1.
Frequency Domain Specifications:
 Gain margin (GM)
 Phase margin (PM)
 Crossover frequency (��ωc)
 Bandwidth
 Resonant peak
 Resonant frequency
 Percent overshoot
2. Locate Poles and Zeros on the S-plane:
 Poles at �=−5,6,−7s=−5,6,−7
 Zeros at �=−2,−4s=−2,−4
3. Formulate Gain Margin and Phase Margin:
 Gain Margin (��GM) = 20log⁡10(GM)20log10(GM)
 Phase Margin (��PM) = ∠�(���)+180∘∠G(jωc)+180∘
4. Comparison of Lag and Lead Compensator:
 Lag compensator improves steady-state accuracy.
 Lead compensator improves transient response.
 Lag introduces phase lag; lead introduces phase lead.
5. Sketch Bode Plot for 11+��1+sT1:
 Plot magnitude and phase using ∣�(��)∣∣G(jω)∣ and ∠�(��)∠G(jω).
 Slope is -20 dB/decade for a pole, +20 dB/decade for a zero.
 Phase changes by -90° for a pole, +90° for a zero.
6. Advantages of Nyquist Plot:
 Provides information on stability and performance.
 Identifies closed-loop stability for various gains.
 Useful for analyzing systems with time delays.
7. Transfer Function vs. State Variable Approach:
 Transfer Function: Describes the system in terms of input and output.
 State Variable: Describes the system using state variables and their derivatives.
8. State Space and State Vector:
 State Space: Mathematical representation of a dynamic system using state variables.
 State Vector: A column vector containing the state variables.
9. Properties of State Transition Matrix (STM):
 STM provides the solution to the state equation.
 It describes the evolution of the system's state over time.
 STM is used to compute the system response at any time.
10. Controllability vs. Observability:
 Controllability: Ability to control the system's state with inputs.
 Observability: Ability to determine the system's state from outputs.
11. Determine Poles, Zeros, and Centroid of OLTF ��(�)=�(�+2)�(�+4)GH(S)=S(S+4)K(S+2):
 Poles: �=0,−4s=0,−4
 Zeros: �=−2s=−2
 Centroid: Average of pole locations.
12. Sketch Root Locus for �(�)=��G(s)=sK:
 As �K increases, poles move towards the origin.
 Root locus includes all possible pole locations.
13. Expression for �β of Lag Compensator:
 Lag Compensator: ��(�)=�+��+�Gc(s)=s+ps+z
 �=��β=pz
14. Effect of Adding Pole to Open Loop Transfer Function:
 Increases system stability.
 Slows down system response.
15. Frequency Response:
 Describes how a system responds to sinusoidal inputs across different frequencies.
16. Polar Plot: a. Order-2, Type-1: One asymptote at −180∘22−180∘ and two break frequencies. b. Order3, Type-0: Three asymptotes at −180∘33−180∘ and no break frequencies.
17. Solution for (0)ϕ(0) and [ϕ(t)]k (State Transition Matrix):
 (0)ϕ(0): Initial conditions of the state vector.

[ϕ(t)]k: Value of ϕ(t) at time t.
18. Block Diagram for State Model of SISO System:
 Single-input, single-output system represented by x˙(t)=Ax(t)+Bu(t) and
 )y(t)=Cx(t)+Du(t).
19. State Equation and Output Equation:
 State Equation: x˙(t)=Ax(t)+Bu(t)
 Output Equation: y(t)=Cx(t)+Du(t)
20. State Space and State Vector:
 State Space: Mathematical representation of a dynamic system using state variables.
 State Vector: A column vector containing the state variables.
21. Calculating K for G(s)=SK(S+100) at s=−10,−150:
 Substitute values into the transfer function and solve for �K.
22. Effect of Addition of Poles in Root Locus:
 Increases damping and stability.
 Slows down the system response.
23. Sketch Bode Plot of �(�)=��G(s)=sK:
 As �K increases, magnitude decreases, and phase becomes more negative.
24. Effects of Phase Lead Compensation:
 Increases bandwidth and speed of response.
 Improves transient response.
25. Advantages of Polar Plot:
 Directly shows gain and phase margins.
 Suitable for analyzing stability.
26. Expression for �α of Lag Compensator:
 Lag Compensator: ��(�)=�+��+�Gc(s)=s+ps+z
 �=11+�α=1+β1, �=��β=zp
27. Block Diagram for State Model of SISO System:
 Single-input, single-output system represented by �˙(�)=��(�)+��(�)x˙(t)=Ax(t)+Bu(t)
and �(�)=��(�)+��(�)y(t)=Cx(t)+Du(t).
28. Solution to Overcome Drawbacks in Transfer Function Mode Approach:
 Use state-space representation for multivariable and time-varying systems.
 Enables handling of initial conditions more effectively.
29. Characteristic Equation for Eigenvalues of System Matrix �A:
 Characteristic equation: det(��−�)=0det(sI−A)=0.
30. Angle of Asymptotes and Centroid for �(�)=�(�+2)(�+3)G(s)=(s+2)(s+3)K:
 Use rules for angle of asymptotes and centroid in the root locus.
31. Advantages of Frequency Response Analysis:
 Provides insight into system behavior at different frequencies.
 Helps analyze stability and transient response.
32. Electrical Network of Lag-Lead Compensator:
 Lag-Lead compensator includes both lag and lead sections in the transfer function.
33. Define Terms: Gain Margin, Phase Margin, ���ωgc, ���ωpc:
 Gain Margin (GM): The amount by which the gain can be increased before instability.
 Phase Margin (PM): The amount by which phase can be increased before instability.
 ���ωgc: Gain crossover frequency.
 ���ωpc: Phase crossover frequency.
34. Drawback of Bode Plot:
 Difficulty in precise determination of gain and phase margins.
35. Magnitude and Phase Diagrams for �(�)=11+��G(s)=1+ST1:
 Draw magnitude and phase diagrams based on the transfer function.
36. State Equation and Output Equation:
 State Equation: �˙(�)=��(�)+��(�)x˙(t)=Ax(t)+Bu(t)
 Output Equation: �(�)=��(�)+��(�)y(t)=Cx(t)+Du(t)
37. Appraisal of State Transition Matrix (STM):
 STM provides a mathematical tool for solving linear time-invariant systems.
 It describes the evolution of the system's state over time.
38. Condition for State Observability:
 A system is state observable if the observability matrix has full rank.
39. Advantages of State Space Model over Transfer Function Model:
 Suitable for multivariable systems.
 Easily accommodates time-varying systems.
 Facilitates modern control techniques.