Non-uniform sampling strategies for digital control

My PhD dissertation focused developing signal processing strategies for non-linear digital control applications. I had argued the traditional approaches of sampling information for real time control and proposed algorithms for processing non-uniformly sampled data. The larger question of why novel approaches are necessary for real-time control applications steered me towards systems of complexity. My efforts have since then been praised for introducing alternative algorithms for the control community to process information. This work optimised the control design, and verified this through an experimental prototype. It focused on design theories from concept to detailed design, mathematical modelling and optimisation algorithms. Due to the success of my studies and the strength of my dissertation, I was offered an industrial opportunity with Thales Transportation as a Systems Engineer. Here, I was able gain useful experience whilst working in teams to design and develop circuit models and analyse fault data. This included an in-depth study on the type of faults occurring on the signalling assets being used, and validating the models. These results were feedback as recommendations to improve the existing industrial design plans.

The alias-free signal sampling technique has successfully been used to identify the underlying spectrum of a signal, even at substantially lower average sample rates as compared to their uniform sampling counterpart requirements. Figure 1 illustrates the concept by sampling an 80 Hz signal with a 100 Hz sample rate. With a uniform sampling scheme, the spectral content of the signal is corrupted due to under-sampling. However, when using a nonuniform sampling pattern, the signal is able to retain its main signal component while the aliases are converted into broadband noise. However, these would be insufficient to deal with loops that experience time variations within the controller actions. Historically, the time-domain formulation is the most commonly used method for performing a time-varying analysis due to its simplicity, although such analysis may not provide an accurate answer since the convolution rule does not exist in the case with time-varying systems.

The primary design methods for discretisation of linear continuous-time transfer functions include the standard z-transformation or impulse invariant method, the bilinear transform and related transformation methods and the matched z-transform techniques. All these methods are essentially algebraic in nature and differ primarily only in the details of the approximation from the continuous-time domain to the discrete. These can be used to yield a recursive digital filter approximation for a continuous-time linear system. As an illustration, Figure 2 shows how the conventional approach of converting discrete time filters is used. Once a uniform sample frequency is selected, the coefficients are calculated (just once) and implemented on a digital compensator in real time. Such techniques on control design are highly developed, and are readily available for LTI systems, both in continuous-time and discretetime cases. The problem with a time-varying sampling frequency poses an obvious question: Can the z-transform equivalent be used in a time-varying sampling frequency?

A modification of the algorithm is shown in Figure 4. In the case with the z-canonic structure (Figure 2), when the coefficients of an already working IIR filter are changed, the output result is calculated using the current input and the stored values (i.e. the internal/ state variables). As the coefficients change, it creates a transient at the output of the state variable. In comparison, the delta-operator replaces the usual ‘shift’ operations of a z-structure, with internal variables that are distinct. According to its definition, its internal variables are no longer dependent on the coefficient values. This reduces the size of the transient being created at the filter output. The analysis carried out in this showed that there exists a broad range of behaviours depending on the implementation structure in use. It is pointed out that correctly choosing the proper structure can significantly reduce the undesirable effects of digital filter reconfigurations. Some important conclusions that are drawn from the analysis are that the: 1) transient phenomenon is caused when the internal variables of the filter are not scaled according to the variations in the sampling period. 2) severity of a transient signal depends on the size of magnitude change in the filter coefficients. This indicates that very large changes in between consecutive sample periods produce worse results. 3) modified canonic delta-filter structure offers a better choice for suppressing the transient phenomenon. This is due to the way the state variables are handled internally by the filter structure.

The focus had been placed upon two issues – the development of the non-uniform sampling algorithm and the frequency response analysis. The work emphasised the importance of choosing a suitable implementation structure for intentional non-uniform sampling, and illustrates that the ‘conventional’ functions using the shift operator are not appropriate, identifying instead the use of the delta-operator, which avoids critical algorithmic problems. In addition, the problem of estimating the frequency response characteristics when utilising the non-uniform sampling algorithm was addressed. Of course, time simulations can be used for control design, even with non-uniform sampling, but frequency domain and root locus are commonly used. The authors have not yet considered how root locus design could be done under the nonuniform sampling conditions, but this article shows how the frequency response can be calculated using an adaptation of the Fourier analysis. It should be noted that uniform sampling is still the most preferable method to execute tasks due to its simplicity and scope, and this research study has not highlighted any distinct advantages of non-uniform sampling that will differentiate the choice of its suitability for a control application. However, this work has brought forward some specific digital control issues and attempted to initiate and expand the scope of intentional non-uniform sampling in this subject for research.

For more information see: Samir Khan , Roger M. Goodall & Roger Dixon (2013) Non-uniform sampling strategies for digital control, International Journal of Systems Science, 44:12, 2234-2254, DOI: 10.1080/00207721.2012.687785