Page 867 - Mechatronics with Experiments

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MATLAB , SIMULINK , STATEFLOW, AND AUTO-CODE GENERATION 853
the tank is equal to the difference between in-flow rate and out-flow rate,
d(Volume in Tank)
= (In-flow rate) – (Out-flow rate) (A.16)
dt
d(A ⋅ y(t))
= Q (t) − Q out (t) (A.17)
in
dt
dy(t)
A = Q (t) − Q out (t) (A.18)
in
dt
The Q is controlled by the valve to be between zero and maximum flow that can go through
in
the valve, [0, Q ]. The Q is a function of the liquid level and the orifice geometry at
max out
the outlet. Let us approximate the relationship as a linear one, that is the higher the liquid
height is, the larger the out-flow rate,
1
Q (t) = ⋅ y(t) (A.19)
out
R
where R represent the orifice restriction as the resistance to flow. Then, the tank dynamic
model can be expressed as
dy(t) 1
A + ⋅ y(t) = Q (t) (A.20)
in
dt R
Let us consider a practical ON/OFF type controller with hysteresis. The controller either
fully turns ON or turns OFF the valve depending on the error between the actual and
measured liquid level. In order to make sure the controller does not switch the valve
ON/OFF at high frequency due to small changes in the liquid level, a small amount
of hysteresis is added in the control function. This type of controller is called a relay
with hysteresis and is commonly used in many automatic control systems such as home
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temperature control, liquid level control ([e , e ] range). In Simulink , the controller
max max
function is implemented with a hysteresis block. In mathematical terms, the controller
function is
e(t) = y (t) − y(t) (A.21)
d
i(t) = Relay (e) (A.22)
Hysteresis
The relay control function with hysteresis where the hysteresis band is [e max , e max ] range.
Flow rate can vary linearly between zero and maximum flow rate as a function of current
signal. Since current signal is either zero or maximum value, the flow rate will be either
zero or maximum flow.

Q (t) = K valve ⋅ i(t) (A.23)
in
= Q max ; when i(t) = i max (A.24)
= 0 ; when i(t) = 0 (A.25)

Let us simulate the liquid level control system for the following conditions. The system
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parameters are e max = 0.05, i max = 1.0A, Q max = 1200 l∕min = 20 l∕s = 0.02 m ∕s, A =
2
3
0.01 m , R = 500 m∕m ∕s. Consider the case that the desired liquid height is y (t) = 1.0m
d
which is commanded as a step function, and the initial height of the liquid is zero (empty
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tank). Figure A.28 shows the Simulink model and simulation results.
Example Consider a room or furnace temperature control system (Figure 1.8). We
need to consider room temperature, outside temperature (cold) and a heater. The heater is
controlled by a relay type controller with hysteresis. The room temperature is initially at
the same temperature as the outside temperature. The controller is set to increase the room
temperature to a higher level. The heater is controlled to regulate the room temperature. As