Hierarchical H2 Control of Large-Scale Network Dynamic Systems

TL;DR

This paper introduces a hierarchical H2 control design for large-scale network systems that ensures convexity, reduces communication links, and maintains performance, enabling scalable control solutions.

Contribution

It proposes a special class of structured projections satisfying quadratic invariance, allowing convex H2 control design with hierarchical implementation for large networks.

Findings

Successfully applied to a 500-node network simulation.

Achieves reduced communication links while preserving control performance.

Enables scalable control design through convex reformulation.

Abstract

Standard H2 optimal control of networked dynamic systems tend to become unscalable with network size. Structural constraints can be imposed on the design to counteract this problem albeit at the risk of making the solution non-convex. In this paper, we present a special class of structural constraints such that the H2 design satisfies a quadratic invariance condition, and therefore can be reformulated as a convex problem. This special class consists of structured and weighted projections of the input and output spaces. The choice of these projections can be optimized to match the closed-loop performance of the reformulated controller with that of the standard H2 controller. The advantage is that unlike the latter, the reformulated controller results in a hierarchical implementation which requires significantly lesser number of communication links, while also admitting model and controller reduction that helps the design to scale computationally. We illustrate our design with simulations of a 500-node network.

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A Introduction

Driven by recent advances in cyber-physical systems, control synthesis for large-scale network dynamic systems has become an increasingly prevalent topic in the control and network communities [nantac]. Physical networks such as power systems, wireless sensor networks, or the recently emerging internet-of-things consist of numerous heterogeneous subsystems that may be defined across complex topologies and wide geographical spans. The typical numbers of subsystems in these networks can scale from thousands to millions, making the design of tractable control mechanisms very challenging. To tackle the curse of dimension, traditional model reduction-based techniques such as singular perturbations [singular], balanced truncation [moore], and controller reduction [obinata] have been developed decades ago to facilitate the analysis and control of large-scale systems, in general. However, these techniques are not readily applicable to networks as their control schemes are mostly unstructured, and hence agnostic of the network topology that contains important constraints for both design and implementation. One such pertinent constraint is communication. For example, conventional feedback controllers such as $\mathcal{H}_{2}$ optimal controllers are defined over unstructured dense transfer matrices, and thus implementing this control would necessitate an impractically large number of communication links for even a moderately sized network. Starting from the idea of decentralized control [siljak], several seminal papers have developed design tools to impose desired implementation structure on controllers. One of them, the work in [qinv], states that a decentralized optimal control admits a convex reformulation if the structural constraint on the controller is quadratic invariant (QI) under the plant model. Built around quadratic invariance, papers including [qinv], [lamperski] and [sabau] incorporate sparsity structures in the controller, aiming to reduce the communication density and to cope with delays. The restriction of these designs is that the sparsity pattern meeting the QI condition is highly dependent on the sparsity of the open-loop network, and therefore, the choice of sparsity can be limited. Moreover, the design complexities of these controllers are dictated by the order of the open-loop plant, and thus can become unscalable for very large-scale networks. A suboptimal design is proposed in [sparse] to generalize the choice of sparsity structure by relaxation algorithms using $l_{1}$-weighted norm. The design, however, is still computationally demanding. Motivated by these challenges, in this paper we present a special class of structural constraint for designing $\mathcal{H}_{2}$ controllers. The constraint set is defined by two projections over the input and output spaces, which result in a hierarchical implementation of the controller. The implementation mechanism works as follows. Selected network subsystems send their output measurements to a designated set of coordinators. These coordinators take the average of these measurements, and exchange the information between themselves to generate a $\mathcal{H}_{2}$ control law. Each coordinator, thereafter, broadcasts this control signal to its respective set of subsystems. The overall execution, therefore, is completely hierarchical, and requires lesser number of communication links. Preliminary work incorporating this hierarchy has been proposed in our recent paper [nanacc] for a LQG controller built upon the idea of clustering based projection. However, the design there is restricted by the choice of certain design parameters that guarantee closed-loop stability. In this paper we formalize the design in [nanacc] into a general $\mathcal{H}_{2}$ framework, where such restrictions do not exist. The hierarchical structure is proven to meet the QI condition irrespective of the structure of the plant, as a result of which, the control problem is reformulated as a standard unconstrained $\mathcal{H}_{2}$ control. The reformulated problem preserves the hierarchical structure inside its input and output matrices, and thus allows for model and controller reduction techniques to reduce the design complexity without breaking the structure of the controller. The reduction technique adopted in this paper is Hamiltonian-based approximation for solving the underlying algebraic Riccati equations (ARE), as proposed in [are1]. The recent paper [local] also addresses similar goals as ours using receding-horizon control, but the scalability of their controller is subject to the sparsity structure of the open-loop network. Related hierarchical designs have been proposed in [hier1, hier2], however, they do not exploit the convex reformulation provided by quadratic invariance. The remainder of this paper is organized as follows. Section i defines the hierarchical constraint, and formulates the problem of hierarchical $\mathcal{H}_{2}$ control. The same problem is parameterized and reformulated into an unconstrained $\mathcal{H}_{2}$ control via quadratic invariance in Section ii. Section v presents a Hamiltonian-based reduction technique to simplify the design of the reformulated controller. The optimality gap between the hierarchical $\mathcal{H}_{2}$ and unconstrained $\mathcal{H}_{2}$ problems is discussed in Section , where we also propose a design on clustering sets to tighten the gap. Validations of our proposed designs are illustrated via simulations of large-scale networks in Section i. Section ii concludes the paper and presents some future works. **Notation ** The following notations will be used throughout this paper: $|\mathcal{I}|_{c}$: cardinality of a set $\mathcal{I}$, $diag(m)$: diagonal matrix with vector $m$ on its principal diagonal, $diag(M,N)$: block-diagonal matrix with matrices $M$ and $N$ on its principal diagonal, $tr(M)$: trace operation on a matrix $M$, $\|M\|_{F}$: Frobenius norm of a matrix $M$, i.e. $\|M\|_{F}=\sqrt{tr(MM^{T})}$. A transfer function matrix (TFM) is defined as $g(s)=C(sI-A)^{-1}B+D$, with a realization form of g(s)=\left[\begin{array}[]{c|c}A\end{array}&B{\\ }{{\hline}}CD\right]. $\mathbb{R}_{sp}$ and $\mathbb{R}_{p}$ denote respectively the set of real-rational strictly proper TFMs and the set of real-rational proper TFMs. $\mathcal{RH}_{\infty}$ denotes the set of all stable TFMs from $\mathbb{R}_{p}$, and $\mathcal{RH}_{2}$ denotes the set of all stable TFMs from $\mathbb{R}_{sp}$. The $\mathcal{H}_{\infty}$ norm of $g(s)\in\mathcal{RH}_{\infty}$ is defined by $\|g(s)\|_{\mathcal{H}_{\infty}}=sup_{\omega}\ \bar{\sigma}[g(j\omega)]$, where $\bar{\sigma}$ denotes the largest singular value. The $\mathcal{H}_{2}$ norm of $g(s)\in\mathcal{RH}_{2}$ is defined by $\|g(s)\|_{\mathcal{H}_{2}}=\sqrt{\int_{-\infty}^{\infty}tr[g^{*}(t)g(t)]\mathrm{d}t}=\sqrt{\frac{1}{2\pi}\int_{-\infty}^{\infty}tr[g^{*}(j\omega)g(j\omega)]\mathrm{d}\omega}$. \par Furthermore, a Lyapunov equation is a matrix equation in the form of @@amsalign A\Phi+ \Phi A^{T} + BB^{T} = 0, {}$