About Optimal Fractional Hold Circuits for Inter- sample Output Reconstruction in Sampled-data Systems.

The design of fractional order-holds (FROH) of correcting gains β ∈[−1,1] (potentially and possibly including zero-order-holds, ZOH with β=0, and first-order-holds, FROH with β=1) is discussed related to achieving output deviations being close with respect to its sampled values. A squared error time- integral between the current output and its sampled values is minimized to yield the appropriate correcting gain of the FROH in an analytic way.


Introduction
It is very common to obviate the continuous-time viewpoint when dealing with discrete-time or digital systems. However, in a wide class of practical problems, discrete-time systems are obtained by the use of sampling and hold devices which operate on continuous signals in order to facilitate the subsequent technological treatment of signals such as data storage and transmission. Another advantage is that discrete-time controllers are easier to implement while they are much more robust than continuous-time controllers. The implementation of discrete-time models is directly related to the sensor technology. However, it is highly suitable that the outputs of the discrete -time system be as close as possible to those of their continuous-time counterparts in problems where discretization is a technical tool to synthesize controllers or to transmit digitalized data . Another objective of interest is to achieve inter-sample outputs being as close as possible to their values at sampling instants. This design objective allows keeping the ripple deviations within small admissible levels in problems where the tracking performances are analyzed from a discrete-time point of view, i.e. at sampling instants, while some components are of a continuous-time nature. A typical situation within this class is the use of discrete-time controllers for continuous-time industrial plants. It is well-known that properties like positive realness of important usefulness in Circuit Theory and Hypertstability issues are often lost when discretizing a system 2 1− . However, other properties like, for instance, stability degree of unstable discrete-zeros may be improved , and even achieved, by using appropriate discretization techniques as, for instance, multirate sampling with fast input sampling 9 , 4 , 3 . The current technology of sensors is very linked to sampling techniques. For instance, non-periodic sampling may improve the efficiency of sampled data processing 7 5− . This strategy includes both selection of samples at certain sampling instants and choice of transmission or blocking of certain data, what in fact relies directly on nonperiodic sampling as well. On the other hand, the choice of the sampling rates is important in many applications 8 . A very close problem to the choice of the sampling rate is the choice of the sampling and hold device used for discretization. The underlying philosophy is that the choice of the gain of such a device is a degree of freedom which may be used to improve the discretization efficiency as an alternative to the choice of the sampling period or the sequence of sampling rates. The analysis is performed by using a quadratic loss function which minimizes the errors within the inter-sample period.

Continuous-Time Plant
Consider the linear time-invariant single-input single-output plant: . A fractional order -hold generates an input of the form: is the k-th sample of the input with sampling period T. Usually, or β=1 the FROH is a ZOH or a FOH, respectively. Fractional-order-holds have been studied related to the improvement of the stability of the discrete plant zeros which is of interest in problems like pole-placement or model-matching controller synthesis since the reference model might be designed with more relaxed ´a priori´ constraints on the need of include prefixed plant unstable zeros 4 1− . In this paper, we discuss the choice of fractional order-holds as a way to improve the deviations of the inter-sample reconstructed output signals compared to their sampled values compared to the use of standard zero-order and first-order-holds.

Inter-Sample Description
Assume that (1) is discretized with a sampling period T so that the state evolution (1) becomes for any One gets from (3) and using (4): Now, taking time-derivatives in (5) with respect to time in (0, T) via (3) yields:

Approximate Output Description of the Inter-Sample Behavior
, first and second-order Taylor series approximations about using the output equation of (1) and (5)-(6) yield: for a first-order approximation and with I being the n-th identity matrix. Eqs. 8 are obtained after substituting (5)-(6) into (7.a), by using the output equation (1) and then comparing the expressions (7.a)-(7.b) to make them identical. The effect of the parameter ρ and its usefulness will become relevant in the next section to evaluate the inter-sample performance related to the correcting gains either for the whole inter-sample time intervals or for only one portion of them which may be considered relevant by the designer depending on the application. If the designer knows that a subinterval of the inter-sample interval is not relevant because the output deviation is negligible with respect to its previously sampled value, then such a subinterval could be removed from the loss function by appropriately selecting the parameter ρ .

Choice of β for Minimal Output Deviations In-Between Samples
In this section, a strategy is used to achieve closeness in-between of the output signal at sampling instants and its sampled values at sampling instants by the choice of the fractional order-hold correcting gain. The main objective of the design is to minimize the ripple effects in-between consecutive samples. Since first or second-order approximations are used to calculate approximately the output in-between samples, the procedure can be considered as a suboptimization one with associate small or moderate computational cost. For that purpose, consider the inter-sample-output deviation performance function:  . In practice, the above limit may be replace for a sufficiently large time interval taking account of the transient period.

Example 1
Consider the linear and time-invariant continuous closed -loop system: + under unity negative feedback and external unity step reference input 1(t). Such a continuous system is discretized via a FROH placed at its plant input with a sampling period T = 0.1 secs. The optimal fractional order -hold correcting gain according to a second-order approximation (i.e. α =1) of the loss performance (9) Table 1 below: The loss function versus the gain of the fractional order-hold over 100 samples is displayed in Figure 2 below with notches for the values for  Table 1 over 100 samples , it turns out that the absolute variations of the loss function from the correcting gains β= -0.4 to β = 0.51 are 99.32 for 100 samples and 103.632 for 10000 samples. The respective relative variations of the exact loss function are 51.73% and 0.34%. The interpretation is that the variations are very significant during the transients as the correcting gains (and then the type of FROH) change while the variations are small in the steady-state, as expected. Note that a steady-state behavior is attained fast due to the plant stability and the fact that a constant reference input is used. It may be claimed that if the steady-state is achieved very slowly or a fast changing reference input is used then the variations related to used FROH are kept through time. This claim will be confirmed in Example 2 below where a quasi-square reference signal is injected to the plant. On the other hand, it might be noticed that, in general, neither the widely used zero-order-hold or the first-order-hold are the best options for reconstruction of the output in-between sampling instants closely to its sampled values. Note also that such a behavior is highly dependent on the chosen correcting fractional order-hold correcting gain. A The closed-loop plant outputs for the three tested sampling and hold devices are displayed in Figures 4 together with the reference signal injected to the reference model. Note that there is an asymptotic tracking of the reference output by the plant output as expected with a negligible inter-sample ripple deviations, related to their maximum values, for the three sampling and hold devices tested as expected. However, the three behaviors are quite distinct during the adaptation transients. The fractional order-hold with β =0.2 exhibits the better performance transient behavior, the first-order-