6CCS3RSC/7CCSMRTS
Assignment: Linear Systems Control
10th March 2020
For this assignment, you will need the template code assignment3.m. Download this from the module’s
KEATS page and save it to your computer. When you are ready to submit your assignment, you will need
to upload this file so bear this in mind while completing the assignment (further instructions on how to
submit your assignment are given below). Open the file using Matlab, and complete the following exercises.
+ −
V
I
C
LR
Figure 1: The schematic diagram of an R-L-C circuit.
A series resistor-inductor-capacitor circuit (see Fig. 1) can be described as a linear system, in which, for
constant voltage, the current across the components follows the equation
d2
dt2
I(t) +
R
L
d
dt
I(t) +
1
LC
I(t) =
1
L
dV
dt
(1)
where I is the current, R the resistance, L the inductance, C the capacitance and dV/dt the rate of change
of the voltage at the power source.
1. Write (1) in state space formulation, as a continuous time, linear time invariant system. You may
assume that the rate of change of the voltage is the control input (i.e., u = dV/dt) and the system
state is the current and its first time derivative1 (i.e., x = (I, dI/dt)>). Using the template code
assignment3.m, implement the matrices Ac and Bc assuming that L = 20H, C = 0.1F , R = 4 Ω.
[9 marks]
2. Derive the equations for the system in discrete time, such that you can compute xt+1 as a function
of xt and ut. Using the template file, implement a simulation of the system, such that you can
compute the current for 0 ≤ t ≤ 20 s if the voltage increases at a constant rate of 1V/s. Assume
that the current is zero and constant at t = 0 s and use sampling rate δt = 2ms.
[12 marks]
1Throughout the assignment, treat the state and its derivatives as a column vector.
Dr M. Howard Department of Engineering
King’s College London
3. Consider the case that the circuit is equipped with a multimeter that enables measurement of current.
Using the template file, implement the observer matrix C and the observability matrix H. Derive the
transfer function for this system and solve for the poles to four decimal places. Implement a vector z
in the template file that contains the poles in order of increasing size (i.e., ordered from the smallest
z to the largest).
[15 marks]
Completed assignments should be submitted to KEATS on 5pm, 30th March 2020.
To submit your assignment, please follow the following steps:
1. Complete the following lines of the source code by adding your name and student number:
1 % Please complete the following with your details
2 firstname =’’;
3 surname =’’;
4 number =’’; % this should be your ’k’ number , e.g., ’k1234567’
2. Rename the file to include your name and ’k’ number in the format:
assignment3 k000000 firstname surname.m
3. Upload the resultant source code as a single .m file on KEATS.
Automated marking will be used to assess the source code you submit. Therefore, please ensure that your
code runs without errors prior to submission. (Code that does not run will automatically be awarded
zero marks.)
This assignment is worth 12% of the module mark.
Dr M. Howard 2 Department of Engineering
King’s College London