ELE8066 Derive and write the averaged model of the DC-DC Buck-Boost converter, as described in Figure: intelligent control Assignment, CU, UK

Questions:
Part A: Modelling, Linearisation, Discretisation

A1. Derive and write the averaged model of the DC-DC Buck-Boost converter, as described in Figure 5. Consider as the measured output of the system the output voltage, equivalently, the voltage of the capacitor C). [1 point]

A2. Linearise the model around the equilibrium point xeq=[20 0.4]T and derive the linearised state space representation. [1 point]

A3. Provide several simulations of the averaged, nonlinear model and the linearised model you have acquired from questions 1 and 2 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. Plot the time responses of the states for the two models against each other. Also, plot the trajectories of each model in the state space. Observe and report the differences, if any. [2 points]

A4. Assuming a zero-order-hold (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, where the disretised system and the continuous-time linearised system are plot in the same figure. [2 points]

A5. Compare also the state coming from the discrete approximation using Euler forward difference acting on the nonlinear system, i.e., by setting 𝑥̇ (𝑡) ≈ 𝑥(𝑡+𝑇)−𝑥(𝑡) 𝑇 . [1 point]

Part B: Control design and stability analysis

B1. Is the linearised continuous system controllable? [1 point]

B2. Using the continuous-time linearised model, develop a stabilizing state space control law that drives the system to the equilibrium point. The closed-loop system must have a damping factor ζ=0.86 and a damped natural frequency ωd=2000. [2 point]

B3. Simulate the open-loop linearised system and the closed-loop linearised system, and the closed-loop averaged system, for two initial conditions and observe/highlight the differences in the responses. Justify your choice of initial conditions. [2 points]

B4. Using Lyapunov functions, verify whether (i) the continuous-time linearised closed-loop system and (ii) the discretized linear system (with the zero-order-hold) are stable. What can be stated regarding the stability of the averaged nonlinear system? [3 points]

Part C: Observer design
C1. The inductor current iL cannot be measured accurately without an expensive sensor. Thus, the controller designed in Part B cannot be implemented without an additional cost. To avoid this, we can develop an observer that estimates both states of the linearised system. Choose the eigenvalues of the closed-loop error dynamics of the observer and justify your decision. [2 points]

C2. Write down the complete observer equation, that is the closed-loop error dynamics and the state estimate dynamics. [1 points]