QUESTION 7---Here is the extracted content from the image:
1. Compare open loop and closed loop control systems. Give example.
2. Obtain the impulse response of a series RL high pass filter.
3. A unity feedback system has the following forward path transfer function.
`G(s) = 180 / [s(s+6)]`
and `r(t) = 4t`. Determine the corresponding static error coefficient
and the steady state error.
4. Compare the features of transient and steady state part of a system response. Give an
example for a second order control system with natural frequency of 2 rad/s and
damping ratio of 0.5.
5. Explain absolute stability and relative stability of control systems.
6. Compare PD, PI and PID controllers.
7. Obtain the DC gain of a unity feedback control system whose overall transfer
function is given by
`T(s) = 10 / [(s+3)(s+5)]`
8. Starting with the principle of argument, state Nyquist stability criterion.
9. Define state transition matrix. Mention any four properties of it.
10. Define the terms state variable and state space. Mention any four distinct advantages
of state space representation.
PART B
(Answer one full question from each module, each question carries 14 marks)
Module -I
11. a) Obtain the differential equations governing the mechanical system shown below and
draw the force-current electrical analogous circuit.
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Let's solve this control systems problem about finding the DC gain. We have a unity feedback control system with overall transfer function T of s equals 10 divided by the product of s plus 3 and s plus 5. The DC gain represents the steady state gain when we apply a step input to the system.
Now let's solve this step by step. First, we apply the DC gain formula which states that DC gain equals T of 0. Next, we substitute s equals 0 into our transfer function. This gives us 10 divided by 0 plus 3 times 0 plus 5. Simplifying, we get 10 divided by 3 times 5, which equals 10 divided by 15. Finally, reducing this fraction gives us 2 divided by 3.
Let's understand what DC gain means physically. The DC gain represents the steady-state ratio between output and input. For our system with DC gain of 2 thirds, this means the steady-state output will be 2 thirds of the input magnitude. For example, if we apply a step input of 3 units, the steady-state output will be 3 times 2 thirds, which equals 2 units. This graph shows how the system responds to a step input, eventually reaching the steady-state value determined by the DC gain.
Let's verify our solution and discuss its applications. Our calculation is correct: substituting s equals 0 into the transfer function gives us the DC gain of 2 thirds. This dimensionless ratio tells us the steady-state behavior. The DC gain has many applications in control systems, including system design, steady-state error calculation, controller tuning, and performance specification. Therefore, our final answer is that the DC gain equals 2 thirds, or approximately 0.667.
To summarize what we've learned: The DC gain of a unity feedback control system is found by evaluating the transfer function at s equals zero. For our given transfer function, this calculation yields two-thirds. This DC gain represents the steady-state ratio between output and input, which is essential for control system analysis and design.