PERFORMANCE ANALYSIS OF HARMONICALLY FORCED NONLINEAR SYSTEMS

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1 PERFORMANCE ANALYSIS OF HARMONICALLY FORCED NONLINEAR SYSTEMS A.Yu. Pogromsky R.A. vn den Berg J.E. Rood Deprtment of Mechnicl Engineering, Eindhoven University of Technology, P.O. Box 513, 56 MB, Eindhoven, The Netherlnds {A.Pogromsky, R..v.d.Berg, Abstrct: The pper dels with the performnce nlysis of hrmoniclly forced nonliner systems of Lur e type. Keywords: hrmonic lineriztion, describing functions, periodic solutions, direct Lypunov method 1. INTRODUCTION It is well known tht ny solution of stble liner time-invrint (LTI) system with hrmonic input converges to unique hrmonic limit solution tht depends only on the input nd not on the initil conditions. Nonliner systems with such property re referred to s convergent systems. Solutions of the convergent systems forget their initil conditions nd fter some trnsient time depend on the system input tht cn be commnd or reference signl. For LTI systems such hrmonic limit solution cn be esily derived while similr problem for nonliner systems is hrd to tckle. Some recent results in this field cn be found, e.g. in (Jönsson, et. l., 3). In this pper we consider the hrmonic lineriztion (Rozenwsser, 1969; Khlil, ) of nonliner systems of Lur e type. We formulte frequency domin condition which ensures tht the hrmonic lineriztion is well-posed nd derive upper bounds for its pproximtion error. We sy tht the procedure of hrmonic lineriztion is well-posed if for given mplitude nd frequency of the excittion signl, the corresponding lgebric hrmonic blnce eqution hs unique solution for clss of nonlinerities described by incrementl sector condition. It turns out tht this problem cn be tckled with the frequency inequlity of the incrementl circle criterion verified for the frequency of the externl excittion.. HARMONIC LINEARIZATION OF LUR E SYSTEMS Consider the following system of differentil equtions { ẋ = Ax Bφ(y) + F u (1) y = Cx + Du

2 where x R n is the stte, u R is the input, y R is the output, φ is continuous sclr function nd the mtrices A, B, C, D, F re of corresponding dimensions. We will ssume tht the nonliner function φ stisfies the following incrementl sector condition φ(y 1) φ(y ) y 1 y µ () for some positive µ. We do not consider the specil cse µ = + here, but focus only on finite µ. A more generl result tht includes lso µ = + cn be derived from our results using the stndrd methods of bsolute stbility theory. Suppose u(t) is periodic function of time with period T nd system (1) hs unique T -periodic solution x(t). Let us pproximte nonliner system (1) by liner system { ξ = Aξ BKζ + F u (3) ζ = Cξ + Du If the mtrix A BKC does not hve eigenvlues on the imginry xis then for the periodic input u(t) the system hs unique periodic solution ξ(t). Let us choose the gin K to minimize the following criterion J := 1 T T [φ( ζ(t)) K ζ(t)] dt, where ζ(t) = C ξ(t) + Du(t). This optiml gin cn be derived from the condition dj dk = nd is given by ( ) 1 T T K = ζ (t)dt φ( ζ(t)) ζ(t)dt. Now, suppose the input u(t) is hrmonic function of period T = π/ω so the output ζ is given by with some ψ. u(t) = b sin ωt, (4) ζ(t) = sin(ωt + ψ), > (5) In this cse the optiml gin is given s function of the mplitude of the output ζ(t): K() = 1 1π π φ( sin θ) sin θdθ. If φ is n odd function, K() is its describing function. Exmples of clcultions of K() for vrious φ cn be found in mny textbooks on the describing function method. From now on we ssume tht φ is n odd function nd, consequently, K() = π φ( sin θ) sin θdθ. Consider system (3). Let s = d dt. Then one cn write ζ(t) = C(sI n A) 1 BK() ζ +(C(sI n A) 1 F + D)u(t). Since the mplitude of ζ(t) is, the following reltion is vlid if A does not hve pure imginry eigenvlues ±iω 1 + K()G(iω) = C(iωI n A) 1 F + D b (6) where G(iω) = C(iωI n A) 1 B. The eqution (6) is referred to s the hrmonic blnce eqution. For this eqution one cn pose the following problem: given b >, ω >, does eqution (6) hve unique positive rel solution (b, ω)? If so, then substituting K((b, ω)) in (3) insted of K one cn esily compute the solution ξ(t) nd then to consider the question of how ccurte the solution ξ(t) pproximtes the solution x(t). Before we study eqution (6) let us chrcterize the function K(). If the nonliner function φ stisfies either the sector or the incrementl sector condition, it is possible to chrcterize the function K() s given by the following results. Lemm 1. Assume tht for ll y R, y there is µ > such tht Then φ(y) y µ K() µ,. Proof: See (Khlil, ), pge 85. By nlogy, Lemm. Assume tht for ll y 1, y R, y 1 y there is µ > such tht

3 Then φ(y 1) φ(y ) y 1 y µ K( 1) 1 K( ) µ, 1,, 1. Proof: Denote Then L K := K( 1) 1 K( ) ( 1 π L K = φ( 1 sin θ) sin θdθ π ) φ( sin θ) sin θdθ π = π µ π (φ( 1 sin θ) φ( sin θ)) sin θdθ 1 sin θ sin θ sin θdθ = µ The left inequlity is proven in the sme wy. (7) Now consider gin eqution (6). The following result is vlid. Theorem 3. Suppose the mtrix A does not hve pure imginry eigenvlues ±iω nd the following frequency inequlity Re G(iω) > 1 µ (8) is fulfilled. Then for ny function K() stisfying (7) nd for ny b > there is unique positive rel solution (b, ω) of eqution (6). Conversely, if Re G(iω) < 1 µ, then there is function K() stisfying (7) such tht eqution (6) hs multiple distinct positive rel solutions for some b >. Proof: Consider the left hnd side of eqution (6) π() = + K()G(iω) The ide of the proof is to show tht if the frequency inequlity (8) holds then π() is strictly incresing function. The sector condition (7) implies tht π() is (Lipschitz) continuous function. Since π() = nd π( ) =, existence nd uniqueness of the positive rel solution (b, ω) of (6) follow. For the ske of simplicity, we will prove the theorem under ssumption tht K() is differentible function of. The generl cse cn be deduced from (7) fter some ε-δ work, for exmple, tking into ccount tht lim inf 1 K( 1 ) 1 K( ) lim sup 1 K( 1 ) 1 K( ) µ. Differentiting π() with respect to yields π() = (1 + (K) G)(1 + KG ) +(1 + KG)(1 + (K) G ) (1 + (K + (K) ) ReG +K(K) [Re(G)] ) (9) Tking into ccount (7), it follows tht K() µ nd (K()) µ. Together with (8) it implies tht the qudrtic expression (9) is positive. To prove the second prt of the theorem, notice tht π() = Re [(1 + (K) G)(1 + KG )] nd therefore if we choose the function K() nd point such tht (K( ) ) is sufficiently close to µ, while K( ) is sufficiently close to zero (or wy round), the derivtive of π becomes strictly negtive for such K( ). However, π() = nd π( ) =, therefore, one cn choose b so tht eqution (6) hs multiple distinct positive rel solutions. It is worth noting tht the frequency inequlity from the theorem hypothesis is the sme inequlity imposed by the incrementl circle criterion yet verified only for the frequency of the externl excittion. The previous results llow one to complete the procedure of hrmonic lineriztion for system (1). Indeed, if the frequency condition holds, there is unique positive rel solution (b, ω), given

4 b, ω. Then substituting K((b, ω)) in (3) gives system liner in ξ. For such system one cn clculte the unique periodic solution ξ(t) using only lgebric clcultions. Like in the stndrd describing function method, one cn expect tht ξ(t) is sufficiently close to x(t). In the next section we derive bound tht estimtes this difference in L -norm. 3. ACCURACY OF HARMONIC LINEARIZATION In this section we study how ccurte the procedure of hrmonic lineriztion is. As before, to formulte the problem sttement, we ssume tht for given hrmonic input u(t) = b sin ωt the system (1) hs unique π/ω-periodic solution x(t). Together with system (1) consider the output z(t) = H x(t), z R (1) with n pproprite mtrix H. We further ssume tht the frequency inequlity (8) holds nd thus, ccording to the results of the previous section there is unique positive rel solution (b, ω) of the hrmonic blnce eqution (6). The problem ddressed here is to find n upper bound for ( ) 1 ω π/ω [ z(t) η(t)] dt π with η(t) = H ξ(t). (11) Let e be the difference x ξ. Then where ė = Ae B [ φ(ȳ) φ( ζ) ] + B (t) ȳ = C x + Du (1) ζ = C ξ + Du (t) = K((b, ω)) ζ(t) φ( ζ(t)) Substituting (5) in the previous expression gives (t) = K((b, ω)) sin(ωt + ψ) φ( sin(ωt + ψ)) Let ( 1 v() = π π [ π φ( sin ϑ)] dϑ φ( sin θ) sin θdθ sin ϑ ) 1. This integrl cn be clculted using the sme technique s in clcultion of K(). Notice tht ( ) 1 ω π/ω v((b, ω)) = (t)dt π where (b, ω) is the solution of (6). If one mkes technicl ssumption tht the pir (A, B) is controllble nd (A, C) is observble nd A does not hve pure imginry eigenvlues then the fulfillment of the frequency condition (8) for ll ω R implies tht for ny ε > there is symmetric mtrix P = P such tht e P ( Ae B [ φ(ȳ) φ( ζ) ]) εe e Tking the derivtive of the qudrtic form V = e P e long the solutions of (1) yields V εe e + e P B (t) Completing the squres with n pproprite ε one cn write V ( z η) + γ (t) with some finite γ >. Integrting the lst inequlity from to π/ω nd using periodicity of V (e(t)) one gets ( ω π/ω [ z(t) η(t)] dt π ) 1 γv((b, ω)). (13) To find the best possible (smllest) γ one cn pose nd numericlly solve the following optimiztion problem: Problem 4. Minimize γ such tht i) P is symmetric ii) A P + P A + H H P B + µ C P B B P + µ C 1 B P γ Now we cn summrize the bove rguments in the following result. Theorem 5. Consider system (1,, 4) under the following ssumptions

5 (A, B) is controllble nd (A, C) is observble. mtrix A does not hve eigenvlues on the imginry xis. the frequency inequlity (8) is stisfied for ll ω R. Along with system (1,, 4) consider its pproximtion (3, 4) with K = K((b, ω)) nd (b, ω) being the unique positive rel solution of the hrmonic blnce eqution (6). Let γ be the solution to Problem 4. Then there is unique π/ωperiodic solution x(t) of (1,4) nd the estimte (13) holds for z nd η defined in (1), (11). To complete the proof one hs to show tht there is unique π/ω-periodic solution. Tht follows from the frequency domin inequlity vi contrction mpping rgument. The previous result requires fulfillment of the incrementl circle criterion (frequency domin inequlity) which is sufficient condition tht for given b > nd ω > the corresponding coefficient of hrmonic lineriztion is uniquely determined. The frequency domin inequlity is lso necessry condition in the sense of Theorem 3: it ensures solvbility of the lgebric hrmonic blnce eqution for the clss of functions K( ) stisfying condition (7). However it is possible tht for given nonlinerity φ the corresponding hrmonic blnce eqution hs unique positive rel solution (b, ω) while the frequency domin inequlity does not hold. In this cse it is still possible to estimte the ccurcy of the method of hrmonic lineriztion. We hope tht n LMI-bsed procedure of checking frequencydomin inequlity for (semi)-finite rnge of frequencies cn be derived with recent generliztion of Klmn-Ykubovich-Popov lemm due to A. Frdkov (Frdkov, 6). It is importnt to note tht the procedure given by Theorem 5 is numericlly efficient. Once the incrementl gin γ is found by solving Problem 4, the limit solution of the liner pproximtion nd the upper bound on the error of this solution (13) cn be esily computed for vrious pirs of (b,ω) from certin domin of interest. The previous theorem cn be further generlized if one tkes into ccount tht is T -periodic signl nd its Fourier trnsform does not contin the first hrmonic. Denote ρ 1 := ρ := sup C(ikωI n A + µ k=3,5,... BC) 1 B sup H(ikωI n A + µ k=3,5,... BC) 1 B Theorem 6. Consider system (1,, 4) under the following ssumptions (A, B) is controllble nd (A, C) is observble. the hrmonic blnce eqution hs unique positive rel solution (b, ω) ρ 1 µ < φ is n odd function Along with system (1,, 4) consider its pproximtion (3, 4) with K = K((b, ω)) nd (b, ω) being the unique positive rel solution of the hrmonic blnce eqution (6). Let γ be defined s γ = ρ µρ 1 Then there is unique π/ω-periodic solution x(t) of (1,4) nd the estimte (13) holds for z nd η defined in (1), (11). 4. ILLUSTRATIVE EXAMPLE Consider system (1) with [ ] [ ] 1 A =, B =, K i K i K K p K i K [ ] C = [ 1], F = f with positive K i, K p, f nd nonnegtive K nd sturtion nonlinerity φ(y) = sign(y) min{1, y} This system corresponds to PI-controlled integrtor with sturtion nd nti-windup (if K >

6 ). The describing function of the sturtion nonlinerity is given by ( 1, 1 ( ) ) K() = 1 sin π, > 1 The squre root of the left hnd side of the hrmonic blnce eqution π() s function of is depicted in Fig. 1 with K i =, K p = 1, K =, ω = 1 (no nti-windup). The shpe of this curve 4 π() π() Fig. 1. The squre root of the left hnd side of the hrmonic blnce eqution π() versus, K =. suggests tht for reltively smll b the system hs one periodic solution nd then, with increse of b the system hs three periodic solutions, nd then, gin if b further increses, the system gin hs only one periodic solution. This quntittive conclusion is supported by numericl simultion. It cn be proved (vn den Berg et. l., 6) tht if K K p > 1 (with nti-windup) then the system hs exponentilly stble (yet not qudrticlly) periodic solution for rbitrry b, ω. This cn be illustrted with Fig. - the function π() becomes monotoniclly incresing for K = 1/K p =.1, ω = Fig.. The squre root of the left hnd side of the hrmonic blnce eqution π() versus, K = 1/K p. 5. REFERENCES Jönsson, U.T., Chung-Yo Ko nd A. Megretski (3), Anlysis of periodiclly forced uncertin feedbck systems, IEEE Trnsctions Aut. Contr., vol. 5(), pp Rozenwsser E.N. (1969) Oscilltions of Nonliner Systems, (Nuk, Moscow), in Russin. Khlil, H. K. () Nonliner systems, third edition (Prentice Hll, New Jersey). vn den Berg, R.A., A.Yu. Pogromsky, G.A. Leonov nd J.E. Rood (6), Design of convergent systems, in Group Coordintion nd Coopertive Control, K. Petersen, H. Nijmeijer (eds), (Springer, Berlin). Frdkov, A.L. (5), Conic S-Procedure nd Constrined Dissiptivity. ArXiv.org, mth/59718, 3 Sept 5. ACKNOWLEDGEMENT This work ws prtilly supported by the Dutch- Russin progrm on interdisciplinry mthemtics Dynmics nd Control of Hybrid Mechnicl Systems (NWO grnt ).