(1) In this problem, you will implement Kalman filters to estimate the core-temperature of cylindrical batteries under unknown cooling conditions. As show in Fig. 1, three estimation approaches (an open-loop observer, a Kalman filter, and a dual Kalman filter) are included in DEKF_h_est_sin 2020.51x. Unzip the provided file Estimation_HW7.zip. The thermal dynamics of the considered cylindrical cell are based on the two-state thermal model as given by: dr -Az + Bu, y-Cx+Du where x = 5, u — [ġ T…]³ and y = (T. T.] are states, inputs and outputs respectively. System matrices A, B, C, and D are defined as follows: -48ah R(24k+ Rh) -150h 24k+Rh A- -320ah -120a(4k+Rh) R²(24k+ Rh) 48ah R²(24k+ Rh) a B- kVR(24k+Rh) 320ah 0 C- 24k+Rh R²(24k+ Rh) 24k-3Rh 120Rk,+15R²h 8(24k+ Rh) 24k 24k+Rh 15Rk 48k +2Rh 4Rh 0 D- 24k+ Rh Rh 0 24ks + Rh Since Kalman filters are to be implemented in the discrete-time domain, the dynamics are discretized via the forward Euler approxiamtion, resulting in the following system: 2+1 - Ash + Belt -Cz + Du where A-I+AAt and B₁ - BAt with a sampling time At. Using the Kalman filter algorithm provided in lecture slides, fill the blanks in the two Matlab function blocks: Thermal DEKF and Thermal KF. The file HW5 DEKF_init.m initializes the simulation including the thermal parameters of the considered battery, the initial values of state and parameter estimates, and loading measurement data. Submit a figure with three subplots showing the convection coefficient, core and surface temperature trajectories (vs. time) obtained from the submodules and actual data.

(a) The first-order system has the transfer function T(s)=\frac{A}{1+s T} Find the system output response in time domain when a unit ramp signal, r(t)=t, is applied to its input. (b) The transfer function of a second-order system is: T(s)=\frac{4}{s^{2}+4 s+4} Find the system output response in time domain when a unit ramp signal, r(t)=t, is applied to its input. \text { (Note: The inverse Laplace transform of } F(s)=\frac{1}{(s+a)^{2}} \text { is } f(t)=t \cdot e^{-a t} \text {.) }

E1.13 Consider the inverted pendulum shown in Figure E1.13. Sketch the block diagram of a feedback control system. Identify the process, sensor, actuator, and controller. The objective is keep the pendulum in the upright position, that is to keep 0 = 0, in the pres- ence of disturbances. Optical encoder to measure angle 7/ 7, torque m, mass

1. Determine whether the following systems are linear systems or not, and show (or explain) why [note that x represents the input and y represents the output]: \text { System 1: } y=x+1 \text { System 2: } y=x+x^{3}

Problem 2. The sine input u = v2 sin 2t and the corresponding output of the stable linear dynamic system are shown in Fig. 1. During steady-state, they have the same frequency, but different amplitudes (the output amplitude is 1) and phase angles. The phase delay turns out to be one eighth (1/8) of the input period. Based on this observation, determine the frequency response function value at the input frequency, in other words G(2j).

E1.4 An automobile driver uses a control system to main- tain the speed of the car at a prescribed level. Sketch a block diagram to illustrate this feedback system.

P1.8 The student-teacher learning process is inherently a feedback process intended to reduce the system error to a minimum. Construct a feedback model of the learning process and identify each block of the system.

1. Reduce the block diagram shown in Figure P5.1 to a single transfer function, T(s) = C(s)/R(s). Use the following methods: R(s) a. Block diagram reduction [Section: 5.2] b. MATLAB 50 s+1 2 S FIGURE P5.1 S 2 MATLAB ML C(s)

4. Let I=c=4 for the PI controller shown. The performance specifications require that τ= 0.2 (a) Compute the required gain values for each case, 1) <=0.707 2) <=1 (b) Use matlab to plot the unit-step command responses for each of the cases in (a). Compare.

B. The transfer function of a system is: T(s)=\frac{s(s+2)}{s^{2}+3 s+1} Find the steady-state response (output) of this system for: (a) Unit step input, (b) Ramp input with a slope of 2 units.

P1.5 A light-seeking control system, used to track the sun, is shown in Figure P1.5. The output shaft, driven by the motor through a worm reduction gear, has a bracket attached on which are mounted two photocells. Complete the closed-loop system so that the system follows the light source./nMotor S Gears € Photocell tubes * Light source FIGURE P1.5 A photocell is mounted in each tube. The light reaching each cell is the same in both only when the light source is exactly in the middle as shown.