Discrete-Time Modal State Reconstruction for Infinite-Dimensional Systems Using Generalized Sampling

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1 Discrete-Time Modal State Reconstruction for Infinite-Dimensional Systems Using Generalized Sampling Christian Harkort and Joachim Deutscher Chair of Automatic Control, University of Erlangen-Nuremberg, Cauerstraße 7, D-9158 Erlangen, Germany ( Abstract: In this contribution a novel approach for discrete-time state reconstruction for infinite-dimensional systems is presented. It is shown that an exact reconstruction of a finite number of modal states of the infinite-dimensional system can be obtained at the sampling time instances. This is achieved by aid of generalized sampling which generates the output samples by weighted averaging over one sampling interval. The reconstruction is done in finite time and is thus non-asymptotic. Since the method makes it possible to implement state feedback controls without involving a Luenberger observer, the spillover problem arising from the early-lumping approach can be avoided. Keywords: non-asymptotic state reconstruction, generalized sampling, sampled-data control, state feedback control, infinite-dimensional system 1. INTRODUCTION The implementation of state feedback controls commonly involves a Luenberger observer for the state reconstruction. If the system to be controlled is infinite-dimensional, the observer is often designed on the basis of an approximation in order to yield a finite observer order. An overview of this so-called early-lumping approach is given in Balas (1982). This approach bears the inherent problem of spillover since the neglectedsystem dynamics may affect the closed-loop dynamics in an undesired way. In the case of sampled-data control this problem can be overcome by the presented approach. It is shown that discrete-time state reconstruction for infinite-dimensional systems can be done by generalized sampling as an alternative for the Luenberger observer. This hybrid approach, where the output samples are obtained by weighted averaging over the last sampling interval, has been studied in Logemann et al. (25), wherein the existence of stabilizing output feedback controls has been analyzed. In the present article generalizedsampling is used to reconstructa finite number of modal states of the infinite-dimensional system at the sampling time instances, whereas it is required that the corresponding eigenvalues are modal observable. If these are also modal controllable, they can then be shifted arbitrarilybystate feedback, whereas no Luenbergerobserver is needed. Hence, the spillover problem is overcome. Another advantage is that the reconstruction is achieved nonasymptotically in finite time. The paper is organized as follows. In Section 2 the discretetime state feedback controlofinfinite-dimensional systems is briefly reviewed. In Section 3 the concept of state reconstructionby use of generalizedsampling is introduced. It turns out that the problem to determine a suitable sampling function used for the sampling process can be reduced to a certain feedforward control problem. This leads to the design procedure developed in Section 4. In order to present the basic ideas without technical complications the class of state linear systems is considered in the Sections 2 4which is restrictedto distributed controland distributed measurement. For the case ofboundary control and boundary measurement the approach is extended to the large class of regular linear systems in Section 5. Finally, the method is demonstrated for the control of a one-dimensional heat conductor in Section DISCRETE-TIME STATE FEEDBACK CONTROL Consider a state linear system described by ẋ(t) = Ax(t) + Bu(t), t >, x() D(A) (1) y(t) = Cx(t) (2) with scalar input u(t) R, scalar output y(t) R and state x(t) X, where the state space X is a Hilbert space with the inner product, and induced norm. For details about suchsystems see Curtain andzwart (1995). The input operator B and output operator C are bounded linear operators, i.e., B L(C, X) and C L(X, C), whereas L(V, W) denotes the space of linear operators from the Banach space V to the Banach space W that are bounded with respect to the operator norm, and have the representations Bv = b v, b X, v C (3) Ch = h, c, c X, h X (4) (for (4) see the Riesz representation theorem in e.g. Curtain and Zwart (1995)). The system operator A : D(A) X X is the infinitesimal generator of the C -semigroup (S(t)) t. Copyright by the International Federation of Automatic Control (IFAC) 13311

2 The approach in this contribution will be developed under the following assumptions: (A1) The spectrum σ(a) of A consists solely of isolated simple eigenvalues so that σ(a) is discrete. (A2) Only n u eigenvalues λ 1, λ 2,...,λ nu lie in the closed right half-plane while all the remaining eigenvalues λ i, i > n u, satisfy sup i>nu Re λ i <. (A3) The eigenvalues λ 1, λ 2,..., λ n, n n u, are modal controllable and modal observable (see Curtain and Zwart (1995)). (A4) A is boundedly invertible, i.e., belongs to the resolvent set ρ(a). A consequence of A1 is that for each eigenvalue λ i there is a single eigenvector φ i of A which is biorthonormal to the eigenvectors ψ j of the adjoint operator A that correspond to the eigenvalues λ j, i.e., φ i, ψ j = δ ij with δ ij denoting the Kronecker delta. Note, that there is no need that the eigenvectors φ i form any sort of basis for X. Assumptions A2 and A3 assure that system (1) is stabilizable by output feedback. Finally, A4 could be relaxed but is assumed for the sake of a clear presentation. In the sequel, system (1) (2) shall be controlled by discrete-time state feedback. The discrete-time system representation and the design of discrete-time state feedback controlare well-knownsubjects that willbe reviewedonly briefly. For details see Rosen and Wang (1992); Logemann (1992). The continuous-time input signal u : [, ) R is generated by a zero-order hold according to u(t) = u k for t [t k, t k+1 ), (5) where u k are instances of the control sequence (u k ) k N, and t k = kt constitutes the sampling time sequence (t k ) k N with sampling constant T. Solving (1) under use of the zero-order hold (5) leads to +1 x(t k+1 ) = S(T)x(t k ) + S(t k τ)bdτ u k, (6) t k which can be written as the difference equation x(t k+1 ) = A d x(t k ) + B d u k (7) with A d = S(T) (8) B d = S(T τ)bdτ = (S(T) I)A 1 B, (9) where B d can be identified with b d X by B d u(t) = b d u(t). As is known from the finite-dimensional case, the eigenvalues λ d,i of A d are related to the eigenvalues λ i of A by λ d,i = e λit, i N (see (Luo et al., 1999, Thm. 3.4)). Thus, the eigenvalues λ 1, λ 2,..., λ nu of A in the closed right half-plane are mapped onto eigenvalues λ d,1, λ d,2,..., λ d,nu of A d outside the unit circle. The discrete-time system (7) is called power stable if there exist M 1 and γ (, 1) such that A k d Mγk k N (1) (see Logemann (1992); Curtain and Zwart (1995)). This implies that x(t k ) decays exponentially toward zero for (u k ) k N = because in this case x(t k ) = A k dx(), k N, follows from (7) so that x(t k ) A k d x(). It is a known fact that (1) is satisfied if and only if sup λ σ(ad ) λ < 1 holds (see (Logemann, 1992, Lem. 1)). For the stabilization of (7) it is therefore necessary to shift those eigenvalues λ d,1, λ d,2,..., λ d,nu that are located outside the unit circle into its interior by the controller. In fact, it follows from (Rosen and Wang, 1992, Lem. 3.1) that this is also sufficient for the discrete-time closed-loop system to be power stable. Moreover, if the discrete-time system (7) is power stabilized, then the continuous-time system (1) is stabilized exponentially so that x(t) for t (see (Rebarber and Townley, 1997, Lem. 2.3)). From the continuous-time case it is well-known that in order to shift the eigenvalues λ 1, λ 2,...,λ n, which contain the ones in the closed right half-plane, by state feedback, only the corresponding modal states ξ i (t) = x(t), ψ i, (11) i = 1, 2,...,n, are needed for the feedback (see Bradshaw and Porter (1972)). It is straightforward to check that the same is true for the discrete-time control so that the control law u k = k T ξ(t k ) (12) with ξ(t) = [ξ 1 (t)... ξ n (t)] T and k C n is applied in the sequel. The remaining eigenvalues λ d,i, i > n, are then not affected by the feedback (12). Introducing Ψ : X C n by, ψ 1 Ψ =. (13), ψ n the feedback law can be expressed as u k = k T Ψx(t k ) (see (11) (13)) which, inserted into (7), yields x(t k ) = à d x( ) with the closed loop system operator Ãd = A d B d k T Ψ. For the solution of the eigenvalue assignment problem, namely the computation of k such that the eigenvalues λ d,1, λ d,2,...,λ d,n of A d are shifted to the desired locations λ d,1, λ d,2,..., λ d,n σ(ãd) while the remaining eigenvalues λ d,i, i > n, appear unaffected also in σ(ãd), the following statement is helpful. Theorem 1. Suppose that the eigenvalues λ d,1,..., λ d,n of A d are mutually different and different from all λ d,i, i > n. Then, the spectrum of the closed-loop system operator à d = A d B d k T Ψ is given by σ(ãd) = σ(a dn b dn k T ) σ(a d ) \ {λ d,1,...,λ d,n } (14) with where A dn = e AnT, b dn = e Anτ b n dτ, (15) b, ψ 1 A n = diag(λ 1,..., λ n ), b n =.. (16) b, ψ n The proof is straightforward and omitted due to a lack of space. Although the eigenvalues λ 1, λ 2,...,λ n are assumed to be modal controllable (see A3) it may happen that (A dn, b dn ) is not controllable for a finite number of discrete values ofthe sampling constantt. Necessaryandsufficient conditions for (A dn, b dn ) to be stabilizable are provided in Rosen and Wang (1992). If (A dn, b dn ) has this property, the discrete-time system (7) can be stabilized by an appropriate choice of k because {λ d,i, i > n} is contained in the inner of the unit circle as implied by Assumption A2. Thus, Theorem 1 shows that the eigenvalue assignment problem in the infinite-dimensional space X can be carried 13312

3 out in the space C n by assigning the desired eigenvalues λ d,1, λ d,2,..., λ d,n to A dn b dn k T C n n. This allows to compute k by the common tools for controller synthesis of lumped parameter systems. 3. STATE RECONSTRUCTION BY MEANS OF GENERALIZED SAMPLING It is apparent from (12) that the vector ξ(t) of modal states is needed at the sampling time instances t k, k N, for the implementation of the state feedback control. As explained in the introduction, the classical state reconstruction by a Luenberger observer is often unsatisfactory due to the problems caused by spillover (see Balas (1982)). In this section it is shown that a finite number of modal states ξ i (t k ) can be reconstructed by aid of the generalized sampling law ỹ k = g(τ )y(τ)dτ (17) so that no Luenbergerobserveris needed. In order to gain insight how the sampling function g(t) = [g 1 (t)... g n (t)] T with g i : [, T] C is to be chosen for this purpose, the output y(t) is expressed explicitly as t y(t) = CS(t )x( )+C S(t τ)bu(τ)dτ (18) for t [, t k ] (compare to (6)). By aid of (5) it follows t S(t τ)bu(τ)dτ = S(t τ)a 1 Bu k 1 t = A 1 Bu k 1 + S(t )A 1 Bu k 1 (19) (see (Curtain and Zwart, 1995, Thm )), where A is invertible by Assumption A4. Using this in (18) gives y(t) = CS(t )(x( ) + A 1 Bu k 1 ) CA 1 Bu k 1, t [, t k ]. (2) The sampled output ỹ k (17) reads with (2) ỹ k = where G = g(τ ) CS(τ )dτ (x( ) + A 1 Bu k 1 ) G u k 1, (21) g(τ )dτ CA 1 B = By defining C : X C n as C = (21) becomes g(τ)dτ CA 1 B. (22) g(τ ) CS(τ )dτ (23) ỹ k = Cx( ) + ( CA 1 B G )u k 1. (24) Since it is the aim to reconstruct the modal states ξ i (t) = x(t), ψ i, i = 1, 2,...,n, it is desired to have C = Ψ (25) because due to Cx(t) = Ψx(t) = ξ(t) (see (11) and (13)) it then follows ξ( ) = ỹ k (ΨA 1 B G )u k 1 (26) from (24). Since ỹ k and u k 1 in (26) are known variables, this allows to compute the vector ξ of the modal states at the time instances, k N. In fact, the sampling function g will be designed in Section 4 such that (25) holds. Since ξ(t k ) is needed for the feedback law (12) rather than ξ( ), the relation between ξ( ) and ξ(t k ) is derived next. Projecting x(t) onto the eigenspace span{φ 1,...,φ n } and application of, ψ i on both sides of (1) gives ξ i (t) = λ i ξ i (t) + b, ψ i u(t) when the eigenvalueeigenvector equation Aφ i = λ i φ i is taken into account. For i = 1, 2,...,n this can be written as ξ(t) = A n ξ(t) + b n u(t) (27) with A n and b n from (16). Consequently, it follows ξ(t k ) = e AnT ξ( ) + e Anτ b n dτ u k 1, (28) where u(t) = u k 1 for t [, t k ) due to the zero-order holdhas been used. Thus, insertionof (15) into (28) gives ξ(t k ) = A dn ξ( )+b dn u k 1, which, by aid of (26), leads to ξ(t k ) = A dn ỹ k + (b dn A dn (ΨA 1 B G ))u k 1, (29) where the right-hand side can be computed because ỹ k and u k 1 are available. Thus, the modal states ξ(t k ), that are needed for the state feedback control (12), are readily obtained from (29), whereas the mentioned relation (25) has to hold. It is shown in the next section how the sampling function g has to be designed such that (25) is satisfied. 4. DESIGN OF THE SAMPLING FUNCTION The design of the sampling function g(t) can be formulated as a trajectory planning problem. This becomes obvious when the adjoint equation C = S (τ ) C g (τ )dτ (3) of (23) is considered. Therein and throughout the paper T denotes the adjoint of a densely defined operator T. In view of (13) it can be verified easily that Ψ = [ψ 1... ψ n ] (31) holds. Thus, by introducing g T (t) = g (T t) = [g 1 (T t)... g n (T t)] equation (3) becomes with (25) [ψ 1... ψ n ] = S (τ ) C g T (t k τ)dτ, (32) where S (t) is the C -semigroup that is generated by A. Finally, applying the substitution τ = τ yields [ψ 1... ψ n ] = S ( τ) C g T (T τ)d τ. (33) When (33) is read column-wise under use of g T (t) = [ g 1 (t)... g n (t)] one can regard the right hand-side of (33) as an input-state map with input g i, input operator C, and system operator A. This means that g i has to be determined such that the adjoint system ż(t) = A z(t) + C g i (t) (34) (see Bensoussan et al. (27)) has a solution z satisfying z() =, z(t) = ψ i, (35) where C v = cv for v C because of (4). Thus, the problem to find g(t) = g T (T t) has been transfered to the design of the feedforward controls g i, i = 1, 2,...,n, that steer z from to ψ i. The results derived so far are summarized in the following theorem

4 Theorem 2. Suppose that the functions g i : [, T] C, i =1, 2,...,n, are chosensuchthat (34) (35) hold. When the sampling function g(t) = g 1 (T t).. g n (T t) is used for the generalized sampling ỹ k = (36) g(τ )y(τ)dτ, (37) then the vector of modal states ξ(t k ) = [ξ 1 (t k )... ξ n (t k )] T is given by ξ(t k ) = A dn ỹ k + Gu k 1, k N (38) with G = b dn + A dn (G ΨA 1 B). Note, that this means that the modal states ξ(t k ) are reconstructed in finite time, i.e., non-asymptotic reconstruction is achieved. In what follows, g i is designed by differentially parameterizing the system variables of (34) by a basic variable (see Rudolph andwoittennek (28)). When z(t) is decomposed into z(t) = ζ i (t)ψ i + z r,i (t) with the modal state ζ i (t) = z(t), φ i and the residual state z r,i (t) = z(t) ζ i (t)ψ i, then it follows from (35) ζ i () =, ζ i (T) = 1 (39) z r,i () =, z r,i (T) =. (4) For the construction of a basic variable the transfer behavior from g i to z r,i is determined in the frequency domain. Application of the Laplace transform L to (34) for the homogeneous initial condition (35) and a simple rearrangement give (si A )ẑ(s) = C ˆ g i (s), Re(s) > β, (41) where ẑ = L {z} and ˆ g i = L { g i } (for β see Curtain and Weiss (1989)). Solving (41) leads to the transfer function F(s) = ẑ(s)/ˆ g i (s) = (si A ) 1 C, s ρ(a ), which has the following form. Lemma 3. F(s) = N(s) D(s), s ρ(a ), (42) where N : C X and D : C C are entire functions. The proof is given in Appendix A. For the case where (41) is a boundary value problem with one spatial variable it is demonstrated in Rudolph and Woittennek (28) how to determine N and D. The dynamics of ζ i is described by ζ i (t) = λ i ζ i (t) + c, φ i g i (t) (43) which is obtained by projecting z(t) onto the eigenspace span{ψ i } and application of, φ i on both sides of (34) when A ψ i = λ i ψ i and C g i (t) = c g i (t) are taken into account. Note, that c, φ i, i = 1, 2,...,n, due to the Assumption A3 that λ i is modal observable. Application of the Laplace transform to (43) for the homogeneous initial condition (39) yields ˆζ i (s) = F ζ,i (s)ˆ g i (s) with F ζ,i (s) = N ζ,i D ζ,i (s) = c, φ i (44) s λ i and thus ẑ r,i (s) = ẑ(s) ˆζ i (s)ψ i = (F(s) F ζ,i (s))ˆ g i (s), (45) whereas the transfer function therein can be written as F(s) F ζ,i (s) = N r,i(s) D r,i (s) = N(s)D ζ,i(s) N ζ,i D(s) (46) D(s)D ζ,i (s) (see (42) and (44)). Introduction of the fictitious output p i (s) = ˆ g i (s)/d r,i (s) yields the parameterizations ˆ g i (s) = D r,i (s)p i (s) (47) ẑ r,i (s) = N r,i (s)p i (s) (48) in view of (45) (46) which shows that p i is a basic variable (see Rudolph and Woittennek (28)). Since N, D and D ζ,i are analytic (see Lemma 3 and (44)) the same is valid for N r,i and D r,i due to (46) so that the Taylor series D r,i (s) = j= a ijs j with a ij C and N r,i (s) = j= b ijs j with b ij X converge on all of C. Consequently, (47) (48) read after use of the inverse Laplace transform g i (t) = i (t), a ij C (49) z r,i (t) = j= a ijp (j) j= b ijp (j) i (t), b ij X. (5) For the series in (49) (5) to converge the trajectory p i must be sufficiently smooth. For a large class of systems a suitable trajectory is given by the Gevrey bump function ( ) 1 η i exp p i (t) = [ t T (1 t, t (, T) T )]ν (51), t {, T }, with ν > 1 (see (Rudolph and Woittennek, 28, 4)). In this case p i (t) and all its derivatives vanish for t = and t = T so that z r,i () = z r,i (T) = is satisfied according to (5) as required (see (4)). The constant η i in (51) has to be chosen such that in addition ζ i (T) = 1 holds (see (39)). The pathological case ζ i (T) = can be avoided by a slight change of ν in (51). The control g i, that corresponds to p i, is then obtained from (49). When the related sampling function g (see (36)) is used for the generalized sampling (37), the desired sequence (ξ(t k )) k N of modal states is reconstructed by (38). 5. EXTENSION OF THE APPROACH TO UNBOUNDED INPUT AND OUTPUT OPERATORS In this section the approach will be extended to the class of regular linear systems which contains the majority of linear time-invariantsystems that are of practicalinterest (for details see e.g. Curtain and Weiss (1989); Weiss (1989a,b, 1994)). Particularly, systems with boundary control and point observation are captured. In this case the control operator B and the observation operator C are unbounded. In order to describe the properties of these operators adequately the spaces X 1 = D(A) with the norm x 1 = (αi A)x, α ρ(a), and X 1 being the completion of X under the norm x 1 = (αi A) 1 x are introduced. X 1 and X 1 are Banach spaces independent from α ρ(a) and satisfy X 1 X X 1 (52) with continuous dense injections (see (Nagel and Schlotterbeck, 1986, 3.5)). It is well-known that the control operator B satisfies B L(C, X 1 ) (see Weiss (1989b)) so that b in (3) has to be considered as an element of X 1 instead of X. Since b may therefore have the character of a generalized function, x(t) D(A) is not assured any 13314

5 more. For (1) to make sense nevertheless, A has to be regarded as an operator A : D(A) = X X 1 instead of A : D(A) = X 1 X. This is possible because S(t) has a continuous extension to a C -semigroup on X 1 whose generator is an extension of A with domain X (see Nagel and Schlotterbeck (1986)). With these generalizations (1) has a continuous state strong solution x( ) C([, ], X) for any u L 2 loc ([, ), R) which means that (1) is satisfied with respect to 1 for a.e. t (see Weiss (1989b)). It is a known fact that the observation operator C satisfies C L(X 1, C) (see Weiss (1989a)). Since x(t) X 1 = D(C) in general, C is commonly replacedby a suitable extension such that x(t) is in its domain. An often used extension is the Lebesgue-extension C L x = lim C S(τ) I A 1 x = lim τ τ λ + Cλ(λI A) 1 x (53) with D(C L ) = {x X both limits in (53) exist} so that x(t) D(C L ) for a.e. t (see Weiss (1994)). C L is a bounded functional on X 1 due to C L C L(X 1, C) and can thus be regarded as an element of X1, the dual space of X 1. Taking into account that X is a Hilbert space so that X can be identified with X, it follows that X1 can be identified with X 1 with respect of the inner product of X, i.e., X1 = {, h h X 1 } (see (Weiss, 1989a, 6)). Hence, one has the representation C L h = h, c, c X 1, h D(C L ) X 1. (54) Summing up, these considerations show that b and c in Sections 2 4 have to be considered as elements in X 1, and the operators S(t), A, and C have to be replaced by suitable extensions. In fact, it turns out that under these generalizations all relations in the preceding sections remain valid. 6. NUMERICAL EXAMPLE Consider the one-dimensional heat conducting system w t (z, t) = w zz (z, t), z (, 1), t > (55) w(, t) =, t (56) w z (1, t) = u(t), t (57) w(z, ) = w (z), z [, 1] (58) y(t) = w(1, t), t (59) with boundary control and point observation at the right end. Therein, w(z, t) describes the temperature of a rod along the spatial coordinate z [, 1], which is fixed at the left end while the heat flow at the right end is determined by the input u (see (56) (57)). Analog to the example in Byrnes et al. (1999) it can be verified that this system has the representation (1) (2) on the state space X = L 2 (, 1) with the state x(t) = w(, t) X, t. Thereby, the operators B and C L are given by (Bu(t))(z) = δ 1 (z)u(t) (6) C L x(t) = x(t), δ 1 = x(1, t), (61) where δ 1 denotes the Dirac function centered at z = 1. Thus, b = c = δ 1 (see (3), (54)), and the generator A of the C -semigroup S(t) on X is given by Ah = h for h D(A) (62) D(A) = { h L 2 (, 1) h, h abs. continuous, (63) h L 2 (, 1), h() = h (1) = }. (64) The spectrum of A satisfies σ(a) = {λ i, i N} with the eigenvalues λ i = π 2 (i.5) 2. Thus, the spectrum is t g1 Fig. 1. Sampling function g 1 (t) ξ t Fig. 2. Comparison of the modal state ξ 1 (t) ( ) and the reconstructed sequence (ξ 1 (t k )) k N ( o ) discrete and entirely contained in the open left half-plane so that the Assumptions A1, A2, and A4 are satisfied. The corresponding eigenvectors of A are given by φ i = 2sin((i.5)πz) and form an orthonormal basis for L 2 (, 1). Since A is a Sturm-Liouville operator it follows A = A so that ψ i = φ i, i N. For T =.15 the most dominant eigenvalue of A d is λ d,1 =.69. Since the remaining ones satisfy λ d,i <.4, i 2, only λ d,1 shall be shifted to the new location λ d,1 =.1 by state feedback so that the modal state ξ 1 (t k ) needs to be reconstructed, and n = 1. Due to b, ψ 1 = c, φ 1 = δ 1, 2sin(zπ.5) = 2 Assumption A3 is satisfied, and (Adn, b dn ) with A dn =.691, b dn =.178 (65) is controllable. The feedback gain k = 3.32 assigns the desired eigenvalue λ d,1 to A dn b dn k which appears also in the dynamics of the closed-loop system (see Theorem 1). For the computation of the sampling function g 1 the trajectory tracking problem (34) (35) for i = 1 has to be solved, wherein A = A and CL v = δ 1v for v C, following from (61), can be used. The suggested design procedure for the feedforwardcontrolrequires to solve (41) yielding ẑ(s) = sinh( sz)/ s cosh( ˆ g 1 (s). (66) s) This reveals D(s) = cosh( s) and N(s) = sinh( sz)/ s, where the singularity of N(s) can be removed by N() = 1. Insertion of the Taylor series of sinh( ) and cosh( ) leads to the parameterization (49) for g i. Since the parameter η 1 in the trajectory (51) is not known at this stage, it is set to η 1 = 1 and ν = 1.3 is used, for which (49) (5) converge uniformly following from (Rudolph and Woittennek, 28, 4.2) and (Lynch and Rudolph, 22, Lem. 1). With the resulting control g 1 (43) is solved yielding ζ 1 (T) = Comparison with (39) shows that η i has to be chosen η 1 = 1/( ) and g 1 adapted appropriately. The resulting sampling function g 1 is depicted in Figure 1. In order to complete the design it remains to compute G =.148. The modal state sequence (ξ 1 (t k )) k N is then obtained from (38) which allows to realize the state feedback control u(t k ) = kξ 1 (t k ). A comparison of ξ 1 (t) and the reconstructed sequence is shown in Figure 2 for w ( ) = φ 1 +φ 2 and u(t) = 1(t 1) with 1(t) denoting the unit step at t =

6 7. CONCLUDING REMARKS The approach has been presented for SISO systems only in order to avoid technical complications. However, the MIMO case can be treated in an analog way. While the method allows only for state feedbacks with finitely many states, it is also possible to generate feedbacks of the whole infinite-dimensional state that are needed, e.g., for optimal control. This is subject to current research. Appendix A. PROOF OF LEMMA 3 First, observe that (si A ) 1 > holds for all s ρ(a ) because (si A ) 1 is injective following from (si A ) =. Therefore, D(s) = (si A ) 1 1, D( D) = ρ(a ) (A.1) can be introduced. In order to yield the expression (42) it will be shown that the numerator and denominator in F(s) D(s) F(s) = (A.2) D(s) can be continued to entire functions N and D, respectively. Let λ denote an arbitrary eigenvalue λ σ(a ) and ψ the corresponding eigenvector of A. The fact that (si A )ψ for s λ implies that (si A ) 1 for s λ, s ρ(a ). Thus, it follows D(s) for s λ so that { D(s), s ρ(a ) D(s) =, s σ(a (A.3) ) is a continuous extension of D on all of C due to ρ(a ) σ(a ) = C. Moreover, D is an entire function because D(s) is analytic on ρ(a ) as (si A ) 1 is, and σ(a ) is discrete (see (Kato, 198, Thm. III 6.7)). So, comparison of (A.2) with (42) reveals that the statement holds if F(s) D(s) can be analytically continued. Since D has a zero for any λ σ(a ), it can be factorized as D(s) = (s λ) ˆD(s) (A.4) with ˆD being entire. The fact that F(s) is analytic except at the eigenvalues of A (see (Weiss, 1994, Thm. 5.8)), which are isolated and simple, has the consequence that F(s) has a Laurent series about λ given by F(s) = f 1 s λ + f j (s λ) j = f 1 s λ + ˆF(s) (A.5) j= for s ρ(a ) (see (Kato, 198, I 5.3, III 6.5)) with ˆF(s) = j= f j(s λ) j. Using (A.4) (A.5) yields F(s)D(s) = f 1 ˆD(s) + ˆF(s)(s λ) ˆD(s). (A.6) It is easy to check that the principal part of the related Laurent series vanishes so that F(s)D(s) can be continued to an entire function N(s) on all of C with N(s) = F(s)D(s) = F(s) D(s) for s ρ(a ) (see (Kato, 198, I 1.7)). So, (A.2) gives (42) with N and D entire as stated. REFERENCES Balas, M.J. (1982). Towarda more practicalcontroltheory for distributed parameter systems. In C.T. Leondes (ed.), Control and Dynamic Systems, volume 18, Academic Press, New York. Bensoussan, A., Prato, G.D., Delfour, M.C., and Mitter, S.K. (27). Representation and Control of Infinite Dimensional Systems. Birkhäuser, Boston. Bradshaw, A. and Porter, B. (1972). Modal control of a class of distributed-parameter systems: Multi-eigenvalue assignment. International Journal of Control, 16(2), Byrnes, C.I., Gilliam, D.S., and Shubov, V.I. (1999). Example of output regulation for a system with unbounded inputs and outputs. In Proceedings of the 38 th Conference on Decision & Control, Curtain, R.F. and Weiss, G. (1989). Well posedness of triples of operators. In F. Kappel, K. Kunisch, and W. Schappacher (eds.), Control and Estimation of Distributed Parameter Systems, Birkhäuser, Basel. Curtain, R.F. and Zwart, H.J. (1995). An Introduction to Infinite-Dimensional Linear Systems Theory. Springer, New York. Kato, T. (198). Perturbation Theory for Linear Operators. Springer, Berlin. Logemann, H. (1992). Stability and stabilizability of linear infinite-dimensional discrete-time systems. IMA Journal of Mathematical Control and Information, 9, Logemann, H., Rebarber, R., and Townley, S. (25). Generalized sampled-data stabilization of well-posed linear infinite-dimensional systems. SIAM Journal on Control and Optimization, 44(4), Luo, Z.H., Guo, B.Z., and Morgül, O. (1999). Stability and Stabilization of Infinite Dimensional Systems with Applications. Springer, New York. Lynch, A.F. and Rudolph, J. (22). Flatness-based boundary control of a class of quasilinear parabolic distributed parameter systems. International Journal of Control, 75(15), Nagel, R. and Schlotterbeck, U. (1986). Basic results on semigroups on Banach spaces. In R. Nagel (ed.), One-Parameter Semigroups of Positive Operators, Springer, Berlin. Rebarber, R.L. and Townley, S. (1997). Stabilization of distributed parameter systems by piecewise polynomial control. IEEE Transactions on Automatic Control, 42(9), Rosen, I.G. and Wang, C. (1992). On stabilizability and sampling for infinite dimensional systems. IEEE Transactions on Automatic Control, 37(1), Rudolph, J. and Woittennek, F. (28). Motion planning and open loop control design for linear distributed parameter systems with lumped controls. International Journal of Control, 81(3), Weiss, G. (1989a). Admissible observation operators for linear semigroups. Israel Journal of Mathematics, 65, Weiss, G. (1989b). The representation of regular linear systems on Hilbert spaces. In K.K. F. Kappel and W. Schappacher (eds.), Control and Estimation of Distributed Parameter Systems, Birkhäuser, Basel. Weiss, G. (1994). Transfer functions of regular linear systems. part i: Characterizations of regularity. TransactionsoftheAmerican Mathematical Society, 342,