1) Consider the plantwhere 3 is the nominal value of and 0.5 is the nominal value of. In this problem we will investigate the sensitivity of closed loop systems with various types of controllers to these two parameters. We will assume we want the settling time of our sys...a) (ITAE Model Matching) Since this is a first order system, we will use the first order ITAE model,i) For what value of will we meet the settling time requirements and the steady state error requirements?ii) Determine the corresponding controller.iii) Show that the closed loop transfer function (using the parameterized form of and the controller from part ii) isiv) Show that the sensitivity of to variations in is given byv) Show that the sensitivity of to variations in is given byb) (Proportional Control) Consider a proportional controller, with .i) Show that the closed loop transfer function isii) Show that the sensitivity of to variations in is given byiii) Show that the sensitivity of to variations in is given byc) (Proportional+Integral Control) Consider a PI controller with and .i) Show that the closed loop transfer function isii) Show that the sensitivity of to variations in is given byiii) Show that the sensitivity of to variations in is given byd) Using Matlab, simulate the unit step response of each type of controller. Plot all responses on one graph. Use different line types and a legend. Turn in your plot and code.g) Using Matlab and subplot, plot the sensitivity to for each type of controller on one graph at the top of the page, and the sensitivity to on one graph on the bottom of the page. Be sure to use different line types and a legend. Turn in your plot...In particular, these results should show you that the model matching method, which essentially tries and cancel the plant, are generally more sensitive to getting the plant parameters correct than the PI controller for low frequencies. However, for h...Hint: If , plot the magnitude of the frequency response using:T = tf([2 0],[1 2 10]);w = logspace(-1,1.7,1000);[M,P]= bode(T,w);Mdb = 20*log10(M(:));semilogx(w,Mdb); grid;xlabel('Frequency (rad/sec)');ylabel('dB');3) For the planta) If the plant input is and the output is, show that we can represent this system with the differential equationb) Assuming we use states and , and the output is , show that we can write the state variable description of the system asorDetermine the A, B, C and D matrices.c) Assume we use state variable feedback of the form , where is the new input to the system, is a prefilter (for controlling the steady state error), and is the state variable feedback gain vector. Show that the state variable model for the closed loo...ord) Show that the transfer function (matrix) for the closed loop system between input and output is given byand if is zero this simplifies toe) Assume and . Show that, in order for , we must haveNote that the prefilter gain is a function of the state variable feedback gain!If matrix is given asthenand the determinant of is given by . This determinant will also give us the characteristic polynomial of the system.4) For each of the systems below:- determine the transfer function when there is state variable feedback- determine if and exist () to allow us to place the closed loop poles anywhere. That is, can we make the denominator look like for any and any . If this is true, the system is said to be controllable.a) Show that forthe closed loop transfer function with state variable feedback isb) Show that forthe closed loop transfer function with state variable feedback isc) Show that forthe closed loop transfer function with state variable feedback is1 ECE-320: Linear Control Systems Homework 4 Due: Tuesday January 10 at the beginning of class 1) Consider the plant 013()0.5pGsssαα= =++ where 3 is the nominal value of 0α and 0.5 is the nominal value of1α. In this problem we will investigate the sensitivity of closed loop systems with various types of controllers to these two parameters. We will assume we want the settling time of our system to be 0.5 seconds and the steady state error for a unit step input to be less than 0.1. a) (ITAE Model Matching) Since this is a first order system, we will use the first order ITAE model, ()oooGssωω=+ i) For what value of oωwill we meet the settling time requirements and the steady state error requirements? ii) Determine the corresponding controller()cGs. iii) Show that the closed loop transfer function (using the parameterized form of ()pGs and the controller from part ii) is 0108( 0.5)3()8( ) ( 0.5)3osGsss sααα+=++ + iv) Show that the sensitivity of ()oGsto variations in 0α is given by 008GsSsα=+ v) Show that the sensitivity of ()oGsto variations in 1α is given by 120.58.5 4oGsSssα−=++ b) (Proportional Control) Consider a proportional controller, with 2.5pk =. i) Show that the closed loop transfer function is 0102.5()2.5oGssααα=++2 ii) Show that the sensitivity of ()oGsto variations in 0α is given by 000.58GsSsα+=+ iii) Show that the sensitivity of ()oGsto variations in 1α is given by 10.58oGSsα−=+ c) (Proportional+Integral Control) Consider a PI controller with 4pk = and 40ik =. i) Show that the closed loop transfer function is 0104 ( 10)()( ) 4 ( 10)osGsss sααα+=++ + ii) Show that the sensitivity of ()oGsto variations in 0α is given by 002( 0.5)12.5 120GssSssα+=++ iii) Show that the sensitivity of ()oGsto variations in 1α is given by 120.512.5 120oGsSssα−=++ d) Using Matlab, simulate the unit step response of each type of controller. Plot all responses on one graph. Use different line types and a legend. Turn in your plot and code. g) Using Matlab and subplot, plot the sensitivity to 0α for each type of controller on one graph at the top of the page, and the sensitivity to 1α on one graph on the bottom of the page. Be sure to use different line types and a legend. Turn in your plot and code. Only plot up to about 8 Hz (50 rad/sec) using a semilog scale with the sensitivity in dB (see below). Do not make separate graphs for each system! In particular, these results should show you that the model matching method, which essentially tries and cancel the plant, are generally more sensitive to getting the plant parameters correct than the PI controller for low frequencies. However, for higher frequencies the methods are all about the same. Hint: If 22()2 10sTsss=++, plot the magnitude of the frequency response using: T = tf([2