Page 124 - Modern Control Systems

98 Chapter 2 Mathematical Models of Systems

where AH and AQi are small deviations from the equilibrium (steady-state) values.
The Taylor series expansion about the equilibrium conditions is given by

H=f(H tQ{)=f{H\Q*) + ^ (H - H*) (2.116)
<?.=<?*

+ // = H (Gi - e*) +

where
dikiVn + k 2Q{) 1 k x
dH // = //* dH //=w* 2
0.=0- 0i=0* tf*
and
_#1 dfaVH + k 2Q x) = k?
dQi //=H» dQi
Qi=Q*
Using Equation (2.114), we have
Q*
H* =
P v 2gA 2
so that
2
A 2 gp
dH / / = / / • ' A x Q*
<?.=o*
It follows from Equation (2.115) that

H = AH,
/
since H* is constant. Also, the term ( / / * , Q*) is identically zero, by definition of
the equilibrium condition. Neglecting the higher order terms in the Taylor series ex-
pansion yields
2
A 2 gp 1
AH = - AH + •AQi. (2.117)
PA
Equation (2.117) is a linear model describing the deviation in water level AH from
the steady-state due to a deviation from the nominal input mass flow rate A ^ .
Similarly, for the output variable Q 2 we have

Qi = Q*2 + AQ 2 = h(H,Q 1) (2.118)

h(H*,Q*) + ^ AH + — - AQi,
1 // = //* X
01=0- ^
where AQ 2 is a small deviation in the output mass flow rate and
dh
dH //=//*
a=0*