{d} (8 points) Assume that due to wind disturbances, there is some noise that is added to the velocity dynamics at each time h: 0 3H1 = Airs + Bus + [0.01 1 wt 330 = [0,0111 (4) where wk ~ N(0,1). For la: 2 D, ..., 50, simulate system 4 and plot on the same graph the UAV's height { only the rst component of the state vector) when under openloop and closedloop control. Which one is more robust? Make sure you. use the same noise sequences for the open-loop and the closed-loop. Use the command mgf) before starting generating the random sequence. {useful JFunction normrnd) 3. {1'2 points) Optimal Cmtml The LQR control offers an effective way to tune the feedback control law. By carefully selecting matrices Q. R, we can control the tradeo between how small the state is and how large the control effort is without damaging stability. Here, we will study the innitehorizon LQR control with cost.- 00 J = (a: some: to + uiRu (5) k= where :cd 2 [2,0]T {remember we want the system to hover id = U at the desire height .230: = 2). (a) (7 points) Simulate system (2) under the innitehorizon LQR control (5}, with Q 2 111R 2 1 for h = D,...,15 {useful function dlqr, be careful of the sign). Repeat for Q : 10011 and R : 1. Plot the states my and the input ink for both cases. Compare the plots and explain any dierences. {l1} {5 points} Assume that the actuator cannot achieve forces larger than 10.5. What LQR parameters should you tune to limit the control eort? y to tune them such that up; <_: w: e report q and plot the states :33 input up. this is a classic model predictive control problem i.e. constrained optimal problem. you will nd more about those topics in ese61.9. points make sure include matlab code. z f mg figure uav moves zaxis direction under gravitational force actuator f. objective to hover at height consider one dimensional of i which moving only zomis direction. its state comprises velocity hence equations motion are: mizfmg where m mass g acceleration. initial we study space methods for design. particular desired with zero velocity. discretization prior controlling system need discretize continuoustime dynamics given above using sampling period t5. assume that piece wise constant intervals prove discretized satises dierence equation: t5 l now t="1," apply mu obtain n an ba. :30="[0," find numerical ualues b. design controllers enable around height. open-loop us closed-loop following questions keep assumption ll use verify controllable. minimum norm sequence : .t.="2,0" openloop controller. feedback law kt :cd forces provided poles bk are strictly inside unit circle. then controller k so closedloop it="0,..,50," simulate repeat closed loop evolution each method on same uelocity compare two responses>