nhcho91
commented
2 years ago - (A,B) is controllable iff (A, BB') is controllable (according to Lancaster and Rodman)
- (A,B) is controllable iff (A', B') is observable (according to duality)
- For p.s.d. A, the principal square root of A (denoted by sqrt(A)) which is p.s.d. (hence symmetric) is unique.
Let Q = Q' is a p.s.d. matrix, as usual in the LQ optimal control problems.
Then,
- (A, sqrt(Q)) is observable iff (A', sqrt(Q)')=(A', sqrt(Q)) is controllable (according to logic 2)
- (A', sqrt(Q)) is controllable iff (A', sqrt(Q)sqrt(Q)')=(A',Q) is controllable (according to logic 1)
- (A',Q) is controllable iff (A,Q')=(A,Q) is observable
Therefore, (A, sqrt(Q)) is observable iff (A, Q) is observable.
위와 같이 생각이 되는데, Lancaster & Rodman 책의 전개가 어떻게 된 것인지가 명확히 와닿지 않네요.
seong-hun
선형 시스템의 최적 제어 문제를 다룬 논문들을 보다보면 다음과 같은 square-root을 사용한 obsevability 조건들이 보입니다.
Examples of the square-root observability condition
From Jiang and Jiang (2012)
From Lee et al. (2014)
For linear systems
하지만 선형 시스템에 대해서 다음과 같은 정리가 있습니다 (Lancaster & Rodman, 1995).
따라서 제 생각에는 적어도 선형 시스템에 대해서는
(\sqrt{Q}, A)obervability와(Q, A)observability가 동치인 것 같은데 맞을까요?PS. 이 질문과 관련이 있는 내용인데, 저는 충분조건인 줄 알았는데, 위 정리에 따르면 동치관계가 있는 것 같습니다.
References
Jiang, Y., & Jiang, Z.-P. (2012). Computational adaptive optimal control for continuous-time linear systems with completely unknown dynamics. Automatica, 48(10), 2699–2704. https://doi.org/10.1016/j.automatica.2012.06.096
Lee, J. Y., Park, J. B., & Choi, Y. H. (2014). On integral generalized policy iteration for continuous-time linear quadratic regulations. Automatica, 50(2), 475–489. https://doi.org/10.1016/j.automatica.2013.12.009
Lancaster, P., & Rodman, L. (1995). Algebraic Riccati equations. Clarendon Press ; Oxford University Press.