6CCS3RSC/7CCSMRTS

6CCS3RSC/7CCSMRTS

Assignment: Linear Systems Control

16th March 2021
For this assignment, you will need the template code assignment.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.-L-C circuit.
A series resistor-inductor-capacitor circuit (see Figure 1) can be described as a linear system, in whichthe
current across the components follows the equation
d
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
   implement the matrices A and B assuming that L = 20 H, C = 0.1 F, R = 4 Ω.
   [5 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 1 V /s. Assume
   that the current is zero and constant at t = 0 s and use sampling rate δt = 2 ms.
   [10 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
   at the same sampling rate. Using the template file, implement the observer matrix C and the
   observability matrix H using your answer to Question 2. 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.
   [15 marks]
   Completed assignments should be submitted to KEATS on 5pm, 5th April 2021.
   To submit your assignment, please follow the following steps:
4. Complete the following lines of the source code by adding your name and student number:
5. % Please complete the following with your details
6. firstname = ’ ’;
7. surname = ’ ’;
8. number = ’ ’; % this should be your ’k ’ number , e . g . , ’k1234567’
9. Upload the resultant source code as a single .m file on KEATS.
   Important notes on submitting:
10. Please ensure that your code runs cleanly prior to submission. Marks will be deducted from
    submissions that do not run without errors or warnings.
11. Please do not change the variable names in the template.
    This assignment is worth 15% of the module mark.
    Dr M. Howard 2 Department of Engineering
    King’s College London