Stability analysis of model-based networked distributed control systems
Highlights
► A model for the NDCS subject to both random packet loss and induced delay in its communication links was developed. ► By providing a 3-step interaction estimating algorithm, the closed-loop system was formulated as a time-dependent impulsive system. ► An LMI-based sufficient condition for the stability analysis of the overall closed-loop impulsive system was proposed. ► A design guideline was also given to synthesize a stabilizing controller for each subsystem. ► An illustrative example of a network of interconnected chemical reactors with recycle was provided to show the effectiveness and applicability of the proposed approach.
Introduction
Economical reasons and technological developments in manufacturing and industrial processes have led to arising large-scale systems with interacting subsystems. Stability analyses and the design of suitable controllers for these complex systems pose numerous challenges. To date, several control structures have been presented; including centralized, decentralized, and distributed control structures. In the last few years, there has been an increased interest in the use of distributed control structures [1], [2], [3], [4], [5], [6], [7], [8], [28], [29]. This is because the centralized control structure suffers from heavy computational burdens, and problems with robustness and reliability [9], and those systems controlled with the decentralized controlled strategy have weaker performance and stability margins, especially when interactions between subsystems are strong [9], [10].
In the distributed control structure, coupled variables of interacting subsystems provide supplemental feedbacks for the local controllers to meet the stability and performance requirements, whilst avoiding the complexity and inflexibility of centralized control structure [10]. However, implementation of supplemental feedbacks in distributed systems by means of traditional point-to-point wired connection is impractical. Recent developments in communication networks make information exchange between these distributed subsystems technically possible and, moreover, economically efficient. This new control structure for large-scale interconnected systems has been considered for example in [1], [2], [3], [4], [5], [6], [7], [8]. All mentioned works consider ideal and unlimited data exchange between subsystems over communication networks.
Networks in control systems enable remote data transfer; improve flexibility and efficiency, in terms of reduced system wiring, and finally, increase system agility and ease of the system's installation, reconfiguration, maintenance, and diagnosis with a smaller timeframe and reduced cost [11], [12], [13], [14]. Despite all these merits, using communication networks to exchange information makes the analysis and synthesis of a control system complex because of two major problems of packet loss and network induced delay [12]. Although references [10], [11], [12], [13], [14], [15], [16] and references therein have devoted their work to the stability analysis and controller design of networked control systems, they all consider non-ideal networks between instrumentations and the controller for a single plant. Therefore, their results cannot be directly applied to linked subsystems of a networked distributed control system (NDCS).
In [17], the construction of stabilizing controllers for spatially distributed interconnected systems in the face of small communication delays was discussed. Ref. [18] developed the distributed control of interconnected systems over failing communication networks with stochastic packet dropout. Ref. [19] analyzed the effect of random information loss in communication links between subsystems on the stability of spatially interconnected systems and designed stabilizing structured distributed controllers. Ref. [20] dealt with the H∞ control of spatially interconnected systems; where communications between neighbors was directed through packet dropping networks. In [21], the distributed output feedback control of dynamically decoupled systems with a common goal to optimize through a packet dropping network was presented. Analyzing stability and designing cooperative controllers for a team of distributed agents in the presence of delayed-information exchange between local controllers was presented in [22]. A quasi-decentralized control framework for plants with distributed, interconnected subsystems was developed in [8]. In their structure, every local control system communicates with the plant supervisor and with other local control systems through a shared communication link. Local observers are implemented in each subsystem to estimate the states of other subsystems in order to reduce the need for data transfer on ideal communication links. None of the literatures mentioned above have considered simultaneous random packet loss and delay which can be induced by a connecting network.
In this paper, the problem of stability analysis and stabilization of a NDCS, in the presence of random network-induced delay and packet loss is studied. The NDCS consists of discrete-time subsystems interconnected through their states. Each subsystem has a local controller. Local controllers of interacting subsystems are connected using a non-ideal communication network, which faces both random delay and random packet loss. In Section 2, this structure is modeled as a discrete-time system that has bounded time-varying delays in its interconnections. Serious degradation of stability margins and performance indices of the NDCS is possible because of induced delay and packet loss. To compensate for these adverse effects, a model-based control strategy is utilized to estimate the states of interacting subsystems, modify local control laws when information is not available, and finally stabilize the overall system. So, each local control unit is composed of two parts: an interaction estimator, and a main controller. The interaction estimator consists of a set of embedded models of other subsystems which provides an approximation of the interactions between subsystems when direct information is not available. Therefore, this control framework is named a model-based networked distributed control system (MB-NDCS). The main controller is a state feedback. After formulating this proposed control strategy in Section 3, the interaction estimation algorithm is presented in Section 4. Then in Section 5, the closed-loop MB-NDCS is modeled as a time-dependent impulsive system. Afterward, a linear matrix inequality (LMI) based sufficient condition is provided to analyze the stability by constructing a quadratic Lyapunov function. A design guideline is also given to synthesize a stabilizing controller for each subsystem. Finally, in Section 6, an example of a network of interconnected chemical reactors with recycle is presented to illustrate the effectiveness of the proposed approach.
Throughout this paper, represents the n-dimensional Euclidean space and is the set of all matrices. stands for non-negative integer sets of numbers. WT and W−1 denote transpose and inverse of any square matrix, respectively. W > 0, or (W < 0) denotes symmetric positive (negative) definite matrix. ||·|| denotes Euclidean norm. Matrices, if not explicitly mentioned, are of appropriate dimensions for algebraic operations. In symmetric block matrices, the symbol ‘*’ represents a term that is induced by symmetry. diag{Pi} for P1, …, PN, is used to denote a block-diagonal matrix with matrices P1, …, PN on the main diagonal and zeros elsewhere. row{Ai} for i = 1, …, N, denotes the matrix [A1, …, AN].
Section snippets
Modeling of NDCSs with both packet loss and transmission delay
Due to limited bandwidth and possible data collisions, transmission delay and packet loss in a networked control system is unavoidable. In order to model a NDCS with both packet loss and transmission delay but without loss of generality; two discrete-time subsystems in which the controllers interact through a non-ideal communication link are considered as shown in Fig. 1. The jth controller sends the state of its corresponding subsystem (xj(k)) at every k time-step, k = 0, …, ∞, with a period of Ts
Proposed control strategy
In order to realize the desired model based networked distributed control structure, each local controller is composed of two connected functional blocks; an interaction estimator and a main controller.
Interaction estimating algorithm
In this section, an algorithm is proposed to estimate the interaction among subsystems. This algorithm is implemented in each local estimator to compensate for the adverse influences of time delay and packet loss on the stability and performance of the closed-loop system. In the sequel of this paper, the following assumptions are made:
- 1.
All local controllers have access to all those states of other subsystems needed for the estimation process through a non-ideal communication network and receive
Closed-loop impulsive system formulation
In this subsection, we provide a sufficient condition to investigate the stability of the overall closed-loop MB-NDCS with the control law (2). Applying this control law to the open-loop subsystem (1) results in the closed-loop dynamical system (3) and the estimation dynamic (4) described below:where for i = 1, …, N.
Defining estimation error as for i = 1, …, N and
Case study
In this section, the example of interconnected chemical reactors with recycle, borrowed from [8], is provided to show the effectiveness of the proposed model-based networked distributed control methodology. The plant composes of two well-mixed, non-isothermal continuous stirred-tank reactors (CSTRs) with interconnections between reactors. Fig. 3 shows the process flow diagram of two interconnected CSTR units. As it can be seen, the feed to CSTR 1 consists of two streams, one containing fresh
Conclusions
This paper is concerned with the problem of stability analysis and stabilization of NDCSs, featuring both random delay and random packet loss in their communication networks. These NDCSs consist of discrete-time interconnected subsystems, in which non-ideal communication networks are employed to connect local controllers of interacting subsystems. Each local control unit consists of two parts: an interaction estimator, utilized to estimate the evolution of neighboring subsystems states when
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