Lectures on Dynamic Systems and  Con trol  Mohammed Dahleh Mun ther A
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Lectures on Dynamic Systems and Con trol Mohammed Dahleh Mun ther A

Dahleh George erghese Departmen of Electrical Engineering and Computer Science Massac uasetts Institute of ec hnology brPage 2br Chapter 26 Balanced Realization 261 In tro duction One opular approac for obtaining minimal realization is kno wn as Ba

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Lectures on Dynamic Systems and Con trol Mohammed Dahleh Mun ther A




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Lectures on Dynamic Systems and  Con trol  Mohammed Dahleh Mun ther A. Dahleh George erghese Departmen of Electrical Engineering and Computer Science Massac uasetts Institute of ec hnology
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Chapter 26 Balanced Realization  26.1 In tro duction One opular approac for obtaining minimal realization is kno wn as Balanc alization In this approac h, new state-space description is obtained so that the reac habilit and observ abilit gramians are diagonalized. This de nes new set of invariant parameters kno wn as Hank el singular alues. This approac pla ys ma jor role

in mo del reduction whic ted in this hapter. 26.2 Balanced Realization Let us start with system with minimal realization As seen in an earlier lecture, the con trollabilit gramian and the observ abilit gramian are obtained as solutions to the follo wing Ly apuno equations AP QA and are symmetric and since the realization is minimal they are also ositiv de nite. The eigen alues of the pro duct of the con trollabilit and observ abilit gramians pla an imp ortan role in system theory and con trol. de ne the Hank el singular alues, as the square ro ots of the eigen alues of )) ould lik to obtain co

ordinate transformation, that results in realization for whic the con trollabilit and observ abilit gramians are equal and diagonal. The diagonal en tries of the transformed con trollabilit and observ abilit gramians will the Hank el singular alues. With the co ordinate transformation the new system realization is giv en AT
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and the Ly apuno equations in the new co ordinates are giv en  QT QT Therefore the con trollabilit and observ abilit gramian in the new co ordinate system are giv en QT are lo oking for transformation suc that   the relation )( QT QT (26.1) Since and is

ositiv de nite, can factor it as where is an in ertible matrix. can write equation 26.1 as RT hic is equiv alen to RT RP RT (26.2) Equation 26.2 means that RP is similar to and is ositiv de nite. Therefore, there exists an orthogonal transformation suc that RP (26.3)  By setting RT arriv at de nition for and as R: With this transformation it follo ws that ( ( )( )( and ( )( RR )(
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26.3 Mo del Reduction Balanced runcation Supp ose start with system where is asymptotically stable. Supp ose is the transformation that con erts the ab realization to balanced realization, with

and diag ). In man applications it ma ene cial to only consider the subsystem of that corresp onds to the Hank el singular alues that are larger than certain small constan t. or that reason, supp ose partition as where con tains the small Hank el singular alues. can partition the realization of accordingly as 11 12 21 22 Recall that the follo wing Ly apuno equations hold whic can expanded as 11 12 11 21 21 22 12 22 11 21 11 12 12 22 21 22 rom the ab matrix equations get the follo wing set of equations 11 (26.4) 11 21 12 (26.5) 22 (26.6) 22 11 11 (26.7)
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21 12 (26.8) 22 22

(26.9)  rom this decomp osition can extract subsystems 11   22  Theorem 26.1 is an asymptotic al ly stable system. If and do not have any ommon diagonal elements then and ar asymptotic al ly stable. Pro of: Let us sho that the subsystem 11   is asymptotically stable. Since 11 satis es the Ly apuno equation 11 11   then it immediately follo ws that all the eigen alues of 11 ust in the closed left half of the complex plane; that is, Re 11 0. In order to sho asymptotic stabilit ust sho 11 has no  purely imaginary eigen alues. Supp ose is an eigen alue of 11 and let an eigen ector asso

ciated with  11 0. Assume that the Kernel of 11 is one-dimensional. The general case where there ma sev eral indep enden eigen ectors asso ciated with can handled sligh mo di cation of the presen argumen t.  Equation 26.7 can written as  11 11   By ultiplying the ab equation on the righ on the left get 11 11 whic implies that 0, and this in turn implies that (26.10)  Again from equation 26.7 get 11 11  whic implies that 11 (26.11)  No ultiply equation 26.4 from the righ and from the left to get 11 ) 11 
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This implies that )( 0, and By ultiplying equation 26.4 on he

righ get 11 ) 11 and hence 11 ) (26.12) Since that the ernel of 11 is one dimensional, and oth and are eigen ectors, it follo ws that where is one of the diagonal elemen ts in No ultiply equation 26.5 on the left and equation 26.8 on the left get 12 21 (26.13) and 12 (26.14) 21 >F rom equations 26.13 and 26.14 get that 21 21 whic can written as 21 Since assumption and ha no common eigen alues, then and ha no common eignev alues, and hence 21 0. ha 11 21 whic can written as 11 12 21 22 This statemen implies that is an eigen alue of tradicts the assumption of the theorem stating that is

asymptotically stable.
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MIT OpenCourseWare http://ocw.mit.edu 6.241J / 16.338J Dynamic Systems and Control Spring 20 1 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .