## Question

###### 3. ^ car cnvironmcntal controllcr maintains thc tcmpcraturc at a constant levcl Too. Thc passcngcr compartmcnt dissipates (loscs) hcat according to Newton's law. Thc car' s onboard systcm utilizes PI-controllcr to hold thc tcmpcraturc at the desired setting Too: According to Our class notes this meansd[' +kT = kR dtwith initial condition T(0) To and R satisfiesR = K 'Tx-T+d f( -Tlu)Jdu) with the controller parameters as the gain K (dimensionless) and integral time ti (units o

3. ^ car cnvironmcntal controllcr maintains thc tcmpcraturc at a constant levcl Too. Thc passcngcr compartmcnt dissipates (loscs) hcat according to Newton's law. Thc car' s onboard systcm utilizes PI-controllcr to hold thc tcmpcraturc at the desired setting Too: According to Our class notes this means d[' +kT = kR dt with initial condition T(0) To and R satisfies R = K 'Tx-T+d f( -Tlu)Jdu) with the controller parameters as the gain K (dimensionless) and integral time ti (units of time): Diffcrcntiatc our diffcrcntial cquation for T and rcwritc this pair of cqua- tions as singlc, sccond-ordcr diffcrcntial cquation for thc tcmpcraturc Note that wC have kR on the RHS while wC had R in the notes_ This choice means that k has no effect on R and R is an external temperature _ Write down the rules that distinguish when the systern is overdamped underdamped; and critically damped. What arc thc units of R? Let' $ assume that = 1, & = 1, and Tx = 1_ Choose the gain K that makes Our controller critically damped. Solve the problem for the temperature using the parameters in (3d). Use thc four-stcp approach: Stcp 1: Gct Yh, Stcp 2: Gct Yp; Stcp 3: Satisfy conditions using y = Yh + Yp, Stcp 4: Sketch: