UNIVERSITI MALAYSIA PERLIS Pusat Pengajian Kejuruteraan Bioproses
ERT 210/4 Process Dynamics and Control Tutorial 4: Transfer Function and State-Space Models
4.2 Consider the following transfer transfer function: function: G ( s )
=
Y ( s ) U ( s )
=
5 10 s +1
(a) What is is the steady-s steady-state tate gain? gain? =5
(f) If u (t ) δ (t ) , that is, the unit impulse at t = 0, what is the output when t → ∞ ? =
(b) What is is the time time constant? constant? = 10 s) = 2/ s s, what is the value of the (c) If U ( s output y(t) when t → ∞ ?
s), what is the value of (d) (d) For the the same same U ( s the output when t = 10? What is the output when expressed as a fraction of the new steady-state value?
s) = (1-e-s)/ s s, that is, the unit (e) If U ( s rectangular pulse, what is the output when t → ∞ ?
(g) If u (t ) = 2 sin 3t , what is the value of the output when t → ∞ ?
4.5 For the process modeled by
2
dy1
dt dy 2 dt
y1
= −2
=
−
3 y 2
4 y1 − 6 y 2
+
+
2u1
2u1
+
4u 2
Find the four transfer functions relating the outputs ( y1, y2) to the inputs ( u1, u2). The u and y are deviation variables.
4.8 A surge tank in Fig. E4.8 is designed with a slotted weir so that the outflow rate, w, is proportional to the liquid level to the 1.5 power; that is, w
= Rh
1.5
where R is constant. If the single stream enters the tank with flow rate wi, find the transfer function H' ( s)/W i( s). Identify the gain and all time constants. Verify units. The cross-sectional area of the tank is A. Density ρ is constant.
