American International University- Bangladesh Faculty of Engineering (EEE) Control Systems Laboratory

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1 Experiment 1 American International University- Bangladesh Faculty of Engineering (EEE) Control Systems Laboratory Title: Introduction to design and simulation of open loop and close loop control systems using a Computer Aided Design (CAD) tool---matlab. Abstract: This experiment introduces students to the widely used simulation software MATLAB and how it is used to describe control system related problems and functions. It shows a few basic techniques such as - 1. Creating polynomial functions, finding roots of a polynomial function, evaluating polynomials, calculating product of two polynomial functions in MATLAB. 2. Obtaining overall transfer functions of open loop and close loop control systems. 3. Observing the step responses of open-loop and closed-loop control systems. Introduction: The objectives of this experiment are to complete the above mentioned tasks with the knowledge of using all the commands associated with the tasks. Basic mathematical skills such as Laplace transformation in control system and understanding of open loop and close loop systems and their differences is necessary for this experiment. As all the tasks are associated with MATLAB, some basic commands and their various uses along with some user guideline for this software is necessary. This entire experiment is based on MATLAB codes in 3 groups. First group of code shows polynomial and its mathematical manipulation, second group shows a simulation of open loop control system and the third group of codes shows a simulation of close loop control system. Doing these in MATLAB allows a accurate solution for students instead doing them using mathematically. It shows them another way of doing these tasks and it saves time. Basic parameters used in MATLAB: Table 1: Managing workspace and file commands

2 Command cd clc clear (all) clear x copyfile delete dir exist help lookfor mkdir movefile pwd rmdir type what which who whos Description Change current directory Clear the Command Window Removes all variables from the workspace Remove x from the workspace Copy file or directory Delete files Display directory listing Check if variables or functions are defined Display help for MATLAB functions Search for specified word in all help entries Make new directory Move file or directory Identify current directory Remove directory Display contents of file List MATLAB files in current directory Locate functions and files Display variables currently in the workspace Display information on variables in the workspace Table 2: Elementary matrices and arrays Command eye linspace ones rand zeros Description Identity matrix Generate linearly space vectors Create array of all ones Uniformly distributed random numbers and arrays Create array of all zeros Table 3: Arrays and Matrices: Basic information Command disp isempty isequal length ndims numel size Description Display text or array Determine if input is empty matrix Test arrays for equality Length of vector Number of dimensions Number of elements Size of matrix

3 Precautions: Make sure that all the m-files are saved in the current directory of MATLAB. Otherwise, the m-files will not be found and executed when they are called from the command window. Theory and Methodology: A combination of components that act together is called system. There are two types of systems- open loop and close loop control systems. Systems in which the output quantity has no effect on the input quantity are open-loop control systems. A generic open-loop system is shown in Fig. 1.1 below. Fig. 1.1: Block diagram of an Open-Loop System The input transducer converts the input into the form that is used by the controller. The controller then drives a process or plant. Other signals such as disturbances are added to the controller and process outputs via summing junctions. The drawback of open-loop system is that it cannot compensate for any disturbances that add to the controller driving signal or the output. Systems in which the output quantity has an effect upon the input quantity---primarily to maintain the desired output value---are close-loop control systems. In close loop control systems feedback of the output quantity is a must. The unit that provides the means for feeding back the output quantity or a function of the output to compare it with the reference input is called a feedback element. The block diagram of closed-loop system is given in fig The output transducer or sensor measures the output response and converts it into the form used by the controller. The first summing junction algebraically adds the signal from the input to the signal from the output, which arrives via the feedback path. The result from this is generally called error signal or actuating signal. If the output signal is directly compared with the reference input then this type of closed-loop system is called a unity feedback system. If a portion of the output signal is compared with the reference input signal then this type of closed-loop system is called a non-unity feedback system. Fig. 1.2: Block diagram of Closed-Loop System

4 The advantage of closed-loop system over open-loop system is that, it can compensate for the disturbance by measuring the output response, feeding the measurement back through a feedback path and comparing that response to the input at the summing junction. Closed-loop systems are less sensitive to noise, disturbances and changes in the environment. In the 1960s, control engineers turned to linear algebra techniques of state space modeling for these systems. The matrix-based mathematics of state space analysis and design drove the need for more sophisticated computer analysis. C. B. Moler of the University of New Mexico wrote an interactive program called MATLAB. The introduction of MATLAB was a phenomenal success, as it revolutionized computer-aided analysis and design of control systems. Open loop control system block diagram: The transfer unction for open loop - Ge(s) = C(s)/R(s) Close loop control system block diagram: Closed-loop, transfer function shown in the Figure right below,

5 The product, G(s)H(s), in equation is called the open-loop transfer function, or loop gain. POLYVAL means Evaluate polynomial. Y = POLYVAL(P,X) returns the value of a polynomial P evaluated at X. P is a vector of length N+1 whose elements are the coefficients of the polynomial in descending powers. Y = P(1)*X^N + P(2)*X^(N-1) P(N)*X + P(N+1) PRINTSYS Print system in pretty format. PRINTSYS is used to print state space systems with labels to the right and above the system matrices or to print transfer functions as a ratio of two polynomials. ROOTS Find polynomial roots. ROOTS(C) computes the roots of the polynomial whose coefficients are the elements of the vector C. If C has N+1 components, the polynomial is C(1)*X^N C(N)*X + C(N+1). POLY Convert roots to polynomial. POLY(A), when A is an N by N matrix, is a row vector with N+1 elements which are the coefficients of the characteristic polynomial, DET(lambda*EYE(SIZE(A)) - A). POLY(V), when V is a vector, is a vector whose elements are the coefficients of the polynomial whose roots are the elements of V. For vectors, ROOTS and POLY are inverse functions of each other, up to ordering, scaling, and roundoff error. CONV Convolution and polynomial multiplication. C = CONV(A, B) convolves vectors A and B. The resulting vector is length LENGTH(A)+LENGTH(B)-1. If A and B are vectors of polynomial coefficients, convolving them is equivalent to multiplying the two polynomials. STEP Step response of LTI models. STEP(SYS) plots the step response of the LTI model SYS (created with either TF, ZPK, or SS). For multi-input models, independent step commands are applied to each input channel. The time range and number of points are chosen automatically. [Y,T,X] = STEP(SYS) also returns, for a state-space model SYS, the state trajectory X, a LT-by- NX-by-NU array if SYS has NX states. FEEDBACK Feedback connection of two LTI models. SYS = FEEDBACK(SYS1,SYS2) computes an LTI model SYS for the closed-loop feedback system Negative feedback is assumed and the resulting system SYS maps u to y. To apply positive feedback, use the syntax SYS = FEEDBACK(SYS1,SYS2,+1).

6 p1=[ ] means defining the function pl = s 3 + 0s 2 + 5s 1 + 6s 0 The row vector is defined in left to right descending order. using polyval, putting s=1; pl is evaluated = (1) + 6 = 12 roots of pl = s 3 + 0s 2 + 5s + 6 = s(s+2)(s+3) For open loop, overshoot is somewhat and steady state value has reached after 1.5 seconds

7 For close loop, overshoot is somewhat (reduced in comparison to open loop) and steady state value has reached before 1.5 seconds Pre lab homework: Students should study - 1. Laplace transform and its mathematical manipulations, 2. Basic idea and block diagram of control systems, 3. Basic idea and block diagram of open loop control system, 4. Basic idea and block diagram of close loop control system. Equipments : Computer Precautions: Students should take the following precautions while conducting the experiment - 1. check the PC if it is working; otherwise inform lab instructor, 2. be careful while handling the power supply buttons of computers and to keep them off when they are not needed, 3. Ckeck if MATLAB is working correctly, Experimental Procedure: Working in MATLAB environment: 1. To start MATLAB software package, click on the START menu in the desktop, then find the MATLAB icon from the program group. This will open a MATLAB Command Window for you. You can find the MATLAB software package in the desktop as well. You can just double-click it. 2. When the MATLAB Command Window pops up, you will get a prompt (>>). You can write your MATLAB commands after this prompt and MATLAB will execute them line by line. Working in the MATLAB environment is generally straightforward as most commands are entered as you would write them mathematically. For example, entering the following sample expression >> a = 4/3 yields the MATLAB response as follows a = However, it is much more convenient to use an M-file (with.m extension) than to enter commands line by line at the MATLAB prompt. In MATLAB Command Window, click New- >M-file to bring up a new M-file. 4. In M-files, statements that begin with a % are considered to be comments and are ignored by MATLAB. The commands in these files are executed in the MATLAB environment simply by entering the name of the file without the.m. For this lab, an M-file Lab1.m was written in C:\Spring 2012 folder. This same file has been loaded in all the computers in the lab. To run this file first you have to change your working directory by writing the following commands at your MATLAB prompt as: >>C:\ Spring 2012

8 Now write the file name at MATLAB prompt to run your file from the directory like following: >>Lab1 and then press ENTER. You will get the answers like following: p1 = num/den = s^3 + 5 s ans = With the help of the teacher, run the M-file on a PC in MATLAB environment. Write down the answers from the MATLAB command prompt in a your notebook and analyze them. Remember, in MATLAB, if the semicolon at the end of a command is not used, it will display computed answer for that command in the MATLAB command window. 6. After running the M-file, two figures will pop up. Go to Figure No. 2. Click on Tools-Axes Properties. In Edit Axes Properties window, click the Manual in Tick Step option. Enter 0.2 for time in seconds. Keep the default value for y(t). Click Manual in the Limits option. Enter 1.8 in the right cell for time in seconds. Keep the left cell as 0. Click Apply and then Ok. This will allow you to analyze both the figures within the same zone. Simulation: See the MATLAB Code in appendix section Command window Output: p1 = num/den = s^3 + 5 s ans = 12 r1 =

9 i i p2 = num/den = s^ e-015 s^2 + 5 s exp = OL_TRF_num = 1 1 OL_TRF_den = num/den = s s^2 + 8 s + 15 G = 1 1 H = CL_TRF_num = 0 1 1

10 CL_TRF_den = num/den = s s^2 + 9 s + 16

11 Report: Write a report on this experiment. With the approval and help of the teacher paste the Matlab results (from the Matlab Command Window), Matlab codes (from the M-file) and Matlab plots in an MS Word file. Only one group should print out this file. Members of the group and other groups should make copies of this printout at some other time. These printouts should be attached in the Body of everyone s lab reports. Comment on the step responses for different control systems. Did you find any difference in the step responses of open loop and closed loop control systems? Provide your comments and recommendations for it. Reference: Norman S. Nise, Control Systems Engineering, available Edition, John Wiley & Sons Inc.

12 Appendices %Lab 1 %MATLAB codes clc; close all; clear all; %EEE 4101 (Control Systems) %Clears the MATLAB Command Window %Closes all figures %Clears all variables %Part 1 %Creating equations, Calculating roots, Outputting transfer functions etc. p1=[ ] printsys(p1,[ ],'s') polyval(p1,1) r1=roots(p1) p2=poly(r1) printsys(p2,[ ],'s') %Defines the coefficients of a linear equation %Outputs the polynomial of the equation %as a transfer function and also in %symbolic form, where the variable %is labelled as "s" %Evaluates the polynomial of the equation %with a value of 1 %Calculates the roots of the equation %Creates an equation from the roots (r1). %This should match with the original equation(p1). %But it will not match exactly in this case. %This is because the imaginary portion of the %complex root has fractions and MATLAB cannot %incorporate all digits to the right of a %fraction in computations. %Outputs the equation as a transfer function %and also in symbolic form, where the variable %is labelled as "s". This should match with %the previous transfer function exp1=[1 1]; exp2=[1 2]; exp=conv(exp1,exp2) %Define an expression called exp1 %Define an expression called exp2 %Multiply these two expresions and store the %result in exp

13 %Part 2 %Simulating an open loop control system clear all; OL_TRF_num = [1 1] OL_TRF_den = conv([1 3], [1 5]) %Clears all variables (i.e. p1,r1,p2 etc.) %Define the numerator of a transfer function %Define the denominator of a transfer function printsys(ol_trf_num, OL_TRF_den, 's') %Outputs the transfer function (open-loop) in %symbolic form, where the variable is "s" [y, x, t]=step(ol_trf_num, OL_TRF_den); %Simulate this system for a step input.it will %store the simulation results in certain %variables like y,x,t figure (1) plot(t, y) grid %Label the figure %Display the simulation in a 2-D plot %Add gridlines in the figure title('unit step responce for open loop') ylabel('y(t)') xlabel('time in seconds') %Add a title to the figure %Label y axis %Label x axis %Part 3 %Simulating a close loop control system %This control system has unity feedback clear all; G = [1 1] H = conv([1 3], [1 5]) %Clears all variables (i.e. p1,r1,p2,gnum, %Gden,y,x,t etc.) %Define the feedforward transfer function %Define the feedback transfer function [CL_TRF_num, CL_TRF_den] = cloop(g, H) %Form the resulting close-loop transfer function printsys(cl_trf_num, CL_TRF_den, 's') %Outputs the transfer function (open-loop) in %symbolic form where the variable is "s" [y, x, t]=step(cl_trf_num, CL_TRF_den);

14 %Simulate this system for a step input.it will %store the simulation results in certain %variables like y,x,t figure (2) plot(t, y) grid %Label the figure %Display the simulation in a 2-D plot %Add gridlines in the figure title('unit step responce for close loop') %Add a title to the figure ylabel('y(t)') xlabel('time in seconds') %Label y axis %Label x axis