Instrumentation and Control Assignment -

Learning Outcomes -

1. Develop and apply knowledge and understanding of scientific principles and methodology necessary to underpin their education in mechanical and related engineering disciplines, to enable appreciation of its scientific and engineering context and to support their understanding of future developments and technologies.

2. Demonstrate an understanding of and ability to apply a systems approach to engineering problems.

3. Apply knowledge of characteristics of particular equipment and processes.

4. Appraise technical literature and other information sources.

QUESTION 1 - Consider the following differential equation:

d²y/dt² + 2 dy/dt + 4y = 4u

a) Convert the 2nd order differential equation into transfer function form.

b) Using your transfer function in part a) and applying a unit step input, use the final value theorem to determine the steady state of the open loop system and comment on the steady state error.

c) Determine the systems damping ratio ζ, natural frequency ω_n, natural frequency and the system poles.

d) Sketch the complex s-plane, indicating the pole locations and corresponding lines of constant damping ratio and lines of natural frequency. Keeping the same natural frequency, indicate on the sketch where the poles would be located for a faster response whilst retaining the same natural frequency. Comment on the form of damping for this faster response. Relate this to a practical system of your choice.

e) Draw by hand the corresponding phase canonical variable form Simulink block diagram of the 2nd order differential equation (including the labels for the input and outputs of the system).

f) The open loop system is now configured as a closed loop control system with a disturbance, as illustrated in Figure Q1. Derive the following transfer function based on Figure Q1:

Y(s) = R(s)[C(s)G(s)/1+C(s)G(s)H(s)] + D(s)[G(s)/1+C(s)G(s)H(s)]

g) Use the final value theorem to determine the steady state error of the closed loop control system and comment on the steady state error. In this example, H(s) = 1, R(s) = 1/s, D(s) = 3/s and C(s) = 80 (i.e. a proportional controller).

h) Draw by hand the corresponding Simulink block diagram of the closed loop control system, with the system G(s) being in phase canonical variable form.

QUESTION 2 - The acceleration of an automotive vehicle can be modelled through the use of a first order transfer function, given by:

G(s) = k/(τs+1)

where τ = 3 and K = 5.

a) To test the system i.e. mathematical model, apply a unit step input at t = 0. Using the Tables of Transform, determine the solution for the system. Draw by hand the graphical output for the first five time constants i.e. τ, 2τ, 3τ, 4τ & 5τ.

Note: to solve this question do not use MATLAB/Simulink.

The transfer function in part a. is used to model the acceleration of a Ford Focus automotive vehicle i.e. medium performance.

b) Approximately determine typical transfer functions of the acceleration for the following two vehicles:

• Fiat Cinquecento (low performance).
• Mclaren P1 (high performance).

c) Simulate your two determined transfer functions using MATLAB/Simulink. For this, the input is the same as in part a, i.e. a unit step. Plot the graphical outputs on the same graph, including a labelled legend.

Note: attach only a copy of the graphical output as evidence for this question.

QUESTION 3 - A dynamic system is represented by the following standard second order transfer function:

G(s) = 8/(s²+6s+8) (3.1)

Assuming initial conditions are zero, obtain Y(s) i.e. the Laplace transform of the system output, when the system is subjected to a unit step input i.e. U(s) = 1/s, applied at time t = 0.

a. By obtaining the inverse Laplace transform of Y(s), find an expression for the response in the time domain, i.e. obtain y(t). You may assume without verification that the partial fraction representation is as follows:

∝/s(as²+βs+γ) = A/s + B/(s+β) + C/(s+γ)

Note: you may wish to check your answer with a friend i.e. MATLAB

b. Simulate the dynamic model given by Equation (3.1) for 5 seconds, when the system is subjected to a unit step input i.e. U(s) = 1/s, applied at time t = 0.

Note: provide evidence within this answer box of your Simulink model and/or MATLAB code and graphical output.

QUESTION 4 - Consider the following closed loop control system in Figure Q4:

where m = 1, c = 2 and k = 5.

a. For the closed loop system configuration in Figure Q4, obtain an expression for the overall closed loop transfer function.

b. Hence or otherwise write down the closed loop characteristic equation.

c. Use the Routh-Hurwitz stability criterion to find the values of K for which the system is stable.

d. Simulate the closed loop control system in Figure Q4. Demonstrate values of K that would make the system:

i. Stable

ii. Unstable

iii. Marginally stable

Note: attach only a copy of the graphical output as evidence for this question.

QUESTION 5 - a) Given the following open loop transfer function:

G(s) = (s+6)/((s+2)(s+4))

Sketch the root loci by hand.

b) Given the following open loop transfer function:

G(s) = (s+4)/((s+3)(s+2)(s+6))

Sketch the root loci by hand.

c) Given the following open loop transfer function:

G(s) = (s+6)/(s(s+2)(s+2)(s+4))

Sketch the root loci by hand.

Note - The work must be original and written with your own wording and interpretation.