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A computationally efficient robust model predictive control framework for uncertain nonlinear systems -- extended version

Johannes Köhler, Raffaele Soloperto, Matthias A. Müller, Frank Allgöwer

arXiv:1910.12081v2eess.SY

TL;DR

The paper addresses robust MPC for incrementally stabilizable nonlinear systems with general state- and input-dependent disturbances, where stronger robustness methods can be computationally demanding. It constructs an online tube using offline scalar bounds for incremental stability and disturbance uncertainty, achieving robust guarantees with a moderate online-computation increase. A nonlinear benchmark demonstrates applicability against state-of-the-art robust MPC approaches.

  • Problem

    Robust MPC for nonlinear systems with general state- and input-dependent disturbances requires constraint tightening and guarantees without the large online computational demand typical of advanced alternatives.

  • Method

    The framework constructs a tube online, using scalar bounds for an offline incremental Lyapunov function and stabilizing feedback plus an offline neighborhood-based disturbance upper bound.

  • Results

    The scheme ensures robust recursive feasibility, robust constraint satisfaction, and practical asymptotic stability, with online computational demand moderately increased over nominal MPC.

  • Takeaways & Limitations

    The approach offers an efficient implementation for robustly stabilizing complex nonlinear systems while exposing a trade-off between computational demand and conservatism.

  • Takeaways & Limitations

    The stability analysis and extension to economic MPC with predicted-uncertainty stage costs are beyond the paper’s scope.

Abstract

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In this paper, we present a nonlinear robust model predictive control (MPC) framework for general (state and input dependent) disturbances. This approach uses an online constructed tube in order to tighten the nominal (state and input) constraints. To facilitate an efficient online implementation, the shape of the tube is based on an offline computed incremental Lyapunov function with a corresponding (nonlinear) incrementally stabilizing feedback. Crucially, the online optimization only implicitly includes these nonlinear functions in terms of scalar bounds, which enables an efficient implementation. Furthermore, to account for an efficient evaluation of the worst case disturbance, a simple function is constructed offline that upper bounds the possible disturbance realizations in a neighbourhood of a given point of the open-loop trajectory. The resulting MPC scheme ensures robust constraint satisfaction and practical asymptotic stability with a moderate increase in the online computational demand compared to a nominal MPC. We demonstrate the applicability of the proposed framework in comparison to state of the art robust MPC approaches with a nonlinear benchmark example. This paper is an extended version of [1], and contains further details and additional considers: continuous-time systems (App. A), more general nonlinear constraints (App. B) and special cases (Sec. IV).

I. INTRODUCTION

The paper develops a nonlinear robust MPC framework for incrementally stabilizable systems with general state- and input-dependent uncertainty. It targets rigorous robustness with lower online complexity by representing nonlinear stability and disturbance information through scalar bounds.

  • Motivation: Robust MPC must preserve feasibility, constraint satisfaction, and stability despite disturbances, but existing min-max, scenario, and stochastic schemes often increase online computational demand.Nominal MPC may have an arbitrarily small robustness margin under hard state constraints.
  • Contribution: The proposed framework applies tube-based constraint tightening to incrementally stabilizable nonlinear systems with general nonlinear state- and input-dependent disturbances.The tube size and corresponding tightening are computed online along the nominal predicted trajectory.
  • Contribution: Instead of explicitly using a potentially complex incremental Lyapunov function and feedback, the method uses scalar bounds that characterize incremental stabilizability.These scalar quantities can be computed numerically, avoiding explicit online evaluation of the nonlinear functions.
  • Results: The framework provides robust recursive feasibility and constraint satisfaction while keeping online computational demand only moderately above nominal MPC.The paper compares computational complexity and conservatism with competing robust MPC approaches in a nonlinear example.

C. Efficient disturbance description

The framework evaluates state- and input-dependent disturbance bounds through an offline function and predicts tube size online using scalar variables, enabling tightened constraints with moderate computational overhead.

  • Efficient disturbance description: The offline function ˜wδ upper-bounds uncertainty within a neighbourhood of each nominal trajectory point and satisfies a monotonicity property needed for valid online bounds.For quadratic Vδ, larger neighbourhood sets require no smaller uncertainty bounds; the construction details are given offline.
  • Efficient disturbance description: The MPC predicts a tube size s online, then uses it to tighten state and input constraints for robust constraint satisfaction.The tube shape is based on sublevel sets of the incremental Lyapunov function Vδ.
  • Efficient disturbance description: The optimization augments nominal state and input variables with scalar tube-size s and disturbance-bound w variables governed by additional nonlinear dynamics and constraints.This avoids explicitly including the nonlinear functions Vδ and κ in the online optimization.
  • Efficient disturbance description: The numerical example reports that online robustification moderately increases computational demand compared with nominal MPC, unlike several competing approaches that can increase it by orders of magnitude.The comparison concerns the proposed online tube construction and constraint tightening for general nonlinear uncertainty.
  • Efficient disturbance description: Because uncertainty depends on state and input, predicted tube sizes need not increase monotonically along the prediction horizon.It is possible to have sk+1|t < sk|t.

B. Theoretical analysis

Under its stated assumptions, the proposed scheme is recursively feasible, satisfies the original constraints, and yields practical asymptotic stability through terminal ingredients and tube-based recursive arguments.

  • Theoretical analysis: Assumptions on terminal controller, cost, set, and uncertainty bound provide the terminal ingredients required by the theoretical analysis.The terminal set must satisfy tightened state and input constraints, and its positive invariance is verified through the terminal conditions.
  • Theoretical analysis: Theorem 1 states that initial feasibility implies recursive feasibility, satisfaction of constraints (2), and practical asymptotic stability of the closed-loop system.These properties hold when Assumptions 1–6 are satisfied.
  • Theoretical analysis: The proof constructs a shifted candidate solution, bounds tube size and disturbance variables, verifies tightened constraints, and preserves terminal-set feasibility.The incremental stabilizing feedback κ bounds the cost increase and supports recursive feasibility.
  • Theoretical analysis: The value function satisfies a practical Lyapunov decrease inequality, which supports the conclusion of practical asymptotic stability.The analysis bounds VN above and below by class-K∞ functions and includes an uncertainty-dependent residual term.
  • Theoretical analysis: A larger prediction horizon N improves the region of attraction and the bound on maximal uncertainty, while terminal-free guarantees are presented only as an expectation.The paper suggests that sufficiently long horizons may permit analogous guarantees without terminal ingredients.

C. Discussion

The discussion positions the method as a simpler, more computationally efficient alternative for general nonlinear uncertainty, while identifying restrictions on dynamic uncertainty and cost treatment.

  • C. Discussion: The framework handles state- and input-dependent uncertainty but cannot directly use dynamic model-mismatch bounds.Dynamic uncertainty may require additional compact input constraints.
  • C. Discussion: Compared with an online ellipsoidal-tube method, the proposed scheme uses n+1 states and m+1 input variables rather than matrix-valued tube and feedback variables.The paper characterizes this as simpler and more computationally efficient, but also more conservative.
  • C. Discussion: The implemented objective minimizes nominal predicted cost rather than explicitly accounting for prediction uncertainty.A worst-case stage-cost modification is proposed to incentivize cautious operation; for additive disturbances, it does not change the optimal open-loop trajectory.
  • C. Discussion: The stability analysis and extension of the uncertainty-aware stage cost to economic MPC are beyond the paper’s scope.The discussion notes that explicit predicted-uncertainty costs become more relevant for economic MPC.
  • C. Discussion: Extending the approach to output-feedback and distributed or hierarchical MPC is identified as future work.The stated main focus remains constraint satisfaction and recursive feasibility despite disturbances or uncertainty.

D. Offline and Online Implementation

The framework provides offline procedures for bounding disturbances and designing terminal ingredients, while reducing online optimization to scalar bounds rather than explicit nonlinear functions.

  • Nonlinear uncertainty bound: A continuity-based construction defines a function ˜wδ that upper-bounds disturbance effects near nominal trajectories.The construction uses the disturbance magnitude at the nominal point and a continuity condition within a bounded Vδ neighbourhood.
  • Nonlinear uncertainty bound: For quadratic incremental Lyapunov functions, Proposition 3 computes the least conservative admissible ˜wδ using the shapes of Vδ and W.The same triangular-inequality reasoning also applies to specified polytopic forms of Vδ.
  • Nonlinear uncertainty bound: The Proposition 3 construction does not apply to arbitrary nonlinear Vδ and κ because the required monotonicity property may fail.This limits the direct construction method, not the broader scalar-bound framework.
  • Nonlinear uncertainty bound: Propositions 1–2 and Corollary 1 compute disturbance bounds and tightening coefficients using scalar incremental-stabilizability bounds instead of explicitly using Vδ.For quadratic Vδ, additional procedures exploit its shape to reduce conservatism.
  • Terminal ingredients: Terminal ingredients are constructed from a terminal controller, terminal cost, terminal set, and conditions ensuring the required robust properties.The paper presents a procedure for computing suitable terminal ingredients under relatively general nonlinear-system conditions.

with some positive constants

The terminal-set construction supports large initial uncertainty when the prediction horizon is sufficiently long, and the offline/online workflow separates computation from repeated MPC solves.

  • Terminal ingredients: A terminal-set proposition constructs suitable terminal ingredients for relatively general nonlinear systems using a bound on uncertainty near the origin.The maximal tube size sf is chosen as large as possible while satisfying tightened terminal constraints.
  • Terminal ingredients: Arbitrarily large uncertainty w can be accommodated when N is sufficiently large, provided its effect is confined to the initial part of the predicted trajectory.The practical implication is operation in regions with large uncertainty for sufficiently long horizons.
  • Implementation: The offline workflow verifies incremental stabilizability, computes ˜wδ and terminal ingredients, and selects the prediction horizon N.The online workflow measures the state, solves the MPC problem, applies the first input, and repeats.
  • Implementation: Online computation consists of measuring xt and solving the MPC optimization problem, then applying the first optimized control input.The next iteration begins after incrementing time.

IV. SPECIAL CASES

The special cases cover additive disturbances and LPV systems, preserving robust feasibility and stability while exposing trade-offs among conservatism, complexity, and uncertainty modeling.

  • IV-A. Additive disturbances: For additive disturbances, offline constraint tightening is the only difference from nominal MPC, yielding equivalent computational complexity.Using constant additive bounds can nevertheless introduce substantial conservatism compared with state- and input-dependent uncertainty descriptions.
  • IV-A. Additive disturbances: Under feasibility at t = 0, the additive-disturbance scheme is recursively feasible, satisfies the original constraints, and is practically asymptotically stable.These properties are stated by Theorem 2 for the resulting closed-loop system.
  • IV-A. Additive disturbances: The additive-disturbance formulation extends linear tube-based tightening to nonlinear uncertain systems using incremental stabilizability, with a simpler implementation than more elaborate offline methods.Its tightening can be viewed as an overapproximation because it uses scalar stabilizability inequalities rather than the full nonlinear uncertain dynamics.
  • IV-B. Linear parameter varying systems: For LPV systems, a contractive polytope and linear feedback define a polytopic incremental Lyapunov function and enable linear computations of the tightening terms.The contraction factor is obtained by an LP, and disturbance bounds use the vertices of the parameter polytope.
  • IV-B. Linear parameter varying systems: The LPV formulation can be implemented as a quadratic program, while simpler disturbance bounds reduce constraints at the cost of greater conservatism.Using a quadratic incremental Lyapunov function may instead require conservative overapproximations or produce a QCQP.
  • IV-B. Linear parameter varying systems: The LPV terminal construction requires ρ + Lw < 1, which ensures stability with a common Lyapunov function and prevents tube size from increasing arbitrarily.This contrasts with cited competing approaches where tube size can grow exponentially with the prediction horizon.
  • Special-case implications: The approximations trade conservatism for simplicity and support linear, nonlinear, additive, parametric, and general mixed uncertainty settings.Further approximations can reduce complexity for higher-dimensional systems.

V. CASE STUDY: NONLINEAR QUADROTOR

The nonlinear quadrotor case study applies the framework to continuous-time uncertain nonlinear dynamics and compares efficiency, performance, and uncertainty conservatism with existing approaches.

  • V. Case study: nonlinear quadrotor: The example demonstrates applicability to an uncertain nonlinear continuous-time system and compares the framework with min-max differential inequalities and Lipschitz-bound methods.It also examines whether state- and input-dependent uncertainty reduces conservatism relative to constant uncertainty bounds.

System model:

The benchmark uses a 10-state quadrotor with additive disturbances and compares offline incremental-stability-based tube construction against competing robust MPC approaches.

  • System model:: The continuous-time quadrotor model has ten states describing position, velocity, pitch, roll, and their rates.Inputs control pitch angle, roll angle, and vertical thrust.
  • System model:: The controller stabilizes xr = [3, 3, 10, 0]⊤ from x0 = [0, 2, 0]⊤ under a quadratic stage cost.The prediction horizon is T = 3 s with sampling time h = 0.3 s and N = 10.
  • System model:: The additive-disturbance scenario assumes w ∈ R^3 with ∥w∥2 ≤ 1 and uses a quadratic incremental Lyapunov function with linear feedback computed offline.The resulting feedback renders the pre-stabilized dynamics incrementally exponentially stable.
  • System model:: The continuous-time disturbance bound is wc = 0.1646, while the corresponding discrete-time bound is 0.48.The discrete-time contraction rate is ρ = 0.944.
  • System model:: The proposed approach has complexity equivalent to, lower than, while its conservatism is similar to and lower than.For N = 10, the Lipschitz-based alternative would require reducing disturbance magnitude by 99% to remain feasible.

Parametric uncertainty:

With ±10% parametric uncertainty, the framework characterizes uncertainty along the predicted trajectory and reduces conservatism through state- and input-dependent tube bounds.

  • Parametric uncertainty:: The parameters n0 and d0 vary by ±10%, while additive disturbances are neglected.The uncertain nonlinear dynamics are treated as parametric uncertainty.
  • Parametric uncertainty:: The parametric case uses an offline quadratic incremental Lyapunov function and linear feedback, with uncertainty vertices θpar,i ∈ {−1, 1}^2.Symmetry permits implementing the uncertainty bound with only two inequality constraints.
  • Parametric uncertainty:: A longer horizon T = 4.5 s with N = 15 is required to find a safe trajectory under the increased pitch and roll uncertainty.The continuous-time disturbance bound is wc = 0.6.
  • Parametric uncertainty:: Peak φ1, φ2, u1, and u2 values decrease by 30–70% relative to the additive-disturbance scenario.The state- and input-dependent bound avoids maneuvers whose uncertainty would make them too risky.
  • Parametric uncertainty:: The scheme requires roughly 33% more decision variables and 2·N additional nonlinear inequalities, producing about 4.5 times nominal-MPC computation time.Replacing the detailed bound reduces online demand by 32% but increases conservatism.

APPENDIX

The appendix extends the main results to continuous-time systems and general continuous nonlinear constraints.

  • APPENDIX: The appendix extends the main results to continuous-time systems and general continuous nonlinear constraints.These extensions are presented in Appendix A and Appendix B, respectively.

A. Continuous-time systems

The continuous-time formulation extends the robust tube MPC framework to nonlinear systems with state- and input-dependent disturbances, under incremental stabilizability assumptions.

  • A. Continuous-time systems: The continuous-time setup models a perturbed nonlinear ODE with compact state- and input-dependent disturbance sets.The model mismatch satisfies dw(x, u, d) ∈ W(x, u).
  • A. Continuous-time systems: The framework assumes local incremental stabilizability through a feedback κ and incremental Lyapunov function Vδ.The transformed dynamics fκ are exponentially incrementally stable.
  • A. Continuous-time systems: An offline function ˜wδ upper-bounds disturbance effects near nominal trajectories and enters the MPC constraints through scalar tube variables.Monotonicity of ˜wδ is required for the disturbance bound.
  • A. Continuous-time systems: The continuous-time MPC optimizes nominal states, inputs, disturbance bounds, and tube size while tightening nonlinear constraints by cjsτ|t.The input is parameterized piecewise constantly for implementation.
  • A. Continuous-time systems: Continuous-time formulations require discretization for tube prediction, and assuming constant state and input over each sampling interval is an approximation.Piecewise-constant terminal controllers also restrict the formulation, while tube propagation can pose numerical challenges.
  • A. Continuous-time systems: Under the stated assumptions and initial feasibility, the scheme is recursively feasible, satisfies the constraints, and yields practical asymptotic stability.These properties are established by Theorem 3.

B. General nonlinear constraints

The framework extends robust constraint satisfaction from Lipschitz-type constraints to general continuous nonlinear constraints using continuity bounds and additional tube-size constraints. The resulting scheme remains recursively feasible and practically asymptotically stable, with computational demand scaling with the number of nonlinear constraints.

  • Optimization constraints: Robust satisfaction for the general nonlinear constraint set ˜Zg requires additional optimization constraints governing the nonlinear constraint bounds and terminal tube size.The terminal conditions extend the existing tube-size requirements and ensure satisfaction at the final prediction step.
  • Continuity assumptions: The extension permits nonlinear constraints ˜gj that need not satisfy the original Lipschitz assumption, provided they obey a continuity bound through functions αj ∈ K∞.The bound limits changes in each constraint function using the distance between points and can be instantiated by polynomial functions with positive coefficients.
  • Continuity assumptions: Using the more general continuity bound instead of the Lipschitz bound can reduce conservatism.
  • Guarantees: Under the stated theorem and assumptions, the augmented optimization problem is recursively feasible, satisfies the constraints, and yields practical asymptotic stability.
  • Computational demand: For ˜αj(r) = ˜cjr^λj, the additional constraints can be replaced by an equivalent formulation with q nonlinear-constraint states and one additional input.With q general nonlinear and p Lipschitz continuous constraints, the computational demand is equivalent to nominal MPC with n + 1 + q states and m + 1 inputs.
  • Computational demand: The robust tube MPC complexity increases with the number q of nonlinear constraints that are not Lipschitz continuous, while robust collision avoidance requires additional dual decision variables.
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