1 ECE-320: Linear Control Systems Homework 5 Due: Thursday April 7 at the beginning of class 1) Consider the plant 013()0.5pGsss where 3 is the nominal value of 0 and 0.5 is the nominal value of1. 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, ()oooGss i) For what value of owill 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)3osGss s s   iv) Show that the sensitivity of ()oGsto variations in 0 is given by 008GsSs v) Show that the sensitivity of ()oGsto variations in 1 is given by 120.58.5 4oGsSss b) (Proportional Control) Consider a proportional controller, with 2.5pk . i) Show that the closed loop transfer function is2 0102.5()2.5oGss ii) Show that the sensitivity of ()oGsto variations in 0 is given by 000.58GsSs iii) Show that the sensitivity of ()oGsto variations in 1 is given by 10.58oGSs 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)osGss s s   ii) Show that the sensitivity of ()oGsto variations in 0 is given by 002( 0.5)12.5 120GssSss iii) Show that the sensitivity of ()oGsto variations in 1 is given by 120.512.5 120oGsSss 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 0 for each type of controller on one graph at the top of the page, and the sensitivity to 1 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.3 Hint: If 22()2 10sTsss, 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'); 2) For the following two circuits, show that the state variable descriptions are given by  11( ) ( ) ( )( ), ( ) 0 [0] ( )111( ) ( ) ( )bL L Lin b inc c caaRi t i t i tLdLLV t y t R V tv t v t v tdtRCC R C                   1( ) ( ) ( )()()( ), ( ) 0 [ ] ( )( ) ( ) ( )100abaL L La b aababin inc c ca b a bRRi t i t i tR R RL R R LdL R RV t y t V tv t v t v tdt R R R RRC                   4 3) For the plant 22()121pnnKGsss a) If the plant input is()ut and the output is()xt, show that we can represent this system with the differential equation 212( ) ( ) ( ) ( )nnx t x t x t Ku t   b) Assuming we use states 1( ) ( )q t x tand 2( ) ( )q t x t, and the output is ()xt, show that we can write the state variable description of the system as    112222120 1 0( ) ( )()2( ) ( )()( ) 1 0 0 ( )()n n nq t q tdutKq t q tdtqty t u tqt                     or ( ) ( ) ( )( ) ( ) ( )q t Aq t Bu ty t Cq t Du t Determine the A, B, C and D matrices. c) Assume we use state variable feedback of the form ( ) ( ) ( )pfu t G r t kq t, where ()rtis the new input to the system, pfGis a prefilter (for controlling the steady state error), and kis the state variable feedback gain vector. Show that the state variable model for the closed loop system is ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )pfpfq t A Bk q t BG r ty t C Dk q t DG r t     or ( ) ( ) ( )( ) ( ) ( )q t Aq t Br ty t Cq t Dr t d) Show that the transfer function (matrix) for the closed loop system between input and output is given by 1()( ) ( )( ( ))()pf pfYsG s C Dk sI A Bk BG DGRs      and ifD is zero this simplifies to 1()( ) ( ( ))()pfYsG s C sI A Bk BGRs    e) Assume( ) 1rt  and 0D . Show that, in order for lim ( ) 1tyt, we must have5 11()pfGC A Bk B Note that the prefilter gain is a function of the state variable feedback gain! Preparation for Lab 5 4) You will be using this code and the following designs in Lab 5, so come prepared! This prelab is really pretty mindless, so just follow along The one degree of freedom Simulink model (Basic_1dof_State_Variable_Model.mdl) implements a state variable model for a one degree of freedom system. This model uses the Matlab code Basic_1dof_State_Variable_Model_Driver.m to drive it. Both of these programs are available on the course website. a) Get the state variable model files for the systems you modeled in lab. Since you will be implementing these controllers during lab 5, if you have any clue at all you and your lab partner will do different systems! You will need to have Basic_1dof_State_Variable_Model_Driver.m load the correct state model into the system! b) You need to set the saturation_level to the correct level for the rectilinear (model 210) or torsional (model 205) system. Assume we have an input step of 1 cm or 15 degrees (be