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L/ ~ I NACA TIM 1406 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1406 ON THE USE OF THE HAR~MOM~IC LIH~EARIZAiTIOIJ METHOD IN THIE AUTOMATIC CONTROL THEORY By E. P. Popov The method of harmonic linearization (harmonic balance), first proposed by N6. M. Krylov and N. N~. Bogolyubov (ref. 1) for the approxi mate investigation of nonlinear vibrations, has been developed and re eived wide practical application to problems in the theory of automatic control refss. 3 to 6). Recently, some doubt has been expressed on the legitimacy of application of the method to these problems, and assertions were made on the absence in them of a small parame~ter of any kind. Never theless, the method gives practical, acceptable results and is a simple and powerful means in engineering computatations. Benee, the importance of questions arises as to its justification. The underlying principle of the method. is the replacement of the given nolinear equation by a linear equation. In establishing the method, a small parameter is considered whose presence makes it possible to speak, with some degree of approxi mation, of the solution of this new equation to the~ solution of the~ given nonlinear equation. In an article by the author (ref. 7), certain con siderations were given on the presence of the small, parameter, but this question has not as yet received a final answe~r. In the present report, a somewhat different approach to the problem is applied that permits: (a) establishing, in the clearest manner, the form of the presence of the small parameter in nonlinear problems of control theory, solvable by the method of harmonic linearization; (b) connecting it with previous intuitive physical concepts (with the "filter property") an~d extending the class of problems possessing this property; and (c) discussing various generalizations of the method. The free motion (transition process and autovibration) for a very wide class of nonlinear systems of automatic control (ref. 7) are described by differential equations of the form Q(p)x + B(p)F(x,px) = O pa(1 ~t"K voprosu o primenenii metoda harmonicheskoi linearizatsii v teorii regulirovaniya." Doklady Akademii Nauk (SSSE), vol. 106, no. 2, 1956, pp. 211214. NACA TIM 1406 where Q(p) and B(p) are polynomials of any degree, of which some re q~uired properties will be established in the following paragraphs, and F(x,px) is a given nonlinear function possible only with respect to as sumptions of the most general character. However, in problems of the theory of control, no assumptions must be made as to the smallness of the nonlinear function F(x,px) or to its small difference from a linear function. In order to render explicit the form in which it would be possible to write a small parameter in equation (1), we shall proceed as follows. Let equation (1) have a periodic solution or a solution approximately periodic differing slightly from the sinusoidal. We write th~is solution in the form ~x = + ey(t) X Sna (2> where E denotes a, small parameter, and y(t) denotes an unknown bounded function of time. In the case of the existence of a periodic solution, we~ write cy(t) = sC ak sin(iwt + 9k) (3) k=2 We represent the gi:en nonlinear function F(x,px) in the form F(x,p) = F(x*C,pxQ) + [F(xlC + esy, px" + cpy) F(x ,px )l (4) Expanding the two components separately in a Fourier series, we obtain F(xlpx) = qO + A + p i a k k+ (5) where qO, A., and B are the coefficients of the initial term? the sine, and cosine terms, rexpectively, the cosine being replaced by p sin at of the expansion of the function F(x",px") in a Fourier series. EFk denotes all the higher harmonics of the expansion of F (x',px*) in a Fourier series (they must not be considered small, since the nonlinearity is not small) where we write Fk = bk sin (kaut + Jlk) (k = 2, 3, .., b) (6) ez'P denotes all the terms of the expansion in a Fourier series of all the expressions shown in brackets in formula (4). This entire expression NACA 00 1_406 is written with a small parameter which, according to equation (4), is small if the derivatives aF/ax and aF/apx are finite. This expression is also computed as smarll in the case of certain discontinuous nonlinear characteristics (e.g., the Raleigh type where the preceding derivatives at the points of discontinuity are delta functions). We may write Sk = ECk sin(kwt + zlk) ( ,1 ,...= 7 We substitute equations (2) and equattion (4) in the given equation (1), so that Q(p)x* + Q(p)sy + R(p)qq + R(p) A +,, p)snat (p)eI Fk + R(p)ef ,@k O~I= O (8) k=2 k=0 Since the equation must be satisfied identically, we separately equate to zero all coefficients with the same order harmonic. We note that formula (8), in the case of the existence of a periodic solution, is exact. From the equating of the zeroth harmonics of equation (8), there is obtained with an accuracy up to a: the relation e3 =y 26F(a sin u, amu cos u) du = O (u = cut) (9) which is a certain general requirement for F(x,px). From the equating of the first harmoics of equation (8), taking account of equations (2) and equation (7), we have a si at= A2 B2 i) Isin (rt + r + B)  Scl s ~c)(in)(i (rat + d1 + 8) (10) where y and. 8 are arguments of th expressions A + iB and R(io)/Q(ioj). On the basis of the3 exact eq~uatiojn (10), we obtain the! following approximation (with an accuracy up to E): a I I/.A2;~Bi)~w I 42 d + P a re (11) NACA TM 1406 It is here assumed that the polynomial. Q(]p) in equation (1) does not hae purely imaginar roots. From the equating of the higher harmonics of equation (8), consider ing equations (3), (6), and (7), we have sak sin(kwt + 9k) =bk sinfSI"( kat t +k k)ec sinai~~fkat~u +d 4k ) (12) where Bk denotes thre argument of the expression Rim/~k) It isthus seen that if bk is not small, the magnitude IR(ikmo)/Qiku)l should be of the order of e. The last component in equation (12) will then be of the order e2. From the exact equation (12), we obtain the following approximate equation (with an accuracy up to e): se =bkl~~ Qlk) k ek = (13) Comparing formulas (13) and (113 we see that, for example, the wish to have in the solution (see eqs. (2) and (3)) (sk2<< a2 (14) leads to the need of satisfying, in the given equation (1), the following requirement: b~kw R 344 (A2 + B2 2IRiw (15) where its satisfying in the concrete system can be checked after a is obtained. The degree of Q(p) should, in, any case, be higher than that of R(p). A particular case of the general expression (15) is intuitively the earlier introduced "filter property" of the linear part. Thus, condition (15) has been obtained anrd must be satisfied by the coefficients of the given differential equation (1) in order that a pe riodic solution, if it exists, may be approximately determined in the form of sinusoids in thre presence of a "strong" nonlinearity of F(x,px). NACA TMl 14106 Th equation for its approximate~ determination according to equation (8) with the substitution of sin wt =x*t/a, assumes the :form Q(p) + R(p) q + p~ r xjt = 0 (16) where A na1 2xr~ i ,acsusnud Q1 0 na JF(a sin u, am cos u)cos u du (17) The replacement of equation (1) by equation (1_6) with its subsequent investigation by the linear methods is called harmonic line~arizattion. This is formally equivalent to following the mode of writing the initial eqution (1) with a small parameter, so tha~t Q(p)x + R(p) + p~ + f(x~p) = O (18) where, according to equation (8), we may write ef(x,p) =~, Rp Fk +s # ~ q+9p)ey where the first term of the three terms shown in brackets, takn sepa rately, is not small. The smallness of the function ef(x,p) is acquired only with the factor R(p) owing to the property of equation (15). There has thus been found the form of the presence of the small parameter a in the equations of nonlinear systems of automatic control required for the application of the method of harmonic linearization (this also refers to the first approximation of the method of small pa rameter which, according to reference 7, coincides with the~ given method). The writing of equation (1) with small parameter in the form of equation (18) is valid in the region of the existence of eqcuation (2). For definiteness we shall assume that the plnmasRp n Q(p) are such that the characteristic equation (eq. (16)) has allpoitv coefficients, and that the quotient of the division of the entire left side of equation (16) by p2 + w2 satisfies the criterion of Burwits. It may then be said that the given nonlinear system (eq. (1)) for the existence of the periodic solution (eq. (2)) is close to the equivalent linear system which is on the boundary of stability, except for the NACA TM~ 1406 circumstance that the linear system is different for various periodic solutions (because of a change in the coefficients q and ql from one periodic solution to another, since q and ql depend on the amplitude a and the frequency a of the required solution). Thus, evidently, the equation of the~ first approximation (eq. (16)) is nonlinear if one speaks of the combination of all possible periodic solutions for different values of the coefficients of the polynomials R and Q (i.e., for different parameters of the system), and is linear only for each given periodic solution (along the given solution). Such is the property of the method of harmonic linearization in its application to systems with "strong" nonlinearity. By writing the given equation (1) in .t;he form of equation (8) all the variants of the method of harmonic balance in control theory and all generalizations of the method also become easily understandable. A similar reasoning may also be developed for the application of the method of N. N. Bogolyubov (ref. 2) to the investigation of transient vibrational processes in certain nonlinear systems of control. REF EREH C E S 1. K~rylov, N. M., and Bogolyubov, N. N.: NJew Methods of Nonlinear Mechan ics. K~iev, 1934; Introduction to Nonlinear Mechanics. Kiev, 1937. 2. Bogolyubov, N. N.: Sborn. tr. Inst. stroiteln. mekh, AN~ Uk~rSSR, 10, K~iev, 1949. 3. Goldfarb, L;. S.: Avtomat i telemekh, no. 5, 1947; M~ethod of Investi gation of Nonlinear Systems of Control Based on the Principle of BHarmnonic Balance. Ch. XXXIII of Fundamentals of Automatic Control, V. V. Solodovnikova, ed., 1954. 4. Popov, E. P.: Avtomat. i telemekh., no. 6, 1953; Izv. ANJ SSSR, OTNd, no. 5, 1954; The Dynamics of Systems of Automatic Control, 1954, pp. 593663; 704710; 716721. 5. Popov, E. P.: DAN, 98, no. 3, 1954. 6. Popov, E. P.: DAN, 98, no. 4, 1954. 7. Po~povt E. P.: Izvi. AN SSSR, OTN, no. 2, 1955. Translated by S. Reiss National Advisory Committee for Aeronautics NACA Langley Futld, Va ~ .C~3 F Nld,., ~N ~oo E m ac~o, mo c mL~ 3 E'* C " ~e: P~o oru 3 Um ~ O ~D~P~z? 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