Question

Consider the system shown in figure 1 below. r(s) S+2 32-35 y(s) Controller Plant Figure 1: Simple proportional control system TASK 5: Suppose we want the final value of our response to be within 0.01% of the final value of our input (step) signal. That is, the steachy state error is 0.01%. Further, let us assume the final value is achieved for t > 10 seconds. In this case, we say that our tracking goal has been achieved. USING YOUR SIMULINK BLOCK DIAGRAM, determine for what values of K is the steady state error 0.01% (for t 10 seconds)? You don't have to analytically solve for the K value (yet).

For the system shown here in tasks 1-4 I have determined that for the value of K<3 the system is unstable, for K=3 the system is in oscillation and K>3 the system is stable. However i am unsure how to calculate the value of K to achieve a steady state error of 0.01% for T ≥ 10.

Many Thanks for the assistance

Answer 1

General Formula to find Steady-State Error:

eloo) = lim SR(S) 1+G(S) 30

Let our input is impulse:

$e(\infty)=\lim_{s\rightarrow 0} \left \{ \frac{s}{1+\frac{K(s+2)}{s^2-3s}} \right \}$

$e(\infty)=\lim_{s\rightarrow 0} \left \{ \frac{s-3}{{s^2-3s+Ks+2K}} \right \}$

$e(\infty)=\frac{3}{2K}$

We are given that, the steady-state error is less than 0.01%.

$\frac{3}{2K}<\frac{0.01}{100}$

Response to a ramp input:

Scope - File Tools View Simulation Help O. + x 104 Output Ramp 10! 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 103 Ready Sample based Offset=0 T=0.001 2:39 AM 7/11/2020