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CONTROL SYSTEMS

Answer the following questions:

(a) Define tracking control using an example.

Ans. Tracking and regulation refer to the ability of a control system to track/reject a given family of reference/ disturbance signals modelled as solutions of a differential/ difference equation. For example Dynamic control of industrial robots.

 

(b) Define transfer function and relate impulse response with transfer function.

Ans. A transfer function a convenient way to represent a linear, time-invariant system in terms of its input-output relationship. If the transfer function of a system is given by H(s), then the impulse response of a system is given by h(t) where h(t) is the inverse Laplace Transform of H(s).

 

(c) Define underdamped, overdamped and critically damped systems.

Ans. (1) Underdamped: The system oscillates (at reduced frequency compared to the undamped case) with the amplitude gradually decreasing to zero.
(2)Overdamped: The system returns (exponentially decays) to equilibrium without Oscillating. (3)Critically damped :The system returns to equilibrium as quickly as possible without oscillating.

 

(d) Find sensitivity of overall transfer function with respect to forward path transfer function.

Ans. Sensitivity to Changes in Parameters With a control system. the transfer functions of elements may drift with time and thus we need to know how such drift will affect the overall performance of the system.

 Overall transfer function = G(s)/ 1 +G(s)H(s).

 

(e) Define and find the slope of Bode plot in case of complex poles.

Ans. A Bode plot is simply a plot of magnitude and phase of a transfer function as frequency varies. However, we will want to be able to display a large range of frequencies and magnitudes, so we will plot vs the logarithm of frequency. use a logarithmic (dB, or decibel) scale for the magnitude as well.

 

(f) Find sensitivity of overall transfer function with respect to feedback path transfer function.

Ans : The forward path transfer function. G(s) changes then the overall transfer function Gorerall(s) will change. We can define the sensitivity of the system to changes in the transfer function of the forward element as the fractional change in the overall system transfer function Goverall(s) divided by the fractional change in the forward element transfer function G(s). i.e.

(Goverall/Goverall)/( G/G)

where Goverall is the change in overall gain producing a change of G in the forward element transfer function.

 

(g) Explain absolute and relative stability and name two methods for each.

Ans.

Relative Stability: it is measure of how fast the transient dies out in the system relative stability is related to settling time. a system having poles away the left half of imaginary axis is considered to be relatively more stable compared to a system having poles closed to imaginary axis, two method (a) Routh Hurwitz criteria (b) Bode plots.

Absolute Stability: If the system returns to it equilibrium state after the inputs given to the system are removed, two method (a) Euler's method, (b) Trapezoidal method.

 

(h) Explain similarity transformation. Why is it used?

Ans. The term "similarity transformation" is used either to refer to a geometric similarity, or to a matrix transformation that results in a similarity are called similar matrices. Similarity transformations transform objects in space to similar objects.

 

(i) What is state transition matrix ? Explain its significance.

Ans. The state-transition matrix is a matrix whose product with the state vector x at the time to gives x at a time t where to denotes the initial time. This matrix is used to obtain the general solution of linear dynamical systems

 

(j) Define phase-plane technique.

Ans. The phase plane method refers to graphically determining the existence of limit cycles in the solutions of the differential equation. The solutions to the differential equation are a family functions. Graphically, this can be plotted in the phase plans like a two-dimensional rector field