A New Control Theory for Dynamic Data Driven Systems

Transcription

1 A New Control Theory for Dynamic Data Driven Systems Nikolai Matni Computing and Mathematical Sciences Joint work with Yuh-Shyang Wang, James Anderson & John C. Doyle

2 New application areas 1

3 New application areas 2

4 Reflex Distributed Rigid Fast (offline computation) Unstable real dynamics Planning Centralized Flexible Slow (online computation) Stable virtual dynamics 3

5 Designing reflex layers for large-scale cyber-physical systems 4

6 , NO. XX, SEPTEMBER ]. to ic al d - - s r (14) nt the arized rly as xpeneters, each n or a inspired from the second-order dynamics of a power network or mechanical system see our companion paper [16] for a detailed discussion. Power grid: frequency regulation Topology (a) (a) Interconnected topology topology i i j (b) Subsystem Interaction between interaction neighbor- (b) Interaction between neighboring ing subsystems Fig. 3. Simulation example interaction graph. 5 Fig. 3. Simulation Example j

7 Centralized and dense control Controller 6

8 Sparse, distributed & localized control w/ delayed communications C C C C C C C C C C C C C C C C C C C 7

9 new challenges info exchange constraints NP-hard problems 6

10 new challenges info exchange constraints comms channels 7

11 new challenges info exchange constraints comms channels infrastructure co-design S A S S S A A A S A S A S S A 10

12 new challenges info exchange constraints comms channels ~125 million homes powered ~100 billion neurons, ~100k miles of axons infrastructure co-design large (huge) scale billions of nodes billions of devices 9

13 A unified theory for system design robustness efficiency resilience + info exchange comms in the loop co-design + scale to millions, billions, trillions of nodes 11

14 time (s) 10k 100 Distributed state of the art Centralized Localized (serial) 1.01 system level synthesis Localized (parallel) k 100k # States 10

15 DDDAS + SLS for Automated Control System Design and Redesign 14

16 Sys ID Controller Synthesis Controller Experts Needed One Shot ID/Design 15

17 2 The analytic solution to (13) is given by the vectorial softthresholding operator [22]: vec(m 2 ) k +1 =(1 λ j / inj,d (M k +1 ) k ) inj,d H 2 vec(m k k ) inj,d (14) Sys ID task. The dynamic interaction between two neighboring subsystems is shown in Fig. 3(b), in which state x i,1 can affect state x j,2 if i and j are neighbors. This interaction model is inspired from the second-order dynamics of a power network or mechanical system see our companion paper [16] for a detailed discussion. in which (c) + = max(0,c) for any scalar c. E TRANSACTIONS ON AUTOMATIC Thus using CONTROL, these analytic VOL. expressions XX, NO. toxx, implement SEPTEMBER the iterate updates (11a) and (11b), the actuator regularized LLQR optimization problem (8) can be solved nearly as quickly as the state-feedback problem, as the most expen- problem Consider the LLQGsive step is proposed to compute the in [25]. update equation parameters (a) Interconnected topology (b) Interaction between neighboring subsystems al optimization problem (F a is decomposed into oj,d,fb ) as defined in (12) once this is done, each oj,d lems, each subproblem update is cana beconvex implemented quadratic via a matrix multiplication or a Fig. 3. Simulation i Example j soft-thresholding. ted on an affine set. In this case, the proximal affine function [20], [25], [30]. We only need The diagonal and off-diagonal entries of A are drawn affine function once. 1 If we only considered subsystems in the d-outgoing set, the localized from auniform distribution over [0.4, The iteration in (30a) - i 0.8] and [ j 0.4, 0.2][ constraint L would have no effect on the synthesized controllers, thus we extend theregion by 1 to incorporate boundary subsystems that must have [0.2, 0.4], respectively. The open loop plant isunstable, with e carried out using multiple a zero response matrix to the disturbance multiplicauced dimension, which significantly improves subsystems at subsystem j. the spectral radius of A given by in the example that (a) Interconnected topology (b) Interaction between neighboring putation time. Consider the LLQR problem with actuator Fig. 3. Simulation example interaction graph. 22], which is the state feedback version of n-wise separable part isidentical to the LLQG swing equation for power network. Consider the swing dynamic equation update (30b) can be computed using matrix Suppose that we use (32) as the actuator norm. [22] that the row-wise separable part can be m i i + d i i = X k ij ( i j )+ w i + u i (40) multiple unconstrained optimization problems, j 2 N i operators given by vectorial soft-thresholding where s an efficient way to update (30a). i, i, m i, d i, w i, u i frequency deviation, inertia, damping, external disturbance, M, there exists other distributed algorithms and control action of the controllable load of bus i. The d to solve the optimization problem (29) in scalable way. For instance, if either g (r ) coefficient k ( ) or ij j. We let x ly convex, we can use the alternating minimiza- i := [ i i ] > be the state of bus i and use e AMA) [31] to simplify the ADMM algorithm. A t I + A t to discretize the swing dynamics. Equation (40) can then be expressed in the form of (2) with ernatives include the over-relax ADMM [20] ted version of ADMM and AMA proposed in 1 t 0 0 A ii = k, A i t 1 di t ij = k ij, Controller Synthesis are the phase angle deviation, Experts Needed is the coupling term between buses i and Ever Changing System Controller 16

18 Need a theory for local, scalable and automated Adaptive Control and Sys ID Fault & Change Detection Controller (re)design Actuation/sensing (re)design 17

19 Need a theory for local, scalable and automated Adaptive Control and Sys ID Fault & Change Detection Controller (re)design Actuation/sensing (re)design 18

20 A System Level Approach

21 A System Level Approach Novel parameterizations of stabilizing controllers

22 A System Level Approach Novel parameterizations of stabilizing controllers (Approximate) Locality = scalability

23 controlled output disturbance measurement control input 22

24 controlled output disturbance measurement control input info sharing constraints 23

25 control input measurement act sns act controller sns sns 24

26 act sns act controller sns sns 25

27 act act comm delay sns sns sns 26

28 act act comm delay sns sns sns 27

29 controlled output disturbance measurement control input info sharing constraints 28

30 System level approach design entire closed loop response 29

31 What we care about System level approach design entire closed loop response 30

32 What we design System level approach design entire closed loop response 31

33 What we design System level approach design entire closed loop response Key idea: maintain the notion of state 32

34 What we design System level approach design entire closed loop response state structure 33

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39 state structure 38

40 parameterize & design entire system response 39

41 need to ensure achievability of 40

42 need to ensure that there exists. such that the above holds 41

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45 in affine space 44

46 Theorem [implementation] 45

47 sys response structure controller internal structure 46

48 controller simulates full system response 47

49 system response and controller internal structure 48

50 (Convex) System level synthesis any (convex) functional SLC: any (convex) set Convex SLCs: system performance, controller structure & architecture, locality, etc. Unified framework for custom controller synthesis 49

51 State-Feedback System Level Parameterization response implementation achievability 50

52 Localizability Sensing / actuation delay: 1 Comm speed: 2 (compared to plant speed) 51

53 Localizability 52

54 Localizability State deviation 53

55 Localizability activate propagate 54

56 Localizability 55

57 Localizability 56

58 Localizability 57

59 Localizability 58

60 Idealized centralized control Control Space State - Comm Speed: Inf Time 59

61 Distributed LQR (Lamperski, Doyle 2013) Control State action Space Comm speed: 2 Comm ~Global speed: optimal Inf Global optimal Time 60

62 LLQR Control State action Space Comm speed: 2 ~Global optimal Time 61

63 LLQR Control State action Space Comm range: 48 3 Time 62

64 LLQR Control State action Space Space-time region Comm speed: 2 ~Global optimal Time 63

65 , NO. XX, SEPTEMBER ]. to ic al d - - s r (14) nt the arized rly as xpeneters, each n or a inspired from the second-order dynamics of a power network or mechanical system see our companion paper [16] for a detailed discussion. Power grid: frequency regulation Topology (a) (a) Interconnected topology topology i i j (b) Subsystem Interaction between interaction neighbor- (b) Interaction between neighboring ing subsystems Fig. 3. Simulation example interaction graph. 64 Fig. 3. Simulation Example j

66 time (s) 10k 100 Distributed state of the art Centralized Localized (serial) 1.01 system level synthesis Localized (parallel) k 100k # States 10

67 Need a theory for local, scalable and automated Adaptive Control and Sys ID Fault & Change Detection Controller (re)design Actuation/sensing (re)design 66

68 Need a theory for local, scalable and automated Adaptive Control and Sys ID Fault & Change Detection Controller (re)design Actuation/sensing (re)design 67

69 LLQR Control action Space Space-time region Actuation Failure Comm speed: 2 ~Global optimal Time 68

70 LLQR Control action LOCAL Fault Det & Redesign Space Space-time region Comm speed: 2 ~Global optimal Time 69

71 LLQR Control action Space Space-time region Comm speed: 2 ~Global optimal Time 70

72 Just scratching the surface Novel parameterization: new approach to constrained optimal control new perspective on closed loop sys ID/adaptive? Locality = scalability: game changer for optimal controller synthesis consequences for DDDAS? Goal: automated control (re)design for DDDAS 71

73 Select references Nikolai Matni Y.-S. Wang, N. Matni and J. C. Doyle, System level parameterizations, constraints and synthesis, IEEE ACC Y.-S. Wang, N. Matni and J. C. Doyle, A system level approach to controller synthesis, IEEE TAC 2017, Submitted. N. Matni and V. Chandrasekaran, Regularization for design, IEEE TAC N. Matni and J. C. Doyle, A theory of dynamics, optimization and control in layered architectures, IEEE ACC 2016.