dual-envelope-mpc

name: dual-envelope-mpc description: Nonlinear Model Predictive Control with dual-envelope constraints for distributed drive systems. Use when: vehicle dynamics control, MPC implementation, envelope-based stability analysis, autonomous drifting, distributed drive electric vehicles, phase plane analysis, saddle point modeling, or constrained nonlinear control systems.

Dual-Envelope Constrained Nonlinear MPC

Overview

This skill provides methodology for implementing dual-envelope constrained Nonlinear Model Predictive Control (NMPC) for distributed drive electric vehicles. The approach uses phase plane analysis with saddle point modeling to define stability envelopes that guide controller design for extreme maneuvers like autonomous drifting.

Key Innovation: Addresses the issue where control input coupling reshapes phase plane topology and shifts saddle point location, invalidating open-loop equilibrium-based envelope structures when used for closed-loop control.

Core Concepts

1. Saddle Point Coordinate Model

Combines nonlinear tire model with handling diagram to establish saddle point coordinates that explicitly account for:

Road adhesion coefficient (μ)
Longitudinal velocity (vx)
Front wheel steering angle (δf)
Additional yaw moment (Mz)

Key Insight: Saddle points define the transition between stable and unstable regions in the phase plane (slip angle β vs yaw rate ωz).

2. Dual Envelope Framework

Outer Envelope:

Defines recoverable set under bounded control inputs
Based on convergence tendency toward saddle points
Constraints: front wheel steering angle + additional yaw moment
Purpose: Ensures state remains recoverable during drifting maneuvers

Inner Envelope:

Characterizes non-drifting stability region
Associated with unsaturated tire forces
Normal vehicle operation stability boundary
Purpose: Defines normal driving stability region

3. Nonlinear Tire Model

Essential for capturing:

Tire force saturation
Coupling between longitudinal and lateral forces
Relationship between slip angle and tire forces

4. Phase Plane Analysis

Phase plane (β, ωz) topology:

Saddle points mark stability transition
Control input coupling reshapes topology
Envelope boundaries define operational regions
Closed-loop vs open-loop dynamics differ

Implementation Workflow

Step 1: Establish Tire Model

Use nonlinear tire model capturing:

Combined slip (longitudinal + lateral)
Force saturation behavior
Road adhesion effects

Common models:

Pacejka tire model
Dugoff tire model
Brush tire model

Step 2: Construct Handling Diagram

Plot tire force characteristics:

Lateral force vs slip angle
Account for longitudinal force coupling
Identify saturation regions

Step 3: Calculate Saddle Point Coordinates

Solve for equilibrium points considering:

Current road adhesion coefficient μ
Vehicle velocity vx
Control inputs (δf, Mz)
Tire force balance equations

Mathematical formulation:

β* = saddle point slip angle coordinate
ωz* = saddle point yaw rate coordinate

Depends on: μ, vx, δf, Mz

Step 4: Build Dual Envelope

Outer Envelope Construction:

Simulate state trajectories under bounded control inputs
Identify convergence regions toward saddle points
Define envelope boundary from recoverable state set
Ensure envelope respects control constraints

Inner Envelope Construction:

Identify tire saturation boundaries
Define region where tire forces remain unsaturated
Characterize normal stability region
Set boundary at transition to drifting regime

Step 5: Implement NMPC Controller

Formulate optimization problem:

minimize: tracking error + control effort
subject to:
  - Vehicle dynamics constraints
  - Outer envelope constraint (recoverability)
  - Inner envelope constraint (if needed)
  - Control input bounds (δf, Mz)
  - State constraints

Use sequential quadratic programming (SQP) or other NMPC solvers.

Step 6: Hardware-in-the-Loop Validation

Validate controller performance:

Compare with NMPC without envelope constraints
Measure tracking errors: speed, sideslip angle, yaw rate
Verify smoother convergence toward drift saddle point
Test under varying road adhesion conditions

Key Benefits

Metric	Improvement vs NMPC without envelopes
Steady-state speed tracking error	-33.07%
Steady-state sideslip angle error	-71.18%
Steady-state yaw rate error	-31.27%
Peak tracking error	-63%
Convergence smoothness	Improved

Control Input Coupling Effects

Critical Issue: Open-loop envelopes derived from equilibrium analysis become invalid for closed-loop control because:

Control input coupling reshapes phase plane topology
Saddle point location shifts under closed-loop dynamics
Envelope structure changes with active control
Boundaries must account for control constraints

Solution: Extended dual envelope framework explicitly accounts for:

Bounded steering angle constraints
Bounded yaw moment constraints
Control-induced topology changes
Closed-loop convergence behavior

Distributed Drive Advantages

Distributed drive electric vehicles provide:

Superior yaw moment control (individual wheel torque)
Faster response than traditional steering-based control
Additional yaw moment Mz as control input
Suitable for extreme maneuvers (drifting)

Phase Plane Topology

State Variables

β: vehicle sideslip angle
ωz: yaw rate

Equilibrium Points

Stable equilibria: attract nearby states
Saddle points: bifurcation points between stability regimes
Unstable equilibria: repel nearby states

Envelope Boundaries

Outer envelope: recoverability boundary
Inner envelope: normal stability boundary
Saddle points: transition markers

Applications

Autonomous Drifting

Controlled extreme maneuvers
Stable drift state maintenance
Smooth transition to/from drift

Vehicle Stability Control

Active safety systems
Envelope-based intervention triggers
Prevention of unrecoverable states

Advanced Driver Assistance

Performance driving assistance
Stability margin monitoring
Predictive control intervention

Mathematical Framework

Vehicle Dynamics

Simplified 2-DOF model:

dβ/dt = -(vx/lf+lr)[lf*Cf*cos(δf) + lr*Cr + Mz/vx]
       / (m*vx) + ωz
       
dωz/dt = [lf*Cf*cos(δf) - lr*Cr - Mz] / Iz

Where:

lf, lr: front/rear axle distances
Cf, Cr: front/rear tire cornering stiffnesses
m: vehicle mass
Iz: yaw moment of inertia
vx: longitudinal velocity

Envelope Constraint Formulation

Outer envelope: g_outer(β, ωz, δf, Mz) ≤ 0 Inner envelope: g_inner(β, ωz) ≤ 0

Constraints in NMPC optimization:

# Check recoverability
if g_outer(current_state, control_inputs) > 0:
    # State outside recoverable envelope
    # Need to steer back toward saddle point
    
# Check normal stability (optional)
if g_inner(current_state) > 0:
    # Operating in drifting regime
    # Ensure outer envelope constraint active

Implementation Considerations

Computational Requirements

NMPC requires real-time optimization
Envelope checking adds computational overhead
Hardware-in-the-loop testing essential
Suitable solvers: qpOASES, ACADO, HPIPM

Tuning Parameters

Envelope boundary margins
Control weight vs tracking weight
Prediction horizon length
Sample time for discretization

Safety Margins

Conservative envelope boundaries recommended
Account for sensor noise and delays
Consider model uncertainty
Validate across operating conditions

Research Paper Reference

Paper: "Dual-Envelope Constrained Nonlinear MPC for Distributed Drive Electric Vehicles Drifting Under Bounded Steering and Direct Yaw-Moment Control"

Authors: Yurun Gan, Ziyu Song, Jing Yang, et al.
arXiv ID: 2604.07342
Published: April 8, 2026
Categories: eess.SY (Systems and Control)
Link: https://arxiv.org/abs/2604.07342

Related Skills

control-systems: General control systems theory
vehicle-dynamics: Vehicle dynamics modeling
mpc-basics: Fundamentals of MPC implementation
phase-plane-analysis: Stability analysis techniques