Explain how the performance of pH level control system can be compliant with environmental regulations and the treatment of wastewater. Provide two examples to explain the importance of a stable pH and its role in minimizing pollution in our ecosystem.

Control design of a pH neutralization plant for Sustainable Wastewater Treatment.

Explain how the performance of pH level control system can be compliant with environmental regulations and the treatment of wastewater. Provide two examples to explain the importance of a stable pH and its role in minimizing pollution in our ecosystem. [5 Marks]

For a neutral pH = 7 , using Laplace transforms and assuming zero initial
conditions, show that
𝑦̅ (𝑠) = 1
𝑠𝑇 + 1 [𝐶𝐴
∗ − 𝐶∗
𝑞∗ ∙ 𝑞𝐴𝑚𝑎𝑥
𝐶𝑚𝑎𝑥
𝑑̅ (𝑠) + 𝐶𝐵
∗ − 𝐶∗
𝑞∗ ∙ 𝑞𝐵𝑚𝑎𝑥
𝐶𝑚𝑎𝑥
𝑢̅ (𝑠)]
where the time constant is 𝑇 = 𝑉/𝑞∗
. [7 Marks]

Construct a block diagram depicting an open loop arrangement for the signals and transfer functions defined in 2). [3 Marks]

Assume zero initial conditions and a step input with magnitudes 𝛼 and 𝛽 for each of 𝑑̅ (𝑠) and 𝑢̅ (𝑠) respectively. Find the concentration output 𝑦(𝑡). [10 marks]

Using MATLAB, produce a unit step response for the output 𝑦(𝑡) and verify the result by comparing it with the analytical result derived in 4). Select the time scales so that both the transients and the steady state output are visible. [10 marks]

Assuming 𝑑̅ (𝑠) = 0, specify the parameter values that needs to be changed for the speed of the response to increase. Explain and justify your reasoning using appropriate mathematical functions and step response plots? [5 marks]

Assuming a unity negative feedback loop, derive the following transfer functions
a. 𝐺𝑟𝑦(𝑠)
b. 𝐺𝑑𝑦(𝑠)
c. 𝐺𝑟𝑒(𝑠)
d. 𝐺𝑑𝑒(𝑠)
[8 marks]

Verify that the closed-loop system is stable by graphically computing the poles and zeros. [4 marks]

Analytically calculate the steady state error due to the disturbance and the reference signal. What can you infer from the values obtained? [7 marks]

Prove that the output 𝑦(𝑡) will only track steady-state targets if there is an integrator in either a feedforward controller 𝐶(𝑠) or the plant 𝐺(𝑠). What other condition is required? In addition, using a mathematical derivation, specify the requirement for disturbance signals to be totally rejected, that is to have no steady-state impact on the output? [12 marks]

Use MATLAB to investigate how offset and performance varies as you change the scalar controller gain 𝐾𝑝. Give some generic conclusions based upon what you observe. [7 marks]