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Analysis of unconstrained nonlinear MPC schemes with time varying control horizon

Lars Grüne, Jürgen Pannek, Martin Seehafer, Karl Worthmann

arXiv:1006.2529v1math.OC

TL;DR

The paper addresses how to analyze stability and performance of nonlinear MPC without stabilizing terminal constraints when control horizons may vary. It derives an explicit suboptimality estimate from controllability assumptions and uses it to study horizon and terminal-weight effects. The analysis establishes performance bounds and asymptotic stability for admissible time-varying control-horizon sequences under the stated assumptions.

  • Problem

    The paper studies stability and closed-loop performance for nonlinear MPC schemes without stabilizing terminal constraints, including the effect of possibly time-varying control horizons.

  • Method

    It analytically solves a linear program to obtain an explicit suboptimality estimate based on a K L 0 controllability function, then analyzes optimization horizons, control horizons, and terminal weights.

  • Results

    The resulting performance bound implies asymptotic stabilization for all admissible time-varying control-horizon sequences when the theorem’s controllability and stability assumptions hold.

  • Takeaways & Limitations

    The explicit index α provides a basis for comparing horizon choices and assessing MPC performance and stability without terminal constraints.

  • Takeaways & Limitations

    Tightness is established for the classical case m⋆=1, while the corresponding statement for m⋆≥2 remains a conjecture.

Abstract

from arXiv · show

For discrete time nonlinear systems satisfying an exponential or finite time controllability assumption, we present an analytical formula for a suboptimality estimate for model predictive control schemes without stabilizing terminal constraints. Based on our formula, we perform a detailed analysis of the impact of the optimization horizon and the possibly time varying control horizon on stability and performance of the closed loop.

1 Introduction

The paper studies stability and closed-loop performance for nonlinear MPC without stabilizing terminal constraints. It extends prior analysis to time-varying control horizons and derives explicit tools for examining horizon choices and final-cost weighting.

  • Motivation: MPC analysis centers on whether closed-loop trajectories converge to and remain near the reference, and how well the optimized objective is achieved.These objectives include specifications such as energy minimization or output maximization.
  • Prior analysis: Suboptimality estimates compare the infinite-horizon closed-loop functional with the optimal infinite-horizon value and can imply stability.For nonlinear or large-scale systems, the finite-horizon value function used in some approaches is usually not computable.
  • Contribution: The paper extends prior analysis to time-varying control horizons, motivated by networked systems where network performance determines the interval between optimizations.It investigates how different, possibly time-varying control horizons affect closed-loop behavior.
  • Contribution: An analytic solution of the prior linear program yields an explicit suboptimality formula based on the K L 0 controllability function.The paper uses this formula to analyze optimization horizons, prove conjectures about minimal stabilizing horizons, and study final-cost weights.
  • Organization: The paper organizes its analysis around controllability, stability, an explicit suboptimality index α, and the effects of optimization and control horizons.It also examines the influence of an additional final weight in the finite-horizon cost.

2 Setup and Preliminaries

The setup formulates infinite-horizon optimal control through receding-horizon finite-horizon problems and generalizes classical MPC by applying multiple optimized controls between successive optimizations. This produces a multistep feedback framework for variable control horizons, including networked implementations.

  • System setup: The framework considers discrete-time systems, including sampled-data systems where discrete time n corresponds to continuous time t=nT.State and control constraints can be represented by restricting the state and control spaces.
  • Finite-horizon MPC: Because infinite-horizon optimization is generally computationally infeasible, the controller uses a finite optimization horizon N and its associated optimal value function.The finite-horizon problem supplies the control sequence used to construct the receding-horizon controller.
  • Multistep implementation: After solving the finite-horizon problem, the controller applies the first m0 controls, reaches a new state, and repeats the construction indefinitely.The control horizon satisfies m0∈{1,...,N−1}.
  • Variable horizons: An admissible control-horizon sequence requires each mi to lie in a prescribed set M, while transmission times identify the latest completed control-horizon interval.The notation ϕ(n) records the most recent transmission time not exceeding n.
  • Feedback formulation: Multistep MPC uses a feedback law that maps the current state and within-horizon time to a control, with classical MPC recovered when only the first optimized control is used.The generalized formulation supports packet-dropout compensation by transmitting whole control sequences between successful network transmissions.
  • Performance framework: The analysis excludes terminal costs and terminal constraints and uses relaxed dynamic programming to relate a decrease condition to infinite-horizon performance.The resulting estimate bounds the scaled infinite-horizon closed-loop cost between the optimal value and an auxiliary function.

3 Controllability and performance bounds

The paper assumes asymptotic controllability expressed through running-cost bounds and uses these bounds to derive computable suboptimality estimates for multistep MPC. The resulting performance index α supports analysis across optimization horizons, control horizons, and final weights.

  • Controllability assumption: Assumption 3.1 requires a control whose running cost is bounded by β(l⋆(x0),n), where β belongs to the class K L 0.The class permits decay in time while allowing the state argument to be either K∞ or identically zero at each time.
  • Controllability models: The framework includes exponential controllability and finite-time controllability with linear overshoot as particular controllability models.A separate property constrains how β evolves over combined time intervals.
  • Intermediate bounds: Bellman’s principle and the controllability assumption yield inequalities for optimal finite-horizon stage costs and shifted optimal values.These inequalities provide the ingredients for the suboptimality theorem.
  • Computation of α: The computable approach determines α through an optimization problem over positive sequences λ0,...,λN−1 and ν satisfying the derived constraints.The analytic construction can be performed separately for each admissible control horizon and then minimized across horizons.
  • Performance bound: Theorem 3.5 guarantees αV∞(x) ≤ αV∞^µN,m(x) ≤ VN(x) for α∈(0,1] under the stated inequalities.Thus α measures how well the multistep MPC strategy approximates the infinite-horizon problem.
  • Terminal weighting: Adding a terminal weight ω≥1 preserves the analysis after adapting the finite-horizon functional and the associated formulas.The original cost is recovered with ω=1.

4 Asymptotic stability

The stability analysis extends the suboptimality criterion to variable control horizons and establishes asymptotic stabilization under a common finite-horizon value function. The criterion applies to every admissible horizon sequence when the controllability and detectability-type assumptions hold.

  • Stability setting: Stability is analyzed for the zero-cost set A, which must be invariant under a zero-cost control and satisfy K∞ bounds relating cost to distance from A.These conditions support asymptotic stability of A for the infinite-horizon optimal feedback when β is summable.
  • Main stability result: If α⋆:=minm∈M{αω_N,m} satisfies the theorem’s criterion, multistep MPC asymptotically stabilizes A for all admissible time-varying control-horizon sequences.The finite-horizon value function VN acts as a Lyapunov function at transmission times.
  • Scope and limitation: For the classical case m⋆=1, the stability criterion is tight under the stated controllability property; the analogous tightness for m⋆≥2 remains a conjecture.The paper therefore does not establish necessity of the criterion for the general multistep case.
  • Proof mechanism: The proof uses a common Lyapunov function for every admissible fixed control horizon, enabling the stability conclusion for variable horizons.The resulting estimates extend from transmission times to the intervening closed-loop trajectory.

5 Calculation of αω

The paper reduces the suboptimality calculation to a linear optimization problem under controllability functions linear in their first argument, then derives an explicit formula for α^ω_N,m.

  • Linear reformulation: For β linear in its first argument, Problem 3.6 has the same optimal value as a linear optimization problem in λ.The reformulation uses γ_k := B_k(r)/r and nonnegative λ variables subject to linear constraints.
  • Linear reformulation: The relaxed Problem 5.3 minimizes 1−(γ_m+1−ω)λ_N−1 over nonnegative λ subject to Aλ ≤ b̄.Its solution is used to analyze the original optimization problem.
  • Explicit formula: Theorem 5.4 gives the optimal value α^ω_N,m explicitly for a K L 0-function linear in its first argument.The result applies for given optimization horizon N, control horizon m, and terminal weight ω.
  • Explicit formula: The proof establishes feasibility of the relaxed problem’s optimum for Problem 3.6, making the relaxed optimum equal to the original optimum.This requires showing the additional inequalities are dispensable at the optimum.
  • Scope and interpretation: For linear K L 0-functions, the α^ω_N,m value from Theorem 5.4 remains a lower bound for Problem 3.6 when exact equality is not established.The paper also illustrates exponential controllability using suitably chosen stage costs for a nonlinear scalar system.
  • Scope and interpretation: The controllability assumption and Formula (26) can be used to compare closed-loop performance for different stage costs and inform cost-function design.The paper cites applications to controlled wave and parabolic PDEs.

6 Characteristics of αω

The explicit α^ω_N,m formula supports quantitative analysis of how optimization horizon, control horizon, overshoot, decay rate, and terminal weight affect stability and performance. Increasing N eventually yields α^ω_N,m = 1 for fixed m, while special cases reveal asymmetric parameter effects and control-horizon independence.

  • Optimization horizon: For each fixed control horizon m, lim N→∞ α^ω_N,m = 1, so sufficiently large optimization horizons ensure asymptotic stability.This holds for β of type (6) or (7) and ω ≥ 1.
  • Parameter effects: The analysis quantifies how the overshoot C and decay rate σ jointly determine stability regions for a given optimization horizon.This parameter-combination analysis is identified as an advantage over the cited prior results.
  • Optimization horizon: As the optimization horizon N increases, the guaranteed stability region grows; doubling N from 2 increases the considered area by 129.4 percent.For a fixed decay rate σ, some overshoot C always guarantees stability.
  • Parameter effects: For C = 1, α^ω_N,m = min{1, 1−(1+σω−ω)σ^(N−1)} > 0.In this special case, the resulting α^ω_N,m does not depend on the control horizon m.
  • Parameter effects: For any σ ∈ (0,1) and sufficiently small C > 1, α^ω_N,m > 0 and asymptotic stability follows, whereas small σ cannot generally guarantee stability for fixed C > 1.The required bound on C depends on N, m, and ω.
  • Finite-time controllability: For finite-time controllability with m = 1, the sufficient horizon bound is N ≥ 2 + ln(γ−ω)/(ln γ−ln(γ−1)) =: f(γ).The bound is an upper estimate for the minimal stabilizing horizon under the stated linearity conditions.
  • Finite-time controllability: For m = ⌊N/2⌋, the required horizon estimates behave asymptotically like 2 ln 2 · γ.The even- and odd-N estimates agree with numerical results reported in prior work.

7 Qualitative characteristics of αω

The section characterizes how the suboptimality index αω_N,m changes with the control horizon, establishing symmetry and monotonicity results under exponential or restricted finite-time controllability assumptions. These properties yield stability conclusions for time-varying control horizons, while finite-time cases require narrower conditions.

  • Motivation: The analysis asks whether varying the control horizon m creates additional stability difficulties because the minimal stabilizing horizon depends sensitively on m.The section studies αω_N,m with respect to m after earlier analysis showed that the optimization horizon N must be chosen appropriately.
  • Symmetry analysis: For exponential controllability and certain finite-time controllability cases, αω_N,m has symmetry relating control horizons m and N−m.The finite-time symmetry result requires the coefficients c_n to vanish for n ≥3, while the exponential result is more general.
  • Stability consequence: If the stability criterion holds for m⋆ = 1, then it also holds for m⋆ = N − 1 under the stated exponential or restricted finite-time assumptions.The result uses symmetry and monotonicity to show that classical MPC stability conditions extend to the proposed time-varying-horizon setting.
  • Monotonicity properties: For exponentially controllable systems, increasing the control horizon improves the optimal value αω_N,m in the analyzed ranges, with monotonicity depending on the final weight.The section distinguishes sufficiently large final weights, η = 1 + σω − ω ≤ 0, from the case η > 0, and also treats ω = 1.

8 Quantitative characteristics of αω

The section characterizes how optimization horizon, control horizon, controllability model, and final-term weighting affect the stability index αω_N,m. It shows that finite-time controllability can provide sharper guarantees than exponential upper bounds, while final weights and control horizons interact nontrivially.

  • Optimization and control horizons: A sufficiently large optimization horizon N ensures stability, but increasing N raises computational cost more than changing the control horizon m or final-term weight.The stability condition is determined by the sign of αω_N,m.
  • Optimization and control horizons: For exponential controllability, Theorem 5.4 determines the maximal overshoot C for a decay rate σ that preserves a positive αω_N,m.The stability region comprises parameter pairs C ≥ 1 and σ ∈ (0,1).
  • Optimization and control horizons: For N = 7 and N = 11, stability regions are illustrated over control horizons m up to ⌊N/2⌋.Corollary 7.4 permits restricting the analysis to these control horizons.
  • Finite-time versus exponential controllability: Using an exponential upper bound can substantially worsen αω_N,m: stability is not guaranteed for m ∈ {2,3,4,12,13,14} in the example.The finite-time controllability coefficients are approximated by C = 5/2 and σ = 4/5 for the exponential case.
  • Finite-time versus exponential controllability: Finite-time controllability is generally preferable because positivity of the required αω_N,m values can be checked directly with Theorem 5.4.Symmetry and monotonicity may still occur even when the theorem’s structural assumptions do not hold.
  • Final-term weighting: A sufficiently large final weight can enable stability when c_m < 1, but an excessively large weight can invalidate the stability condition.In one example, increasing the control horizon compensates for this drawback.

9 Example

Numerical MPC simulations compare the analytical estimates with linear and nonlinear systems. They show that longer control horizons can improve suboptimality estimates, while final weights and open-loop duration introduce system-dependent trade-offs.

  • Simulation setup: The simulations use a linear inverted pendulum, a nonlinear inverted pendulum, and an arm–rotor–platform model to compare analytical results with numerical MPC behavior.The examples vary control horizons, optimization settings, and final weights.
  • Linear inverted pendulum: For the linear inverted pendulum, the closed loop is asymptotically stable for every tested m, and m_i > 1 can improve the suboptimality bound.The simulated α-curves approximately resemble the analytical curves, and larger final weights make α grow faster for small m.
  • Nonlinear inverted pendulum: For the nonlinear inverted pendulum, acceptable α values can be computed for m = 3 through 20 using repeated optimization at each sampling point.For ω = 1, α is negative for m = 1,...,4, whereas larger control horizons yield positive α values.
  • Nonlinear inverted pendulum: For the nonlinear inverted pendulum, increasing the final weight raises α across all considered control horizons.The paper identifies this as consistent with the stabilizing effect of terminal costs.
  • Arm–rotor–platform model: For the arm–rotor–platform model, increasing ω improves the suboptimality estimate, whereas increasing m gives little improvement for small m and decreases α for large m.The corresponding closed-loop cost decreases with ω but rises with m.
  • Arm–rotor–platform model: Longer control horizons keep the system open loop longer, which may be harmful under modelling errors or external perturbations.A detailed quantitative analysis of these effects remains under investigation.

10 Appendix

The appendix supplies technical lemmas and polynomial arguments used to derive the explicit suboptimality formula and its symmetry and monotonicity properties. It also establishes positivity and structural results under the controllability assumptions.

  • Auxiliary results for Section 5: The appendix proves technical lemmas and a corollary used to derive formula (26).These results include induction arguments and algebraic identities involving the coefficients and control-horizon parameters.
  • Auxiliary results for Section 5: When γ_m+1 > ω, the optimal solution of Problem 5.3 satisfies Aλ = b and λ > 0 componentwise.The proof uses the negative objective coefficient associated with λ_N−1 and feasibility-preserving perturbations.
  • Symmetry and monotonicity: Polynomial root arguments establish the symmetry properties needed for the αω_N,m analysis.The proof compares positive roots and uses induction in N, beginning with N = 2m + 1.
  • Symmetry and monotonicity: For exponential controllability, αω_N,m ≥ 0 when m < N − m, under the stated conditions on N and m.This result follows from Lemma 7.8 and the preceding polynomial analysis.
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