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Time-optimal elevator control with higher-order state constraints: analysis and computation of boundary contacts
Dmitry Karamzin, Aleksandra Zhukova, Roman Chertovskih, Pedro A. Aguiar
TL;DR
The paper addresses the challenge of finding time-optimal elevator trajectories under higher-order state constraints, where boundary contacts can be numerous or accumulate. It derives specialized optimality conditions and a finite-dimensional computational method, then analyzes contact behavior and demonstrates the approach on a fourth-order elevator model.
Problem
Higher-order state constraints make the structure and efficient computation of time-optimal trajectories challenging, despite the possibility of arbitrarily many or accumulating boundary contacts.
Source: §I · p. 1Method
The paper specializes a higher-order constrained maximum principle and converts its optimality conditions into finite-dimensional nonlinear equations for standard solvers.
Source: §I · pp. 1–2Results
The analysis characterizes boundary-contact behavior, and the computational approach is demonstrated on a fourth-order elevator model with solutions independently checked by direct IPOPT optimization.
Source: §I · p. 2, §III · pp. 4–5, §V · p. 6Takeaways & Limitations
The formulation provides a procedure for efficiently computing candidate extremals using standard nonlinear equation solvers.
Source: §I · p. 2
Abstract
from arXiv · showhide
In this paper, we consider the classical time-optimal elevator problem in the presence of higher-order state constraints. The main contribution is a constructive indirect solution framework based on a Pontryagin Maximum Principle specifically formulated for higher-order constrained systems. First, we derive specialized optimality conditions and analyze the resulting structure of extremal trajectories. Second, we show that the infinite-dimensional optimal control problem can be transformed into a finite-dimensional system of nonlinear algebraic equations involving switching times, boundary-contact times, and multiplier parameters. This provides a computationally efficient procedure for calculating candidate optimal trajectories using standard nonlinear equation solvers. Third, we characterize the geometry of boundary contacts and explain the emergence of contact-chattering phenomena in higher-order constrained systems. In particular, we show that normal extremals cannot evolve along the constraint boundary for a positive amount of time, although sequences of boundary contacts may accumulate indefinitely. The proposed approach is demonstrated on a fourth-order elevator model, and the computed solutions are independently verified using a direct IPOPT-based optimization method.
I. INTRODUCTION
The paper studies minimum-time elevator motion under physical and operational constraints, including constraints on higher-order derivatives. It develops optimality conditions, analyzes boundary contacts, and proposes a finite-dimensional computational approach.
I. INTRODUCTION
- The elevator problem transfers a cabin between floors in minimum time while respecting physical and operational limits, in a broader class of motion-planning problems with derivative bounds.The model starts and ends at rest and controls motion through a bounded third derivative.Source: §I · p. 1
- A state constraint on x_i has order k+1−i, so the constraint on x_1 is order k and requires substantially more analysis than a first-order constraint.Source: §I · p. 1
- Higher-order constrained trajectories may have arbitrarily many boundary contacts, including accumulating contacts, while their structure and efficient computation remain challenging.Source: §I · p. 1
- The paper specializes a higher-order state-constraint maximum principle to the elevator problem and studies qualitative properties of the boundary-contact set.The analysis addresses finite and accumulating contact sequences.Source: §I · p. 1
- The computational method converts optimality conditions into finite-dimensional nonlinear equations for standard solvers and is independently validated against direct IPOPT optimization.Source: §I · p. 2
II. GENERAL THEORY
The general framework formulates time-optimal control with state constraints of differing orders and gives corresponding maximum-principle conditions. It also states existence and sufficiency results under specified assumptions.
II. GENERAL THEORY
- The framework minimizes final time subject to system dynamics, endpoint constraints, admissible controls, and state inequalities, with endpoint constraints assumed strictly interior to each state constraint.A feasible process satisfies the dynamics, endpoint conditions, and state inequalities; an optimal process has minimal final time.Source: §II · p. 2
- For each state constraint, Γ denotes time derivatives of its defining function along system trajectories, up to the order used in the conditions.Source: §II · p. 2
- The maximum principle uses a scalar λ, a costate function, and constraint multipliers; λ > 0 defines a normal extremal.The multiplier functions have specified regularity, and the conditions include a maximum condition and multiplier relations.Source: §II · pp. 2–3
- Every optimal process satisfies the stated maximum principle, derived by transforming the problem to one with twice the state-space dimension.Source: §II · p. 3
- A feasible process guarantees existence of a solution when the control set and endpoint set are compact and the dynamics satisfy the stated affine and growth conditions.Source: §II · p. 3
- Under convexity of the control, endpoint, and state-constraint sets, a normal extremal is a solution when the dynamics are linear and the constraint functions are convex.Source: §II · p. 3
III. ANALYSIS OF OPTIMALITY CONDITIONS IN THE ELEVATOR PROBLEM
Applying the higher-order optimality conditions identifies normal extremals as solutions to the elevator problem and constrains how their boundary contacts can be structured. The analysis distinguishes cases with finite contacts from a possible accumulating-contact regime.
III. ANALYSIS OF OPTIMALITY CONDITIONS IN THE ELEVATOR PROBLEM
- Any normal extremal represents a solution to the elevator problem, so computation can focus on normal extremals.The paper states that the assumptions of its solution theorem hold for Problem (1).Source: §III · p. 3
- For a normal extremal, the set Σ is nowhere dense, ruling out evolution along it over any interval.The proof excludes an interval in Σ by contradiction with the conservation law.Source: §III · p. 4
- Accumulating distinct contacts have a limit in Σ0; for s = 1, Σ is finite or countably many isolated points plus one limit, and Σ0 is empty or a singleton.The stated finite-or-accumulating structure is specific to s = 1; the paper notes it need not hold for s > 1.Source: §III · p. 4
- In Case A, extremals do not exist for s = 1; when k and s exceed 1, only abnormal extremals on a zero-measure set of boundary values may exist.The analysis concludes that these extremals require specific boundary conditions.Source: §III · p. 4
- In Case B, normal extremals generally exist when k − s > 1 and s > 1, while abnormal extremals are limited to a zero-measure set of boundary values.For N contacts, the construction counts 2N + k variables and the same number of algebraic equations.Source: §III · p. 5
- Case C requires Condition (17), and the paper’s stated conclusion is that only finite chattering is feasible within a normal situation.The accompanying Proposition 7 places Case C among abnormal extremals and restricts it to specific boundary values of zero measure.Source: §III · p. 5
IV. COMPUTATIONS
The computations turn the extremal conditions into a finite algebraic system for solving candidate trajectories, then apply the scheme to a fourth-order elevator example. The computed solution has three boundary contacts and is independently checked against IPOPT.
IV. COMPUTATIONS
- In Case B, normal extremals exist for almost all feasible boundary-value pairs, while pairs admitting abnormal extremals form a closed, measure-zero subset.Source: §IV · p. 5
- The construction integrates piecewise-constant controls from boundary values, with zeros of Δ(t) determining control switching points.The variables include the contact times and additional switching times, which satisfy contact and switching conditions.Source: §IV · pp. 5–6
- For fixed contact count N, the scheme solves a finite system with k + N(k + 4) + 1 equations and the same number of unknowns.The unknowns include switching and contact times, multiplier parameters, and trajectory values at contacts; the scheme is tried across j = 0,...,N.Source: §IV · pp. 5–6
- For symmetric boundary values and odd N, the scheme collapses; the two central intervals must be joined with all roots r_i placed there.Source: §IV · p. 6
- About 0.01 less than the indirect solution time, IPOPT found a similar solution structure; the indirect computation gives T = 8.516928415 and three boundary contacts.The stated equation-solving accuracy is approximately 10^-9.Source: Figure 3 · p. 6, §IV · p. 6
V. CONCLUSIONS
The paper presents a theoretical framework and optimality conditions for the time-optimal elevator problem, then uses the extremal-structure analysis to suggest a computational procedure. It applies that scheme to a four-dimensional elevator case with constrained velocity.
V. CONCLUSIONS
- The computational scheme is applied to a four-dimensional elevator problem with constrained velocity, and the resulting solutions are visualized.Source: §V · p. 6