TY - GEN
T1 - The Minimal Directed Information Needed to Improve the LQG Cost
AU - Sabag, Oron
AU - Tian, Peida
AU - Kostina, Victoria
AU - Hassibi, Babak
N1 - Publisher Copyright:
© 2020 IEEE.
PY - 2020/12/14
Y1 - 2020/12/14
N2 - We study a linear quadratic Gaussian (LQG) control problem, in which a noisy observation of the system state is available to the controller. To lower the achievable LQG cost, we introduce an extra communication link from the system to the controller. We investigate the trade-off between the improved LQG cost and the consumed communication (information) resources that are measured with the conditional directed information. The objective is to minimize the directed information over all encoding-decoding policies subject to a constraint on the LQG cost. The main result is a semidefinite programming formulation for the optimization problem in the finite-horizion scenario where the dynamical system may have time-varying parameters. This result extends the seminal work by Tanaka et al., where the direct noisy measurement of the system state at the controller is assumed to be absent. As part of our derivation to show the optimality of an encoder that transmits a Gaussian measurement of the state, we show that the presence of the noisy measurements at the encoder can not reduce the minimal directed information, extending a prior result of Kostina and Hassibi to the vector case. Finally, we show that the results in the finite-horizon case can be extended to the infinite-horizon scenario when assuming a time-invariant system, but possibly a time-varying policy. We show that the solution for this optimization problem can be realized by a time-invariant policy whose parameters can be computed explicitly from a finite-dimensional semidefinite program.
AB - We study a linear quadratic Gaussian (LQG) control problem, in which a noisy observation of the system state is available to the controller. To lower the achievable LQG cost, we introduce an extra communication link from the system to the controller. We investigate the trade-off between the improved LQG cost and the consumed communication (information) resources that are measured with the conditional directed information. The objective is to minimize the directed information over all encoding-decoding policies subject to a constraint on the LQG cost. The main result is a semidefinite programming formulation for the optimization problem in the finite-horizion scenario where the dynamical system may have time-varying parameters. This result extends the seminal work by Tanaka et al., where the direct noisy measurement of the system state at the controller is assumed to be absent. As part of our derivation to show the optimality of an encoder that transmits a Gaussian measurement of the state, we show that the presence of the noisy measurements at the encoder can not reduce the minimal directed information, extending a prior result of Kostina and Hassibi to the vector case. Finally, we show that the results in the finite-horizon case can be extended to the infinite-horizon scenario when assuming a time-invariant system, but possibly a time-varying policy. We show that the solution for this optimization problem can be realized by a time-invariant policy whose parameters can be computed explicitly from a finite-dimensional semidefinite program.
UR - http://www.scopus.com/inward/record.url?scp=85099884030&partnerID=8YFLogxK
U2 - 10.1109/CDC42340.2020.9304490
DO - 10.1109/CDC42340.2020.9304490
M3 - Conference contribution
AN - SCOPUS:85099884030
T3 - Proceedings of the IEEE Conference on Decision and Control
SP - 1842
EP - 1847
BT - 2020 59th IEEE Conference on Decision and Control, CDC 2020
PB - Institute of Electrical and Electronics Engineers
T2 - 59th IEEE Conference on Decision and Control, CDC 2020
Y2 - 14 December 2020 through 18 December 2020
ER -