HPMPC - A new software package with efficient solvers for Model Predictive Control

Transcription

1 - A new software package with efficient solvers for Model Predictive Control Technical University of Denmark CITIES Second General Consortium Meeting, DTU, Lyngby Campus, May 2015

2 Introduction Model Predictive Control and Moving Horizon Estimation

3 Introduction Model Predictive Control + optimal control signal + easy incorporation of forecasts + predictive adaptation to setpoint changes + natural handling of constraints and MIMO + generalization to non-linear systems - need for a model - optimization problem solved on-line

4 Introduction Application: smart grid

5 Linear MPC problem min x,u s.t. N 1 n=0 1 2 [u n x n 1] x n+1 = A n x n + B n u n + b n [ ] [ ] Rn S n s n un S n Q n q n x n + 1 [ ] [ ] P p s n q n ρ n 1 2 [x N xn 1] p π 1 x 0 = ˆx 0 u n u n u n, n = 0,..., N 1 x n x n x n, Problem size n = 1,..., N n x : state vector dimension n u : control vector dimension N: control horizon length

6 Linear MPC problem min x,u s.t. N 1 n=0 1 2 [u n x n 1] x n+1 = A n x n + B n u n + b n [ ] [ ] Rn S n s n un S n Q n q n x n + 1 [ ] [ ] P p s n q n ρ n 1 2 [x N xn 1] p π 1 x 0 = ˆx 0 u n u n u n, n = 0,..., N 1 x n x n x n, n = 1,..., N general and flexible formulation sub-problem in stochastic and non-linear MPC in smart energy grids, N can be very large (plants with very different dynamics)

7 IP methods - some general theory General QP program & KKT system min x,u s.t. 1 2 x Hx + g x Ax = b Cx d Hx + g A π C λ = 0 Ax b = 0 Cx d t = 0 λ t = 0 (λ, t) 0 Newton method (2nd order method) for the KKT system H A C 0 x Hx k A π k C λ k + g A π C 0 0 I λ = Aπ k b Cx k t k d 0 0 T k Λ k t Λ k T k

8 IP methods - some general theory structured system, can be condensed as [ H + C (T 1 k Λ k )C A ] [ ] xk = A 0 π k [ g C = (Λ k e + T 1 k Λ k d + T 1 ] k σµ k e) b KKT system of an equality constrained QP

9 Linear-Quadratic Control Problem Linear MPC Problem: min x,u s.t. N 1 n=0 1 2 [u n x n 1] x n+1 = A n x n + B n u n + b n x 0 = x 0 [ ] [ ] Rn S n s n un S n Q n q n x n + 1 [ ] [ ] P p s n q n ρ n 1 2 [x N xn 1] p π 1 used as routine to compute the search direction in IPM most computationally expensive part of IPM backward Riccati recursion used to factorize the KKT matrix computational cost linear in N

10 Implementation - current approaches Riccati solver implementation code-generated triple-loop linear algebra works well for small problems optimized BLAS 2 works well for large problems 0 Gflops OpenBLAS Triple Loop N=10, n u = n x

11 Implementation - our approach Naive approach use code-generated triple loop for small instances use BLAS for large instances Better approach: combine the best of the two high-performing code (key routines (gemm) in optimized assembly: SIMD vectorization, hide operation latency) portability (most code is architecture-independent) small overhead (focus on small-scale problems) partial code generation (avoid branches & index computation)

12 Implementation - our approach We got the best of the two approaches close-to-peak performance on all architectures performance increases quickly for small matrices Gflops OpenBLAS Triple Loop Kernel AVX N=10, n u = n x

13 Test machine # 1 Intel Core i7 3520M 2 cores, 4 threads AVX SIMD 346$ (processor only) 18 W per core

14 Intel Core i7 - MPC solver Comparison with state-of-the-art solver for linear MPC FORCES n x n u N double single double single Table : Test problem: mass-spring system with box constraints. Run times [ms] for 10 IP iterations. Test processor: Intel Core i7 3.6 GHz.

15 Test machine # 2 ARM Cortex A9 4 cores NEON SIMD 120$ (entire board) < 1 W per core

16 ARM Cortex A9 - MPC solver Comparison with state-of-the-art solver for linear MPC FORCES n x n u N double single double single Table : Test problem: mass-spring system with box constraints. Run times [ms] for 10 IP iterations. Test processor: ARM Cortex 1.0 GHz.

17 Test machine # 3 ARM Cortex A7 2 cores NEON SIMD 50$ (entire board) < 0.5 W per core

18 ARM Cortex A7 - MPC solver Comparison with state-of-the-art solver for linear MPC FORCES n x n u N double single double single Table : Test problem: mass-spring system with box constraints. Run times [ms] for 10 IP iterations. Test processor: ARM Cortex 1.0 GHz.

19 Conclusion Application of HPC techniques to MPC : state-of-the-art solver for embedded MPC on the algorithmic side: structure-exploiting solver on the implementation side: hardware-exploiting code MPC and MHE problems, box and soft constraints Target architectures: x86, x86 64, ARM, PowerPC