Learning for Control – Viability Kernel Approximation (MPC Safety Constraints)

This repository contains the implementation developed for the "Learning for Control" course project.

The goal of this project is to learn an approximate control-invariant set (viability kernel) of a dynamical system and use it as a terminal constraint inside a Model Predictive Control (MPC) framework to ensure recursive feasibility and improve closed-loop safety.

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Project Overview

The main idea is to approximate the N-step backward reachable set of a target safe set S, and use a learned representation of this set to enforce terminal constraints in MPC problems.

Instead of computing the set analytically (intractable for nonlinear systems), we learn a neural network classifier that predicts whether a state belongs to the viability kernel.

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Methodology

Data Generation (Reachability Analysis)

For each sampled initial state x₀, we solve an optimal control problem with horizon N:

• system dynamics constraints
• input and state constraints
• terminal constraint x_N ∈ S

Each sample is labeled:

• 1 → feasible (x₀ can reach S in N steps)
• 0 → infeasible

This produces a dataset of state-label pairs (x, label).

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Learning the Viability Kernel

A neural network classifier is trained to approximate the feasibility function:

• Input: system state x
• Output: probability of being feasible

The model learns the boundary of the N-step backward reachable set.

Implementation:

• PyTorch / TensorFlow MLP
• nonlinear activation functions
• supervised classification

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MPC with Learned Constraint

The trained network is used as a terminal constraint inside MPC:

• terminal state x_N must satisfy NN(x_N) ≈ 1

This enforces:

• recursive feasibility
• improved stability
• safer trajectories

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Systems

Dynamics are implemented using CasADi with URDF-based models for the double pendulum.

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Experiments

We compare three MPC formulations:

1. MPC without terminal constraint
2. MPC with learned viability constraint

Metrics:

• feasibility rate
• constraint violations
• trajectory quality
• robustness under perturbations

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Technologies

• Python
• CasADi (optimal control)
• PyTorch
• NumPy
• nonlinear control & optimization tools

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Objectives

• Learn a control-invariant set from data
• Approximate viability kernel using neural networks
• Improve MPC recursive feasibility
• Compare learned vs classical MPC formulations

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Author

Marco Misseroni
MSc Mechatronics Engineering – Electronics and Robotics University of Trento