Nested functions & Local functions

Transcription

1 Lab 5

2 Nested functions & Local functions Nested functions are functions defined within functions. They are helpful when: - We wish to write small or temporary functions which do not merit creation of a new m-file - When we wish to share some information between the outer function and the nested function. A nested function is able to access all the variables defined in its enclosing function. We cannot define nested functions in a script. We can convert any script to a function by creating a function declaration for it. It need not receive or return any variables. Here is an example converting the main.m file to a function: function main() the contents of your existing main.m A local function is a function that is defined in the same file as another function, but defined outside it's body. function func1() function func() % a local function in the same file Example: function test_function( ) a = 5; % (1) Defining an anonymous function f exp(a*x.^); % vs % () Defining a nested function - this can have multiple statements! function y = f(x) u = exp(a*x.^); y = u*10; x = 0:0.01:1; % call the anonymous function y = f(x); plot(x,y); xlabel('x'); ylabel('f(x)'); % call the nested function y = f(x); figure;

3 plot(x,y); xlabel('x'); ylabel('f(x)'); % call the local function y = f3(x, a); figure; plot(x,y); xlabel('x'); ylabel('f3(x)'); % vs % (3) Defining a local function - this can have multiple statements but cannot % access data from other functions directly. So a needs to be a parameter to the function. function y = f3(x, a) u = exp(a*x.^); y = u*10; 1) Write a main function that declares the variables x1, y1, x, y, x3, y3 to describe three points in space: (0, 0), (5, 5), and (3, 4). Write 3 functions in the same file: 1) an anonymous function eucledian_distance1, ) a nested function eucledian_distance, 3) and a local function eucledian_distance3 Each one should have the input arguments ( x1, y1, x, y) and should return the Eucledian distance between the points (x1, y1) and (x, y). Call eucledian_distance1to calculate the distance between point 1 and, call eucledian_distance for the distance between points 1 and 3, and eucledian_distance3 for the distance between points and 3.

4 Performing Finite Integrals To perform integrals, we define the integrand as a function(normal/anonymous/nested/local). You must ensure that your function is capable of handling vector inputs. Example: Integrate y = (a x) from x = 0 to 1. function test_function( ) % some constant a a = ; % an anonymous function defining the integrand f (a*x).^; % computation of the finite integral from x=0 to x=1 y = quad(f, 0, 1) % MATLAB versions 01 and later can use 'integral' instead of 'quad' When using a local function, the local function does not have access to the parameter a defined in the main function. It needs to be supplied to the function as arguments: function f_local(x, a) function body The problem here is the function handle we use with 'quad' cannot have more than one parameter. We will need to define a temporary anonymous function that takes nly one argument, but calls the local function with all the required parameters: f_temp f_local(x,a); The anonymous function has access to a, and simply calls the local function with that value. We then use this anonymous function in the quad statement: y = quad(f_temp, 0, 1); The energy of a signal y(t)= A cos( π t T ) in the interval 0 to T is given by: y= 0 T ( Acos( π t T )) Use A = and T = 1. Compute this by defining y(t) using 1) anonymous functions and ) local functions.

5 Indefinite Integrals with the Symbolic Math toolbox We have already seen how the symbolic math toolbox give MATLAB the ability to understand and evaluate expressions. We will now see how it can also perform complex tasks such as indefinite integrals. We know that sin (x)cos(x)=sin (x). Lets see if MATLAB knows that too: syms x; y = *sin(x)*cos(x); z = simplify(y); disp(z); In general, the results from 'simplify' are based upon certain rules. They may or may not be what we are looking for, but most of the time it will work out. Now lets throw some integrals at it: z= sin( x)cos(x)dx= sin(x)dx= cos(x) cos( x)cos( x) +c= +c= cos (x)+c syms x; y = *sin(x)*cos(x); z = int(y); % the 'int' function integrates symbolic expressions disp(z); We may have been looking for the cos(x) expression, but this is a very subjective problem. Definite integrals The 'int' function handles definite integrals the same way as 'quad'. syms x; y = *sin(x)*cos(x); z = int(y, x, 1, ); % we can be explicit that we are integrating y with respect to x pretty(z); % MATLAB keeps the result in symbolic form z_value = double(z); % we can find the numerical value by converting it to double disp(z_value); If the limits are +/-, we can use +Inf and -Inf. There is an equivalent 'diff' function to compute derivatives. Please read about it yourself. We will find the expression for the energy in a sinusoid: x(t)=c cos(w 0 t +θ) The energy is defined as: E= 1 T +T / T / x (t) You will need to define the following as symbols: C, w0, theta, t. Then compute T = *pi/w0. Define x(t)=c cos(w 0 t+θ) and then compute E using the 'int' function.

6 Solving Ordinary Differential Equations We can use the ode45 (or ode3) function to solve ordinary differential equations. The differential equation needs to be converted to a set of first order differential equation. Examples: 1) dy =x+ - this is already first order dt ) d y dy = 3 d t dt y+x - this is second order and needs to be converted Let us define y 1 = y, y = ẏ. Then ẏ 1 = y and ÿ= 3 ẏ y +x becomes ẏ = 3y y 1 +x. So we get first order equations, one for ẏ 1 and one for ẏ. 3) Recall that for RLC circuit in the problem sets, we wrote state space equations like: q 1 =q + q 3 q = x q 1 q 3 =3 x 3q 1 9q 3 We now need to define function 'qdot' that takes the input a vector for time t, and a vector for the state variable q=[q 1 q q 3 ] T, and returns the value of q=[ q 1 q q 3 ] T (all must be column vectors). We do that as follows: function qdot = calc_qdot(t, q) x(t) = sin(5*t); qdot = zeros(3,1); % force qdot to be a column vector qdot(1) = *q() + *q(3); qdot() = x(t) - q(1); qdot(3) = 3*x(t) - 3*q(1) 9*q(3); We then invoke the ode45 function from main as: t0 = 0; tf = 5; % initial and final time for the simulation q0 = [0 0 0]'; % Initial conditions for q1, q, q3 [t,q] = ode45(@calc_qdot,[t0 tf],q0); plot(t,q); xlabel('time'); ylabel('state variables'); leg('q1 - V C', 'q - I L1', 'q3 - I L'); Simulate the ODE in example above for t = 0 to 5. Use zero initial conditions.