Time-optimal control for the induction phase of anesthesia

Abstract

This paper deals with the control of the induction phase of anesthesia. The objective during this first phase is to bring the patient from its awake state to a final state corresponding to some given depth of anesthesia, measured by the BIS (Bispectral index), within a minimum time. This optimal time control strategy is addressed by means of the maximum principle of Pontryagin. Furthermore, since the anesthesia model presents multiple time scale dynamics that can be split in two groups : fast and slow and since the BIS is a direct function of the fast ones, we only consider the optimal control of fast states. The final state of the slow dynamics is let free. The synthesis approach of the optimal control is detailed then tested in the case of a nominal patient.

Asymmetric State Feedback for Linear Plants With Asymmetric Input Saturation

“…Finally, we may rearrange (19a) to obtain (9a), by using again (20) and also using (15b) as follows:…”

“…Most existing works dealing with the design of a state feedback (locally or globally) stabilizing the origin of a linear system with input saturation address the simplified problem of symmetric saturation (see, e.g., the extensive surveys in the books [9], [21], [17]). Nevertheless, in many practical situations (see, e.g., the applications in [18], [20]), the two limits are substantially different and the typical solution adopted in the literature is to disregard achievable control performance by conservatively focusing on the smallest limit.…”

Switched control strategy including time optimal control and robust dynamic output feedback for anaesthesia

“…The characterisation of the optimal control u * is detailed in [39]. Denote λ ∈ R n a co-state vector used to define the Hamiltonian H associate to problem (18):…”

“…where λ 1 * is the first component of vector λ*. From a practical point of view, the initial values of the co-state vector λ* being unknown, an iterative approach has to be used [23]. On the other hand, considering that the Hamiltonian should be equal to zero at any point of the optimal trajectory and that u(t = 0 + ) = U max (such that the trajectory is increasing from x(0) = − x e < 0 to x f (T f ) = 0), it follows that…”

“…Indeed, in a previous work [40], we presented a strategy to design a robust state feedback controller for the hypnosis maintenance phase taking into account the saturation of the actuator, the multi-time scale dynamics in the anesthesia model and the variability of the patient. On the other hand, the control problem of the induction phase has been addressed in [39], using an optimal control strategy. In the current paper, the objective is to propose a full strategy mimicking the medical practice by designing a switched controller involving a first (optimal) open-loop control for the induction phase followed by a closed-loop control for the maintenance phase.…”