1. [+2, 15 min] You release a ball from an initial height of h(t = 0) = 22a + 100 meters. The acceleration due to gravit

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1 2 15 Min You Release A Ball From An Initial Height Of H T 0 22a 100 Meters The Acceleration Due To Gravit 1 (63.21 KiB) Viewed 31 times

1. [+2, 15 min] You release a ball from an initial height of h(t = 0) = 22a + 100 meters. The acceleration due to gravity is g = -9.8m/s². You may ignore air resistance. Let x₁ = h be the height of the ball and x2 = h be its velocity. Write a state-space model for the dynamics of the falling ball, along with an initial con- dition (0). (b) At what time T does the ball hit the ground and with what velocity?
2. [+2, 15 min] Consider a system i = Ax + Bu, y = Cx+ Du, where B is a (b+1)x(c+1) matrix. (a) How many inputs does the system have? (b) How many outputs does the system have? (c) How many states does the system have? (d) How many transfer functions does the system have? (e) What is the size of the controllability matrix?
3. [+2, 15 min] Consider the single-input single-output system -a b x = x + Bu с -d y = Cx Find a B matrix (of appropriate size) such that the system is controllable. (b) Find a C matrix (of appropriate size) such that the system is unobservable. AND Give two states that are now unobservable with your B.
4. [+2, 15 min] Consider the single-input single-output system -a b x = x + U₂ C y = Cx, where is your solution to Problem 3a (or if you skipped problem 3, any C that makes the system observable). Design an observer-based controller that places the closed-loop controller poles at -1, -2, and places the observer poles at -3, -4. Give your final answer in terms of the control gain K, the observer gain L, and the final transfer function of your controller C(s).
5. [+2, 15 min] Consider the scalar system x = (d+1)x+u (a) Design a controller u(x) that minimizes the objective function J = = √ x(t)² + pu(t) ², where p = c + 1. (b) The case when p→ 0 could be called the 'cheap control' case. Explain this terminology and what it means on the resulting controller. (c) The case when p→∞ could be called the 'expensive control' case. Explain this terminology and what it means on the resulting controller.
6. [+2, 15 min] The dynamics of your blimp robot from a bird's eye view is given by x = v cos 0 ỷ=usin 0 8 = w where v is the forward thrust speed (back propeller) and w is the turning rate (rudder). Your robot broke so the propeller is stuck on full throttle v = 1, but you have access to control the rudder u = w. = (a) Let the initial condition of the blimp be x(0) y (0) trajectory of the blimp for u(t) = 1 for all t > 0. 0 (0) 0. Compute and draw the (b) Design a controller u(x, y, 0) that drives the blimp directly to the North (i.e., 0 = π/2).