A unified event-based control approach for FOPTD and IPTD processes based on the filtered Smith predictor

https://doi.org/10.1016/j.jfranklin.2016.11.017Get rights and content

Highlights

  • A novel unified architecture for event-based control based on the FSP is analyzed.

  • The event-based approach is valid both for self-regulating and integral processes.

  • A simple tuning method to provide robust stability to limit cycles is described.

  • Robustness, disturbances and event-based performance can be simultaneously handled.

  • Simulations and experimental results are provided as illustrative examples.

Abstract

A new unified design of an event-based PID control architecture for self-regulating and integral processes is investigated in this work. The design is based on the symmetrical-send-on-delta (SSOD) sampling technique and on the filtered Smith predictor (FSP). In particular, the conditions to achieve robust stability to limit cycles are studied from the point of view of the filter parameters through the describing functions theory. In this context, a simple frequency domain-based tuning methodology to address the cases of first-order-plus-dead-time and integrator-plus-dead-time processes is proposed. With this method, the properties of the filter parameters are also explored to cope with the set-point tracking and the load disturbance rejection tasks. The effectiveness of the general approach is evaluated by simulations and experimental tests.

Introduction

Until now, the industrial applications of process control have been practically handled by time-based feedback control systems, most of them governed by standard proportional-integral-derivative (PID) controllers. A significant factor of their success is the existence of numerous tuning rules and design techniques [1], which allow the user to provide a satisfactory performance for many processes with a relatively easy design [2].

Despite PID control is well established as a technology, the advancement of industrial processes continuously demands for new developments. A significant challenge has arisen in process control because of the more and more use of wireless sensors and actuators, which conform the well-known networked control systems (NCS) (see [3], [4]). Such systems are a valuable example of distributed architectures that suffer from the lack of efficient sampling techniques in order to minimize the effect of data dropout and stochastic time delays, while saving energy and bandwidth (essentially in battery-powered systems) [5]. In this context, it makes sense to transmit information between agents (sensors, actuators controllers, networks coordinators, etc.) when something significant has occurred and this calls for the techniques based on events [6], [7]. As demonstrated in recent works in the field of event-based PID control [8], [9], [10], [11], [12], a satisfactory trade-off between information exchange and the closed-loop performance can be achieved when a tight tracking control is not of main concern and a small steady-state error is acceptable.

In event-based control it is not the progress of time but the occurrence of events which forces agents to transmit information. Although this is conceptually simple, the design of an efficient control scheme and of a systematic tuning method poses a challenging problem. Note that the event condition may be any mathematical function (see [13], [14] for typical event-based conditions), which can be combined with different techniques (event-based, self-triggered, sporadic, etc.), obtaining different architectures and nonlinear responses [15], [16]. In particular, when wireless communications are considered, the protocol constraints and the energy consumption need to be considered as a part of the control loop design [17].

Even so, some relevant consensus has been encountered. The send-on-delta (SOD) algorithm is the most employed in practice (see [4], [18], [19], [20]) because of its simplicity of implementation. It consists in computing a new sample when a signal changes more than a predefined threshold. Additionally, the use of hybrid sampling schemes, that is, those that combine time-based and event-based sampling schemes in agents involved in the closed-loop, have become one of the most extended design approaches. By this way, the event-based behavior can be characterized using the well-known automatic discrete control theory. Based on the synergy between both frameworks, many theoretical results for event-based PID control have been recently presented and tested on real plants, but unfortunately they are based on particular solutions either for self-regulating or non-self-regulating (integral) processes. Indeed, the case of integral processes has actually not been properly addressed until now.

In this work, a unified framework for self-regulating and integral processes is proposed. It is based on the general model discussed in [21] under the assumption that the event-based sampling technique is applied on the error signal [9], [10], [22]. Note that, although such an approach has been effectively applied to self-regulating processes with delay [23], [24], [25], it is not inherently suitable to be applied to integral processes.

In this sense, a new implementation is proposed in this paper. This results from the redesign of the architecture proposed in [24] in order to obtain a unified approach able to cope with the most common (both self- and non-self-regulating) industrial processes. The new scheme is based on the structure of the filtered Smith predictor (FSP) [26]. The advantages of this structure in comparison with the original Smith predictor one (implemented in [24]) rely on the two additional filters considered, which allow the control system to use a common design to simultaneously handle the performance and the robustness properties for both kind of processes. Under this scenario, the performance and robustness of the proposed event-based structure is analyzed for a significant set of self-regulating and integral processes by providing a tuning methodology. In particular, the describing function (DF) approach [27] is used to determine the design conditions of the filters that robustly guarantee the avoidance of limit cycles for each particular process.

In this context, some theoretical analyses are provided, which are supported by simulations, and experimental results.

The paper is organized as follows. In Section 2, the event-based control system is presented and its behavior is described. In Section 3, the robust stability and limit cycles problems are analyzed. In Section 4, the tuning procedure is described. In Section 5, practical issues on the compensation of load disturbances are introduced. The results from simulation are presented in Section 6 while experimental results are shown in Section 7. Finally, Section 8 presents the main conclusions.

Section snippets

Control architecture

The scheme of the event-based control system (EBCS) analyzed in this work is shown in Fig. 1. In line with the general model [21], the architecture considers three functional/main blocks: the controller, the process and the event generator. Their tasks are described in the following part.

The dynamic behavior of the stable and integral processes under study can be described (or approximated), respectively, by a first order plus time delay (FOPTD) transfer function as:Ps(s)=Kτs+1eLs=Gs(s)eLsor

Robust stabilization of the EBCS

The stability properties of an EBCS can be derived from the analysis of the behavior of its equilibrium points [10]. The analysis is based on the properties of the describing function (DF) of the nonlinearity introduced by the SSOD algorithm and it will be exploited for the tuning of the controller parameters.

The describing function is an approximate method, classically used to analyze nonlinear systems by characterizing the limit cycles experimented in the presence of nonlinear elements [29],

Tuning procedure

As for all industrial control systems, in addition to the achievement of a satisfactory performance, it is important to ensure that the controller is easy to design and for this reason a clear (robust) tuning procedure has to be sought.

Because of the nonlinear nature of the EBCS, the tuning of the filter can make the system unstable, to reach an equilibrium point without limit cycles, or admit a stable limit cycle. For this reason, the proper knowledge of system properties is essential to

Disturbance compensation/estimation algorithm

The event-based control of integral processes as proposed in this work requires an additional (feedforward) compensation action to deal with step load disturbances. Due to the quantized nature of control actions derived from the use of a P controller in the scheme of Fig. 2, the unquantized part of a load disturbance cannot be compensated and, consequently, a bimodal limit cycle surely arises. To clarify this point, we can consider the simplified version of the scheme in Fig. 4 where it is

Simulation results

In this section, simulation results are provided. In particular, the examples illustrate the robust design of the filter for a significant range of different process dynamics (both self-regulating and integral) in order to avoid the presence of limit cycles and the compensation of exogenous signals.

Experimental results

The proposed approach has also been implemented in a real application. For this purpose, a laboratory-scale setup of the Computer Science Department of the University of Cordoba has been used. In particular, the experimental setup consists in the speed control of a brushless DC motor as described in [24]. Since the system has an apparent dead time that is very small in comparison with the dominant time constant, a time delay of 4 s has been added via software at the process output. The estimated

Conclusions

In this paper, a unified structure for an event-based control system has been analyzed. The methodology is based on the integration of the filtered Smith predictor in the event generator.

The robust stability has been addressed from the perspective of the limit cycles introduced by the nonlinear element. A describing function approach has been proposed to this aim, where two of the most common process structures of industrial relevance have been considered for this analysis, namely, FOPTD and

Acknowledgment

This work has been supported in part by Project DPI2012-31303 (financed by the Spanish Ministry of Economy and Competitiveness).

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