1 State-Space Canonical Forms

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1 State-Space Caoical Forms For ay give system, there are essetially a ifiite umber of possible state space models that will give the idetical iput/output dyamics Thus, it is desirable to have certai stadardized state space model structures: these are the so-called caoical forms Give a system trasfer fuctio, it is possible to obtai each of the caoical models Ad, give ay particular caoical form it is possible to trasform it to aother form Cosider the system defied by () y + a ( ) y + + a ẏ + a y b () u + b ( ) u + + b u + b u where u is the iput, y is the output ad () y represets the th derivative of y with respect to time Takig the Laplace trasform of both sides we get: Y (s) ( s + a s + + a s + a ) U(s) ( b s + b s + + b s + b ) which yields the trasfer fuctio: Y (s) U(s) b s + b s + + b s + b s + a s + + a s + a () Give the a system havig trasfer fuctio as defied i () above, we will defie the cotrollable caoical ad observable caoical forms Cotrollable Caoical Form The cotrollable caoical form arrages the coefficiets of the trasfer fuctio deomiator across oe row of the A matrix: x + u a a a a x y [ b a b b a b b a b + b u

2 The cotrollable caoical from is useful for the pole placemet cotroller desig techique Example Cosider the system give by U(s) Y (s) s + 3 s + 3s + Obtai a state space represetatio i cotrollable caoical form By ispectio, (the highest expoet of s), therefore a 3, a, b, b ad b 3 Therefore, we ca simply write the state space model as follows: [ x [ 3 y [ 3 [ x [ x + [ u Observable Caoical Form The observable caoical form is defied i terms of the trasfer fuctio coefficiets of () as follows: x a a b a b b a b + x a u b x a b x y [ + b u Note the relatioship betwee the observable ad cotrollable forms: A obs B obs C obs D obs A T cot Ccot T Bcot T D cot

3 Example Give the system trasfer fuctio of example, fid the observable caoical form state space model Recall that by ispectio, we have (the highest expoet of s), ad therfore a 3, a, b, b ad b 3 Thus, we ca write the observable caoical form model as follows: [ [ [ [ x x 3 + u 3 y [ [ x 3 Diagoal Caoical Form The diagoal caoical form is a state space model i which the poles of the trasfer fuctio are arraged diagoally i the A matrix Give the system trasfer fuctio havig a deomiator polyomial that ca be factored ito distict (p p p ) roots as follows: Y (s) U(s) b s + b s + + b s + b (s + p )(s + p ) (s + p ) The deomiator polyomial ca be rewritte by partial fractio expasio as follows: b + c + c + + c s + p s + p s + p The the diagoal caoical form state space model ca be writte as follows: p x p + u p x y [ c c c + b u Example 3 Give the system trasfer fuctio of example, fid the diagoal caoical form state space model 3

4 The trasfer fuctio of the system ca be re-writte with the deomiator factored as follows: U(s) Y (s) s + 3 s + 3s + s + 3 (s + )(s + ) therefore p ad p Trivially, the partial fractio expasio of the deomiator gives c ad c ad the diagoal from model ca be writte as: [ [ [ [ x x + u y [ [ x 3 Time Domai Solutio of Diagoal Form A useful result of the diagoal form model is that the state trasitio matrix Φ(t) e At is easily evaluated without resortig to ivolved calculatios: p p A p e p t e At e p t e p This greatly simplifies the task of computig the aalytical solutio to the respose to iitial coditios 4 Jorda Form The Jorda form is a type of diagoal form caoical model i which the poles of the trasfer fuctio are arraged diagoally i the A matrix Cosider the case i which the deomiator polyomial of the trasfer fuctio ivolves multiple repeated roots: Y (s) U(s) b s + b s + + b s + b (s + p ) 3 (s + p 4 )(s + p 5 ) (s + p ) 4

5 The deomiator polyomial ca be rewritte by partial fractio expasio as follows: b + c (s + p ) 3 + c (s + p ) + c s + p + c s + p + + c s + p The the Jorda caoical form state space model ca be writte as follows: p p 3 p 4 p 4 p x y [ c c c + b u 5 State-Space Modelig with MATLAB x x 3 x 4 + u MATLAB uses the cotrollable caoical form by default whe covertig from a state space model to a trasfer fuctio Referrig to the first example problem, we use MATLAB to create a trasfer fuctio model ad the covert it to fid the state space model matrices: >>um [ 3; % umerator polyomial >>de [ 3 ; % deomiator polyomial >> [A,B,C,D tfss(um,de) % Matrices of SS model A -3 - B 5

6 C 3 D Note that this does ot match the result we obtaied i the first example See below for further explaatio No we create a LTI state space model of the system usig the matrices foud above: >> ctrb_sys ss(a,b,c,d) a x x x -3 - x b u x x c x x y 3 d u y Cotiuous-time model NOTE: MATLAB uses a differig cotrollable caoical format to ours whe it derives the model from the trasfer fuctio The state order is reversed from our defiitio, thus they are ot directly comparable If we use our ow defiitio of the cotrollable form from Example, 6

7 we ca geerate the observable ad cotrollable models as follows: >>A [ ; - -3; >>B [;; >>C [3 ; >>D ; % our cotrollable form >>csys ss(a,b,c,d); % create the cotrollable caoical model >>osys ss(a,c,b,d); % create the observable caoical model Cautio should be take whe usig the MATLAB cao() commad, which is a method for covertig amogst the caoical forms MATLAB produces valid alterative caoical forms, but they are ot the same as the defiitios used i our textbook I MATLAB the compaio form is similar to the observable caoical form, ad the modal form is similar to the diagoal form They will all produce exactly the same iput to output dyamics, but the model structures ad states are differet 7

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