Problem 2 Consider the following model of an electrical heating system, used to heat a room: x(t) = [-0.5 1 2 x(t) + u(t) + 0(t), y(t) = [10] x(t), where y is the room temperature, u is the power to the heater, and is the outdoor temperature. The state variable x is the room temperature, and x2 is the temperature of the heater. Assume that the outdoor y is measured and that the control system should cope with a typical winter in Uppsala, i.e., assuming (t) = 0C. Let's indicate by r(t) the desired room temperature. a) Design L and lo so that the closed-loop eigenvalues are -1.5 and -6, and the static error is zero when r(t) = step(t). Don't forget to check the conditions for designing a state-feedback controller! [1pt] b) Design the innovation gain K of the observer so that its poles are in -10 and -20. Don't forget to check the conditions for designing a state observer! [1pt] c) Finally, write this controller as a general linear feedback. [1pt]