TOTAL MASS CONTROL IN UNCERTAIN COMPARTMENTAL SYSTEMS Claudia Sousa *,1 Teresa Mendonca ** Paula Rocha *** * Escola Superior de Educacao Jean Piaget, V. N. Gaia, Portugal ** Faculdade de Ciencias da Universidade do Porto, Porto, Portugal *** Universidade de Aveiro, Aveiro Abstract: In this paper we analyse the control of the total mass of compartmental systems, un- der the presence of uncertainties. We consider a control law that has a very good performance when applied to compartmental systems without uncertainties and show that, even when the system parameters are not exactly known, that good performance is maintained. In fact, for a wide class of compartmental systems of R 3 , it is possible to prove that, when we apply that control law to the real system, the total mass of the system converges to a positive constant value, which depends on the parameter uncertainties and that can be made arbitrarily close to the desired mass, provided that the uncertainties in the parameter values are sufficiently small. The obtained results are illustrated by simulations for the control of the administration of the neuromuscular relaxant drug atracurium to patients undergoing surgery. Keywords: Compartmental systems, positive control, uncertain systems, neuromuscular blockade control, full outflow connectedness. 1. INTRODUCTION Compartmental models have been successfully used to model biomedical and pharmacokinetical systems, see, for instance, (Godfrey, 1983) or (Jacquez, 1993). This kind of systems consist of a finite number of sub- systems, the compartments, which exchange matter with each other and with the environment. Such sys- tems are positive systems (i.e., systems for which the state and output variables remain nonnegative when- ever the input is nonnegative) and, as is well-known, in this case, the design of suitable control laws is more delicate, since one has to guarantee the positivity of the control input. In this framework, a nonnegative adaptive control law is proposed in (Haddad, 2003), in order to guarantee the partial asymptotic set-point stability of the closed loop system and a positive feed- back control law is proposed in (Bastin, 2002), in order to stabilise the total system mass at an arbitrary 1 This work was partially supported by FCT through the Unidade de Investigacao Matematica e Aplicacoes (UIMA), Universidade de Aveiro, Portugal. set-point. In (Magalhaes, 2005) the same positive con- trol law proposed in (Bastin, 2002) was used for the control of the neuromuscular blockade (see (Lemos, 1991), (Linkens, 1994) and (Mendonca, 1998)) of pa- tients undergoing surgery, but no analysis was made of the effect of parameter uncertainty in its performance. In this paper, we consider the control law proposed in (Bastin, 2002) and used in (Magalhaes, 2005), and analyse its performance for the control of the total mass of a wide class of compartmental systems, when the system parameters are not exactly known. Thus, we consider that the control law is tuned for a nominal process model that contains an additive uncertainty with respect to the real model, and analyse the be- havior of the total mass in the controlled system. It turns out that, in this case, bounds for the asymptotical mass offset can be easily expressed in terms of the system uncertainties. Moreover, the total system mass converges to a positive constant value that depends on the value of the desired mass and on the uncertainties of the system parameters.