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Hi all, got a Control question here, and I'm struggling with what I assume is a simple algebraic step. Thanks in advance!
1. Homework Statement
A closed loop control system governs the level of water in a tank (H(s)) to meet a target height (Hi(S)). The flow of water into the tank is controlled by a transducer that feeds the current level of the tank into a differencing junction that works out the error (H(s)Hi(s)). The flow rate of water pumped in is proportional to this error, with gain K.
The flow out of the tank is also constrained by a linearized flow restrictor, with flow out equal to the height/constant (Qd=H(s)/R).
There is also an additional flow into the tank from a separate pipe, with flow rate Qd.
The question is to find the transfer function, time constant and steady state gain. I've attached a diagram.
2. Homework Equations
3. The Attempt at a Solution
So far I've gotten as far as the governing equation:
Qi + Qd  Qo = A.dH(t)/Dt
Laplace: Qi(s) + Qd(s)  Qo(s) = A.s.H(s)
Where Qi = Flow in
Qd = Additional disturbance flow
Qo = Flow out
A = XSection area of tank
Using the information about the individual components this goes to:
K.Hi(s)  K.H(s) + Qd(s)  H(s)/R = A.s.H(s)
The correct way to describe transfer function (Checked with answer booklet):
H(s)=(R.K.Hi + R.Qd(s)) / (R.k +R.A.s + 1)
But I can't get the hang of expressing it in a way that would allow me to get the specific time function, ie I can't arrange it into form H(s)/Hi(s)=u/(1+Ts) where u is the SS Gain and T is the time constant.
Any help with this step would be fab!
Regards
1. Homework Statement
A closed loop control system governs the level of water in a tank (H(s)) to meet a target height (Hi(S)). The flow of water into the tank is controlled by a transducer that feeds the current level of the tank into a differencing junction that works out the error (H(s)Hi(s)). The flow rate of water pumped in is proportional to this error, with gain K.
The flow out of the tank is also constrained by a linearized flow restrictor, with flow out equal to the height/constant (Qd=H(s)/R).
There is also an additional flow into the tank from a separate pipe, with flow rate Qd.
The question is to find the transfer function, time constant and steady state gain. I've attached a diagram.
2. Homework Equations
3. The Attempt at a Solution
So far I've gotten as far as the governing equation:
Qi + Qd  Qo = A.dH(t)/Dt
Laplace: Qi(s) + Qd(s)  Qo(s) = A.s.H(s)
Where Qi = Flow in
Qd = Additional disturbance flow
Qo = Flow out
A = XSection area of tank
Using the information about the individual components this goes to:
K.Hi(s)  K.H(s) + Qd(s)  H(s)/R = A.s.H(s)
The correct way to describe transfer function (Checked with answer booklet):
H(s)=(R.K.Hi + R.Qd(s)) / (R.k +R.A.s + 1)
But I can't get the hang of expressing it in a way that would allow me to get the specific time function, ie I can't arrange it into form H(s)/Hi(s)=u/(1+Ts) where u is the SS Gain and T is the time constant.
Any help with this step would be fab!
Regards
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