1 / 46

EEE8013 Revision lecture 1 Ordinary Differential Equations

EEE8013 Revision lecture 1 Ordinary Differential Equations. Modeling. Model : Ordinary Differential Equations (ODE):. Dynamics : Properties of the system, we have to solve/study the ODE. First order systems: Study approaches. First order ODEs :. Analytic :

emory
Download Presentation

EEE8013 Revision lecture 1 Ordinary Differential Equations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. EEE8013 Revision lecture 1 Ordinary Differential Equations Autumn 2008

  2. Modeling Model: Ordinary Differential Equations (ODE): Dynamics: Properties of the system, we have to solve/study the ODE. Autumn 2008

  3. First order systems: Study approaches First order ODEs: Analytic: Explicit formula for x(t) (a solution – separate variables, integrating factor) which satisfies INFINITE curves (for all Initial Conditions (ICs)). Autumn 2008

  4. First order linear equations First order linear equations - (linear in x and x’) General form: Numerical Solution: k=5, u=0.5 Autumn 2008

  5. Analytic solution: Step input Autumn 2008

  6. Response to a sinusoidal input Autumn 2008

  7. Response to a sinusoidal input Autumn 2008

  8. Second order ODEs Second order ODEs: So I am expecting 2 arbitrary constants u=0 => Homogeneous ODE Let’s try a Autumn 2008

  9. 1.5 Overall solution 1 x 2 0.5 0 x 1 -0.5 0 1 2 3 4 5 6 Overdamped system Roots are real and unequal Autumn 2008

  10. Critically damped system Roots are real and equal A=2, B=1, x(0)=1, x’(0)=0 => c1=c2=1 Autumn 2008

  11. Underdamped system Roots are complex Underdamped system r=a+bj Theorem: If x is a complex solution to a real ODE then Re(x) and Im(x) are the real solutions of the ODE: Autumn 2008

  12. Underdamped system, example A=1, B=1, x(0)=1, x’(0)=0 => c1=1, c2=1/sqrt(3) Autumn 2008

  13. Undamped Undamped system A=0, B=1, x(0)=1, x’(0)=0 =>c1=1, c2=0: Autumn 2008

  14. Stability In all previous cases if the real part is positive then the solution will diverge to infinity and the ODE (and hence the system) is called unstable. Autumn 2008

  15. is the damping factor and is the natural frequency of the system. Natural frequency, damping frequency, damping factor Case 1: => Overdamped system implies that Autumn 2008

  16. Natural frequency, damping frequency, damping factor Case 2: => Critically damped system implies that Case 3: => Underdamped systems implies called damped frequency or pseudo-frequency Case 4: No damping the frequency of the oscillations = natural frequency Autumn 2008

  17. Summary Autumn 2008

  18. If then cases 1-3 are the same but with Stability revised Autumn 2008

  19. NonHomogeneous (NH) differential equations • u=0 => Homogeneous => x1 & x2. • Assume a particular solution of the nonhomogeneous ODE: xp If u(t)=R=cosnt => Then all the solutions of the NHODE are So we have all the previous cases for under/over/un/critically damped systems plus a constant R/B. If complementary solution is stable then the particular solution is called steady state. Autumn 2008

  20. Example x(0)=1, x’(0)=0 => c1=-1, c2=-1/sqrt(3) Autumn 2008

  21. State Space Very difficult to be studied => so we use computers Computers are better with 1st order ODE 1 nth => n 1st Powerful tools from the linear algebra Use sensors: Output = x => Autumn 2008

  22. State Space Autumn 2008

  23. Block Diagram • X is an n x 1 state vector • U is an q x 1 input vector • Y is an p x 1 output vector • A is an n x n state matrix • B is an n x q input matrix • C is an p x n output matrix • D is an p x m feed forward matrix (usually zero) Autumn 2008

  24. State space rules The state vector describes the system => Gives its state => The state of a system is a complete summary of the system at a particular point in time. If the current state of the system and the future input signals are known then it is possible to define the future states and outputs of the system. • The choice of the state space variables is free as long as some rules are followed: • They must be linearly independent. • They must specify completely the dynamic behaviour of the system. • Finally they must not be input of the system. Autumn 2008

  25. State space The system’s states can be written in a vector form as: • A standard orthogonal basis (since they are linear independent) • for an n-dimensional vector space called state space. Matlab definition Autumn 2008

  26. Solution Autumn 2008

  27. Assume is a parameter => A homogeneous linear system Solution II How can we solve that ??? (This last equation is the characteristic equation of the system, why???). Autumn 2008

  28. Solution III I assume that a2=1 so a1=2 I assume that a1=2 so a2=-2 Matlab example Autumn 2008

  29. General Solution The roots of this equation are called eigenvalues negative eigenvalues => stable positive eigenvalues => unstable repeated eigenvalues => eigenvectors are not linearly independent. Complex eigenvalues => conjugate and the eigenvector will be complex =>solution will consists of sines, cosines and exponential terms Autumn 2008

  30. Properties of general solution If we start exactly on one eigenvector then the solution will remain on that forever. Hence if I have some stable and some unstable eigenvalues it is still possible (in theory) for the solution to converge to zero if we start exactly on a stable eigenvector. Determines the nature of the time response (stable, fast..) Determines the extend in which each state contributes to Determines the extend in which the IC excites the To find the eigenvalues and eigenvectors use the command eig() Autumn 2008

  31. Example Autumn 2008

  32. Example Autumn 2008

  33. Example Autumn 2008

  34. State Transition Matrix Until now the use of vector ODEs was not very helpful. We still have special cases => No special cases are needed then 4 ways to calculate it!!! Use the command expm (not exp)! X(0)=[1 2] X(5) =? Autumn 2008

  35. State Transition Matrix Autumn 2008

  36. State Transition Matrix Autumn 2008

  37. Solution Autumn 2008

  38. SS => TF??? TF Response to ICs Autumn 2008

  39. SS => TF??? CE of the TF The TF is a matrix Autumn 2008

  40. SS => TF??? Example: Find the TF of Autumn 2008

  41. Basic properties of state space State space transformations State space representations are not unique Same input/output properties, => same eigenvalues T is an invertible matrix Z is the new state vector Autumn 2008

  42. Basic properties of state space Do these two systems have the TF? Matlab example Autumn 2008

  43. Observability - Controllability Observability - Controllability Notice the structure of A and C Autumn 2008

  44. Observability - Controllability There is a pole zero cancellation The cancellation is due to C=[3 0]. pzmap(ss_model) Matlab verification We can influence x2 through U but we cannot observe how it behaves and hence there is no way to feedback that signal to a controller!!! Autumn 2008

  45. Observability - Controllability In this case we can see how both states behave but we can not change U in any way so that we can influence x2 due to the form of B. Unobservable & uncontrollable Minimal realisation. Difficult task if the system is nonlinear!!!! Autumn 2008

  46. Observability - Controllability Check the rank >> rank(obsv(A,C)) >> rank(ctrb(A,B)) Autumn 2008

More Related