performance analysis and distributed control design for systems interconnected over an arbitrary graph☆
Introduction
Over the past few years, large-scale interconnected system has received a great deal of research due to the wide range of applications in web-transport systems, automated highway systems, multi-agent system, interconnected chemical reaction control and so on. When dealing with such systems, the traditional centralized control approach [1], [2] is not always appropriate as its cost might be too high and might lead to models of very high order, which renders controller synthesis and implementation impractical due to computational complexity. So, some scholars have begun to focus on the analysis and explore distributed control architecture for the interconnected system from various perspectives. C. Langbort et al. developed a framework for analysis the well-posedness, stability and contractiveness of systems interconnected over an arbitrary graph. The state-space description of such a system was introduced and an linear matrix inequality based distributed controller synthesis algorithm was developed for this class of interconnected systems [3]. F. Mazen et al. considered the discrete-time interconnected system and introduced a framework in which to consider distributed control over infinite graphs and provided synthesis and analysis conditions for design problems in [4]. A scheduled state-feedback controller synthesis algorithm was developed for a discrete-time, linear parameter-varying system constrained with input saturation and the result was oriented to the distributed control of large-scale systems in [5]. An algorithm called “D-K Synthesis Algorithm” was proposed in [6] for the synthesis of robust distributed controllers for uncertain discrete-time interconnected systems. Some necessary and sufficient conditions were obtained in [7] for the controllability of an interconnected system.
In addition, the performance index holds a special place among all performance indexes known to control engineering. The objective of optimal control is to minimize the error energy of the system when the system is subject to an impulse input or, equivalently, a white noise input of unit variance. Because of its effectiveness in dealing with the random noise and disturbance, the optimal control theory has been heavily studied for decades. A lot of results and conclusions have been made for continuous and discrete time systems [8], [9]. For an interconnected system, because of its special interconnection structure and distributed control architecture, it will face more disturbance than other systems. A. S. M. Vamsi et al. studied the optimal control problem of the discrete-time, strictly causal interconnected system which has a special structure [10]. However, among all the papers which we can get hold of, none of them studies the performance and distributed control of the continuous-time interconnected system. To fill this gap, in this paper we will present the effective methods for evaluating the performance and optimal distributed control of the continuous-time interconnected system.
Motivated by these discussions, in this paper, we consider the continuous-time interconnected system which consists of heterogeneous subsystems connected arbitrarily over an undirected graph. The contribution of this paper is threefold and is summarized as follows. Firstly, we give the definition of the norm of the continuous-time interconnected system and derive a sufficient condition for the well-posedness, asymptotic stability and performance of the interconnected system. Secondly, a sufficient condition to guarantee the existence of the distributed controller is developed. Thirdly, for minimizing the norm of the closed-loop system and solving the synthesis problem, a modified cone complementarity linearization (CCL) algorithm is given to deal with the nonlinear part in the condition and acquire the optimal performance. Moreover, the construction of the distributed controller involves solving quadratic matrix inequalities and in the general case is not simple. A generalized singular value decomposition (GSVD) based method is then given to deal with this problem and the procedure of constructing the distributed controller is shown in steps.
This paper is organized as follows. The system description and the problem formulation are given in Section 2. The analysis on well-posedness, stability and evaluation of performance of the interconnected system is shown in Section 3. In Section 4, we present the design method and implementation of distributed controller. A practical example is included in Section 5, and conclusions are found in Section 6.
: The notation denotes the set of real numbers; denotes the set of positive real numbers. The notation will be used to denote real valued, finite vectors whose size is either clear from context or not relevant to the discussion. denotes the space of by real matrices; denotes the space of symmetric by real matrices. The identity and zero matrix are denoted by and , respectively, or just and . The notation represents the transpose of the matrix . Given real symmetric matrix , denotes property for all . We will use to express concisely and to denote the vector . The triplet is the inertia of a symmetric matrix , where , and are the numbers of negative, zero, and positive eigenvalues of , respectively. Also, given matrices and , we define and by For a square integrable continuous time signal with , is said to belong to the -space if where is the norm of .
Section snippets
Preliminaries and problem formulation
Consider the interconnected system which consists of heterogeneous subsystems connected arbitrarily over an undirected graph and each subsystem is captured by the following state-space equation [3]: with initial state for all and , where is the state variable, is a disturbance acting on subsystem , is a performance output, and are the overall
Analysis on Performance
In this section, we will analyze the well-posedness, asymptotic stability and evaluate the performance of the interconnected system . Before that, some preparations are needed. First, we give the following lemma which provides an alternative way to check the well-posedness of the interconnected system .
Lemma 1 The interconnected system is well-posed if and only if is invertible.
Proof First, recall the ideal interconnection relationship in (2) that , it can be verified that the sets
Distributed Control Design
In this section, we will give a sufficient condition to guarantee the existence of the distributed controller firstly, and then, propose a method to construct the distributed controller. First, by applying Theorem 1 to the closed-loop system , it is easy to get the following result:
Lemma 4 Given a scalar . The closed-loop system is well-posed, asymptotically stable and has the specified performance if and there exist scalars and matrices ,
A practical example
In this section, a practical example of interconnected chemical reactors with recycle, borrowed from [17], is presented to demonstrate the effectiveness and applicability of the proposed distributed control design methodology. The interconnected chemical reactors consist 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 we can see from Fig. 3, the input
Conclusions
In this paper, we have investigated the performance and distributed controlling of the interconnected system which consists of heterogeneous subsystems connected arbitrarily over an undirected graph. The main contributions of this paper could be summarized as follows. (1) The classical definition of the performance specification for one-dimensional temporal system has been extended to the interconnected system for the first time. (2) A sufficient condition which helps evaluate the
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2021, ISA TransactionsCitation Excerpt :However, some restrictions limit the utilization of the results obtained in the above literatures. For example, the interconnection relation among subsystems are assumed to subject to a fixed interconnection subspace (e.g., see [7–9] and the references therein); The modified cone complementary linearization algorithm in [10] is not applicable for LSSs due to the heavy computational cost; A connection matrix is used to model the subsystems interconnection in [11,12], however, some restraints are required on the connection matrix; The authors of [13–15] only considered a special class of LSSs called “decomposable systems”. It is a challenging and significant task to derive computationally attractive stability conditions and controller design methods of LSSs without many constrains on the system models.
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This work is supported by National Natural Science Foundation of China (61673218, 11701281) and Natural Science Foundation of Jiangsu Province, China (BK20170817).