2 Solving Ordinary Differential Equations Using MATLAB

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1 Penn State Erie, The Behrend College School of Engineering E E 383 Signals and Control Lab Spring 2008 Lab 3 System Responses January 31, 2008 Due: February 7, 2008 Number of Lab Periods: 1 1 Objective The objective of this laboratory exercise is to learn how to use MATLAB to solve ordinary differential equations (ODEs) and find the responses of LTI systems for impulse, step and general inputs. 2 Solving Ordinary Differential Equations Using MATLAB MATLAB has the capability to solve a wide variety of problems involving differential equations. In the following, we introduce how to use MATLAB to solve initial value ordinary differential equations. We consider the following initial value problem (IVP) which is a set of first-order differential equations in state space representation along with initial conditions. ẋ = f(x, t), x(t 0 )=x 0. (1) Here x(t) =[x 1 (t),,x n (t)] T is the state variable in R n space; x 0 =[x 10,,x n0 ] T is the initial state. Note: When an initial value problem is not specified as a set of first-order differential equations, it can be written as one by using the technique that you will learn in EE BD 410 (It will also be briefly explained during the lab). MATLAB offers seven IVP solvers: ode23, ode23s, ode23t, ode23tb, ode45, ode113 and ode15s. Among then, ode45 is typically the first solver to try on a new problem. We will use ode45 in the followings. An example of using the IVP solver to solve the classic van der Pol equation ÿ μ(1 y 2 )ẏ + y =0 (2) 1

2 is presented as follows. Here μ is a parameter greater than zero. If we choose x 1 = y, x2 =ẏ, then the van der Pol equation becomes ẋ 1 = x 2, (3) ẋ 2 = μ(1 x 2 1)x 2 x 1. (4) Now we begin solving the equations. Note that the related m files vdpol.m and solution.m can be downloaded from the course web page (or you can type them in). To solve the equations (3)-(4), they must be coded in a function m file as xdot = odefile(t,x) (see Figure 1 for the function when μ = 2). That is, the file must accept a time t and a solution x and return values for the derivatives. Note that the input arguments are t and x even when the function does not use t explicitly. Note also that the output xdot must be a column vector. function xdot=vdpol(t,x) %VDPOL van der Pol equation. % Xdot = VDPOL(t,X) % Xdot(1) = X(2) % Xdot(2) = mu*(1-x(1)^2)*x(2)-x(1) % mu = 2 mu = 2; xdot = [x(2); mu*(1-x(1)^2)*x(2)-x(1)]; Figure 1: The MATLAB function vdpol.m. Given the above ODE file, this set of ODEs is solved using the following script m file(seefigure 2) (we assume the span of t is [0, 20] and x(0) = [2 0] T ). %solution.m %Solving the van der Pol equation %by using vdpol.m. tspan = [0 20]; % time span to integrate over x0 = [2; 0]; % initial conditions (must be a column) [t,x] = ode45( vdpol,tspan,x0); % solving the equation by calling % the ODE file name and using default % settings of ode45.m Figure 2: The MATLAB function solution.m. After you have run solution.m, you can view the sizes of t and x by using size(t) and size(x) respectively. You can also plot the two states on the same plot by typing in plot(t,x(:,1),t,x(:,2), -- ). Note: In solution.m, you can specify the solution time points you desired by simply adding them to tspan, for example tspan = linspace(0,20,100);. If you are not satisfied with the default 2

3 tolerances and options given by ode45, you can use the function odeset to specify your own options. For example, in solution.m, you can use the following lines options=odeset( AbsTol,1e-8, RelTol, 1e-6); [t,x]=ode45( vdpol,tspan,x0,options); 3 Responses of LTI Systems for Impulse, Step and General Inputs Responses Using Transfer Functions From the systems point of view, an LTI system is often represented by its transfer function H(s). Note that the impulse response h(t) is simply the inverse Laplace transform of the system transfer function, H(s). As such, being able to determine and plot the system impulse response will allow one to completely characterize a system in the time domain. MATLAB has the ability to plot the system impulse response by using the impulse command. If one examines the syntax for the impulse command via the MATLAB help facility (i.e., help impulse), one sees that there are numerous ways that one can describe a system so that its impulse response can be found. In this laboratory exercise, the system will be described by the coefficients of the numerator and denominator of the transfer function entered in vector form. As an example, consider a system represented by its transfer function 3 s The m file shown in Figure 3 will then allow one to plot the impulse response in the time interval 0 t 10. Note that if the fourth line in the m file is changed to impulse(num,den,t), then the remaining lines can be eliminated since MATLAB will automatically generate its standard labeled impulse response plot if one uses impulse(num,den,t). (The complete technique in Figure 3 is only necessary if one wants to customize the way the data is presented.) t=0:0.01:10; num=[3]; den=[1 0.5]; imp_resp=impulse(num,den,t); plot(t,imp_resp) xlabel( Time, t ) ylabel( Amplitude, h(t) ) title( Example impulse response ) %define time vector %define numerator coefficients %define denominator coefficients %store impulse response data in imp_resp %plot impulse response data versus t %x axis label %y axis label %plot title Figure 3: MATLAB m file used to plot the system impulse response for 3 s+0.5. In addition to the system impulse response, MATLAB can also determine the system step response. This is important as one is often required to characterize the system step response as underdamped, overdamped or critically damped. The MATLAB command for the step response is 3

4 step and the syntax for using this command can be obtained by using help step. In the context of the impulse response m file shown in Figure 3, one only needs to replace all occurrences of the word impulse with the word step to obtain the properly labeled system step response. Finally, MATLAB can plot the response of a system for some general input. Some common general inputs, which can be directly generated by MATLAB, include sinusoids (see help sin or help cos), square waves (see help square) and various types of pulse trains (see help pulstran, help rectpuls or help tripuls). To plot the response for these general inputs, the lsim command is used (see help lsim). As an example, the m file shown Figure 4 illustrates how one can plot the response of the previously defined H(s) for a 1 Hz, 75% duty cycle pulse train input that has a peak amplitude of 1 over the time interval 0 t 10. t=0:0.01:10; num=[3]; den=[1 0.5]; in_signal=(square(2*pi*t,75)+1)/2; out_signal=lsim(num,den,in_signal,t); plot(t,out_signal, k-,t,in_signal, k: ) xlabel( Time, t ) ylabel( Amplitude, y(t) and u(t) ) title( Example pulse train response ) legend( y(t), u(t) ) %define time vector %define numerator coefficients %define denominator coefficients %define the 1Hz, 75% duty cycle pulse %train of amplitude=1 %find system output %plot system output & input versus t %x axis label %y axis label %plot title %put legend to indicate y(t) and u(t) Figure 4: MATLAB m file used to plot the response of 3 s+0.5 pulse train input with an amplitude of 1. for a 75% duty cycle, 1 Hz Responses Using Convolution Another method to obtain the response of an LTI system is by using convolution. In EE BD 326, you have learned that the response for an LTI system can be represented by the convolution y(t) = h(τ)u(t τ)dτ (5) where u(t) is the system input (do not be confused with the unit step function 1(t)) and h(t) is the impulse response of the system. The continuous-time convolution (5) can be approximated by a discrete-time convolution lim h(nδτ)u(t nδτ)δτ. (6) Δτ 0 n= Stated in words, one creates a discrete-time vector, evaluates the two functions of interest along the time vector, convolves the two vectors and then multiplies the convolution by Δτ. IfΔτ is chosen to be small enough, n= h(nδτ)u(t nδτ)δτ can provide us with a good approximation of the system response. In particular, consider the case of a causal system with input u(t) which satisfies 4

5 u(t) = 0 for any t<0. Assume we want to compute the system response y(t) for0 t T. We can divide the interval [0,T]intoN equal intervals, each with length Δτ (i.e., NΔτ = T ). The system response at the grid point t = kδτ can then be approximated by y(kδτ) = k h(nδτ)u(kδτ nδτ)δτ. (7) n=0 It can be observed that y(kδτ)asexpressedin(7)isexactlythe(k+1)th element in the convolution of the following two vectors h = [h(0),h(δτ),h(2δτ),,h(nδτ)], (8) u = [u(0),u(δτ),u(2δτ),,u(nδτ)] (9) multiplied by Δτ. Figure 5 shows a MATLAB m file which computes and plots the response (for t [0, 6]) of a system whose impulse response is h(t) =3e 0.5t 1(t) for the input signal u(t) =1(t) 1(t 2). deltatau=0.005; %define deltatau to be a small number t=0:deltatau:6; %define time vector N=length(t); %the total number of grid points h=3*exp(-0.5*t); %the impulse reponse for t in [0,6] %since we want to know the system %response for t in [0,6] and u(t)=0 %for t<0, knowing h(t), t in [0,6] suffices %our needs for the solution of this example u=zeros(1,n); %this and next lines define u(1:(n-1)/3)=ones(1,(n-1)/3); %the input u(t)=1(t)-1(t-2) %as a row vector z=conv(h,u); %z is the convolution of h and u y=z(1:n)*deltatau; %for our problem, we only need to know %the first N elements in z multiplied by %deltatau plot(t,y) %plot the response data versus t xlabel( Time, t ) %t axis label ylabel( Output, y(t) ) %y axis label title( Example system response ) %plot title Figure 5: MATLAB m file used to plot the system response using convolution. 4 Exercises All exercises are to be completed by creating, saving and running the necessary MATLAB script m files and/or function m files. 5

6 1. Determine the state solution of the following linear system [ ] [ ẋ = x, x(0) = for t [0, 6]. Plot the state trajectory (phase plot) by using the command plot(x(:,1),x(:,2)) and put appropriate labels and title. 17 points. 2. A spring-mass-damper system has the differential equation mÿ + bẏ + ky = u. By defining x 1 = y, x2 = ẏ, it can be written as a set of first order ordinary differential equations ], ẋ 1 = x 2, (10) ẋ 2 = k m x 1 b m x u. (11) m Determine x 1 (t) andx 2 (t) fort [0, 100]. Here we assume that m = 10, k =4,b =2and u(t) =0.5cos(0.5t), for t 0. The initial condition is y(0) = ẏ(0) = 0 (i.e., x 1 (0) = x 2 (0) = 0). Plot the input u(t) and the output y(t) versus time on the same plot. Observe what is the relationship between their frequencies (be sure to clearly mention this in your lab report). (Hint: you can put the expression of u(t) directly into the system state equation so that the right hand side of the system equation only depends on t and x). 17 points. 3. Plot the step response and the impulse response (use separate graphs) for 200 s 2 +30s over the time interval 0 t 1 with a step size of overdamped system.) 17 points. (This system is an example of 4. Repeat Problem 3 for 29 s 2 +4s +29 but use a time interval of 0 t 3 with a step size of (This system is an example of underdamped system.) 16 points. 5. For the system defined by s s s , use the lsim command to determine and plot the output (plot on the same graph with the input) if the input to the system is a 3 Hz square wave with a peak to peak amplitude of 2. Use a time interval of 0 t 2andastepsizeof0.01. (Assume zero initial condition for the system.) 17 points. 6

7 6. Repeat Problem 5 but use 100 cos(20t) as the input. Use a time interval of 0 t 3witha step size of (Assume zero initial condition for the system.) 16 points. 7. (Optional Problem) Use the convolution method to find and plot the response of the system (for 0 t 1) in Problem 3 when the input is a unit step input. Show your results for Δτ = 0.01 and Δτ = Compare your results with the one you obtained from Problem 3 by showing both results on the same plot and making comments. (Hint: you should first find out the system impulse response.) 10 points. Final Report Expectations for Lab 3 For this laboratory report, submit the information noted below. m files and/or items entered into MATLAB and the plots generated by MATLAB should be included in the laboratory report in computer generated form. Items that require analysis/interpretation on your part can be presented in handwritten form. Also, include a title page with the course name and number, the date the exercise was performed and your name along with the name of your lab partner(s). 1. For Problems 1-2 of the laboratory exercise, include the script m files, the function m files and the plots (labeled appropriately and titled, each curve clearly indicated). For Problem 2, be sure to also clearly indicate the relationship between the frequencies of the input and the output. 2. For Problems 3-6 of the laboratory exercise, include the script m files and the plots (labeled appropriately and titled, each curve clearly indicated). 3. For Problem 7 (if you choose to do it), include the script m files, the plots (labeled appropriately and titled, each curve clearly indicated), and your comments on the comparisons. 7