw-Hill. Kang, L., Moharir, M., Almansoori, A., & Daoutidis, P. (2017). Submitted to Amer. Contr. Conf. Jogwar, S. S. & Daoutidis, P. (2016). Submitted to Chem. Eng. Sci. 39

40 Examples - Hydrodealkylation (HDA) Plant Hydrodealkylation (HDA) Plant 5 communities detected in system digraph Separator Physically located in the unit operations Consistent with previous studies on decentralized control of HDA plant 40

41 Concluding Remarks Community detection powerful tool to systematically address decomposition of complex plants with different degrees of decentralization - Flexible: can be combined with different interaction measures Connectivity Connectivity + Sensitivity Connectivity + Sensitivity + Finite time response - Scalable to large networks Efficient and mature algorithms - Promising framework for addressing distributed and hierarchical control 41

42 Future Outlook Community Detection Robustness of decomposition to uncertainty Model reduction Distributed state estimation Fault detection and isolation Hierarchical community structure detection 42

43 Future Outlook Communities / Decomposition Control Performance Controller Communication Distributed MPC Algorithms 43

44 Acknowledgement 44

45 Example Gas Sweetening Plant 6 inputs, 6 outputs System digraph: 32 nodes, 100 edges 45

46 Example Gas Sweetening Plant 2 communities detected on either the bipartite graph or the system digraph Units sharing a common liquid solvent stream are more strongly interacting than units connected by gaseous streams. The absorption and desorption which dictate the variations of concentrations and temperatures of various streams occur in the liquid solvent. 46

47 Example Reactor-Separator Process Steady State Block RGA F R F f1 V 1 F f2 V 2 V 3 x V 1 V V 3 A1 x x A2 A3 2 F 1 F 2 F 3 x B1 x B1 x B2 x B2 T 1 T 2 T 3 x B3 x B3 Q 1 T 1 T 2 Q 2 Q 3 T 3 Communities are not clustered topologically 47

48 Example Reactor-Separator Process Community detection in bipartite graph without sensitivity F R F f1 V 1 F f2 V 2 V 3 x V 1 V V 3 A1 x x A2 A3 2 F 1 F 2 F 3 x B1 x B1 x B2 x B2 T 1 T 2 T 3 x B3 x B3 Q 1 T 1 T 2 Q 2 Q 3 T 3 Communities are distributed around units 48

49 Efficient Algorithm of Modularity Maximization Fast Unfolding (Louvain) Algorithm Initialize Each node in a community. Modularity optimization For each node: compute Q when moving each node from its community into other communities. Choose the move with max Q. Community aggregation internal edge self-loop external edge edge between nodes Blondel, V. D. et al. (2008). J. Stat. Mech. Theor. Exp., 2008(10), P