MINIMIZATION OF FREQUENCY-WEIGHTED l 2 -SENSITIVITY FOR MULTI-INPUT/MULTI-OUTPUT LINEAR SYSTEMS

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1 Automatique et ordinateurs MINIMIZAION OF FREQUENCY-WEIGHED -SENSIIVIY FOR MULI-INPU/MULI-OUPU LINEAR SYSEMS AKAO HINAMOO, OSAMU ANAKA, AKIMISU DOI Key words: MIMO inear disrete-time systems, Frequeny-weighted -sensitivity. he probem of minimizing a frequeny-weighted -sensitivity measure subjet to -saing onstraints is onsidered for muti-input/muti-output (MIMO) inear disretetime systems. he onstrained optimization probem is onverted into an unonstrained optimization probem by using inear-agebrai tehniques. An effiient quasi-newton agorithm with osed-form formua for gradient evauation is then appied to sove the unonstrained optimization probem. Finay, the optima system struture is onstruted by empoying the resuting oordinate transformation matrix that minimizes the frequeny-weighted -sensitivity measure subjet to the saing onstraints. A numeria exampe is aso presented to iustrate the utiity of the proposed tehnique.. INRODUCION he synthesis of a muti-input/muti-output (MIMO) inear disrete-time system with a given transfer funtion matrix is an important researh topi, sine the state-spae equations orresponding to the transfer funtion matrix are not unique. Naturay, among the infinite number of reaizations of the transfer funtion matrix, it is often desirabe to identify a state-spae reaization that minimizes a suitabe sensitivity measure. When reaizing a fixed-point state-spae desription with finite word ength (FWL) from a presribed transfer funtion matrix with infinite auray oeffiients, the oeffiients in the state-spae desription must be trunated or rounded to fit the FWL onstraints. his oeffiient quantization usuay aters the harateristis of the system. For instane, a stabe system may be turned to an unstabe one. his motivates the study of the oeffiient sensitivity minimization probem. In [ ], two main asses of tehniques have been expored for onstruting state-spae desriptions Graduate Shoo of Engineering, Hiroshima University, Higashi-Hiroshima , Japan, E-mai: {hinamoto, o tanaka}@hiroshima-u.a.jp Fauty of Appied Information Siene, Hiroshima Institute of ehnoogy, Hiroshima , Japan, E-mai: doi@.it-hiroshima.a.jp Rev. Roum. Si. ehn. Éetrotehn. et Énerg., 54, 4, p , Buarest, 9

2 396 akao Hinamoto, Osamu anaka, Akimitsu Doi that minimize the oeffiient sensitivity: / sensitivity minimization [ 6] and sensitivity minimization [7 ]. It has been argued in [7 ] that the sensitivity measure based on the norm is more natura and reasonabe reative to that based on the / sensitivity minimization. Aternativey, it is we known that the use of saing onstraints an be benefiia for suppressing overfow osiations [3,4]. Either sensitivity minimization probem or frequeny-weighted sensitivity minimization probem subjet to saing onstraints for SISO (singeinput/singe-output) state-spae digita fiters has been soved iterativey by onverting it into an unonstrained optimization probem with an appropriate inear transformation [5, 6]. However, to our best knowedge, there is no study on the minimization of frequeny-weighted sensitivity subjet to saing onstraints for MIMO inear disrete-time systems. In this paper, the probem of minimizing a frequeny-weighted sensitivity measure subjet to saing onstraints for MIMO inear disrete-time systems is investigated. First, an expression for evauating the frequeny-weighted sensitivity is introdued and the frequeny-weighted sensitivity minimization probem subjet to saing onstraints is formuated. Next, the onstrained optimization probem is onverted into an unonstrained one by using inear agebrai tehniques. An effiient quasi-newton agorithm [7] is then appied to sove the unonstrained optimization probem. A numeria exampe is aso presented to iustrate the utiity of the proposed tehnique.. PROBLEM FORMULAION Consider a stabe, ontroabe and observabe MIMO inear disrete-time system (A,B,C,D) n desribed by: x( k + ) = Ax( + Bu(, y( = Cx( + Du( () x( is an n state-variabe vetor, u( is a q input vetor, y( is a p output vetor, and A, B, C and D are rea onstant matries of appropriate dimensions. he transfer funtion of the inear system in () is given by: H z ) = C( zi A) B + D, () ( n whose (i, j) th eement is desribed by i n b j H ( z) = ( zi A) + d, (3)

3 3 Minimization of frequeny-weighted -sensitivity 397 B = [ b b L bq ], d d L d q d d d L q C =, D =. M M M O M p d p d p L d pq he frequeny-weighted sensitivity of the inear system in () is defined as foows. Definition. Let X be an m n rea matrix and et f(x) be a saar ompex funtion of X, differentiabe with respet to a the entries of X. he sensitivity funtion of f(x) with respet to X is then defined as: (4) f f S X =, ( S X ) =. (5) X x Definition. Let X(z) be an m n ompex matrix-vaued funtion of a ompex variabe z, and et x pq (z) be the (p, q)th entry of X(z). he norm of X(z) is then defined as: () d tr X () X () z X z z z. j (6) z z From (3) and Definitions and, the overa frequeny-weighted -sensitivity measure for the inear system in () is defined as: p q p q H ( z) H ( z) A( ) B( ) i= j= A i= j= bj S = W z + W z + + p q H ( z) WC z i= j= i p q i= j= ( ), = WA( z) f j( z) gi( z) + p q B( ) gi ( ) ( ) ( ), f j i= j= + q W z z + p W z z (7) f ( z) = ( zi A) b and j n j g ( z ) = ( zi A) i i n. Sine term D in () and the sensitivities with respet to its eements are independent of the state-spae

4 398 akao Hinamoto, Osamu anaka, Akimitsu Doi 4 oordinate, they are negeted in (7). he frequeny-weighted -sensitivity measure in (7) an be expressed as: p q S tr MA( In) qtr WB ptr K (8) i j M A (I n ), W B, and W C are obtained by the foowing genera expression: with dz X Y() z Y () z j (9) z z Y( z) = W ( z) f ( z) g ( z) for X = M ( I ) * * B n B Y( z) = W ( z) C( zi A) for X = W n Y( z) = W ( z)( zi A) B for X = K. A j i A n () he matries K C, W B, and M A (I n ) an be omputed using: = C B = B B = = K F () F (), W G () G (), A n = A A = M ( I ) H ( ) H ( ), k = C B = B k= k= F () w ( A B, G () w ( CA, k k j i A A k= k= k H () = A b A, H () = w ( H (, () () with w A (, w B (, and w C ( denoting the unit-puse responses of frequenyweighting funtions W A (z), W B (z), and W C (z), respetivey. If a oordinate transformation is defined by: x( = x( (3) and if it is appied to the inear system in (), then we obtain a new reaization ( A, B, C, D haraterized by: ) n

5 5 Minimization of frequeny-weighted -sensitivity 399 A A, B B, C C W W, K K B B. (4) From () and (4), it is ear that the transfer funtion H(z) is invariant under the oordinate transformation in (3). For the new reaization, the frequenyweighted sensitivity measure in (8) is hanged to p q ( ) tr A( ) tr B tr, i j S M q W p K (5) M H H (6) A( ) A() A(). If -saing onstraints are imposed on the new state-variabe vetor x (, then it is required that: ( ) ( ) for,,, K ii K ii i n (7) K is the ontroabiity Gramian of the state-spae mode in (), defined by: dz K ( zin A) BB ( z In A ) j, (8) z z whih an be obtained by soving the Lyapunov equation: K = AKA + BB. (9) As a resut, the minimization probem of a frequeny weighted sensitivity measure subjet to saing onstraints is now formuated as foows: Given oeffiient matries A, B and C, obtain an n n nonsinguar matrix whih minimizes (5) subjet to the -saing onstraints in (7). 3. PROBLEM SOLUION When the inear system in () is stabe and ontroabe, the ontroabiity Gramian K is symmetri and positive-definite [8]. his impies that K / satisfying K = K / K / is aso symmetri and positive-definite. Defining: the -saing onstraints in (7) an be expressed as ˆ = K, ()

6 4 akao Hinamoto, Osamu anaka, Akimitsu Doi 6 ˆ ˆ for i,, n. ii he onstraints in () simpy state that eah oumn in vetor. If matrix ˆ is assumed to have the form: () ˆ must be a unity ˆ t t tn =,, L,, () t t tn then () is aways satisfied. From () it foows that (5) is hanged to p q ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ o( ) tr A( ) tr B tr i j J M q W p K, (3) ˆ ˆ ˆ ˆ ˆ ˆ A( ) A() A() ˆ () () HA K HA K ˆ ˆ WB K WBK, K K KK M H H. (4) From the foregoing arguments, the probem of obtaining an n n nonsinguar matrix whih minimizes (5) subjet to the saing onstraints in (7) an be onverted into an unonstrained optimization probem of obtaining an n n nonsinguar matrix ˆ whih minimizes (3). Now we appy a quasi-newton agorithm [7] to minimize (3) with respet to matrix ˆ given by (). Define: n x t, t,, t. (5) hen J ˆ o ( ) is a funtion of x, denoted by J(x). he agorithm starts with a trivia initia point x obtained from an initia assignment ˆ = In. hen, in the kth iteration a quasi-newton agorithm updates the most reent point x k to point x k+ as [7]. xk xk αkd k, (6)

7 7 Minimization of frequeny-weighted -sensitivity 4 S d S J( x ), α k k k arg min J( x α d ), k k k k α γ S γ δ δ δ γ S S γ δ, k k k k k k k k k k k k Sk γkδk γkδk γkδk S I, δ x x, γ J( x ) J( x ). k k k k k k (7) Here, J (x) is the gradient of J(x) with respet to x, and S k is a positive-definite approximation of the inverse Hessian matrix of J(x). his iteration proess ontinues unti J( x ) J( x ) ε (8) k is satisfied ε > is a presribed toerane. If the iteration is terminated at step k, then x k is viewed as a soution point. he impementation of (6) requires the gradient of J(x). Cosed-form expressions for J (x) are given beow. k J ˆ ˆ ˆ o ( ) Jo( ξζ ) Jo( ) im, t ξζ ( β β β β ), 3 4 (9) ˆξζ is the matrix obtained from ˆ with its (, ξζ) th omponent perturbed by [8, p. 655] g ˆ e ˆ ˆ ˆ, e ˆ M ( ˆ) ˆ g ˆ, p q ξζ ζ ξζ β ˆ ζ A ξζ eg ˆ ζ ξζ i j β β p q ˆ ˆ ˆ ˆ ˆ ζ A A i j e H () H () g, qewg ˆ ˆ ˆ ˆ, β pe ˆ Kg ˆ, 3 ζ B ξζ 4 ζ ξζ ξζ (3) g ξζ t t ζ ζ / tξζ 3 tξζtζ tζ eζ, t ζ e denotes an n unit vetor whose ξ th eement equas unity. ζ

8 4 akao Hinamoto, Osamu anaka, Akimitsu Doi 8 4. NUMERICAL EXAMPLE Consider a two-input/three-output inear disrete-time system (A o,b o,c o,d) n speified by A C = - 9, B = = , D= (3) he frequeny-weighting funtions used in this exampe were given by a FIR digita ow-pass fiter with the foowing unit-sampe response: ω () i =ω () i =ω () i = A B C = 563exp[ 33( i 4) ] (3) for i, and zero ese. After arrying out the saing for the above system with a diagona oordinate transformation matrix, the frequeny weighted sensitivity of the saed system was omputed from (8) as: S = Appying the quasi- Newton agorithm in (6) to the saed reaization, after 3 iterations, we obtained matrix ˆ as ˆ = or equivaenty, matrix was derived as (33) = (34)

9 9 Minimization of frequeny-weighted -sensitivity 43 hen, the frequeny-weighted -sensitivity measure in (3) beome S ( P 3 ) = 99787, (35) and the profies of the frequeny-weighted sensitivity during the first 3 iterations of the agorithm are shown in Fig.. Fig. Profies of frequeny-weighted sensitivity. 5. CONCLUSIONS We have investigated the probem of minimizing the frequeny-weighted sensitivity measure subjet to saing onstraints for MIMO inear disretetime systems. After onverting the minimization probem of the frequenyweighted sensitivity subjet to saing onstraints into an unonstrained optimization probem, an effiient quasi-newton agorithm has been appied to sove the unonstrained optimization probem. he resuting oordinate transformation matrix has then been empoyed to onstrut the optima MIMO system struture. Our simuation resuts have demonstrated the vaidity and effetiveness of the proposed tehnique. It is noted that the sensitivity measure in [9] onsiders the sensitivity behavior of the transfer funtion at one frequeny point to be as important as at another frequeny point. In other words, a frequeny-weighted sensitivity measure has not yet been onsidered in [9]. On the other hand, in this paper we onsider a frequeny-weighted sensitivity measure whih orresponds to the more genera ase. Notie that soutions for frequeny-weighted sensitivity minimization woud

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