Universal laws and architectures: Theory and lessons from grids, nets, brains, bugs, planes, docs, fire, bodies, fashion,
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1 Universal laws and architectures: Theory and lessons from grids, nets, brains, bugs, planes, docs, fire, bodies, fashion, earthquakes, turbulence, music, buildings, cities, art, running, throwing, Synesthesia, spacecraft, statistical mechanics John Doyle 道陽 Jean-Lou Chameau Professor Control and Dynamical Systems, EE, & BioE Ca # 1 tech
2 Efficiency/instability/layers/feedback All create new efficiencies but also instabilities Needs new distributed/layered/complex/active control Sustainable infrastructure? (e.g. smartgrids) Money/finance/lobbyists/etc Industrialization Society/agriculture/weapons/etc Bipedalism Maternal care Warm blood Major transitions Flight Mitochondria Oxygen Translation (ribosomes) Glycolysis (2011 Science)
3 Compute Comms for Comp/Cntrl/Bio Info Thry Optimization Statistics Control, OR Orthophysics (Eng/Bio/Math) Physics
4 slow fragile efficient fast robust flexible inflexible waste
5 Compute Research progress Slow Fast Insight Decidable Pspace Flexible NP P analytic Inflexible From toys to real systems Control, OR
6 Compute Research progress Slow Fast Insight Decidable Pspace Flexible NP P analytic Inflexible Improved algorithms Control, OR
7 Control of cyberphysical systems? Physical: Efficient, therefore less stable Computing: Distributed with delays Communication: With latency Therefore Control: Distributed with sparse actuation (but add sensing) with delays in computing and communications but free memory/bandwidth (for now) Fundamental tradeoffs? Scalability?
8 Control of cyberphysical systems? Physical: Efficient, therefore less stable Computing: Distributed with delays Communication: With latency Therefore Control: Distributed with sparse actuation and sensing with delays in computing and communications but free memory/bandwidth (for now) Fundamental tradeoffs? Scalability?
9 Localized (distributed) control Localizable control: Wang, Matni, You and Doyle ACC 14 Localized LQR control: Wang, Matni, and Doyle CDC 14
10 Another extremely toy model Concretely illustrate important new ideas Minimal complexity otherwise Familiar, intuitive circuit dynamics Instability mechanism is artificial but challenging
11 LC circuit Each node = grounded capacitance Each link = inductance
12 System Model Assuming each L and each C has unit value, the dynamics of the system are where x(t) is states of node voltage and link current, M is the incidence matrix of the circuit graph. (Will reorder for plotting later.)
13 Discrete Time System Model A first order (Euler) approximation is With step = 0.2, the maximum eigenvalue of Ad is Artificially create a very unstable system
14 Simplified diagram (2 states per node) Actuated and sensed Only sensed Only sensed Actuated and sensed Only sensed
15 Simplified diagram (2 states per node) Actuated and sensed Only sensed Only sensed Actuated and sensed Only sensed
16 Actuated and sensed Only sensed Simplified diagram (2 states per node) Actuated and sensed Only sensed
17 Nominally each has delay 1. Expensive? 0. Physical 1. Actuation Simplified diagram (2 states per node) Actuated and sensed Only sensed
18 Controller Physical plant
19 Sense, comm/comp, act. Expensive? 0. Physical 1. Actuation 2. Comms speed 3. Comp speed 4. Sensing Actuated and sensed Only sensed
20 Controller Physical plant
21 Controller plane Data plane SDN/ODP
22 Controller plane Cyber Data plane Physical
23 Sense, comm/comp, act. Nominally each has delay 1. Actuated and sensed Only sensed
24 Expensive: physical plant passive stability actuation low delay (comms and comp) Cheap: comms bandwidth compute memory sensing True for cells, nets, grids, brains, but not in general Actuated and sensed Only sensed
25 System Model The discrete time system equation is Example: 30 C, 29 L
26 Simplified diagram Open loop dynamics
27 Open loop xt () t xt ()
28 Open loop xt () t
29 Threshold to x(t) < 1 xt () 1 0 v () t t
30 xt () 1 0 Open loop v () t 2 v () t 9 v () t 15 Disturbance propagation Threshold to x(t) < 1 t
31 xt () 1 0 Open loop v () t 2 v () t 9 v () t v () t 22 log xt ( ) v () t 29 t
32 log xt ( ) xt () Spacetime cone log xt ( ) Open loop plot log xt ( ) t
33 log xt ( ) log xt ( ) xt () 1 0 v () t 15 Threshold to x(t) < Open loop plot log xt ( ) t
34 log xt ( ) xt () Spacetime cone log xt ( ) Open loop log xt ( ) t
35 t xt () Space-time state cone State
36 Controller Design Critical Issues 1. Transient LQ (H2) cost: (x x+u u) 2. Actuator Density 3. Communication (vs plant) Speed 4. Locality/Scalability (Computation) 5. Time/space horizon
37 Actuator Density Standard (centralized) optimal H2 control No delay (initially) Defer other issues ( comm, comp, sense) Objective: min sum (x x+u u) Actuator density = # actuators / # states Trade-off: actuator density vs norm Example: 30 C, 29 L
38 Norm - Actuator Density (normalized) Opt H2 norm Artificially unstable system 1 3 Normalized by undelayed centralized.2.5 Actuator Density 1
39 Standard control (circa 1970) norm Comm speed = 0 delay Actuator Density 1 Actuated and sensed Only sensed
40 Opt undelay central state Optimal Controller + Norm H2 optimal Communication undelayed Design/model global/huge P Implementation local/huge P Sparse actuation Opt undelay central ctrl
41 xt () log xt ( ) log ut ( ) Color code? ut () t
42 Expensive? 0. Physical 1. Actuation 2. Comms speed 3. Comp speed 4. Sensing Actuated and sensed Only sensed
43 Nominally delay 1. Communication speed Expensive? 0. Physical 1. Actuation 2. Comms speed 3. Comp speed 4. Sensing Versus plant speed
44 undelay central ctrl undelay central state Communication speed =
45 undelay central ctrl undelay central state Communication speed =
46 Norm Distributed Localized Undelayed central Normalized by undelayed centralized Communication Speed
47 undelay central ctrl undelay central state delay distr ctrl delay distr state
48 undelay central ctrl central stat delay distr state delay distr ctrl
49 Distributed (QI) Controller + Norm (H2) small + Optimal for constraints + Communication delayed Design/model global/huge P Implementation local/huge P delay distr ctrl delay distr state
50 delay local ctrl delay local state delay distr ctrl delay distr state
51 delay distr state delay local delay distr ctrl
52 delay local ctrl delay local state Localized Controller + Norm (H2) small + Optimal for constraints + Communication delayed + Design/model local/small + Implementation local/small + State local Everything is scalable.
53 Norm Distributed Localized Undelayed central Normalized by undelayed centralized Communication Speed
54 Linear equations delay local ctrl delay local state xt [ ] 0 x X u U
55 t xt () Space-time state cone State
56 t xt () Communication and Control Cone Space-time state cone State finite impulse response (FIR)
57 Control t xt () xt [ ] 0 x X u U Space-time state cone finite impulse response (FIR) Local space-time controllability
58 Communications t xt () Past Past delayed state needed to compute control
59 Control t xt () Past Local space-time controllability xt [ ] 0 x X u U Space-time state cone finite impulse response (FIR) This can linearly constrain any optimization
60 State Optimal undelayed centralized state (old) State Optimal delayed distributed (newish) (but not scalable) Optimal delayed localized (very new, scalable)
61 AWGN in C2, L26, C29 undelay central ctrl undelay central state delay local ctrl delay local state
62 xt () Local space-time controllability Control Localized Controller t + Norm (H2) small + Past Optimal for constraints + Communication is delayed + Design/model local/small + Implementation local/small + State local xt [ ] 0 x X u U Space-time state cone finite impulse response (FIR) This can linearly constrain any optimization
63 Localized Controller + Norm (H2) small + Optimal for constraints + Design/model is local + Implementation is local + State stays local - Bandwidth is? Output feedback? Mostly good? Approximately local? news, but? Layering? incomplete? Nonlinear, MPC, etc?? Comms codesign? See also Javad s new relaxations
64 Extensions Scalable optimal control Localizable control: Y.-S. Wang, N. Matni, S. You and J. C. Doyle ACC 14 Localized LQR control: Y.-S. Wang, N. Matni, and J. C. Doyle CDC 14 Output feedback progress Dealing with varying-delays (jitter) Two player LQR with varying delays: N. Matni and J. C. Doyle CDC 13, N. Matni, A. Lamperski and J C. Doyle IFAC 14
65 More Nikolai Matni Distributed/scalable system identification Low-rank + Low-order decompositions: N. Matni and A. Rantzer, CDC 14 Structured Robustness Distributed Controllers Satisfying an H norm bound: N. Matni CDC 14 Regularization for Design Topology/interconnection design: N. Matni, CDC 13 (best student paper), TCNS 14 More broadly (including actuator/sensor placement): N. Matni and V. Chandrasekaran, CDC 14
66 More Extensions/Apps Apps: neuro, smartgrid, CPS, cells IMC/RHC, etc (all of centralized control theory) Cyber theory: Delay jitter (uncertainty) Cyber: Comms co-design (CDC student prize paper) Physical: Robustness (unmodeled dynamics, noise) Cyber-phys: System ID, ML, adaptive SDN (Software defined nets, OpenDaylight) Revisit layering as optimization? Poset causality (streamlining)? Quantization and network coding?
67 Revisit layering as optimization decomposition Chiang, Low, Calderbank, Doyle, 2007 Controller Physical plant
68 Controller plane Data plane SDN/ODP
69 Layered Architectures Controller plane Cyber Data plane Physical
70 Conjecture: Norm bad before method breaks Norm norm Centralized 1 3 Actuator Density Distributed Localized Undelayed central Tradeoffs Communication Speed
71 1.15 norm Delayed Centralized Decentralized Localized Error Local control theory Communication Speed 0 Speed (=1/delay)
72 Local control theory 1.15 norm Delayed Centralized Decentralized Localized Communication Speed Error 0 Speed (=1/delay)
73 Info theory Local control theory Error 0 Rate (BW) Error 0 Speed (=1/delay) Due to quantization, loss, noise Communications
74 Info theory Local control theory Error Error Speed (=1/delay) Rate (BW) 0 0 Error Communications
75 Local control theory Info theory Error Speed (=1/delay) Rate (BW) 0 0 Error Communications
76 Local control theory Info theory Error Speed Rate 0 0 Error Communications
77 Local control theory Error Info theory Error Error Rate 0 Speed Communications
78 Local control theory Error Info theory Error Error Rate ideal 0 Speed Communications
79 Error Error Error Rate ideal 0 Speed Local control
80 Error Error Error Resource 1 ideal 0 Resource 2 Tradeoffs
81 Control over limited channels (Martins et al) a) e=d-u - decode/ actuate d delay source/ disturbance 1 f d f d 0 Plant (P) decode/ control/ encode C A Channels C S sense/ encode b) c) S p 0 P p j E j D j d) e) log S d p C min 0,log f) Pz 0 S S d p C A z 1 z p p ln S j d ln if p z 2 2 z 2 z p z
82 Universal laws and architectures (Turing) Special Architecture (constraints that deconstrain) General ideal Fast Speed Slow
83 Memory is cheap, reusable, powerful. Time is not. Special Memory General Fast Speed Slow
84 Error Error Error Resource 1 ideal 0 Tradeoffs Resource 2
85 Cheap: memory, bandwidth, sensors Not : time (1/speed), actuators Brains/bodies, cells, CyberPhySys, Error Error Actuation ideal 0 Speed (comms, comp) Critical Tradeoffs
86 All costs are ultimately physical. Cost Actuation ideal 0 Speed (comms, comp) Critical Tradeoffs
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https://www.cds.caltech.edu/~doyle John Doyle 道陽 Jean-Lou Chameau Professor Control and Dynamical Systems, EE, & BioE
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