Robust explicit MPC design under finite precision arithmetic

Abstract

We propose a design methodology for explicit Model Predictive Control (MPC) that guarantees hard constraint satisfaction in the presence of finite precision arithmetic errors. The implementation of complex digital control techniques, like MPC, is becoming increasingly adopted in embedded systems, where reduced precision computation techniques are embraced to achieve fast execution and low power consumption. However, in a low precision implementation, constraint satisfaction is not guaranteed if infinite precision is assumed during the algorithm design. To enforce constraint satisfaction under numerical errors, we use forward error analysis to compute an error bound on the output of the embedded controller. We treat this error as a state disturbance and use this to inform the design of a constraint-tightening robust controller. Benchmarks with a classical control problem, namely an inverted pendulum, show how it is possible to guarantee, by design, constraint satisfaction for embedded systems featuring low precision, fixed-point computations.

Explicit MPC: Hard constraint satisfaction under low precision arithmetic

“…It should be noted that, running extensive simulations we have been able to verify that the preliminary results presented in [9] where misleading. In most of the cases the numerical error does not converge.…”

“…In this paper, we extend the basic idea presented in [9] and we propose a method to guarantee hard constraint satisfaction of an explicit MPC scheme [10,11] when the algorithm is implemented on a platform using finite and low 40 precision arithmetic. Compared to [9], this paper adds a detailed algorithmic presentation, unveiling the machinery required for the robust controller design, adds a detailed implementation on a FPGA platform and provides experimental results.…”

“…Compared to [9], this paper adds a detailed algorithmic presentation, unveiling the machinery required for the robust controller design, adds a detailed implementation on a FPGA platform and provides experimental results. Furthermore, this paper uses the design tools presented in [12] and [13].…”

Explicit robustness and fragility margins for linear discrete systems with piecewise affine control law

“…Note however that under Assumptions 1-4 these sets are equivalent. Also, the fragility margin obtained by the above results can be used in the context of explicit MPC design under finite precision arithmetic discussed in Suardi et al (2014).…”

Closed loop analysis of control command software

“…However, the fact that floating-point usually only induce tiny errors, makes the proof search 7 then proof checking scheme, used in this section, practical. Fixed-point arithmetic may introduce larger errors which may require the use of more involved proof search techniques, actually taking rounding errors into account [24,27].…”