Source-linked AI summary
Predictive safety filter using system level synthesis
Antoine P. Leeman, Johannes Köhler, Samir Benanni, Melanie N. Zeilinger
TL;DR
Learning-based controllers can perform well, but missing safety guarantees limit their use in some industrial applications. This paper combines system level synthesis with predictive safety filtering to optimize backup control under disturbances, and also proposes an explicit variant. The SL-MPSF enlarges the safe set and reduces intervention relative to existing MPSFs, while the explicit variant requires no online optimization but is more conservative.
Problem
Learning-based controllers’ potential in many industrial applications remains limited by a lack of safety guarantees in terms of constraint satisfaction.
Source: §1 · p. 1Method
Results
The SL-MPSF provides a larger safe set and lower control intervention than existing MPSF formulations, while the explicit filter avoids online optimization but is more conservative.
Source: §1 · p. 2Takeaways & Limitations
The explicit filter trades greater conservatism for avoiding online optimization, while the SL-MPSF reduces interference with potentially unsafe controllers.
Source: §1 · p. 2, §6 · p. 10The stated safety setup assumes a square, invertible disturbance-shaping matrix.
Source: §2 · p. 3
Abstract
from arXiv · showhide
Safety filters provide modular techniques to augment potentially unsafe control inputs (e.g. from learning-based controllers or humans) with safety guarantees in the form of constraint satisfaction. In this paper, we present an improved model predictive safety filter (MPSF) formulation, which incorporates system level synthesis techniques in the design. The resulting SL-MPSF scheme ensures safety for linear systems subject to bounded disturbances in an enlarged safe set. It requires less severe and frequent modifications of potentially unsafe control inputs compared to existing MPSF formulations to certify safety. In addition, we propose an explicit variant of the SL-MPSF formulation, which maintains scalability, and reduces the required online computational effort - the main drawback of the MPSF. The benefits of the proposed system level safety filter formulations compared to state-of-the-art MPSF formulations are demonstrated using a numerical example.
1. Introduction
Safety filters can address the safety-guarantee gap that limits learning-based controllers in industrial applications, but existing approaches trade off conservativeness, design complexity, and online computation. The paper proposes two SLS-based safety filters: one enlarges the safe set while reducing intervention, and an explicit variant avoids online optimization.
1. Introduction
- Safety-filter approaches trade off offline design complexity, online computation, and conservativeness; larger safe sets extend operational range and can support safer exploration.Interventions can affect learning-controller performance, including stability, operational cost, or tracking error.Source: §1 · p. 2
- Existing MPSFs use a fixed offline tube-controller to mitigate disturbances and uncertainties, but this simplification can make them overly cautious.The resulting conservativeness can limit practical performance, especially in learning settings.Source: §1 · p. 2
- The proposed SL-MPSF jointly optimizes the safe backup trajectory and tube-controller, enlarging the safe set and reducing intervention at the cost of increased computational complexity.Its formulation uses a general terminal set for recursive feasibility and does not require a finite-impulse-response constraint.Source: §1 · p. 2
- The explicit filter designs a safe set and backup control law offline, requiring no online optimization but being more conservative than SL-MPSF.Its simple safe-set shape is intended to support scalability; related methods still require online optimization for robust constraint satisfaction.Source: §1 · p. 2
- The paper illustrates both proposed filters by comparing them with state-of-the-art safety filters in a numerical example.The introduction identifies missing safety guarantees as a barrier to exploiting learning-based controllers in many industrial applications.Source: §1 · pp. 1, 3
2. Preliminaries
The preliminaries set up robust safety filtering for a constrained linear system with bounded disturbances. The filter seeks to certify arbitrary candidate inputs while intervening as little as possible, and the proposed SLS approach optimizes both a backup trajectory and its disturbance-mitigating controller.
2. Preliminaries
- The system is modeled with nominal state and input trajectories constrained by state, input, and terminal-set requirements.The setup considers a linear system with bounded disturbances and, for simplicity, a box disturbance set.Source: §2 · pp. 3–4
- The setup imposes state and input constraints, assumes a compact input set, and requires the disturbance-shaping matrix to be square and invertible.The safety-filter definition requires robust satisfaction of state constraints under all modeled disturbances.Source: §2 · p. 3
- The safety filter must ensure robust constraint satisfaction for all future steps and disturbances while returning the candidate input whenever possible.The modular filter can certify arbitrary candidate inputs, including those from learning-based controllers or humans.Source: §2 · pp. 3–4
- The system-level formulation optimizes the nominal backup trajectory, tube-controller, and disturbance reachable set, guaranteeing robust constraint satisfaction with reduced conservatism.The online optimization may be computationally demanding, motivating an explicit alternative.Source: §2 · p. 4
3. System Level Model Predictive Safety Filter
The SL-MPSF parameterizes affine feedback policies with system responses, then optimizes a backup trajectory and controller under robust constraints. Its feasible set is safe under the terminal-set assumption, and optimizing affine feedbacks with tight reachability analysis reduces conservativeness relative to state-of-the-art MPSF schemes.
3.1. Parameterisation of affine controllers
- Affine feedbacks are represented by system responses from disturbances to closed-loop state and input errors, enabling robust constraint checks on disturbance reachable sets.The state and input reachable sets are constrained to lie within their respective constraint sets.Source: §3.1 · p. 5
3.2. System level predictive safety filter
- The SL-MPSF optimizes a nominal backup trajectory and an affine tube-controller while enforcing robust state, input, and terminal constraints.Its quadratic-program cost modifies the candidate input only as much as necessary, and the applied control is the first element of the optimal sequence.Source: §3.2 · p. 6
- Under Assumption 1, the feasible-state set XN is a safe set for the SL-MPSF.Feasibility is robustly preserved for any potentially unsafe input and disturbance in W, independently of the cost function.Source: §3.2 · pp. 6–7
- Optimizing affine feedbacks and using tight reachability analysis makes SL-MPSF less conservative than state-of-the-art MPSF schemes.The explicit alternative requires only one offline LP and reduces online computation, but may require more frequent and stronger control interventions.Source: §3.2 · p. 7
4. Explicit system level safe set
The explicit system-level safe set is designed offline by optimizing a safe-set shape and affine backup controller. Its safety filter checks whether a learned input keeps the state in that set, while the backup controller provides a robust return path.
4. Explicit system level safe set
- A single offline LP constructs the explicit safe set and backup law, avoiding the per-step QP required by the online filter.The optimization uses the parameterization introduced in Section 3.1.Source: §4 · pp. 7–8
- Unlike prior explicit safe-set approaches based on robust positive invariance and offline controller design, this method needs no controller tuning and uses a weaker periodic-invariance condition.The weaker condition permits optimization over a simple safe-set shape, such as a unit infinity-norm ball.Source: §4 · p. 7
- The offline optimization maximizes safe-set volume while enforcing robust constraint satisfaction and return to the safe set within N steps.The constraints cover disturbances and initial conditions; the resulting affine backup controller is used when the learned input fails the safety check.Source: §4 · pp. 7–8
- The explicit filter verifies set containment before applying a learned input, and Theorem 6 identifies its control law and safe set as a safety filter and corresponding safe set.The affine backup controller robustly satisfies state and input constraints under disturbances and returns the system to the safe set within N steps.Source: §4 · pp. 8–9
5. Numerical example
In a double-integrator example, the SL-MPSF safe set is nearly maximal and larger than the MPSF safe set, while requiring less aggressive intervention. The explicit filter has much lower evaluation time than SL-MPSF in the reported implementation.
5. Numerical example
- The comparison uses a double integrator with disturbance bound ∥w(t)∥∞≤1 and prediction horizon N = 10.The methods are compared with the maximal RCI set as a theoretical size reference, although it is described as not practically relevant because of poor scalability.Source: §5 · p. 9
- The SL-MPSF safe set is nearly maximal and significantly larger than the MPSF safe set; the explicit set also covers regions outside both the maximal RPI and MPSF sets.In general, the maximal RPI set is not contained in the explicit set because that set uses a simple scalar parameterization.Source: §5 · pp. 9–10
- SL-MPSF requires less aggressive modification of potentially unsafe inputs while ensuring safety over a larger set of states.The comparison shows maximal intervention for MPSF and SL-MPSF; lower intervention is preferable.Source: §5 · p. 10, Figure 1 · p. 10
- 0.031ms ± 0.028ms explicit-filter evaluation time compares with 65.7ms ± 20.0ms for SL-MPSF, while MPSF takes 4.8ms ± 1.8ms.The explicit-filter code was not optimized for speed.Source: §5 · p. 10
6. Conclusion
The paper presents two SLS-based predictive safety filters: an online-optimization SL-MPSF and an explicit filter. The authors expect the approaches can be extended to nonlinear settings using tools developed in Leeman et al. (2023).
6. Conclusion
- The SL-MPSF enlarges the safe set through online optimization over a robustness-ensuring tube-controller, allowing safety verification for more states with less intervention.The reduced intervention is described as reducing the impact on the potentially unsafe controller's performance.Source: §6 · p. 10
- The second proposed filter is explicit and requires neither online optimization nor offline controller tuning.The paper presents this as an efficient explicit formulation.Source: §6 · p. 10
- The authors expect both approaches can extend to nonlinear settings using tools developed in Leeman et al. (2023).This is presented as an expected extension, not as a demonstrated result.Source: §6 · p. 10