Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

Transcription

1 Lecture 0 Coversio Betwee State Space ad Trasfer Fuctio Represetatios i Liear Systems II Dr. Radhakat Padhi Asst. Professor Dept. of Aerospace Egieerig Idia Istitute of Sciece - Bagalore

2 A Alterate First Compaio Form (Toeplitz first compaio form) H ie.. ( s) bs + b s + + bs + b 0 y( s) = = s + a s + + as + a0 u( s) ( ) ( ) y + a y + + a y + a y 0 0 ( ) ( ) = bu+ bu + + b u + bu Ref: B. Friedlad, Cotrol System Desig Mc Graw Hill,

3 A Alterate First Compaio Form (Toeplitz first compaio form) Defie the state variables x,, x such that y = x+ p0u x = x2 + pu x = x + p u x = x + p u x = a x a x + p u 0 3

4 A Alterate First Compaio Form (Toeplitz first compaio form) From the above defiitio, we have y = x + p u ( ) ( ) ( ) ( 2) ( ) ( ) 0 y = x + p u = x + p u + p u y = x + p u + p u = x + p u + p u + p u y = x + p u + p u + + p u + p u 2 0 y = a x a x a x + p u 2 0 ( ) ( ) + p u + p u + + p u + p u 2 0 4

5 A Alterate First Compaio Form (Toeplitz first compaio form) ( ) ( ) y + a y + + a y + a y 0 ( 0 0) + ( p + + a p + a p ) u = p + a p + + a p + a p u ( ) ( p a p u ) pu 0 ( ) 0 (from the TF) ( ) ( ) = bu+ bu + + b u + bu 5

6 A Alterate First Compaio Form (Toeplitz first compaio form) Equatig the coefficiets: p = b p + a p = b p + a p + + a p = b p + a p + + a p = b Now, we eed to solve for p0, p,, p ( ) 6

7 A Alterate First Compaio Form (Toeplitz first compaio form) I matrix form: p0 b a 0 0 p b a 2 a 0 = a0 a a p b 0 Note : Toeplitz Matrix Toeplitz matrix is a osiular matrix ( determiat = 0) Hece, the solutio always exists! 7

8 A Alterate First Compaio Form (Toeplitz first compaio form) State space form (from defiitio of state & output variables): x x p0 x x 2 p = + u x a a a x p 0 [ 0 0] [ ] y = X + p u 0 Note : The physical meaig of state variables i differet compaio froms are differet. 8

9 Block Diagram for realizatio of Toeplitz first compaio form 9

10 Alterate/Toeplitz first compaio form: Some commets Toeplitz realizatio also requires itegrators Extesio of Toeplitz realizatio is straightforward to MISO systems Oe eed to solve m-system of liear equatios, where m is the umber of iput (cotrol) variables. However, oly itegrators are eeded. 0

11 Secod Compaio Form (Observable Caoical Form) bs + b s + + bs + b 0 ys () H() s = = s + a s + + as + a0 u() s ie.. ( + ) s a s as a y() s ( ) 0 = bs + b s + + bs+ b us () Rearragig the terms: [ ] () () [ () ()] + + s y s bu s s a y s b u s [ ays bus] + () () = 0 0 0

12 Secod Compaio Form (Observable Caoical Form) Simplify: s s [ ys () bus ()] = [ b us () a ys ()] + + [ bus () ays ()] Solve for y(s): 0 0 ys () = bus () + b us () a ys () + + bus 0 () ays 0 () s s [ ] [ ] = bus () + [ b us () a ys ()] + [ b 2us () a 2ys ()] s s + x ( s) 2 x ( s) 2

13 Block Diagram Realizatio 3

14 Secod Compaio Form (Observable Caoical Form) The equatios ca be writte as: y = x + bu x = x a y + b u = a x + x + b a b u ( ) ( ) 2 2 x = x a y + b u = a x + x + b a b u ( ) ( ) x = x a y+ bu = a x + x + b ab u ( ) ( ) x = b u a y = a x + b a b u ( )

15 Secod Compaio Form (Observable Caoical Form) I vector-matrix form, we ca write it as x a x b a b x 2 a x 2 b 2 a 2b = + u x a x b ab x a x b0 a0b X A X B y = [ 0 0] X + [ b ] u C D 5

16 Secod Compaio Form (Observable Caoical Form) () I secod compaio form, the coefficiets of the deomiator of the trasfer fuctio appear i a colum, whereas i the first compaio form they appear i a row. (2) Extesios of secod compaio form to SIMO, MISO cases etc. are possible (but beyod the scope this course) (3) Cotrollable caoical form is good for cotrol desig, whereas observable caoical form is good for observer/filter desig 6

17 Commet o miimal realizatio (for MIMO systems) We have see realizatio of: SIMO systems: itegrators MISO systems: itegrators Questio: How about MIMO systems? Will it still be possible to realize it with itegrators? Aswer: No! However, oe way of realizig MIMO systems will be to use a umber of structures (of either SIMO or MISO form) i parallel; i.e. m p if U ad Y the it is always possible to realize such a MIMO system with o more tha x mi (m, p) itegrators. 7

18 Commet o miimal realizatio (for MIMO systems) Questio: How about fewer itegrators? Aswer: This leads to the questios of miimal realizatio ; a subject of cosiderable research durig 970s. Why ecessary? Because a o-miimal realizatio is either o-cotrollable or o-observable (or both). It may cause theoretical ad computatioal problems too. Solutio: Several algorithms exist for fidig a miimal realizatio ad correspodig A,B,C,D matrices. However, these are beyod the scope of this course! Further readig: T. Kailath, Liear Systems, Pretice Hall,

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