Time-delayed feedback control optimization for quasi linear systems under random excitations

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1 Acta Mech Sin (2009) 25: DOI /s RESEARCH PAPER Time-delayed feedback control optimization for quasi linear systems under random excitations Xueping Li Demin Wei Weiqiu Zhu Received: 28 April 2008 / Revised: 13 November 2008 / Accepted: 13 November 2008 / Published online: 10 January 2009 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH 2009 Abstract A strategy for time-delayed feedback control optimization of quasi linear systems with random excitation is proposed. First, the stochastic averaging method is used to reduce the dimension of the state space and to derive the stationary response of the system. Secondly, the control law is assumed to be velocity feedback control with time delay and the unknown control gains are determined by the performance indices. The response of the controlled system is predicted through solving the Fokker Plank Kolmogorov equation associated with the averaged Itô equation. Finally, numerical examples are used to illustrate the proposed control method, and the numerical results are confirmed by Monte Carlo simulation. Keywords Quasi linear system Time-delayed feedback Stochastic averaging method Random excitation 1 Introduction The mathematical theory of stochastic optimal control is quite well developed [1 4. The two principal approaches to stochastic optimal control problems are Pontryagin s The project was supported by the National Natural Science Foundation of China ( ) and Specialized Research Fund for the Doctoral Program of Higher Education of China ( ). X. Li (B) D. Wei School of Civil Engineering and Transportation, South China University of Technology, Guangzhou, China @zju.edu.cn W. Zhu Department of Mechanics, State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou, China maximum principle and Bellman s dynamical programming principle. For a long period of time, the theory of stochastic optimal control was mainly applied to economics, especially finance. In engineering field, only the linear quadratic Gaussian control strategy has been widely used. In the last few years, a nonlinear stochastic optimal control strategy has been proposed for the control of mechanical and structural systems under random excitations [5 7 based on the stochastic averaging method for quasi Hamiltonian systems [8 10 and the stochastic dynamical programming principle. The feedback control optimization of nonlinear systems under random excitations was studied by Elbeyli et al. [11 and the feedback minimization of first-passage failure of quasi integrable Hamiltonian systems was studied by Deng et al. [12. In the implementation of control, time delay is usually unavoidable due to the time spent in measuring and estimating the system state, calculating and executing the control forces, etc. The time delay often leads to instability or poor performance of the controlled system, and thus it has drawn much attention in the control community. The time-delayed controlled systems under deterministic excitation were studied in [13,14, and the effect of time delay on the controlled linear systems under Gaussian random excitation was studied by Di Paola and Pirrotta using a Taylor expansion-based approach [15. By means of the frozen time approach and the Kronecker product, two criteria of asymptotic stability were derived for the linear, time variant dynamic systems with short time delays [16. A stochastic averaging method for quasi intergrable Hamiltonian systems with time time-delayed feedback control under Gaussian white noise excitation was proposed by Liu and Zhu [17, time-delayed feedback control optimization was developed for quasi linear systems. The present paper is organized as follows. In Sect. 2, the system equations are transformed into differential equations

2 396 X. Li et al. without time delay and the stochastic averaging method is applied. The control problem is then formulated in the averaged domain and the optimal time-delayed control law is determined. Two examples of quasi linear systems subject to Gaussian white noise excitation are studied in Sect. 3 to validate the proposed technique. 2 Control design formulation 2.1 Quasi linear systems with time-delayed feedback control Consider an n-degree-of-freedom quasi linear system with time-delayed feedback control governed by the following equations: Q i (t τ) = A i (t τ)cos i (t τ) A i (t) cos[ω i (t τ)+ Ɣ i(t) = A i (t){cos[ω i t + Ɣ i(t) cos ω i τ + sin[ω i t + Ɣ i(t) sin ω i τ} = Q i (t) cos ω i τ P i ω i sin ω i τ, (4) P i (t τ) = A i (t τ)ω i sin i(t τ) A i (t)ω i sin[ω i (t τ)+ Ɣ i(t) = A i (t)ω i {sin[ω i t + Ɣ i(t) cos ω i τ cos[ω i t + Ɣ i(t) sin ω i τ} = P i cos ω i τ + Q i(t)ω i sin ω iτ. (5) Q i = P i, Ṗ i = ω 2 i Q i εc ij ( Q, P)P j + εf i ( Q τ, P τ ) + ε 1/2 f ik ( Q, P)W k (t), i, j = 1, 2,...,n; k = 1, 2,...,m, (1) The numerical results for the two example described in Sect. 3 will show that Eqs. (4) and (5) are acceptable even for large delay time τ. 2.2 Stochastic averaging and stationary solutions Q i and P i are generalized displacements and momenta, respectively; ε is a small positive parameter; Q =[Q 1, Q 2,...,Q n T, P =[P 1, P 2,...,P n T ; εc ij ( Q, P) represent the coefficients of quasi linear damping of uncontrolled system; ω i are the natural frequencies of uncontrolled system; ε 1/2 f ik ( Q, P) represent the amplitudes of stochastic excitations; εf i ( Q τ, P τ ) denote the time-delayed feedback control forces; Q τ = Q(t τ), P τ = P(t τ); τ is delay time; W k (t) are Gaussian white noises in the sense of Stratonovich with correlation functions E[W k (t)w l (t + T ) =2D kl δ(t ), k, l = 1, 2,...,m. (2) When ε = 0, system (1) is degenerated into n-dof linear Hamiltonian system. It has a family of periodic solutions around the origin. The solution to Eq. (1) can be expressed by By substituting Eqs. (4) and (5) into Eq. (1), the equations in system (1) can be converted into the following equations Q i = P i, Ṗ i = ω 2 i (τ)q i εc ij ( Q, P; τ)p j + ε 1/2 f ik ( Q, P)W k (t), (6) i, j = 1, 2,...,n; k = 1, 2,...,m, ω i (τ) are the natural frequencies of controlled system; εc ij ( Q, P; τ)are the coefficients of quasi linear damping of controlled system. The solution to Eq. (6) can be expressed as Q i (t) = A i cos i (t), P i (t) = A i ω i sin i (t), (7) i (t) = ω i t + Ɣ i (t). Q i (t) = A i cos i (t), P i (t) = A i ω i sin i(t), i (t) = i (t) + Ɣ i (t), (3) Equation (7) can be regarded as a transformation from Q i, P i to A i,ɣ i. Equation (6) is then transformed into the following differential equations according to Itô differential formula: when ε is small, amplitudes A i (t) and phase Ɣ i (t) are slowly varying processes. For some finite τ, A i (t τ)and Ɣ i (t τ) can be approximated by A i (t) and Ɣ i (t), respectively. Then, the following approximate expressions can be obtained: Ȧ i = εf 1i (A, Ɣ; τ)+ ε 1/2 G 1ik (A, Ɣ; τ)w k (t), Ɣ i = εf 2i (A, Ɣ; τ)+ ε 1/2 G 2ik (A, Ɣ; τ)w k (t), (8) i = 1, 2,...,n; k = 1, 2,...,m,

3 Time-delayed feedback control optimization for quasi linear systems under random excitations 397 F 1i (A, Ɣ; τ) = sin i c ij (A, ; τ)a j ω j sin j, ω i F 2i (A, Ɣ; τ) = cos i c ij (A, ; τ)a j ω j sin j, A i ω i G 1ik (A, Ɣ; τ) = sin i f ik (A, ), ω i G 2ik (A, Ɣ; τ) = cos i f ik (A, ). A i ω i The stochastic averaging method for quasi Hamiltonian systems has been well developed [8,9. The dimension and form of the averaged Itô and FPK equations depend upon the internal resonance of the associated linear system with modified natural frequencies. In non-resonant cases, the averaged Itô differential equations are of the form da i = m i (A; τ)dt + σ ik (A; τ)db k (t), i = 1, 2,...,n; k = 1, 2,...,m. (10) The averaged FPK equation is of the form { p = [m i (a; τ)p+ 1 2 } [b ij (a; τ)p. (11) t a i 2 a i a j In Eqs. (10) and (11) m i (a; τ) = ε F 1i + D kl G 1ik a j G 1ik G 1 jl + D kl G 2 jl, γ j b ij (a; τ) = σ ik (a; τ)σ jk (a; τ) = ε 2D kl G 1ik G 1 jl, (12) i, j = 1, 2,...,n; k = 1, 2,...,m, in which D kl is the intensity of the Gaussian white noise W k (t). The exact stationary solution of FPK Eq. (11) with vanished probability potential flow at the boundary is of the form p(a; τ) = C exp[ λ(a; τ), (13) C is a normalization constant and λ(a; τ) is the socalled probability potential which is governed by equations λ b rs = b rs 2m r, r, s = 1, 2,...,n. (14) a s a s If the diffusion matrix B = [b rs is not singular, i.e., its inverse matrix B 1 = G =[g rs exists, then Eq. (14) can be converted into ( ) λ brs = g ir 2m r. (15) a i a s Furthermore, if the following compatibility conditions ( ) brs g ir 2m r = ( ) brs g jr 2m r (16) a j a s a i a s (9) are satisfied, then the probability potential is a s λ λ(a; τ) = λ 0 + da s, (17) a s 0 λ 0 = λ(0,τ) and the second term is a summation of line integrals over s = 1, 2,...,n. The exact stationary solution p(a; τ) of the averaged FPK Eq. (11) is obtained by substituting Eq. (17) into Eq. (13). Note that a is related to H =[H 1, H 2,...,H n T with H i = (Pi 2 + ωi 2Q2 i )/2 by H i = ωi 2a2 i /2, i = 1, 2,...,n. The stationary probability density of H is p(h; τ) = p(a; τ) a H, (18) (a)/ ( H) is absolute value of the Jacobian determinant of the transformation from a to H. The approximate stationary probability density of system (6)isthen[18 p( Q, P; τ) = p(h; τ)/t (H; τ) H=H(Q,P), (19) T (H; τ) = (2π) n / n i=1 ω i. (20) If the compatibility conditions (16) are not satisfied, the stationary solution of the FPK Eq. (11) can be solved by numerical method. 2.3 Determination of the optimum control gains The form of the performance index depends upon the objective of the control, for example, minimization of response, stabilization or maximization of the reliability. Here, the control objective is to minimize the response. So the performance index is of the form: n J = [E(H i ) + R i E(ui 2 ), (21) i=1 R i is a positive weighting factor. In this paper, the cost is function of the steady state responses only. Assume the control force is a general function of the state variables with undetermined coefficients. These coefficients are determined in such a way that the cost function J in Eq. (21) is minimized. This is a manageable task since the steady state solutions of the response and its probability density function are often available. Two examples will be used to demonstrate the design procedure. 2.4 Evaluation of controller Two measures are adopted to evaluate the controller here. The first is the fractional reduction of the standard deviation

4 398 X. Li et al. (SD) of displacements and velocity, namely K h = ( σ u h σ c h ) /σ u h, (22) h denoted the displacements or velocity; K h is the effectiveness which measures the capacity of the controller; σh u is the SD of the uncontrolled system; σ h c is the SD of the controlled system. The second is the controller efficiency µ h = K h /σ u, (23) which measure the SD of the optimal control force per unit of the square roof of excitation intensity. Where σ u is the SD of the control force. Obviously, the higher the values of K h and µ h, the better the controller. 3 Examples 3.1 Example 1 Consider a van der Pol oscillator with time-delayed feedback controls subjected to random excitation. The equation of motion is Ẍ + ω 2 X ε(α β X 2 )Ẋ = ε(λ 1 X τ + λ 2 Ẋ τ ) + ε 1/2 W (t), (24) ε is a small positive parameter; u = ε(λ 1 X τ + λ 2 Ẋ τ ) is the control force, X τ1 = X (t τ) and Ẋ τ2 = Ẋ(t τ) are delayed system state variables; ελ 1 and ελ 2 are feedback control gains; W (t) is Gaussian white noise with intensity 2D. Note that there is no Wong Zakai correction term for this example. Let X = Q, Ẋ = P, the time-delayed state variables can be approximately expressed in terms of system state variables without time delay, i.e., Q τ = Q cos ω τ 1/ω P sin ω τ, P τ = Qω sin ω τ + P cos ω τ. (25) Equation (24) can be converted into the following equations: Q = P, (26) Ṗ = ω 2 (τ)q + ε(c(τ) β Q 2 )P + ε 1/2 W (t), ω(τ) = ( ω 2 + ελ 1 cos ω τ + ελ 2 ω sin ω τ) 1/2, c(τ) = α λ 2 cos ω τ + λ 1 /ω sin ω τ. (27) The modified Hamiltonian is of the form [ H = P 2 + ω 2 Q 2 /2. (28) The modified frequency is ω. Applying the stochastic averaging method to the modified system leads to the following averaged Itô differential equation: da = m(a; τ)dt + σ(a; τ)db(t). (29) The corresponding averaged FPK equation is of the form: { p = t a [m(a; τ)p+1 2 } [b(a; τ)p, (30) 2 a2 m(a; τ) = ca 2 βa3 8 + D 2aω 2, b(a; τ) = D (31) ω 2. The exact stationary solutions to FPK Eq. (30) is p(a; τ) = C 2 a exp[ λ(a; τ), (32) λ(a; τ) = ca2 ω 2 2D + βa4 ω 2 16D. (33) The stationary probability density of Hamiltonian is p(h; τ) = p(a; τ) g(a) a= 2H ω = C 2 exp[ λ(h; τ), (34) λ(h; τ) = cω2 H D + β H 2 4ω 2 D. (35) The stationary probability density of original system (24) is then p(h; τ) p(q, P; τ) = T (H; τ), (36) H=(P 2 +ω 2 Q 2 )/2 T (H; τ) = ω(τ)/2π. Consider the time-delayed feedback control as u = ε(λ 1 Q τ + λ 2 P τ ), (37) and the a cost function is as follows J = E(H) + RE(u 2 ). (38) By using the proposed method in Sect. 2.3, the feedback control gains with different delay time and weighting factor are given in Table 1. The probability density p(q) and p( P) of system (24) with both displacement and velocity time-delayed feedback controls are shown in Figs. 1, 2, 3, and 4 by using the proposed technique and Monte Carlo simulation. The theoretical result is denoted by solid line ( ) and the simulation result is denoted by symbols ( ). The parameters are ω = 1.2,α = 1.0,β = 1.0,ε = 0.01 and D = 1. It is seen that the results obtained by using the proposed technique agree well with those from numerical simulation even for longer delay time. To examine the effectiveness and efficiency of the proposed control strategy, the result is listed in Table 1. It is seen from the table that the proposed control strategy performs well if the weighting

5 Time-delayed feedback control optimization for quasi linear systems under random excitations 399 Table 1 SD displacements, SD velocity, SD control force and controller efficiency versus delay time and weighting factor τ R λ 1 λ 2 σ q σ p σ u K q K p µ q µ q Uncontrolled , , Fig. 1 The probability density p(q) of system (24) with different weighting factor when delay time τ = 1 Fig. 3 The probability density p(q) of system (24) with different weighting factor when delay time τ = 3 Fig. 2 The probability density p(p) of system (24) with different weighting factor when delay time τ = 1 Fig. 4 The probability density p(p) of system (24) with different weighting factor when delay time τ = 3 factor is properly selected on condition that the delay time of the control system is definitive. 3.2 Example 2 Consider two linear oscillators coupled by linear and nonlinear damping subjected to external excitations of two random excitations and time-delayed feedback controls. The equations of motion of the system are of the form Ẍ 1 + ε [α 11 Ẋ1 + α 12 Ẋ 2 + β 1 Ẋ 1 (X1 2 + X 2 2 ) + ω 2 1 X 1 = u 1 + ε 1/2 W 1 (t), (39) Ẍ 2 + ε [α 21 Ẋ 1 + α 22 Ẋ2 + β 2 Ẋ 2 (X1 2 + X 2 2 ) + ω 2 2 X 2 = u 2 + ε 1/2 W 2 (t),

6 400 X. Li et al. α ii,α ij,β i,η i,ω i are constants; u 1 = εη 1 Ẋ 1τ,u 2 = εη 2 Ẋ 2τ are the time-delayed control forces; Ẋ iτ = Ẋ i (t τ) are time-delayed velocity; W k (t) are independent Gaussian white noises with intensities 2D k.letx i = Q i, Ẋ i = P i, the Hamiltonian system associated with Eq. (39)is linear and the sub-hamiltonians are of the form H i = [ P 2 i + ω 2 i Q2 i /2, i = 1, 2. (40) The time-delayed system state variables in system (39) can be approximately expressed in terms of those without time delay as P iτ = P i cos ω i τ + Q iω i sin ω iτ, i = 1, 2. (41) The modified sub-hamiltonians are of the form H i =[Pi 2 + ωi 2 Q2 i /2, i = 1, 2, (42) ω 2 i (τ) = ω 2 i + εη i ω i sin ω iτ. Also, the linear damping coefficients become α ii (τ) = α ii + εη i ω i sin ω i τ.introducing transformation (7), Eq. (39) can be transformed into the following differential equations: Ȧ i = F 1i (A, Ɣ; τ)+ G 1ik (A, Ɣ; τ)w k (t), Ɣ i = F 2i (A, Ɣ; τ)+ G 2ik (A, Ɣ; τ)w k (t), (43) i = 1, 2; k = 1, F 11 = ε [ α 11 P 1 + α 12 P 2 + β 1 P 1 (Q 2 1 ω + Q2 2 ) sin 1, 1 F 12 = ε [ α 22 P 2 + α 21 P 1 + β 2 P 2 (Q 2 1 ω + Q2 2 ) sin 2, 2 F 21 = F 22 = ε [ α 11 P 1 + α 12 P 2 + β 1 P 1 (Q 2 1 A 1 ω + Q2 2 ) cos 1, 1 ε [ α 22 P 2 + α 21 P 1 + β 2 P 2 (Q 2 1 A 2 ω + Q2 2 ) cos 2, 2 G 111 = ε1/2 ω 1 sin 1, G 112 = ε1/2 ω 2 sin 2, G 221 = ε1/2 A 1 ω 1 cos 1, G 222 = ε1/2 A 2 ω 2 cos 2. (44) Assume rω 1 + sω 2 = 0(ε), r, s are integers. In this case, the averaged FPK equation is of the form of Eq. (11) with the following drift and diffusion coefficients m 1 (a 1, a 2 ; τ) = α 11 2 a 1 β 1 8 a3 1 β 1 4 a 1a2 2 + D 1 2a 1 ω1 2, m 2 (a 1, a 2 ; τ) = α 22 2 a 2 β 2 8 a3 2 β 2 4 a 2a1 2 + D 2 2a 2 ω2 2, b 11 (a 1, a 2 ; τ) = D 1 ω1 2, (45) b 22 (a 1, a 2 ; τ) = D 2 ω2 2, b 12 (a 1, a 2 ; τ) = b 21 (a 1, a 2 ; τ) = 0. The stationary solution of the averaged FPK equation is of the form of Eq. (13), λ/ a s satisfy the following equations: λ b 11 = 2m 1, a 1 (46) λ b 22 = 2m 2. a 2 If (β 1 /D 1 )(ω 1 /ω 2 ) = (β 2 /D 2 )(ω 2 /ω 1 ), the averaged FPK equation has an exact stationary solution p(a 1, a 2 ; τ) = C 3 exp[ λ(a 1, a 2 ; τ), (47) λ(a 1, a 2 ; τ) = ω2 1 D 1 + ω2 2 D 2 [ α 11 2 a2 1 + β 1 16 a4 1 + β 1 4 a2 1 a2 2 D 1 ω1 2 ln a 1 [ α 22 2 a 22 + β 2 16 a4 2 D 2 ω2 2 ln a 2. (48) The stationary probability density of H 1, H 2 is p(h 1, H 2 ; τ)= p(a 1, a 2 ; τ) (a 1, a 2 ) (H 1, H 2 ) a1 =, (49) 2H 1 /ω 1 a 2 = 2H 2 /ω 2 (a 1, a 2 )/ (H 1, H 2 ) is absolute value of the Jacobian determinant of the transformation from H 1, H 2 to a 1, a 2.The approximate stationary probability density of the displacements and velocities of the original system (39) is then p(q 1, Q 2, P 1, P 2 ; τ) = 1 4π 2 p(h 1, H 2 ; τ). (50) Hi =(Pi 2+ω2 i Q2 i )/2 Consider the time-delayed feedback controls as u i = ε(a i P iτ ). (51) The cost function is as follows 2 J = [E(H i ) + R i E(ui 2 ). (52) i=1 By using the method proposed in Sect. 2.3, the feedback control gains with different delay time and weighting factor

7 Time-delayed feedback control optimization for quasi linear systems under random excitations 401 Table 2 SD displacements, SD velocity and SD control force versus delay time and weighting factor τ R i η 1 η 2 σ q1 σ p1 σ q2 σ p2 σ u1 σ u2 Uncontrolled Fig. 5 The probability density p(q 1, P 1 ) of system (39) without control Fig. 6 The probability density p(q 1, P 1 ) of system (39) with time-delayed velocity feedback control when τ = 3, R 1 = 20, R 2 = 20 Fig. 7 The probability density p(q 1, P 1 ) of system (39) with time-delayed velocity feedback control when τ = 3, R 1 = 10, R 2 = 10 Table 3 Controller efficiency versus delay time and weighting factor τ R i η 1 η 2 K q1 K p1 K q2 K p2 µ q1 µ q1 µ q2 µ q are given in Table 2. The probability density p(q 1, P 1 ) of the system (39) with velocity delayed feedback controls are shown in Figs. 5, 6, and 7. The results obtained by using the proposed technique are shown in Figs-a and the results obtained by using Monte Carlo simulation are shown in Figs-b. Some parameters are ε = 0.01,ω 1 = 1.0,α 11 = 5.0, α 12 = 5.0,ω 2 = 1.0,α 22 = 5.0,β 1 = 5.0, D 1 = 6, D 2 = 4,α 21 = 5.0 and β 2 = β 1 D 2 ω1 2/ω2 2. It is seen that the results obtained by using the proposed technique agree well with those from digital simulation even for longer delay time. To examine the effectiveness and efficiency of the proposed control strategy, the result is listed in Tables 2 and 3. Itis seen from the table that the proposed control strategy performs well if the weighting factor is properly selected.

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