$H_2$ performance analysis and $H_2$ distributed control design for systems interconnected over an arbitrary graph

Abstract

This paper is concerned with the $H_2$ performance analysis and $H_2$ distributed controller design problem of the interconnected system which consists of heterogeneous subsystems connected arbitrarily over an undirected graph. The classical definition of the $H_2$ performance of one-dimensional temporal system is extended to the interconnected system for the first time. A sufficient condition for the well-posedness, asymptotic stability and evaluating the $H_2$ performance of the interconnected system is presented in this paper and this condition is in terms of individual subsystems. Based on this condition, a sufficient condition is given for the existence of the $H_2$ distributed output-feedback controller and a constructive method is then presented for the design of optimal $H_2$ distributed controller. A practical example is included to illustrate the applicability and effectiveness of the proposed $H_2$ distributed control design methodology in this paper.

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 $H_{\infty }$ 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 $H_2$ performance index holds a special place among all performance indexes known to control engineering. The objective of $H_2$ 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 $H_2$ 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 $H_2$ 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 $H_2$ performance and $H_2$ distributed control of the continuous-time interconnected system. To fill this gap, in this paper we will present the effective methods for evaluating the $H_2$ performance and $H_2$ 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 $H_2$ norm of the continuous-time interconnected system and derive a sufficient condition for the well-posedness, asymptotic stability and $H_2$ performance of the interconnected system. Secondly, a sufficient condition to guarantee the existence of the $H_2$ distributed controller is developed. Thirdly, for minimizing the $H_2$ 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 $H_2$ 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 $H_2$ performance of the interconnected system is shown in Section 3. In Section 4, we present the design method and implementation of $H_2$ distributed controller. A practical example is included in Section 5, and conclusions are found in Section 6.

$Notations$: The notation $\mathcal{R}$ denotes the set of real numbers; ${\mathcal{R}}^{+}$ denotes the set of positive real numbers. The notation $x\in {\mathcal{R}}^{•}$ will be used to denote real valued, finite vectors whose size is either clear from context or not relevant to the discussion. ${\mathcal{R}}^{n\times m}$ denotes the space of $n$ by $m$ real matrices; ${\mathcal{R}}_S^{n\times n}$ denotes the space of symmetric $n$ by $n$ real matrices. The $n\times n$ identity and $n\times m$ zero matrix are denoted by $I_n$ and $0^{n\times m}$, respectively, or just $I$ and $0$. The notation $A^T$ represents the transpose of the matrix $A$. Given real symmetric matrix $A$, $A<0(\le 0)$ denotes property $v^{T}Av<0(\le 0)$ for all $v\ne 0$. We will use ${(\star )}^{T}AB$ to express $B^{T}AB$ concisely and $(x_1;x_2)$ to denote the vector ${\left[x_1^T{x}_2^T\right]}^T$. The triplet $in\left(A\right)=(i{n}_{-}\left(A\right),i{n}_0\left(A\right),i{n}_{+}\left(A\right))$ is the inertia of a symmetric matrix $A$, where $i{n}_{-}\left(A\right)$, $i{n}_0\left(A\right)$ and $i{n}_{+}\left(A\right)$ are the numbers of negative, zero, and positive eigenvalues of $A$, respectively. Also, given matrices $A_1,A_2,\dots ,A_L$ and $1\le k\le l\le L$, we define ${diag}_{k\le i\le l}{A}_i$ and ${col}_{k\le i\le l}{A}_i$ by

$${diag}_{k\le i\le l}{A}_i≔\left[\begin{array}{cccc}\hfill A_k\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill A_{k+1}\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill A_l\hfill \end{array}\right],{col}_{k\le i\le l}{A}_i=\left[\begin{array}{c}\hfill A_k\hfill \\ \hfill A_{k+1}\hfill \\ \hfill ⋮\hfill \\ \hfill A_l\hfill \end{array}\right].$$

For a square integrable continuous time signal $f$ with $f\left(t\right)\in {\mathcal{R}}^n$, $t\in {\mathcal{R}}^{+}$ is said to belong to the ${\mathcal{L}}_2$-space if

$${\Vert f\Vert }_2^2={\int }_0^{\infty }{f}^T\left(t\right)f\left(t\right)dt<\infty ,$$

where ${\Vert \cdot \Vert }_2$ is the ${\mathcal{L}}_2$ norm of $f$.