Fakultät für Mathematik und Naturwissenschaften. Institut für Mathematik, Fachabteilung Analysis und Systemtheorie DIPLOMARBEIT

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1 Fakultät ür Mathematik un Naturwissenschaten Institut ür Mathematik, Fachabteilung Analysis un Systemtheorie DIPLOMABEIT A normal orm or time-varying linear systems zur Erlangung es akaemischen Graes Diplom-Mathematiker vorgelegt von Markus Müller betreut von Pro. Dr. Achim Ilchmann Ilmenau, 3. September 25

2 Zusammenassung Für zeitinvariante Eingangs/Ausgangssysteme existieren verschieene Charakterisierungen ür en sogenannten elativgra es Systems. Man unterscheiet zwischen linearen un nichtlinearen Eingangs/Ausgangssystemen. In ieser Arbeit wir eine verallgemeinerte Deinition es elativgraes ür zeitvariante nicht-lineare Systeme angegeben un arau aubauen eine Charakterisierung es elativgraes zeitvarianter linearer Systeme begrünet. Diese Charakterisierung stützt sich ausschließlich au ie zeitabhängigen Systemmatrizen un eren Ableitungen. Es wir gezeigt, ass in ieser Darstellung es elativgraes zeitvarianter linearer Systeme ie bekannte Darstellung ür zeitinvariante lineare Systeme augehoben ist un, interpretiert man as zeitvariante lineare System als zeitinvariantes nichtlineares System, ie Deinition ebenso augehoben ist. Die Darstellung es elativgraes zeitvarianter linearer Systeme wir nun genutzt, um bezüglich einer zeitvarianten linearen Transormation eine Normalorm ür zeitvariante lineare Systeme zu entwickeln, welche in ihrer Struktur er Normalorm zeitinvarianter linearer Systeme gleicht. Die Normalorm ür zeitvariante lineare Systeme wir schließlich als Grunlage er Betrachtung er sogenannten Nullynamik ieser Systeme verwenet. Dazu wir Beschränktheit sowie Stabilität er Nullynamik einiert un es wir bewiesen, ass ie Nullynamik eines zeitvarianten linearen Systems genau ann beschränkt ist, wenn ie Nullynamik er korresponierenen Normalrom beschränkt ist. Entsprechene Behauptungen weren auch ür asymptotische un exponentielle Stabilität gezeigt. Abstract There are several types o characterisations o the so-calle relative egree or time-invariant input output systems. Well known concepts or relative egree exist or time-varying linear an nonlinear systems. This iploma thesis gives a generalise einition o relative egree or time-varying nonlinear systems. This einition is use to establish a characterisation o relative egree or time-varying linear systems in terms o the time-varying system matrices an their erivatives. It will be prove that i time-invariant linear systems are consiere, then the escription or time-varying linear systems results in the well known characterisation o relative egree or time-invariant linear systems, an, i the time-varying linear system is interprete as time-invariant nonlinear system, then the einition o relative egree is assure. The main result o this work is, that the characterisation o relative egree or time-varying linear systems will be use to erive a normal orm or this systems concerning time-varying linear coorinate transormation. These normal orm is similar to the normal orm o time-invariant linear systems. Finally the normal orm or time-varying linear systems is use to consier the so-calle zero ynamics o this systems. Bouneness an stability or the zero ynamics will be eine an it will be shown that a time-varying linear system has boune zero ynamics i, an only i, the zero ynamics o the corresponing normal orm are boune. 2

3 List o symbols N,, C natural, real an complex numbers note that is not eeme to be a natural number N Gl n C CU, W C pw U, W C l U, W I n xx, y partial := N {}, non-negative integers := {x x }, non-negative real numbers general linear group o invertible matrices A n n := {s C e s < }, set o complex numbers with strict negative real part vector space o continuous unctions : U W, U an W are open sets vector space o piecewise continuous unctions : U W, U an W are open sets vector space o l-times ierentiable unctions : U W, U an W are open sets an l N { } ientity matrix o imension n N erivative with respect to x o C1 U V, W, U, V an W are open sets L λ Lie-erivative o λ C 1 n, concerning a vectoriel C n, n, eine by L λ := λ x, see, or example, [7, Sect. 1.2] L k h k-th Lie-erivative o λ Ck n, concerning a vectoriel C k 1 n,, eine by L k λ := L L k 1 λ, an L λ := λ, see also, or example, [7, Sect. 1.2] i j binomial coeicient lim s i i A S n 1 := lim sup s j, upper limit o a sequence si i N U, U is an open set i j i := 2, Eucliian norm on n, n N := sup Ax or A n m, matrix norm inuce by the norm x =1 := sup x or a unction : U W, U an W are open sets x U := {x n x = 1}, unit circle in n 3

4 L U, W A > := { : U W < }, vector space o boune unctions, U an W are open sets : A n n is positive einite δ A 1 δ : η S n 1 : δ η T Aη 1 δ, δ, 1 4

5 1 Introuction This iploma thesis eals with ynamical systems, which are use in many iels o engineering, physics an economics. There are several categories o ynamical systems. The main results o this paper will concern time-varying linear systems o the orm } ẋ = Atx + Btu 1.1 yt = Ct xt, with A C l, n n, B, C T C l, n m, n, m N an l N. The input an output o system 1.1 are vector value unctions, i.e. u, y : m. Hence we call 1.1 a multi-input multi-output time-varying linear system. I m = 1 we call 1.1 a single-input single-output time-varying linear system an write bt := Bt n an ct := Ct 1 n. In control theory, the concept o relative egree goes back to single-input single-output linear systems escribe in the requency omain o the orm qsy s = psus, 1.2 or p, q [s] where Y, U are the Laplace transorm o unctions y, u :. The unction gs = ps/qs is calle transer unction. The ierence r = eg q eg p is calle relative egree o the system 1.2. To erive a characterisation in the time omain, take any realization o ps/qs, say } ẋt = A xt + but 1.3 yt = c xt with A n n, b, c T n. System 1.3 is equivalent to t I n A xt = but yt = c xt. Then gs = ps/qs = csi A 1 b = ca k bs k+1. Now it is easy to see that r = eg q eg p i, an only i, k= k =,...,r 2 : ca k b =, ca r 1 b. 1.4 This characterisation o relative egree or single-input single-output time-invariant linear systems leas to a normal orm or this systems. Ilchmann et al., see [6, Lemma 3.5], erive a normal orm or multi-input multi-output time-invariant linear systems. This gives obviously the normal orm or single-input single-output time-invariant linear systems. I 1.3 has relative egree r, then it is c ca. ca r 1 [ b, Ab,...,A r 1 b ] = ca r 1 b Gl r, ca r 1 b an thereore C := [ c T, ca T,...,cA r 1 T] T an B := [ b, Ab,...,A r 1 b ] have ull rank, i.e. rk C = rk B = r. For completing C to a basis-transormation, a matrix V n r o ull rank... 5

6 ξ with imv = ker C is chosen. The coorinate transormation := Ux with ξ η r, η n r, [ ] C U := an N := V N T VV T I BCB 1 C] converts 1.3 into t ξ η = r S P... Q ξ y = [1,,...,] η ξ η +. ca r 1 b where 1,..., r, S 1 n r, P n r an Q n r n r. u We assert ξ 1, ξ 2,..., ξ r = y, y 1,..., y r 1. It ollows, that the irst r 1 erivatives o y are inepenent on u an y r = r i y i 1 + ca r 1 bu. i=1 Thus, the normal orm 1.5 or the linear system 1.3 is a useul instruments to simply rea o the zero ynamics o time-invariant linear systems, see [7, Sec. 4.3] an Section In the ace o the characterisation o relative egree or time-invariant linear systems 1.3, the next step is to expect that conition 1.4 is also applicable to time-varying linear systems 1.1. It is obvious, that or time-varying unctions A, B, C only a local einition o relative egree is wise. For example, i Bt = n m or t T, where T is an open set, CtAt k Bt = or all t T an every k N an so system 1.1 has no relative egree on T. Suppose nevertheless, that a single-input single-output time-varying linear system 1.1 has relative egree r N on an open set T i, an only i, Then we obtain t T, k {,...,r 2} : ctat k bt =, ctat r 1 bt. 1.6 y = ctx ẏ = t ctx = ċtx + ctatx + btu = ċt + ctatx + ctbt u }{{} y 2 = t + ctatx = tċt ċt + ctat x + + ctat Atx + btu = = tċt + ctat x + ċtat + ctat 2 x + ċtatbt + ctatbt }{{} = Hence the secon erivative o y may epen on u or any r N, which is a unamental ierence to time-invariant linear systems. Qualities concerning the zero ynamics o timevarying linear systems woul be lost. So, the einition o relative egree as in 1.6 is inapplicable on time-varying linear systems. We have to look or another way to eine relative egree or time-varying systems, see Section 2. u. 6

7 Isiori [7, p. 137] generalise the concept o relative egree to single-input single-output timeinvariant nonlinear systems, aine in the control, o the orm } ẋ = x + gxu 1.7 yt = hxt, with, g C l n, n, h C l n, an l N: the system 1.7 has relative egree r {1,...,l} at x n i, an only i, there exists an open neighbourhoo U o x, such that, x U k {,...,r 2} : L g L k hx =, L gl r 1 hx ; 1.8 see [7, Sect. 4.1]. Setting x = Ax, gx = b an hx = cx gives that L hx = h xx = cax an thereore L g L k hx = cak b. Thus, the characterisation o relative egree o time-invariant linear systems is capture by 1.8. The importance o the relative egree or non-linear systems is, that it also leas to a normal orm or time-invariant nonlinear systems, see [7, Sec. 4.1]. I system 1.7 has relative egree r at x, then there exists a ieomorphism Φ, eine in a neighbourhoo o x, which transorms 1.7 uner ξ T, η T T = Φx to ξ 1 = ξ 2 ξ 2 = ξ 3 ξ r 1. = ξ r ξ r = L r hφ 1 ξ, η + L g L r 1 hφ 1 ξ, ηu η = qξ, η y = ξ 1, 1.9 or some q C l n, n r. This gives ξ = y,..., y r 1 T an y r = ξ r in a neighbourhoo o x. Thus, the normal orm or time-invariant non-linear systems gives immeiately that, or xt = x, the relative egree r is exactly equal to the number o times one has to ierentiate the output yt at time t = t in orer to have the value ut o the input explicitly appearing [7, p. 139]. Moreover, u enters only in a single ierential equation in 1.9 irectly an it is possible to rea o the zero ynamics, see [7, Sec. 4.3] an Section 4. Consequently the relative egree or time-invariant systems linear an non-linear can by characterise by the erivatives o the output y. The purpose o the present note is to generalize the concept o relative egree to time-varying linear systems 1.1, i.e. ẋ = Atx + Btu yt = Ctxt, with A C l, n n, B, C T C l, n m an n, m, l N, an to erive a time-varying linear coorinate transormation which takes system 1.1 to a normal orm. To this en, we generalize 7

8 the concept o relative egree to time-varying nonlinear systems. The paper is organize as ollows. In Section 2 the relative egree or time-varying nonlinear systems is eine, relate to time-invariant nonlinear systems. Moreover, the relative egree or time-varying linear systems is characterise in terms o the system matrices A, B, C an its erivatives. A normal orm or time-varying linear systems with relative egree r is erive in Section 3. In Section 4 the zero ynamics o time-varying linear systems are consiere. Furthermore, einitions or bouneness an stability o zero ynamics are given an characterise using the normal orm or time-varying linear systems. 8

9 2 elative egree: einition an characterisation In this section, we will consier the concept o relative egree or time-invariant an time-varying systems. We will generalize the existing einition or time-invariant nonlinear systems, to timevarying nonlinear systems. With the generalise einition a characterisation o relative egree in terms o A, B, C will be given or time-varying linear systems. 2.1 Time-invariant nonlinear systems First, look at the einition o relative egree or time-invariant nonlinear multi-input multioutput systems, aine in the control, o the orm } ẋ = x + gxu 2.1 yt = hxt, with C l n, n, g C l n, n m, h C l n, m an l N. The strict relative egree is eine as ollows. Deinition 2.1 Let U n be open an r {1,...,l}. The time-invariant nonlinear system 2.1 has strict relative egree r on U i, an only i, i ξ U k {,...,r 2} : L g L k hξ = m m, ii ξ U : L g L r 1 hξ Gl m. This einition is ue to Isiori [7, p. 22] who eines it more general or a vector relative egree; we consier only the strict relative egree, that is ii. The original einition in [7, p. 22] eines the relative egree at a point ξ n an some neighbourhoo; however, this is equivalent to Deinition 2.1, the latter is technically easier to eal with in the ollowing. In the single-input single-output case, the notion o strict is reunant an Isiori showe that the relative egree r is exactly equal to the number o times one has to ierentiate the output yt at time t = t in orer to have the value ut o the input explicitly appearing [7, p. 139]. 2.2 Time-varying nonlinear systems Isiori s characterisation o relative egree concerning the erivatives o the output, i.e. the relative egree r N o system 2.1 is equal to the number o times one has to ierentiate the output yt at t = t in orer to have an explicitly epenence on ut, was ormalize by Liberzon et al. [1]. We exten their notion o unctions H k to time-varying nonlinear systems o the orm } ẋ = Ft, x, u 2.2 yt = Ht, xt where F C l n m, n, H C l n, m, l N, 9

10 an eine recursively, or k =, 1, 2,...,l 1, the unctions H t, x := Ht, x an H k+1 : n m k+1 m t, x, u,...,u k H k t + H k x Ft, x, u + k 1 j= H k u j u j This einition allows to express the k-th erivative o yt at t in terms o t, xt an ut,...,u k 1 t. We give the ollowing lemma. Lemma 2.2 t k {,...,l 1} : y k t = H k t, xt, ut,...,u k 1 t. 2.4 Proo: With the einition o H k, i.e. H t, x := Ht, x an 2.3, it ollows that t : y t = Ht, xt = H t, xt. Suppose that 2.4 hols or all k {,...,N}, where N l 2. Then or all t it ollows that y N+1 t = t yn t = t H Nt, xt, ut,...,u N 1 t = H N t t, xt, ut,...,u N 1 + H N xt = H N+1 t, xt, ut,...,u N t. N 1 Ft, xt, ut + j= H N u j t uj+1 t This proves 2.4. Lemma 2.2 shows the relationship between the unctions H k an the erivatives o the output y or system 2.2. Thereore, we introuce the einition o relative egree or time-varying nonlinear systems. Deinition 2.3 Let T an U n be open sets an r N. Then a system 2.2 is sai to have strict an uniorm relative egree r N on T U i, an only i, i t, x T U k {1,...,r 1} i {,...,k 1} : H k u i t, x, u,...,u k 1 = m m ; ii t, x, u T U m : H r u t, x, u,...,u r 1 Gl m. The notion strict reers to the multivariable case where we o not allow or a relative egree vector with ierent components r 1,...,r m N m, see [7, Sect. 5.1], but assume that the matrix u H r t, x, u,...,u r 1 = u H r t, x, u is invertible or all t, x, u T U m. To prove the generality o this characterisation o relative egree, it is require to show that Deinition 2.3 applie to a time-invariant nonlinear system is equivalent to Deinition 2.1 applie to the same system. We have to give a technical proo or the statement Isiori mae about the equity o relative egree an the number o times one have to ierentiate the output to have an explicitly epenence on the input. 1

11 Proposition 2.4 Let U n be an open set, r, l N with r l, an consier the timeinvariant nonlinear system 2.1, aine in the control. Then 2.1 has relative egree r on U in the sense o Deinition 2.1 i, an only i, 2.1 has relative egree r on U in the sense o Deinition 2.3. Proo: : Suppose that 2.1 has relative egree r on U in the sense o Deinition 2.1. We show by inuction on k that the ollowing hols on U omitting the arguments: k {1,...,r 1} i {1,...,k 1} : H k = L k h, H k u i = m m. 2.5 All ollowing statements will be consiere on U. We have, in view o i in Deinition 2.1, H 1 = H x + g u + H u u 1 = L h + L g hu = L h, an so H 1 u L h = m m. I 2.5 hols or k r 2, then, again in view o i in Deinition 2.1, u = H k+1 = H k x + g u k 1 H k + u j+1 = L k+1 u h + L g L k hu = L k+1 h, j= j H k+1 u i an so, or all i {1,...,k}, o Deinition 2.3 hols. To prove ii in Deinition 2.3, note that, in view o 2.5, = u i L k+1 h = m m. This proves 2.5, an thereore i H r = H r 1 x + g u r 2 H r 1 + u j+1 = L r u h + L gl r 1 hu, j= j an thus, invoking ii in Deinition 2.1, Hr u = L g L r 1 h Gl m. This proves ii in Deinition 2.3. : Suppose that 2.1 has relative egree r on U in the sense o Deinition 2.3. We show irst by inuction on k that the ollowing hols on U omitting the arguments: k {,...,r 2} : L g L k h = m m, H k+1 = L k+1 h. 2.6 Again, we consier all ollowing statements on U. We have H 1 = H x + g u + H u u 1 = h x + h x g u = L h + L g hu, an so, in view o i in Deinition 2.3, m m = H 1 u = L g h, which gives H 1 = L h. I 2.6 hols or k r 3, then, again in view o i in Deinition 2.3, H k+2 = H k+1 x + g u + = x Lk+1 h + k j= x Lk+1 H k+1 u j u j+1 hg u = L k+2 h + L g L k+1 hu, 11

12 an urthermore m m = H k+2 u = L g L k+1 h, which yiels H k+2 = L k+2 h. This proves 2.6. Finally, by 2.6, i in Deinition 2.1 hols. Applying 2.6 again gives H r = H r 1 x + g u r 2 H r 1 + u j+1. = L r u h + L gl r 1 hu, j= j an thus, invoking ii in Deinition 2.3, Hr u = L g L r 1 h Gl m. This proves ii in Deinition 2.1 an completes the proo o the proposition. emark 2.5 It ollows rom the proo o Proposition 2.4 that the relative egree o the timeinvariant system 2.1 oes not epen on t: i its relative egree is eine on T U or some open set T, then it is eine on U. Proposition 2.4 shows that it is wise to eine the relative egree o time-varying input output systems concerning the erivatives o the output y. Thus, Deinition 2.3 is a ine way to upgrae the concept o relative egree or time-varying nonlinear systems. Now, we can use this general einition, to get a characterisation o relative egree or time-varying linear systems, which is similar to the characterisation o relative egree o time-invariant linear systems, see Time-varying linear systems Consier the time-varying linear system 1.1. To give an characterisation o relative egree or this systems, we have to consier H k or any k {,...,l 1}. Applying the einition o H k to system 1.1, that is Ft, x, u = Atx+Btu an Ht, x = Ctx, gives, that omitting the arguments o A, B, C H t, x = Cx H 1 t, x = H t + H x A x + B u = Ċ + CAx + CB u H 2 t, x = H 1 t + H 1 x A x + B u + H 1 u u 1 Ċ + CAx + CB u + Ċ + CAA x + B u + CB u 1 = t = C + ĊA + CȦx + ĊB + CḂu + ĊA + CA2 x + ĊB + CABu + CB u 1 = C + 2ĊA + CȦ + CA2 x + 2ĊB + CḂ + CABu + CB u 1 hols or every t, x n. Computing H 3 t, x explicitly in terms o A, B, C is alreay complex. I we o not want to have a recursively notation or H k or any k {,...,l 1}, but a notation explicitly in terms o A, B, C, the notation above is not applicable or large k. To characterise the relative egree or time-varying linear systems 1.1 in terms o the matrixunctions A, B, C an their erivatives, we introuce the ollowing right operator. 12

13 Deinition 2.6 For l N, A C l, n n, an C C l, m n set t : t k {1,...,l} : t + At Ct := Ċt + CtAt, t + At k Ct := t + At t + At k 1 Ct. The sub-script inicates that A acts on C by multiplication rom the right. This operator is the basic tool to characterise the relative egree o time-varying linear systems. The ollowing theorem is the irst o the main results o this work. Theorem 2.7 Let r, l N with r l, an T be an open set. Then the time-varying linear system 1.1 has relative egree r on T n i, an only i, A, B, C satisy t T k =,...,r 2 : t + At k Ct Bt = m m t T : t + At r Ct Bt Gl m. To prove Theorem 2.7, we have to show that Deinition 2.3 i an ii is equivalent to 2.7 on T n. Thereore, we have to consier H k applie to the time-varying linear system 1.1 or any k {1,...,l 1}. Our irst purpose has to be a representation o H k in terms o A, B, C an their erivatives. For this, we present the ollowing lemma. Lemma 2.8 Let r, l N with r l. Then the unctions H k eine in 2.3 an applie to the linear time-varying system 1.1 satisy, or all k {,...,l 1} an all t, x, u,...,u k n m k+1, H k+1 t, x, u,...,u k = t + At k+1 k Ct x + j= Proo: Applying the einition o H k to we have k i=j i j i j [ t t + At k i ] Ct Bt u j. 2.8 Ft, x, u := Atx + Btu an Ht, x := Ctx or t, x, u n m, H 1 t, x, u = t Ctx + x Ctx[Atx + Btu ] = t + At Ct x + CtBtu = t + At 1 Ct x + j= i=j i j i j [ t t + At i ] Ct Bt u j, 13

14 which shows 2.8 or k =. To prove 2.8 by inuction over k {,...,l 1}, assume that 2.8 hols or k {,...,l 2}. Then, omitting the arguments o H k an A, B, C, we calculate H k+1 = t H k + x H k [Ax + Bu ] + k 1 u l H k u l+1 [ = t t + A k [ t + A k + x + k 1 l= u l [ k 1 k 1 C x + k 1 j= i=j l= k 1 j= i=j i j C x ] [Ax + Bu ] i j ] i j [ t t + A k 1 i ] C B u j ] i j [ t t + A k 1 i ] C B u j u l+1. Dierentiating with respect to t, x, u l or every l {,...,k 1} results in H k+1 = t t + A k + t + A k + k 1 k 1 l= i=l i l C x + k 1 k 1 j= i=j i j C A x + t + A k C B u ] C B t i l [ t + A k 1 i i j+1 [ t t + A k 1 i ] C B u j u l+1. Now, we substitute l := j + 1 in the sum in the thir line o the latter equation, which gives [ H k+1 = t t + A k C + t + A k ] C A x + t + A k C B u k 1 i j j= i=j k 1 + k 1 + k j=1 i=j 1 i j+1 [ t t + A k 1 i ] C B u j i i j+1 [ j 1 t t + A k 1 i ] C B u j. We extract the irst summan o the sum in the secon line an the last summan o the sum in the thir line o the latter equation, which allows us to merge these both sums. Thus, we obtain H k+1 = t + A k+1 C x + t + A k C B u ] C B + k 1 + k 1 i i= j=1 + k 1 i=k 1 t i+1 [ t + A k 1 i k 1 i=j [ i j + i j 1 ] t + j 1 j 1 t i k 1 t u i j+1 [ t + A k 1 i ] C B u j j 1 j+1 [ t + A k 1 j+1 ] C B u j i k+1 [ t + A k 1 i ] C B u k. Aing the coeicients at u in the irst an secon line o the latter equation, an aing the binomial coeicients, using the well known act, that i j + i j 1 = i+1 j hols or every i, j N, 14

15 see, or example, [11, Th ], gives that H k+1 = t + A k+1 k 1 C x + + k 1 j=1 k 1 i=j i+1 j C B ] i i+1 [ t t + A k 1 i u i= 1 }{{} = k È i= i t i t +Ak i i+1 j [ t t + A k i+1 C B uj + t + A k j + t + A C B uk. C B ] Substituting i := i 1 in the sum in the secon line o the latter equation yiels H k+1 = t + A k+1 k C x + i= + k 1 k j=1 i=j+1 i j i t i j [ t + A k i + j j j j t t + A k j + t + A C B uk, t i [ t + A k i C B ] C B uj u j C B u j u C B ] an thus, we combine the summans in the sum in the secon line o the latter equation, such that H k+1 = t + A k+1 k C x + i i [ t t + A k i ] C B u i= k ] C B + k 1 + k j=1 i=j k t The latter equation irectly yiels H k+1 i j i j [ t t + A k i k k t + A k k C B uk. = t + A k+1 k C x + k j= i=j i j u j u i j [ t t + A k i ] C B u j, which shows 2.8 or k + 1 an thereore the proo o the lemma is complete. Proo o Theorem 2.7: : Suppose that Deinition 2.3 hols. We show the irst conition in 2.7 by inuction over k {1,...,r 2} omitting the arguments o A, B an C. All ollowing statements will be consiere on T. For N = k = we have, by 2.8, H 1 = t + A C x + CB u, an so, by Deinition 2.3, m m = H 1 u = CB. Suppose that the irst conition in 2.7 hols or all k =,...,N, where N r 3. Then 2.8 yiels, H N+2 = t + A N+2 C x + t + A N+1 C B u, 15

16 an so, invoking i o Deinition 2.3 again, m m = H N+2 u This proves the irst conition in 2.7. = t + A N+1 C B. To see the secon conition in 2.7, note that 2.8 an the irst conition in 2.7 yiel an so, by ii o Deinition 2.3, H r = t + A r This completes the proo o 2.7. C x + t + A r 1 C B u, t + A r 1 H r C B = Gl m,. u : Suppose that 2.7 hols. Then 2.8 yiels k =,...,r 2 : H k+1 = t + A k+1 C x, an thus i in Deinition 2.3 ollows. Finally, the secon statement in 2.7 together with 2.8 gives H r = u u t + A r C x + t + A r 1 C B u = t + A r 1 C B Glm. This proves ii o Deinition 2.3 an completes the proo o the theorem. With Theorem 2.7 we have a characterisation o relative egree or time-varying linear systems, which is close in structure to the characterisation o relative egree o time-invariant linear systems. We o not nee to compute H k or every k {1,...,r} an then ierentiate the H k s with respect to u i or every i {,...,k}. We only have o compute erivatives o A an C, what is an avantage to computing H k, too equation 2.8 claims to compute erivatives o B. emark 2.9 i I ollows rom the proo o Theorem 2.7 that the relative egree o the time-varying linear system 1.1 oes not epen on x: i its relative egree is eine on some T U, where U n is open, then it is eine on T n. We thereore omit, at most places in the ollowing, the secon component in T n. ii I A, B, C are real analytic matrices an the linear system 1.1 has relative egree r on T n or some open T, then the Ientity Theorem or analytic unctions, see, or example, [5, Sec. 3.5], implies that 1.1 has relative egree r on \D n, where D enotes a iscrete set. iii I the linear system 1.1 is time-invariant, then Theorem 2.7 yiels that 1.1 has relative egree r N on n i, an only i, CA k B = m m or all k =,...,r 2 an CA r 1 B Gl m. This is the well known characterisation o strict relative egree, see [7, em ] or single-input single-output systems. 16

17 Instea o eining the relative egree o the linear time-varying system 1.1 as in Deinition 2.3, we may consier the equivalent escription o 1.1 as a time-invariant nonlinear system an etermine the relative egree accoring to Deinition 2.1. In the ollowing we will show that both einitions coincie. Introucing an aitional variable z with initial conition z =, 1.1 is equivalent to t x z = Azx 1 y = Czx. + Bz ut Proposition 2.1 Let T an U n be open sets, r, l N with r l. The time-varying linear system 1.1 has relative egree r on T U as in Deinition 2.3 i, an only i, the equivalent nonlinear time-invariant system 2.9 has relative egree r on U T as in Deinition Proo: Writing Azx x, z = 1, gx, z = Bz, hx, z = Czx, 2.1 we show by inuction over N {,...,l} that x, z U T k {,...,l} : L k hx, z = z + Az k Cz x For k = N =, we obviously have L hx, z = Czx. Suppose that 2.11 hols or all k {,...,N} or some N {,...,l 1}. Then L N+1 hx, z = L z + A N Cx = x z + A N Cx = z + A N CAx + z = z + A N+1 Cx., z z + A N z + A N Cx Cx Ax 1 This completes the proo o 2.11 an gives, or all x, z U T an all k {,..., l}, L g L k hx, z = x z + Az k Czx, z z + Az k Czx Bz = z + Az k Cz Bz Finally, Theorem 2.7 is a consequence o 2.11 an With the latter proposition we have shown an alternative way to get Theorem 2.7. In oing so we o not nee a einition o relative egree or time-varying nonlinear systems. The proo o Proposition 2.1 also gives, that the relative egree oes not epen on U, i.e. i system 2.9 has relative egree r on U T as in Deinition 2.1 or an open set U, then system 2.9 has relative egree r on T as in Deinition 2.1. This ollows rom the special structure o the unctions, g an h. With equation 2.1 ollows, that, t an h, t are linear unctions an g is inepenent on x, i.e. x gx, z = or every x n an every z T. Hence, the term L g L k x, z oes not epen on x or every k l an every z T. 17

18 3 Normal orm In this section, we erive a normal orm or time-varying linear systems 1.1 which is similar to the normal orm o time-invariant linear systems 1.5, see Section 1. Theorem 2.7 may alreay inicate that the matrix unctions t + A k C, k =,...,r 1 are caniates or a new basis, however, this potential basis nees to be complete. We introuce the ollowing matrix unctions which will serve to erive a time-varying linear transormation. Let r, l N with r l. Consier the system 1.1 an eine, or r N an t, Ct t Ct := + At Ct rm n. t + At r 1 Ct [ Bt := Bt, t At Bt,..., t At r 1 ] Bt n rm Γt := t + At r 1 Ct Bt m m. The ollowing proposition presents two more characterisations when 1.1 has relative egree r. They are rather technical but essential to esign the coorinate transormation or the normal orm. Proposition 3.1 Let r, l N with r l, T be an open set. Then the ollowing conitions are equivalent: i The system 1.1 has relative egree r on T. ii a t i, j {i, j N 2 i + j r 2} : t + At i Ct t At j Bt = m m ; iii b t i, j {i, j N 2 i + j = r 1} : t + At i Ct t At j Bt Glm ; t T : CtBt = 1 r 1 Γt Γt... Gl rm. The proo o Proposition 3.1 epens crucially on the ollowing lemma. 18

19 Lemma 3.2 The linear time-varying system 1.1 satisies, or all i, j N with i + j l an all t, an t + At i Ct t At j Bt [ = t t + At i Ct t At j 1 ] Bt t + At i Ct t At j Bt t + At i+1 Ct t At j 1 Bt, 3.1 = j 1 µ [ j µ t j µ t + At i+µ ] Ct Bt. 3.2 µ= Proo: First, we show statement 3.1. Omitting the arguments o A, B an C we have t + A i C t A j B = t + A i C t A t A j 1 B = t + A i C [ t t A j 1 B A ] t A j 1 B. [ Aing zero, i.e. we a an subtract ] t t + A i C t A j 1 B to latter equation, results in t + A i C t A j B = t + A i C t t A j 1 B A t A j 1 B [ ] t + A i C t A j 1 B + t t = t [ t = t [ ] t + A i C t A j 1 B t + A i C [ t + A i C t A j 1 B ] t + A i C + ] t + A i CA }{{} = t +A t i +A C t A j 1 B [ ] t + A i C t A j 1 B t + A i+1 which shows 3.1 an completes the irst part o the proo. [ ] t t A j 1 B C t A j 1 B, We prove 3.2 by ixing i N an inuction over j =,...,l i. For j =, 3.2 is obvious. Suppose that 3.2 hols or j l i 1. Then, invoking 3.1 an omitting the arguments o 19

20 A, B an C, it ollows that t + A i C t A j+1 B [ = t t + A i C ] t A j B t + A i+1 j = t 1 µ j j µ [ µ t t + A i+µ ] C B µ= j 1 µ j µ= µ t j µ [ t + A i+1+µ ] C B. We substitute µ := µ 1 in the secon sum, which gives t + A i C t A j+1 B = j 1 µ j µ= µ t j+1 1 µ 1 j µ 1 t µ=1 j+1 µ [ t + A i+µ ] C B C t A j B j µ+1 [ t + A i+1+µ 1 ] C B, an now, we extract the irst summan o the irst sum an the last summan o the secon sum o the latter equation, which allows us to concatenate both sums. This results in t + A i C t A j+1 B = j j+1 [ t t + A i+ ] C B + j [ 1 µ j µ 1 µ 1 j µ 1 ] j+1 µ [ t t + A i+µ ] C B µ=1 1 j+1 1 j j+1 1 t j j 1+1 [ t + A i+j+1 ] C B. Aing the binomial coeicients by using [11, Th ] yiels t + A i C t A j+1 B = 1 j j+1 [ t t + A i+ ] C B = + j 1 µ j+1 j+1 µ [ t t + A i+µ ] C B µ=1 µ + 1 j+1 j+1 j+1 [ j+1 t t + A i+j+1 ] C B j+1 1 µ j+1 j+1 µ [ t t + A i+µ ] C B, µ= which completes the proo o the lemma. µ 2

21 Proo o Proposition 3.1: Using Theorem 2.7, we show the equivalence i ii. : Let i {,...,r 1}. With 2.7 ollows t T µ {,...,r 2 i} : Thus, equation 3.2 yiels ii. t T µ = r 1 i : : Setting j = in ii we obtain 2.7. CB = t + At i+µ Ct Bt = m m t + At i+µ The equivalence ii iii ollows rom 3.2 an the act that C t + A C =. t + A r 1. t + A r 1 C Ct Bt Glm [ B, t A B,..., t A r 1 B ] CB... C t A r 1 B... C B... t + A r 1.. C t A r 1 B The equivalent characterisations o relative egree o time-varying linear systems mae in Proposition 3.1 show, that we are able to involve the erivatives o B. Nevertheless, we have to compute the same erivatives o C an A as it is require or characterisation 2.7. Corollary 3.3 I the linear time-varying system 1.1 has relative egree r N on T, T an open set, then a irect consequence o Proposition 3.1 is the ollowing: i rk Ct = rm an rk Bt = rm or all t T ; ii the two sets o matrix unctions C, t + At C,..., t + At r 1 C an B, t A B,..., t A r 1 B are both linearly inepenent over T ; iii rm n. Statement iii o Corollary 3.3 ollows rom statement i o Corollary 3.3. It is obvious, that i a matrix M rm n has rank rm, then the inequality rm n must be true. Statement ii o Corollary 3.3 shows that we are allowe to choose the matrix unctions t + A k C, k =,...,r 1 as substructure or a new basis. Because the inequality rm n may be strict, we have to complete the basis. We use Doležal s Theorem, see [13], to erive the basis. The ollowing theorem is base on [13, Theorem 2]. We prove some more results which are use in Section 4. 21

22 Theorem 3.4 Doležal s Theorem re-revisite Let M C l, n n, l N an r {1,...,n} t : rkmt = r. Then there exists a matrix unction T C l, Gl n such that t : MtTt = [Ft, n n r ], β > t : Tt β, 3.3 ε, 1 t : ε T T ttt 1 ε, 3.4 where, obviously, rkft = r or all t. Moreover, or any m {1,..., n} an any partition we have Tt = [Xt, Vt], Xt n n m, Vt n m γ > t : Vt γ, 3.5 δ, 1 t : δ V T tvt 1 δ. 3.6 To prove Theorem 3.4 we present the ollowing lemma. Lemma 3.5 Let P C l, n n an l N. Suppose that Pt is symmetric positive einite or every t. Then ε, 1 t : ε etpt an Pt 1 ε 3.7 i, an only i, δ, 1 t : δ Pt 1 δ, 3.8 Proo: ecall that δ Pt 1 δ is a short syntax or η Sn 1 : δ η T Ptη 1 δ. : Conition 3.7 gives that η S n 1 t : η T Ptη η T Pt η Pt 1 ε, which yiels the secon inequality in 3.8. It remains to show the irst equality in 3.8, i.e. δ, 1 t η S n 1 : δ η T Ptη. Since Pt is symmetric positive einite or all t there exists an orthonormal matrix Ut n n with U T tut = I n or all t such that p 1 t t : Pt = Ut... U T t. 3.9 p n t 22

23 Since Ut is orthonormal or all t the secon inequality in 3.7 yiels that p 1 t β > t :... = U T tptut p n t U T t Pt Ut β, an since Pt is positive einite or all t it ollows that β > i {1,...,n} t : < p i t β. 3.1 With the irst inequality in 3.7 we obtain p 1 t t ε etpt = et Ut... U T n t = p i t, p n t i=1 an thus, with 3.1 it ollows that ε, 1 t i {1,...,n} : ε p i t 1 ε Seeking a contraiction, suppose that η i i N S n 1 t i i N : lim i η T i Pt i η i =. It is p 1 t i = lim ηi T Pt i η i = lim ηi T Ut i... U T t i η i, i i p n t i an since Ut is orthonormal or all t it ollows that η i := U T t i η i S n 1 or all i N. With inequality 3.11 we get that = lim i η T i p 1 t i... p n t i η i lim i η T i ε... ε η i = lim i ε η T i η i = ε >, a contraiction. This shows the irst inequality in 3.8 an completes this part o the proo. : Set [e 1,...,e n ] := I n. With 3.8 we get that δ, 1 i, j {1,...,n} t : Pt i,j = e T i Pte j 1 δ, which yiels that ε, 1 t : Pt 1 ε. Consier again an orthonormal matrix Ut n n as in 3.9. It is η S n 1 t : Ut T η S n 1, 23

24 an with 3.8 it ollows that p 1 t i δ, 1 η S n 1 t : δ η T UtPtU T tη = η T... η, p n t i which gives that δ, 1 i {1,...,n} t : δ p i t an inally ε, 1 t : ε etpt. This shows 3.7 an completes the proo. Proo o Theorem 3.4 Sen et al., see [13, Theorem 2], show that there exists a matrix unction T C l, n n such that MtTt = [Ft, n n r ] or all t, where rkft = r or all t. Moreover, they show that α, 1 t : α ettt an Tt 1 α, which yiels 3.3 an thus, it ollows that T T ttt n n is positive einite or all t an obviously symmetric. Furthermore, we get that ett T ttt = ett T tettt α 2 an T T ttt = T T t Tt 1 α 2 or every t, i.e. ε, 1 t : ε ett T ttt an T T ttt 1 ε. Thus, Lemma 3.5 shows 3.4. Consier the partition T = [X, V] with X C l, n n m an V C l, n m an any m {1,...,n}. The inequality 3.5 is a irect consequence o 3.3. With 3.4 it ollows that It remains to show the irst inequality in 3.6. Seeking a contraiction, suppose that Then it ollows that = lim i η T i V T t i Vt i η i δ, 1 t : V T tvt 1 δ. η i i N S m 1 t i i N : lim i η T i V T t i Vt i η i =. = lim i 1 n m, η T i [ X T t i Xt i X T t i Vt i V T t i Xt i V T t i Vt i ] = lim i 1 n m, η T i η i T T t i Tt i η i an this contraicts 3.4. Thereore, we have shown 3.6 an this completes the proo o the theorem., 24

25 emark 3.6 Suppose that the time-varying linear system 1.1 has relative egree r N on T, T an open set. By Corollary 3.3, the rows in C qualiy as new basis vectors but the basis nees to be complete. By Doležal s Theorem, i.e. Theorem 3.4, we may choose T = [t 1,...,t n ] C 1 T, Gl n such that [ ] [ ] C F T = C 1 T, n n with F C 1 T, Gl rm. For we have For U := it ollows rom V := [t rm+1,...,t n ] C 1 T, n n rm t T : im Vt = ker Ct an rk Vt T Vt = n rm [ ] C C 1 T, n n an N := V T V 1 V T [ I BCB 1 C ] C 1 T, n rm n N that U C 1 T, Gl n an has inverse [ ] C [BCB N 1, V ] = I n, U 1 = [ BCB 1, V ] C 1 T, Gl n. We are now in a position to erive the main result o this note, that is a normal orm o the time-varying linear system 1.1. Theorem 3.7 Let r, l N with r l an T be an open set. Suppose the time-varying linear system 1.1 has relative egree r on T an choose U C 1 T, Gl n, V C 1 T, n n r, an N C 1 T, n rm n as in emark 3.6. Then the coorinate transormation ξ := Ux, ξt = yt, T...,y r 1 t T T rm, ηt n rm η converts 1.1 on T into t ξ = Ât + η ˆBtut ξ y = Ĉt η ξ η 3.13 where Ât = I... m m I, ˆBt = m m, 1 t 2 t... r t St t + At r 1 Ct Bt Pt... Qt n rm m Ĉt = [ I... ], 25

26 an 1,..., r CT, m m, S CT, m n rm, P CT, n rm m, as well as Q CT, n rm n rm. Proo: The special orm o U an U 1 gives immeiately [ ˆB = UB, ĈU = C, an  = UA + U ] U 1 = t + A U U 1. Thereore, it remains to show that t + A C. t + A r 1 C t + A r C t + A N = I [ ] C I N r S P 1 P 2... P r Q 3.14 with P 2 =... = P r =. Since equality o the upper blocks in 3.14 is immeiate, it remains to show that t + A N BCB 1 = [P 1,,...,] Writing we see that CB = [ η 1,...,η r ], CB 1 = ψ 1. ψ r, B = [ ] β 1,...,β r BCB 1 C t A B = BCB 1 C [ β 2,...,β r, t A r B ] ψ 1 = [ ] β 1,...,β r. ψ r [ η 2,...,η r, ]... = [ I... ] β 1,...,β r. I.... = [ β 2,...,β r, ], I an thus [ BCB 1 C I n ] t A B = [ β 2,...,β r, ] I n [ β2,...,β r, t A r B ] which, by invoking t + A N = [,...,, ], 3.16 = t V T V 1 V T I BCB 1 C + V T V 1 V T A t + A BCB 1 C,

27 implies t + A N B = [ t V T V 1 V T I BCB 1 C + V T V 1 V T A t + A BCB 1 C ] B = V T V 1 V T [ A t + A BCB 1 C ] B [ = V T V 1 V T A BCB 1 C B t CB 1 C BCB 1 C BCB 1 CA ] B [ = V T V 1 V T A BCB 1 C + BCB 1 t CBCB 1 C ] BCB 1 C BCB 1 CA B [ = V T V 1 V T A BCB 1 C + BCB 1 CBCB 1 C + BCB 1 CBCB ] 1 C BCB 1 C BCB 1 CA B [ = V T V 1 V T AB B + BCB 1 CB + BCB 1 CB ] BCB 1 CB BCB 1 CAB = V T V 1 V T [ BCB 1 C I n ] t A B 3.18 = V T V 1 V T [,...,, ] Finally, CB 1 = 1 r 1 Γ 1... Γ 1, applie to 3.19 yiels This completes the proo o the theorem. Exploiting the special structure o the coorinate transormation, the matrices P an Q in 3.13 may be expresse in terms o A, B, C an the matrix unction V eine in emark 3.6. Furthermore, we will express S in terms o A, B, C an V. Proposition 3.8 Uner the assumptions o Theorem 3.7 an using the notation introuce in emark 3.6, the normal orm 3.13 may also be written as y r [ ] ξ = t η η = Qtη + Γtut + Pty,

28 where := [ 1,..., r, S] [ = [ m rm m, I m ] t + A r C BCB 1, t + A r ] C V [ = t + A r C BCB 1, t + A r ] C V 3.21 Γ = t + A r 1 C B, 3.22 Q = t + A N V = [ V T V 1 V T t A V BΓ 1 ] t + A r C V, 3.23 I m P = t + A N B CB 1. = 1 r 1 V T V 1 V T [BCB 1 C I] t A r BΓ Proo: First, we prove Since  = t + A U U 1 we get an with the einition o C ollows = [ m rm m, I m ] t + A C [ BCB 1, V ], The irst equalities in 3.23 an 3.24 ollow immeiately rom the normal orm To see the secon equality in 3.23 note that 3.17 yiels t + A NV = [ [ t V T V 1 V T][ I n BCB 1 C ] V T V 1 V T [ t BCB 1 C ] = t = t + V T V 1 V T A V T V 1 V T BCB 1 CA ] V [ V T V 1 V T][ V BCB 1 ] }{{} CV + V T V 1 V T AV = V T V 1 V T [ t BCB 1 C + BCB 1 C A ] V [ V T V 1 V T] V + V T V 1 V T AV }{{} = t +A VT V 1 VV V T V 1 V T[ BCB 1 }{{} CV = +B t CB 1 }{{} CV +BCB 1 CV + BCB 1 C ] AV, = which gives t + A NV = t + A V T V 1 V V V T V 1 V T BCB 1 t + A C V. 28

29 Invoking t + A V T V 1 V T V = t V T V 1 V T V + V T V 1 t VT V + V T V 1 V T AV = V T V 1 t V T V V T V 1 V T V + V T V 1 t VT V + V T V 1 V T AV = V T V 1 t VT V V T V 1 V T t V + VT V 1 t VT V + V T V 1 V T AV = V T V 1 V T t A V, we arrive at t + A N V = V T V 1 V T t A V V T V 1 V T BCB 1 t + A C V = V T V 1 V T t A V V T V 1 V T BCB 1 = V T V 1 V T t A V This proves the secon equality in t + A r C V... Γ 1 V T V 1 V T B 1 r 1 Γ 1. t + A r C V = V T V 1 V T t A V [ Γ 1 V T V 1 V T B,..., t ] + A r C V t A r 1 B. [ = V T V 1 V T t A V B Γ 1 ] t + A r C V. To see the secon equality in 3.24, recall the equation 3.18 which gives Set B = [β 1,...,β r ]. Thus, we get [P,,...,] = V T V 1 V T [ BCB 1 C I n ] t A B. [P,,...,] = V T V 1 V T [ BCB 1 C I n ] [ β2,...,β r, t A r B ] 29

30 an, in the view o 3.16, [ P = V T V 1 V T,...,, [ BCB 1 ] ] C I n t A r B. 1 r 1 Γ 1 = 1 r 1 V T V 1 V T [ BCB 1 C I n ] t A r BΓ 1. This completes the proo o the proposition. Corollary 3.9 I the system 1.1 is time-invariant an o relative egree r, then all matrices erive in emark 3.6 are also time-invariant an the normal orm is y r = CA r [ BCB 1, V ][ ] ξ + CA η r 1 B ut η = N A V η + 1 r 1 V T V 1 V T [ BCB 1 ] C I n A r B CA r 1 B y. Proo: The normal orm 3.25 or time-invariant linear systems is a irect consequence o Proposition 3.8. I we consier a time-invariant linear system an compute the normal orm using the timevarying results, i.e. Theorem 3.7, then Corollary 3.9 shows that we arrive at the well known normal orm or time-invariant linear systems, see, or example, [6, Sec. 3]. 3

31 4 Zero ynamics In this section we analyze an characterise the concept o zero ynamics or ynamical systems. First, consier single-input single-output time-invariant non-linear systems. 4.1 Zero ynamics o time-invariant nonlinear systems Consier system 1.7. Suppose that 1.7 have relative egree r l at some points o interest x, e.g. an equilibrium point o 1.7. Isiori shows, see [7, Sec. 4.1], that there exists a neighbourhoo U n o x an a ieomorphism Φ : U n, which transorms 1.7 uner ξ T, η T T = Φx to the normal orm 1.9, i.e. ξ r 1 ξ 1 = ξ 2 ξ 2 = ξ 3 ξ r. = ξ r = L r hφ 1 ξ, η + L g L r 1 hφ 1 ξ, ηu η = qξ, η y = ξ 1, or some q C l n, n r. The ieomorphism Φ = φ 1,...,φ n T ulils ξ 1 ξ 2 = φ 1 x = hx = φ 2 x = L hx ξ r. = φ r x = L r 1 hx or the irst r new coorinates. Furthermore, Φ has a Jacobian matrix unction, which is nonsingular at U, i.e. or all x U it is x Φx Gl n. Thereore the value o the aitional n r unctions φ r+1,...,φ n can be chosen arbitrarily. Moreover, it is always possible to choose φ r+1,...,φ n in such a way that see [7, Sec. 4.1]. x U i {r + 1,...,n} : L g φ i x =, 4.1 I x is such that x = an hx =, then ξ = at x. Since we may choose arbitrarily the value o η at x, set η = at x. Thereore, without loss o generality, we assume that ξ = an η = at x an thus, i x is an equilibrium or system 1.7, its corresponing point ξ, η =, is an equilibrium or the normal orm 1.9. With x = ollows L r hφ 1, = L r hx = an with 4.1 we get q, =. Now, we want to in pairs x, u C 1 T, n CT,, solving 2.1 on an open set T such that the corresponing output y o system 1.7 is ientically zero on T. Isiori calls this problem: the Problem o Zeroing the Output. We will call the tuples consisting o solution an input o the system, which solve the problem o zeroing the output Zero ynamics o the associate system. 31

32 O course, the trivial pair x = x, u =, where x is an equilibrium o system 1.7, i.e. x = an hx =, solve the problem. However, we are intereste in ining not only these trivial solution. ecalling the normal orm 1.9 o system 1.7, we euce rom = yt = ξ 1 t or all t T, that ξ i t = ξ i 1 t = or all t T an all i {2,...,r}. Thus, we have ξt = or all t T. Thereore, the normal orm gives, that ut must be the unique solution o the equation = L r hφ 1, ηt + L g L r 1 hφ 1, ηtut. Since we assume that system 1.7 has relative egree r at xt = x, there exists an open set T with t T, such that L g L r 1 hφ 1, ηt or all t T. Furthermore, the normal orm 1.9 o system 1.7 gives that η solves the ierential equation η = q, η. From this analysis we obtain the ollowing: a pair x, u C 1 T, n CT, which ulils that the output y is zero on T, has to satisy x = Φ 1, η, i.e. ξt = or every t T an where η solves ut = Lr hφ 1, ηt L g L r 1 hφ 1, ηt, t T η = q, η, t T. These results or time-invariant nonlinear systems are well known, see, or example, [7, Sec. 4.3]. Next we will give a einition o zero ynamics in a more general way or any input output system, time-invariant or time-varying. 4.2 Zero ynamics o time-varying systems Consier the time-varying nonlinear system 2.2, i.e. ẋ = Ft, x, u yt = Ht, xt, where F C l n m, n, H C l n, m an l N. We eine the zero ynamics o system 2.2 as ollows. Deinition 4.1 Consier system 2.2 an an open set T. We call the set { } ZD T F, H := x, u C 1 T, n CT, m x, u satisies 2.2 an t T : yt = the zero ynamics o system 2.2 on T. 32

33 emark 4.2 i In Deinition 4.1 we consier x C 1, n an u C,. It is also possible to consier piecewise continuous inputs, i.e. u C pw, m. Then the irst erivative o the solution is also piecewise continuous, what yiels, that x / C 1, n, but piecewise continuous ierentiable. However, we are not intereste in the smoothness o input o system 2.2, an assume, that x an u are suiciently smooth. ii Subject to the system one consiers, we write or example ZD T, g, h or a system o the orm 1.7, or ZD T A, B, C or a system o the orm 1.1. Now, consier the time-varying linear system 1.1. Suppose that 1.1 have relative egree r l on an open set T. With the coorinate transormation ξ := Ux, ξt = yt, T...,y r 1 t T T rm, ηt n rm, η where U C 1 T, n n is eine by emark 3.6, an Theorem 3.7 we have a normal orm o system 1.1, i.e. I ξ ξ t = η... I + η ut t 2 t... r t St Γt Pt... Qt yt = [ I... ] ξt ηt where 1,..., r CT, m m, S CT, m n rm, P CT, n rm m, an Q CT, n rm n rm an Γt = t + At r 1 Ct Bt m m. The normal orm 4.2 o system 1.1 gives the ollowing proposition or the zero ynamics o time-varying linear systems. Proposition 4.3 Assume that system 1.1 has relative egree r l on an open set T. Consier the matrix unctions U = V T V 1 V T [ I BCB 1 C ] an V C 1 T, n n rm which are eine by emark 3.6, an let Γ = t + A r 1 C B, S = t + A r C V an [ Q = V T V 1 V T t A V BΓ 1 t + A r ]. C V Then x, u ZD T A, B, C x, u = U 1, Γ 1 S η, η where η C 1 T, n rm solves η = Qtη, t T. 33

34 Proo: Since system 1.1 has relative egree r on T, Theorem 3.7 in combination with the ξ coorinate transormation = Ux yiels the normal orm 4.2. A tuple x, u η ZD T ξt A, B, C ulils = yt = Ctxt = CtUt 1 = ξ ηt 1 t or all t T. With 4.2 ollows ξt = ξt = or all t T. Thereore, x = U 1. With ξ = η ollows, that the input u must solve the equation t T : = Stηt + Γtut, where S an Γ are given by the normal orm 4.3, see Proposition 3.8. This gives ut = Γt 1 Stηt or all t T. Since ξt = or all t T, equation 4.2 gives, that η solves η = Qtη, t T, an Proposition 3.8 completes the proo. Suppose that x, u = U 1, Γ η 1 S η hols or the conitions mae in the proposition. Then yt = Ctxt = CtUt 1 = or all t T. Thereore ηt x, u ZD T A, B, C. Proposition 4.3 gives a characterisation or x, u ZD T A, B, C that contains the solution o the corresponing normal orm 4.2. We have x = U 1 T, η T on T where η solves η = Qtη or all t T. Thus, the solution o system 1.1 can be receive rom the solution o the timevarying linear ierential equation η = Qtη. This is an avantage or urther analysis. 4.3 Stability o zero ynamics o time-varying linear systems This section eals with the stability-behaviour o zero ynamics o time-varying linear systems. First, we eine various concepts o stability or zero ynamics or any time-varying system, linear an non-linear. Deinition 4.4 Consier a time-varying system 2.2 an an open set T with [t, T. The zero ynamics o 2.2 on T are calle i boune i, an only i, x, u ZD T F, H : x L T, n, ii asymptotically stable i, an only i, the zero ynamics o 2.2 on T are boune an iii exponentially stable at t i, an only i, lim xt =, t x, u ZD T F, H λ, c > t t : xt c e λ t t xt. 34

35 Consier system 4.2 eine on an open set T with [t, T or t, i.e. consier the normal orm o a time-varying system as stan-alone system. The zero ynamics o this system, i.e. all tuples ξ T, η T T, u ZD T Â, ˆB,Ĉ, where Â, ˆB,Ĉ are eine by Theorem 3.7, ulil t T : ξt =, ut = Γt 1 Stηt, η = Qtη. Thereore, the zero ynamics o system 4.2 are boune i, an only i, the solution o the time-varying ierential equation η = Qtη is boune or every t T. Next we recall the einition o stability or time-varying linear ierential equations. Deinition 4.5 See, or example, [12, Sec. 6]. The time-varying linear ierential equation o the orm η = Qtη 4.3 where Q C 1, m m or m N is calle i stable i, an only i, there exists γ > such that any solution o 4.3, satisies t t t : ηt γ ηt. ii asymptotically stable i, an only i, it is uniormly stable an or every δ > exists T > such that any solution o 4.3 satisies t t t + T : ηt δ ηt. iii exponentially stable i, an only i, there exists c, λ > such that any solution o 4.3 satisies t, t, t t : ηt c e λt t ηt. Suppose that system 1.1 has relative egree r l on an open set T with [t, T or t. With Proposition 4.3 we get xt = Ut 1 = [ BtCtBt ηt 1, Vt ] = Vtηt, η an thereore ηt = V T tvt 1 Vt T xt. Now, we will show, that the zero ynamics o system 1.1 are boune i, an only i, the zero ynamics o its normal orm 4.2 are boune. We nee to recall Doležal s Theorem, see Section 3. Theorem 3.4 shows that the matrix unction V T V C 1, n rm n rm eine by emark 3.6 is boune below an above on any open set T, i.e. there exists δ, 1 such that δ V T tvt 1 δ or all t T an there exists γ > such that Vt γ or all t T. We present the ollowing theorem. Theorem 4.6 Consier the time-varying linear system 1.1 an an open set T with [t, T or t an Q as in equation Suppose, that system 1.1 has relative egree r l on T. Then, the zero ynamics o system 1.1 on T are boune asymptotically stable, exponentially stable, respectively i, an only i, the time-varying linear ierential equation η = Qtη is stable asymptotically stable, exponentially stable, respectively. 35

36 Proo: Deinition 4.5, Proposition 4.3 an the ollowing proposition prove the corollary. Proposition 4.7 Consier the time-varying linear system 1.1 an an open set T. Suppose that system 1.1 has relative egree r l on T. Then the zero ynamics o system 1.1 on T are boune asymptotically stable, exponentially stable, respectively i, an only i, the zero ynamics o the corresponing normal orm 4.2 are boune asymptotically stable, exponentially stable, respectively. Proo: With Proposition 4.3 an the coorinate transormation [ ξ T, η T] T = U x it ollows that x = Vη, η = V T V 1 V T x, on T. We have to show that V an V T V 1 V T are boune on T, i.e. V L T, n n rm an V T V 1 V T L T, n rm n. Theorem 3.4 gives that there exists γ > such that Vt γ or every t T, which yiels that V L T, n n rm. Moreover, Theorem 3.4 gives that there exists δ, 1 such that δ V T tvt 1 δ or all t T. Since VT tvt is positive einite or all t T there exists an orthnormal matrix Wt n rm n rm such that It ollows that t T : W T tv T tvtwt = v 1 t... v n rm t ε, 1 i {1,...,n rm} t T : ε v i t 1 ε,. an thus, it is This gives t T which yiels that i {1,...,n rm} t T : ε v i t 1 1 ε. v 1 t 1 : V T tvt 1 = Wt... W T t, v n rm t 1 ε, 1 t T : ε V T tvt 1 1 ε. Thereore we have shown that V T V 1 V T L T, n rm n which completes the proo o the proposition. Theorem 4.6 gives a characterisation o stability or time-varying systems, which mainly epens on the matrix unction Q. Because o that Q contains erivatives o A, C, V an the inverse o Γ = t + A r 1 C B, there is no simple rule to choose A, B, C an V to have a asymptotically or exponentially stable system η = Qt η. Thus, this work oes not give more etaile inormation how to chose A, B, C an the transormation V to have boune or stable 36