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Robust Closed-Form Control for MIMO Nonlinear Systems under Conflicting Time-Varying Hard and Soft Constraints (extended version)

Farhad Mehdifar, Charalampos P. Bechlioulis, Dimos V. Dimarogonas

TL;DR

This work tackles enforcing time-varying hard safety constraints and soft performance constraints on uncertain high-relative-degree MIMO nonlinear systems. It develops a robust closed-form velocity-level controller by consolidating constraints into two scalar functions using Log-Sum-Exp, employing reciprocal barrier functions, and applying a dynamic relaxation to resolve hard-soft conflicts, all within a low-complexity backstepping-like framework that does not require uncertainty bounds. A semi-global stability result guarantees forward invariance of the hard-constraint set intersected with a relaxed soft-set, with a global stability extension achieved by a shifting function that decouples tuning from initial conditions. The approach is demonstrated on a unicycle robot with moving obstacles, showing safe constraint enforcement, finite-time soft-constraint satisfaction when feasible, and resilience to constraint degeneracies, with potential impact on safe-by-design control for complex mechanical systems.

Abstract

This paper introduces a novel robust closed-form control law to handle time-varying hard and soft constraints in uncertain high-relative-degree nonlinear MIMO systems. These constraints represent spatiotemporal specifications in mechanical systems' operational space, with hard constraints ensuring safety-critical requirements and soft constraints encoding performance or task objectives. Initially, all constraints are consolidated into two separate scalar time-varying hard and soft constraint functions, whose positive level sets define feasible regions. A closed-form control law is developed to enforce these constraints using appropriately designed reciprocal barriers and nonlinear transformation functions. When conflicts between hard and soft constraints arise, the control law prioritizes hard constraints by virtually relaxing soft constraints via a dynamic relaxation law. Notably, the proposed control law maintains low complexity by avoiding approximation schemes for coping with system uncertainties. Simulation results confirm the effectiveness of the proposed method.

Robust Closed-Form Control for MIMO Nonlinear Systems under Conflicting Time-Varying Hard and Soft Constraints (extended version)

TL;DR

This work tackles enforcing time-varying hard safety constraints and soft performance constraints on uncertain high-relative-degree MIMO nonlinear systems. It develops a robust closed-form velocity-level controller by consolidating constraints into two scalar functions using Log-Sum-Exp, employing reciprocal barrier functions, and applying a dynamic relaxation to resolve hard-soft conflicts, all within a low-complexity backstepping-like framework that does not require uncertainty bounds. A semi-global stability result guarantees forward invariance of the hard-constraint set intersected with a relaxed soft-set, with a global stability extension achieved by a shifting function that decouples tuning from initial conditions. The approach is demonstrated on a unicycle robot with moving obstacles, showing safe constraint enforcement, finite-time soft-constraint satisfaction when feasible, and resilience to constraint degeneracies, with potential impact on safe-by-design control for complex mechanical systems.

Abstract

This paper introduces a novel robust closed-form control law to handle time-varying hard and soft constraints in uncertain high-relative-degree nonlinear MIMO systems. These constraints represent spatiotemporal specifications in mechanical systems' operational space, with hard constraints ensuring safety-critical requirements and soft constraints encoding performance or task objectives. Initially, all constraints are consolidated into two separate scalar time-varying hard and soft constraint functions, whose positive level sets define feasible regions. A closed-form control law is developed to enforce these constraints using appropriately designed reciprocal barriers and nonlinear transformation functions. When conflicts between hard and soft constraints arise, the control law prioritizes hard constraints by virtually relaxing soft constraints via a dynamic relaxation law. Notably, the proposed control law maintains low complexity by avoiding approximation schemes for coping with system uncertainties. Simulation results confirm the effectiveness of the proposed method.

Paper Structure

This paper contains 14 sections, 3 theorems, 58 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Main Results
  4. Smooth Consolidated Hard and Soft Constraint Functions
  5. Hard and Soft Constraints Satisfaction Strategy
  6. Low-Complexity Controller Design
  7. Stability Analysis
  8. From Semi-Global to Global Stability Results
  9. Simulation Results
  10. Conclusions
  11. Definition of $\mathcal{C}^1$ switch function
  12. Gradients and partial time derivatives of $\alpha_{h}$ and $\alpha_{s}$
  13. Proof of Theorem \ref{['th:main']}
  14. On Properties of $\rho_{s}^{\text{r}}(t)$

Key Result

Theorem 1

Consider the MIMO nonlinear system eq:sys_dynamics_mimo subject to time-varying hard and soft constraints eq:hard_soft_constraints. Let $\rho_{s}(t)$ in eq:soft_const_set_lower_bound be designed according to Subsection subsec:control_strategy (i.e., Step 1-a) and eq:rho_relax_dyn. Moreover, suppose

Figures (6)

  • Figure 1: Simulation results of Example 1: Initial condition dependent tuning of the control law.
  • Figure 2: Simulation results of Example 1: Initial condition free control law and a larger $\delta_\gamma$.
  • Figure 3: Simulation results of Example 2.
  • Figure 4: Simulation results for Example 3: existence of an undesired equilibrium in the closed-loop system under stationary constraints.
  • Figure 5: Simulation results for Example 3: escaping the undesired equilibrium under a mildly time-varying soft constraint
  • ...and 1 more figures

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • proof
  • Remark 4
  • Remark 5
  • Remark 6
  • Corollary 1
  • proof
  • ...and 3 more