A Variable Structure Parallel Observer System for Robust State Estimation of Multirate Systems with Noise

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1 A Variable Structure Parallel Observer System or Robust State Estimation o Multirate Systems with Noise May-Win L. Thein Department o Mechanical Engineering University o New Hampshire Durham, New Hampshire mthein@cisunix.unh.edu Abstract This paper proposes the Variable Structure Parallel Observer System or Noisy Measurements (VSPOSN), a multirate estimation technique or discrete time systems, to compensate or output measurements that are noisy and only periodically available. The VSPOSN is an extension o the original Parallel Observer System. The VSPOSN is based upon a stable and robust Discrete-Time Sliding Mode Observer with a set o attractive boundary layers driving a Luenberger observer running in parallel. The resulting observer system provides stable state estimates or each control step despite incomplete and noisy measurements. The VSPOSN is applied to a second order resonance model with nonlinear uncertainties/disturbances as proo o concept, and a stability proo and error bounds are provided. I. INTRODUCTION It is known that the perormance o observer-based control techniques rely heavily on the accuracy o the applied observer. Thus, the need to investigate methods o state estimation becomes critical. In discrete time systems, the control input is calculated and determined based upon system state estimates at each control time step. In turn, the state estimator updates its estimates at the same rate via output measurements. However, in many cases, as in disk drive applications, the output measurements are not only corrupted with noise, but are not even available every control time step. That is, these noisy measurements are only available every k control sample periods. Thus, a multirate estimation method that is robust against noise is needed to provide stable state estimates or each control time step. In Phillips and Tomizuka s work 1, the linear modelbased techniques uses a Prediction/Correction type method which perorms at two sampling rates. Lu and Fisher s work 2 uses an adaptive inerential algorithm or systems with no external disturbances. Ramachandran, Young, and Misawa 3 proposed a polynomial t prediction method and another numerical technique, which is based on a Taylor Series t prediction method. The works o Kando, Aoyama, and Iwazumi s 4 and Shousse and Taylor 5 show a multirate observer design that is used or singularly perturbed systems. They, also, as with Phillips and Tomizuka 1, use a slow and ast sampling observer together. In Hara and Tomizuka s work o 6, a multirate estimation technique is developed primarily or the purpose o obtaining smooth estimation output. In addition, Thein and Misawa 7 proposed the Parallel Observer System (POS) which provides stable state estimates and is comprised o an augmented system, so as to allow or matched uncertainty estimation. Thein and Misawa 8 also developed the Adaptive Parallel Observer System (APOS), a modi ed orm o the POS, which provides stable state and system parameter estimates, despite the presence o aliased resonant requencies and time-varying system parameters. In turn, the stable system parameter estimates enable the extraction o the actual system damping ratios and aliased resonant requencies. More recently Thein introduced the Variable Structure Parallel Observer (VSPOS) 9, which incorporated a Discrete Sliding Mode Observer in the original POS network. It should be noted that none o the above estimation techniques are eectively robust against unmatched uncertainties/disturbances, except the VSPOS. However, there was no guarantee o perormance in the presence o noise. In this paper, the original Double Boundary Layer Discrete Sliding Mode Observer (DBL-DSMO) is reviewed. In addition, a modi ed orm o the Parallel Observer System (POS), the Variable Structure Parallel Observer System or Noisy Measurements (VSPOSN) with the same orm as the original VS- POS and incorporating the DBL-DSMO, is presented. Here, stable state estimates are provided in the presence o noisy measurements and unmatched uncertainties/disturbances. In addition, theoretical development and stability proo are provided. Finally, the simulation results o the Variable Structure Parallel Observer System or Noisy Measurements (VSPOSN) implemented on a resonance model are shown. All vector norms in this paper reer to the Euclidean (L 2 ) norm, and all matrix norms are one o the induced p-norms (p 1, 2, ), whichever is the most convenient to apply. Both vector norms and matrix norms are denoted by. II. THE DBL-DSMO Studies in continuous time sliding mode control have been very well developed within the past several decades. Only recently has discrete-time sliding mode structure been applied to observer design. Thein and Misawa 1 proposed a discrete-time sliding mode observer (DSMO) with proven stability in the presence o bounded matched and unmatched disturbances/nonlinearities. The DSMO had an attractive boundary layer, outside o which the DSMO had convergent quasi-sliding mode dynamics, and within which the estimate errors had a nite bound. Most recently, Thein urther analysis o the DSMO resulted in the Double Boundary Layer DSMO (DBL-DSMO) 11, where the estimate error, despite the presence o noise, is guaranteed not to chatter back and orth along the outer boundary layer. On the contrary, the DBL-DSMO state estimate error trajectory converges to within the smallest existing inner boundary layer that satis es given stability requirements, despite the original choice o the boundary layer used in implementation. The author considers the ollowing system: x(k + 1) A + A x(k) + B + B u(k) +B + B u(k) + Γ(k) y(k) Cx(k) + δ n (k) (1) where x(k) R N is the state, y(k) R is the scalar output such that C 1 1 (N), and δ n (k) R represents measurement noise. Uncertainties are denoted by A, B,

2 and u(k). Any nonlinearities and/or external disturbances are denoted by Γ(k). The given system is stable and (A, C) is completely observable. The system in Equation (1) may be written as x(k + 1) Ax(k) + Bu(k) + δ(k) y(k) Cx(k) + δ n (k) (2) where all matched and unmatched disturbances/uncertainties are lumped together in δ(k) such that δ(k) Ax(k) + B u(k) + Bu(k) + u(k) + Γ(k). The DBL-DSMO is o the ollowing standard orm: ˆx(k + 1) Aˆx(k) + Bu(k) + Hy(k) ŷ(k) ( ) y(k) ŷ(k) +K sat φ ŷ(k) C ˆx(k) (3) the same as that o its continuous-time counterpart. The observer gain matrix is H R Nx1, and the variable structure component sat( ) is the saturation unction with gain K R Nx1 and boundary layer φ R. Note that y(k), as in Equation (2), still includes measurement noise δ n (k). Under certain conditions, it has been shown that i H, K, and φ are chosen appropriately, then the DSMO is stable with the boundary layer φ acting as a demarkation parameter dierentiating the area o quasi-sliding mode behavior rom the smaller area o bounded estimate error. It is shown that an arbitrarily large outer boundary layer φ o can be chosen to yield robust state estimation in the presence o matched and unmatched uncertainties/disturbances and measurement noise. Furthermore, given that certain stability requirements are met, it is seen that the state estimate error trajectory approaches the smallest existing inner boundary layer φ i, within which the estimate errors have a nite bound. III. THE VSPOSN The proposed Variable Structure Observer System or Noisy Measurements (VSPOSN) is based upon the Parallel Observer system (POS) 7. The POS has two separate observers running in parallel. The Slow Observer System perorms at the output measurement period T s (the slower rate), and the Fast Observer System runs at the control input period T (the aster rate). Both systems are Luenberger Observers. The Slow Observer state estimates are used to eed the Fast Observer state estimates. By using a stable Slow Observer to estimate all states during the times o available measurement and using these estimates to drive a Fast Observer, a stable set o all estimates is available during control sample points. This is an added advantage over having just a single output measurement available to drive estimates to actual values. The Parallel Observer System s Fast Observer is highly dependent upon the accuracy o the estimates o the Slow Observer. Thereore, it is crucial to the success o the Parallel Observer System that the Slow Observer be made robust. It is, thereore, decided that the Slow Observer be replaced with a better estimation technique. Previously, the Adaptive Parallel Observer System (APOS) was developed by Thein and Misawa 8. Here, a Discrete Adaptive Observer (DAO) developed by Suzuki et al. 12 replaces the Slow Observer to provide stable state and system parameter estimates to the Fast Observer. The DAO also takes into account time-varying system parameters and requencyaliased vibrations within the system. The overall APOS was proved to provide convergent state estimates during the ON sample periods and stable estimates with nite error bounds during the INTER sample periods. The DAO (and, thus, the APOS) was shown, however, to be highly susceptible to disturbances and uncertainties. It is decided, then, that a Variable Structure System estimation technique be applied or noise-corrupted measurement, in addition to added robustness against disturbances and uncertainties. The variable estimation technique that is used is the DBL-DSMO. This DBL-DSMO allows or bounded matched and unmatched uncertainties/disturbances and is proved in 11 that it is robust against measurement noise. It has an attractive boundary layer, outside o which the DBL- DSMO has convergent quasi-sliding mode dynamics, and an inner boundary layer within which the estimate errors have a nite bound. One may consider the ollowing SISO discrete time system: x(m, n + 1) A + A x(m, n) + (m, n) + B + B u(m, n) +B + B u(m, n) y(m, n) C s x(m, n) + δ n (m, n) (4) The states to be estimated are represented by x R N 1, and the single output is represented by y R. The measurement is subject to noise, denoted by δ n (m, n). Here, A R N N and B R N 1 are the discretized system parameters. The control input u is determined by an observer-based control technique and is applied in discrete time, and the sample time is denoted by T. The output measurement is available once every k sample periods. Thereore, the observer cycle lasts kt. (The term cycle reers to the sampling period o the output measurement and is denoted by T s.) In addition, m reers to the output measurement time step (or cycle), and n reers to the control input time step within each cycle m. Also, A R N N and B R N 1 represent parametric uncertainties, and u represents added input disturbances. There is an added disturbance (m, n) R N 1 which represents any other existing uncertainties or disturbances. Equation (4) may be written in a orm with an augmented state to include the matched uncertainties: x(m, n + 1) w(m, n + 1) A B 1 B + u(m, n) δ (m, n) + x(m, n) w(m, n) y(m, n) C s x(m, n) + δ n (m, n) (5)

3 The matched uncertainties are represented by w(m, n) R, a scalar quantity, such that B w(m, n) A x(m, n) + B u(m, n)+b u(m, n)+ B u(m, n)+ δ w (m, n), where δ w (m, n) R N 1 represents any additional uncertainties that satisy the given matching condition. The term δ (m, n) represents any remaining unmatched uncertainties. The augmented state in Equation (5) is implemented so as to be able to estimate the nominal value o the matched uncertainty w(m, n). The proposed multirate state estimation technique, the Variable Structure Parallel Observer System or Noisy Measurements (VSPOSN), is model based. The VSPOSN method assumes that the control input signal into the plant is available at every sampling period T and that the output measurement is available once every k control sampling periods (or once every cycle). This means that there are k 1 sampling instances each cycle where the output measurement is not available. In addition, it assumed that a bounded level o measurement noise is present in the system. The VSPOSN has two separate observers running in parallel. The Double Boundary Layer Discrete Sliding Mode Observer (DBL-DSMO) perorms at the output measurement period T s (the slower rate), and the Fast Observer System, a Luenberger Observer, runs at the control input period T (the aster rate). The DBL-DSMO state estimates are used to eed the Fast Observer state estimates. By using a stable slower rate DBL-DSMO to estimate the states during the times o available measurement and using these estimates to drive a Fast Observer, a stable set o estimates is available during all control sample points. This multirate architecture has an added advantage in that the DBL-DSMO provides the Fast Observer with stable ull state order estimates, as opposed to just a single, noisy output measurement, used in standard single rate observer structures. In this paper, or any given sample point n, an ON sample point is considered such that n k Z +. For all other cases, n is de ned as an INTER sample point. The structure o the Double Boundary Layer Discrete Sliding Mode Observer in this application is as ollows: ˆxD (m + 1) ŵ D (m + 1) As B s ˆxD (m) 1 ŵ D (m) Bs Hs + u(m) + ỹ(m) h s ( ) Ks ỹ(m) + sat k s φ ŷ D (m) C s ˆx D (m) (6) where ỹ(m) y(m) y D (m) study, discretized with a sample rate o T s (or kt ). Matrices A s R N N and B s R N 1 de ne the system in study, discretized with a sample rate o T s (or kt ). The observer gains are H s R Nx1 and h s R or the state and uncertainty estimates, respectively. The variable structure component sat( ) is the saturation unction with gains K s R Nx1 and k s R and boundary layer φ R. Since the DBL-DSMO updates itsel during the ON sample points, it is a single rate Variable Structure Observer, where the output measurements and control input values are always available. Note that the ollowing equivalent control input may also be implemented in Equation (6) so as to take into account all control signals measured at sample period T : B s u(m) k i Ak i B u (m, i). In this way, although the DBL- DSMO is running at the sample period T s, no control input goes undetected, including input signals that are implemented during INTER sample points. The proposed Fast Observer is a Luenberger Observer o the ollowing orm: A B ˆx (m, n + 1) ŵ (m, n + 1) 1 B + u(m, n) L L + w l l w ˆx (m, n) ŵ (m, n) ɛ (m, n) ɛ w (m, n) ŷ (m, n) ˆx T s (m, ) ŵ s (m, ) T and y (m, n) ˆx T s (m, ), ŵ s (m, ) T. Here, ɛ (m, n) ˆx D (m) ˆx (m, n) i n and otherwise. Similarly, ɛ w (m, n) ŵ D (m) ŵ (m, n) i n and otherwise. Also, ŵ (m, n) is the Fast Observer estimate o the nominal value o the matched disturbance, and L R N N, L w R N 1, l R 1 N and l w R are the corresponding appropriate Luenberger gains. The reader should note that the eedback measurement o the Fast Observer is ˆx T s (m, ) ŵ s (m, ) T, allowing or ull state and uncertainty eedback rom the DBL-DSMO state and uncertainty estimates when they are available. And, since the eedback is taken rom the DBL-DSMO, the Fast Observer is not directly aected by the existing measurement noise. Thus, the Fast Observer takes advantage o the DBL-DSMO s n+1 stable state estimates or eedback (when it is available), as opposed to just a single, noise-corrupted measurement. Thereore, despite the noisy and limited measurements, the Fast Observer is designed to provide estimates at each control sample time step and more e ciently converge to the stable DBL-DSMO s estimates during the ON sample steps. IV. STABILITY ANALYSIS Assumption 1. Realλ i (A ) < 1 and Realλ j (A s ) < 1, where λ( ) represent system eigenvalues. Assumption 2. The requency content o the control input does not exceed the system Nyquist requency and that the sample rate o the Fast Observer System is above that o the Nyquist requency. Assumption 3. The unmatched uncertainty δ s (m) or the ON sample points and the unmatched uncertainty δ (m, n) and or the INTER sample points are bounded such that max i ( δ s (i) ) δ s and max j ( δ (m, j) ) δ. Here, δ s and δ are nite values. Assumption 4. Let K be chosen such that K i δ s. The stability o the VSPOSN in the augmented state implementation, as in Equation (5), ollows that o Section 2 (7)

4 and can be applied appropriately or the augmented state. As such, the proo will not be repeated here. The stability proo o the VSPOSN is given in this section. Here, we assume that all o the stability and convergence requirements or the o the Double Boundary Layer Discrete Sliding Mode Observer (DBL-DSMO), as described in 11, are satis ed. The ollowing terms are used in the development o stability proo or the Variable Structure Parallel Observer System or Measurement Noise (VSPOSN): e x(m, n) ˆx (m, n), e w w(m, n) ŵ (m, n), ɛ (m) ˆx D (m) ˆx (m, ), and ɛ w (m) ŵ D (m) ŵ (m, ). The proo o convergence or the VSPOSN is similar to that o the POS 7, 8, 9 and involves the use o stability analysis based on eigenvalues. De ne S,, A ls, and F such that S As L s C s B s l s C s 1 F, A k A ls L A k l l w A L B l 1 L w, As A k L B s A k L w l (1 l w ) Assumption 5. All o the necessary requirements or the stability and convergence o the Double Boundary Layer Discrete-Time Sliding Mode Observer (DBL-DSMO), as described in the previous section, are satis ed. In the Fast Observer rom Equation (7), ŷ (m, n) C ˆx (m, n). Because measurements are available once every k INTER sample periods, this means a measurement is available when n k, that is, when (m, n) (m + 1, ). Thereore, ɛ (m, n) ɛ (m) i n, and ɛ (m, n) or n. Similarly, ɛ w (m, n) ɛ w (m) i n and ɛ w (m, n) or n. Denote δ (m, n) as the bounded unmatched uncertainties during the INTER sample points as described in Assumption 3. In addition, one de nes the ollowing parameters such that e (m, ), w e w (m, ), D ɛ (m, ), w D ɛ w (m, ), C D L D + L w w D, C wd l D + l w w D, A A, B B, F, C D C D, C wd C wd, ω w, and (as previously) δ n,o such that δ n (m) δ n,o. Theorem 1. Let Assumptions 1 through 5 hold, in addition to requiring the system in study to be stable. Furthermore, restrict k, the number o INTER sample points between two consecutive measurements, to be a nite integer and the matrices in Equation (8) to be stable. Then, given the original system o the problem statement, the Double Boundary Layer Discrete Sliding Mode Observer o Thein 11, and the Fast Observer System described in Equation (7): 1) The Fast Observer state and uncertainty estimates during the ON sample points are stable. 2) The Fast Observer state and uncertainty estimates are stable during the INTER sample points. Furthermore,, (8) at a general time step n, the state and uncertainty estimate errors have nite bounds o, respectively: e (m, n + 1) A n+1 F + A n C D +Φ B (w + C wd ) + δ e w (m, n + 1) ω + C wd (9) where Φ 1 An+1 1 A Proo: On the ON sample points, the Fast Observer may be represented as ˆx (m + 1) As B s ˆx (m) ŵ (m + 1) 1 N 1 ŵ (m) Bs + u(m) This means that ˆx (m + 1) ŵ (m + 1) and that ɛ (m + 1) ɛ w (m + 1) +A ls ɛ (m) ɛ w (m) F ˆx (m) ŵ (m) +A ls ˆxD (m) ŵ D (m) Bs + u(m) ɛ (m) Hs F + ỹ ɛ w (m) h D (m) s ( ) Ks ỹd (m) + sat k s φ where ỹ D y(m) y D (m). One may rewrite the above equation simply as E (m+1) F E (m)+ p (m), where E (m + 1) augments the state estimate error vector ɛ (m) with the disturbance estimate ɛ w (m) and p (m) is composed o the remaining terms in the above equation. De ne E to be E (m ). Then, the propagated error based on E and p (m) may be written as unction o cycle m, such that E (m + 1) F m E + m i F m i p (i). From the restriction on F, lim m F m (N+1) (N+1) m 2 and lim m E (m) lim m i F m i 2 p (i). Hence, given that F F and p p, it is shown m that E (m) lim m i F m i 2 p lim m p 1 F m 1 F p 1 F. Assuming that the given system to be estimated is stable and that the DBL- DSMO state and uncertainty estimates are stable (rom Assumption 5), it can be inerred that p is a nite value, making the overall error dynamics between the DBL-DSMO and the Fast Observer stable. In turn, then, the Fast Observer state and uncertainty estimates are also stable. Thereore, Part (1) o Theorem 1 has been proven. The stability proo or the INTER sample points or the Variable Structure Parallel Observer System uses the error dynamics or the Fast Observer System with uncertainty estimation: e (m, n + 1) A e (m, n) + B e w (m, n) L ɛ (m, n) L w ɛ w (m, n) + δ (m, n) and e w (m, n + 1) e w (m, n) l ɛ (m, n) l w ɛ w (m, n). Using the

5 expressions de ned or Theorem 1, Schwartz s Inequality, triangle inequality, and nite series analysis, one may conservatively calculate the resulting propagated errors as e (m, n + 1) e (m, n + 1) A n+1 F + A n C D + 1 A n+1 1 A B (w + C wd ) + δ and e w (m, n + 1) ω + C w. Remark 1. In the worst case scenario, the error in the state estimates or the Fast Observer System continue to increase during the INTER sample points until the next measurement. Under this condition, then, the greatest maximum bound or any INTER sample point would occur at n k, or such that (m, n + 1) (m, k). Remark 2. Given the results o Theorem 1, an upper bound o the error estimates o the Fast Observer state estimates ˆx (m, k) is e (m, k 1) A k F + A k 2 C + B (w + C w ) + δ, and the upper bound o the uncertainty estimation ŵ is e w (m, k 1) w + C w. The reader should note that the above bounds may be extremely conservative and may not be o practical use except to show that the INTER sample estimate bounds are, indeed, nite. V. EXAMPLE A second order system is used to con rm the validity o the DBL-DSMO and the proposed VSPOSN. A second order resonance model is studied in this example, where the natural requency is.16hz and the damping ratio is.5. The states to be examined are position x 1 and velocity x 2. The system control sampling period is.7s, while the measurement output (position) is available only once every.35s (i.e., k 5). Here, the control input is u(t) sin(.26πt). The unmodeled disturbances/uncertainties are given as 1 (t).2 sin(.18πt) or the position dynamics, and 2 (t).1 sin(.114πt) or that o the velocity dynamics. Measurement noise is introduced into the system simulation using the MATLAB Random Number Generator with a mean o and variance o.1. The VSPOSN observer parameters are H.1 (a 1,2 + a 2,1 ) T, where a is the discretized A matrix o the second order resonance model, K.2.1, and φ.6. The Fast Observer and PRES observer gains are chosen so that both observer poles are.95 ±.6i. For this example, none o the observer states are augmented, as in Equation (5), to allow or uncertainty estimation, and both observers are implemented outside o the control loop. In addition, to show the simplicity o and the ease o application o the DBL-DSMO, the DBL-DSMO parameters were not nely tuned. For example, the outer boundary layer φ was chosen arbitrarily large. Figures 1 and 2 show the position and velocity estimate responses, respectively, or the DBL-DSMO and VSPOSN. Convergence o the DBL-DSMO and VSPOSN to within the boundary layer is ast, despite the large dierence in initial conditions, the presence o unmatched disturbances/uncertainties, and the high level o noise. In act, it is di cult to dierentiate between the DBL-DSMO and Position Actual.. DBL DSMO Estimate... Measurement Noise VSPOSN Estimate Fig. 1. Velocity State and Estimates o x 1 and Measurement Noise... Measurement Noise... Actual.. DBL DSMO Estimate VSPOSN Estimate Fig. 2. State and Estimates o x 2 and Measurement Noise VSPOSN estimates in the gures. A better view is given in Figures 3 and 4, where the estimate errors o the DBL- DSMO and VSPOSN are given or the position and velocity, respectively. The trajectories o the phase plane plots or the DBL- DSMO and the VSPOSN estimate errors (not shown here) quickly converge to within the respective boundary layers, despite the act the outer boundary layer was chosen arbitrarily large. VI. CONCLUSIONS AND FUTURE WORK A modi cation o the Parallel Observer System is proposed to address the issue o noisy, periodic measurements with matched and unmatched uncertainties/disturbances. Here, a Double Boundary Layer Discrete Sliding Mode Observer (DBL-DSMO) System is implemented in place o the Slow Observer. The DBL-DSMO is made robust against the uncertainties/disturbances and against measurement noise. As a result, the improved Parallel Observer System, the Variable Structure Parallel Observer System or Measurement Noise

6 Position Error DBL DSMO VSPOSN Fig. 3. Estimate Errors o x 1 Velocity Error DBL DSMO VSPOSN Fig. 4. Estimate Errors o x 2 (VSPOSN), is proven to yield stable and convergent state estimates, despite the act that measurements are noisy and available only periodically. The VSPOSN is applied to a second order resonance model with unmatched disturbances. The simulations show that the VSPOSN state estimates quickly converge to, and remain within, the observer boundary layer. Future work includes the modi cation o the VS- POSN to account or requency-aliased vibrations and timevarying system parameters. It is also planned to eventually develop a stable observer-based control scheme to which the VSPOSN will be implemented. 2 Weiping Lu and D. Grant Fisher. Multirate adaptive inerential estimation. IEE Proceedings-D, 139(2): , Parthasarathy Ramachandran, Gary E. Young, and Eduardo A. Misawa. Intersample output estimation with multirate sampling. In Proceedings o the IEEE Conerence on Control Applications, pages , Hisashi Kando, Tomohiki Aoyama, and Tetsuo Iwazumi. Multirate observer design via singular perturbation theory. International Journal o Control, 5(5):25 223, Kenneth R. Shousse and David G. Taylor. Discretetime observers or singularly perturbed continuous-time systems. IEEE Transactions on Automatic Control, 4(2): , February Takeyori Hara and Masayoshi Tomizuka. Multi-rate controller or hard disk drive with redesign o state estimator. In Proceedings o the American Control Conerence, pages , May-Win L. Thein and Eduardo A. Misawa. A parallel observer or non-deterministic multirate systems. To be published in the Transactions o the ASME Journal o Dynamic Systems, Measurements, and Control. 8 May-Win L. Thein and Eduardo A. Misawa. An adaptive parallel observer or multirate systems with requency-aliased output measurements. To be published in the Transactions o the ASME Journal o Dynamic Systems, Measurements, and Control. 9 May-Win L. Thein. A variable structure parallel observer system or periodic measurement output. In Proceedings o the American Control Conerence, pages , May-Win L. Thein and Eduardo A. Misawa. A discrete time sliding mode observer with an attractive boundary layer. In Xinghuo Yu and Xu Jian-Xin Xu, editors, Advances in Variable Structure Systems: Analysis, Integration, and Application, pages World Scienti c, May-Win L. Thein. A discrete time variable structure observer or uncertain systems with measurement noise. In Proceedings o the IEEE Conerence on Decision and Control, pages , Takashi Suzuki, Takumi Nakamura, and Masanori Andoh. Discrete adaptive observer with ast convergence. International Journal o Control, 31(6): , 198. VII. REFERENCES 1 Anthony M. Phillips and Masayoshi Tomizuka. Multirate estimation and control under time-varying data sampling with application to inormation storage devices. In Proceedings o the American Control Conerence, pages , 1996.