Describig Fuctio: A Approximate Aalysis Method his chapter presets a method for approximately aalyzig oliear dyamical systems A closed-form aalytical solutio of a oliear dyamical system (eg, a oliear differetial equatio or a oliear map) is usually ifeasible except possibly for special cases that are usually cotrived; therefore, it is a usual practice to carry out approximate aalysis of oliear dyamical systems Describig fuctio (DF) is a aalytical tool for represetatio of a oliear autoomous system as a equivalet liear time-ivariat system, which is the best liear approximatio of the oliear autoomous system i a certai sese his method ca also be used to predict the existece of periodic solutios i feedback systems although it does ot provide ay ecessary or sufficiet coditios
Iterpretatio of Describig Fuctio as a Optimal Quasiliearizatio from a Fuctioal Aalytic Perspective Let C F be the space of almost everywhere cotiuous fuctios that have fiite average power, ie, lim d f ( ) < f CF ad (, ) τ τ [It is implied that the limit exists ad is fiite] Let I = { ψ : C F C F } be the space of causal time-ivariat bouded operators Note that the operators ψ : C F C F could be liear or oliear Remark : C F is a vector space over the real field R with a ier product defied as: f,g dτg( τ) f ( τ) f,g C F[,] ad ay fiite (, ) Accordigly, the orm o this ier product space is defied as: It follows from Cauchy-Schwarz iequality that f dτ f ( τ) for ay fiite (, ) ad f C F f,g Defiitio : he cross-correlatio of two fuctios f g f,g C F f, g C is defied as: Φ τ = ( ) f,g ( ) lim dτ g(t + τ)f (t) Let L be a subspace of I cotaiig oly the liear fiite-dimesioal operators hat is, L cosists of proper Hurwitz ratioal trasfer fuctios For example, let e(t) CF be the iput to the oliear fuctio ψ I so that its output is ψ ( e(t) ) CF If h L is a liear approximatio of ψ I, the the output correspodig to the iput t e(t) C F is ( h e) (t) dτh(t τ)e( τ) ad h L[, ) he problem is to idetify a proper Hurwitz trasfer fuctio ĥ correspodig to the impulse respose h L that approximates the oliear fuctio ψ I i the sese of miimizig the followig error fuctioal: J (h) lim dt ψ(e(t)) (h e)(t) he resultig (causal Hurwitz) impulse respose h(t) or the correspodig trasfer fuctio ĥ (s) ) is called a optimal quasi-liearizatio of the oliear operator ψ I heorem : he error fuctioal J(h) is miimized if ad oly if Φe,h e( τ) = Φe, ψ(e) ( τ) τ Proof: Let h L he objective is to idetify ad optimal idetify a optimal h Let δj ( h,h) J( h) J(h) = lim F t be aother liear approximatio of ψ I he, ( ) τ( h e (t) d h(t τ)e( τ) ) ad h L[, ) h L such that J (h) J( h) h L Let us use the variatioal approach to = lim dt ψ(e(t)) ( h e)(t) ψ(e(t)) (h e)(t) dt ( h e)(t) (h e)(t) ( ( ) h h) e (t) ψ(e(t))
= lim dt Sice h miimizes the error fuctioal J ( ) if ad oly if ( h h) e) (t) + ( h h) e) (t) ((h e)(t) ψ(e(t) ) ( ( h h) e) (t) ((h e)(t) ψ(e(t) )) = lim dt Expressig the covolutio terms as itegrals, we have: dt ( ( h h) e) (t) ((h e)(t) ψ(e(t )) ) ( ) = t dt τ τ ψ dτ ( h h)( )e(t ) (h e)(t) (e(t ) δj ( h,h) h L, we must have ( ( )) = d τ( h h)( τ) τ ψ dt e(t ) (h e)(t) (e(t ) by iterchagig the order of itegratio τ τ herefore, lim dτ( h h)( τ) dt ( e(t τ) ((h e)(t) ψ(e(t) )) = τ possible if lim dt ( e(t τ) ((h e)(t) ψ(e(t) )) = τ lim τ dt implyig that (((h e)(t + τ) ψ(e(t + τ) ) e(t) ) = τ for all fuctios ( h h) L his is Settig i place of τ whe, we have lim ( ) = ( ψ + τ ) dt (h e)(t + τ)e(t) lim dt (e(t )e(t) Hece, Φe,h e( τ) = Φe, ψ(e) ( τ) τ Remark : he optimal approximatio of the oliear fuctio ψ by a liear trasfer matrix ĥ is depedet o the give referece poit However, there is o guaratee of optimality or eve existece of such a trasfer fuctio if the referece poit is altered because the oliear fuctio ψ is also altered Example : Let the iput sigal be a o-zero costat, ie, e(t) = c t he, ψ ( e(t)) = ψ(c) t ad Φe, ψ (e)( τ) = cψ(c) which is a costat τ By the optimality coditio, the liear approximatio h(t) must satisfy Φe,h e ( τ) = cψ(c) that leads to: lim dt ((h e)(t )c) = cψ(c) lim dt (h e)(t) = ψ(c) + τ If the oliearity is memoryless, the ψ(c) is a real costat he a possible choice is ( h e)(t) = ψ(c) t (c) his implies that h(t) = ψ ψ(c) δ(t) or ĥ(s) = However, this choice may ot be uique c c Example : Let e(t) = ESi( Ω t) t he, ψ(e(t)) = A + ( AkSi(k Ω t) + BkCos(k Ω t) ) t k= Let ψ be a odd fuctio implyig that A = ad B k = k N he, Φ τ = e, ψ(e) ( ) lim dt AkSi(k Ω (t + τ ) E Si(k Ω t) k=
π / Ω Sice dtsi(k Ω t) Si( Ω t) = π δ Ω k, we have A Φ ψ ( τ) = lim e, (e) ( Ω + τ ω ) dt Si( (t ) ESi( t) A Now, if we choose ĥ(s) = A h(t) = (t), E E δ t A the (h e)(t) = d τ δ(t τ)esi( Ω t) = A Si( t) E Ω he, Φ ( ) = ( Ω + τ Ω ) ( τ) = lim Ω + τ ω dt Si( (t ) ESi( t) lim e,h e dt Si( (t ) ESi( t) τ herefore, Φe, ψ (e)( τ) = Φe,h e( τ) τ heorem : Let the iput ad output of the oliear block be e(t) = ESi( Ωt) ad ψ ( e(t)) = z(t) = zss (t) + ztr (t), respectively, where the first harmoic z is cotiuous ad ( π / Ω) periodic ad ztr L([, )) Let the oliear operator ψ ( ) be approximated by the followig trasfer fuctio ĥ (s) such that Re gre( ω) g ( ( ĥ(iω) ) = ad Im( ĥ(iω) ) = im E E where gre( ω ) ad gim ( ω) are eve ad odd fuctios of ω, respectively; ad the first harmoic of z ss (t) is give as: z ss (t) = gre( Ω) Si( Ωt) + gim ( Ω)Cos( Ω t) Proof: We have ztr L([, )), ie, dt ztr (t) = ϑ < ; ad the eergy of the iput sigal over ay fiite is: dt e(t) E dtsi ( t) E = Ω e(t) E for ay fiite Havig e,z tr dt ( ztr (t) e(t) ), it follows from Cauchy-Schwarz iequality that e, ztr e z tr for ay fiite E ϑ Now, the cross-correlatio Φe,z tr lim e,ztr lim e ztr lim = herefore, Φ e,z tr ( τ) = τ he implicatio is that ay trasiet part of the oliear block output is ucorrelated to its iput We cosider oly the periodic part z ss (t) of the output z (t) because Φ e,z ( τ) = Φe,z ( τ) + Φe,z ( τ) = Φe, z ( τ) ss tr ss Sice e(t) = ESi( Ω t), settig = π / Ω where N, we have e(t),si πk(t + θ) = ad k(t ) e(t),cos π + θ for all k N -{ } Lettig, it follows that Φ e,z ) ( ) ( τ = Φ ss e, z τ ss ê(iω ) = iπe δ( ω+ Ω) δ( ω Ω), the output of the liear approximatio block is obtaied as the Fourier Give ( ) iverse of ĥ(iω ) ê(iω) as follows: ω) ( h *e)(t) = ( Re( ĥ(iω ))Si( Ωt) + Im(ĥ(iΩ))Cos( Ωt) )E herefore, (h e)(t) = z ss (t) t Proof is thus complete Defiitio : Followig the otatios i heorem, the describig fuctio K eq (, ) of the oliear operator ψ is the complex-valued fuctio defied as: K eq (E,Ω) = gre ( Ω ) gim ( Ω) + i E E heorem 3: If the oliear operator ψ is memoryless, time-ivariat, ad cotiuous, the the describig fuctio K eq (E, Ω ) is idepedet of Ω Proof: he output i respose to the iput e(t) ESi( Ωt) = is z(t) ψ( ESi( Ωt) ) = that is decomposed as z(t) = zss (t) + ztr (t) as the steady-state ad trasiet compoets Give the operator ψ to be memoryless, time- 3
ivariat, almost everywhere cotiuous, ad bouded, the trasiet part z tr (t) of the output becomes zero g g ( ) herefore, the coditios of heorem prevail ad we have K eq (E,Ω) = re ( Ω ) im Ω + i Now cosider aother E E sigal e(t) = ESi( Ωt) of same amplitude ad differet frequecy Ω Ω so that e ca be obtaied from e by time scalig herefore, ( ψe )(t) = ( ψ e) Ωt implyig that the first harmoics are idetical after time scalig herefore, Ω K (E, ) K (E, eq Ω = eq Ω) Corollary to heorem 3: If, i additio, the operator ψ is odd, the K eq is real for E>, ie, g im ( Ω ) = Proof: For e(t) ESi( Ωt) = ad ψ beig odd, Fourier expasio of ( e) (t) ψ does ot cotai ay cosie term Corollary to heorem 3: Let, i additio to the coditios specified i heorem 3 ad Corollary, ψ be sectorbouded, ie, k,k (, ) such that k σ σψ( σ) k σ σ R he, k Keq (E) k Proof: By Corollary to heorem 3, K eq is real ad idepedet of ω because ψ is memoryless, time-ivariat, (almost everywhere) cotiuous, ad odd herefore, π / Ω Keq (E) = Ω dt πe π = d E θ π ( ψ(esi( Ω t))si( Ω t) ) π πe ( ) = k ( ψ(esi θ)si θ) dθ k ( ESiθ) π πe π πe Similarly, K (E) = dθ ( ψ(esi θ)si θ) dθ k ( ESiθ) eq ( ) = k 4