let Theta=Theta(t)[C] be the temperature of a graphics processing unit (GPU), a component of a computer. the computations performed by the GPU generate heat at a rate q=q(t)[J/s]. to prevent the temperature of the GPU from rising enough to damage its components, a fan is used to blow air, which is at temperature theta_a=theta_a(t)[C], across the GPU at a constant speed. owing to heat transfer from the (hotter) GPU to the (colder) air, the air cools the GPU.

Deviation Variables:

theta_a(t):=theta_a(t) theta_a_bar q(t):=q(t) q_bar theta_(t):=theta(t)theta_bar

Dynamic model:

C(dtheta/dt)=q+UA[theta_a-theta]

C, U, and Aare [constant] parameters.

Question:

1.a suppose you are given the steady state values of the inputs, theta_a_bar and q_bar. write the steady state value of the output theta_bar in terms of these two inputs (and model parameters).

b. write the dynamic model in terms of the deviation variables theta_a(t), q_(t), and theta_(t).

c. derive the transfer functions Gtheta_a(s)and Gq(s)that describe how the temperature theta_ responds to the inputs theta_aand q_, respectively. write both in stan- dard, gain/time-constant form.

let 0 =0(t) [C] be the temperature of a graphics processing unit (GPU), a component of a computer. the computations performed by the GPU generate heat at a rate q=q(t) [J/s]. to prevent the temperature of the GPU from rising enough to damage its components, a fan is used to blow air, which is at temperature a 0.(t) [C], across the GPU at a constant speed. owing to heat transfer from the (hotter) GPU to the (colder) air, the air cools the GPU. le. OK cool air quis fan GPU Figure 1: GPU cooling. two inputs: (1) the temperature of the air, 6.(t), and (2) the heat generated by the computations, q(t). both inputs affect one output: the tem- perature of the GPU, 0(t). a dynamic model for the output (t) in response to the two inputs q(t) and 0 (t) is: de CHE = qt) + UA[0.(t) 0(t)] (1) dt here, C, U, and A are [constant] parameters. define deviation variables: 07(t) == 0,(t)-02 q*(t) := 9(t)- 0*(t) == 0(t) - , with 7, 2, and q nominal steady state values of e(t), 0.(t), and g(t), respectively. ai