**Simultaneous Multi-Parametric Hybrid Model Predictive Control and Estimation with Application to the Intravenous Anaesthesia**

Ioana Nascu^{a,b}, Richard Oberdieck^{a,b}, Efstratios N. Pistikopoulos^{b}*

^{a}Dept. of Chemical Engineering, Centre for Process Systems Engineering (CPSE), Imperial College London SW7 2AZ, Lodnon, U.K

^{b}Artie McFerrin Department of Chemical Engineering, Texas A&M, College Station TX (^{*}stratos@tamu.edu)

**Keywords:** Anaesthesia, Explicit/Multi-Parametric Model Predictive Control, Hybrid Systems, Moving Horizon Estimation

The control of biomedical systems has become a very important and challenging research area during the last decades. One of the important challenges in controlling such systems is the presence of strong nonlinearities. These nonlinearities are included in the pharmacodynamic model, part of the mathematical model of the system which is usually represented by the Hill curve and it describes the relation between the concentration of the drug and the effect observed on the patient. Due to the S – shape characteristic of the Hill curve, a piecewise linearization around three points can be performed, resulting in a piece-wise affine formulation. The control of such a system translates in a hybrid model predictive control (hMPC) problem (Bemporad and Morari 1999) and the solution of a mixed-integer quadratic programming (MIQP) problem formulation. Since the online implementation of hMPC needs the online solution of an MIQP problem, this will introduce a high computational burden. When dealing with biomedical systems a fast control action is of great importance since the lack of it can affect the patient’s safety. A way to deal with this issue is by solving the hMPC problem explicitly offline via the solution of a multi-parametric mixed-integer quadratic programming (mp-MIQP) problem where the initial states are treated as parameters and the optimization problem is solved as a function thereof (Pistikopoulos 2009). For the solution of such problems, recently novel solution procedures have been developed (Oberdieck and Pistikopoulos 2014).

The implementation of hybrid explicit/multi-parametric MPC, and in general, MPC is that it is based on the assumption that accurate, noise-free information is available for all states of the system. However, in reality, state information needs to be inferred from the available, likely noisy, output measurements using a state estimator. Dealing with system constraints entails the simultaneous use of moving horizon estimation (MHE) which can be implemented in a multi-parametric fashion (Darby and Nikolaou 2007, Voelker, Kouramas et al. 2010). By using mp-MHE we will be able to estimate the state of each individual patient when dealing with biomedical processes. This is very important since it will be able to overcome the challenge of inter- and intra- patient variability that is found in such systems. Since most biomedical process can be treated as a hybrid systems due to the piece affine formulation of the Hill function, the mp-MHE will be reformulated into a novel hybrid mp-MHE formulation that will be implemented simultaneously with hybrid multi parametric model based predictive control.

This new development is integrated in PAROC, a comprehensive step-by-step framework and software solution for the general design, operational optimization and control of process systems (Pistikopoulos, Diangelakis et al. 2014). The framework includes a high-fidelity model of the system that is approximated if the model is too complicated using discrete time models in state space form via model order reduction techniques or system identification techniques. The system model in the state space form is used to formulate an optimization problem subject to the state space model and constraints. The state space model used by the hybrid mp-MPC controller is estimated using the newly developed hybrid mp-MHE. The resulting hybrid multi-parametric programming problem is solved with state-of-the-art techniques and the solution is validated against the original high fidelity model.

In this paper the presented framework is applied on the intravenous anaesthesia process. Intravenous is an important part of surgery and intensive care unit and it represents the reversed pharmacological state of the patient where hypnosis, analgesia and muscle relaxation are guaranteed (Bailey and Haddad 2005). During the last decades, there has been a lot of research on the development of PID tuning techniques, model based strategies using predictive, (Ionescu, Machado et al. 2014), (Hodrea, Morar et al. 2012), robust (Caiado Daniela, Lemos João et al. 2013), adaptive (Haddad, Hayakawa et al. 2003), (Nascu, Nascu et al. 2012) and multi-parametric MPC (Nascu, Lambert et al. 2014), (Krieger and Pistikopoulos 2014) for the control of depth of anaesthesia. Nevertheless, control of anaesthesia, is still not a trivial task and poses a variety of challenges that need to be addressed; from the presence of nonlinearities, inter- and intra- patient variability, multivariable characteristics, variable time delays, model analysis variability.

In order to control the depth of anaesthesia and for the first step of the framework, a model that captures the dynamical response of the patient is required. The PK-PD models most commonly used for Propofol are the 4^{th} order compartmental model described by (Schnider, Minto et al. 1998), (Minto, Schnider et al. 1997). The pharmacokinetic (PK, the linear part) – pharmacodynamic (PD, the nonlinear part) blocks denote compartmental models, used to represent the distribution of drugs in the body, i.e. mass balance. In each compartment the drug concentration is assumed to be uniform, assuming perfect and instantaneous mixing. The transport rate of the drug leaving the compartment is assumed to be proportional to the drug concentration. Once we have the state space of the model, the next step of the framework will be focused in designing the simultaneous hybrid control and estimation and their solution via mp MIQP. For this study, the simulations are performed on a set of 12 virtual generated patients under constraints and the presence of noise in the induction and maintenance phase. The performances of the controller are evaluated for the whole set of patients and under the effect disturbances and the existence of constraints. The implementation of this framework on the intravenous anaesthesia process represents an important step in the control of DOA since is able to deal with two major challenges in the process of anaesthesia: the nonlinearity by using the piece-wise linearization of the Hill curve and the inter and intra patient variability by using estimation techniques to obtain the states of each individual patients. Another advantage of the simultaneous hybrid mp-MPC and hybrid mp-MHE is that is able to overcome the noise that can corrupt the required data.

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