Convergence Analysis of Natural Power Method and Its Applications to Control
Abstract
This paper analyzes the discrete-time natural power method, demonstrating its convergence to the dominant $r$-dimensional subspace corresponding to the $r$ eigenvalues with the largest absolute values. This contrasts with the Oja flow, which targets eigenvalues with the largest real parts. We leverage this property to develop methods for model order reduction and low-rank controller synthesis for discrete-time LTI systems, proving preservation of key system properties. We also extend the low-rank control framework to slowly-varying LTV systems, showing its utility for tracking time-varying dominant subspaces.
Summary
This paper investigates the convergence properties of the discrete-time natural power method for general square matrices, demonstrating its convergence to the dominant *r*-dimensional subspace corresponding to the *r* eigenvalues with the largest absolute values. This is a key distinction from the Oja flow, which converges to the subspace corresponding to the eigenvalues with the largest *real* parts. The authors provide a convergence analysis for the natural power method, generalizing existing results that were previously limited to symmetric positive semi-definite matrices. They then leverage this convergence property to develop methods for model order reduction (MOR) and low-rank controller synthesis for discrete-time linear time-invariant (LTI) systems, proving preservation of key system properties like reachability and observability. Furthermore, the paper extends the low-rank control framework to slowly-varying linear time-varying (LTV) systems. The authors illustrate the utility of their approach for tracking time-varying dominant subspaces in an LTV system, using an observer-based controller with gains designed instantaneously via pole assignment. While the method performs well for systems with slower variations, performance degrades when the convergence rate of the natural power method slows down due to closely spaced eigenvalues. This work provides a foundation for applying the natural power method in control applications, particularly in scenarios where extracting the dominant subspace based on eigenvalue magnitudes is crucial.
Key Insights
- •The discrete-time natural power method converges to the *r*-dominant subspace corresponding to the *r* eigenvalues with the largest *absolute* values, unlike the Oja flow which targets eigenvalues with the largest *real* parts.
- •Theorem 3 provides a convergence proof for the natural power method applied to general square matrices, expanding beyond the previously analyzed case of symmetric positive semi-definite matrices.
- •The paper proposes Algorithm (9), a modified version of the natural power method, which provably converges to an element of the target subspace *U(d)r* under certain conditions, providing a stationary point for the algorithm.
- •Lemma 6 demonstrates that the complement subspace *U⊥* obtained from the natural power method corresponds to the fast subspace of the system, making it useful for analyzing singularly perturbed systems.
- •Proposition 7 shows that the proposed model order reduction scheme preserves reachability and observability of the reduced system and maintains the same transfer function as the original system.
- •Numerical experiments (Figure 1) illustrate that the convergence rate of the natural power method is affected by the proximity of eigenvalues, with slower convergence observed when |λm[A]| and |λm+1[A]| are close. For example, convergence is slow when *α = 0.9* in the example matrix *Aα*.
- •The application to LTV systems highlights a limitation: the controller's performance degrades when the system varies too quickly, causing the natural power method to struggle to track the dominant subspace accurately.
Practical Implications
- •The MOR method can be used to simplify complex discrete-time LTI systems while preserving key properties like reachability, observability, and the transfer function, making it useful for control design and analysis.
- •The low-rank controller synthesis approach for LTV systems offers a potential method for controlling time-varying systems with reduced computational complexity, particularly when the system variations are slow.
- •The convergence analysis of the natural power method provides theoretical guarantees for its use in extracting dominant subspaces in various applications, including signal processing, system identification, and control.
- •Future research can focus on developing error bounds for the MOR method, exploring more sophisticated control design techniques for LTV systems based on the natural power method, and investigating applications in areas like adaptive control and online system identification.
- •Practitioners can use the natural power method and the proposed MOR techniques to design controllers for high-dimensional discrete-time systems, potentially leading to more efficient and implementable control solutions.