A Pontryagin Principle-Based Frequency Governor for Constrained Computing Systems

Abstract

Management of power is a crucial problem in computing systems where power is finite, processor performance and energy needs are high, and thermal constraints have to be respected. The trade-off between performance and energy expenditure is well recognized. To satisfy these conflicting requirements, in this article, a dynamic system framework is adopted, and results from optimal control theory, notably Pontryagin’s Minimum Principle (PMP), are applied to derive an energy-optimal (EO) time-varying processor speed law, or frequency governor, to execute assigned tasks. PMP is chosen as it allows for system input constraints as well as thermal and power budget constraints to be considered; the PMP-based governor is also compared with a Model Predictive Controller (MPC) implemented by following the Explicit-MPC framework using a linear model. The main contributions of this article are 1) determining an empirical time-invariant nonlinear dynamic model of an Intel CPU with task execution rate, power consumption, and temperature as the outputs, and clock frequency as the input; 2) the Linux implementation of a PMP-based clock frequency governor on the CPU based on a linear model as well as the nonlinear model; and 3) hardware implementation of an Explicit-MPC on the same platform using the frequency schedule derived from linear model simulations. Limits on task completion times and energy savings achieved in the execution of three benchmark tasks: MiBench, LINPACK, and Sorting positive integers, are presented. Experimental results show that it is possible to reduce energy consumption with an increase in task execution time while executing these benchmark tasks; it is also shown that it is possible to tune the PMP and MPC parameters to obtain similar performances. The approach presented in this article can be applied to design optimal controllers for other types of stand-alone or heterogeneous computing systems.