If A(k ; ), A(k ; 2),..., A(`) are all nvertble, then one could use the state transton matrx to obtan x(k) from x(`) even when k < `, but we shall typ

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1 Unversty of Calforna Department of Mechancal Engneerng Lnear Systems Fall 999 (B. Bameh) Lecture 4: Dscrete-Tme Lnear State-Space Models In the prevous lecture we showed how dynamc models arse, and studed some specal characterstcs that they may possess. We focused on state-space models and ther propertes, presentng several examples. In ths lecture we wll contnue the study of state-space models, concentratng on solutons and propertes of DT lnear state-space models, both tme-varyng and tme-nvarant. 4. Tme-Varyng Lnear Models A general nth-order dscrete-tme lnear state-space descrpton takes the followng form: x(k +) = A(k)x(k)+B(k)u(k) y(k) = C(k)x(k)+D(k)u(k) (4.) where x(k) 2 R n. Gven the ntal condton x(0) and the nput sequence u(k), we would lke to nd the state sequence or state trajectory x(k) aswell as the output sequence y(k). Undrven Response Frst let us consder the undrven response, that s the response when u(k) = 0 for all k 2 Z. The state evoluton equaton then reduces to x(k +) =A(k)x(k) : (4.2) The response can be derved drectly from (4.2) by smply teratng forward: x() = A(0)x(0) x(2) = A()x() = A()A(0)x(0) x(k) = A(k ; )A(k ; 2) :::A()A(0)x(0) (4.3) Motvated by (4.3), we dene the state transton matrx, whch relates the state of the undrven system at tme k to the state at an earler tme `: x(k) =(k `)x(`) k `: (4.4) The form of the matrx follows drectly from (4.3): (k `) = ( A(k ; )A(k ; 2) A(`) k>`0 I k = ` : (4.5) These notes are part of a set under development by Mohammed Dahleh (UC Santa Barbara), Munther Dahleh (MIT), and George Verghese (MIT). Comments and correctons would be greatly apprecated. Please do not copy for or dstrbute to people not enrolled n ECE230A/ME243A wthout permsson from one of the coauthors.

2 If A(k ; ), A(k ; 2),..., A(`) are all nvertble, then one could use the state transton matrx to obtan x(k) from x(`) even when k < `, but we shall typcally assume k ` when wrtng (k `). The followng propertes of the dscrete-tme state transton matrx are worth hghlghtng: (k k) = I x(k) = (k 0)x(0) (k + `) = A(k)(k `): (4.6) Example (A Sucent Condton for Asymptotc Stablty) The system (4.) s termed asymptotcally stable f, wth u(k) 0, and for all x(0), we have x(n)! 0 (by whch we mean kx(n)k! 0) as n!. Snce u(k) 0, we are n eect dealng wth (4.2). Suppose ka(k)k < (4.7) for all k, where the norm s any submultplcatve norm and s a constant (ndependent of k) that s less than. Then and hence k(n 0)k n kx(n)k n kx(0)k so x(n)! 0 as n!, no matter what x(0) s. Hence (4.7) consttutes a sucent condton (though a weak one, as we'll see) for asymptotc stablty of (4.). Example 2 (\Lftng" a Perodc Model to an LTI Model) Consder an undrven lnear, perodcally varyng (LPV) model n state-space form. Ths s a system of the form (4.2) for whch there s a smallest postve nteger N such that A(k + N) = A(k) for all k thus N s the perod of the system. (If N =,the system s actually LTI, so the cases of nterest here are really those wth N 2.) Now focus on the state vector x(mn) fornteger m,.e., the state of the LPV system sampled regularly once every perod. Evdently x(mn + N) = h A(N ; )A(N ; 2) A(0) x(mn) = (N 0) x(mn) (4.8) The sampled state thus admts an LTI state-space model. The process of constructng ths sampled model for an LPV system s referred to as lftng. Drven Response Now let us consder the drven system,.e., u(k) 6= 0 for at least some k. Referrng back to (4.), we have x() = A(0)x(0) + B(0)u(0) x(2) = A()x() + B()u() = A()A(0)x(0) + A()B(0)u(0) + B()u() (4.9) 2

3 whch leads to X k; x(k) = (k 0)x(0) + (k ` +)B(`)u(`) `=0 = (k 0)x(0) + ;(k 0)U(k 0) (4.0) where ;(k 0) = h (k )B(0) j (k 2)B() j j B(k ; ) U(k 0) = 0 u(0) u(). u(k ; ) C A (4.) What (4.0) shows s that the soluton of the system over k steps has the same form as the soluton over one step, whch sgven n the rst equaton of (4.). Also note that the system response s dvded nto two terms: one depends only on the ntal state x(0) and the other depends only on the nput. These terms are respectvely called the natural or unforced or zero-nput response, and the zero-state response. Note also that the zero-state response has a form that s remnscentofaconvoluton sum ths form s sometmes referred to as a superposton sum. If (4.0) had been smply clamed as a soluton, wthout any sort of dervaton, then ts valdty could be vered by substtutng t back nto the system equatons: x(k +) = (k + 0)x(0) + kx `=0 X (k + `+)B(`)u(`) k; = (k + 0)x(0) + (k + `+)B(`)u(`)+B(k)u(k) = A(k) " `=0 (k 0)x(0) + k; X `=0 (k ` +)B(`)u(`) # + B(k)u(k) = A(k)x(k) +B(k)u(k) : (4.2) Clearly, (4.2) satses the system equatons (4.). It remans to be vered that the proposed soluton matches the ntal state at k = 0. We have x(0) = (0 0)x(0) = x(0) (4.3) whch completes the check. If Y(k 0) s dened smlarly to U(k 0), then followng the sort of dervaton that led to (4.0), we can establsh that Y(k 0) = (k 0)x(0) + (k 0)U(k 0) (4.4) for approprately dened matrces (k 0) and (k 0). We leave you to work out the detals. Once agan, (4.4) for the output over k steps has the same form as the expresson for the output at a sngle step, whch sgven n the second equaton of (4.). 4.2 Lnear Tme-Invarant Models In the case of a tme-nvarant lnear dscrete-tme system, the solutons can be smpled consderably. We rst examne a drect tme-doman soluton, then compare ths wth a transform-doman soluton, and nally return to the tme doman, but n modal coordnates. 3

4 Drect Tme-Doman Soluton For a lnear tme-nvarant system, observe that A(k) =A B(k) =B where A and B are now constant matrces. Thus ) for all k 0 (4.5) (k `) =A(k ; ) :::A(`) =A k;` k ` (4.6) so that, substtutng ths back nto (4.0), we are left wth X k; x(k) = A k x(0) + A k;`; Bu(`) `=0 h = A k x(0) + A k; B j A k;2 B j j B 0 u(0) u(). u(k ; ) C A (4.7) Note that the zero-state response n ths case exactly corresponds to a convoluton sum. Smlar expressons can be worked out for the outputs, by smplfyng (4.4) we leave the detals to you. Transform-Doman Soluton We know from earler experence wth dynamc lnear tme-nvarant systems that the use of approprate transform methods can reduce the soluton of such a system to the soluton of algebrac equatons. Ths expectaton does ndeed hold up here. Frst recall the denton of the one-sded Z-transform : Denton 4. The one-sded Z-transform, F (z), of the sequence f(k) s gven by F (z) = k=0 z ;k f(k) for all z such that the result of the summaton s well dened. The sequence f(k) canbeavector or matrx sequence, n whch casef (z) s respectvely a vector or matrx as well. It s easy to show that the transform of a sum of two sequences s the sum of the ndvdual transforms. Also, scalng a sequence by a constant smply scales the transform by the same constant. The followng shft property of the one-sded transform s crtcal, Z and not hard to establsh. Suppose that f(k)! F (z). Then. 2. g(k) = ( f(k ; ) k 0 k =0 =) G(z) =z ; F (z): g(k) =f(k +) =) G(z) =z [F (z) ; f(0)] : 4

5 Convoluton s an mportant operaton that can be dened on two sequences f(k), g(k) as (f g)(k) = kx m=0 g(k ; m)f(m) whenever the dmensons of f and g are compatble so that the products are dened. The Z transform of a convolutons of two sequences satsfy Z(f g) = = = = = k=0 k=0 z ;k (f g)(k) z ;k m=0 k=m m=0 k=0 m=0 z ;m = F (z)g(z): kx m=0 f(k ; m)g(m) z ;k f(k ; m)g(m) z ;(k+m) f(k)g(m) k=0 z ;k f(k)!! g(m) Now, gven the state-space model (4.), we can take transforms on both sdes of the equatons there. Usng the transform propertes just descrbed, we get Ths s solved to yeld zx(z) ; zx(0) = AX(z)+BU(z) (4.8) Y (z) = CX(z)+DU(z): (4.9) X(z) = z(zi ; A) ; x(0) + (zi ; A) ; BU(z) Y (z) = zc(zi ; A) ; x(0) + h C(zI ; A) ; B + D {z } Transfer Functon U(z): (4.20) To correlate the transform-doman solutons n the above expressons wth the tmedoman expressons n (4.0) and (4.4), t s helpful to note that (zi ; A) ; = z ; I + z ;2 A + z ;3 A 2 + (4.2) as may be vered by multplyng both sdes by (zi ; A). The regon of convergence for the seres on the rght s all values of z outsde of some sucently large crcle n the complex plane. What ths seres establshes, on comparson wth the denton of the Z-transform, s that the nverse transform of z(zi ; A) ; s the matrx sequence whose value at tme k s A k for k 0 the sequence s 0 for tme nstants k<0. That s we can wrte I A A 2 A 3 A 4 ::: Z! z(zi ; A) ; 0 I A A 2 A 3 ::: Z! (zi ; A) ; : Also snce the nverse transform of a product such as(zi ; A) ; BU(z) s the convoluton of the sequences whose transforms are (zi ; A) ; B and U(z) respectvely, weget x(0) Ax(0) A 2 x(0) A x(0) ::: 3 Z! z(zi ; A) ; x(0) 0 B AB A 2 B A 3 Z B ::: (u(0) u() u(2) u(3) :::)! (zi ; A) ; BU(z): 5

6 Puttng the above two peces together, the parallel between the tme-doman expressons n (4.0) and (4.4), and the transform-doman expressons n (4.20) should be clear. The Transfer Functon Matrx It s evdent from (4.20) that the transfer functon matrx for the system, whch relates the nput transform to the output transform when the ntal condton s zero, s gven by H(z) =C(zI ; A) ; B + D: (4.22) For a mult-nput, mult-output (MIMO) system wth m nputs and p outputs, ths results n a p m matrx of ratonal functons of z. In order to get an dea of the nature of these ratonal functons, we express the matrx nverse as the adjont matrx dvded by the determnant, as follows: H(z) = C [adj(zi ; A)] B + D: det(zi ; A) The determnant det(zi ; A) n the denomnator s an n th -order monc (.e. coecent of z n s ) polynomal n z, known as the characterstc polynomal of A and denoted by a(z). The entres of the adjont matrx (the cofactors) are computed from mnors of (zi ; A), whch are polynomals of degree less than n. Hence the entres of the matrces and (zi ; A) ; = H(z) ; D = adj(zi ; A) det(zi ; A) Cadj(zI ; A)B det(zi ; A) are strctly proper,.e. have numerator degree strctly less than ther denomnator degree. Wth the D term added n, H(z) becomes proper that s all entres have numerator degree less than or equal to the degree of the denomnator. For jzj %, H(z)! D. The polynomal a(z) forms the denomnators of all the entres of (zi ; A) ; and H(z), except that n some, or even all, of the entres there may be cancellatons of common factors that occur between a(z) and the respectve numerators. We shall have a lot more to say later about these cancellatons and ther relaton to the concepts of reachablty (or controllablty) and observablty. To compute the nverse transform of (zi ; A) ; (whch s the sequence A k; ) and the nverse transform of H(z) (whch s a matrx sequence whose components are the zero-state unt sample responses from each nput to each output), we need to nd the nverse transform of ratonals whose denomnator s a(z) (apart from any cancellatons). The roots of a(z) also termed the characterstc roots or natural frequences of the system, thus play a crtcal role n determnng the nature of the soluton. A fuller pcture wll emerge as we proceed. Multvarable Poles and Zeros You are famlar wth the dentons of poles, zeros, and ther multplctes for the scalar transfer functons assocated wth sngle-nput, sngle-output (SISO) LTI systems. For the case of the p m transfer functon matrx H(z) that descrbes the zero-state nput/output behavor of an m-nput, p-output LTI system, the dentons of poles and zeros are more subtle. We nclude some prelmnary dscusson here, but wll leave further elaboraton for later n the course. 6

7 It s clear what we would want our eventual dentons of MIMO poles and zeros to specalze to n the case where H(z) s nonzero only n ts dagonal postons, because ths corresponds to completely decoupled scalar transfer functons. For ths dagonal case, we would evdently lke tosay that the poles of H(z) are the poles of the ndvdual dagonal entres of H(z), and smlarly for the zeros. For example, gven H(z) = z+2 0 (z+0:5) 2 z 0 (z+2)(z+0:5) we would say that H(z) has poles of multplcty 2 and at z = ;0:5, and a pole of multplcty at z = ;2 and that t has zeros of multplcty at ;2, at z = 0, and at z =. Note that n the MIMO case we can have poles and zeros at the same locaton (e.g. those at ;2 n the above example), wthout any cancellaton! Also note that a pole or zero s not necessarly characterzed by asngle multplcty we may nstead have a set of multplcty ndces (e.g. as needed to descrbe the pole at ;0:5 n the above example). The dagonal case makes clear that we do not want to dene a pole or zero locaton of H(z) n the general case to be a frequency where all entres of H(z) respectvely have poles or zeros. For a varety of reasons, the approprate denton of a pole locaton s as follows: Pole Locaton: H(z) hasapole at a frequency p 0 f some entry of H(z) hasapole at z = p 0. The full denton (whchwe wll present later n the course) also shows us how to determne the set of multplctes assocated wth each pole frequency. Smlarly, t turns out that the approprate denton of a zero locaton s as folows: Zero Locaton: H(z) has a zero at a frequency 0 f the rank of H(z) drops at z = 0. Agan, the full denton also permts us to determne the set of multplctes assocated wth each zero frequency. The determnaton of whether or not the rank of H(z) drops at some value of z s complcated by thefactthat H(z) mayalsohave a pole at that value of z however, all of ths can be sorted out very ncely. Smlarty Transformatons Suppose we have characterzed a gven dynamc system va a partcular state-space representaton, say wth state varables x x 2 x n. The evoluton of the system then corresponds to a trajectory of ponts n the state space, descrbed by the successon of values taken by the state varables. In other words, the state varables may be seen as consttutng the coordnates n terms of whch wehave chosen to descrbe the moton n the state space. We are free, of course, to choose alternatve coordnate bases.e., alternatve state varables to descrbe the evoluton of the system. Ths evoluton s not changed by the choce of coordnates only the descrpton of the evoluton changes ts form. For nstance, n the LTI crcut example n the prevous chapter, we could have used L ; v C and L + v C nstead of L and v C. The nformaton n one set s dentcal wth that n the other, and the exstence of a state-space descrpton wth one set mples the exstence of a statespace descrpton wth the other, as we now show more concretely and more generally. The exblty to choose an approprate coordnate system can be very valuable, and we wll nd ourselves nvokng such coordnate changes very often. Gven that we have a state vector x, suppose we dene a constant nvertble lnear mappng from x to r, as follows:! r = T ; x x = Tr: (4.23) 7

8 Snce T s nvertble, ths maps each trajectory x(k) to a unque trajectory r(k), and vce versa. We refer to such a transformaton as a smlarty transformaton. The matrx T embodes the detals of the transformaton from x coordnates to r coordnates t s easy to see from (4.23) that the columns of T are the representatons of the standard unt vectors of r n the coordnate system of x, whch s all that s needed to completely dene the new coordnate system. Substtutng for x(k) n the standard (LTI verson of the) state-space model (4.), we have Tr(k +) = A Tr(k) + Bu(k) (4.24) y(k) = C Tr(k) + Du(k): (4.25) or r(k +) = (T ; AT ) r(k)+(t ; B) u(k) (4.26) = b Ar(k)+ b Bu(k) (4.27) y(k) = (CT) r(k) +Du(k) (4.28) = b Cr(k)+Du(k) (4.29) We now have a new representaton of the system dynamcs t s sad to be smlar to the orgnal representaton. It s crtcal to understand, however, that the dynamc propertes of the model are not at all aected by ths coordnate change n the state space. In partcular, the mappng from u(k) to y(k),.e. the nput/output map, s unchanged by a smlarty transformaton. Soluton n Modal Coordnates The proper choce of a smlarty transformaton may yeld a new system model that wll be more sutable for analytcal purposes. One such transformaton brngs the system to what are known as modal coordnates. We shall descrbe ths transformaton now for the case where the matrx A n the state-space model can be dagonalzed, n a sense to be dened below we leave the general case for later. Modal coordnates are bult around the egenvectors of A. To get a sense for why the egenvectors may be nvolved n obtanng a smple choce of coordnates for studyng the dynamcs of the system, let us examne the possblty of ndng a soluton of the form for the undrven LTI system x(k) = k v v 6= 0 (4.30) x(k +)=Ax(k) (4.3) Substtutng (4.30) n (4.3), we nd the requste condton to be that (I ; A) v =0 (4.32).e., that be an egenvalue of A, and v an assocated egenvector. Note from (4.32) that multplyng any egenvector by a nonzero scalar agan yelds an egenvector, so egenvectors are only dened up to a nonzero scalng any convenent scalng or normalzaton can be used. In other words, (4.30) s a soluton of the undrven system s one of the n roots of the characterstc polynomal a(z) = det(zi ; A) =z n + a n; z n; + + a 0 (4.33) 8

9 and v s a correspondng egenvector v. A soluton of the form x(k) = k v s referred to as a mode of the system, n ths case the th mode. The correspondng s the th modal frequency or natural frequency, and v s the correspondng modal shape. Note that we can excte just the th mode by ensurng that the ntal condton s x(0) = 0 v = v. The ensung moton s then conned to the drecton of v, wth a scalng by at each step. It can be shown farly easly that egenvectors assocated wth dstnct egenvalues are (lnearly) ndependent,.e. none of them can be wrtten as a weghted lnear combnaton of the remanng ones. Thus, f the n egenvalues of A are dstnct, then the n correspondng egenvectors v are ndependent, and can actually form a bass for the state-space. Dstnct egenvalues are not necessary, however, to ensure that there exsts a selecton of n ndependent egenvectors. In any case, we shall restrct ourselves for now to the case where because of dstnct egenvalues or otherwse the matrx A has n ndependent egenvectors. Such ana s termed dagonalzable (for a reason that wll become evdent shortly), or non-defectve. There do exst matrces that are not dagonalzable, as we shall see when we examne the Jordan form n detal later n ths course. Because (4.3) s lnear, a weghted lnear combnaton of modal solutons wll satsfy t too, so x(k) = nx = v k (4.34) wll be a soluton of (4.3) for arbtrary weghts, wth ntal condton x(0) = nx = v (4.35) Snce the n egenvectors v are ndependent under our assumpton of dagonalzable A, the rght sde of (4.35) can be made equal to any desred x(0) by properchoce of the coecents, and these coecents are unque. Hence specfyng the ntal condton of the undrven system (4.3) speces the va (4.35) and thus, va (4.34), speces the response of the undrven system. We refer to the expresson n (4.34) as the modal decomposton of the undrven response. Note that the contrbuton to the modal decomposton from a conjugate par of egenvalues and wll be a real term of the form v k + v k. Before proceedng to examne the full response of a lnear tme-nvarant model n modal terms, t s worth notng that the precedng results already allow us to obtan a precse condton for asymptotc stablty of the system, at least n the case of dagonalzable A (t turns out that the condton below s the rght one even for the general case). Recallng the denton n Example, we see mmedately from the modal decomposton that the LTI system (4.3) s asymptotcally stable j j < for all n,.e. all the natural frequences of the system are wthn the unt crcle. Snce t s certanly possble to have ths condton hold even when kak s arbtrarly greater than, we see that the sucent condton gven n Example s ndeed rather weak, at least for the tme-nvarant case. Let us turn now to the LTI verson of the full system n (4.). Rather than approachng ts modal soluton n the same style as was done for the undrven case, we shall (for a derent pont of vew) approach t va a smlarty transformaton to modal coordnates,.e., to coordnates dened by the egenvectors fv g of the system. Consder usng the smlarty transformaton x(k) =V r(k) (4.36) where the th column of the n n matrx V s the th egenvector, v : V = v v 2 v n (4.37) 9

10 We refer to V as the modal matrx. Under our assumpton of dagonalzable A, the egenvectors are ndependent, so V s guaranteed to be nvertble, and (4.36) therefore does ndeed consttute a smlarty transformaton. We refer to ths smlarty transformaton as a modal transformaton, and the varables r (k) dened through (4.36) are termed modal varables or modal coordnates. What makes ths transformaton nterestng and useful s the fact that the state evoluton matrx A now transforms to a dagonal matrx : V ; AV = dagonal f n g = n = (4.38) The easest way to verfy ths s to establsh the equvalent condton that AV = V, whch n turn s smply the equaton (4.32), wrtten for = nand stacked up n matrx form. The reason for callng A \dagonalzable" when t has a full set of ndependent egenvectors s now apparent. Under ths modal transformaton, the undrven system s transformed nto n decoupled, scalar equatons: r (k +)= r (k) (4.39) for = 2 ::: n. Each of these s trval to solve: we have r (k) = k r (0). Combnng ths wth (4.36) yelds (4.34) agan, but wth the addtonal nsght that = r (0) (4.40) Applyng the modal transformaton (4.36) to the full system, t s easy to see that the transformed system takes the followng form, whch s once agan decoupled nto n parallel scalar subsystems: r (k +) = r (k)+ u(k) = 2 ::: n (4.4) y(k) = r (k)++ n r n (k)+du(k) (4.42) where the and are dened va V ; B = h 7 5 CV = 2 n (4.43) n The scalar equatons above can be solved explctly by elementary methods (compare also wth the expresson n (4.7): r (k) = k r (0) {z } ZIR X k; + k;`; u(`) 0 {z } ZSR (4.44) where \ZIR" denotes the zero-nput response, and \ZSR" the zero-state response. From the precedng expresson, one can obtan an expresson for y(k). Also, substtutng (4.44) n (4.36), we can derve a correspondng modal representaton for the orgnal state vector x(k). We leave you to wrte out these detals. 0