System MLD Model

A Discrete Hybrid Automata (DHA) is a connection of a finite state machine (FSM) and a switched affine system (SAS) through a mode selector (MS) and an event generator (EG) [373].

A set of linear dynamic systems defining the cooling system has been integrated into a state-space model. Their interconnection has been defined as a set of states, defined as an automaton (see Fig. 7.31) and the transition among these states has been defined according to a set of logical rules according to Table 7.5. This allows a DHA for the system to be defined. From this representation, an abstract repre­sentation in a set of constrained linear difference equations involving mixed-integer and continuous variables may be found that yield the equivalent Mixed Logical Dy­namical (MLD) model. Using the composed state-space model as defined in (7.22), the system was defined in HYSDEL, a tool for modeling used with the Matlab hy­brid toolbox for obtaining the MLD model for simulation and control. More infor­mation about HYSDEL can be consulted in [373].

The MLD modeling framework has been chosen from among others hybrid sys­tem techniques due to its extensive use in modeling hybrid systems and because the resulting model can easily be used in the future to develop controllers. From this framework, it is possible to model the evolution of continuous variables through linear dynamic discrete-time equations, the discrete variables through propositional logic statements and automata and the interaction of both. The main idea of this approach is to embed the logical part in the state equations by transforming boolean variables into 0-1 integers and by expressing the relationships as mixed-integer lin­ear inequalities. The MLD model, which is introduced in [373], related to the pre­vious state-space model, is

x(k + 1) = Ax(k) + Bi u(k) + B3z(k)

y(k) = Cx(k) (7.26)

E2S(k) + E3z(k) < Eiu(k) + E5

where x(k) є R" x {0,1}"’ is a vector of continuous and binaries states, g(k) є {0, 1}n, z(k) є Rrc represent auxiliary binary and continuous variables, respectively, y(k) є R^c x {0, 1}pi is the output (in this case, the absorption machine inlet tem­perature) and u(k) є R“c x {0,1}m’ are the inputs, including both discrete (V1, V3 and V4 valves and gas heater state signals) and continuous ones (irradiance, ambient temperature, flow and the system inlet temperature). Finally, A, B1, B3, C, E1, E2, E3, E5 are matrices of suitable dimensions.