Elsevier

Journal of Process Control

Volume 22, Issue 10, December 2012, Pages 1865-1877
Journal of Process Control

RST controller design for a non-uniform multi-rate control system

https://doi.org/10.1016/j.jprocont.2012.09.010Get rights and content

Abstract

In this work, a non-uniform multi-rate controller which includes an RST control stage is introduced. Due to several issues, in some systems the use of non-uniform (irregular) multi-rate sampling becomes inevitable. When using a uniform (regular) multi-rate controller in this kind of situations, control performance is usually degraded (if it is compared to that obtained in the nominal, uniform sampling context). Thus, the design of a non-uniform controller proper for this sampling scheme is needed to keep the performance. A sequence of different non-uniform sampling frames can be considered, and different controllers must be designed for each frame using appropriate methods. When switching among these controllers, stability problems can appear. So, the control system stability will be assured in terms of Linear Matrix Inequalities (LMIs). To achieve some advantages at the design step, the discrete-time input–output representation will be used. But, since the classical, uniform zi delay operator could not be able to represent non-uniform sampling situations, the so-called non-uniform operator will be required. Simulation results illustrate that this control proposal is able to keep control system performance and preserve stability.

Highlights

► A theorem for the design of a non-uniform multi-rate controller which includes an RST control stage is introduced. ► Another theorem presents the so-called non-uniform operator to face non-uniform frames at the model stage. ► The control system is designed to deal with a sequence of different non-uniform sampling frames. ► Both robustness and control performance benefits are derived. ► Stability is assured by means of LMIs. Sensitivity to modeling errors is studied.

Introduction

There are some systems where the non-uniform (irregular) sampling appears in a natural way. It can be shown in hard disk drive head position control, where the location of damaged servo sectors and the collision in the self-servo track writing process make the feedback position error signal unavailable, resulting in a non-uniform sampling rate [1]; in real-time operating systems, where applications are implemented by decomposing them into several tasks in such a way that, due to task priorities and resource sharing, pre-emption and blocking may appear, arising irregularities in the sampling [2], [3], [4]; in networked control systems where sensors, controllers and actuators are connected through a common communication bus, which introduces timing variations in the control loop due to network induced delays, packet dropouts, and packet disordering [5], [6], [7]; in event-based systems where the sampling is triggered by the occurrence of some events, which usually appear following non-uniform patterns [8]; in many chemical processes, where variables which indicate product quality by means of chemical analyzers are infrequently and irregularly sampled [9], [10], [11]; and so on.

As a consequence of most of these sampling irregularities, control updates and output feedback measurements work at different rates. So, using multi-rate control techniques becomes a natural solution. In this work, feedback data is assumed to be scarce (which can be a realistic assumption in most of the previous environments). Then, a Multi-Rate Input Control (MRIC) scenario will be considered, where the control signal updating is faster than the output one (as known, actuating at a faster rate than measuring may provide robustness advantages [12]). But some restriction will be imposed to the MRIC approach: all the process inputs and outputs must be necessarily available at the slow rate time instants (that is, at the initial time of the sampling frame). This is not a hard restriction in most of the previous applications, except in networked control systems as a consequence of disposing no direct link among devices. Loss of packets, communication delays, etc. could appear, and hence all the samples could not be available at the required moments. In this particular case, some additional solutions can be adopted depending on the problem to be treated. For example, in [7] and [13] gain scheduling approaches were proposed to deal with delayed slow rate process inputs (that is, control actions). In [14], a non-uniform observer was introduced to estimate lost slow rate process outputs. These are two of the possible scenarios faced in previous authors’ works, but other situations can be found in [5], [6], and literature therein.

Whereas the uniform multi-rate sampling case has been widely studied (since the seminal work by Kranc [15] until current works such as [16], [17], [18], [19], [20]), the non-uniform case has been less treated (despite existing a lot of situations where it could be applied). To the best of the authors’ knowledge, few research groups gather the majority of works that deal with this kind of sampling, where usually the non-uniform sampled-data system is represented as a Linear Periodically Time-Varying (LPTV) system. Refs. [1], [3], [21], [22], [23] are some examples of the referred works. Only in few of these works (such as [3], [21]) the system is modeled by means of the input–output representation. Nevertheless, most of the works use the state–space representation, and more concretely, the well-known lifting technique to model and design the multi-rate control system. The main reason is because it facilitates the modeling step. Nevertheless, this technique introduces two main drawbacks at the design step: it could become more complex, and the consequent controller presents time-varying gains, which worsen the inter-sample behavior [24]. In the present work, these drawbacks are solved by using a specific algebra based on the so-called skip-expand operators [17], [18], [25]. As a consequence of using these operators, three benefits are derived:

  • The closed-loop system is not represented by an augmented lifted system, otherwise by a transfer function, which facilitates the design step.

  • When using the expand operator followed by a desired linear stationary filter (in our case a digital Zero Order Hold, ZOH), the slow-rate signal of the multi-rate control system can be approximated at a fast-rate signal, which enables to design the fast-rate side of the multi-rate controller.

  • When considering skip-expand properties, the multi-rate design complexity is reduced.

The non-uniform sampled-data modeling proposed in [3] is now revised, and its notation is improved and adapted to be used when designing the non-uniform multi-rate controller. The non-uniform modeling is based on the non-uniform operator, whose aim is to introduce some replacements on the uniform, classical operator zi in order to adapt the latter one to a non-uniform sampling pattern.

The multi-rate controller is designed via a model-based procedure. In [17] the multi-rate design for the uniform sampling case is introduced. In the present work, the non-uniform sampling scenario is studied and, in addition, an RST control stage is included in the multi-rate design. The RST controller is very popular due to its good compromise between performance and complexity. It is a two-degree-of-freedom controller obtained via an input–output model-based pole placement method, which implies the resolution of a diophantine equation. R–S–T is the names of each one of the polynomials to be deduced by the design procedure. Its two degree of freedom consists of a feedforward side defined by T/R, and of a feedback side defined by S/R. In [26], the general design procedure for the RST controller is introduced. In our work, this general method is adapted so as to include this controller in the multi-rate control system. The RST stage is designed to not cancel the numerator of the process transfer function. This fact, together with the use of the digital ZOH (commented previously), assures steady-state ripple-free closed-loop response to step reference signal [17], [18]. Including the RST stage in the multi-rate controller simplifies the multi-rate control design. In [26], more details about this controller can be found.

An interesting benefit of the present work is the consideration of a sequence of different non-uniform sampling patterns (say, the non-periodic case). That means designing several LPTV controllers (one for each non-uniform sampling pattern) and switching among them according to the current frame. As known, it is possible to get instability by switching among asymptotically stable systems [27]. Then, to assure stability, some Lyapunov conditions will be formulated in terms of Linear Matrix Inequalities (LMIs).

The present paper is organized as follows: in Section 2 preliminaries and notation are introduced. The input/output modeling for non-uniform multi-rate sampled-data systems is reviewed. Its notation is improved and adapted to be used in the design proposal, yielding a theorem for the modeling step in Section 3. Section 4 presents another theorem, this one related to the design step, which describes how the RST control stage is included in the non-uniform multi-rate controller. Control system stability is studied from two points of view: sensitivity to modeling errors and stability for switching systems (which is enunciated in terms of LMIs). In Section 5, a simulation example is shown. Results enable to observe control performance degradation when using a (unique) uniform multi-rate controller in a non-periodic environment. Nevertheless, if different non-uniform multi-rate controllers (and the switching among them) are considered, the performance is kept (compared to a nominal, uniform sampling context) and stability preserved (according to the appropriate LMI analysis). Finally, in Section 6, the main conclusions are exposed.

Section snippets

Preliminaries

Since notation used in [17], [18] is becoming a standard fashion to represent multi-rate sampled-data systems, this work will take this notation to introduce preliminary concepts.

Two different Z-transforms can be expressed according to the considered sampling and updating periods over a continuous time signal x(t). So, if the sampling or updating is carried out each T time unitsXT(z)=ΔZT[x(t)]=k=0x(kT)zkwhere X will be the sampled signal, and the variable z−1 represents the T-unit delay

Input/output non-uniform modeling for multi-rate sampled-data systems

In this section, a theorem which enunciates the input/output modeling for non-uniform sampled-data systems is presented. This theorem is defined after reviewing some previous results obtained in [3], and improving and adapting notation to that presented in Section 2.

RST controller design for a non-uniform multi-rate control system

In this section the classical RST controller [26] is adapted to be included in the multi-rate control system. As a consequence of this inclusion, the multi-rate controller design can be simplified. Fig. 2 shows a block diagram in order to easily appreciate how the control system is defined by two parts: a first side, where the RST control action is generated, and a second side that uses this action to calculate the non-uniform multi-rate control updating.

Assumption: As shown in Fig. 2, input

Simulation example

One of the aims of this example is to show how, when a process is sampled in a non-uniform way, an accurate model for it can be reached following the steps provided by theorem of Section 3. Despite its reliability, sensitivity to modeling errors will be studied too.

Another goal of the example is to compare three different situations. The first of them is the uniform sampling case. In this context a uniform multi-rate controller is designed in order to achieve some performance (the nominal one).

Conclusions

An approach to face the appearance of a sequence of different non-uniform sampling patterns in multi-rate sampled-data systems is introduced. This approach is based on including an RST control stage in the multi-rate control system. This inclusion simplifies the multi-rate controller design, which is model-based. The overall multi-rate control system is designed to avoid ripple closed-loop responses to step references.

To develop the design proposal, a suitable input/output modeling for

Acknowledgments

The authors A. Cuenca and J. Salt are grateful to the Spanish Ministry of Education research Grants DPI2011-28507-C02-01 and DPI2009-14744-C03-03. In addition, A. Cuenca is grateful to Generalitat Valenciana Grant GV/2010/018, and J. Salt to the financial support of the Spanish Ministry of Education, “Salvador de Madariaga” Program PR2010-0310.

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