Question

Problem 10. If a continuous-time system's transfer function is given by G[s] = (3+3) P (2+5.55+25)(3+2and one wants to control the system with a discrete-time controller without changing the system's bandwidth, what is a reasonable sample period? 1. T = 0.001 seconds 2. T=0.1 seconds 3. T = 0.4 seconds 4. T=0.04 seconds 3. None of the above.

Answer 1

In theory, in order for discrete-time signal to retain all the frequencies and all the information contained in the continuous-time signal, we need to use a sampling period of T <= 1/(2*fm), fm is the maximum frequency of continuous-time signal.

For example, if maximum frequency contributing to the spectrum of continuous-time signal is 50 Hz, then sampling period

Ts <= 0.01 s

If we choose a sampling period > 0.01 s, we will not be able to fully reconstruct your original continuous-time signal from the discrete-time signal, and the spectrum of discrete-time signal will appear distorted, i.e., not the same as the spectrum of the continuous-time signal. This effect is known as "aliasing".

For Stable System: Both the margins should be positive or phase margin should be greater than the gain margin.

The greater the Phase Margin (PM) and Gain Margin (GM), the greater will be the stability of the system. The phase margin refers to the amount of phase, which can be increased or decreased without making the system unstable. It is usually expressed as a phase in degrees.

Using this Criteria we should choose which will be the reasonable sample period

Now I have attached Bode, Nyquist, Nicholas plots for your reference

We will now analyse using bode plot at each of the given T in sec and determine reasonable one.

The same analysis can be done on Nyquist and Nicholas

at T=0.001s we can see that Phase Margin and Gain Margin are high so it is reasonable sample period

3 + Bode plot transfer function (2+5) (2.5 +5.5 s +s) sampling period 0.001 Bode plot: magnitude (dB) 1000 phase (degrees) 1000 100 frequency - Ys U(s)

Nyquist plot: Re -1.0 -0.8 -0.6 -0,4 -0.2 Nichols plot: magnitude (dB) -200 -150 0 -100 -50 phase (degrees) - Yis U15

Bode plot 3+ transfer function (2 + 5)(2.5 +5.5 s +s?) sampling period 0.01 Bode plot: magnitude (dB) 10 100 phase (degrees) 10 frequency

Nyquist plot: Re -1.0 -0.8 -0.6 -0.4 -0.2 -0.4 - YS Nichols plot: magnitude (dB) -200 -150 -100 -50 phase (degrees) -

Bode plot 3+ transfer function - (2 + 5) (2.5 +5.5 s + S2 sampling period 0.1 Bode plot: magnitude (dB) 0.1 phase (degrees) - 100 -120 frequency - Ys US)

Nyquist plot: -1.0 -0.8 -0.6 -0.4 -0.2 (5) US Nichols plot: magnitude (dB) -200 100 - 100 0 100 phase (degrees) 200 - Ys) U(5)

Bode plot 3+5 transfer function (2+) (2.5 +5.5 s +s) sampling period 0.04 Bode plot: magnitude (dB) phase (degrees) 0.1 - Ys frequency Us

Nyquist plot: Re -1.0 -0.8 -0.6 -0,4 -0.2 157 Nichols plot: magnitude (dB) - 200 -150 -500 -100 -50 phase (degrees) - Ys) US

Bode plot 3 + transfer function (2 + 5)(2.5 +5.5 s + s2 sampling period 0.4 Bode plot: magnitude (dB) phase (degrees) - 120 0.01 0.1 Ys) frequency Us

Nyquist plot: 0.4 -1.0 LLLLL Re Re -0.8 -0.6 -0.4 -0.2 -0.4 Nichols plot: magnitude (dB) -200 100 200 -100 0 phase (degrees) - YS US)