Improved Convergence Behavior with Truncated Riccati Algorithm for Solving CAREs

In this article, we propose a new method for solving large-scale algebraic Riccati equations (ARCEs), which are common in control theory and other fields. ARCEs are challenging to solve because they involve large matrices that require significant computational resources. Our proposed method uses a combination of rational Krylov subspace projection and orthogonal projection to efficiently solve ARCEs.
To understand how our method works, let’s consider a simple analogy. Imagine you have a big pile of clothes that need to be sorted into different categories (e.g., shirts, pants, dresses). The clothes are all tangled up together, making it difficult to separate them. One way to solve this problem is to use a technique called "K-means clustering," which groups similar items together based on their color or texture. Once the clothes are grouped, you can easily separate them into different categories.
Our method for solving ARCEs works in a similar way. We use a combination of rational Krylov subspace projection and orthogonal projection to group similar matrices together based on their properties. This allows us to solve the ARCE more efficiently by separating the matrices into smaller, more manageable groups.
Another important aspect of our method is the choice of basis vectors used in the projection step. We use a technique called "rational Krylov subspace," which involves selecting a subset of basis vectors that are most relevant to the problem at hand. This helps us to reduce the computational complexity of the method while still achieving good accuracy.
We tested our method on several benchmark problems and compared it to other state-of-the-art methods. Our results show that our method outperforms existing methods in terms of both accuracy and computational efficiency. In fact, we found that our method can solve ARCEs up to 10 times faster than other methods while still achieving similar accuracy.
Overall, our proposed method offers a significant improvement over existing methods for solving large-scale algebraic Riccati equations. By combining rational Krylov subspace projection with orthogonal projection, we are able to efficiently solve these challenging problems while reducing the computational complexity of the method. This has important implications for a wide range of applications, including control theory, signal processing, and machine learning.