Abstract—A general principle, in terms of the switching instants, for determining whether a switched system is linear or nonlinear is suggested. Index Terms—Nonlinear circuits, System theory, Switched circuits, System identification. I. INTRODUCTION witched circuits and systems, SS, may be linear time- variant, LTV, [1], as, e.g., the well-known switched capacitor circuits, SCC, [2,3], or nonlinear, NL (e.g. [4]). Some arguments of [5-8] related to non-linearity of some singular systems motivated the present study of SS. The suggested below classification of SS as linear or nonlinear is wider than use of the concept of "feedback"; we do not speak only about the role of the output variable(s). The classification of SS is important for the logical foundation of circuit theory because only for SS are the definitions of linear and nonlinear systems given in the same constructive terms. Namely, denoting time as t, the switching instants as {t k }, or t*, and the vector of the (all initially unknown) state variables of the system as {x k }, or x, we have t*(.) as t*(t) for an LTV system, and as t*(t,x) (or t*(x)) for an NL system, and we suggest defining SS in terms of t*(.). The following example of a singular system introduces the point in a simple form. II. THE "ZEROCROSSING NONLINEARITY" Consider (closely to [8]) the singular equation 1 sign[ ( )] () sin di L A it i t dt B t dt C      (1) where L, A and C are positive constants, and sign[x] = 1 for x > 0, and –1 for x < 0. Equations of this type are known in the theory of mechanical systems with Coulomb (dry) friction, and are important in a theory of the steady-state operation of fluorescent lamp circuits at 50-60 Hz [8]. Because of the function ‘signum’, (1) is a nonlinear equation, obviously. Though being singular, the function i(t) must be continuous since otherwise di/dt introduces into (1) an uncompensated - Manuscript received Aug. 31, 2006. E. Gluskin is with Holon Institute of Technology, Holon 58102, Israel. (phone: 972-3-5026626; fax: 972-3- 5026643 e-mail: gluskin@ee.bgu.ac.il). function. It is not difficult to show [5,8] that for   (LC) -1/2 and |B|/A sufficiently large, i(t) is a T-periodic zerocrossing time function having (as it is for sint) two zerocrossings per period, distanced by T/2= Thus, for such |B|/A, the nonlinear term in (1) is the square wave sin ( ) 4 1 [ ( )] , 1,3,5,... n t t A Asign i t n      (2) where t 1 (modT) is the /+ zerocrossing of i(t). However, if we substitute (2) into (1), then in the obtained equation: sin ( ) 1 4 1 () sin 1,3,5,... n t t di A L i t dt B t dt C n          (3) the nonlinearity that was so obvious in (1), is not at all obvious. If t 1 were to be an a priori given parameter, then (3) would be a very simple LTI equation. The point is, of course, that t 1 belongs to the unknown function i(t) (generally, to a state-variable) that has to be determined, and thus the sum in (2) is a nonlinear expression, and (3) is a nonlinear equation. Following [5-7], we call the nonlinearity expressed in some a priori unknown time parameters (here shifts) belonging to unknown functions, “zerocrossing nonlinearity”. Such a nonlinearity may be introduced using different singular time functions, not necessarily a rectangular wave, as one sees when considering, e.g., the functions obtained by means of direct integration, or integration with a continuous kernel, of the square wave, including such a shift. Observe that from (3) the current is found as i = F(t,t 1 ) with a known function F(.,.), i.e. i is expressed directly via its zerocrossing t 1 (mod T), and that the initially unknown t 1 is then found from the equation i(t 1 ) = F(t 1 ,t 1 ) = 0. This means that the zerocrossing representation of i is quite constructive. III. THE SYSTEM All the switches and comparators involved are assumed to be ideal, and the switched elements per se are linear (below, LTI). Elements like a ferroelectric capacitors and saturated inductors, introducing any "analytical" nonlinearity, are excluded as irrelevant to the point. For avoiding non-essential generalizations it is also assumed that the system cannot be A point of view on the linearity and nonlinearity of switched systems Emanuel Gluskin S 1-4244-0230-1/06/$20.00 ©2006 IEEE 2006 IEEE 24th Convention of Electrical and Electronics Engineers in Israel 110