Experiment 81 – ELEC207 coursework
Design of a Stable Martian Segway
Report template
1. Mathematical Modelling
A) Please define the values for l, m and ts that you will use for your coursework. [1E]
B) Now derive the transfer function, H(s)=θ(s)/T(s), of the Segway in terms of l, m and g. [1E]
C) Using your values for l and m along with g=3.711 ms-2, write the transfer function with the denominator and numerator of your transfer function in polynomial form. [1E]
D) Calculate the position of the poles for your Segway and plot the poles on the complex plane. [1E]
2. Validating that the Open-loop System is Unstable
E) Insert a picture of the time-response of your Segway to the unit-step. [2E]
F) Comment on whether this time-response indicates that the open-loop system is stable. [1M]
3. Ensuring that the Closed-loop System is Stable Using PID Control
G) Write the closed-loop transfer function for your Segway in terms of Kp, KI and KD as a ratio of polynomials in s. Ensure that the highest order term in s in the denominator has a coefficient of unity. [3M]
H) What is the characteristic polynomial that would result in these pole positions? [1M]
I) By equating the coefficients in the closed-loop transfer function’s denominator and this characteristic function, deduce values for Kp, KI and KD which will ensure that the closed-loop system is stable. [3M]
4. Validating That the Closed-loop System is Stable
J) Insert a picture of the time-response of your closed-loop system to the unit-step. [2M]
5. Optimising the Time-Response Using Root Locus
K) Calculate the positions of the open-loop zeros (ie the zeros of C(s)H(s)) for the values of l, m, Kp, KI and KD that you have used. [1M]
L) State the positions of the open-loop poles (ie the poles of C(s)H(s)) for the values of l and m that you have used. [2E]
M) Sketch the root locus for C(s)H(s) and identify the points on the root locus that are such that Re(s) = — 4/ts. [3M]
N) Write the open-loop transfer function, C(s)H(s), as a ratio of polynomials in s. [1H]
O) Write as a polynomial in s involving K. [1H]
P) Write as a polynomial in involving K. [1H]
Q) Complete a Routh table for Deduce the value of k that is such that Re(s) = — 4/ts [3H]
6. Validating the Response of Optimised System
R) Insert a picture of the time-response of your improved closed-loop system to the unit-step. [2H]