High-Speed Integer Optimal Control using ADP Bartolomeo Stellato and Paul Goulart
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1 High-Speed Integer Optimal Control using ADP and Paul Goulart EUCCO /09/2016 Leuven, Belgium
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 Transients Torque Time[ms] Inputs Time[ms] 22
54 Transients Torque Time[ms] Inputs Time[ms] 0.35 ms 22
55 Transients Torque Time[ms] Inputs Time[ms] 0.35 ms 3.5 ms 22
56 Transients Torque Time[ms] Inputs Time[ms] Very Fast 0.35 ms 3.5 ms Transient Times! 22
57 Conclusions - New Approach to Direct MPC 23
58 Conclusions - New Approach to Direct MPC Meaningful formulation ( ) THDvsf sw 23
59 Conclusions - New Approach to Direct MPC Meaningful formulation ( THDvsf sw ) Complexity reduction with good performance (ADP) 23
60 Conclusions - New Approach to Direct MPC Meaningful formulation ( THDvsf sw ) Complexity reduction with good performance (ADP) Real-Time FPGA implementation (< 25 µs ) 23
61 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) 23
62
63 Backup - Transient Currents = ( ) Time[ms] 25
64 Backup - Dense Formulation Integer Program + ( ) ( ) {,, } 26
65 Backup - SDP Formulation Quadratic Cost Function X ( ) ( )= ( )+ + SDP Formulation ( )+ + ( ), M, =,..., = S, R, R, =,..., 27
66 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 28
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