Analysis and Design Project

ELEC 3500 – Fall 2019

Due date: Frdiay, Dec. 6

1 Background

The ball-on-beam is a classical problem in control systems study. A ball rolls

along a beam as sketched in Fig. 1, and its distance from the pivot point

is measured as x. The objective is to move the ball to a desired position

xd (subscript d stands for “desired”) by adjusting the beam angle θ. The

beam angle is changed by an electric servomotor whose command input is

a signal named θm. Dynamics of the ball-on-beam system can be highly

nonlinear, but very good control systems can be designed using the linear

system techniques studied in this course.

x

휃

Figure 1: The ball-on-beam process: change the ball position x by adjusting

beam angle θ.

A more general two-dimensional problem is the ball-on-plate, where the

ball’s (x, y) position is controlled. Several experiments can be found on

YouTube:

• One built by 2 Master’s students in Iran (listen to the mechanical

sounds!): https://www.youtube.com/watch?v=gO4dPVd7bw8

• Another one built in the Netherlands: https://www.youtube.com/

watch?v=uERF6D37E_o

The design project for this term focuses on one-dimensional motion of the

ball-on-beam system, as illustrated in Fig. 1.

1

Table 1: Signals and blocks to be shown in the control system model.

Symbol Description

x measured position of the ball

xd desired position of the ball

e the difference xd − x, i.e. the error

θm servomotor input signal

θ beam angle

G1(s) the model from servomotor input θm to beam angle θ

G2(s) the model from beam angle θ to ball position x

C(s) the compensator that is to be designed

Table 2: Desired control system performance specifications.

Specification value

Time for the ball to finish a step change in

desired position

< 5 time units

Overshoot in step response < 20%

Final error in ball position e(t→∞) = 0

Phase margin > 60◦

2 The Plant Description

The dynamic relationship between the beam angle θ and the servomotor in-

put signal θm can be described by a second-order, linear ordinary differential

equation (ODE):

θ¨ + 4θ˙ + 2aθ = bθm. (1)

The model parameter a equals the number of letters in your family (last)

name, and parameter b equals the number of letters in your first (given)

name. The linear acceleration of the ball x¨ is proportional to the beam

angle θ, and that relationship can be modeled described by another second-

order, linear ODE:

x¨ = cθ. (2)

Parameter c equals the number of letters in your second (middle) name.

(If you do not have a second name, then c = a). The goal is to design a

feedback system that controls ball position x.

2

3 Assignment

The entire assignment has 85 points of credit. Where it is convenient, use

software tools (e.g. MATLAB) to help study and design the feedback system,

and to generate figures for the report.

1. (2 pts) Draw a block diagram model of the control system. The dia-

gram should show the signals and blocks listed in Table 1.

2. (2 pts) Using the plant description in Sec. 2, find the transfer functions

G1(s) and G2(s).

3. (2 pt) The plant input is the servomotor command input θm, and the

plant output is the measured ball position x. Find the plant model,

and call it G(s). Explain how G(s) is related to G1(s) and G2(s).

4. (3 pts) What is the dc gain of the plant model G(s)? What does the

dc gain say about the steady-state behavior of the ball? Explain your

analysis in layman’s terms.

5. (2 pts) Does the model G(s) represent a BIBO stable plant? Explain

your analysis method(s). Note: Study the model G(s) alone, NOT in

any feedback control system. No figures are required for this problem;

just analyze the model G(s).

6. (7 pts) Next, consider the feedback system with proportional control

C(s) = 1. Generate the Nyquist plot. Apply the Nyquist criterion

(Ncw =?, P =?, Z =?), and determine if the feedback system (closed-

loop) is stable. Determine the gain and phase margins on the Nyquist

diagram, and explain how these are found (not on the Bode diagram).

7. (2 pts) Continue the analysis of a proportional control system, but let

C(s) = K. Sketch the root locus diagram for the system. Describe

the model that is used to generate the root locus diagram.

8. (2 pts) Can the feedback system be stable using a proportional control

K? Explain using the root locus diagram.

9. (8 pts) Can the feedback system with proportional control simultane-

ously meet all of the performance specifications listed in Table 2? If

so, find a suitable value of K that meets all performance specification.

If not possible, explain which of the specifications cannot be met, and

why.

3

10. If necessary:

(a) (16 pts) Redesign the compensator transfer function C(s) so that

the feedback system simultaneously meets all specifications listed

in Table 2. Additionally:

i. (4 pts) The compensator model C(s) should be “proper,”

which means C(s) cannot have more zeros than poles.

ii. (2 pts) The compensator C(s) should not have any poles or

zeros in the right half of the s-plane, or on the imaginary

axis. For example, the compensator should NOT have a zero

at s = 0.

(b) (5 pts) Explain how you design the compensator. Specifically, de-

scribe the tool(s) that you choose (root locus, frequency response,

etc), and describe how you use the tools.

11. (6 pts) Verify the design by simulating the feedback system step re-

sponse. Create plots of the step response for ball position x(t), and

the beam angle θ(t). Label axes appropriately.

12. (2 pts) Generate the root locus diagram that includes the compensator

C(s). Explain what model is used to generate the diagram.

13. (5 pts) On a Bode plot, show the compensated gain and phase margins,

and the associated phase and gain crossover frequencies. Describe the

model that is used to determine the stability margins.

14. (5 pts) Describe how computer-based design tools (e.g. MATLAB)

are used in this project. Describe how the project might have been

approached if these tools were not available.

Submit work that is neatly presented and well-organized. A typed report is

NOT required. Follow these guidelines:

• (6 pts) Answers should be complete, yet concise. Discussion should be

sufficiently detailed to have some archival value for you, or a younger

sibling soon coming to Auburn. Write in complete, well-formed sen-

tences.

• (2 pts) Figures and tables must be clearly labeled and captioned.

Study the figure and tables in this assignment for examples of labels

and captions. Captions should be informative.

4

• (2 pts) Clearly label axes on graphs. Design graphs carefully – be sure

that labels are legible. Illegible figures will receive zero credit.

• Staple paper sheets together, but do not submit a binder. Reports

torn from a spiral bound notebook will not be accepted (zero credit).

Alternatively, submit the report electronically on AU Canvas.

5