Question

A speed control for a gasoline engine is shown in Fig. 5 below. riguies (a) Determine the necessary gain K if the steady-state error is required to be less than 10% of the reference input R(s)=

s
R
​
, where R is a positive constant. (b) With the smallest gain satisfying part (a): - determine closed-loop stability of this system using Nyquist criterion (generate Nyquist plot with MATLAB - determine gain and phase margins of the system using Nyquist plot - What is the value of K that destabilizes this system (if such exists)? NOTE: part (b) can be solved graphically by clicking on the appropriate points on the generated graph make sure that you attach the Nyquist plot with shown clicked points

Answer 1

The given question requires us to determine the necessary gain K for a speed control system for a gasoline engine, and then analyze the stability of the system using the Nyquist criterion.

(a) To determine the necessary gain K, we need to find the gain value that ensures the steady-state error is less than 10% of the reference input R(s) = s/R, where R is a positive constant.

To achieve this, we can use the steady-state error formula for a system with a unity feedback configuration:

Error(s) = R(s) - C(s)

where C(s) represents the output of the system. In this case, the output is the speed of the engine.

The steady-state error for a system with unity feedback can be calculated using the formula:

Error_ss = 1 / (1 + K)

where K is the gain of the system. Since we want the steady-state error to be less than 10% of the reference input, we can set up the following inequality:

Error_ss < 0.1

Substituting the formula for Error_ss, we get:

1 / (1 + K) < 0.1

Simplifying the inequality, we find:

1 + K > 10

K > 9

Therefore, the necessary gain K for the system to have a steady-state error less than 10% of the reference input is K > 9.

(b) With the smallest gain satisfying part (a), we need to determine the closed-loop stability of the system using the Nyquist criterion. The Nyquist plot can be generated using MATLAB.

By analyzing the Nyquist plot, we can determine the gain and phase margins of the system. The gain margin is the amount by which the system gain can be increased before the system becomes unstable. The phase margin is the amount by which the phase of the system can be increased before the system becomes unstable.

To determine the gain margin and phase margin, we look at the Nyquist plot and identify the point where the Nyquist curve intersects the critical point (-1,0) on the real axis. The distance from this point to the origin gives us the gain margin, and the angle from the negative real axis to the intersection point gives us the phase margin.

To destabilize the system, we need to find the value of K that causes the Nyquist curve to encircle the critical point (-1,0) in a clockwise direction. This indicates instability in the system.

Please note that to provide more accurate and specific guidance, it would be helpful to see the actual Nyquist plot with the clicked points.