To days Outline

1. To days Outline • Linear Equations • ODE • Integration • Handle Graphics. • Exercises on this days topics Lecture 5

2. Linear Equations • Linear equations • Using left division • MATLAB uses Gaussian elemination • Has the solution in MATLAB • X=A\Y • Example: Solve the system of linear equation: Lecture 5

3. ODE • MATLAB can solve system of first order ordinary differential equations with known initial condition. • MATLAB offers a number of different solvers • ode45 • This is the typical first solver to try on a new problem. • ode15s • This is typical to try if ode45 fails or are too inefficient. Solves stiff problems. • Stiff problem are described as problem in which the time constants vary a lot. Lecture 5

4. ODE • General form • [time,x]=solver(fh,t,x0) • time: Time values • x: contains the solution for each time value • solver: one of MATLAB’s ode solvers • fh: A function handle to the function that describes the differential equations • t: is the time span T0 to TFINAL • x0: Initial condition Lecture 5

5. ODE Example: Solve the differential equation: Lecture 5

6. ODE Lecture 5

7. ODE • Higher order differential equations must be rewritten into system of first order differential equations. y b m F k Lecture 5

8. Integration • Trapezoidal numerical integration. • z = trapz(x,y) • computes the integral of y with respect to x using the trapezoidal method. • The trapezoidal technique is used when we only know the integrand in a number of specific points. • With a smaller delta-x the numerical integration becomes more accurate Lecture 5

9. Handle Graphics • A set of low level functions that control the characteristic of a graphic object. • Changing grids, line color etc. that is not supported by the standard LinSpec option in the plot command. • MATLAB graphics system is based on a hierarchical system of Graphical Objects Lecture 5

10. Handle Graphics • Each graphical object are known by a unique name: Handle • Each graphical object has special data: properties • A handle is automatically returned by any command that creates a graphic object. • Hndl=figure Lecture 5

11. Handle Graphics Lecture 5

12. Handle Graphics • When an object is created all of its properties are initialized to default values. • plot(y) uses the default line color, line style, line width etc. • All properties can be examined using: • get(Handle,’PropertyName’) ; • All properties can be changed using: • set(Handle,’PropertyName’,Value1’,….); • If value are left out MATLAB display a list of possible property values for that actual property name Lecture 5

13. Handle Graphics • To view all properties: • x=[0:0.1:2]; • y=x.^2; • Hndl=plot(x,y) %Handle to the line. • result=get(Hndl) • result will be a structure containing all properties to the line Lecture 5

14. Handle Graphics Lecture 5

15. Handle Graphics • To change the line width from default 0.5 to 5: • set(Hndl,’LineWidth’,5) Lecture 5

16. Handle Graphics • Functions that return handles • gcf • Get current figure. • gca • Get current axes in the current figure. • gco • Get current object in the current figure. • findobj • Finds a graphics object with a specific property value • The current object is defined as the last object clicked on with the mouse. Lecture 5

17. Handle Graphics • Position of figure Objects • [left bottom width height] • Units can be: pixels, inches, cm, points and normalized coordinates. • Normalized coordinates are between 0-1. • (0,0) = Lower left corner • (1,1) = Upper right corner • Using normalized coordinates will make the figure to appear in the same relative position regardless of screen resolution. • Normalized coordinates are recommended to use if possible Lecture 5

18. Handle Graphics • Position of axes and uicontrol Objects • Also a 4-element vector. • [left bottom width height] • Position is specified relative to the figure that contains the object. • Default units: • Normalized coordinates within the figure. Lecture 5

19. Handle Graphics • Position of text Objects • The position of a text object refers to the actual axes. • x,y in 2D • x,y,z in 3D • The position of the text object relative to a specified point is controlled by: • HorizontalAlignment • {Left},Center,Right • VerticalAlignment • {Middle}, Top, Cap, Baseline, Bottom Lecture 5

20. Handle Graphics Example: - Construct a dartboard with 10 circles. - Add text in each circle what the point would be if the dart should hit within that circle. - Construct a frame around the dartboard. - Simulate that a person throw 10 darts at the board with the standard deviation 4 in x and y direction. Lecture 5

21. Lecture 5

22. Exercises on this days topics Lecture 5