Piecewise continuous hybrid systems based observer design for linear systems with variable sampling periods and delay output
Introduction
This work is devoted to observer design for a class of linear systems with variable sampled and delayed measurements. In the last few decades with recent advances in communications and computing technologies, networked visual servoing systems (NVSS) have received a considerable amount and increasing attentions, especially motivated by more flexible contactless wiring, improved signal/noise ratio and common communication network connection [1]. Its have been widely implemented in industrial production, building automation, remote video communicating networks, education, and aerospace exporting. Compared with traditional point-to-point control systems, the advantages of NVSS are great particularly when a system becomes more and more communicable, intelligent and flexible by requiring more sensors, more actuators and more complicate controllers. However despite their numerous advantages and wide applications, the communication networks (Ethernet/Internet) and vision based systems in the control loops makes the control design more complicated. The main concern is the signal processing, coding, decoding and the networks across transmission induced delay which degrade inevitably the systems performance and possible cause systems instability. Therefore, the design of continuous undelayed state observer for NVSS is an very important issue.
In NVSS, there are two types of delays which are generally from two parts: the time delay Tsc between the sensor and the controller and the event based delay Tca between the controller and actuator. Many methods have been reported to compensate the Tca, such as through a Lyapunov–Krasovskii functional derived observer in [2], [3], a discrete Lyapunov functional method based full dimension state observer in [4], a feedback control method based on forecast for state compensation [5], a Grey theory based prediction compensation method [6], and a dual dynamics based periodic observer [7], etc.
In present paper, we focus on the compensation of the variable and unknown delay Tsc which is introduced by sensor sampling effect and information coding, decoding, processing and networks communication induced delay. This compensation is particularly important for NVSS [8], [9], [1], [10] because of their limited communication ability and image shoot speed where the application of the classical sampled-data control runs into serious difficulties due to important sampling periods. In the literature, there exist four main approaches dealing with this problem:
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The first one is refereed as continuous approach means to based on the continuous-time system to design its continuous-time observer (its sampling effects in the original controlled systems have been ignored), and then discretized the derived continuous-time observer for applications; the problem is that the resulting discretized observer works only under the sampling period which is small enough to meet the closed-loop systems stability [11], [12], [13].
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The second is referred as discrete approach, a discrete observer is designed on the discretized plant model. The main advantages of this approach are that the resulting conditions are tight and less conservative than other methods [14]. However, it suffers from the inaccuracy of the discretized model comparing to its continuous-time model, the complexity of the conditions and the difficulty to include uncertainties in the original system [15].
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The third is generally referred as the input delay approach which is introduced firstly in [16], [17]. This method allows for using the continuous-time Lyapunov–Krasovskii framework in order to take into account the sampling as a particular type of delay. The main advantages of the method are the possibility to take into account time varying uncertainties both in the sampling period and in the system parameters. However the resulting conditions are generally more conservative than the previous method, and necessary to solve complex linear matrix inequalities which are not always existing and easy.
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And the last one is the most recent developed methods referred as hybrid systems based approach [18], [1], [19], [20], [21]: such as in [18], the introduction of an impulsive system approach which refines the previous input delay method; and in [19], [20], the introduction of a particular type of piecewise continuous observer (PCO) which is developed on a piecewise continuous hybrid systems (PCHS) and used to reconstruct the visual servoing control systems continuous un-delayed state.
Recalling that the referred proposed PCO observer which is derived by using PCHS, has a very simple structure and can be implemented easily for real-time applications. The used PCHS, which is firstly proposed in [22] and then developed in [19], [23], [20], is a hybrid system characterized by autonomous switchings and controlled impulses. Further on, to show the proposed observer performances, a comparison with the most effective and widely accepted input delay approach of Lyapunov–Krasovskii observer (LKO) [16], [17] is effectuated.
The following paper is organized as follows: Section 2 presents the problem formulation of NVSS. Section 3 introduces the observer design preliminaries of PCHS and NVSS time diagrams. In Section 4, a PCO observer which is based on the PCHS theory and the sampled and delayed measurements whose sampling periods and delayed values are variable and unknown is designed and analyzed. For comparison sake, in Section 5 a LKO observer and their corresponding numerical results are illustrated. Finally, some conclusion remarks are followed in Section 6.
Section snippets
Problem statement
From the analysis of NVSS, we study the case when the only available plant information is delivered from the plant output via a numerical camera sensor introducing a delay with variable value which corresponds to the time needed to sample, code, decode, process and communicate the information (see Fig. 1). Further on, we consider that the camera sampling period is also Ti to prevent out of memory in micro-processor or network congestion problem.
Their dynamics can be described
Preliminaries of piecewise continuous systems
The observer which is proposed in this paper for linear systems with sampled and delayed output whose sampling period and delayed value are variable and unknown is based on the theory of a particular class of hybrid systems (HS) called piecewise continuous hybrid systems (PCHS). As it is well known, HS is a dynamic system that exhibits both continuous and discrete dynamic behavior – a system that can both flow (described by a differential equation) and jump (described by a difference equation
Piecewise continuous observer design with variable sampled and delayed measurements
In this section, a piecewise continuous observer (PCO) with five step construction is designed by means mainly of PCHS. This new type PCO observer whose stability is analyzed and demonstrated uses only the sampled and delayed output with variable sampling periods and delayed values. With this developed PCO observer, other adaptations for continuous undelayed state estimation methods from the other kind feedbacks such as sampled output yi, delayed output , sampled and delayed state ,
Validations
In this section, a Lyapunov–Krasovskii observer (LKO) which is introduced in [16], [17] is briefly presented and the performances of PCO and LKO are compared by a numerical example of NVSS.
Conclusion remarks
This paper deals with the design and analysis of a new class of state observers, called PCO observers, which make possible to estimate the system continuous undelayed state using sampled and delayed output measurements with variable periods. This type PCO observer can be also adapted easily according to other kind feedbacks, such as variable sampled output, variable delayed output, variable sampled and delayed state, variable sampled state, or variable delayed state.
These observers are based on
Acknowledgments
This work was partially supported by the National Natural Science Foundation of China (61304077 and 61203115), by the Natural Science Foundation of Jiangsu Province (BK20130765), by the Specialized Research Fund for the Doctoral Program of Higher Education of China (20123219120038), by the Chinese Ministry of Education Project of Humanities and Social Sciences (13YJCZH171), by the Fundamental Research Funds for the Central Universities (30920130111014), and by the Zijin Intelligent Program of
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