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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!

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.

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

## 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.

## Homework Equations

## 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