Technical Summary: Feedback Linearisation with State Constraints

Problem Formulation

Feedback Linearisation (FBL) is a mature technique for transforming input-affine nonlinear control systems into linear systems, enabling the use of linear design methods like pole placement. However, a significant limitation arises when dealing with state constraints. While FBL successfully linearizes the system dynamics, it often transforms simple state constraints in the original nonlinear system into complex, nonlinear constraints in the linearized domain. This transformation diminishes the primary advantage of FBL, as controlling the resulting linear system under these complex constraints becomes as difficult as controlling the original nonlinear system.

Existing approaches to this problem typically rely on optimization-based methods (e.g., Model Predictive Control) or Linear Matrix Inequalities (LMIs), which impose significant computational burdens unsuitable for online applications. Other methods are often restricted to simple "box" constraints. The paper addresses the challenge of designing an FBL controller that guarantees state constraint satisfaction without requiring online optimization, applicable to general time-varying constraints.

Methodology

The proposed methodology follows a three-step workflow: State Augmentation, Integral Control Modification, and Switching FBL Implementation.

1. State Augmentation and Constraint Capture

The core of the approach involves augmenting the system dynamics to encapsulate state constraints before applying FBL.

• Slack Variables: A time-varying slack variable $Z(t)$ is introduced such that the constraint $\phi(t, x(t)) \leq 0$ is satisfied if $Z(t) \geq 0$. Specifically, the constraint is rewritten as $\phi(t, x(t)) + \frac{1}{2}z^2(t) = 0$.
• Analytical Derivation: The paper derives an analytical expression for the time derivatives of this augmented equation. By taking derivatives until the system input appears explicitly (determined by the relative degree), the original input $u$ is expressed as a function of the slack variable's derivatives (treated as a pseudo-input $w$).
• Sequential Augmentation: For systems where the number of constraints exceeds the number of inputs (MIMO with $r > m$), a sequential augmentation strategy is employed. Constraints are captured one subset at a time, iteratively expanding the state space.
• Theoretical Guarantee: Theorem 1 establishes that for any integrable pseudo-input $w$, the resulting trajectory of the original system satisfies the state constraints. This is because as the system approaches the constraint boundary ($z \to 0$), the actuation authority on the constraint vanishes, preventing violation.

2. Integral Controller Structure

To ensure the pseudo-input $w$ is integrable (a requirement for Theorem 1) and to handle steady-state errors, the augmented system is further modified with an integral controller structure.

• An integrator state $\xi$ is added, where $\dot{\xi} = \tilde{w}$ (a new virtual input).
• The pseudo-input is defined as a bounded function of the integral state, $w = s_\beta(\xi)$, where $s_\beta$ is a saturation-like function.
• This modification ensures that the input to the augmented system remains bounded and integrable, preserving the constraint satisfaction guarantees.

3. Handling Ill-Defined Relative Degrees and Switching

A critical challenge identified is that the augmented system exhibits an ill-defined relative degree at the boundaries of the state constraints (where slack variables become zero). This singularity prevents the direct application of standard FBL, which requires an invertible decoupling matrix.

• $\epsilon$-Numerical Relative Degree ($\epsilon$-NRD): To address this, the paper introduces the concept of $\epsilon$-NRD. Instead of relying on exact zero/non-zero Lie derivatives, the method uses a threshold $\epsilon$ to determine the relative degree based on whether Lie derivative coefficients are numerically significant.
• Switching Strategy: The controller is synthesized based on the local $\epsilon$-NRD. As the system trajectory evolves and crosses regions where the $\epsilon$-NRD changes (typically near constraint boundaries), the controller switches to a new FBL law synthesized for the current local relative degree.
• Back-Substitution: To ensure practical implementation, the augmented states (slack variables) are algebraically eliminated from the final controller expression using the derived relationships. The final controller depends only on the original system states and the integral states.

Key Contributions

1. Scalable Constraint Handling: The paper extends previous work on state augmentation to cases where the constraint dimension exceeds the input dimension. It provides a novel sequential augmentation method and derives the first analytical expression for the transformation of constraints into system dynamics (Lemma 1).
2. Fundamental Design Principles (Theorem 1): The authors establish that the integrability and invertibility of virtual controllers are key to certifying constraint satisfaction. They prove that the proposed augmentation method guarantees constraint satisfaction for any integrable input.
3. Resolution of Ill-Defined Relative Degrees: The paper theoretically proves that the augmentation leads to ill-defined relative degrees at constraint boundaries (Proposition 1). It overcomes this by integrating a switching controller strategy based on $\epsilon$-NRD, allowing FBL to be applied even when the relative degree changes dynamically.
4. Practical Implementation Workflow: Unlike previous theoretical results, the paper provides a definitive, closed-form implementation method (Algorithm 1 and 2) that bridges theory and practice. It details how to remove augmented state dependencies via back-substitution, enabling the controller to run on the original system with integral states.

Results

The paper validates the methodology through numerical experiments on Single-Input Single-Output (SISO) systems:

• Example 1: A system with a time-varying quadratic constraint. The authors manually derived the controller, demonstrating that the proposed method stabilizes the system to the origin while strictly satisfying the constraint. In contrast, a classical FBL controller (without augmentation) violated the constraint. The experiment also highlighted the effect of pole placement on convergence speed and the necessity of controller switching near the boundary.
• Example 2: A Lorenz-like chaotic system with multiple constraints and a time-varying reference. The proposed switching FBL controller successfully tracked the reference while remaining within the feasible region. The classical FBL controller, ignoring constraints, exhibited significant overshoot and violated the state constraints.

Significance and Claims

The paper claims to offer a general, optimization-free FBL method capable of handling state constraints. Its primary significance lies in:

• Avoiding Online Optimization: Unlike MPC, the proposed controller is synthesized offline (or via simple switching logic) and does not require solving nonlinear programming problems online, making it suitable for real-time applications.
• Handling General Constraints: The method is not limited to simple box constraints but can accommodate general time-varying state constraints.
• Robustness to Singularities: By utilizing the $\epsilon$-NRD and switching logic, the method addresses the fundamental issue of ill-defined relative degrees at constraint boundaries, a known limitation in applying FBL to constrained systems.

The authors note that while the underlying augmentation theory applies to MIMO systems, the practical implementation involving the switching logic is currently restricted to SISO systems due to the tractability of the switching conditions. Future work is identified as extending this switching framework to MIMO systems.