JOURNAL OF THEORETICAL AND APPLIED MECHANICS 43, 3, pp. 675-693, Warsaw 2005 APPLICATION OF POLI-CRITERIAL LINEARIZATION FOR CONTROL PROBLEM OF STOCHASTIC DYNAMIC SYSTEMS Lesław Socha Department of Mathematics and Sciences, Cardinal Stefan Wyszynski University in Warsaw e-mail: leslawsocha@poczta.onet.pl The problem of the determination of response characteristics and quasi- optimal control for nonlinear stochastic dynamic systems by using a multi-criteria linearization technique is presented in this paper.This idea was first introduced in previous author’s paper (Socha, 1999a) for a sim- ple dynamic system. In this paper, it is extended, and detailed analysis is given for a nonlinear oscillatorwithGaussian external excitations and for a few criteria of statistical linearization. The obtained results are illustrated by a numerical example for Duffing’s oscillator. Key words: Stochastic control of nonlinear systems, LQG control pro- blem, stochastic linearization, Pareto optimal solution 1. Introduction Linearization methods are the most versatile methods for analysis of non- linear systems and structures under stochastic excitations. Different criteria of linearization connectedwithwell known linearizationmethods such as statisti- cal linearization, equivalent linearization or exact linearization in three basic spaces were separately considered in the literature. They were discussed in: • space of moments of stochastic processes – Elishakoff (2000) • space of probability density functions of stochastic processes – Socha (1999b) • space of spectral density functions of stochastic processes – Bernard and Taazount (1994). 676 L.Socha The number of different linearization criteria is large (more than 100). For details see, for instance, a survey papers Elishakoff (2000) or Socha and Soong (1991). It is clear that the choice of a group of criteria of lineariza- tion depends on a considered problem but in one group, for instance, in the space of moments of stochastic processes the problem is open and the choice requires analysis. To study multicriteria problems, special approaches called multicriteria optimization methods were developed in the literature. In the field of mechanics, it was reviewed in Stadler (1984). The objective of this paper is to show a relationship between these criteria using two approaches of poly-criterial optimization techniques, namely the scalarization method and Pareto-optimal solution (Sawaragi et al., 1985; Skulimowski, 1996) in appli- cation to the determination of the response characteristics and quasi-optimal control for nonlinear stochastic dynamic systems. This idea was first introdu- ced in previous author’s paper (Socha, 1999a) for a simple dynamic system. In this paper, it is extended and detailed analysis is given for a nonlinear oscil- lator with Gaussian external excitations and for a few criteria of statistical linearization. The obtained results are illustrated by a numerical example for Duffing’s oscillator. Consider a nonlinear stochastic model of a dynamic system described by the Ito vector differential equation dx(t)=Φ(x) dt+ M ∑ k=1 Gk dξk(t) (1.1) where x= [x1, . . . ,xn] > is the state vector, Φ= [Φ1, . . . ,Φn] > is a nonlinear vector function such that Φ(0) = 0, Gk = [Gk1, . . . ,Gkn] > are deterministic vectors, ξk are independent standard Wiener processes. We assume that a unique solution to equation (1.1) exists. 2. Linearization techniques for stochastic systems 2.1. Statistical and equivalent linearization There are two basic groups of linearization methods for stochastic dy- namic systems, namely the statistical (or local) linearization and equivalent linearization. In the case of statistical linearization, the objective is to find for a nonlinear vector Φ= [Φ1, . . . ,Φn] > an equivalent one ”in the sense of a linearization criterion”, i.e., replacing Y =Φ(x,t) (2.1) Application of poli-criterial linearization... 677 in equation (1.1) by a linearized form Y =Φ0(mx,Θx, t)+K(mx,Θx, t)x 0 (2.2) where mx =E[x] Θx = [θij] =E[x 0 ix 0 j] (2.3) with x0i =xi−mxi being the centralized stochastic process Φ0 = [Φ01, . . . ,Φ0n] > is a nonlinear vector function of the moments of x and K= [kij] is a n×n matrix of statistical linearization coefficients. In the case of equivalent linearization, the objective is to find for nonlinear dynamic system (1.1) an equivalent one in the sense of a linearization criterion based on response properties for nonlinear system (1.1) and for the following linearized system dx(t)= [A(t)x+C(t)]dt+ M ∑ k=1 [Dkx+Gk]dξk(t) (2.4) where A = [aij], Dk = [dkij], i,j = 1, . . . ,n, k = 1, . . . ,M are matrices and C = [C1, . . . ,Cn] >, Gk = [Gk1, . . . ,Gkn] > are vectors of linearization coefficients. 2.2. Basic linearization criteria for stochastic dynamic systems Criterion 1a – Criterion of Equality of the First and Second Moments of Nonlinear and Linearized Variables (Kazakov, 1956) E[Yi] =Φi0 (2.5) E[(Yi−E[Yi])(Yj −E[Yj])] = n ∑ i=1 n ∑ j=1 kikjθij Criterion 1b – Criterion of Moment Error of Approximation E [( Φ 2p i (x,t)− ( Φ0i+ n ∑ j=1 kjx 0 j )2p)2] p=0,1,2, . . . (2.6) where for p = 0 and p = 1 it is known in the literature as Criterion of Mean-square Error of Approximation (Kazakov, 1956) and Energy Criterion (Elishakoff and Zhang, 1984) 678 L.Socha Equivalency of response probability densities (Socha, 1999b): Criterion 2a – Pseudomoment metric I2a = +∞ ∫ −∞ . . . +∞ ∫ −∞ x [2r]|gN(x)−gL(x)| dx1 . . .dxn (2.7) where x = [x1, . . . ,xn] >, x[2r] = x 2r1 1 x 2r2 2 . . .x 2rn n , r1, . . .rn ∈ N, ∑n i=12ri =2r. Criterion 2a – Square probability metric I2b = +∞ ∫ −∞ . . . +∞ ∫ −∞ [gN(x)−gL(x)]2 dx1 . . .dxn (2.8) where gN(x) and gL(x) are probability density functions of solutions to nonlinear (1.1) and linearized systems (2.4), respectively. 2.3. Poly-criteria optimization methods In this section,we quote somebasic definitions and facts frommulticriteria optimization theory (Socha, 1999a,b). A subset Θ of a linear space B is called a convex cone if and only if ∀α1 ­ 0 ∀α2 ­ 0 ∀x1x2 ∈Θ (α1x1+α2x2)∈Θ (2.9) To every convex cone Θ, there corresponds an ordering relation R in B defi- ned by x1 ¬x2 ⇐⇒x2−x1 ∈Θ (2.10) The relations of partial order induced by convex cones are generalization of the natural order in Rn defined as follows x¬y⇐⇒∀i=1, . . . ,n xi ¬ yi (2.11) where x = [x1, . . . ,xn] > and y = [y1, . . . ,yn] >. This is equivalent to the relation y−x ∈ Rn+. The positive orthant Rn+ satisfies all properties of the convex cones. The general problem of multicriteria optimization is (F : Ud →B) → min(Θ) (2.12) where the set of admissible controls Ud is a subset of a linear space U, the goal space B is partially ordered Banach space with a closed convex cone Θ. Moreover, it is assumedthat theadmissible set F(Ud) is non-emptyandclosed. Application of poli-criterial linearization... 679 2.4. Pareto-optimal approach Acontrol uopt is said to benondominated or Pareto-optimal, or Θ-optimal if and only if (F(uopt)−Θ)∩F(Ud)= {F(uopt)} (2.13) Condition (2.13) means that no element of the admissible set is better than uopt in the sense of the partial order relation. Relation (2.10) plays the fundamental role in classical problems of multi- criteria optimization which can be reduced to the simultaneous minimization of scalar functions (F1,F2, . . . ,Fm)→min (2.14) 2.5. Scalarization methods Themost frequently used scalarization method for the problem (F : Ud →Rn) → min(Rn+) (2.15) is a positive convex combination of the criteria Fw(u)= N ∑ i=1 wiFi(u) (2.16) where u ∈ Ud, wi > 0 for 1 ¬ i ¬ N and ∑N i=1wi = 1. The parameters wi > 0, 1¬ i¬N are weight coefficients. The scalarization by distance Fd(u)= d(q,F(u)) (2.17) where d is a metric in the goal space, q is a fixed unattainable element of the goal space which dominates at least one point from F(Ud), for instance Fp(u)= ‖q−F(u)‖pp (2.18) where ‖·‖pp is a p-th power of the norm in the Lp space. For instance, as the scalarizing family for thefinite-dimensionalmulticriteria optimization problem with respect to the natural partial order in Rn, one can consider the following family of functionals Np(u,w) = N ∑ i=1 wi(Fi(u)−qi)p w∈Rn+ \{0} 1¬ p¬∞ (2.19) 680 L.Socha In a particular case, when n= 2, Ud =R 1 = {k : −∞< k <+∞} and Fn = In, n = 1,2 an illustration of a dominated point q and relation (2.17) is given in Fig.1. In this case, I1 and I2 are two criteria (for instance line- arization criteria) and the convex curve is parametrized by k (for instance, a linearization coefficient). The points of the curve are defined by (I1(k),I2(k)). Fig. 1. A geometric illustration of condition (2.17) 3. Applications for single degree of freedom systems Consider a single degree-of-freedom system described by dx1 =x2 dt (3.1) dx2 = [−f(x1)−2hx2] dt+σ dξ where h and σ are constant parameters, f is a nonlinear function such that f(0)= 0. Then, the mean value of the stationary solution is equal to zero, i.e E[x1] = 0. An equivalent linearized system has the form dx1 =x2 dt (3.2) dx2 = [−kx1−2hx2] dt+σ dξ where k is a linearization coefficient. Application of poli-criterial linearization... 681 The most frequently used scalarization method is a positive convex com- bination of considered criteria, i.e. Iopt(k)= N ∑ i=1 αiIi(k) (3.3) where Iopt and Ii, i=1, . . . ,N aremultiobjective criteria of the linearization, and partial criteria of the linearization, respectively, αi > 0, i=1, . . . ,N are weight coefficients such that ∑N i=1αi =1. The idea of finding thePareto-optimal solution is to determine a nondomi- nated point q whose coordinates are defined byminimal values of considered criteria, i.e. q(k)= q(Iimin(k)) (3.4) where Iimin(k)=min k Ii(k) (3.5) The scalarization distance dw is defined, for instance,by dw = √ √ √ √ N ∑ i=1 αi(Ii(k)− Iimin(k))2 (3.6) where αi > 0, i=1, . . . ,N are weight coefficients such that ∑N i=1αi =1. To illustrate an application of the Pareto-optimal approach and scalari- zation method in the determination of response characteristics, we use two criteria of the statistical linearization (N = 2). In further consideration, we analyse two cases of themoment criteria and criteria in the probability density space. The corresponding criteria and linearization coefficients have the following forms. 3.1. Statistical linearization criteria in state space • Mean-square Criterion IMS = E[f2(x1)]E[x 2 1]− (E[f(x1)x1])2 E[x21] (3.7) kMS = E[f(x1)x1] E[x21] 682 L.Socha • Energy Criterion IE = E [(x1 ∫ 0 f(x) dx )2] 1 4 E[x41] 1 4 E[x41] − ( E [ x2 1 2 x1 ∫ 0 f(x) dx ])2 1 4 E[x41] (3.8) kE = E [ x2 1 2 x1 ∫ 0 f(x) dx ] 1 4 E[x41] 3.2. Probability density linearization criteria • Pseudomoment metric IPMM = +∞ ∫ −∞ . . . +∞ ∫ −∞ x 2p 1 x 2q 2 |gN(x1,x2)−gL(x1,x2)| dx1dx2 (3.9) where p+q= r, p,q=0,1, . . .. • Square probability metric IPSM = +∞ ∫ −∞ . . . +∞ ∫ −∞ [gN(x1,x2)−gL(x1,x2)]2 dx1dx2 (3.10) where gN(x) and gL(x) are probability density functions of solutions to nonlinear (3.1) and linearized systems (3.2), respectively. gN(x1,x2) is defined by the Gramm-Charlier expansion (Pugacev and Sinicyn, 1985) gN(x1,x2)= gG(x1,x2) [ 1+ N ∑ k=3 ∑ σ(ν)=k cν1ν2Hν1ν2(x1,x2) ν1!ν2! ] (3.11) where gG(x1,x2)= 1 2π √ k11k22−k212 exp [ −k11x 2 2−2k12x1x2+k22x21 2(k11k22−k212) ] (3.12) Application of poli-criterial linearization... 683 and cν1ν2 = E[Gν1ν2(x1,x2)], are quasimoments ν1,ν2 = 0,1, . . . ,N, ν1+ν2 =3,4, . . . ,N, Hν1ν2(x1,x2) and Gν1ν2(x1,x2) are Hermite’s po- lynomials defined by Hpq(x1,x2)= (−1)p+q exp (1 2 A ) ∂p+q ∂x p 1∂x q 2 exp ( −1 2 A ) (3.13) Gpq(x1,x2)= (−1)p+q exp (1 2 A ) · · [ ∂p+q ∂y p 1∂y q 2 exp ( −1 2 (k11y 2 1 +2k12y1y2+k22y 2 2) )] y=Vx where A= v11x 2 1+2v12x1x2+v22x 2 2 K= [ k11 k12 k21 k22 ] K −1 =V = [ v11 v12 v21 v22 ] (3.14) kij =E[xixj] i,j =1,2 For the stationary probability density function the corresponding mo- ments are as follows k12 = k21 =0 k22 = 1 v22 v12 = v21 =0 k11 = 1 v11 (3.15) Themoment k11 has to be found frommoment equations. 4. Example Consider the Duffing oscillator described by dx1 =x2 dt (4.1) dx2 = [−ω20x1−εx31−2hx2] dt+σ dξ where ω20, ε, h and σ are constant parameters. The parameters selected for calculations are: ω20 =0.5, ε=0.1, h=0.05 and σ 2 =0.2. The corresponding linearized system has form (3.2). 684 L.Socha 4.1. Moment criteria in statistical linearization • Mean-square Criterion IMS =(ω 2 0 −k)2E[x21]+15ε2(E[x21])3+6ε(ω20 −k)(E[x21])2 (4.2) • Energy Criterion IE = 3 4 (ω20−k)2(E[x21])2+105 (ε 4 )2 (E[x21]) 4+ 15 4 ε(ω20−k)(E[x21])3 (4.3) where E[x21] = σ2 4hk (4.4) Then, the set of dominating points is presented in Fig.2 Fig. 2. A graphical illustration of the set of dominating points determined by (4.2)-(4.4) • Convex combination Iopt =αIMS(k)+(1−α)IE 0¬α¬ 1 (4.5) The characteristics kmin = kmin(α) and Iopt = Iopt(α) determined by relation (4.5) are shown in Fig.3a,b, respectively. The characteristics dE = dE(k) and dw = dw(α) obtained by Scalariza- tion by distance have the form dE(k)= √ [IMSmin − IMS(k)]2+[IEmin − IE(k)]2 (4.6) dw(α)= √ α[IMSmin − IMS(k)]2+(1−α)[IEmin − IE(k)]2 for 0 ¬ α ¬ 1. A graphical illustration of these characteristics is given in Fig.4a,b, respectively. Application of poli-criterial linearization... 685 Fig. 3. (a) Characteristics kmin = kmin(α) determined by relation (4.5); (b) characteristics Iopt = Iopt(α) determined by relation (4.5) Fig. 4. (a) Characteristics dE = dE(k) determined by relation (4.6)1; (b) chracteristics dw = dw(α) determined by relation (4.6)2 4.2. Probability density linearization criteria in equivalent linearization • Pseudomoment metric IPMM = +∞ ∫ −∞ . . . +∞ ∫ −∞ x 2p 1 x 2q 2 |gN(x1,x2)−gL(x1,x2)| dx1dx2 (4.7) where p+q= r, p,q=0,1, . . .. • Square probability metric IPSM = +∞ ∫ −∞ . . . +∞ ∫ −∞ [gN(x1,x2)−gL(x1,x2)]2 dx1dx2 (4.8) 686 L.Socha where gN(x) and gL(x) are probability density functions defined by gN(x1,x2)= 1 cN exp [ − 2h q21 (ω20x 2 1+ ε 2 x41+x 2 2) ] (4.9) gL(x1,x2,k) = 1 cL 4h √ k q21 exp [ −2h q21 (kx21+x 2 2) ] where cN and cL are normalized constants. Then, the set of dominating points for parameters ω20 = 0.5, h = 0.05, ε=0.1, σ2 =0.2, p=1, q1 =0 is presented in Fig.5. Fig. 5. A graphical illustration of the set of dominating points determined by (4.7) and (4.8) 5. Applications in control problems The multicriteria analysis was also used in the determination of optimal control for linear stochastic systems by Piunovski (1996) and by Radievski (1993). In this Section, we extend these results for a class of nonlinear systems combining the multicriteria approach, statistical or equivalent linearization with the LQG technique. Consider the following optimal control problem. The nonlinear stochastic model of a dynamic system is described by dx(t)= [Ax(t)+Φ(x)+Bu(t)] dt+ M ∑ k=1 Gk dξk(t) (5.1) Application of poli-criterial linearization... 687 where x ∈ Rn and u ∈ Rm are the state and control vectors, respec- tively. A and B are time invariant matrices of appropriate dimensions, Φ= [Φ1, . . . ,Φn] > is a nonlinear vector function such that Φ(0)=0 are time invariant deterministic vectors, ξk are independent standard Wiener proces- ses for k=1,2, . . . ,M.We assume that the unique solution to equation (5.1) exists and the system is controllable. The control strategy is designed to minimize the criterion I =E[x>Qx+u>Ru] (5.2) where Q and R are time-invariant positive definite symmetric matrices. 5.1. Quasi-optimal control We assume that the nonlinear vector Φ(x) can be substituted by a linearized form Φ(x)=Aex (5.3) where Ae is a n×nmatrix of linearization coefficients such that (A+Ae,B) is stabilizable anddetectable. Then, the optimal control for the linearized system dxL(t)= [(A+Ae)xL(t)+Bu(t)] dt+ M ∑ k=1 Gk dξk(t) (5.4) can be foundby a standardmethod (Kwakernak andSivan, 1972) in the linear feedback form u=−KxL K=R−1B>P (5.5) where K is the gainmatrix and P is a positive solution to the algebraicRiccati equation P(A+Ae)+(A+Ae) > P−PBR−1B>P+Q=0 (5.6) Substituting (5.5) into equation (5.4) yields dxL(t)= [(A+Ae−BK)xL(t)] dt+ M ∑ k=1 Gk dξk(t) (5.7) The corresponding covariance equation and criterion have the form (A+Ae−BK)VL+VL(A+Ae−BK)>+ M ∑ k=1 GkG > k =0 (5.8) and IL =E[x > L(Q+K > RK)xL] = tr[(Q+K > RK)VL] (5.9) 688 L.Socha where the subindex L corresponds to the linearized problem, tr denotes the trace of the matrix VL =E[xLx > L] (5.10) In the case of statistical linearization, the elements of the nonlinear vector Φ(x) have to be replaced by corresponding equivalent elements ”in the sense of a given criterion” in a linear form. The following two moment criteria for scalar functions are considered: Criterion 1. Mean-square error of displacements (Kazakov, 1956) E[(c1x−φ(x))2]→min (5.11) Criterion 2. Mean-square error of potential energies (Elishakoff and Zhang, 1984) E [( x ∫ 0 [c2y−φ(y)] dy)2 ] →min (5.12) In the case of application of the linear feedback gain obtained for the linearized system to the nonlinear system, the state equation and the corre- sponding criterion have the form dxN(t)= [(A−BK)xN(t)+Φ(xN(t))] dt+ M ∑ k=1 Gk dξk(t) (5.13) and IN = tr[(Q+K > RK)VN] VN =E[xNx > N] (5.14) where the subindex N denotes the original nonlinear problemwith VN =E[xNx > N] (5.15) In general, the covariance matrix VN can be found approximately. To obtain the linearization matrix Ae and quasi-optimal control, one of the four proposed criteria should be selected and used with an iterative procedure. 5.2. Iterative procedure for multicriteria nonlinear control problem The following procedure is an extended version of a standard one given in (Yoshida, 1984): Step 1. Choose a parameter of the nonlinear system (an element ofEq. (5.1)) and criteria of linearization, then select Ae =0 in (5.3). Next, for every criterion repeat Steps 2-11. Application of poli-criterial linearization... 689 Step 2. Solve (5.6). The solution to (5.7) is P . Step 3. Substitute P obtained in Step 2 into (5.6) and find the matrix K. Next, substitute K and Ae = 0 into equation (5.8) and solve the equ- ation. The solution of equation (5.8) is VL. Step 4. Substitute P obtained in Step 2 into (5.9) and find IL. Step 5. For each nonlinear element find the linearization coefficient which minimizes the selected criterion, for instance, (5.11) or (5.12). Step 6. Substitute the matrix of linearization coefficients Ae(VL) obtained in Step 5 into equation (5.8) and then solve the equation. Step 7. If the error is greater than a given parameter ε1, then repeat Steps 3-6 until VL converges. Step 8. Substitute thematrixof linearization coefficients Ae(VL) intoRiccati equation (5.6) and then solve the equation. Step 9. Substitute thematrix P obtained in Step 8 into covariance equation (5.8) and then solve the equation. Step 10. If the error is greater than a given ε2, then repeat Steps 3-9 until VL and P converge. Step 11. Calculate criteria IL and IN given by (5.9) and (5.14), respectively. Step 12. Calculate ameasure of themulticriteria optimizationproblembased on criteria calculated in Step 11 for different linearization criteria chosen in Step 1. 5.3. Example (Duffing oscillator) Consider the Duffing oscillator described by dx1 =x2 dt (5.16) dx2 = [−ω20x1−εx31−2hx2+ bu] dt+σ dξ where ω20,ε,h,b and σ are constant parameters, u is a scalar control, ξ is the standardWiener process, and the mean-square criterion is I =E[x>Qx+ru2] (5.17) 690 L.Socha where x= [x1,x2] >,Q= diag[Qi], i=1,2; Qi,r are positive constants. The linearized system has the following form dx1 =x2 dt (5.18) dx2 = [−ω20x1−εcx1−2hx2+ bu] dt+σ dξ where c is a linearization coefficient. The coordinates of solutions to algebraic Riccati and covariance equations denoted by P = [pij] and VL = [vLij], respectively, for i,j =1,2 are the following p11 =2hp12+ cp22+βp12p22 p12 = 1 β (−c+ √ c2+Q1β) (5.19) p22 = 1 β √ 4h2+β(Q2+2p12) and vL22 = g2 2(2h+βp22) vL12 =0 vL11 = v22 γ+βp12 (5.20) where β = b2/r. The optimal value of the criterion for linearized system is IL =(Q1+βp 2 12)vL11 +(Q2+βp 2 22)vL22 (5.21) Applying the obtained linear feedback control to nonlinear systemwe ob- tain the state equation and the corresponding criterion dx1 =x2 dt (5.22) dx2 = [−2hx2−ω20 −εx31−β(x1p12+x2p22)] dt+σ dξ and INopt =(Q1+βp 2 12)vN11 +(Q2+βp 2 22)vN22 (5.23) where the second ordermoments vN11 and vN22 can be found in an analytical form from vNii = +∞ ∫ −∞ +∞ ∫ −∞ x2igN(x1,x2) dx1dx2 i=1,2 (5.24) Application of poli-criterial linearization... 691 where gN(x1,x2)= 1 cN exp [ − 2h+βp22 σ2 (ω20 +βp12)x 2 1+ε x41 2 +x22 ] (5.25) and cN is a normalized constant. One can show (Elishakoff and Zhang, 1984; Kazakov, 1956) that the line- arization coefficients for two considered criteria have the form c1 =3E[x 2 1] c2 =2.5E[x 2 1] (5.26) To obtain the quasi-optimal controls and corresponding mean-square cri- teria I1 and I2 depending on the choice of linearization coefficients c1 and c2, one can use the iterative procedure proposed in the previous section. To illu- strate the obtained results, a comparison of the considered criteria is discussed. The set of dominating points for parameters ω20 = 1, h= 0.05, b= 1, ε= 1, σ=1, Q1 =Q2 =1, r=100 is presented in Fig.6. Fig. 6. A graphical illustration of the set of dominating points Figure 8 shows that the considered mean-square criteria I1 and I2 are linearly dependentand there are nodominatedpoints. Itmeans that the quasi- optimal controls obtained for both criteria are the same. This confirms an earlier observation presented in an earlier author’s paper (Socha, 2000). 6. Conclusions Numerical studies show that in the response analysis there are significant differences between the obtained linearization coefficients and corresponding 692 L.Socha criteria in contrast to control problemswhere for a givenmean-square criterion of minimization (5.2) there are no significant differences between the applied linearization methods. It means that the mean-square criteria corresponding to different linearization criteria are linearly dependent. This linear depen- dence also appears in application of linearization techniques with criteria in probability density space. References 1. BernardP.,TaazountM., 1994,Randomdynamics of structureswith gaps: simulation and spectral linearization,Nonlinear Dynamics, 5, 313-335 2. Elishakoff I., 2000,Multiple combination of the stochastic linearization cri- teria by the moment approach, J. Sound Vib., 237, 550-559 3. 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Control, 39, 279-291 Zastosowanie wielokryterialnej linearyzacji w problemie sterowania stochastycznych nieliniowych układów dynamicznych Streszczenie W pracy przedstawiono problem wyznaczania quasi-optymalnego sterowania w nieliniowych stochastycznych układach dynamicznych za pomocą wielokryterial- nej metody linearyzacji stochastycznej. Pomysł wielokryterialnej linearyzacji został zasygnalizowanywewcześniejszej pracy autora (Socha, 1999a).Wniniejszym artyku- le jest on rozwinięty i zastosowanydo problemu sterowania, a szczegółowaanaliza jest przeprowadzona dla nieliniowego oscylatora z addytywnymwymuszeniemGaussa. Manuscript received December 20, 2004; accepted for print, March 18, 2005