3.1 Regular Basic Block

Regular Basic Blocks (RBB) can have a continuous state x and a discrete state z. If it does have an x and if u denotes its regular input, then, when the block is active over an interval of time, x evolves continuously according to

 

dx/dt = f(t,x,z,u,p,n_e) (1)

where f is a vector function, p is a vector of constant parameters and n_e is the activation code which is an integer designating the port(s) through which the block is activated. In particular, if activating input ports are $i_1,i_2,\dots,i_n$, then

On the other hand, activated by an event, the states x and z jump instantaneously according to the following equations:

  

x(t_e) = g_c(t_e,x(t_e^-),z(t_e^-),u(t_e),p,n_e) (2)

z(t_e) = g_d(t_e,x(t_e^-),z(t_e^-),u(t_e),p,n_e) (3)

where t_e denotes the event time. The discrete state z remains constant between any two successive events so z(t_e^-) can be interpreted as the previous value of z.

During activation times, the regular output of the block is defined by

 

y(t)=h(t,x(t^-),z(t^-),u(t),p,n_e) (4)

and is constant when the block is not active.

Finally, RBB's can generate activation signals of event type. If it is activated by an event at time t_e, the time of each output event is given by

 

t_evo = k(t_e,z(t_e),u(t_e),p,n_e) (5)

where t_evo is a vector of time, each entry of which corresponds to one activation output port. The absence of event corresponds to a time smaller than the current time. Event generations can also be pre-scheduled. Pre-scheduling of events can be done by setting the "initial firing variables of blocks with event output ports.