PC Seminar 1 Summary
By- 110122066(Navya)
Seminar Summary: Height Control in Non-Linear Conical Tank System(2,66,94,112)
Introduction This seminar explores height control for a non-linear conical tank system,
addressing challenges introduced by the tank’s non-linear dynamics and examining effective
control strategies.
Non-Linearity Analysis The seminar begins by analyzing the conical tank’s non-linear flow rate
and height relationship, highlighting the need for advanced control to maintain stability and the
desired response.
Controller Design Focus areas include:
Root Locus Analysis: Demonstrating how controller gain influences system stability.
Control Strategies: Examining Proportional (P), Proportional-Integral (PI), Proportional-
Derivative (PD), and Proportional-Integral-Derivative (PID) control methods.
Disturbance Response The seminar evaluates how each control approach handles
disturbances, focusing on settling time, overshoot, and stability.
Gain-Scheduled PID: Where controller parameters adjust dynamically based on operating
conditions, improving system performance.
Comparative Analysis: All controllers are compared to assess stability, robustness, and
response to disturbances, highlighting strengths and areas for potential improvement.
Conclusion: The seminar concludes by emphasizing insights into the effectiveness of various
controllers, with a particular focus on Gain-Scheduled PID’s potential in optimizing performance
in non-linear control systems.
Seminar Summary of gate questions by Ram, Gautam and haneesh.(56,104,120)
1.bode plot
The question is about finding the transfer function of a system given its Bode magnitude plot.
poles at each frequency. Using these values, the gain constant 𝐾.
The slopes of different sections of the plot are analyzed to determine the number of zeros and
K is calculated, and the transfer function is derived.
2.pid controller
The question asks for the PID controller parameters kp,ki and kd based on a given
characteristic equation. A transfer function with unity feedback is provided, and the solution
involves deriving the characteristic equation for the closed-loop system. By comparing the
derived characteristic equation with a target polynomial, the PID parameters are determined.
3.pi converter
The question involves a controller that outputs a 4-20 mA signal to control motor speed, ranging
linearly from 140 to 600 rpm. It asks for (a) the current corresponding to 310 rpm and (b) the
control output as a percentage of the 4-20 mA range. The solution includes calculating the linear
relationship and expressing the output as a percentage.
Seminar Summary of Non Interacting tank system by Denham (34) , Muthu (40) , Rupak
(86) and Avinash (96) :
The presentation focuses on non-interacting tank systems, commonly used to model processes
like the human respiratory system. A first-order system involves energy flowing through a single
capacity, while a second-order system has two capacities in series, where one tank influences
the other's dynamics, creating a more complex behavior.
Key definitions include deviation variables, representing the difference from steady-state values,
and transfer functions, which relate output and input deviations in Laplace terms. The problem is
to determine the transfer function for a liquid level system with nonlinear resistances in non-
interacting tanks.
MATLAB plots illustrate the step responses of individual tanks and a combination of the three
systems. Additionally, the presentation compares linear and nonlinear system plots, highlighting
their behavioral differences.
Gate questions by 12, 28, 50, 52
1 Stability of Control Systems
This problem involves analyzing the stability of a given control system using specific
parameters. The focus is on determining whether the system remains stable under various
conditions.
2 Control System Response to Disturbances
This problem examines how a control system responds to external disturbances. It requires
evaluating the system's output and stability when subjected to sudden changes in input.
3 Parameter Tuning in PID Controllers
This problem focuses on tuning the proportional (Kp) and integral (Ki) gains of a PID controller.
The objective is to find optimal values that enhance system performance and stability.
4 Effect of Control Parameters on System Behavior
This problem presents scenarios with different values of Kp and Ki (e.g., Kp = Ki = 1 and Kp =
Ki = -1) to analyze how these parameters affect the overall behavior and stability of the control
system.
Gate questions by 04,84,88,98
1.Minimum Phase System Analysis
The document discusses a system with specific corner frequencies (5 rad/s, 40 rad/s, and 100
5❑
10
rad/s), calculates poles, and derives an open-loop transfer function, G(s)= . It
(1+ 0.2 s) ¿ ¿
includes phase angle calculations for different frequencies and MATLAB code for transfer
function and Bode plot analysis.
2.Laplace Transform Problem
Another section works through solving for constants AAA and BBB in the expression
A B
Y (s)= + , followed by finding the inverse Laplace transform and calculating a specific
s 1+ s
time-domain response value.
3.Peak Overshoot and Damping Ratio Calculations
The assignment includes solutions for GATE 2021 questions on calculating damping ratios (ζ)
based on peak overshoot values and relating them to transfer function parameters.
4.Transfer Function and Disturbance Response
It covers the derivation and simplification of a transfer function, followed by simulations to
illustrate system responses to input and output disturbances.
Gate question based on Servo Problem by 72
According to the question, the disturbance transfer function Gd can be neglected because the
problem is focused on the system's response to a unit step change in the setpoint, which makes
it a servo problem.
We compute the transfer function of the process by simplifying the block diragram, and it
comes out to be = 2/15 divided by (s+4/5). Now since there is a unit step change, we can write
the output h(s) = Transfer function multiplied by 1/s. On applying the final value theorem, we get
the required output h to be 1/12 . When we put this h in the output equation h(t) (after taking the
inverse laplace transform of h(s)), we get the final time = 0.87 minutes, which is the required
answer. For further analysis the problem can be made a regulatory problem if the unit step
change is made in the disturbance instead of setpoint, and after performing the calculations in
similar fashion, we get t = infinity.
Gate problem(08,16,76,90)-
1)The question involves analyzing a position control system with a given block diagram. In Part
(a), we need to determine the damping ratio and steady-state error with no derivative feedback.
In Part (b), we must find suitable values of parameters and to achieve a damping ratio of 0.7
without affecting the steady-state error.
2)(a) Frequency for 90° phase lag: Find the frequency where the phase of G_p(s) = 1/{(s+1)
(2s+1)} is -90°. Substitute s = j*omega and calculate the phase angle.
(b) Determine k for gain margin: Use the open-loop transfer function G(s) = G_c(s)G_p(s) with
G_c(s) = k/s.Calculate k such that the open-loop gain margin is 2.5.
(c) Steady-state errors: Use the final value theorem with the closed-loop transfer function for
unit-step and unit-ramp inputs to determine the steady-state errors.
3)The question involves finding transfer function from the given block diagram using block
diagram reduction and finding a stable range using Routh Hurwitz table.
4)
i). Determine the asymptote as omega tends to 0:
- Substitute s = j omega in G(s) = 1/{s(2s+1)(s+1)}
- For small omega (i.e., as omega tends to 0 , analyze the behavior of G(j omega) by
approximating each term in the denominator.
- The asymptotic behavior will show how G(j omega) approaches zero as omega tends to 0.
ii). Find the range of k for stability in terms of omega_{PC} :
- For stability, the Nyquist plot must not encircle the critical point (-1,0).
- Identify the phase crossover frequency omega_{PC} , where the phase of G(j omega)
reaches -180°.
- Calculate the gain k such that the magnitude |G(j\omega_{PC})| at this frequency keeps the
system stable (typically ensuring k. |G(j\omega_{PC})| < 1
Gate Question (64,68,48)
Q2-The problem focuses on analyzing a cascade control system to evaluate its disturbance
rejection capabilities, specifically addressing the offset caused by a unit step change in the
secondary disturbance, the system’s inner and primary loops are analyzed for stability and
offset. Through simplifications and the use of characteristic equations, the ultimate gain for
marginal stability in the primary loop is determined, revealing a significant difference between
cascade control and conventional feedback; the final value theorem is employed to calculate the
offset
Q3- The solution uses the Ziegler-Nichols method to tune a PID controller for the transfer
function. A proportional controller is applied, and the gain is adjusted to 7.5 to induce sustained
oscillations, with an oscillation period of 1.975 seconds. Using Ziegler-Nichols formulas, the PID
parameters are calculated as Kp = 4.5, Ki = 4.557, and Kd = 1.111. An initial simulation with
these values results in oscillations, which are minimized by increasing the derivative gain Kd to
2.111, thereby reducing overshoot and improving system stability.
Q4- The solution includes the calculation of gain and phase margins for a control system with an
open-loop transfer function. The gain crossover frequency ωgωg is found to be 1.26 rad/sec by
setting the magnitude to 1. The phase is calculated to be -266.5°, leading to a phase margin
(PM) of -86.5°, indicating instability. The phase crossover frequency is determined as 0.63
rad/sec, and the gain at this frequency reveals a gain margin (GM) of -7.08 dB, confirming that
any increase in gain would further destabilize the system. Thus, the system is concluded to be
unstable due to negative gain and phase margins.
PD Controller velocity error constant question(46)
This problem involves determining the gains of a Proportional-Derivative (PD) controller in a
control system to achieve specific dynamic performance, particularly the velocity error constant
Kv and the damping ratio zetaζ. The PD controller provides a control signal proportional to both
the error (through KP) and the rate of change of error (through KD), affecting the system's
stability and response speed.
The velocity error constant Kv defines the system's ability to track a ramp input with
minimal steady-state error, and it's calculated using the open-loop transfer function
by Kv=lims→0sG(s)H(s)
K_v = \lim_{s \to 0} s G(s) H(s)
Kv=lims→0sG(s)H(s). Additionally, the damping ratio zetaζ affects the oscillatory
behavior and overshoot of the system's transient response. By equating the
system's characteristic equation to the standard second-order form, we determine
values for KP and KD that meet the required Kv and zetaζ, ensuring optimal system
performance.
GATE Question(110)
For the given open-loop transfer function G(s)=1/s(s−1)(s+2)(s+3), the system has
one pole in the right half-plane, making it potentially unstable. Using the Nyquist
stability criterion, the system's stability is determined by whether the Nyquist plot
encircles the point −1.
As the proportional gain K increases, the plot scales accordingly. The critical value
for stability is when the plot just touches the point −1. Hence, the closed-loop
system remains stable as long as the proportional controller gain K is less than or
equal to 6. For K≤6, the Nyquist plot may encircle −1, leading to instability.
The closed-loop system will be stable for proportional controller gain 0<K≤6.
Beyond this value, the system may become unstable.
