Abraham Goldsmith

Efficient Convex Optimization on GPUs for Embedded Model Predictive Control

GPU applications have traditionally run on PCs or in larger scale systems. With the introduction of the Tegra line of mobile processors, NVIDIA expanded the types of systems that can exploit the massive parallelism offered by GPU computing architectures. In this paper, we evaluate the suitability of the Tegra X1 processor as a platform for embedded model predictive control. MPC relies on the real time solution of a convex optimization problem to compute the control input(s) to a system. Relative to traditional control techniques such as PID, MPC is very computationally demanding. Quadratic programming algorithms for the solution of convex optimization problems generally lend themselves to parallelization. However, until the introduction of the Tegra, there has never been an off-the-shelf embedded processor that would enable a massively parallel embedded implementation. We investigate two different gradient based algorithms, ADMM and PQP, for solving the QP that occurs in a large class of MPC problems. The performance of these algorithms is dominated by the performance of matrix-matrix and matrix-vector products. Our work focuses on maximizing the performance of these operations for relatively small matrices of 100 to 1000 elements per dimension, which are common in the MPC control implementations found in automotive and factory automation applications. Modern BLAS libraries for CPUs and GPUs are quantitatively evaluated. We create SGEMV kernels that can outperform the state-of-the-art cuBLAS by 2.3x on TX1. Different kernel fusion schemes utilizing concurrent kernel execution and zero copy mechanisms are investigated. For ADMM, our implementation achieves 46.6x speedup over the single threaded CPU version and 2.7x speedup over the optimized OpenBLAS version. For PQP, we achieve 41.2x speedup over the single threaded CPU version and 4.2x speedup over the OpenBLAS version.

Control of dual-stage processing machines by bounded tracking-error MPC

We consider a dual-stage precision manufacturing machine where a worktool is actuated via a motion system consisting of a “fast” stage with large bandwidth but small operating range, and a “slow” stage with smaller bandwidth but larger operating range. We design a controller based on a recently developed tracking method for constrained systems that guarantees enforcement of constraints and of an assigned bound on the tracking error. For satisfying the controller assumption, we design a reference trajectory generation algorithm that is simple and can also be executed offline. The proposed control system guarantees correct processing of the pattern and finite processing time, for which bounds can be easily computed.

Extended command governors for constraint enforcement in dual-stage processing machines

This manuscript presents a scheme for the constrained control of a dual-stage system used in precision manufacturing. The system consists of two stages, a fast and a slow stage, whose actuators have different bandwidths. The fast stage is primarily constrained in its range of operation, and the slow stage is primarily constrained in allowable velocity and acceleration. The constrained control is based on the extended command governor, which is a constraint-enforcement scheme used for closed-loop systems subject to state and control constraints. A method of dividing the motion between fast and slow stages is presented which is based on tracking a minimal-motion reference for the slow stage. The extended command governor scheme is modified to simultaneously ensure constraint-admissible tracking of the minimal-motion reference and machining of the desired manufacturing pattern. Numerical simulation results are reported, showing successful tracking and constraint enforcement.