The eigenvalue derivatives of linear damped systems

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1 Control and Cybernetcs vol. 32 (2003) No. 4 The egenvalue dervatves of lnear damped systems by Yeong-Jeu Sun Department of Electrcal Engneerng I-Shou Unversty Kaohsung, Tawan 840, R.O.C e-mal: Abstract: In ths note, the dervatves of egenvalues wth respect to the model parameters for lnear damped systems s proposed by means of kronecker algebra and matrx calculus. A numercal example s also provded to llustrate the use of the man result. Keywords: dervatves of egenvalues, lnear damped systems. 1. Introducton In recent years, the stablty and performance of lnear damped systems has been wdely nvestgated. In the past, there have been a number of nterestng developments n responses, ampltude bounds, or egenvalue bounds for lnear damped systems. For nstance, the ampltude bounds of lnear damped systems have been proposed n Hu and Schehlen (1996) and Schehlen and Hu (1995). Furthermore, the response bounds of lnear damped systems have been nvestgated n Hu and Eberhard (1999), Ncholson (1987a), and Yae and Inman (1987). Besdes, the egenvalue bounds of nonclasscally damped systems have been derved n Ncholson (1987b). There exst varous applcatons demonstratng that an approprate choce of the egenvalues of a closed-loop system does not necessarly gve a relable ndcaton as to the suffcency of the stablty margn snce the dervatves of egenvalues wth respect to the model parameters may be very large. Consequently, the dervatves of egenvalues wth respect to the model parameters for lnear damped systems are of mportance. Furthermore, when explorng the dynamc behavor and ts dervatves of egenvalues wth respect to the model parameters, kronecker algebra and matrx calculus frequently furnsh shorter and more concse calculatons. It s the purpose of ths note to estmate the dervatves of egenvalues wth respect to the model parameters usng the kronecker algebra approach and the matrx calculus approach.

2 736 YEONG-JEU SUN 2. Problem formulaton and man results For convenence, we defne some notatons that wll be used throughout ths note as follows: R m n := the set of all real m by n matrces, C m n := the set of all complex m by n matrces, I r := the unt matrx wth r rows and r columns, Q > 0 := the matrx Q s a symmetrc postve defnte matrx, λ := λ a real-valued functon λ dfferentated wth m j respect to a matrx M wth, M = [m j ] A B := the kronecker product of A and B, E j := the elementary matrx E j s defned as the matrx (of order n n) whch has a unty n the (, j)th poston and all other elements zero, A #L := the left nverse matrx of the matrx A, A := the nduced Eucldean norm of the matrx A. Before presentng the problem formulaton, let us ntroduce some lemmas, whch wll be used n the proof of the man theorem. Lemma 2.1 (Wenmann, 1991) If A C n m and z C s 1, then A z = (I n z)a. Lemma 2.2 (Wenmann, 1991) If A C n n and c C, then det(a) c adj(a) ]. Consder the followng uncertan lnear damped system = tr [ A c J(M)ÿ(t) + B(M)ẏ(t) + K(M)y(t) = τ(t), (1) where y(t) R n s the dsplacement vector, τ(t) R n s the external exctaton, J R n n s the matrx of nertal moment, B R n n s the dampng matrx, K R n n s the stffness matrx, and M R r s s the parameter s matrx. Clearly, the characterstc equaton of system (1) s gven by det[λ 2 J + λb + K] = 0. (2) The objectve of ths note s to estmate the dervatves of egenvalues wth respect to the parameter s matrx wthout the use of egenvectors. Now we present the man result. Theorem 2.1 Assume that λ s a smple egenvalue of system (1). Then the dervatves of egenvalues wth respect to the parameter s matrx M := [m jk ] R r s of the system (1) are gven by

3 The egenvalue dervatves of lnear damped systems 737 { [( λ = tr λ 2 J B + λ + K ) ]} adj(λ 2 J + λ B + K) m jk m jk m jk m jk {tr[(2λ J + B) adj(λ 2 J + λ B + K)] } 1, j r, k s. Proof. At frst, the exstence of soluton to λ s proved as follows. For an egenvalue λ, there exsts a C n 1 wth a 0 satsfyng (λ 2 J + λ B + K)a = 0. Takng partal dervatves of both sdes of the above equaton wth respect to the matrx M R r s yelds (λ 2 J + λ B + K)a Furthermore, one has = (λ2 Ja ) + (λ Ba ) + (Ka ) = 0. (3) (λ 2 Ja ) = λ2 (Ja ) + λ 2 (Ja ) [ λ J = 2λ (Ja ) + λ 2 (I s a ) + (I r J) a ], (4a) (λ Ba ) = λ (Ba (Ba ) ) + λ = λ (Ba ) + λ [ B (I s a ) + (I r B) a ] (4b) and (Ka ) = K (I s a ) + (I r K) a. Combnng (3) and (4) gves [ λ 2λ ] (Ja ) + λ 2 J (I s a ) + λ (Ba ) + λ B (I s a ) + K (I s a ) + λ 2 (I r J) a + λ (I r B) a + (I r K) a = 0. It can be readly obtaned that ( ) λ [2λ Ja + Ba ] = λ 2 J (I B s a ) λ (I s a ) K (I s a ) [ I r (λ 2 J + λ B + K) ] a. (4c)

4 738 YEONG-JEU SUN Applyng Lemma 1 to above equaton, we obtan [ Ir (2λ Ja + Ba ) ] λ = J λ2 (I B s a ) λ (I s a ) K (I s a ) [ I r (λ 2 J + λ B + K) ] a. (5) Furthermore one has det(2λ J+B) 0, whch mples (2λ J+B)a = (2λ Ja + Ba ) 0, a 0. Hence we conclude that the matrx [I r (2λ Ja + Ba )] C rn r has full column rank, whch mples that the left nverse of matrx [I r (2λ Ja + Ba )] exsts. Postmultplyng (5) by [I r (2λ Ja + Ba )] #L gves λ = [I r (2λ Ja + Ba )] #L { λ 2 K (I s a ) + λ 2 B (I s a ) + K (I s a ) +[I r (λ 2 J + λ B + K)] a }, so that the exstence of λ det(λ 2 J + λ B + K) m jk Furthermore, t can be shown that (λ 2 J + λ B + K) m jk s guaranteed. By Lemma 2 wth (2), one has (λ 2 = tr J + λ B + K) adj(λ 2 m J + λ B + K) = 0. jk λ = 2λ J + λ 2 J + λ B B + λ + K m jk m jk m jk m jk m jk = λ (2λ J + B) + λ 2 J B K + λ + λ. (7) m jk m jk m jk m jk Substtutng (7) nto (6), yelds λ tr (2λ J + B) adj(λ 2 m J + λ B + K) jk [ ( = tr λ 2 J B + λ + K ) ] adj(λ 2 m jk m jk m J + λ B + K), jk whch mples that { tr λ m jk = [( λ 2 Ths completes our proof. J B + λ + K ) ]} adj(λ 2 J + λ B + K) m jk m jk m jk {tr [ (2λ J + B) adj(λ 2 J + λ B + K) ]} 1. (6)

5 The egenvalue dervatves of lnear damped systems 739 Remark 2.1 The characterstc equaton of det[λb + K] = 0 wth the assumptons that (A1) B s nonsngular; (A2) Both matrces B and K can be smultaneously dagonalzed has been consdered n Prells and Frswell (2000). Ths s the specal case of (3.) wth J = 0. It s noted that the methodology used n our man result s dfferent from that used n Prells and Frswell (2000). Furthermore, even f both (A1) and (A2) are not satsfed, our man result may be appled to the system (1). 3. Illustratve example Consder the lnear damped system J(M)ÿ(t) + B(M)ẏ(t) + K(M)y(t) = τ(t), (8) where y(t) R 2 s the dsplacement vector, τ(t) R 2 s the external exctaton, J = α 1, B = 0 β 0 1, K = , 1 β and M := [α β] s parameter s matrx wth 1 α 3 and 0.5 β 1.5. We assume that the nomnal values are α = 2 and β = 1. In ths case, from (3.), the egenvalues of the system (8) are λ 1,normal = j, λ 2,normal = j, λ 3,normal = j, λ 4,normal = j. By Theorem 1, the dervatves of egenvalues of system (8) can be estmated by λ 1 = [ j j ], λ 2 = [ j j ], λ 3 = [ j j ], λ 4 = [ j j ]. In the case of α = 2.05 and β = 0.95 (.e., α = 0.05 and β = 0.05 ), the

6 740 YEONG-JEU SUN egenvalues can be approxmately calculated as λ 1 = j = λ 1,normal + λ 1 = j + λ 1 α = j, β λ 2 = j = λ 2,normal + λ 2 = j + λ 2 α = j, β λ 3 = j = λ 3,normal + λ 3 = j + λ 3 α = j, β λ 4 = j = λ 4,normal + λ 4 = j + λ 4 α = j, β 4. Conclusons In ths note, the calculaton dervatves of egenvalues wth respect to the model parameters for lnear damped systems has been proposed by means of kronecker algebra and matrx calculus. A numercal example has also been provded to llustrate the use of the man result. Acknowledgment The author thanks the Natonal Scence Councl of ROC, for supportng ths work under Grant E References Graham, K. (1981) Kronecker Products and Matrx Calculus wth Applcatons. Wley, New York. Hu, B. and Eberhard, P. (1999) Response bounds for lnear damped systems. ASME Journal of Appled Mechancs 66, Hu, B. and Schehlen, B. (1996) Ampltude bounds of lnear forced vbratons. Archve of Appled Mechancs 66, Ncholson, D.W. (1987a) Egenvalue bounds for lnear mechancal systems wth nonmodal dampng. Mech. Res. Commun. 14, Ncholson, D.W. (1987b) Response bounds for nonclasscally damped mechancal systems under transent loads. ASME Journal of Appled Mechancs 54, Prells, U. and Frswell, M.I. (2000) Calculatng dervatves of repeated and nonrepeated egenvalues wthout explct use of egenvectors. AIAA Journal 38,

7 The egenvalue dervatves of lnear damped systems 741 Schehlen, W. and Hu, B. (1995) Ampltude bounds of lnear free vbratons. ASME Journal of Appled Mechancs 62, Wenmann, A. (1991) Uncertan Models and Robust Control. Sprnger-Verlag, New York. Yae, K.H. and Inman, D.J. (1987) Response bounds for lnear underdamped systems. ASME Journal of Appled Mechancs 54,

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