Input/output delay approach to robust sampled-data H control

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1 Systems & Control Letters 54 (5) Input/output delay approach to robust sampled-data H control E. Fridman, U. Shaked, V. Suplin Department of Electrical Engineering-Systems, Tel-Aviv University, Ramat Aviv, Tel-Aviv 69978, Israel Received May 4; received in revised form 9 July 4; accepted August 4 Available online 1 October 4 Abstract Sampled-data output-feedback H control of linear systems is considered. The only restriction on the sampling and hold is that the distances between the sequel sampling times and holding times are not greater than given bounds. A new approach, which was recently introduced to sampled-data state-feedback stabilization, is developed to the H control. The system is modelled as a continuous-time one, where the control input and the measurement output have piecewise-continuous delays. Sufficient linear matrix inequalities (LMIs) conditions for H control of such systems are derived via Lyapunov Krasovskii functionals and descriptor approach to time-delay systems. For the first time the new approach allows to develop different robust control methods for the case of sampled-data H control. 4 Elsevier B.V. All rights reserved. Keywords: Sampled-data control; H control; LMI; Delay model; Lyapunov Krasovskii functionals 1. Introduction Sampled-data H control of systems has been studied in a number of papers (see e.g. 4,11,15,16, and the references therein). Two main approaches have been used. The first one is based on the lifting technique,18 in which the problem is transformed to equivalent finite-dimensional discrete H control problem. The second, more direct, approach is based on the representation of the system in the form of hybrid discrete/continuous model and the solution is obtained in terms of differential Riccati equations with jumps. These approaches give necessary and sufficient conditions and lead to equivalent solutions. To the best of our knowledge, the only LMI solution to sampled-data output-feedback H control was derived by Lall and Dullerod 11 for the lifted discrete system when the sampling and the hold operators are periodic and their rates are commeasurable. This solution is computationally complicated because it includes the evaluation of the matrices of the lifted system. This work was supported by the C&M Maus Chair at Tel Aviv University. Corresponding author. Tel.: ; fax: address: emilia@eng.tau.ac.il (E. Fridman) /$ - see front matter 4 Elsevier B.V. All rights reserved. doi:1.116/j.sysconle.4.8.1

2 7 E. Fridman et al. / Systems & Control Letters 54 (5) 71 8 The hybrid system approach has been applied recently to robust sampled-data H control for the case of equidistant sampling 9. To overcome difficulties of solving differential inequalities with jumps, a piecewise linear in time Lyapunov function has been suggested. As a result in 9 LMIs have been derived, which do not depend on the sampling interval, and thus are very conservative. The sampling interval independent LMI conditions have been also derived recently for the case of robust H filtering under sampled-data measurements 17. Modelling of continuous-time systems with digital control as continuous systems with delayed control input was introduced by Mikheev et al., Astrom and Wittenmark 1,1. The digital control law may be represented as delayed control as follows: u(t) = u d (t k ) = u d (t (t t k )) = u d (t τ(t)), t k t<t k+1, τ(t) = t t k, (1) where u d is a discrete-time control signal and the time-varying delay τ(t) = t t k is piecewise linear with derivative τ(t) = 1 for t = t k. Moreover, τ t k+1 t k. Recently, this input delay approach was applied to robust sampleddata stabilization via Lyapunov Krasovskii technique in 6. It is the purpose of the present paper to develop this approach to the case of sampled-data H control. Bounded real lemmas (BRLs) for systems with time-varying delays were derived for the cases where the derivative of the delay is less than one via Lyapunov Krasovskii functionals (see e.g. 15). For the case of time-varying delay without any restrictions on the delay of the derivative a BRL was obtained in 7. This became possible due to a new descriptor model representation of the delay system introduced in 5. In the present paper we consider the output-feedback sampled data H control problem by finding solution for the continuous-time H control problem for systems with uncertain but bounded (by the maximum sampling and holding intervals) delays in the control input and in the measurement output. We apply the BRL of 7 and derive solution in terms of LMIs. The solution which we obtain is robust with respect to different sampling and holding with the only requirement that the maximum sampling interval and maximum holding interval are not greater than given bounds. The LMI conditions are sufficient only, but they are comparatively simple. For the first time the new approach allows to develop different robust control methods for the case of sampled-data H control. We give a solution to H control of systems with norm-bounded uncertainties. Notation. Throughout the paper the superscript T stands for matrix transposition, R n denotes the n-dimensional Euclidean space with vector norm, R n m is the set of all n m real matrices, and the notation P>, for P R n n means that P is symmetric and positive definite. Let C n a,b denotes the space of continuous functions : a,b R n with the supremum norm c and L, ) be the space of the square integrable functions with the norm L. We also denote x t (θ) = x(t + θ) (θ h, ).. Main results.1. Problem formulation Consider the system ẋ(t) = Ax(t) + B 1 w(t) + B u(t), z(t) = C 1 x(t) + D 1 u(t), () where x(t) R n is the state vector, w(t) R n w is the disturbance, u(t) R n u is the control input and z(t) R n z is the controlled output.

3 E. Fridman et al. / Systems & Control Letters 54 (5) The control signal is assumed to be generated by a zero-order hold function with a sequence of hold times <t 1 < <t k < u(t) = u d (t k ), t k t<t k+1, () where lim k t k =, and u d is a discrete-time control signal. The measurement output y k R n y is assumed to be available at discrete sampling instants < σ 1 < < σ k <, lim k σ k =, and it may be corrupted by w k = w(σ k ): y k = C x(σ k ) + D 1 w k, k =, 1,,... (4) We consider an output-feedback sampled-data H control. Assume that A1. C T 1 D 1 =. A. t k+1 t k h k. A. σ k+1 σ k g k. We define the following performance index for a prescribed scalar γ > : J c (w) = (z T (s)z(s) γ w T (s)w(s)) ds. (5) Our objective is to find a dynamic output-feedback control law of the form u(t) = C c x c (t k ) + D c y k,t k t<t k+1, ẋ c (t) = A c x c (t) + B c y k, which for all sampling and hold times satisfying A and A internally stabilizes the system and leads to J c < for x() =, and for all non-zero w L, )... The input and the output delay model We consider the following piecewise-constant measurement: (6) y(t η(t)) = C x(t η(t)) + D 1 w(t η(t)), η(t) = t σ k,t σ k < σ k+1. (7) From A it follows that η(t) g. The output-feedback controller law is described by ẋ c (t) = A c x c (t) + B c y(t η(t)), x c (t) =, t h,, ū(t) = C c x c (t) + D c y(t η(t)), where the sampled version u(t) =ū(t τ(t)) is applied to (). We represent z in the form x(t) z(t) = C 1 D 1 C c. x c (t) (8a c) (9) In order to restore the transference property of the sample and hold component, namely to recover the filtering property of the sample and hold which filters out the high-frequency part of the sampled signal, and in order to conveniently describe the sampling of y(t) and u(t), we introduce prescribed LTI components that are connected

4 74 E. Fridman et al. / Systems & Control Letters 54 (5) 71 8 in series to y(t) of (7) and ū(t) of (8c) and produce the sampled version of y(t) and u(t). These components will be later added to the controller of (8). We thus consider the following two components: ξ(t) = ρi ny ξ(t) + ρy(t), ξ 1 (t) = ρi nu ξ 1 (t) + C c x c (t) + D c ξ(t), (1a,b) where ξ R n y, ξ 1 R n u, and ρ < 1 is a positive scalar. Denoting ζ(t) = col{x(t), ξ(t), ξ 1,x c (t)} the following closed-loopsystem is obtained: ζ(t) = A ζ(t) + A 1 ζ(t η(t)) + A ζ(t τ(t)) + Bw(t), where A ρi A ny, D c ρi nu C c B c A c ρc A 1, ρb A, B 1 ρd B 1. (11a) (11b e) The corresponding z(t) is given by z(t) = Cζ(t), where C = C 1 D 1. (1).. BRL for linear systems with time-varying delays Consider an auxiliary system ẋ(t) = A x(t) + A 1 x(t η(t)) + A x(t τ(t)) + B 1 w(t), z(t) = Cx(t), (1a,b) with the performance index (5), where x(t) R n, w(t) R n w, z(t) R n z, τ(t) and η(t) are piecewise-continuous delays satisfying τ(t) h, η(t) g, and A i,i=, 1,, B 1 and C are constant matrices. Applying to (1) and (5) the Lyapunov Krasovskii functional of the form V( x(t),ẋ t ) = V 1 ( x(t)) + V (ẋ t ), (14)

5 where and x(t) = col{x(t), ẋ(t)}, E = P1 P = P P V 1 ( x(t)) = x T (t)ep x(t), E. Fridman et al. / Systems & Control Letters 54 (5) In,, P 1 = P T 1 > (15a d) t t V (ẋ t ) = ẋ T (s)r 1 ẋ(s)ds + ẋ T (s)r ẋ(s)ds dθ, g t+θ h t+θ (15e,f) and finding the conditions that (d/dt)v( x(t),ẋ t ) + z T (t)z(t) γ w T (t)w(t) <, we obtain similarly to 7,8 the following BRL: Lemma 1. Consider (1). For a prescribed γ >, the cost function (5) achieves J c (w) < for all non-zero w L q, ) and for all piecewise-continuous delays τ( ), η( ), satisfying inequalities τ(t) h, η(t) g, if there exist n n-matrices P 1 >,P,P and R i = Ri T that satisfy the following LMIs: Ψ 1 P T C T B 1 γ <, I I and Ri A T i P, i= 1,, Z i (16a,b) where P is given by (15c) and Ψ 1 = Ψ + gz 1 + hz +, gr 1 + hr Ψ = P T I i= A i= + T i P. A i I I I.4. Output-feedback H control Consider the closed-loopsystem (11), (1) and the performance index J c. We apply Lemma 1 to system (11), (1) directly, where we replace R i by diag{r i, } with R i R n c n c,we denote n c = n + n y + n u. Pre- and post-multiplying the resulting LMI that is equivalent to (16a) by diag{q T,I,I} and diag{q, I, I}, respectively, and the one equivalent to (16b) by diag{i, Q T } and diag{i,q}, respectively, where P 1 Q1 = Q =. The following is obtained for Ẑ i = Q T Z i Q. Q Q Lemma. Consider (11), (1). For a prescribed γ >, the cost function (5) achieves J c (w) < for all non-zero w L q, ) and for all piecewise-continuous delays τ( ), η( ), satisfying inequalities τ(t) h, η(t) g, if there exist n c n c matrices Q 1 >,Q,Q and n c n c matrices R i = R T i that satisfy the following

6 76 E. Fridman et al. / Systems & Control Letters 54 (5) 71 8 inequalities: Q1 C Φ B T Q T Inc Q T g h Q T Q T γ I nw I nz gr 1 1 hr 1 and Ri I nc A T i, i= 1,, Ẑ i Inc <, (17a,b) where I Φ = i= Q + Q A T i I I i= A T i I + hẑ + gẑ 1. We further denote X M T Q 1 = M U I Y J = N,Q 1 1 = Y N T, N V, and J = diag{j, J} (18) Multiplying (17a) by diag{ J T,I} from the left and by diag{ J,I} from the right, respectively, and (17b) by diag{i, J T } from the left and by diag{i, J } from the right, respectively, we obtain Q + Q T Q Q T + J T Q 1 ( i= A T i )J J Q Q T + h Z + g Z T Q 1 C T Q T 1 J T g Inc Q T B Q T h Inc Q T γ I nw I nz <, gr 1 1 hr 1 and Ri I nc A T i J, i= 1,, Z i (19a,b) where Q i+1 = J T Q i+1 J and Z i = J T Ẑ i J, i = 1, and where it is required that J T X I Q 1 J = >. I Y () It is readily found that A i = diag{ā, }+ B Θ C, i= (1a)

7 where Ā = A ρb ρc ρi ny ρi nu C = Inc Ĉ, and Θ = E. Fridman et al. / Systems & Control Letters 54 (5) ˆB, B = I nc Ac B c C c D c, (1b d) with ˆB = and Ĉ = I ny. I nu Hence ( ) J T Q 1 A T i J i= X M = Ā T T XĈ T I I nc Y + nc Ĉ T Θ T N B T ˆB Y XĀ = T I ˆB Ā T Ā T + Y Ĉ T K T T Ξ, () I where Inu M K = N T Θ +. () Y ˆB Ĉ X I ny Y ĀX Substituting the above result in (19a,b) and choosing R 1 = diag{ε 1 I n, εi ny, εi nu } and R = diag{ εi n+ny, ε I nu }, where < ε tends to zero, and ε 1 and ε are positive tuning parameters, we obtain the following. Theorem 1. Consider system (). For prescribed scalars γ > and < ρ, J c (w) < w(t) L ) under the sampled-data output-feedback controller of (8) and (1) for all holding and sampling times satisfying A and A, respectively, if for some tuning scalar parameters ε 1 and ε there exist 4n c 4n c matrices Z 1, Z,n c n c matrices Q, Q, n c n c matrices X, Y, and a (n c + n u ) (n c + n y )-matrix K that satisfy () and the following three LMIs: Γ + h Z + g Z 1 <, ε1 I n ρc T I nc Y, Z 1 ε I nu ρb T I n I nc Y, Z where Q + Q T Q Q T + Ξ X Q Q Q T I ˆB 1 I Ĉ1 T g T n,ny Q Q Y T I ny h T Q T Γ γ I nw +n v I nz gε 1 1 I n y hε 1 I n u I nu (4a c), (5)

8 78 E. Fridman et al. / Systems & Control Letters 54 (5) 71 8 B1 ˆB 1 = ρd 1, and Ĉ 1 = C 1 D 1. If a solution to the above exists then the output-feedback controller that achieves the required performance is described by the series connection of the pre- and the post-controllers (1a) and (1b), respectively, with the system described in (8), where the matrices of the latter are obtained by solving () for Θ and taking N =I nc, M =I nc YX. Remark 1. In () the matrices N and M can be obtained by factorizing I nc YX=N T M. One of these factorizations yields N =I nc. The transfer function matrix of the obtained controller will not depend on the specific factors chosen. The LMIs in Theorem apply variables that are in fact redundant. Inequalities (4b,c) set lower bounds on Z 1 and Z. The latter matrices can thus be replaced in (4a) by their respective bounds. We thus obtain the following: Corollary 1. Consider system (). J c (w) < w(t) L ) for prescribed scalars γ > and < ρ, under the sampled-data output-feedback controller of (8) and (1) for all holding and sampling times satisfying A and A, respectively, if for some tuning scalar parameters ε 1 and ε there exists a n c n c matrices Q, Q, n c n c matrices X, Y, and a (n c + n u ) (n c + n y )-matrix K that satisfy () and the following inequality: Γ ρb g I ρc h I Y Y gε 1 I n hε I nu <. (6).5. The case of systems with norm-bounded uncertainties The results of the previous section can be easily generalized to the case of systems with norm-bounded uncertainties. Theorem. Consider system () where A, B 1, B, C, and D 1 are replaced by A + ΔA, B 1 + ΔB 1, B + ΔB, C + ΔC and D 1 + ΔD 1, respectively, and where ΔA ΔB 1 ΔB = H Δ(t) E E 1 E, i =, 1, and ΔC ΔD 1 = H c Δ(t) E c E d (7a,b) with Δ and Δ satisfying Δ(t) T Δ(t) I and Δ(t) T Δ(t) I. (8a,b) For prescribed scalars γ > and < ρ, J c (w) < w(t) L ) under the sampled-data output-feedback controller of (8) and (1) for all holding and sampling times satisfying A and A, respectively, if for some tuning scalar parameters ε 1 and ε, r 1 and r there exist 4n c 4n c matrices Z 1, Z,n c n c matrices Q, Q, n c n c

9 E. Fridman et al. / Systems & Control Letters 54 (5) matrices X, Y, and a (n c + n u ) (n c + n y )-matrix K that satisfy () and the following inequalities: Γ + hẑ + gẑ 1 I H E T X X ρe T c I I ρe T I r 1 r Y H Y c E T ρed T 1 <, r 1 I r I r 1 I r I ε 1 I n ρc T I nc Y ρec T Z 1 r H c, r I r I ε I nu ρb T I nc Y ρe T Z r 1 H, (9a c) r 1 I r 1 I where Γ is defined in (5). If a solution to the above exists then the output-feedback controller that achieves the required performance is described by the series connection of the pre- and the post-controllers (1a) and (1b), respectively, with the system described in (8), where the matrices of the latter are obtained by solving () for Θ and taking N =I nc, M =I nc YX. Proof. The result follows by replacing in (4) A, B 1, B, C, and D 1 by A + ΔA, B 1 + ΔB 1, B + ΔB, C + ΔC, and D 1 + ΔD 1 respectively, and by using inequalities of the type rhh T r 1 E T E H ΔE + E T Δ T H T rhh T + r 1 E T E scalars <r and applying Schur complements formula. Similarly to Corollary 1, the three LMIs of Theorem can be combined to one LMI..6. On state-feedback H control Assume A and A1. D 1 =. Our objective is to find a state-feedback controller of the form u(t) = Kx(t k ), t k t<t k+1, ()

10 8 E. Fridman et al. / Systems & Control Letters 54 (5) 71 8 which for all hold times satisfying A internally stabilizes the system and leads to J c <, for x() = and for all non-zero w L, ). Under A1 a simple solution to state-feedback H control will be derived. We represent a piecewise constant control law as continuous time control with time-varying piecewise-continuous delay τ = t t k, as given in (1). We will thus look for a state-feedback controller of the form u(t) = Kx(t τ(t)). (1) Substituting (1) into (), we obtain the following closed-loopsystem: ẋ(t) = Ax(t) + B Kx(t τ(t)) + B 1 w(t), z(t) = C 1 x(t). () From A it follows that τ(t) h since τ t k+1 t k. We will further consider () as the system with uncertain and bounded delay, and we require that J c < holds for all non-zero w L, ) and for zero initial condition x(t) =, t. In order to apply Lemma 1 to (), we denote P = εp, where ε is a scalar, P = P 1, P 1 = P T P 1 P, R = P T RP, Z i = P T Z i P, i=1,, and Ȳ =KP. Multiplying (16a) by diag{ P T, P T,I,I} and diag{ P, P, I, I} and (16b) by diag{ P T, P T, P T } and diag { P, P, P }, from the right and the left, respectively, the following is obtained. Theorem. Assume A1. Consider (). For a prescribed scalar γ >, J c (w) < w(t) L ) under sampled-data state-feedback controller for all holding times satisfying A, if for some tuning scalar parameter ε there exist n n matrices < P 1, P, Z 1, Z, Z, R, and n u n- matrix Ȳ that satisfy P T A T + AP + B Ȳ + Ȳ T B + h Z 1 P 1 P + εp T A T + εȳ T B T + h Z P T C T 1 B 1 εp T εp + h( R + Z ) εb 1 <, I γ I R Ȳ T B T εȳ T B T Z 1 Z. (a,b) Z The state-feedback gain is given by K = Ȳ P 1. (4) Consider now () under the continuous state-feedback u(t) = Kx(t) and the index (5). It is well-known that in the continuous case J c < iff there exist n n-matrix <P 1, and n u n-matrix Y that satisfy P1 A T + AP 1 + B Y + Y T B P 1 C1 T B 1 I <, (5) γ I where K = YP 1 1. If (5) is feasible than for small enough h> () is feasible too (take e.g. the same P 1,Y = Ȳ, and P = P 1, ε, while R and Z are any matrices satisfying (b)). We, therefore, obtain the following result (similar to the one of 1 which was proved by using the lifting technique): Corollary. Consider () under A1, A. If the continuous-time state-feedback u = Kx(t) achieves J c (w) <, then there exists h > such that for all h (,h the sampled-data state-feedback (), with the same gain K, achieves J c (w) <.

11 E. Fridman et al. / Systems & Control Letters 54 (5) The LMIs of Theorem are affine in the system matrices and thus the solution for the system with polytopic type uncertainty readily follows. The results of this subsection may be easily adapted to the case of systems with norm-bounded uncertainties..7. Examples Example 1. We consider an example of 14. The system is ẋ(t) =.8x(t) + w(t) + u(t), 1 z(t) = x(t) + u(t), 1 y(k) = x(σ k ) +.v k, k =, 1,.... (6) We choose ρ =.1 and apply Theorem 1, for ε 1 =.1 and ε =.4 we obtain γ min = 1.5. This result is higher than the minimum attenuation level of γ min =.816 reported in 14 for the case of equidistant sampling with the period g = π/4 and holding with the period h = 1. 1 Consider next the state-feedback sampled-data H control of (6a) with z(t) = x(t). Applying Theorem for holding bound of h = 1 a minimum attenuation level of γ =.8866 is obtained for ε =.74. The corresponding feedback gain is K =.678. The result obtained for holding that tends to zero is γ = Example. We consider here another example of 14. Given the system: 1 ẋ(t) = x(t) + w(t) + u(t), z(t) = x(t) + u(t),.1 y k =1x(σ k ) +.1v k,k=, 1,.... (7) For the output-feedback control with h=1 and g=π/4 we choose ρ=.1 and apply Theorem 1. For ε 1 = and ε = 1. 1 we obtain γ min = This result is close to the minimum attenuation level of γ min = 1. found in 14 for the case of equidistant holding with the period 1 and equidistant sampling with the period π/4. It is noted that the two poles introduced by the components of (8) and (1) at.1 are canceled by zeros of the controller so that the overall feedback controller possesses four poles and two zeros. We assume next that the model of (7) encounters norm bounded uncertainty of type (7) with H = 1 T, E =, and with zero E 1, E and H c. Applying Theorem, for the above value of ρ=.1, we obtain a minimum value of γ min = 1.4 for r 1 = 9, ε 1 =.4 1 4, and ε = Conclusions A new approach, which was recently introduced to sampled-data state-feedback stabilization, is developed to the sampled-data H control. The system is modelled as a continuous-time one, where the control input and the measurement output have piecewise-continuous delays. It is assumed that the maximum holding interval is not greater than h> and the maximum sampling interval is not greater than g>. The h and g-dependent sufficient LMI conditions are derived for output-feedback H control of such systems via Lyapunov Krasovskii functionals

12 8 E. Fridman et al. / Systems & Control Letters 54 (5) 71 8 and descriptor approach to time-delay systems. The results are generalized to the case of systems with norm-bounded uncertainties. The solution of the output-feedback control problem is based on introducing simple filters that precede the sampling of the measurement and the control input. The steady-state gain of these filters is one, and they filter out signal components of frequencies equal to or larger than the corresponding Nyquist frequencies 1. Although the poles of these components are cancelled by the zeros of the controller that is obtained by solving the LMIs of Theorems 1 and, the fact that this controller should possess zeros at prespecified locations is somewhat restrictive. References 1 K. Astrom, B. Wittenmark, Adaptive control, Addison-Wesley, Reading, MA, B. Bamieh, J. Pearson, A general framework for linear periodic systems with applications to H sampled-data control, IEEE Trans. Autom. Control 7 (199) T. Basar, P. Bernard, H Optimal control and related minimax design problems, A dynamic game approach, Systems and Control: Foundation and Applications, Birkhauser, Boston, T. Chen, B. Francis, Optimal sampled-data control systems, Springer, Berlin, E. Fridman, New Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems, Systems Control Lett. 4 (1) E. Fridman, A. Seuret, J.-P. Richard, Robust sampled-data stabilization of linear systems: an input delay approach, Automatica 4 (4) E. Fridman, U. Shaked, Delay dependent stability and H control l: constant and time-varying delays, Int. J. Control 76 () H. Gao, C. Wang, Comments and further results on a Descriptor system approach to H control of linear time-delay systems, IEEE Trans. Automat. Control 48 () L. Hu, J. Lam, Y. Cao, H. Shao, An LMI approach to robust H sampled-data control for linear uncertain systems, IEEE Trans. System Man Cybernet. B () H. Kwakernaak, R. Sivan, Modern Signals and Systems, Prentice-Hall, Englewood Cliff, NJ, 1991 (Chapter 9.). 11 S. Lall, G. Dullerod, An LMI solution to the robust synthesis problem for multi-rate sampled-data systems, Automatica 7 (1) Yu. Mikheev, V. Sobolev, E. Fridman, Asymptotic analysis of digital control systems, Automat. Remote Control 49 (1988) Y. Oishi, A bound of conservativeness in sampled-data robust stabilization and its dependence on sampling periods, Systems Control Lett. (1997) M.S. Sagfors, H.T. Toivonen, H and LQG control of asynchronous sampled-data systems, Automatica (1997) U. Shaked, I. Yaesh, C. de Souza, Bounded real criteria for linear time-delay systems, IEEE Trans. Automat. Control 4 (1998) N. Sivashankar, P. Khargonekar, Characterization of the L -induced norm for linear systems with jumps with applications to sampled-data systems, SIAM J. Control Optim. (1994) S. Xu, T. Chen, Robust H filtering for uncertain impulsive stochastic systems under sampled measurements, Automatica 9 () Y. Yamomoto, New approach to sampled-data control systems a function space method, in: Proceedings of the 9th Conference on Decision and Control, Honolulu, HW, 199, pp