1. An inverted pendulum is mounted on the motor driven cart. It is unstable and may fall. Here we assume it falls only along the plane of the paper. The control force is applied to the cart. Obtain a mathematical model of the system taking as the output and as the input. (All notations have their usual meaning)
<img src='./qimages/10969-1.jpg'>

2. Find the transfer function for the block diagram using Mason's Gain Formula. <img src='./qimages/10969-2.jpg'>

3.(a) A unity feedback system has open-loop transfer function 1/s Find the static error coefficients and dynamic error coefficients.

(b) How does steady state error of a system depend on the type of the system?

4. Using Routh-Hurwitz stability criterion, determine the number of poles in the left half, right half and on the imaginary axis of s-plane for a system whose characteristic equation is s^7 5s^6 11s^5 15s^4 19s^3 15s^2 9s 5 0. Also comment on the stability of the system.

5. For the given system, sketch the root locus in a graph paper as varies from 0 to infinity, showing all relevant steps and calculations.

<br><br> <img src='./qimages/10969-5.jpg'>

6. Consider a unity feedback system having open-loop transfer function (OLTF)
Sketch a suitable Nyquist path (contour) with a mathematical description. Sketch a Nyquist plot on a plain paper showing relevant calculations. From the Nyquist plot, determine the stability of the system.

7. Consider the system given below, where <img src='./qimages/10969-7.jpg'> <br><br>

Using root locus analysis, design a PI controller such that: Damping ratio of Dominant Poles =0·45 Undamped Natural Frequency =3·5 rad/sec Steady state error(ess) =0.

8.Consider a unity feedback system with open-loop transfer function (OLTF) s(s
The system is desired to have the following specifications:
Phase Margin 50° and Kv 20 sec^-1. using the Bode plot method, design a lead compensator such that the above requirements are met.

9.(a) Write the dimension of the D matrices of a state space model of a 3-input 4-output 5th order system.
Write a state space model of a SISO 2nd order system such that it Both controllable and observable
Controllable but not observable

(iii) Observable but not controllable
Calculate the state transition matrix for the following system <img src='./qimages/10969-9c.jpg'> Realize a lag compensator using op-amp, resistances and capacitances.

10.(b) Discuss briefly about Neural based control.

10.(c) Discuss the effect of pole-zero-cancellation in a transfer function.