Optimal Adaptive Control of Linear Stochastic Systems with Quadratic Cost Function
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917775573778432 |
|---|---|
| author | Liu, Nian Zhao, Cheng Tan, Shaolin Lü, Jinhu |
| author_facet | Liu, Nian Zhao, Cheng Tan, Shaolin Lü, Jinhu |
| contents | In this paper, we consider the adaptive linear quadratic Gaussian control problem, where both the linear transformation matrix of the state $A$ and the control gain matrix $B$ are unknown. The proposed adaptive optimal control only assumes that $(A, B)$ is stabilizable and $(A, Q^{1/2})$ is detectable, where $Q$ is the weighting matrix of the state in the quadratic cost function. This condition significantly weakens the classic assumptions used in the literature. To tackle this problem, a weighted least squares algorithm is modified by using random regularization method, which can ensure uniform stabilizability and uniform detectability of the family of estimated models. At the same time, a diminishing excitation is incorporated into the design of the proposed adaptive control to guarantee strong consistency of the desired components of the estimates. Finally, by utilizing this family of estimates, even if not all components of them converge to the true values, it is demonstrated that a certainty equivalence control with such a diminishing excitation is optimal for an ergodic quadratic cost function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_09250 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal Adaptive Control of Linear Stochastic Systems with Quadratic Cost Function Liu, Nian Zhao, Cheng Tan, Shaolin Lü, Jinhu Optimization and Control In this paper, we consider the adaptive linear quadratic Gaussian control problem, where both the linear transformation matrix of the state $A$ and the control gain matrix $B$ are unknown. The proposed adaptive optimal control only assumes that $(A, B)$ is stabilizable and $(A, Q^{1/2})$ is detectable, where $Q$ is the weighting matrix of the state in the quadratic cost function. This condition significantly weakens the classic assumptions used in the literature. To tackle this problem, a weighted least squares algorithm is modified by using random regularization method, which can ensure uniform stabilizability and uniform detectability of the family of estimated models. At the same time, a diminishing excitation is incorporated into the design of the proposed adaptive control to guarantee strong consistency of the desired components of the estimates. Finally, by utilizing this family of estimates, even if not all components of them converge to the true values, it is demonstrated that a certainty equivalence control with such a diminishing excitation is optimal for an ergodic quadratic cost function. |
| title | Optimal Adaptive Control of Linear Stochastic Systems with Quadratic Cost Function |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2409.09250 |