NiiL

Week 14 · Comparing Stability and Performance Indices

English translation

Before practice

01 Why check stability first?

Some systems initially behave well but oscillate more strongly over time. What should you check before evaluating speed or accuracy?

Description
  • Check stability before comparing performance
  • A small error over a short observation interval cannot establish long-term behavior

02 Evaluation suited to the purpose

One response deviates greatly but recovers quickly; another stays slightly off target for a long time. How should we decide which is better?

Description
  • ISE, IAE, and ITAE apply different weights to magnitude and duration
  • Choose metrics suited to the purpose and compare using the same input and observation interval

Practice

1. Calculate the closed-loop model and final value

  • Use G(s)=1s+2G(s)=\frac1{s+2} and H(s)=2s+4H(s)=\frac2{s+4} from the “3. Tracking error and error at the comparison point” code in the practice Live Script, but first calculate the case K=2K=2 by hand.
  • Substitute U=K(R−HY)U=K(R-HY) into Y=GUY=GU. Collect the YY terms on one side to obtain YR\frac{Y}{R}, then find the denominator roots and determine whether all poles have negative real parts.
  • Find the final output and sensor measurement for a unit step, then calculate the final values of r−yr-y and r−Hyr-Hy separately.
  • Then set K to 2, run the code, and compare the two results.
  • Skip the already calculated K=2K=2 and continue with K=4,8K=4,8.
  • For the Routh activity, also derive the first column yourself as previously instructed, then check using roots.

2. Determine stability

  1. Derive the first column of the Routh array for s3+3s2+2s+Ks^3+3s^2+2s+K and compare K∈{1,6,7}K\in\{1,6,7\}.
  2. Find the poles of K=6K=6 and check whether any have real part 0.
  3. Explain how the response differs from the case where all poles have negative real parts.

3. Tracking error and error at the comparison point

  1. For K∈{2,4,8}K\in\{2,4,8\}, calculate r−yr-y, the target minus the actual output, and r−Hyr-Hy, the target minus the sensor measurement, separately.
  2. State which error to use to judge whether the actual output is close to the reference.

4. Finite-horizon performance index

  1. Using the plant G(s)=1/(s2+2s+1)G(s)=1/(s^2+2s+1) and KK from the ISE calculation, calculate ISE with end times of 10 s, 20 s, and 40 s.
  2. Tabulate the KK used, integration end time, and ISE. For each KK, compare how much ISE increases from 10→20 s and from 20→40 s.
  3. Check whether the increments decrease and judge whether the integral approaches a constant. Include the integration interval in your results, and distinguish finite-horizon integrals from infinite-horizon indices.

5. Optimization and Approximation

  1. Keep the K3 search range at 0.1–10, change only the spacing, and rerun the minimum search. Tabulate the search spacing before and after the change, the gain giving the minimum, and the absolute difference between 10\sqrt{10} and that gain.
  2. Create a comparison model by multiplying only one denominator coefficient of GL by 1.1. For example, extract the coefficients with [numL,denL]=tfdata(GL,'v'), apply denL(end)=1.1*denL(end), and build a new model with tf(numL,denL). Record the order of the changed coefficient and its values before and after the change.
  3. Calculate max(abs(yh-yl)) before and after the change at the same t=0:0.01:15. Use this maximum absolute difference to state whether the approximation error for GH increases or decreases.

Submission and saving

Include code, actual outputs and plots, and your own interpretation. Submit through KLAS; check the semester schedule and latest announcements for the deadline.

Filename: Name_StudentID_Week14.mlx.

Restart the runtime, run all code from the top, then save.

Optional practice

Example: Classifying an imaginary-axis pole as asymptotically stable

  1. Find the poles using p=roots([1 0 1]), and record the result of applying all(real(p)<=0) as the decision criterion.
  2. Write this condition beside the asymptotic stability test all(real(p)<-1e-9). Distinguish poles with real part 0 from those with negative real parts, and restore only the conditional expression to test asymptotic stability.