Real-Time Integer Optimal Control on FPGA Bartolomeo Stellato and Paul Goulart

Transcription

1 Real-Time Integer Optimal Control on FPGA and Paul Goulart CDC 2016 Dec 12, 2016 Las Vegas, NV, USA

2 Medium-voltage drives market is worth 4bln $. and is growing 10% every year!

3 Power Distribution Grid Rectifier Inverter Motor Load ~ = ~ = ~ Transformer 3

4 Power Distribution Inverter Motor = ~ 3

5 Power Distribution Inverter Motor = ~ 3

6 Traditional Control Schemes Are Suboptimal Traditional Control Controller u cont Modulator Inverter and Motor = ~ x 4

7 Traditional Control Schemes Are Suboptimal Direct MPC Controller u sw Inverter and Motor = ~ x 5

8 But Direct MPC is Difficult Currents 1 THD Time[ms] Inputs f sw Time[ms] 6

9 But Direct MPC is Difficult Currents 1 THD Time[ms] Tradeoff THD vs f sw Inputs f sw Time[ms] 6

10 But Direct MPC is Difficult Very fast dynamics 7

11 But Direct MPC is Difficult Very fast dynamics Mixed Integer Optimization Problems in 25 µs! 7

12 But Direct MPC is Difficult Control Objectives Timing THD f sw Tradeoff THD vs f sw Mixed Integer Optimization Problems in 25 µs! 8

13 Problem Formulation

14 Total Harmonic Distortion ( ) ( ) = 10

15 Total Harmonic Distortion ( ) ( ) = THD Component in Cost Function = ( ) ( ) 10

16 Total Harmonic Distortion ( ) ( ) = THD Component in Cost Function = ( ) ( ) Internal Motor States 10

17 Total Harmonic Distortion ( ) ( ) = THD Component in Cost Function = ( ) ( ) Internal Motor States Oscillating References 10

18 Switching Frequency = ( ) ( ) = 11

19 Switching Frequency = = ( ) ( ) FIR Filter 11

20 Switching Frequency = = ( ) ( ) Approximate with IIR Filter FIR Filter 11

21 Switching Frequency = = ( ) ( ) Approximate with IIR Filter FIR Filter ˆ ( ) = 11

22 Switching Frequency = = ( ) ( ) Approximate with IIR Filter FIR Filter ˆ ( ) = Frequency Estimate 11

23 Switching Frequency = = ( ) ( ) Approximate with IIR Filter FIR Filter ˆ ( ) = Frequency Estimate Desired Frequency 11

24 Switching Frequency = = ( ) ( ) Approximate with IIR Filter FIR Filter ˆ ( ) = Frequency Estimate Desired Frequency Filter States x iir 11

25 Complete Block Diagram MPC Inverter and Motor = ~ IIR = 12

26 Complete Block Diagram Controller MPC Inverter and Motor = ~ IIR = 12

27 Optimal Control Problem Infinite Horizon minimize γ k l(x(k)) k=0 subjectto x(k+1)=ax(k)+bu(k) x(0)=x 0 x(k) X,u(k) { 1,0,1} 3 13

28 Optimal Control Problem Infinite Horizon minimize γ k l(x(k)) k=0 subjectto x(k+1)=ax(k)+bu(k) x(0)=x 0 x(k) X,u(k) { 1,0,1} 3 where l(x(k))= i(k) i (k) 2 2+δ ˆ f sw (k) f sw 2 2 THD 13

29 Optimal Control Problem Infinite Horizon minimize γ k l(x(k)) k=0 subjectto x(k+1)=ax(k)+bu(k) x(0)=x 0 x(k) X,u(k) { 1,0,1} 3 where l(x(k))= i(k) i (k) 2 2+δ ˆ f sw (k) f sw 2 2 THD 13

30 Optimal Control Problem Infinite Horizon minimize γ k l(x(k)) k=0 subjectto x(k+1)=ax(k)+bu(k) x(0)=x 0 x(k) X,u(k) { 1,0,1} 3 where l(x(k))= i(k) i (k) 2 2+δ ˆ f sw (k) f sw 2 2 THD 13

31 Optimal Control Problem Infinite Horizon minimize γ k l(x(k)) k=0 subjectto x(k+1)=ax(k)+bu(k) x(0)=x 0 x(k) X,u(k) { 1,0,1} 3 where l(x(k))= i(k) i (k) 2 2+δ ˆ f sw (k) f sw 2 2 THD Impossible to Solve Online 13

32 Problem Solution

33 Optimal Control Problem Infinite Horizon minimize γ k l(x(k)) k=0 subjectto x(k+1)=ax(k)+bu(k) x(0)=x 0 x(k) X,u(k) { 1,0,1} 3 ( ( )) 15

34 Optimal Control Problem minimize Short Finite Horizon N 1 k=0 γ k l(x(k))+γ N V(x(k)) subjectto x(k+1)=ax(k)+bu(k) x(0)=x 0 x(k) X,u(k) { 1,0,1} 3 ( ( )) 15

35 Optimal Control Problem minimize Short Finite Horizon N 1 k=0 γ k l(x(k))+γ N V(x(k)) subjectto x(k+1)=ax(k)+bu(k) x(0)=x 0 x(k) X,u(k) { 1,0,1} 3 ( ( )) Very Short Horizon N 15

36 Optimal Control Problem minimize Short Finite Horizon N 1 k=0 γ k l(x(k))+γ N V(x(k)) subjectto x(k+1)=ax(k)+bu(k) x(0)=x 0 x(k) X,u(k) { 1,0,1} 3 ( ( )) Very Short Horizon N Approximate ( ( )) Offline 15

37 Approximate Dynamic Programming ( )= Bellman Equality { ( )+ ( + )} 16

38 Approximate Dynamic Programming ( )= Bellman Equality { ( )+ ( + )} T Bellman Operator T 16

39 Approximate Dynamic Programming ( )= Bellman Equality { ( )+ ( + )} T Bellman Operator T Contractive lim M TM V =V 16

40 Approximate Dynamic Programming ( )= Bellman Equality { ( )+ ( + )} T Bellman Operator T Contractive Monotone lim M TM V =V = T T 16

41 Approximate Dynamic Programming Bellman Inequality V(x) min u {l(x)+γv(ax+bu)} T Bellman Operator T Contractive Monotone lim M TM V =V = T T 16

42 Approximate Dynamic Programming Bellman Inequality V(x) min u {l(x)+γv(ax+bu)} T Bellman Operator T Contractive Monotone lim M TM V =V = T T Sufficient condition for underestimating T = 16

43 Finding the Best Underestimator Infinite Dimensional LP X ( ) ( ) ( ) T ( ) 17

44 Finding the Best Underestimator Infinite Dimensional LP maximize X V(x)c(dx) subjectto V(x) T M V(x) x Iterated Bellman Inequality 17

45 Finding the Best Underestimator Infinite Dimensional LP maximize X V(x)c(dx) subjectto V(x) T M V(x) x Iterated Bellman Inequality Restrict to Quadratic Functions V(x)=x Px+2q x+r 17

46 Finding the Best Underestimator Infinite Dimensional LP maximize X V(x)c(dx) subjectto V(x) T M V(x) x Iterated Bellman Inequality Restrict to Quadratic Functions V(x)=x Px+2q x+r Tractable SDP (Offline) 17

47 FPGA Implementation PROTOIP Toolbox (Suardi et al.) Brute force enumeration Pipelined Evaluation of Switch Sequences Parallelized Matrix Computations

48 Timing Benchmarks =.. =.. [ ] 19

49 Timing Benchmarks =.. =.. [ ] Computation Times under 25 µs! 19

50 Hardware in the Loop Tests

51 Steady-State Performance at = 300 Hz THD [%] ADP State of the art Horizon Length N 21

52 Steady-State Performance at = 300 Hz THD [%] ADP State of the art Horizon Length N Better Performance with Short Horizons 21

53 Conclusions - New Approach to Direct MPC 22

54 Conclusions - New Approach to Direct MPC Meaningful formulation ( ) THDvsf sw 22

55 Conclusions - New Approach to Direct MPC Meaningful formulation ( THDvsf sw ) Complexity reduction with good performance (ADP) 22

56 Conclusions - New Approach to Direct MPC Meaningful formulation ( THDvsf sw ) Complexity reduction with good performance (ADP) Real-Time FPGA implementation (< 25 µs ) 22

57 Conclusions - New Approach to Direct MPC Meaningful formulation ( THDvsf sw ) Complexity reduction with good performance (ADP) Real-Time FPGA implementation (< 25 µs ) High-Speed Finite Control Set Model Predictive Control for Power Electronics B. Stellato, T. Geyer, P. J. Goulart IEEE Transactions on Power Electronics (In Press) 22

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59 Backup - Transient Currents = ( ) Time[ms] 24

60 Backup - Dense Formulation Integer Program + ( ) ( ) {,, } 25

61 Backup - SDP Formulation Quadratic Cost Function X ( ) ( )= ( )+ + SDP Formulation ( )+ + ( ), M, =,..., = S, R, R, =,..., 26

62 Backup - Switch Positions C dc C dc C dc V dc N i x V dc N i x V dc N i x C dc C dc C dc 27