February 2021 1/5 ELE8066 Semester 1 Coursework Intelligent Systems & Control Semester 1 ELE8066 – N. Athanasopoulos Instructions -Preparing a report in Latex is preferable, alternatively prepare in MS Word. -There is a strict 10-page limit, excluding the first title page. Any material after 10 pages will not be graded. -Use sections and subsections that match to each question. -Provide short and precise comments/observations when required. -The report should be a single pdf file. -The simulink and m-files should also be uploaded in a single .zip file. In these files, use commenting to explain what you are doing. -You can use Matlab special functions to answer questions. -In summary, additionally to the report itself containing any derivations/calculations, you must upload a copy of any m-files used. -When running the m-files, generate annotated (i.e. labels on all variables, etc.) plots for the various responses (when comparing different responses, they should be plotted in the same figure.). -The report will be uploaded in Canvas. The deadline for uploading your report is mentioned clearly in the Assignments section in Canvas. February 2021 2/5 ELE8066 Semester 1 Coursework Control of a DC-DC Buck-Boost Converter Digital power converters are used in billions in household electronics, but also in electricity (micro)-grids. The principle of DC-DC converters is simple; switching is enabled by transistors, regulating the output voltage to a constant value that can be larger or smaller than the input voltage, depending on its value and the topology of the circuit. In this assignment, we will study the DC-DC buck-boost converter, which allows the output voltage to be either higher or lower than the input voltage. The schematics of the converter are below. We model each transistor pair as an ideal two-position switch. The resistor RL accounts for the losses in the power stage. The current source Iload corresponds to the current drawn from the load connected to the output of the power converter. Figure
1. The Buck-Boost converter is used to increase or decrease an input
voltage: for example, it can charge a battery from a solar panel, or provide voltage for a lighting system for a vehicle. Figure 2. The Buck-Boost schematics and the equivalent electric circuit where two ideal switches replace the transistors. February 2021 3/5 ELE8066 Semester 1 Coursework There are four possible different configurations for the power converter, accounting for all combinations of the switches pair (q1, q2): (0,0), (0,1), (1,0), (1,1), as shown in Figure 3. For each mode, a linear dynamical system can be derived. For example, for Mode 1, we have The overall system is a switching system of the form = Aix+bi, when Mode = ‘i’ where x=[x1 x2] T=[Vc iL] T, Ai , i=1,2,3,4, are the system matrices for each model, and bi, i=1,2,3,4 are constant vectors. A direct way to control the converter is to control the switches, however this requires advanced tools from control theory, as well as expensive components. Another method is to control the duty cycle ratios d1, d2, of each switch over a sampling period Ts which is constant. The duty cycle ratio is a real scalar between 0 and 1 that corresponds to the time the switch is ON (i.e., ‘1’) over one period, as shown in Figure 4. Mode 1 Mode 2 Mode 3 Mode 4 Figure
3. The different modes / configurations of switches for the Buck-Boost
converter. For each configuration of q1, q2, The Kirchhoff laws can be used to construct the linear model for each of the converter. Figure
4. The waveforms of the switches d1, d2 and the corresponding duty
cycle ratios. The interval where mode i (i=1,2,3,4) is activated are depicted with Mi (i=1,2,3,4). February 2021 4/5 ELE8066 Semester 1 Coursework The equivalent averaged system is produced by averaging the behaviour of the system within one sampling period, as shown in Figure 5. For the averaged system, one can consider two state variables and two inputs, namely x=[x1 x2] T=[Vc iL] T, u=[u1 u2] T=[d1 d2] T. This is a nonlinear model, with the state space representation being of the form = f(x) + g(x) u + a, where x is the two dimensional input vector, g(x) is a 2×2 matrix having as elements functions of the state vector x, u is the two dimensional input vector and a is a constant vector. The function f(x) and the matrix function g(x) can be derived from the averaged circuit by applying Kirchhoff’s laws. The values of the elements of the circuit are L=200μH, RL=0.2Ω, Iload=0.2A, Vs=15V, C=22μF. The sampling time is 10μs. Figure 5. The equivalent averaged circuit. February 2021 5/5 ELE8066 Semester 1 Coursework Questions: 1. Derive and write down the averaged model of the DC-DC Buck-Boost converter, as described in Figure 5. Linearise the model around the equilibrium point xeq=[20 0.4] T and derive the linearised state space representation. [2 points] 2. Simulate the averaged model and the linearised model around the equilibrium point. Start by having as initial condition the equilibrium point, and gradually use initial conditions that are farther away from it. You can use as constant input the input vector corresponding to the equilibrium point. Observe the differences, if any, and report them. [4 points] 3. Assuming a zoh discretisation scheme, derive the discretised system from the linearised system, for a sampling period T=10μsecs. Compare the discretised version with the continuous-time system in a simulation. Compare also the state matrices coming from the discrete approximation using Euler forward difference acting on the nonlinear system, i.e., by setting () ≈ (+) () . [3 points] 4. Using the linearised model, develop a stabilising state space control law that drives the system to the equilibrium point. Is the system controllable The closed-loop system should have a damping factor ζ=0.9 and damped natural frequency ωd=103. Simulate the open-loop linearised system and the closed-loop linearised system, and the closed-loop averaged system, for one or two initial conditions and observe/highlight the differences in the responses. Justify your choice of initial conditions. [5 points] 5. The inductor current iL cannot be measured accurately without an expensive sensor. Thus, the controller designed in question 5 cannot be implemented without an additional cost. To avoid this, we can develop an observer that estimates both states of the linearised system. Is the system observable Consider as the (measured) output of the system the output voltage (equivalently the voltage of the capacitor C). Choose the poles of the error dynamics of the observer and justify your decision. Write down the complete observer equation, that is the closed-loop error dynamics and the state estimate dynamics. [4 points] 6. Using a quadratic Lyapunov function, verify whether the linearised closed-loop system is stable. What can be said for the stability of the averaged nonlinear system [2 points] 7. Consider the controller designed in question 4, using the state estimation instead of the actual states, as designed in question 6. Derive the resulting closed-loop system and write down the state equations and output equations. Is the resulting closed-loop system stable and why Simulate two closed-loop systems with the same controller designed in question 4 with (i) taking the actual state as feedback and (ii) taking the state estimation as feedback, and compare the two time responses. [Bonus]
