LINEARIZATION OF NONLINEAR EQUATIONS By Dominick Andrisani. dg x. ( ) ( ) dx

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1 LINEAIZATION OF NONLINEA EQUATIONS By Domiick Adrisai A. Liearizatio of Noliear Fctios A. Scalar fctios of oe variable. We are ive the oliear fctio (). We assme that () ca be represeted si a Taylor series epasio abot some poit as follows d d ( ) ( ) = ( ) = + = + = ( ) ( ) ( ) d! d + hiher order terms A liear approimatio for () ivolves taki oly the first two terms d( ) ( ) ( ) = + ( ) d This approimatio is most accrate if ( ) order terms are eliible. = is small so that the elected hiher

2 Eample: ( )= Epadi () abot = ives ( ) d ( ) + d ( ) = = + ( ) = = 4+ 4( ) = 4+ 4 Notice that this is a liear fctio of. The simplificatio reslted becase we evalated all oliear terms at the mber riht had side of the eqatio above at = =. Becase we evalated the terms o the = =, the oly term that depeds o is the - term, ad this is a liear term. This approimatio to () is valid ear =. We ca eerate aother approimatio to () by epadi () abot = ives ( ) = + ( ) = + ( ) = = + This secod approimatio is valid ear =. Clearly the liear approimatio depeds o the choice of referece poit. A. Scalar fctio of variables Give the oliear fctio (, ). This fctio ca be represeted by a Taylor series epasio abot, as follows

3 (, ) = (, ) +, ( ) +, ( = = = =! ( ) + ( ) =, = =, = + ( )( ) + h. o. t. =, = A liear approimatio of ca be obtaied by retaii the first three terms above (derlied). The two variables i this problem ca be associated toether i a vector as follows ( ) where = Eample: (, ) = cos ca be approimated abot =, = as follows,, = = = = (, ) = ( cos ) + ( cos ) ( ) ( si ) ( ) =, = = 4+ 4( ) + = 4+ 4 The same fctio ca be approimated abot = =π/ 4 π (, ) = cos( ) + ( cos ) ( ) π 4 =, = 4 π ( si ) ( ) π =, = 4 4 π = + 4( ) = π 4 4 Notice aai how importat the liearizatio poit or referece poit is to the liearized reslt.

4 A.3 Vector fctio of a vector of variables. Let ( ) be a vector of oliear fctios. Let be a vector of variables = = M M, A liear approimatio abot is + = ( ) ( ) where ( ) (, ) (, ) = M L L = vector = M L L L = Jacobia Matri = matri A.4 Accracy of liearized soltio. Whe we approimate ( ) by retaii oly the liear terms, we mst aratee that the deleted terms, i.e., the h.o.t. are eliible. This is tre oly whe is small, i.e. whe the pertrbatios from the referece poit are small. B. Liearizatio o Noliear Differetial Eqatios i First Order Form B. First order form Noliear differetial eqatios i first order form ca be writte as

5 where = M = (, ), ( ) =, M, =, M = M Note that represets specified forci fctios ad ( ) is a specified iitial coditio vector. m Eample B.a =, = + = + B. The referece or trim soltio Whe we were liearizi oliear fctios, we saw how importat the choice of referece poit was. I liearizi oliear differetial eqatios, we are also cocered with the referece abot which we liearize. However, we are ow iterested i obtaii a liearized soltio valid for all time. This reqires that we liearize arod a referece soltio, which is valid for all time. Let () t be a kow soltio to the oliear differetial eqatio with specified forci fctio t. i.e., () ad specified iitial coditio ( ) t ( ) = ( (), t ()) t ( ) () t is said to be the referece soltio to the oliear differetial eqatio. Eample B.b For the differetial eqatios ive i Eample B.a () t =, () t, = () t =

6 Eample B.c Eample B.d is a costat soltio to the oliear differetial eqatio. Verify this fact for yorself by sbstitti this soltio ito the differetial eqatio ive i Eample B.a. Please keep straiht i yor mid the differece betwee a differetial eqatio (e.. = ) ad a soltio to a differetial eqatio (e.. = for = ). For the differetial eqatios ive i Eample B.a () t = () t =, ẋ = [ ] is aother costat soltio to the oliear differetial eqatios. For the differetial eqatios ive i Eample B.a =± = =± = cost = cost ẋ = is a costat soltio to the oliear differetial eqatios for ay costat. B.3 Liearizatio abot a referece soltio Let (), t () t be a referece soltio. We ow wat to fid a liearized soltio to the oliear differetial eqatio abot this referece soltio. We aai epad ( ) i a Taylor series epasio abot ad i.e., = (, ) +, ( ) + =, ( ) = = = +h.o.t. The liear approimatio is obtaied by assri that eoh that the h.o.t. ca be elected. B.4 Defiitio of small distribtio variables ad are small Defie = = = For the liearized soltio to be valid, these pertrbatios mst be small.

7 B.5 Separatio of the liearized differetial eqatios ito two parts Assmi that the pertrbatios are small, we ca write the approimatio to the differetial eqatios as = (, ) + ( ) + ( ) we ca ow sbstitte the small pertrbatio variables + = (, ) + + I the eqatio above we have simplified the otatio with to deote, = =. Notice that the derlied terms are merically eqal from the defiitio of referece soltio. Sice they are eqal, they ca be cacelled ot leavi = + This is a set of liear small pertrbatio differetial eqatios. I smmary, the oriial oliear problem = (, ), ( ) with soltio t () for specified ipt (t) has bee decomposed ito two separate problems. The referece problem r = (, ) with iitial coditio ( ) with soltio () t to ipt () t The small pertrbatio problem with iitial coditio with soltio () t to ipt (). t = + ( ) = ( ) ( ) Fially the total approimate soltio is ive by the etire soltio procedre is show i Fire. t () = () t +t ()

8 B.6 O picki a referece soltio Ay soltio to (, ) = makes a ood referece soltio bt these soltios ca be hard to fid. A easier set of soltios are costat soltios i.e., soltios so that () t = ad ()= t costat for t ()= costat. For costat referece soltios, fidi the referece soltio to a oliear differetial eqatio becomes a problem of fidi the soltio to a oliear alebraic eqatio (, )= B.7 Liearizatio Eample t ( ) = = + a) Choice of eferece Soltio To simplify or choice, assme that the referece soltio is costat, i.e., = =. This reqires that = ad + =. These eqatios ca be satisfied wheever = ad = Vales of ad which satisfy these eqatios are =± where is ay costat =±

9 Eact Noliear Soltio ((t)) (e.., from merical iteratio) Noliear Differetial Eqatios = (, ), ( ) () t = specified Noliear eferece Problem r = (, ) ( ), specified where typically ad are costats Liear Small Pertrbatio Problem = + ( ) = ( ) ( ) () t = t () Noliear eferece Soltio (e.., costat soltio), r Liear Soltio (i.e., from Laplace Trasforms) () t Total Approimate Soltio t () = + t () ( ) = + ( ) t () = + t () Fire Soltio Procedres for Noliear Differetial Eqatios

10 We will cosider two differet referece soltios e.# (), () ( ) e.# (), () ( ) f t t f t t = + + =+ = = = = b) Small Pertrbatio Eqatios of Motio = + where = = A B = = =, = = =, Usi ef. # = =, = + + Usi ef. # = =, = c. The Liear Soltio for eferece # () ẋ =

11 () ẋ = take Laplace trasforms () s () s () = () s s () () s () s () = () s mltiply () by s s () s s () = s () s mltiply () by set these eqal s () s = 4 () s + 4() s () s () s s() = 4() s + 4() s () ( s) [ s + 4]= s( ) ( ) + 4( s) s ( ) ( ) 4 () s = + ( ) s ( ) s + 4 s + 4 The first term o the riht ives iitial coditio respose. The secod term o the riht () s 4 cotais the trasfer fctio s () = s + 4. To fid () t take the iverse Laplace trasform. From () s () s () = () s () s = s() s () [ ] To fid the soltios () t ad () t yo mst be ive the ipt t () ad the iitial coditios ( ( ), ( ) ). The the soltios ca be fod si iverse Laplace trasforms. d) Total Soltio for eferece # () t () t = () t () t + () t () t = + + t () = () t + t () = + t () + ( ) ( ) = + ( ) () t () t

12 e) Commet: For this procedre to be valid the pertrbatios mst be small, i.e., all mst be small. Sppose we have the oliear problem with ( ) =. ( ) =. 99 t ( ) =. +. siωt the we ca se ef.# ad the we ca have ( ) =. ( ) =. t ( ) =. siωt O the other had if for the oliear problem we have ( ) =. ( ) =. 99 t ( ) =. siωt We wold se ef. # with ( ) =. ( ) =. t ( ) =. siωt

13 C. Otpt Eqatios Ofte oliear differetial eqatios are associated with oliear otpt eqatios. This may come abot i modeli the sesors aboard a aircraft. The sesors are ofte oliear fctios of the state vector ad cotrol vector. Otpt eqatios ca be epressed as follows. y = h(, ) where is y a p vector This ca also be liearized abot referece soltio ad as follows. h h y = y + y h(, ) + ( ) + ( ) The derlied terms are eqal by defiitio of y o the referece ad ca be cacelled ot o both sides of the eqatio. That leaves the liear small pertrbatio otpt eqatios i terms of small pertrbatio variables. h h y = + = C + D The total otpt eqatio i liear form is the ive by the followi. yt () y +yt () D. Stability Oe of may possible defiitios of dyamic stability for oliear systems is ive i terms of the eievales of the Jacobia matri, A = = If all the eievales of A have eative real parts we say that the referece soltio, (, ) is stable.., If at least oe of the eievale of A has a positive real part we say that the referece soltio, (, ), is stable. If at least oe eievales of A has a zero real part, ad if all the other eievales have eative real parts, we ca draw o coclsio abot the stability of the referece soltio, (, ). E. Cocldi Commets We have see how the soltio to oliear differetial eqatios ca be fod by decomposi the problem ito two simpler parts. The referece part is simpler becase it is ofte a oliear alebraic problem. The secod small pertrbatio part is simpler becase it ofte ivolves solvi liear differetial eqatios with costat coefficiets. The total approimate soltio to the oriial oliear differetial eqatio was show to be the sm of the two simpler parts.