A(t) L T *(t) t/t t/t

Size: px

Start display at page:

Download "A(t) L T *(t) t/t t/t"

Lambert Jones
6 years ago
Views:

1 he Real Story behind Floquet-Lyapunov Factorizations RAYMOND J. SPIERI Department of Mathematics and Statistics Acadia University Wolfville, NS BP X CANADA raymond.spiteri@acadiau.ca http//ace.acadiau.ca/~rspiteri PIERRE MONAGNIER Department of Mechanical Engineering and Centre for Intelligent Machines McGill University 8 University Street Room, Montreal, QC HA A CANADA pierrem@cim.mcgill.ca Abstract Floquet-Lyapunov theory plays a ubiquitous role in the analysis and control of time-periodic systems. A real linear system with -periodic coecients will generally lead to a complex Floquet factorization. However, it is well known that by regarding the system as having -periodic coecients, it is possible to obtain real factors. In this paper, we show that there exists a simple way to dene real factors using known results for the real matrix logarithm and the knowledge of the state transition matrix of the system on a single period. he proposed procedure also determines whether the real Lyapunov transformation is -periodic, -antiperiodic, or generally -periodic. Key-Words Floquet theory, time-periodic systems, Lyapunov transformation, real matrix logarithm Introduction he Floquet-Lyapunov theorem (see, e.g., [], [], []) is a well-known and celebrated result in the eld of linear, time-periodic (LP) systems. he theorem consists of two main parts the Floquet representation theorem and the Lyapunov reducibility theorem. In the following, we specialize the discussion of these results in terms of the state transition matrix namely, the fundamental solution matrix (t ) that satises ( ) I. he Floquet representation theorem says that the state transition matrix (t ) of the LP system _x(t) A(t)x(t) () where A(t) A(t+) can be written in the form (t ) L(t )e tf () In the classical theory, the factors L(t ) and F are n n matrices such that L(t ) L(t + ) L( ) I n where I n is the n n identity matrix, and F ln () is constant. On the other hand, the Lyapunov reducibility theorem says that the time-dependent change of variables x(t) L(t )z(t) transforms the original LP system into a linear, time-invariant system _z(t) Fz(t) where L(t ) and F are the same as in equation (). In general, the state transition matrix of a real system will have complex factors L(t ) and F. It is well known that it is always possible to obtain real factorizations within the classical theory by consid-

2 A() system matrix F Floquet matrix I n n n identity matrix J Jordan canonical form of ( ) L( ) Lyapunov transformation S n n diagonal matrix U n n n super-diagonal matrix W generalized eigenvectors of ( ) Y n n Yakubovich matrix r i algebraic multiplicity of i s i geometric multiplicity of i ( ) state transition matrix p, i eigenvalues of F i eigenvalues of ( ) able Notation ering the state transition matrix over two periods, F ln () L (t +) L (t ) L( ) I n () his result oers sucient conditions only. he issue of nding necessary and sucient conditions to obtain a real factorization on one period was rst raised in a short note by [8]. His result is summarized in the following theorem heorem In equation (), let A(t) be areal matrix function, where A(t + ) A(t). An arbitrary real matrix X(t ) that is a fundamental solution of system () can be represented in the form () where F is real matrix, L(t ) is a real matrix function, det L(t ), and L(t ) is such that L(t + ) L(t )Y () and Y is a real matrix satisfying Y I n FY YF () Moreover, the converse statement is true as well Let L(t ) F Y be arbitrary real matrices satisfying the above conditions, and let X(t ) be described by (). hen, X(t ) is a real matrix of fundamental solutions of some equation () with a -periodic real matrix function A(t) On the other hand, the existence, uniqueness, and construction of real factorizations using information on only one period are issues that have not been investigated. hese issues are treated in this report. he Matrix Logarithm Because L(t ) satises L(t ) (t )e,tf, where (t ) is real for a real system (), conditions on the real nature of the factors can be focused on conditions for obtaining a real F his leads us to examine which conditions are necessary and sucient for the logarithm of a real matrix to be real.. he Real Matrix Logarithm We have the following results for the existence and uniqueness of a real logarithm of a real matrix [] heorem Suppose M R nn. here is a real X R nn such that e X M if and only if M is nonsingular and has an even number of Jordan blocks of each size for every negative eigenvalue. If X exists, it is uniquely dened if and only if it satises the above conditions and all the eigenvalues of M are real, positive, and have geometric multiplicity. Otherwise, there exist at least a countably innite number of matrices X such that e X M. Remark For real systems (), the state transition matrix (t ) is real and non-singular. herefore, its logarithm always exists. his theorem provides necessary and sucient conditions for the logarithm to be real. Hence, we can focus our discussion on the existence of a real F in terms of the nature of the eigenvalues of the monodromy matrix () (also called the Floquet multipliers), and the structure of their associated Jordan blocks. Remark he classical result () is a straightforward consequence of heorem Because () is real, its eigenvalues must either be real or occur in complex-conugate pairs. Because () (), any negative eigenvalues of () must have come from purely imaginary eigenvalues of (). Because these eigenvalues occur in complex-conugate pairs, the Jordan blocks of () corresponding to negative eigenvalues must occur in pairs.

3 when Floquet multipliers i satisfy real factorizations for log branch k,choose P i R,fg r i s i 8i r i with i for i 8i r i with i for some i 9i r i > i.e., at least one paired PW,,,, i R,fg r i >s i Pdiag (), 8i s i i.e., unpaired Jordan blocks with ln(j ), 9i s i > i.e., paired Jordan blocks I ri, P r i, i R in conugate pairs r i s i simple or multiple P I ri, i U r i r i >s i Pdiag (), able Multiplicity of Real Factorizations. Characterization he matrix logarithm can be constructed by means of the following procedure []. Choose a Jordan form J for the non-singular matrix (). Choose a matrix W that puts () into the Jordan form J (i.e., () WJW, ). Choose a logarithm ln( i ) for each elementary Jordan block J i i I ri + U ri of J. Dene ln (()) W, mx [ln( i )I ri i r i, X,Uri i A W, () It is shown in [] that this approach can produce all the logarithms of matrix () when W is extended to WP, where P is any matrix that commutes with J Summary of Results. Generalized Denition of F he classical denition F ln(()) is not conducive to generating real factors. However, the use of the construction on two periods may result in a loss of information about L( ) by treating it as a -periodic function. In his study of classical periodic dierential equations, [] investigates when solutions of Mathieu's equations are periodic or anti-periodic. As far as we know, no one has attempted to extend these results to general LP systems, with an obvious application to the Floquet representation theorem. Using Yakubovich's result, we adopt a generalized denition for F F ln (Y()) where Y is constant matrix, which we call the Yakubovich matrix. Y satises heorem and can be chosen according to ( ) to give a real factorization. his leads to a novel Real Floquet Representation heorem that supersedes the result on two periods. he results reported are as follows. Existence and Uniqueness It can be veried that ln(y()) is always real for the choice Y WSW, where s ii, if i is real and negative, and s ii otherwise i n. When there are an innite number of real representations, they can be constructed by taking k + k k Z in equation (), with blocks of P chosen as shown in able.. Periodicity We see that Y indicates precisely the periodicity of the corresponding real L(t )

4 If Y I n, L(t ) L (t ) is -periodic If Y,I n, L(t ) L (t ) is -antiperiodic Otherwise, L(t ) L (t ) is generally - periodic. We also note that when real factorizations are not unique, Y k (,) k Y and the above is still true for the corresponding real transformation matrix L k. Illustration he following system rst appeared in []. Let _x(t) A(t)x(t), with A(t) U(t) ZU(t) cos t, sin t, where U(t) Z sin t cos t, and >. he period of A(t) is. One real pair of Floquet factors is given in [] as (t) L (t)e tf () cos t sin t where L (t) (8), sin t cos t and F, he monodromy matrix is thus L e F,e and its eigenvalues satisfy,,, (9),, (),e, p (, ) (). High-Order Unpaired Jordan Blocks he case leads to negative and repeated eigenvalues for the monodromy matrix where, e, e, W e,,e, W J ()W, is the matrix of generalized eigenvectors and J (),e,,e, he matrix J () consists of only one Jordan block associated with a negative real repeated eigenvalue (multiplier). Hence, no real, -periodic factor L (t) is possible. Only an antiperiodic factor L () exists, as given in equation (8), and this is the unique real decomposition. Figure shows the variations of A(t) and L (t) when and Let,e,, ln(,), hen, ln (J ()) F W J F W, as expected. A(t) L *(t),, Figure Periodic quantities when. op -periodic A(t) Bottom -antiperiodic L (t) (Note L (t) L (t)). Complex-Conugate Multipliers In this example, we will consider the case where the monodromy matrix has two complex-conugate eigenvalues, i.e., < Regardless of the sign of the real parts of the multipliers, it is possible to have real, -periodic or -antiperiodic factors... Derivation of F Using equation (), we obtain the following monodromy matrix for and, " e (),, cos( p ),p e, sin( p # p ) e, sin( p ) e, cos( p )

5 From equation (), the eigenvalues are e and e,, where e, (,+ p ) A set of eigenvectors is p p V, he Floquet exponents are (ln(r)+),+(,+p ) and we obtain the Floquet matrix F V V,, ( p, ),( p, ), As expected, F satises () e F and L ( ) is -periodic... Derivation of F We use the theorem to build another real matrix matrix, F corresponding to a -antiperiodic matrix L ( ). o this end, we create a new matrix with eigenvalues f +, i g We obtain F V + i,,,, i V, Note that this is the matrix given in equation (9), and the corresponding L () in equation (8) is cos(t) sin(t) L, sin(t) cos(t) which is -antiperiodic as expected... Derivation of L ( ) We can use the above expression of L ( ) to derive a closed-form expression for L ( ). First, we write the two computed real factorizations of the state transition matrix ( ), (t ) L (t )e tf L (t )e tf Of course, adding any other odd multiple of would have lead to a -antiperiodic factorization. and use it to express L in terms of already known quantities, i.e., L (t) L (t)e tf e,tf Recalling how F was constructing, we can write e tf V e t t F, e V, () and the new matrix L () has components l (t) cos (t)+ p sin (t) l (t) sin(t)(, p ) l (t) sin(t)(p, ) p l (t) cos (t)+ sin (t) with period, as expected. A plot of the variations of the matrices is given in Figure for, and. A(t) L (t) L (t) Figure Periodic quantities when <. op -periodic A(t) Middle -periodic L (t) (Note L (t) L (t)) Bottom -periodic L (t)... Paired Jordan Blocks Since there was a single Jordan block in the example of Section., the real decomposition was

6 unique and L() was -antiperiodic because the multiplier,e, was negative. We modify this example by building a new system matrix A () based on the previous A(), namely A(t) A (t) he monodromy matrix is A(t), e, e, e, e, e, e, with Jordan canonical form ( )W J, J () where J and J () W,e,,e,,e, W, Based on the example in Section., we derive F W ln (,J ) W,,,,, he Yakubovich matrix associated to F is Y,I n, indicating that the corresponding L () will be -antiperiodic, as was the case in Section.. However, since the Jordan blocks are paired, it is possible to derive another real pair ff L ()g where L () is-periodic. We build J, and we de-, rive F W J W, When, F,( +),( +),,,, Using L k (t ) (t )e,tf k we get L (t), where l (t) l (t) cos(t)[cos(t) + sin(t)] l (t),l (t) sin(t)[cos(t) + sin(t)] l (t) l (t),cos(t) sin(t) l (t),l (t),sin (t) l (t) l (t) sin(t) l (t),l (t),sin(t) l (t) l (t) cos(t)[cos(t), sin(t)] l (t),l (t) sin(t)[cos(t), sin(t)] We note that L () is-periodic, as expected. Acknowledgements he work reported here was partially supported by NSERC (Canada's Natural Science and Engineering Research Council) under Strategic Proect Grant # SR9. References [] F. M. Arscott. Periodic Dierential Equations. he Macmillan Company, New York, 9. [] R. E. Bellman. Methods of Nonlinear Analysis. Academic Press, New York, 9. [] R. W. Brockett. Finite Dimensional Linear Systems. Wiley, New York, 9. [] E. A. Coddington and N. Levinson. heory of Ordinary Dierential Equations. McGraw-Hill, New-York, 9. [] W. J. Culver. On the existence and uniqueness of the real logarithm of a matrix. Proceedings of the American Mathematical Society, (){, October 9. [] F. R. Gantmacher. he heory of Matrices. Chelsea, New York, 99. [] H.-O. Kreiss. Dierence methods for sti ordinary dierential equations. SIAM J. on Numerical Analysis, (), January 98. [8] V. A. Yakubovich. A remark on the Floquet- Lapunov theorem. Vestnik Leningrad. Univ., ()88{9, 9. in Russian.

Real Floquet Factors of Linear Time-Periodic Systems

Real Floquet Factors of Linear Time-Periodic Systems Pierre Montagnier 1, Christopher C. Paige 2, Raymond J. Spiteri,3 Abstract Floquet theory plays a ubiquitous role in the analysis and control of time-periodic