*Answer the following in brief (5 x 4)*

### Restrictions on pole and zero locations in the transfer function.

The restrictions on pole and zero locations in the Conditions For Driving Point Function with common factors in P(s) and Q(s) cancelled are listed below.

- The coefficients in the polynomials P(s) and Q(s) of network function N(s) = P(s)/Q(s) must be real and positive.
- Complex poles or imaginary poles and zeros must occur in conjugate pairs.
- (a) The real parts of all poles and zeros must be zero, or negative. (b) If the real part is zero, then the pole and zero must be simple.
- The polynomials P(s) and Q(s) may not have any missing terms between the highest and the lowest degrees, unless all even or all odd terms are missing.
- The degree of P(s) and Q(s) may differ by zero, or one only.
- The lowest degree in P(s) and Q(s) may differ in degree by at the most one.

### Compare the block diagram representation and signal flow graph.

**Block diagrams**consist of

- Blocks - these represent subsystems-typically modelled by, and labelled with, a transfer function
- Signals - inputs and outputs of blocks - signal direction indicated by arrows - could be voltage, velocity, force, etc.
- Summing junctions - points where signals are algebraically summed -subtraction indicated by a negative sign near where the signal joins the summing junction

**Signal flow**graphs consist of:

### What is a dominant pole? What is the location of poles for stable systems?

- A dominant pole is a pole that is near to origin than other poles in the system.
- The poles near the jw axis are called the dominant poles. Or, get the closed-loop TF from Open-loop TF. Determine the poles of the denominators.
- The poles which have very small real parts or are near to the jw axis have a small damping ratio. These poles are the dominant poles of the system.
- A dominant pole is significantly required in a stability analysis because it is that location that gives an idea where the root locus is progressing- towards the right or towards the left. It is also called near poles.

*Location of poles for a stable system*

- Take a simple closed-loop system with the plant (G), feedback path (H) with unity gain, then the transfer function of your system becomes T = G/(1+GH).
- Here, the characteristic equation is 1 + GH. The roots of this equation give you the location of poles. Plot these poles on S plane. Roots of this characteristic equation give the eigenvalues of the system, if the eigenvalues are negative in nature, then the system is stable, otherwise not stable.
- When we plot these poles on the S plane, all poles fall on the left side of the S plane.

### Which is greater between gain and phase cross over frequency?

- The gain crossover frequency is less than the phase crossover frequency for stability.
- The gain crossover frequency is simply the frequency corresponding to which the open-loop system’s transfer function has unity magnitude, the phase angle corresponding to it should obviously be more than -180 degrees.
- This basically implies that the phase crossover phenomenon has not occurred for a lower frequency (as phase angle, being negative, is assumed to decrease with increasing frequency), that is, the phase crossover frequency is more than the gain crossover frequency.

### When a proportional controller is introduced in a system, does its performance improve or not?

- The controller is a device that is used to alter or maintain the transient state & steady-state region performance parameter as per our requirement.
- Proportional control, in engineering and process control, is a type of linear feedback control system in which a correction is applied to the controlled variable which is proportional to the difference between the desired value (setpoint, SP) and the measured value (process variable, PV).
- The P-controller can stabilize a first-order system, can give a near-zero error, and improve the settling time by increasing the bandwidth.
- It can also destabilize the system by using very high gains because it reduces the gain margin.
- Even though the system can be stable by the use of a small gain proportional controller, the performance of the system are generally not so good.
- Example: adaptive cruise control is an example of proportional control. Throttle input is adjusted variably to react to both decreasing slopes and increasing momentum. As well, the throttle is adjusted variably to the proximity of other vehicles in front of the controlled car. A separate proportional control system acts on the brakes gradually if nearing a followed vehicle too closely or dramatically in the case of emergency collision avoidance.

A proportional controller

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