Synthesis of Limit Cycles and Reference Tracking via Switching Affine Systems

Technical Summary: Synthesis of Limit Cycles and Reference Tracking via Switching Affine Systems

Problem Statement
Periodic behavior is ubiquitous in fields ranging from biology to engineering, yet the mathematical analysis of nonlinear oscillators (e.g., Kuramoto, Van der Pol) remains constrained. Characterizing limit cycles, particularly regarding uniqueness and stability, is often feasible only in special cases. Existing data-driven techniques, such as machine learning, can approximate periodic dynamics but typically lack the structural properties required for formal verification. Furthermore, previous approaches to approximating limit cycles using switching or piecewise-affine systems (PAS) have been largely confined to planar ($\mathbb{R}^2$) settings with simple two-region partitions, or they fail to provide guarantees for global stability in higher dimensions.

The primary objective of this work is twofold:

1. To develop a synthesis method that approximates the limit cycles of nonlinear ordinary differential equations (ODEs) using switching affine systems in arbitrary dimensions ($\mathbb{R}^{n_x}$), ensuring the resulting limit cycle is globally stable.
2. To design a reference tracking controller for such systems that achieves asymptotic convergence to a periodic trajectory, even when the reference system itself lacks global stability guarantees.

Methodology

State Space Partitioning and Dynamics Synthesis
The proposed method begins with a set of sampled state points $\mathcal{F}$ collected along a stable limit cycle of a nonlinear system. The state space is partitioned into a finite number of polytopes $P_i$. Unlike previous planar methods, this approach handles arbitrary dimensions ($n_x = 2, 3, >3$) by defining a common center point $x_s$ and a set of boundary planes (or rays in 2D) that intersect at $x_s$.

• For $n_x=2$, lines are drawn from $x_s$ through selected sample points to define regions.
• For $n_x \geq 3$, a hyperplane is first determined to minimize the distance to the sample points, defining an intersection line $\Gamma$ passing through $x_s$. Regions are then formed by planes containing $\Gamma$ and specific sample points.

The dynamics within each region $P_i$ are modeled as $\dot{x} = A_i x + b_i + Bu(t)$. The synthesis is formulated as a constrained numerical optimization problem. The objective is to minimize the error between the model's trajectory and the sampled data points. Crucially, the optimization enforces contractivity to guarantee global stability:

1. Continuity: The vector fields must be continuous across switching boundaries.
2. Common Lyapunov Function: A positive-definite matrix $Q$ must exist such that $A_i^T Q + Q A_i \prec 0$ for all regions. This ensures the system is contractive, guaranteeing that any trajectory converges to a unique limit cycle with the same period as the input signal $u(t)$.

To handle the nonlinearity of the matrix exponential in the cost function, the method employs Taylor series approximations and countermeasures (such as ensuring small transition times between regions and bounding the singular values of $A_i$) to maintain accuracy without excessive computational complexity.

Reference Tracking
For the tracking problem, the paper addresses scenarios where the identified periodic reference trajectory may not be globally stable (e.g., if stability constraints were omitted during identification to improve approximation accuracy). The control law is designed as $u_c(t) = u_r(t) + K_i x_c(t) + w_i - (K_j x_r(t) + w_j)$.
To achieve asymptotic tracking ($\lim_{t\to\infty} ||x_c(t) - x_r(t)|| = 0$), the authors propose a synthesis scheme using multiple Lyapunov functions rather than a single common one.

• A piecewise quadratic Lyapunov function $V_i$ is defined for each region containing the reference trajectory.
• Stability conditions are derived to ensure the Lyapunov function decreases strictly within regions and remains bounded across switching boundaries, provided the dwell time in each region exceeds a minimum threshold $T_{min}$.
• This approach relaxes the requirement for a common matrix $Q$, allowing for less conservative convergence results compared to methods relying on a single Lyapunov function.

Key Contributions

1. High-Dimensional Synthesis: The paper extends limit cycle approximation via switching affine systems from planar, two-region partitions to higher-dimensional state spaces ($\mathbb{R}^{n_x}$) with general partitions.
2. Global Stability Guarantee: By leveraging the contraction principle and enforcing a common Lyapunov function during the synthesis optimization, the method guarantees that the generated limit cycle is globally stable and unique.
3. Reference Tracking with Multiple Lyapunov Functions: The work introduces a tracking controller for periodic switching affine systems that utilizes multiple Lyapunov functions. This is presented as the first application of such a technique to periodic reference tracking in this context, offering a less conservative alternative to common Lyapunov approaches.
4. Constructive Partitioning Rules: The paper provides specific geometric rules for partitioning state spaces in 2D, 3D, and higher dimensions based on sampled data, ensuring the resulting polytopes are compatible with the synthesis constraints.

Results
The effectiveness of the proposed methods is demonstrated through numerical examples:

• 2D and 3D Approximation: The synthesis method successfully reconstructed limit cycles from sampled data in both 2D and 3D spaces. Simulations showed that trajectories starting from various initial conditions (both inside and outside the limit cycle) converged to the unique, globally stable limit cycle. Increasing the order of the Taylor approximation for the matrix exponential significantly reduced the approximation error.
• Reference Tracking: A numerical example in 2D demonstrated the tracking controller's ability to asymptotically follow a periodic reference trajectory. The reference system was explicitly constructed to lack global stability guarantees (violating the common Lyapunov condition), yet the controlled system successfully tracked the reference from a distant initial state, with the tracking error converging to zero.

Significance and Claims
The authors claim that this work addresses a critical gap in the systematic investigation of oscillatory phenomena, particularly in applications like biological circadian rhythms where understanding stability and synchronization is indispensable. By providing a tractable model class (switching affine systems) that preserves essential dynamic properties (stability, uniqueness) while allowing for rigorous verification, the method facilitates the analysis of complex nonlinear oscillators.

The paper modestly notes that while the optimization problem is non-convex and global solutions are not guaranteed a-priori, any feasible solution yields a convergent trajectory. Furthermore, the authors acknowledge that the requirement for a common Lyapunov function can be restrictive for approximation accuracy; thus, they propose a two-step strategy: identify the model without strict stability constraints to maximize data fit, and then synthesize a tracking controller using the multiple Lyapunov function approach to ensure convergence. Future work is identified as exploring partitioning schemes without a common center point and coupling multiple oscillators.