V. Balakrishnan and S. Boyd. (To Appear in Systems and Control Letters, 1992) Abstract

Transcription

1 On Cmputing the WrstCase Peak Gain f Linear Systems V Balakrishnan and S Byd (T Appear in Systems and Cntrl Letters, 99) Abstract Based n the bunds due t Dyle and Byd, we present simple upper and lwer bunds fr the `nrm f the `tail' f the impulse respnse f nitedimensinal discretetime linear timeinvariant systems Using these bunds, we may in turn cmpute the `gain f these systems t any desired accuracy By cmbining these bunds with results due t Khammash and Pearsn, we derive upper and lwer bunds fr the wrstcase `gain f discretetime systems with diagnal perturbatins Keywrds: SISO discretetime LTI systems, cmputatin f `gain, discretetime systems with diagnal perturbatins, wrstcase `gain Ntatin Z +, R, R + and C dente the set f nnnegative integers, real numbers, nnnegative real numbers and cmplex numbers respectively All the sequences in this nte are dened ver Z + The `nrm f a cmplexvalued sequence u is dened as kuk = sup ju(k)j: k0 Thus, the `nrm f a sequence is its peak value The `nrm f a cmplexvalued sequence u is dened as kuk = X k0 ju(k)j: Fr a matrix P R nn, P T stands fr the transpse (P ); (P);:::; n (P) are the singular values f P in decreasing rder (P ) dentes the spectral radius, which is the maximum magnitude f the eigenvalues f P I stands fr the identity matrix, with size determined frm cntext Research supprted in part by AFOSR under cntract F49609J003

2 Bunds fr the `gain Cnsider a stable, nitedimensinal discretetime linear timeinvariant (LTI) system described by the state equatins x(k + ) = Ax(k)+ bu(k); x(0) = 0; y(k) = cx(k)+du(k); () where the input u(k) R, the utput y(k) R and the state x(k) R n We assume that fa; b; c; dg is minimal The impulse respnse f system () is the real sequence given by h(k) = ( d; k =0; ca k, b; k>0: The `gain f system (), which is the largest pssible peak value f the utput y ver all pssible inputs u with a peak value f at mst ne, is just khk : khk = kyk sup : kuk >0 kuk khk is usually apprximated by summing nly a nite, typically large (say N)number f terms: S N = NX k=0 jh(k)jkhk : Obviusly, S N is a lwer bund fr khk, and increases mntnically t khk with increasing N The `errr' khk, S N is just the ` nrm f the tail, P k>n jh(k)j Many simple bunds n this errr are pssible; fr instance, if the ples f the system () are distinct, we may write dwn a residue expansin fr the impulse respnse h(k): h(k) = ( d; k =0; P n i= r ip k, i ; k>0: where p ;p ;:::;p n are the distinct ples f the system and r i are the residues (see fr example, [7], Chapter ) Then, X k>n jh(k)j Similar bunds are pssible when the ples are nt distinct i= jp i j N jr i j,jp i j : ()

3 The rst purpse f this nte is t present mre sphisticated, and in many cases, substantially better bunds fr the `nrm f the tail These bunds are based n Therem f [], which states that fr the system (), where jdj + (W W = X k=0 W c ) khk jdj+ (A T ) k c T ca k and W c = i= X k=0 i (W A k bb T (A T ) k are the bservability and cntrllability Gramians respectively [4] i (W singular values f the system () Wc ); (3) W c ) are just the Hankel We nw bserve that f0;h(n+ );h(n+ );:::g, the tail f the impulse respnse f system (), is just the impulse respnse f the system fa,a N b,c,0g Applying bunds (3) t this system, we have fr any N 0, (W A N W c X ) jh(k)j k>n Thus, we have upper and lwer bunds fr khk : S N + (W A N Wc ) khk S N + i= i= i (W i (W A N Wc ): (4) A N Wc ); 8N 0: (5) The rati between the upper and lwer bunds fr khk in (4) is at mst n, whereas the rati between the residueexpansin based upper bund () and anylwer bund can be arbitrarily large We next shw that with increasing N, the dierence between the upper and lwer bunds cnverges mntnically t zer W satises the Lyapunv equatin A T W A, W + c T c =0; which implies that fr k = ;;:::Therefre, (A T ) k W A k, (A T ) k, W A k, +(A T ) k, c T ca k, =0 (W A k W c ) T (W A k W c ) (W A k, W c ) T (W A k, W c ); k =;;::: 3

4 This immediately means i (W A k Wc ) i (W A k, Wc ); i =;;:::;n and k =;;:::; frm which it fllws that the dierence between the upper and lwer bunds in (5) cnverges mntnically t zer with increasing N The abve argument shws that all f the Hankel singular values f the impulse respnse f the `tail' system fa; A N b; c; 0g decrease mntnically (t zer, since the system is stable) as N! In fact, we can say mre: If we nrmalize the Hankel singular values by dividing them by the rst ne, the number f `nrmalized' Hankel singular values that cnverge t nnzer values as N! equals the number f `dminant' Jrdan blcks f A, that is, the number f Jrdan blcks f A which crrespnd t an eigenvalue f A with maximum magnitude, and which have the largest size amng all Jrdan blcks crrespnding t an eigenvalue with maximum magnitude Thus, fr large N, the number f signicant terms in the sum P n i= i(w A N W c ) is just the `eective rder' f the tail system fa; A N b; c; 0g Finally, we discuss infrmally a scheme fr nding N min = min ( N i= i (W A N W c ), (W A N W c ) < which is the smallest value f N fr which the dierence between the upper and lwer bunds in (5) is less than As a preliminary step, W c and W are cmputed Then: ) ; We nd the smallest psitive integer M such that N min M This is dne iteratively where at the kth iteratin, we frm the matrix A k by squaring A k, and check if (WA k Wc)+ i= i (W A k W c ) <; 4

5 and stp if the cnditin is satised Clearly, M iteratins are needed Each iteratin invlves three n n matrix multiplies and ne cmputatin f singular values Fr use in part (), we stre the matrices fa; A ;:::;A M g By a simple bisectin, N min is then lcated in the set f M, ; M, +;:::; M g We assume that M, since cmputing N min is trivial therwise We start by frming ~A = A (M, + M,) and checking if (W AW ~ c )+ i= i (W ~ AW c ) <: (Nte that since A M, and A M, are bth already available frm step (), and therefre this invlves three n n matrix multiplies and ne cmputatin f singular values) If the answer is yes, then N lies in the set f M, ; M, +;:::; M, + M, g Otherwise, N lies in the set f M, + M, ;:::; M g By cntinuing this prcess (at mst M, times) f halving the set where N lies, we may cmpute N min exactly Once N min is fund, S Nmin can be cmputed t give khk t within an abslute accuracy f (assuming innite precisin arithmetic; we have nt cnsidered the eects f data runding here) The exact determinatin f N min takes apprximately 6M matrix multiplies and M cmputatins f singular values Frming S Nmin takes abut N min matrixvectr multiplies and N min vectrvectr inner prducts (Recall that M, <N min M ) Since cmputing singular values is by far the mst expensive f the abve calculatins, it might prve advantageus t nt cmpute N min exactly, but t instead use an upper bund btained by terminating the bisectin in step () earlier Cmputatin may be further reduced by rst balancing system (), s that the Gramians W c and W are diagnal and equal We nte that fr calculating the H nrm f system () t within a relative accuracy, there exist methds (see []) where the cmputatinal ert invlved depends nly n and the state dimensin n Hwever fr determining khk using the bunds in (5) t within an accuracy f 5

6 (relative r abslute), the number f cmputatins depends n the system matrices A, b, c and d as well We knw f n way t vercme this deciency 3 Bunds fr the wrstcase `gain We nw cmbine the results f the previus sectin with results frm [5] t derive bunds fr the wrstcase `gain f discretetime LTI systems with diagnal uncertainty We cnsider the system shwn in Figure : H is a stable discretetime LTI plant ; ;:::; m are scalar LTI perturbatins that act n the system Nw, fr sme ntatin (indices i; j =;;:::;m): i : Impulse respnse f perturbatin i h 00 : Openlp ( = 0) impulse respnse frm w t z h i0 : Openlp ( = 0) impulse respnse frm w t y i h 0i : Openlp ( = 0) impulse respnse frm u i t z h cl () : Clsedlp impulse respnse frm w t z We assume that k i k and dente by the crrespnding set f all pssible perturbatins The quantity finterest is the wrstcase (ie maximum pssible) `gain frm w t z, which we dene as hlds: L wc = sup kh cl ()k : In [5], Khammash and Pearsn shw that the L wc if and nly if the fllwing cnditin There exists sme nnzer x =[x 0 ; :::; x m ] with x i 0 such that x i mx j=0 kh ij k x j i =0;; :::;m: (COND) Cnditin (COND) may be expressed simply in terms f a matrix whse (i; j)entry is kh ij k, i; j =0;;:::;m 6

7 Fact Cnditin COND hlds if and nly if the spectral radius f the matrix M = is at least ne 6 4 kh 00 k kh 0 k kh 0m k kh 0 k kh k kh m k kh m0k kh mk kh mm k This fact, stated withut prf in Therem f [6], is immediate frm the fllwing characterizatin f the spectral radius f a nnnegative matrix (a matrix with nnnegative entries) M (see, fr example, page 504, crllary 833 f [3]): (M) = (M ij refers t the (i; j)entry f M) that max x0; x6=0 min 0im; xi6=0 x i mx j=0 M ij x j : By simply scaling w and z by = p (>0) as in Figure, and applying Fact, we cnclude L wc = supf j (D MD ) g; (6) where Fr cnvenience, we partitin M as D = M = " " = p 0 0 I # : # M () M () M () M () ; (7) where M () R +, M () R m +, M () R m + and M () R mm + If (D MD ) fr all >0, then we dene L wc = This crrespnds t the case when (M () ), and the system is nt `stable (see [5]) On the ther hand, if (D MD ) < fr all >0, we dene L wc = 0 This crrespnds t the case when either the rst rw (r the rst clumn) f M is identically zer (with (M () ) < ) Then h cl () = 0 fr all Of curse, every entry f M is the `gain f sme LTI system; therefre, the remarks made in Sectin abut cmputing `gains apply here as well We mayhwever use the fact that M is 7

8 w u u m H m y y m z Figure : Linear system with diagnal uncertainty w p u u m H m p y y m z Figure : Uncertain linear system with the impulse respnse frm w t z scaled by = 8

9 nnnegative t derive bunds n L wc based n the bunds fr the entries f M We start with the fllwing fact Fact The spectral radius f a nnnegative matrix is a nndecreasing functin f its entries (See Crllary 89 n page 49 f [3]) Fact implies that (D PD ) is a nndecreasing functin f the entries f the nnnegative matrix P and a nnincreasing functin f >0 These, in turn, mean that the functin (P )fa nnnegative matrix P dened by (P ) = supf j (D PD ) g is nndecreasing with the entries f P We then have the fllwing bunds fr L wc : Therem Let N ij and N ij be lwer and upper bunds fr kh ijk cmputed via equatin (5) fr sme N > 0 Let M N lb and M N ub be matrices with (i; j)entry N ij and N ij respectively (i; j = 0; ;:::;m) Then and L N lb =(Mlb N ) = supf j D Mlb N D g; L N ub =(Mub) N = supf j D Mub N D g; are lwer and upper bunds respectively fr L wc, ie L N lb L wc L N ub Cmputatin f L N lb and LN ub is straightfrward, nce we make the fllwing bservatin: Fact 3 The spectral radius f a nnnegative matrix is als an eigenvalue (See Therem 83 n page 503 f [3]) Givena(m+)(m+ ) matrix P,we rst partitin cnfrmally as with M in equatin (7) as P = " # P () P () P () P () : 9

10 Then, (P )=if (P () ) Otherwise, we nte that (D PD )= D P, and slve fr D Px = x fr sme nnzer (m + )vectr x t btain (P )=P () + P () (I, P () ), P () : The abve frmula shws that if (P () ) <, (P ) is just the unique slutin t the equatin (D PD )= 4 Cnclusin We have presented simple bunds n the `gain f singleinput singleutput linear discrete time systems We have shwn hw tcmbine these bunds with recent results frm [5] t cmpute guaranteed bunds fr the wrstcase ` gain f discretetime LTI systems with diagnal uncertainty The bunds may be easily extended t blck diagnal uncertainties as well as t cntinuus time systems References [] S Byd and V Balakrishnan, A regularity result fr the singular values f a transfer matrix and a quadratically cnvergent algrithm fr cmputing its L nrm, Systems Cntrl Lett 5 (990) {7 [] S Byd and J Dyle, Cmparisn f peak and RMS gains fr discretetime systems, Systems Cntrl Lett 9 (987) {6 [3] R A Hrn and C A Jhnsn, Matrix Analysis (Cambridge University Press, Cambridge, United Kingdm, 985) [4] T Kailath, Linear Systems (PrenticeHall, New Jersey, 980) [5] M Khammash and J B Pearsn, Perfrmance rbustness f discretetime systems with structured uncertainty, IEEE Trans Autmat Cntrl 36 (4) (99) 398{4 0

11 [6] M Khammash and J B Pearsn, Rbustness synthesis fr discretetime systems with structured uncertainty, Prceedings f the 99 American Cntrl Cnference, (99) 70{74 [7] K Ogata, DiscreteTime Cntrl Systems (PrenticeHall, Englewd Clis, New Jersey, 987)