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Model Predictive Control for Safety-Critical Systems Using Taylor's Theorem with Lagrange Remainder
Shuo Liu, Liang Wu, Wei Xiao, Ján Drgoňa, Calin A. Belta
TL;DR
MPC-CBF formulations can require multiple tuning parameters and may be difficult to balance for high-relative-degree constraints. This paper incorporates a numerically approximated Taylor-Lagrange remainder into receding-horizon MPC with a tunable safety margin. In a unicycle obstacle-avoidance comparison, MPC-TLR produces safer trajectories than MPC-DHOCBF and MPC-DC while requiring fewer tuning parameters and being less conservative than MPC-HOCBF.
Problem
MPC-CBF formulations face challenges establishing valid barrier guarantees and balancing safety, feasibility, and conservativeness as high-order constraints add tuning parameters.
Source: §I · p. 1Method
MPC-TLR incorporates a numerically approximated Taylor expansion with Lagrange remainder into receding-horizon MPC and uses a tunable safety margin to account for approximation errors.
Source: §I · pp. 1–2Results
MPC-TLR yields safer trajectories than MPC-DHOCBF and MPC-DC while requiring fewer tuning parameters and being less conservative than MPC-HOCBF.
Source: §VI · p. 8Takeaways & Limitations
In the reported comparisons, MPC-TLR balances safety and conservativeness with fewer tuning parameters than the high-order CBF formulations.
Source: §VI · p. 8The reported comparison is a unicycle obstacle-avoidance example rather than a broader set of application settings.
Source: §I · p. 2
Abstract
from arXiv · showhide
Safety-critical Model Predictive Control (MPC) formulations based on Control Barrier Functions (CBFs) often require multiple tuning parameters and may become conservative or difficult to keep feasible, particularly for high-relative-degree constraints. To address these limitations, in this paper we draw inspiration from the recently introduced Taylor-Lagrange Control (TLC) method, and propose an MPC formulation based on a Taylor expansion with Lagrange remainder (MPC-TLR). To enable digital implementation, we develop a numerical approximation of the Lagrange remainder under zero-order-hold control. The resulting safety constraint includes a tunable margin to account for approximation errors and balance safety, feasibility, and conservativeness. Comparisons with MPC using high-order CBFs (MPC-HOCBF), discrete-time high-order CBFs (MPC-DHOCBF), and direct safety constraints (MPC-DC) on a unicycle obstacle-avoidance problem show that MPC-TLR yields safer trajectories than MPC-DHOCBF and MPC-DC, while requiring fewer tuning parameters and being less conservative than MPC-HOCBF. Its broader control-input coverage under the adopted discretization can improve feasibility and control performance for short prediction horizons.
I. INTRODUCTION
The introduction identifies unresolved safety-guarantee and tuning challenges in MPC-CBF formulations, then motivates a predictive Taylor-Lagrange safety constraint with a numerical remainder approximation. The preliminaries section reviews HOCBF and DHOCBF as comparison baselines.
I. INTRODUCTION
- MPC-CBF formulations may lack an established valid barrier guarantee, while high-order versions add tuning parameters that complicate balancing safety, feasibility, and conservativeness.The introduction also notes that one-step optimization can produce overly aggressive actions because it does not explicitly account for future behavior.Source: §I · p. 1
- MPC-TLR applies a Taylor expansion with Lagrange remainder under zero-order-hold control, approximating the integral remainder using intermediate states and numerical quadrature.The number of quadrature points can be adjusted to improve approximation accuracy.Source: §I · pp. 1–2
- A tunable safety margin compensates for numerical approximation errors and adjusts the trade-off among safety, feasibility, and conservativeness.Source: §I · p. 2
- MPC-TLR yields safer trajectories than MPC-DHOCBF and MPC-DC, requires fewer tuning parameters, and is less conservative than MPC-HOCBF in the reported comparison.The comparison uses a unicycle obstacle-avoidance example under different horizons and parameter settings.Source: §I · p. 2
II. PRELIMINARIES
- The preliminaries review continuous-time HOCBFs and discrete-time DHOCBFs as baseline safety formulations for later comparisons.Source: §II · p. 2
A. High-Order CBF (HOCBF)
The HOCBF section sets up an affine control system, defines safety through forward invariance, and introduces recursive functions and sets used to state the high-order barrier condition. It also gives a forward-invariance guarantee under that condition.
A. High-Order CBF (HOCBF)
- The setup assumes locally Lipschitz dynamics and a bounded control set with finite componentwise input limits.Source: A. High-Order CBF (HOCBF) · p. 2
- Safety means forward invariance of C = {x : h(x) ≥ 0}, and the section builds recursive functions ψ_i and sets C_i from the safety function.Source: A. High-Order CBF (HOCBF) · p. 2
- A HOCBF is defined through the recursive functions and class K functions, and satisfying its inequality keeps the specified sets forward invariant under the theorem's initial-condition and controller assumptions.Source: A. High-Order CBF (HOCBF) · p. 2
B. Discrete-time HOCBF (DHOCBF)
The DHOCBF section derives a discrete-time model and defines relative degree by the number of steps until an input affects the safety function. Its recursive constraints yield a forward-invariance guarantee but are sufficient rather than necessary for safety.
B. Discrete-time HOCBF (DHOCBF)
- Applying a numerical discretization such as forward Euler produces the discrete-time state-transition model used to define DHOCBFs.Discrete-time relative degree is the minimum number of steps required for an input to affect the safety function.Source: B. Discrete-time HOCBF (DHOCBF) · p. 2
- The DHOCBF construction recursively defines discrete-time functions using state increments and class K functions, then uses them to specify the barrier condition.Source: B. Discrete-time HOCBF (DHOCBF) · p. 2
- Under its initial-set and controller assumptions, satisfying the DHOCBF constraint renders the specified sets forward invariant, but the condition is sufficient rather than necessary for safety.A safe trajectory may remain safe while violating the prescribed DHOCBF inequalities.Source: B. Discrete-time HOCBF (DHOCBF) · pp. 2–3
III. PROBLEM FORMULATION AND APPROACH
The problem is to drive a system toward a desired state while satisfying safety, state, and input constraints and minimizing a finite-time objective. The proposed approach places Taylor-Lagrange safety constraints on predicted states within a receding-horizon optimization.
III. PROBLEM FORMULATION AND APPROACH
- The control objective is to reach a desired state while satisfying safety and state and input limitations, with the finite-time cost minimized.Source: §III · p. 3
- Conventional CBF controllers optimize only the current input and can be myopic, whereas MPC optimizes an input sequence over a finite horizon using predicted state evolution.Source: §III · p. 3
- HOCBF and DHOCBF conditions are sufficient rather than necessary and their class K functions add tuning parameters; TLC offers necessary and sufficient safety conditions with fewer parameters but lacks explicit future-state prediction.Source: §III · p. 3
- The approach discretizes the dynamics, approximates the cost over the prediction horizon, and imposes TLR-based constraints on predicted states with a tunable margin.These constraints combine with state and input bounds in a finite-horizon nonlinear, generally nonconvex optimization solved at each sampling instant.Source: §III · p. 3
IV. TAYLOR–LAGRANGE-BASED MODEL PREDICTIVE CONTROL
The section places the general mth-order Taylor–Lagrange representation in a receding-horizon MPC framework using predicted inputs and intermediate states. Its exact safety condition avoids recursive class K functions, but the integral remainder requires numerical approximation for implementation.
IV. TAYLOR–LAGRANGE-BASED MODEL PREDICTIVE CONTROL
- MPC combines finite-horizon prediction with the necessary-and-sufficient safety characterization induced by the Taylor–Lagrange representation.Predicted inputs are applied under zero-order hold, with intermediate states representing the continuous-time trajectory and integral remainder.Source: §IV · p. 3
A. General mth-Order Taylor–Lagrange Formulation
- The general mth-order representation is derived by applying the fundamental theorem of calculus to h and recursively to its derivatives.The resulting expression includes an integral remainder.Source: A. General mth-Order Taylor–Lagrange… · p. 3
- The exact TLR expression makes h(x(t+∆t)) nonnegative if and only if its right-hand side is nonnegative, without recursive class K functions.Evaluating its remainder requires the state trajectory throughout the sampling interval, which is generally unavailable in closed form under zero-order-hold control.Source: A. General mth-Order Taylor–Lagrange… · p. 4
B. ZOH Implementation and Numerical Approximation of TLR
Under zero-order-hold control, the method approximates the integral remainder using intermediate states and numerical quadrature. A tunable safety margin compensates for approximation error, while the number of subintervals trades accuracy against computational cost.
B. ZOH Implementation and Numerical Approximation of TLR
- The approximation uses M + 1 intermediate time points because the state evolves continuously and is generally unavailable in closed form during each sampling interval.The input is held constant over the interval under the zero-order-hold policy.Source: B. ZOH Implementation and Numerical… · p. 4
- The integral remainder is approximated by numerical quadrature using intermediate states obtained by integrating the dynamics over subintervals under the held input.The approximation uses quadrature weights; increasing the number of subintervals generally improves accuracy when the integrand is sufficiently smooth and the quadrature rule converges, but increases computational cost.Source: B. ZOH Implementation and Numerical… · p. 4
- A nonnegative margin δ compensates for numerical approximation error, and a larger δ makes the safety constraint more conservative.δ = 0 corresponds to the nominal numerical approximation.Source: B. ZOH Implementation and Numerical… · p. 4
C. MPC-TLR formulation
MPC-TLR embeds the numerically implementable TLR safety condition in a finite-horizon optimal control problem with predicted dynamics and state and input constraints. The resulting problem is generally nonlinear and nonconvex, with computational complexity increasing with the horizon, quadrature points, and TLR order.
C. MPC-TLR formulation
- Predicted states evolve by applying each predicted control input to the preceding predicted state, with inputs constrained to U.The first element of the optimized sequence is applied to the continuous-time system under zero-order hold before the problem is solved again at the updated state.Source: C. MPC-TLR formulation · pp. 4–5
- The MPC-TLR problem imposes TLR-based safety constraints along the prediction horizon alongside predicted dynamics and state and control-input constraints.The continuous-time integral cost is approximated by a finite sum over the horizon; the optimal input sequence is recomputed at each time step and its first element is applied.Source: C. MPC-TLR formulation · pp. 4–5
- The formulation is generally nonlinear and nonconvex, and its computational complexity increases with prediction horizon N, quadrature points M + 1, and TLR order m.The safety constraint requires approximately N(M + 1) additional nonlinear function evaluations and safety constraints; local nonlinear programming methods generally yield only locally optimal solutions.Source: C. MPC-TLR formulation · p. 5
D. Comparative Analysis with Existing Safety-Critical MPC Formulations
The formulations share a receding-horizon optimization framework but differ in their safety constraints and the prediction-horizon inputs those constraints cover. Their relative control-input coverage depends on the discrete-time relative degree and horizon, while the more complex constraints and discretized predictions have practical limitations.
D. Comparative Analysis with Existing Safety-Critical MPC Formulations
- The methods retain the same receding-horizon objective, initial condition, dynamics, and state and input constraints, differing in the safety condition imposed.MPC-HOCBF imposes its continuous-time condition at each prediction step, while MPC-DHOCBF's forward-difference condition has a shortened index range.Source: D. Comparative Analysis with Existing… · p. 5
- MPC-TLR and MPC-HOCBF safety constraints involve up to N control inputs, versus N − md + 1 for MPC-DC and MPC-DHOCBF when md > 1.MPC-DC constraints are imposed on predicted future states; MPC-DHOCBF uses an md-step forward-difference construction.Source: D. Comparative Analysis with Existing… · p. 5
- When md is close to N, greater control-input coverage may improve feasibility and reduce conservativeness; when N is much larger than md, its performance effect may be negligible.The intermediate points in MPC-TLR approximate the Lagrange remainder under the same zero-order-hold input and add no control decision variables.Source: D. Comparative Analysis with Existing… · p. 5
- MPC-TLR and MPC-HOCBF have more complex safety constraints than MPC-DC and MPC-DHOCBF, which may increase constraint and Jacobian evaluation costs.MPC-TLR evaluates higher-order derivatives and numerically approximates the integral remainder; MPC-HOCBF uses successive barrier-function derivatives.Source: D. Comparative Analysis with Existing… · p. 5
- None of the four formulations gives an error-free safety guarantee for the actual continuous-time implementation unless discretization error is accounted for.A more accurate discretization scheme or smaller sampling interval can reduce the discrepancy.Source: D. Comparative Analysis with Existing… · pp. 5–6
V. IMPLEMENTATION AND RESULTS
The numerical results section evaluates MPC-TLR against MPC-HOCBF, MPC-DHOCBF, and MPC-DC using a unicycle model. The implementation uses a forward-Euler discrete-time model with a 0.1-second sampling time.
V. IMPLEMENTATION AND RESULTS
- The numerical evaluation uses a unicycle model to compare MPC-TLR with MPC-HOCBF, MPC-DHOCBF, and MPC-DC.Source: §V · p. 6
1) System Dynamics:
- The unicycle state comprises 2-D position, heading angle, and linear speed; its inputs are angular velocity and linear acceleration.Source: 1) System Dynamics · p. 6
- The discrete-time model uses forward Euler with a 0.1-second sampling time, and each state component is bounded between −10 and 10.Source: 1) System Dynamics · p. 6
2) System Configuration:
The unicycle obstacle-avoidance setup compares four safety-critical MPC methods across horizons and safety-parameter settings. Under the tested settings, MPC-TLR is safer than MPC-DHOCBF and MPC-DC while less conservative than MPC-HOCBF.
2) System Configuration:
- The unicycle targets [3, 0.01, 0, 0] and avoids a circular obstacle centered at (0, 0) with radius 1.The reference state's y-component is set to 0.01 to avoid an optimization singularity.Source: 2) System Configuration · p. 6
3) Safety Function and Parameters:
- The candidate safety function is squared distance from the obstacle center minus its squared radius, and has relative degree two.HOCBF and DHOCBF use two linear class K functions each; MPC-TLR uses δ and M=3.Source: 3) Safety Function and Parameters · p. 6
4) Cost Function and Parameters:
- The MPC objective uses stage and terminal costs with Q=P=10·I4 and R=I2, with parameters and environment held identical across methods.Source: 4) Cost Function and Parameters · p. 6
5) Solver Configurations and CPU Specs.:
- The methods were implemented in MATLAB with YALMIP and solved using IPOPT; controls were applied to the continuous system under zero-order hold.ode45 integrated dynamics over each 0.1-second sampling interval.Source: 5) Solver Configurations and CPU Specs · p. 6
B. Performance
- Longer horizons trigger earlier, smoother avoidance, while safety-parameter settings trade clearance against conservativeness and trajectory efficiency.At N=12, MPC-HOCBF becomes infeasible and MPC-DC converges to an undesirable local minimum; smaller HOCBF/DHOCBF gains and larger δ for MPC-DC/TLR increase clearance.Source: B. Performance · p. 6
- At N=4, MPC-HOCBF and MPC-TLR stay outside the obstacle boundary while MPC-DHOCBF and MPC-DC partially enter the unsafe region.MPC-TLR inputs are also smoother and closer to zero, indicating lower control effort.Source: Figure 3 · p. 7, B. Performance · p. 7
- MPC-TLR achieves greater minimum clearance than MPC-DHOCBF and MPC-DC, while MPC-HOCBF maintains the largest clearance but is more conservative.The comparison is under the tested settings.Source: B. Performance · pp. 6–7
- For N≥8, feasibility rates are comparable despite longer MPC-TLR/HOCBF computation; at N=4, MPC-TLR has the highest observed feasibility rate.The table reports computation-time mean ± standard deviation and feasibility across 100 initial states.Source: B. Performance · pp. 7–8, Table I · p. 8
VI. CONCLUSION AND FUTURE WORK
The paper proposes MPC-TLR, numerically approximating the Lagrange remainder under zero-order-hold control and adding a tunable safety margin. In the tested comparisons, it yields safer trajectories than MPC-DHOCBF and MPC-DC while using fewer tuning parameters and being less conservative than MPC-HOCBF.
VI. CONCLUSION AND FUTURE WORK
- MPC-TLR numerically approximates the Taylor-Lagrange remainder under zero-order-hold control and incorporates it into receding-horizon MPC with a tunable safety margin.Source: §VI · p. 8
- Under the tested settings, MPC-TLR achieves safer trajectories than MPC-DHOCBF and MPC-DC while requiring fewer tuning parameters and being less conservative than MPC-HOCBF.Source: §VI · p. 8
- Future work will address inter-sampling safety guarantees and computationally efficient MPC-TLR implementations.Source: §VI · p. 8