Lecture Notes. A Note on Extension of the Eigenvalue Concept. J. Jim Zhu. Two
parametrized linear periodic sys- tems are presented here for use in ad-.
Lecture Notes
A Note on Extension of the Eigenvalue Concept J. Jim Zhu
Two parametrized linear periodic systems are presented here for use in advanced senior and graduate control courses to demonstrate the issues arising in attempting to extend the eigenvalue concept for linear time-invariant systems to linear time-varying systems.
Eigenvalue Concept Counterpart In general, the stability analysis and analytical solutions for the class of linear time-varying (LTV) systems of the form X = A(t)x, x(t0) = xo
(1)
are very difficult problems. However, it is well known that the stability of a linear time-invariant (LTI) system, where A ( t )= A = constant, is precisely determined by the left-half-complex-plane (LHCP) confinement of the eigenvalues h, given by the roots of det [kZ- A ] = 0. Moreover, the analytical solution of a LTI system can be written explicitly using the eigenvalues in the form of exponential modes ,'It. Therefore, it is intriguing to ask: What is the LTVcounterpart of the LTI eigenvalue concept.? Since for a LTI system A ( t ) = A , it is tempting to extend the LTI eigenvalue concept in a point-wise fashion to thefmZen-time eigenvalues (FTEs) h;(t) as defined by the roots of det [h(t)Z- A ( t ) ]= 0, and to make the following conjecture.
Conjecture 1 (Well-Known False). A LTV system would be asymptotically stable at tQf a n d only f i t s FTEs hi(t)were to be confined to the open LHCP for all t 2 to.
Unfortunately, in general the FTEs do not constitute analytical solutions for LTV systems. Moreover, Conjecture 1 fails, in general, to be either sufficient or necessary, as has been repeatedly demonstrated by many researchers (see [ l ] and references therein). Despite the general failure of Conjecture 1, it is still plausible that when the magnitude and the rate of parameter variation in A ( t ) about some nominal value Anom are sufficiently small and slow, the stability of the LTV system with A ( t ) should remain the same as that of the LTI system with Anom . This is indeed true, owing to the continuous dependence of the solution of a LTV system on its parameter variation. To date many such perturbational results have been developed based on FTEs of A ( t ) (see [ 11 and references therein), which may be summarized as follows: Theorem 1. LetA(.): R+ 3 Rnxn be continuous and unfor-mly bounded such that ( i ) IIA(t) II S M f o r some M > 0 and for all t 2 0, and (ii)Rehl(t)2 - afor some a > 0 and for all i 5 n. Then there exists a 6 >O for which any of the following conditions guarantees exponential stability of ( I ) :
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IIIA(t)lldt I 6T for all to 2 0; to
e) lim sup IIA(t + T) - A(t)ll = 0 t4 0: The author is with the Electrical and Computer Engineering Department, Louisiana State University, Baton Rouge, LA 70803.
to+T
d ) there exists T > 0 such that 0272- 1708/93/$03.000 1993IEEE
Consequently, the stability of a LP system is equivalent to that of the LTI system ( 2 ) .The eigenvalues of B, denoted by p;,are known as the Floquet characteristic exponents (FCE) of the LP system. The Floquet stability criterion can now be stated as follows. Theorem 2 (Floquet). A LP system is exponentially asymptotically stable if and only if all FCEs pi lie in the open LHCP.
IEEE Control Systems
The application of Theorem 2 is hampered by the fact that, in general, FCEs elude analytical solutions, but numerical techniques do exist for evaluating FCEs (see [2] and references therein). In this note we present two parametrized LP systems, generalized from the wellknown equations of Markus-Yamabe (MY) [3] and Wu [4], which can be used in advanced senior and graduate control courses to demonstrate, in the light of Floquet Theory, that: i) the powerful eigenvalue concept for LTI systems does not, in general, carry over to LTV systems in a point-wise (frozen-time) fashion, ii) perturbational extensions of the LTI eigenvalue concept tend to be overly restrictive, and iii) LTI systems may be stabilized or destabilized by structural perturbations without any control input. A brief summary of current research efforts on extending the LTI eigenvalue concept to LTV systems is given for the inspired students to further explore this challenging research area.
Two Examples
I
(-b+a)+a cos cot b-a sinot A(t)= -b-a sincot (-b+a)-a cos cot
(3) which falls back to the original M-Y equation [3J for a = 0.75, b = l , o = 2. The FIEs of (3) are given by h1,2 = (a k h) k
m,
which are independent of both t and W. Based on Conjecture 1, the (false) domain of asymptotical stability (DOAS) in the a-b parameter plane would be given by a
