Max-plus-linear systems
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1 Max-plus-linear systems Ton van den Boom Introduction week 2018 Introductionweek 2018, DCSC slide 1
2 Algebra Conventional(+, )-algebra: (a+b) c= a c+b c a (b+c)= a b+a c (λ a) (µ b)=(λ µ) (a b) Introduce a different algebra: + = max = + Introductionweek 2018, DCSC slide 2
3 Max-Plus Algebra Introduce notation from max-plus algebra: Matrices: x y=max(x,y) x y=x+y [A B] i j = a i j b i j = max(a i j,b i j ) [A C] i j = n k=1 a ik c k j = max k=1,...,n (a ik+ c k j ) Introductionweek 2018, DCSC slide 3
4 Max-plus-linear discrete-event systems System description: x(k) = A x(k 1) B u(k) y(k) = C x(k) k is an event counter x i (k) is time that state event i occurs in kth cycle. u(k) is time that input event occurs in kth cycle. y(k) is time that output event occurs in kth cycle. Introductionweek 2018, DCSC slide 4
5 Modeling of railway networks Railway networks can be modeled using max-plus linear models. D train 2 A B C train 1 A railway junction Introductionweek 2018, DCSC slide 5
6 x 1 = departure time of train 1 from station A x 2 = departure time of train 2 from station D x 3 = departure time of train 1 from station B d 3 = scheduled departure time of train 1 from station B t 1 = traveling time from station A to station B t 2 = traveling time from station D to station B Time table: x 3 d 3 Continuity: x 3 x 1 +t 1 + w B Connection: x 3 x 2 +t 2 + c B Departure time of train 1 from station B: D x 2 train 2 A B C x 1 x 3 x 3 = max( x 1 +t 1 + w B, x 2 +t 2 + c B, d 3 ) train 1 Introductionweek 2018, DCSC slide 6
7 Departure time of train 1 from station B: x 3 = max( x , x , 56) =( x 1 25) (x 2 33) 56 [ ] = x 1 56 x 2 For a network with departure vector x we obtain: x= A x d For a cyclic timetable this will result in: x(k)= A x(k 1) d(k) Introductionweek 2018, DCSC slide 7
8 Legged robots Aerial phase 6 Ground phase t Introductionweek 2018, DCSC slide 8
9 Legged robots Aerial phase 6 Ground phase t Introductionweek 2018, DCSC slide 9
10 Legged robots Aerial phase 6 Ground phase t Introductionweek 2018, DCSC slide 10
11 Legged robots Aerial phase 6 Ground phase t Introductionweek 2018, DCSC slide 11
12 Legged robots Aerial phase 6 Ground phase t Introductionweek 2018, DCSC slide 12
13 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 13
14 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 14
15 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 15
16 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 16
17 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 17
18 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 18
19 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 19
20 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 20
21 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 21
22 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 22
23 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 23
24 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 24
25 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 25
26 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 26
27 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 27
28 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 28
29 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 29
30 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 30
31 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 31
32 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 32
33 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 33
34 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 34
35 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 35
36 Large-scale Printers 22 STEPS: 3 SHEETS INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 36
37 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 37
38 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 38
39 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 39
40 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 40
41 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 41
42 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 42
43 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 43
44 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 44
45 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 45
46 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 46
47 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 47
48 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 48
49 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 49
50 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 50
51 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 51
52 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 52
53 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 53
54 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 54
55 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 55
56 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 56
57 Large-scale Printers INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 57
58 Large-scale Printers 20 STEPS: 6 SHEETS INPUT PRINTING OUTPUT INVERTER Introductionweek 2018, DCSC slide 58
59 Container terminal QUAY CRANES AGV S YARD CRANES Introductionweek 2018, DCSC slide 59
60 quay crane AGV yard crane 2 2 Introductionweek 2018, DCSC slide 60
61 2 2 Introductionweek 2018, DCSC slide 61
62 2 2 Introductionweek 2018, DCSC slide 62
63 2 2 Introductionweek 2018, DCSC slide 63
64 2 2 Synchronization 1 Introductionweek 2018, DCSC slide 64
65 2 2 Introductionweek 2018, DCSC slide 65
66 2 2 Synchronization 2 Introductionweek 2018, DCSC slide 66
67 2 2 Introductionweek 2018, DCSC slide 67
68 2 2 Introductionweek 2018, DCSC slide 68
69 Palletizer Introductionweek 2018, DCSC slide 69
70 Introductionweek 2018, DCSC slide 70
71 Synchronization 1 Introductionweek 2018, DCSC slide 71
72 Introductionweek 2018, DCSC slide 72
73 Synchronization 2 Introductionweek 2018, DCSC slide 73
74 Introductionweek 2018, DCSC slide 74
75 Introductionweek 2018, DCSC slide 75
76 Synchronization 3 Introductionweek 2018, DCSC slide 76
77 Introductionweek 2018, DCSC slide 77
78 Introductionweek 2018, DCSC slide 78
79 Introductionweek 2018, DCSC slide 79
80 Max-Plus Linear systems Introductionweek 2018, DCSC slide 80
81 Max-plus-linear zero-element and unit-element Max-plus zero: x ( )=max(x, )=x x ( )=x+( )= Max-plus unit: 0 x 0=x+0=x Introductionweek 2018, DCSC slide 81
82 Max-plus-linear eigenvalues and eigenvectors For a cyclic timetable the system description is given by x(k)= A x(k 1) d(k) Max-plus-linear eigenvalue λ and eigenvector v: Properties: A v=λ v λ = natural cycle time (For Dutch railway network: ± 57 minutes). v = natural timetable (For well-defined network: v d). Introductionweek 2018, DCSC slide 82
83 Switching max-plus linear system Railway systems: Change order of trains. Legged robot: Change gait of robot. Paper flow in printers: Change paper size/thickness. Container terminal: Change route of container. Production system: Choose machine for processing. The system can operate in a different modes x(k)= A (l) (k) x(k 1) B (l) (k) u(k) in which A (l) and B (l) are system matrices forl-th mode. Introductionweek 2018, DCSC slide 83
84 Max-min-plus-scaling systems Piecewise affine systems Ton van den Boom Introduction week 2017 Introductionweek 2018, DCSC slide 84
85 Max-min-plus-scaling (MMPS) systems System description x(k+ 1)= f x (x(k),u(k)) y(k)= f y (x(k),u(k)), where entries of f x and f y are MMPS expressions in x(k), u(k). Example MMPS system: x(k +1) = max( 2x+5, 3)+min(x 3, max( x+3, 2x 7)) Introductionweek 2018, DCSC slide 85
86 Max-min-plus-scaling system: x(k+ 1)=max( 2x+5,3)+min(x 3,max( x+3,x 7)) Introductionweek 2018, DCSC slide 86
87 Min-max canonical form: x(k+ 1)=min( max( x+2,x), max( x+6,x 4)) Introductionweek 2018, DCSC slide 87
88 Max-min canonical form: x(k+ 1)=max( x+2, min(x, x+6), x 4) Introductionweek 2018, DCSC slide 88
89 Difference canonical form: x(k+ 1)=max( 2x+3,1,2x 9) max(x 5, x+1) Introductionweek 2018, DCSC slide 89
90 MSc thesis subjects Max-plus linear systems Modeling & control of MPL systems (theory / application) Stochastic MPL systems. Application using MPL models. Max-min-plus-scaling linear systems Modeling & control of MMPS systems (theory / application) Stochastic MMPS systems. Introductionweek 2018, DCSC slide 90
91 Important courses SC Optimization in Systems & Control SC Model Predictive Control SC Modeling & Control of Hybrid Systems WI4062TU - Transport, Routing and Scheduling CIE Railway traffic management Introductionweek 2018, DCSC slide 91
92 Questions?? Introductionweek 2018, DCSC slide 92
Railway disruption management
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