Feedback Control Systems (FCS)

1. Feedback Control Systems (FCS) Lecture-26-27-28-29 State Space Canonical forms Dr. Imtiaz Hussain email: imtiaz.hussain@faculty.muet.edu.pk URL :http://imtiazhussainkalwar.weebly.com/

2. Lecture Outline • Canonical forms of State Space Models • Phase Variable Canonical Form • Controllable Canonical form • Observable Canonical form • Similarity Transformations • Transformation of coordinates • Transformation to CCF • Transformation OCF

3. Canonical Forms • Canonical forms are the standard forms of state space models. • Each of these canonical form has specific advantages which makes it convenient for use in particular design technique. • There are four canonical forms of state space models • Phase variable canonical form • Controllable Canonical form • Observable Canonical form • Diagonal Canonical form • Jordan Canonical Form • It is interesting to note that the dynamics properties of system remain unchanged whichever the type of representation is used. Companion forms Modal forms

4. Phase Variable Canonical form • The method of phase variables possess mathematical advantage over other representations. • This type of representation can be obtained directly from differential equations. • Decomposition of transfer function also yields Phase variable form.

5. Phase Variable Canonical form • Consider an nthorder linear plant model described by the differential equation • Where y(t) is the plant output and u(t) is the plant input. • A state model for this system is not unique but depends on the choice of a set of state variables. • A useful set of state variables, referred to as phase variables, is defined as:

6. Phase Variable Canonical form • Taking derivatives of the first n-1state variables, we have

7. Phase Variable Canonical form • Output equation is simply

8. Phase Variable Canonical form … ∫ ∫ ∫ ∫ ＋ ＋

9. Phase Variable Canonical form

10. Phase Variable Canonical form (Example-1) • Obtain the state equation in phase variable form for the following differential equation, where u(t) is input and y(t) is output. • The differential equation is third order, thus there are three state variables: • And their derivatives are (i.e state equations)

11. Phase Variable Canonical form (Example-1) • In vector matrix form Home Work: Draw Sate diagram

12. Phase Variable Canonical form (Example-2) • Consider the transfer function of a third-order system where the numerator degree is lower than that of the denominator. • Transfer function can be decomposed into cascade form • Denoting the output of the first block as W(s), we have the following input/output relationships:

13. Phase Variable Canonical form (Example-2) • Re-arranging above equation yields • Taking inverse Laplace transform of above equations. • Choosing the state variables in phase variable form + +

14. Phase Variable Canonical form (Example-1) • State Equations are given as • And the output equation is

15. Phase Variable Canonical form (Example-1) • State Equations are given as • And the output equation is

16. Phase Variable Canonical form (Example-1) • State Equations are given as • And the output equation is • In vector matrix form

17. Companion Forms • Consider a system defined by • where u is the input and y is the output. • This equation can also be written as • We will present state-space representations of the system defined by above equations in controllable canonical form and observable canonical form.

18. Controllable Canonical Form • The following state-space representation is called a controllable canonical form:

19. Controllable Canonical Form

20. … … ∫ ∫ ∫ ∫ ＋ ＋ ＋ Controllable Canonical Form

21. Controllable Canonical Form (Example) • Let us Rewrite the given transfer function in following form

22. Controllable Canonical Form (Example)

23. Controllable Canonical Form (Example) • By direct decomposition of transfer function • Equating Y(s) with numerator on the right hand side and U(s) with denominator on right hand side.

24. Controllable Canonical Form (Example) • Rearranging equation-2 yields • Draw a simulation diagram using equations (1) and (3) -2 -3 3 U(s) Y(s) 1/s 1/s P(s) 1

25. Controllable Canonical Form (Example) • State equations and output equation are obtained from simulation diagram. -2 -3 3 U(s) Y(s) 1/s 1/s P(s) 1

26. Controllable Canonical Form (Example) • In vector Matrix form

27. Observable Canonical Form • The following state-space representation is called an observable canonical form:

28. Observable Canonical Form

29. Observable Canonical Form (Example) • Let us Rewrite the given transfer function in following form

30. Observable Canonical Form (Example)

31. Similarity Transformations • It is desirable to have a means of transforming one state-space representation into another. • This is achieved using so-called similarity transformations. • Consider state space model • Along with this, consider another state space model of the same plant • Here the state vector , say, represents the physical state relative to some other reference, or even a mathematical coordinate vector.

32. Similarity Transformations • When one set of coordinates are transformed into another set of coordinates of the same dimension using an algebraic coordinate transformation, such transformation is known as similarity transformation. • In mathematical form the change of variables is written as, • Where T is a nonsingular nxn transformation matrix. • The transformed state is written as

33. Similarity Transformations • The transformed state is written as • Taking time derivative of above equation

34. Similarity Transformations • Consider transformed output equation • Substituting in above equation • Since output of the system remain unchanged [i.e. ] therefore above equation is compared with that yields

35. Similarity Transformations • Following relations are used to preform transformation of coordinates algebraically

36. Similarity Transformations • Invariance of Eigen Values

37. Transformation to CCF • Transformation to CCf is done by means of transformation matrix P. • Where CM is controllability Matrix and is given as and W is coefficient matrix Where the ai’s are coefficients of the characteristic polynomial s+

38. Transformation to CCF • Once the transformation matrix Pis computed following relations are used to calculate transformed matrices.

39. Transformation to CCF (Example) • Consider the state space system given below. • Transform the given system in CCF.

40. Transformation to CCF (Example) • The characteristic equation of the system is

41. Transformation to CCF (Example) • Now the controllability matrix CM is calculated as • Transformation matrix P is now obtained as

42. Transformation to CCF (Example) • Using the following relationships given state space representation is transformed into CCf as

43. Transformation to OCF • Transformation to CCf is done by means of transformation matrix Q. • Where OM is observability Matrix and is given as and W is coefficient matrix Where the ai’s are coefficients of the characteristic polynomial s+

44. Transformation to OCF • Once the transformation matrix Qis computed following relations are used to calculate transformed matrices.

45. Transformation to OCF (Example) • Consider the state space system given below. • Transform the given system in OCF.

46. Transformation to OCF (Example) • The characteristic equation of the system is

47. Transformation to OCF (Example) • Now the observability matrix OM is calculated as • Transformation matrix Q is now obtained as

48. Transformation to CCF (Example) • Using the following relationships given state space representation is transformed into CCf as

49. Home Work • Obtain state space representation of following transfer function in Phase variable canonical form, OCF and CCF by • Direct Decomposition of Transfer Function • Similarity Transformation • Direct Approach

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