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Dynamic Tube MPC for Nonlinear Systems

Brett T. Lopez, Jean-Jacques E. Slotine, Jonathan P. How

arXiv:1907.06553v1eess.SYcs.SY

TL;DR

Nonlinear tube MPC can be suboptimal when its tube geometry is fixed, and calculating nonlinear tubes can impede the use of state-dependent uncertainty. DTMPC jointly optimizes tube geometry and the nominal trajectory using boundary layer sliding control. Simulations demonstrate robust obstacle avoidance with tube geometry adapted to obstacle proximity and uncertainty.

  • Problem

    Nonlinear tube MPC often fixes tube geometry, and the complexity of calculating nonlinear tubes can limit use of state-dependent uncertainty.

    Source: §I · p. 1

  • Method

    DTMPC augments nominal MPC with tube geometry dynamics derived using boundary layer sliding control, jointly optimizing the tube and open-loop trajectory.

    Source: §I · p. 1, C. Discussion · p. 4, A. Overview · p. 4

  • Results

    Simulations show DTMPC robustly avoids obstacles while changing tube geometry through control bandwidth or state-dependent uncertainty.

    Source: A. Optimized Tube Geometry · p. 7, §IX · p. 8

  • Takeaways & Limitations

    State-dependent uncertainty and combined tube and tracking-error dynamics can reduce conservativeness and improve optimization feasibility.

    Source: §IX · p. 8

Abstract

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Modeling error or external disturbances can severely degrade the performance of Model Predictive Control (MPC) in real-world scenarios. Robust MPC (RMPC) addresses this limitation by optimizing over feedback policies but at the expense of increased computational complexity. Tube MPC is an approximate solution strategy in which a robust controller, designed offline, keeps the system in an invariant tube around a desired nominal trajectory, generated online. Naturally, this decomposition is suboptimal, especially for systems with changing objectives or operating conditions. In addition, many tube MPC approaches are unable to capture state-dependent uncertainty due to the complexity of calculating invariant tubes, resulting in overly-conservative approximations. This work presents the Dynamic Tube MPC (DTMPC) framework for nonlinear systems where both the tube geometry and open-loop trajectory are optimized simultaneously. By using boundary layer sliding control, the tube geometry can be expressed as a simple relation between control parameters and uncertainty bound; enabling the tube geometry dynamics to be added to the nominal MPC optimization with minimal increase in computational complexity. In addition, DTMPC is able to leverage state-dependent uncertainty to reduce conservativeness and improve optimization feasibility. DTMPC is demonstrated to robustly perform obstacle avoidance and modify the tube geometry in response to obstacle proximity.

I. INTRODUCTION

The introduction motivates nonlinear tube MPC that adapts tube geometry alongside the nominal trajectory, while the related work and problem formulation identify why existing approaches make this difficult. DTMPC uses boundary layer sliding control to make online tube optimization practical and to exploit state-dependent uncertainty.

I. INTRODUCTION

  • Existing nonlinear tube MPC methods can produce overly conservative tubes, while designing tube geometry online is often too complex for real-time optimization.The proposed approach provides a simple, exact relationship among tube geometry, control parameters, and uncertainty.Source: §I · p. 1, §II · p. 2
  • DTMPC jointly optimizes tube geometry and the open-loop reference trajectory for nonlinear systems under uncertainty.Boundary layer sliding control links tube geometry, control parameters, and uncertainty, allowing tube dynamics to be added to the optimization with minimal computational increase.Source: §I · p. 1

III. PROBLEM FORMULATION

  • Boundary layer sliding control supports online tube optimization and state-dependent uncertainty, addressing fixed-tube and conservatism limitations in nonlinear tube MPC.The formulation assumes bounded model error and bounded disturbances in the span of the control input matrix.Source: §II · p. 2, §III · p. 2

IV. BOUNDARY LAYER SLIDING CONTROL

This section motivates time-varying boundary layer sliding control as an ancillary controller for DTMPC. Its system assumptions limit the stated scope to systems that are feedback linearizable or minimum phase.

A. Overview

  • Sliding mode control offers strong robustness, but discontinuous high-frequency control can be impractical, motivating continuous boundary layer variants.The section reviews time-varying boundary layer sliding control as an ancillary controller for DTMPC.Source: A. Overview · p. 3
  • The assumed system has the same number of outputs to be controlled as inputs.The system dynamics are expressed in the form required by the control design.Source: A. Overview · p. 3
  • The stated system assumption requires feedback linearizability or minimum-phase dynamics; extending DTMPC to more general nonlinear systems is future work.The authors describe the assumption as covering many systems, while identifying more general nonlinear systems as outside the current scope.Source: A. Overview · p. 3

B. Sliding Control

Sliding control defines a manifold whose convergence drives tracking error to zero, while a time-varying boundary layer smooths the control input. The section relates the boundary-layer dynamics to a cutoff frequency and describes the resulting control law.

B. Sliding Control

  • Reaching the sliding manifold makes tracking error converge to zero exponentially, but discontinuous control can excite unmodeled dynamics and shorten actuator life.A boundary layer around the switching surface is introduced to smooth the control input.Source: B. Sliding Control · p. 3
  • A time-varying boundary layer is made attractive by satisfying a differential equation, with η_i setting the convergence rate to the sliding surface.The layer is defined by |s_i| ≤ Φ_i.Source: B. Sliding Control · p. 3
  • The controller uses a saturation function, and outside the boundary layer its parameters must satisfy an attractiveness condition.The control design is presented in vector form using the stacked dynamics and input matrix for the output variables.Source: B. Sliding Control · p. 3
  • For |s| ≤ Φ, sliding-variable dynamics inside the boundary can be written as a first-order filter with cutoff frequency involving K(x*) and Φ.The passage introduces α as the desired cutoff frequency and states that the final control law is given by (8), (9), and (13).Source: B. Sliding Control · p. 3

C. Discussion

Boundary-layer sliding control bounds tracking error within a robust control invariant tube whose geometry is tied to the boundary-layer thickness. That relationship lets the controller account for state-dependent uncertainty and vary tube geometry with operating conditions.

C. Discussion

  • The boundary-layer thickness uniquely describes the robust control invariant tube’s geometry, and bounded error establishes that the tube is invariant.The tube geometry is obtained by taking the element-wise absolute value of the error vector and using the sliding-variable bound.Source: C. Discussion · p. 4
  • The tube geometry can be tighter when calculated from current rather than predicted tracking error.The paper identifies this as an additional option for the tube construction.Source: C. Discussion · p. 4
  • The tube geometry’s relation to uncertainty supports less-conservative tubes by allowing the controller and optimizer to use state-dependent uncertainty.The discussion notes that uncertainty bounds are often chosen as large constants when this relation is unavailable.Source: C. Discussion · p. 4
  • Increasing bandwidth α reduces boundary-layer thickness but also increases uncertainty’s influence, so bandwidth should respond to objectives and constraint proximity.The control bandwidth α shapes the sliding variable’s response to uncertainty and influences tube geometry and control gain.Source: C. Discussion · p. 4

A. Overview

The DTMPC overview introduces constraint tightening to account for uncertainty and avoid constraint violations. The tightened state and actuator constraints are guaranteed to satisfy the true constraints for all uncertainty realizations.

A. Overview

  • The overview identifies constraint tightening as necessary to prevent uncertainty from causing constraint violations.The stated state and actuator constraints are assumed to have a specified form, which the paper describes as not overly restrictive for many physical systems.Source: A. Overview · p. 4

B. Constraint Tightening

  • The modified state constraint guarantees satisfaction of the true state constraint for every uncertainty realization.The tightened constraint follows from bounding the state by the nominal state plus the tracking-error bound.Source: B. Constraint Tightening · p. 4
  • The modified actuator constraint guarantees satisfaction of the true actuator constraint for every uncertainty realization.The actuator bound accounts for feedback input in addition to the feedforward decision variable.Source: B. Constraint Tightening · p. 5

C. Optimized Tube Geometry

DTMPC makes tube geometry an optimization decision so it can respond to changing operating conditions. Its augmented tube dynamics keep bandwidth and tube thickness smooth.

C. Optimized Tube Geometry

  • Making tube geometry an optimization decision lets DTMPC adapt it to changing conditions, such as obstacle density.The paper contrasts this capability with other nonlinear tube MPC algorithms that lack an explicit relation between tube geometry, control bandwidth, and uncertainty.Source: C. Optimized Tube Geometry · p. 5
  • An artificial input v is added to the augmented tube dynamics, which are stable so long as α remains positive.The augmentation is intended to keep α and Φ smooth, maintaining continuity in the control signal.Source: C. Optimized Tube Geometry · p. 5

D. Complete Formulation

The complete DTMPC formulation jointly optimizes nominal motion and tube geometry subject to nonlinear dynamics and tightened constraints. It can use state-dependent uncertainty when selecting the open-loop trajectory.

D. Complete Formulation

  • The optimization includes nominal dynamics, bandwidth and tube-geometry dynamics, and tightened state and actuator constraints.The tube geometry is represented by Ω with matrices Ac and Bc, while the optimization uses internal variables marked with ˇ.Source: D. Complete Formulation · p. 5
  • DTMPC outputs an optimal open-loop feedforward input, trajectory, and control bandwidth.The formulation includes a quadratic state cost and terminal cost.Source: D. Complete Formulation · p. 5
  • Problem 1 jointly optimizes the open-loop trajectory and tube geometry, reducing conservativeness by using state-dependent uncertainty and constraint proximity.The formulation combines tube geometry and error dynamics and is non-convex because of the nonlinear dynamics.Source: D. Complete Formulation · pp. 5–6

VI. COLLISION AVOIDANCE MODEL

Collision avoidance tests whether DTMPC can use peak tracking error and state-dependent uncertainty to preserve safety in changing obstacle environments. The model and optimization account for nonlinear drag, disturbances, obstacle geometry, and tube dynamics.

A. Overview

  • Obstacle avoidance tests whether the optimizer can use peak tracking error for collision margins and adapt tube geometry to changing obstacle densities.The section identifies state-dependent uncertainty as another capability this test is intended to demonstrate.Source: A. Overview · p. 6

B. Model

  • The system is modeled as a double integrator with nonlinear drag, an unknown bounded drag coefficient, and bounded disturbance.The tracked state is inertial position; gravity, drag, and disturbance are included in the dynamics.Source: B. Model · p. 6

C. Collision Avoidance DTMPC

  • The collision-avoidance problem minimizes control effort while incorporating obstacle shape, location, and size into the DTMPC optimization.Its optimization dynamics include control bandwidth and tube-related variables.Source: C. Collision Avoidance DTMPC · p. 6

VII. SIMULATION ENVIRONMENT

Simulation evaluates optimized tube geometry in known and partially known obstacle environments, then tests state-dependent uncertainty. The optimized bandwidth changes along the trajectory, and Monte Carlo trials verify that the changing tube remains invariant.

VII. SIMULATION ENVIRONMENT

  • The three test cases vary environment knowledge and uncertainty: known obstacles, newly observed obstacles under receding-horizon planning, and state-dependent uncertainty with constant bandwidth.The study notes that optimizing bandwidth and leveraging state-dependent uncertainty together in a receding-horizon setting is possible, but separates them for clarity.Source: §VII · p. 6
  • The non-convex problem was solved by sequential convex programming and converged in three to four iterations in the tested cases.The optimization used a straight-line initial solution; a better initial guess might be needed for large perturbations to find feasibility.Source: §VII · p. 6

A. Optimized Tube Geometry

  • High bandwidth produces a compact tube near narrow gaps and slaloms, while low bandwidth produces a larger tube in open space.The trajectory and tube geometry are optimized together in the known obstacle environment.Source: A. Optimized Tube Geometry · p. 7
  • Across 1,000 disturbance-profile simulations, the closed-loop state stayed inside the tube as its geometry changed.Disturbance profiles were sampled uniformly from the disturbance set.Source: A. Optimized Tube Geometry · p. 7

B. Receding Horizon Optimized Tube Geometry

DTMPC adapts its trajectory and tube geometry as obstacles are revealed, and uses state-dependent uncertainty to alter speed and tube size. The conclusions identify simultaneous trajectory and tube optimization, uncertainty-aware feasibility, and reduced conservatism as supported capabilities.

B. Receding Horizon Optimized Tube Geometry

  • As obstacle sets are observed, DTMPC replans and shifts to high bandwidth near newly discovered obstacles while retaining low bandwidth in open space.The receding-horizon figures show successive plans as additional obstacles become known.Source: B. Receding Horizon Optimized Tube… · p. 7, Figure 4 · p. 7

C. State-Dependent Uncertainty

  • For the tested obstacle field, treating velocity-dependent uncertainty as merely bounded made the optimization infeasible with the same straight-line initialization.With state-dependent uncertainty, speed falls near obstacles, reducing uncertainty and tightening the tube, while speed and tube size increase in open regions.Source: C. State-Dependent Uncertainty · p. 8

IX. CONCLUSIONS

  • The conclusions report that DTMPC jointly optimizes trajectory and tube geometry, uses state-dependent uncertainty to reduce conservatism and improve feasibility, and can combine tube and error dynamics to further reduce conservatism.These properties are attributed to boundary layer sliding control; future work is to extend DTMPC to more general nonlinear systems.Source: §IX · p. 8
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