Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  present a non-model based approach for asymptotic, locally exponentially stable attainment of the optimal open-loop control sequence for unknown, discrete-time linear systems with a scalar input, where not even the dimension of the system is known. This control sequence minimizes the finite-time horizon cost function, which is quadratic in the measured output and in the input.  make no assumptions on the stability of the unknown system, but  do assume that the system is reachable. The proposed algorithm employs the multi-variable discrete-time extremum seeking approach to minimize the cost function, extending results established for the scalar discrete-time extremum seeking method. Simulation results show that the Hessian's condition number, used as a measure of the optimization problem's level of difficulty, increases with both the system's level of instability and the length of the finite horizon for a scalar system. Thus,  suggest solving ll-conditioned, shorter time horizon optimal control problems to obtain good initial control estimates for longer time horizon problems.  also show that the algorithm accommodates input constraints by employing the projection operator.
['Paul Frihauf', 'Miroslav Krstic', 'Tamer Basar']
I am writing a research paper on a idea Finite-horizon LQ control for unknown discrete-time linear systems via extremum seeking
[{"from": "human", "value": "I am writing a research paper on a idea Finite-horizon LQ control for unknown discrete-time linear systems via extremum seeking"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  present a non-model based approach for asymptotic, locally exponentially stable attainment of the optimal open-loop control sequence for unknown, discrete-time linear systems with a scalar input, where not even the dimension of the system is known. This control sequence minimizes the finite-time horizon cost function, which is quadratic in the measured output and in the input.  make no assumptions on the stability of the unknown system, but  do assume that the system is reachable. The proposed algorithm employs the multi-variable discrete-time extremum seeking approach to minimize the cost function, extending results established for the scalar discrete-time extremum seeking method. Simulation results show that the Hessian's condition number, used as a measure of the optimization problem's level of difficulty, increases with both the system's level of instability and the length of the finite horizon for a scalar system. Thus,  suggest solving ll-conditioned, shorter time horizon optimal control problems to obtain good initial control estimates for longer time horizon problems.  also show that the algorithm accommodates input constraints by employing the projection operator."}]