Probabilistic performance validation of deep learning-based robust NMPC controllers

Abstract

Solving nonlinear model predictive control problems in real time is still an important challenge despite of recent advances in computing hardware, optimization algorithms and tailored implementations. This challenge is even greater when uncertainty is present due to disturbances, unknown parameters or measurement and estimation errors. To enable the application of advanced control schemes to fast systems and on low-cost embedded hardware, we propose to approximate a robust nonlinear model controller using deep learning and to verify its quality using a posteriori probabilistic validation techniques.We use a probabilistic validation technique based on finite families, combined with the idea of generalized maximum and constraint backoff to enable statistically valid conclusions related to general performance indicators. The potential of the proposed approach is demonstrated with simulation results of an uncertain nonlinear system.

Stability and feasibility of neural network-based controllers via output range analysis

“…which define the piece-wise affine function that the neural network describes. By viewing the neurons as hyperplanes [33], an activation pattern can be defined, which assigns a binary value to every neuron in the hidden layer to model the ReLU function (8). This limits the maximum possible number of different activation patterns to…”

“…The parametric description of a ReLU network described in this section implies that the network can be seen as an affine function of the input, which includes additional binary variables to describe the ReLU function (8). This fact will be exploited in the next subsection to formulate a mixed-integer linear program that can compute the set of possible outputs of the network for a given set of inputs.…”

“…Neural networks derived via imitation learning are usually obtained by minimizing the error between the output of the network and an optimal control input, which can be obtained by solving an optimization problem. This strategy has been shown to achieve comparable performance to exact optimal control schemes, but usually only probabilistic safety certificates can be derived including information of the dual problem [7] or using probabilistic verification approaches approaches [8], [9], [10], [11], [12]. A different alternative to achieve guarantees of neural network-baesd approximate controllers is to perform an additional projection onto the feasible set [13], [14] or use the network to compute the {benjamin.karg, sergio.lucia}@tu-berlin.de) initial guess for an optimization solver [15] instead of using it as a controller.…”

A probabilistic validation approach for penalty function design in Stochastic Model Predictive Control

“…Property 1 has been already used in the context of probabilistic scaling and validation (see Alamo et al (2019) and Karg et al (2019)). See also Tempo et al (1997) for the particularization of the result to the case r = 1 and a single constraint.…”

“…The first use of probabilistic validation is to compute offline a constraint tightening, following the stochastic tube-based MPC approach proposed in (Lorenzen et al, 2016), but using probabilistic validation setting instead of the scenario approach. Secondly, to guarantee recursive feasibility, we relax the constraints using a penalty function method Kerrigan and Maciejowski (2000) and, following ideas presented in Karg et al (2019), we perform an offline probabilistic design of the penalty parameter, selected among a finite-family of values, so that the desired probabilistic guarantees of the closed-loop constraint satisfaction are fulfilled.…”