DESIGN FOR A BRAIN 19/17 



where the z's are the selected x's, and z's the corresponding as's. 

 These have only to be solved for %,..., z n in terms of 

 »!, . . . , Z n and the equations are in canonical form. So the 

 new system is also state-determined (by S. 19/9). 



This transformation implies that in a state- determined system we 

 can avoid direct reference to some of the variables provided we use 

 derivatives of the remaining variables to replace them. 



Example: x x = x x — x 2 \ 



x 2 = oXi -\- x 2 J 



can be changed to omit direct reference to x 2 by using x x as a new 

 independent variable. It is easily converted to 



dx 1 /dt = ij 



dxjdt = — 4>x ± + 2a?j 



which is in canonical form in the variables x, and x* 



» 



19/17. Systems which are isolated but in which effects are 

 transmitted from one variable to another with some finite delay 

 may be rendered state-determined by adding derivatives as 

 variables. Thus, if the effect of x 1 takes 2 units of time to reach 

 x 2 , while x 2 s effect takes 1 unit of time to reach x v and if we write 

 x(t) to show the functional dependence, 



then dx x (t)/dt =/iK(0, oc 2 (t - 2)Y 



dx 2 (t)/dt=f 2 {x 1 (t- 1), x 2 (t)} 



!} 



This is not in canonical form; but by expanding x x (t — 1) and 

 x 2 {t — 2) in Taylor's series and then adding to the system as 

 many derivatives as are necessary to give the accuracy required, 

 we can obtain a state-determined system which resembles it as 

 closely as we please. 



19/18. If a variable depends on some cumulative effect so that, 

 say, x 1 =f< (f>(x 2 )dt I, then if we put <j>(x 2 )dt = y, we get the 



equivalent form 



dxjdt=f(y) 

 dy/dt = <f>(x 2 ) 

 dxjdt = . . . etc. 



which is in canonical form. 



250 



